chi stop her norris badiou

BADIOU’S BEING AND EVENT

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A Reader’s Guide

CHRISTOPHER NORRIS

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© Christopher Norris 2009

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British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

ISBN-10: HB: 0-8264-9828-0 PB: 0-8264-9829-9 ISBN-13: HB: 978-0-8264-9828-1 PB: 978-0-8264-9829-8

Library of Congress Cataloging-in-Publication DataNorris, Christopher, 1947-

Badiou’s Being and event: a reader’s guide/Christopher Norris.p. cm.

Includes index. ISBN-13: 978-0-8264-9828-1 (HB) ISBN-10: 0-8264-9828-0 (HB) ISBN-13: 978-0-8264-9829-8 (pbk.) ISBN-10: 0-8264-9829-9 (pbk.)

1. Badiou, Alain. Etre et l’événement. 2. Ontology. 3. Set theory. 4. Events (Philosophy) I. Title.

B2430.B273E875 2009111–dc22

2008045238

Typeset by Newgen Imaging Systems Pvt Ltd, Chennai, IndiaPrinted and bound in Great Britain by MPG Books Ltd,

Bodmin, Cornwall

For Wendy Lewis and Ray Davies

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CONTENTS

Acknowledgements viii

Author’s Note x

1. Context 1

2. Overview of Themes 14

3. Reading the Text 37

4. Further Reading 284

Notes 289

Index 307

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ACKNOWLEDGEMENTS

My thanks to colleagues in the Philosophy Section at Cardiff – especially Robin Attfield, Pat Clark, Rhian Rattray, Alison Venables and Barry Wilkins – for their friendship and encour-agement over the past few years. I am more than happy to acknowledge the regular fix of lively talk and intellectual stimu-lus – as well as good social cheer – that came from my weekly informal seminars with M.A. and Ph.D. students. My interest in Badiou’s work was reinforced by meeting and hearing him at the splendid conference ‘Wandering with Spinoza’, hosted by the Centre for Ideas at the Victorian College of the Arts, Melbourne, Australia in September 2006 and organized by Elizabeth Presa and Dimitris Vardoulakis. This was also, as it turned out, some-thing of a Badiou occasion and confirmed my already strong sense that Being and Event is itself one of those major – ‘evental’ – occur-rences in the history of thought that his book so impressively documents. Since then my writing of this Reader’s Guide has been helped, sustained and occasionally nudged in some new direc-tion by discussions with or readings of (among others) Gideon Calder, Sam Gillespie, Paul Gorton, Gerald Gould, Theo Gram-menos, Peter Hallward, Terry Hawkes, Vesna Main, Billy McMurtrie, Scott Newton, Laurence Peddle, Robert Reay-Jones, Alison Scott-Baumann, Rob Stradling and Rea Wallden. Man-uel Barbeito Varela invited me to Santiago de Compostela where I was able to present an impromptu version of some of this material and benefited greatly from the students’ stimulating company during a week of intensive but none the less enjoyable exchange. Ricky Sebold read through a draft version and offered some exceptionally well-informed and pertinent comments. Reg Coates, Dave Hilldrup, Dave Hume and Dick James – stalwart companions of the control-line handle – did much to keep me on an even keel and to ward off the perils of writerly solitude, as did Tim Andrews, Richard Evans, John Mealing, Gordon May,

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Digby Perriam and other friends in the South Bristol Model Air-craft Club. Sarah Campbell at Continuum Books was quick to come up with the idea of this project when I first mentioned my interest in Badiou, and has provided much welcome help and advice as the work went along. Wendy Lewis, Ray Davies, Di Corker, Margaret Innocent, Lyn Mererid, Beaty Smith and all my comrades in Cor Cochion Caerdydd (the Cardiff Reds Choir) were a great source of hope and a constant inspiration through some bad political times, although I realize now that I’ve been prefacing books with sentences to much the same syn-copated upbeat/downbeat effect for the past two decades and more. To Alison, Clare and Jenny – as usual – my love and gratitude.

CardiffOctober 2008

ACKNOWLEDGEMENTS

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AUTHOR’S NOTE

I should mention that there will have been published, by the time this book appears, an English translation by Alberto Toscano of Badiou’s Logiques des mondes (Paris: Seuil, 2006; Logics of Worlds, London: Continuum, 2009). It is a work fully compara-ble to Being and Event in scale and philosophical ambition, as well as marking a number of significant (at times quite extraor-dinary) new departures in Badiou’s thinking about issues of politics and ontology vis-à-vis developments and formal pro-cedures in the logico-mathematical domain. In fact there were already clear signs of this new direction in his thinking by the time of Badiou’s Briefings on Existence: A Short Treatise on Transitory Ontology (original French publication 1998). This explains the confusion regarding its place in Badiou’s oeuvre because the new work has been described, since it first appeared and pending its English publication, either as ‘Volume Two of’ or as ‘a sequel to’ Being and Event. What is clear enough – I should say – from a perusal of Logiques des mondes is that it stands as a largely self-sufficient text which takes up many lines of argument from that earlier work but does so from an angle (or variety of angles) so distinctive as truly to constitute a sequel rather than a supplement, continuation, integral part or concluding statement. Given my remit here this is just as well since Being and Event is in itself something of an Everest – not to say a veritable K2 – for the commentator wishing to convey both the rigour and the sheer exhilarating sweep of Badiou’s philosophical enterprise.

Since my book is a running commentary with just a few short-lived proleptic swerves from Badiou’s carefully chosen sequence of argument I have decided, for clarity’s sake, to stick with his descriptive (often lengthy) section titles for my own headings from Chapter 3 on and use them as convenient signposts along the way. References to Being and Event are given by page number

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only in the text, while other works by Badiou are referred to by bare title in the chapter endnotes and then with full publication details in the ‘Further Reading’ section. The latter also contains a selective bibliography of writings on Badiou for readers with an interest in various specific aspects of his work. All other sources are fully referenced in the chapter endnotes.

AUTHOR’S NOTE

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CHAPTER 1

CONTEXT

Of the making of books (and of books about books) there is seemingly no end so this one needs at least a few words by way of self-justification. In fact the case for writing it is also the case for considering Badiou’s Being and Event a work that thoroughly deserves such treatment and which makes exceptional demands of its reader.1 These go beyond a willingness to stretch one’s mind around a great range of subject-areas including mathema-tics, politics, ethics, aesthetics and psychoanalysis along with philosophy of logic, language, epistemology and – central to Badiou’s project – ontology as a constant point of reference for all those other disciplines of thought. It also requires an effort to grasp the unusual combination of a highly speculative mind-set very much in the ‘continental’ (i.e. post-Kantian mainland-European) line of intellectual descent with an approach that strives for the utmost degree of conceptual and logical precision. Thus his book is very clear from the outset that it will have no truck with any version of the standard Manichean account according to which – and depending on one’s own allegiance – it is either a case of the ‘analytic’ virtues contra the typically ‘conti-nental’ vices of obscurity, pretension and stylistic self-indulgence or else of ‘continental’ creativity and intellectual flair versus the tedium, arid technicality and narrowly professionalized ethos of much ‘analytic’ or mainstream Anglo-American philosophy. Badiou’s work overleaps the great rift that is supposed to have opened up between these two intellectual cultures, both of which have their proximate source in certain issues and unresolved problems bequeathed by Kant.

One reason for this is his adamant refusal to accept that Kantian agenda – whether in epistemology, ethics, aesthetics or political thought – whose effect (so he argues) has been to impose a variety of false and actively misleading dualisms like those between subject and object, mind and world, sensuous intuitions

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and concepts of understanding, the phenomenal and the nou-menal or mere inclination (however benign) and the dictates of absolute moral law.2 In each case, that refusal goes along with his leading claim that ontology – rather than epistemology – must be the starting point for any project that would seek to throw off this inherited burden of strictly (on their own terms) insoluble problems and instead make room for the kind of constructive or creative-exploratory thinking that is philosophy’s true vocation. Not that Badiou is for one moment setting the philosopher up as sovereign dispenser of truth, knowledge and wisdom or wishing to re-live past philosophical glories from a time before the other disciplines (the natural, social and human sciences) began to assert their distinctive claims. On the contrary, he never ceases to emphasize that philosophy is not and in truth has never been such a self-sufficient enterprise possessed of its own intellectual authority or juridical power whereby it presumes to lay down standards of validity or truth for those other, supposedly less well-regulated subject-domains. In fact it is a central precept of Badiou’s work – fully borne out by his own practice in a text like Being and Event – that philosophy can make progress or achieve worthwhile results only when it acknowledges its crucial depend-ence on certain extra-philosophical ‘conditions’ of thought, that is to say, its constant need for involvement with debates and activities outside what counts (on a typecast division of aca-demic labour) as philosophy pure and proper.

For Badiou the most essential conditions are science, politics, art and love, the latter understood largely (though by no means exclusively) in psychoanalytic terms. It is in relation to develop-ments in these four chief areas of knowledge and experience that philosophy is able to reflect upon the kinds of conceptual or ethical breakthrough that can then be seen to mark a genuine and lasting – rather than merely notional or short-term – stage of advance. Of course the precise nature of that relationship and the different forms that it can take in each case are topics that need a lot more detailed explanation and to which we shall return at various points in this book. For now, what I want to stress is the fact that Badiou seeks neither to aggrandize philosophy’s juridical role as was once (not so long ago) the fashion nor yet to play it down – like some present-day postmodernists, neo-pragmatists and followers of Wittgenstein – to the point of

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recommending that philosophers abandon their trade altogether, or else give up their delusions of epistemological grandeur and content themselves with being just one more voice in the ‘cul-tural conversation of mankind’.3 This latter idea is one that Badiou has most emphatically denounced and which he sees as threatening not only the fine-tuned balance of interests between philosophy and its fourfold conditions but also its capacity for critical reflection on issues in the wider intellectual, ethical, cultural and socio-political spheres. On his account there is something highly suspect about the widespread ‘linguistic turn’ across various philosophic schools of thought, from the echt-analytic (Frege-Russell) mode and the reactive ‘ordinary lan-guage’ approach with its sources in Wittgenstein or Austin to hermeneutics, post-structuralism, postmodernism and the type of ‘strong’-descriptivist or ultra-constructivist thinking espoused by someone like Richard Rorty. Whatever their deep-laid differ-ences these schools have one major premise in common, namely their belief – the hallmark of linguistic philosophy in general – that only through recourse to language as the basis of all enquiry could philosophy escape from those false dilemmas induced by the epistemological ‘way of ideas’ from Descartes to Kant and beyond.

For Badiou, the ‘linguistic turn’ is not – as its proponents would have us believe – a means of liberation from outworn ideas or misconceived pseudo-dilemmas but rather a means of distracting attention from problems that would otherwise occupy the forefront of any philosophical project meriting the name.4 These latter are best thought of as ontological in character since they have to do directly with the question of being – in its vari-ous kinds or modalities – as distinct from the epistemological question with regard to our knowledge thereof or the linguistic question with regard to what we can say, describe, or justifiably assert concerning it. Those who take the language-first approach should be seen not so much as philosophers in any genuine or properly applicable sense of that term but rather as standing squarely in line of descent from the ancient sophists, or skilled all-purpose rhetoricians, whom Plato charged with promoting mere eloquence above the interests of truth and justice. In treat-ing language – for which read alternatively ‘discourse’, ‘para-digm’, ‘descriptive framework’ or ‘conceptual scheme’ – as their


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ultimate court of appeal these thinkers more or less overtly embrace a pragmatist, conventionalist or communitarian stance whereby ‘truth’ cashes out for all practical purposes as what’s ‘good in the way of belief ’, or best suited to convince those (the relevant target-group) whose agreement or good opinion one seeks to win. What is distinctive about Badiou’s critique is the fact that he locates this sophistical strain not only in thinkers like Rorty who would happily subscribe to such a characterization of their views but also right across that large tract of present-day philosophic country where the dominant approach is one or another version of the linguistic turn.

Above all, as we shall see, it is Wittgenstein’s later thinking and its pervasive influence in so many quarters of current aca-demic debate that Badiou identifies as the single most potent source of a loftily dismissive attitude to real, even urgent philo-sophical problems, one that would have us regard those prob-lems as involving no more than a regrettable failure to observe the proprieties of this or that ‘language-game’ or communal ‘form of life’.5 As against such derelictions of its proper role Badiou is at pains to emphasize his point: that although philo-sophy cannot be pursued to any worthwhile, constructive or humanly beneficial purpose except through exposure to issues and challenges thrown up by one or other of its four ‘condi-tions’, nevertheless it – and they – will suffer a lapse into inertly orthodox or passively conformist ways of thought unless philo-sophy retains a sense of its own relative autonomy and proper (reflective and critical) role. Where this function is exercised to most decisive effect is through the readiness to raise questions or engage issues that directly concern those other disciplines yet which, given various practical constraints, may not rank high (or figure at all) on the list of priorities for anyone doing special-ist work in the field. Such is most strikingly the case, he argues, with regard to mathematics where there exists a great distance between working mathematicians who mostly take little interest in what philosophers might have to say and philosophers – espe-cially those in the analytic mainstream – whose approach and whose choice of topics for debate all too often invite just such a negative response. This they do mainly by focusing on problems amenable to their sorts of treatment – prototypically the issue about rule-following – and hence failing to think mathematically

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in a way that might put them more closely in touch with the sources of mathematical creativity.6 What sets Badiou’s work so decidedly apart is its refusal to engage in suchlike familiar (since intra-philosophical) kinds of debate and its insistence that any ‘philosophy of mathematics’ worthy of the name will have to manifest not only a strong intellectual command of the subject but also the capacity for conceptual work of a sufficiently high order.

In Being and Event this work has to do with developments in post-Cantorian set theory and their potentially transformative effect – as Badiou seeks to show – on disciplines, practices and modes of social being as well as of intellectual enquiry far beyond the mathematical as normally (i.e. restrictively) defined. Most significant here is Cantor’s radically innovative way of conceiving infinity not, in quasi-mystical or crypto-theological terms, as that which by very definition transcends the utmost powers of human reckoning or calculative grasp but rather as belonging to a realm of transfinite numbers, quantities and ratios which can indeed be reckoned with and placed in precisely calcu-lable relationship one with another.7 This discovery – as Badiou most emphatically considers it, rather than a mere paradigm change or shift of working method among mathematicians at a certain point in time – is the heart of his entire philosophical project and will therefore be the focus of detailed commentary later on. For the moment, there are three points that chiefly need explaining. First is the fact that, following Cantor, it became possible to define an infinite set as a set whose members are equinumerous (i.e. stand in a one-to-one equivalence relation) with one of its proper subsets. Thus the infinite set of natural numbers (integers) can be paired off with the infinite set of odd numbers even though the latter is a proper subset of the former or a set that on any ‘normal’ (finite) method of counting would be missing one integer for every two counted. It follows, sec-ondly, that there must exist multiple orders or ‘sizes’ of infinite set whose cardinality or proportion one to another can best be thought of by analogy with that between the integers and even numbers, but which all have the same constitutive feature of numerical equivalence with one of their proper subsets. Thirdly, this means that the development of set-theoretical concepts, procedures and hypotheses can be expected to proceed through

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a series of further limit-point encounters like that which moti-vated Cantor’s original discovery. In other words, it is a mode of enquiry that has typically achieved its most notable advances at the point of coming up against some obstacle, paradox or hitherto unnoticed anomaly which threatens to undermine its conceptual foundations.8

Such was the problem that Bertrand Russell famously uncov-ered with regard to self-referring or self-predicative expressions, that is, the dilemma or logical contradiction that results if one tries to make sense of phrases such as ‘the set of all sets that are not members of themselves’.9 Russell purported to resolve this problem by adopting a purely stipulative rule, one which decreed – as a matter of fiat – that such expressions were illicit since they contained terms belonging to two logically distinct (object-language and meta-linguistic) levels and were therefore certain to create all manner of confusion. On the other hand, some set theorists – those who rate higher in Badiou’s esteem – have rejected any fall-back ‘solution’ along these lines and opted instead for a highly conjectural, even adventurous but none the less rigorous and consequent mode of hypothesis formation which requires the truth of certain as-yet unprovable statements or theorems. What Badiou brings out very clearly is the way that mathematicians from Cantor to the present have responded to such challenges by allowing their thought to be drawn ahead of itself, so to speak, by that which exceeds its present-best powers of assured set-theoretical grasp. Advances of this sort may come about either through the impact of some currently recognized but so far recalcitrant problem or else through the anticipatory awareness of a problem that has not yet emerged distinctly to view but which they are able to discern as a kind of presumptive anomaly, or a sense of possibilities at present blocked off by the limits of existing methods and proof-procedures. Hence the cen-trality to Badiou’s project of certain set-theoretical methods developed by Paul Cohen, in particular those of ‘forcing’ and ‘the generic’, which between them provide a means of explaining how such developments occur.10

Here again, I must defer a more adequate treatment of these complicated matters until we reach the relevant stage of my detailed commentary on the text. The same goes for those other main topic-areas of Being and Event – politics, science, art and

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love – set forth by Badiou as the principal ‘conditions’ that should or must inform any philosophically substantive project of thought. The political dimension is directly linked to his writing on mathematics since the latter provides Badiou not only with the rudiments of a social ontology – that is, a conception of the various structures and modes of interaction that character-ize human collectivities – but also, more to the point, with a closely analogous means of explaining the modus operandi of various socio-political orders. Thus, in present-day terms, there is a large and precisely specifiable gap between the claims put forward on behalf of liberal democracy – claims concerning inclusiveness, equality, justice, universal rights, freedoms guaran-teed by rule of law and so on – and the reality of various blatant exceptions to each and every one of these notional achievements. According to Badiou, set theory provides by far the most ade-quate conceptual resources when it comes to accounting for this democratic deficit and locating those oppressed or excluded social elements that lack recognition or pass unacknowledged under dominant (official or consensual) ideas of social member-ship. Thus he sets out to show – utterly improbable as this might seem to readers brought up in orthodox disciplinary ways of thought – that there exists a close structural homology between, on the one hand, such pressing issues of social justice or political representation and, on the other, the sharply distinguished set-theoretical concepts of part and member, belonging and inclusion, or inconsistent and consistent multiplicity.

This latter distinction plays a crucial role in Badiou’s case for mathematics as the basis of social and political as well as scien-tific or physical ontology. For it is precisely through the ever-present excess of inconsistent over consistent multiplicity – of the parts whose sum total constitutes the truth of a given situ-ation over the members legally or socially acknowledged as belonging to it – that a politics of radical participant democracy might hope to gain a strengthened critical purchase. In so doing it would fasten on precisely those excluded, disenfranchised or oppressed minorities (such as the sans-papiers or ‘paperless’, legally unrecognized immigrants in France) who simply don’t count as persons for official, electoral or social-welfare purposes. That is to say, they don’t figure anywhere in the legal-administra-tive ‘count-as-one’, another of those key concepts in Badiou


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that enable him to posit a direct equivalence – not just a loose analogy or suggestive structural kinship – between the two domains of set theory and political philosophy. Thus it is always at those particular locations or ‘evental sites’ on the margins of the recognized body politic that the dominant order is likeliest to come under strain through a forced encounter with that which exceeds and challenges its sovereign or juridical powers. What then emerges with (at times) revolutionary force is the discrep-ancy between what Badiou terms the ‘state of the situation’ and the ‘situation’ itself. That is to say, there suddenly appears a non-negotiable rift between the socio-political order as viewed from a legitimist, reformist or social-democratic perspective and that which so far exceeds its grasp as to pose a constant (if hitherto suppressed or dissimulated) threat to its authority. Just as in the case of mathematics, logic and the formal sciences – where signal advances most often occur through the encounter with some deep-laid obstacle or paradox – so likewise in the political con-text it is always at these points of maximal stress that society first manifests the signs of imminent structural change. Here also it is a matter of decisive events (‘singularities’ as Badiou terms them, again in a specific mathematical sense) which could not possibly have been predicted since they found no place in the pre-existent social ontology, yet which afterwards – follow-ing this singular event – necessitate a likewise radical change in our grasp of what had indeed been possible in just that prior situation.

Here we should recall the title of this book and its pointed juxtaposition of two terms which between them stake out the philosophic territory that Badiou aims to traverse. Being is the domain of ontology, that is, of whatever can truly be said con-cerning the existence of entities (whether physical or abstract) which might always transcend the scope of current-best know-ledge – of empirical verification or formal proof – while none the less possessing an objective truth-value that may yet be discov-ered at some further, more advanced stage of enquiry. Badiou will have much to say about this cardinal distinction between truth and knowledge, one that runs strongly counter to the thinking of assorted empiricists, logical positivists and anti-realists not to mention post-structuralists, neo-pragmatists, social constructiv-ists and others who will have no truck with any such (as they

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take it) hopelessly outmoded ‘metaphysical’ conception.11 Hence – to repeat – his insistence on the power of mathematics, unique in this regard, to take us into speculative regions of thought where knowledge may be forced up against its limits at any given time and indeed drawn beyond those limits by a sense of what requires some more or less radical re-thinking of current truth-claims, methods and procedures. Events, on the other hand, are just those strictly unforeseeable and – as they appear at the time in question – wholly contingent irruptions of the new that may turn out to exert a uniquely powerful and lasting effect but which elude ontological specification precisely insofar as they belong to no existing (i.e. up-to-now thinkable) order of things. To this extent their nature and modes of occurrence cannot be captured by any account that adopts a straightforwardly progressive or developmental conception of truth as that to which knowledge constantly aspires or upon which it steadily converges at the end of enquiry. Rather, what results from a grasp of this evental character – as opposed to the cumulative process of knowledge acquisition – is a sharpened sense of how far it can exceed or transcend the scope of knowledge at any given stage in that process.

Such was, for instance, the advent of Cantorian set theory with its totally transformed conception of the infinite and its opening up of a conceptual terrain – a new-found range of onto-logical resources – whose discovery was not so much a further consequence of previous advances or successes but rather a product of the leap beyond a previous state of logical impasse. Thus it offers a striking example of the way that events – in Badiou’s strictly defined sense of that term – have the twofold character of arriving, so to speak, out of the blue or without the kind of partial foreknowledge that attends more regular modes of occurrence and also of requiring that their further develop-ment be followed through with a rigorous fidelity to their logical or indeed their ethico-political consequences. What marks the genuine event as distinct from its various false surrogates – among which, he thinks, must be counted a great many so-called ‘major’, ‘epochal’ or ‘world-historical’ episodes – is the fact that it exerts this intransigent demand on those who come after and whose elec-tive or self-imposed task it is to press so far as possible in working out its further (presently obscure or unguessed-at) implications.

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This is why Badiou insists so resolutely on the return to ontology as first philosophy in place of what he sees as the disastrous wrong turn that thinking has taken, first towards epistemology (thus privileging issues of the scope and limits of knowledge over questions of truth) and then – worst of all – towards lan-guage as that which purportedly constitutes the ultimate horizon of human understanding. It is also why he lays such unremitting stress on mathematics as the basis of all ontological enquiry and, accordant with this, on the essentially formal character of those various projects or truth procedures that constitute the otherwise diverse spheres of human scientific, political, ethical and artistic endeavour.

Hence Badiou’s idea of the ‘militant’ for truth who seizes – or is seized by – some such intimation of that which lies beyond their present-best powers of attainment or epistemic grasp yet which none the less exerts an intransigent demand upon their will and capacity to seek it out. Perhaps the hardest thing to grasp about his work is how he can maintain that crucial distinc-tion between being and event, or the domain of ontology and that which intrinsically eludes any pre-established mode of onto-logical grasp. After all, it is Badiou’s chief contention – not only as concerns mathematics and the formal as well as physical sci-ences but likewise across the whole range of subject-areas where truth is in question – that advances of this order are always pre-ceded by a formal discovery or the invention of a new, ontologi-cally ground-breaking procedure which then makes room for their admission to the realm of knowable truths, provable theo-rems or accredited matters of fact. In which case, it might be asked, how could such advances ever come about were it not that there always already existed a space of possibility opened up for them by some foregoing venture onto new ontological ground? That is to say, how can Badiou think to square these two appar-ently conflicting claims, first with regard to the prior status of ontology as a matter of absolute philosophic principle, and sec-ond with regard to the event as, by very definition, that which falls outside and beyond the sphere of any knowable ontology? For there would seem to be a clearly marked logical tension – if not an outright contradiction – between the idea of truth as intrin-sically a matter for ontological enquiry (since pertaining to the ‘being of beings’ and whatever can be thought concerning them)

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and the idea of truth as strictly a matter of post hoc fidelity to events whose occurrence could never be envisaged (let alone aimed for or knowingly sought) by any such project.

It is no exaggeration to say that this issue touches the quick of his work not only in philosophy of mathematics but also in each of those other main fields which Badiou takes to comprise phi-losophy’s fourfold enabling conditions. On the one hand he is compelled to insist, albeit with regret, that philosophy cannot itself claim to be one of those generic truth-procedures since it is neither (like them) a means of access to particular, context-specific instances of how truth emerges in the course of human endeavour or enquiry nor again (like mathematics) a first-order discipline engaged in the discovery of hitherto unexplored onto-logical terrain. Thus the best that philosophy can do is keep a close eye on the distinction between being and event – or onto-logy and that which unpredictably emerges to throw those explo-rations off their preconceived course – while seeking to articulate and clarify the process through which such events come to pass and to exert a profoundly transformative effect on one or other of its own elemental conditions. Still Badiou makes it clear that he is far from relegating philosophy to the kind of strictly subordinate or ‘under-labourer’ role to which it was confined by thinkers – from Locke to the logical positivists – for whom the natural sciences figured as a model of soundly based, rationally conducted and knowledge-conducive enquiry. Indeed it is his contention with regard to the scientists, political theorists, artists and psychoanalysts (the latter here figuring as those who, qua ‘experts’ in the subject, should have most to tell us concerning love) that they are all of them crucially reliant on philosophy when it comes to distinguishing knowledge from truth, or that which lies within the compass of their present-best conceptual grasp from that which, while currently eluding such grasp, none the less makes its absence felt through the pressure of unresolved problems, dilemmas or failures of demonstrative proof. More-over – and despite his unequivocal belief that mathematics must constitute the basis for any regional ontology remotely adequate to the task in hand – Badiou also counts the mathematicians (even the most eminent among them) as standing in need of philosophic assistance when drawing out the kinds of further implication that their thinking truly warrants.


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This helps to explain his at times quite waspish criticism of those analytic philosophers who focus on their own sorts of intra-disciplinary dispute and thereby avoid any serious engage-ment with real mathematical challenges. However he also takes issue with ‘working mathematicians’ – including some whose achievements he holds in the highest regard – who dismiss the philosophers’ efforts tout court as so much irrelevant or futile since (as they presume) mathematically under-informed intellec-tual pretension.12 In his own case (as I have indicated here and will go to argue in more detail) this charge can be shown to miscarry when confronted with Badiou’s meticulously reasoned presentation of set-theoretical developments from Cantor to Cohen and also his manifest depth of engagement with issues that can scarcely be considered beneath or outside the mathe-maticians’ concern. Yet by far the most convincing defence of Badiou’s position is that which he himself provides in the course of Being and Event, that is, his demonstration – in the strictest (mathematical) sense of that term – that set theory can offer the basis for some highly productive and innovative thinking in subject-areas from which it might seem (most of all to analytic philosophers) as remote as could be. This is why it is one of the very few philosophic works published during the past 50 years that merit the kind of close-focused exegetical commentary afforded to such acknowledged classics as Plato’s Republic, Aristotle’s Metaphysics, Descartes’ Meditations, Spinoza’s Ethics, Kant’s Critique of Pure Reason or (although Badiou might strongly dissent) Wittgenstein’s Philosophical Investigations. Nor is it in any way coincidental that those same classic texts – Wittgenstein’s apart – are among his chief points of reference in Being and Event, whether treated from a broadly sympathetic and supportive or a mainly critical and diagnostic standpoint. For among the many striking and, I would say, intellectually heartening aspects of Being and Event is the way that Badiou cuts clean across the supposed genre-divide between recent ‘continental’ thought with its frequent focus on the detailed exegesis of canonical texts and mainstream analytic philosophy with its typical assumption that problems are best, most produc-tively engaged as if de novo and with minimal reference to what-ever has been written about them in the past.

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Hence my seemingly extravagant claim for Being and Event as a work that bids fair for admission to the elevated company men-tioned above. Badiou is a truly original thinker in the sense of that term that these comparisons properly imply, and not just the debased and currently overworked sense which often means either someone who has invented a minor new twist on well-worn philo-sophic themes or else someone who has come up with a genuinely new line of argument but only by reason of its utter perversity or the sheer unlikelihood that anyone would think that.13 No doubt there will be some analytic philosophers whose first response will be to turn the argument around and deploy it as a tu quoque rejoinder against Badiou or those who put the case for his pre-eminent status in this regard. Thus they will ask what could possi-bly be more perverse than a project of thought that claims to derive substantive truths with respect not only to natural-scientific but also to political, social, ethical and even aesthetic understand-ing from an abstract ontology grounded in the even more abstract resources of set-theoretical thought. Any adequate answer will have to take the form of a commentary on Being and Event that keeps this challenge constantly in view while also acknowledging just how far – and in just what specific or decisive ways – Badiou’s thinking exceeds the limits of that analytic paradigm.

Such will be my aim in what follows, an aim that will have been best achieved if the reader eventually casts my book aside and returns to the original text with a sharpened sense of its extraordinary range and depth of philosophical insight. For their convenience – but also because Badiou pursues such a complex yet rigorously consequent order in the working-out of his case – I have designed this volume mainly as a serviceable vade mecum and therefore tried, so far as possible, to key my commentary closely to the text of Being and Event. The approach throughout is a mixture of close exegesis (always with an eye to concepts or arguments that are likely to put up resistance, especially for those unacquainted with the rudiments of set theory) and passages of more generalized comment where I draw out the further implica-tions of Badiou’s work. Along the way I also seek to clarify its relation to those many other thinkers, past and present, with whom he has engaged in a running debate of often quite remark-able intensity and scope.

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CHAPTER 2

OVERVIEW OF THEMES

1. POINTS OF DEPARTURE: AGAINST THE CULTURAL TIDE

By 2005 when the English translation appeared Alain Badiou was able to look back over the 15 years since L’être et l’événement was published and reflect, ‘not without pride’, that here he had fulfilled his desire to write a ‘great’ book, to ‘inscribe his name in the history of philosophy’ and moreover ‘in the history of those philosophical systems which are the subject of interpretations and commentaries throughout the centuries’ (BE xi). Not the most modest of self-estimates, but one that he places squarely before us in the knowledge that so far as the prospective reader is concerned it will stand or fall on what his work is able to achieve over the next 500 pages of intensive philosophical argument. For this is indeed, as he is pleased to announce, a text that invites – even requires if it is to have any hope of gaining a well-equipped readership – the kind of close-focused scholarly and critical ‘interpretation and commentary’ that have normally been lav-ished on works by the great (and mostly long dead) philosophers from Plato down.

Badiou’s Preface to the English translation goes on to make a number of retrospective points about the book’s genesis and, in particular, the historico-political and socio-cultural context of its writing. Thus he stakes out his distance from a number of positions and movements of thought which he thinks to have characterized that earlier period and still to be a source of much that is wrong with present-day intellectual culture. Most promi-nent among them is the kind of ‘free-world’ liberalism – in his view the thinly veiled adoption of a rhetoric with its source in old-style Cold War propaganda – very often taken up as a badge of ideological conformity by those who had once belonged to various groupings of the radical left but had switched allegiance

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in the wake of les événements (the failed revolution) of May 1968. Badiou has nothing but scorn for these purveyors of a doctrine of ‘human rights’ that was highly selective in its application as well as philosophically and politically bankrupt and which marked, as he writes, a culture ‘in full intellectual regression’ and ‘moral philosophy disguised as political philosophy’ (p. xi). Along with this went a widespread return to Kant – often in pointed opposition to Marx – for his conception of politics (or his idea of how philosophers should properly view politics) not as a matter of active engagement on the part of those with a practical stake in its outcome but rather as a topic for contem-plative judgement from the standpoint of one ideally uninvolved in any such premature rush to take sides.1 That is to say, Kant’s political philosophy – especially when read (as was mostly the case) through a postmodernist lens – offers plentiful resources for anyone intent upon reversing the thrust of Marx’s famous dictum that philosophers had hitherto interpreted the world, but that henceforth their task was to change it. Hence, Badiou thinks, the very marked revival of interest in Kant among vari-ous twice-born (i.e. ex-leftist) recruits to the banner of ‘free-world’ democracy and ‘human rights’, these latter conceived in ideologically loaded terms. Such was – and continues to be – the ‘abstract universality of our epoch’, this ‘alliance between the market and parliamentarianism’ which ‘functioned as if the only possible doctrine, and on a worldwide scale’ (p. xii). Moreover, it found a willing echo in those varieties of cultural-relativist thinking that had lately migrated across sundry disciplines from its home-ground in anthropology and cultural studies to other, less likely fields such as philosophy of science, epistemology, his-toriography and even (through the so-called ‘strong programme’ in sociology of knowledge) logic and mathematics.2

These developments were further reinforced by the ‘linguistic turn’ that became a very prominent feature of philosophy during the second half of the twentieth century, whether in the main-stream analytic tradition or the broadly ‘continental’ (mainland-European) line of descent.3 More precisely, the former lent support to this wider cultural-relativist trend through that influ-ential branch of the analytic enterprise that found its chief inspi-ration in the later Wittgenstein and his idea of ‘language-games’ or cultural ‘life-forms’ as the end point of all philosophical as

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likewise of all ethical, historical, anthropological or even scien-tific enquiry.4 Badiou is far more in sympathy with the other main branch that goes back to the founding figures Gottlob Frege and Bertrand Russell, and which holds on the contrary that ‘ordinary language’ stands in need of analysis – of revision, correction and logical refinement – if it is to serve as an adequate vehicle for thought and not create all sorts of conceptual confu-sion.5 Indeed, his work not only cuts clean across the supposed great rift between the ‘analytic’ and ‘continental’ modes of phi-losophizing but also engages deeply with issues that have divided the analytic community. These include the debate between real-ists and anti-realists of various stripe, especially with regard to philosophy of mathematics, and also – as I have said – that between thinkers who counsel a wise acceptance of the guidance to be had from ordinary language and thinkers (Badiou among them) who consider this a counsel of passive acquiescence in the kind of tenacious ‘common-sense’ illusion that such an outlook is prone to encourage, whether as concerns the physical sciences, politics, ethics or art. Thus he stands very squarely and vigorously opposed to the linguistic turn in whatever guise from Wittgen-stein, J. L. Austin and other analytically approved variants to Heideggerian hermeneutics, post-structuralism and the mélange of these and other (often ill-assorted) ideas that makes up the discourse of postmodernism. Moreover he rejects it on ethico-political as well as on ‘purely’ philosophical grounds since here too it tends to produce an ideologically conformist outlook or an uncritical willingness to fall in with orthodox beliefs, values or presuppositions. Such, after all, is the cultural-relativist upshot of any philosophy that holds truth in some given subject-domain to be specifiable only in terms that would meet with the agree-ment – or at any rate the sympathetic understanding – of those who share the same language-game or belong to the same com-munal form of life.

Whence Badiou’s chief aim in Being and Event: to put the con-trary case with maximum emphasis across the whole range of disciplines where this doctrine has left its mark and to do so, moreover, in a way that – contra the relativists, postmodernists, ‘strong’ textualists and other purveyors of latter-day sophistical fashion – places truth firmly at the centre of all his work. Only thus could he have any hope of refuting that prevalent liberal-pluralist

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creed to the effect, in his own acidulous words, ‘that all cultures were of the same value, that all communities generated values, that every production of the imaginary was art, that all sexual practices were forms of love, etc’ (p. xii). This passage will give some idea of the extent to which Badiou’s thinking goes against the current grain, at least as defined by majority opinion on the so-called ‘cultural left’. That is to say, it is unlikely to go down well with those for whom any talk of truth – let alone any talk of universally applicable standards by which certain cultures or sub-cultures might be found decidedly wanting – must be the product of a deeply conservative, authoritarian or neo-imperialist mindset. For Badiou, conversely, it is they – the cultural-linguistic relativists and adepts of this fashionable turn against truth – who have opened the way to a situation where values such as free-dom, democracy and justice can be so far corrupted by false or mendacious usage as to signify just the opposite of what they could (and should) properly mean. Much of his attention in Being and Event will be devoted to offering a detailed diagnosis of how this situation has come about and, more constructively, a set of proposals for turning it around through the renewed engagement with questions that have either languished unasked or received confused or inadequate answers.

More specifically, the book will raise those questions in each of the four principal contexts – science, politics, art and love – where relativism has exerted its most pernicious influence. It will do so by placing them in another context which, according to Badiou, must be the main point of reference for any enquiry that would seek to go beyond the deliverances of commonsense, intu-itive or ideological belief. This is the domain of ontology – of beings in their various forms, kinds or distinctive modes of exis-tence – and on his account our only means of access to it is by way of mathematics or of set theory as the formal science whose vocation it is to explore that domain insofar as it lies within the compass of thought or conceptualization. In his Preface Badiou merely canvasses these themes with a view to later development but they are here laid out with exceptional concision and clarity. Thus he gives us to understand, first, that ‘situations are nothing more, in their being, than pure indifferent multiplicities’, that is, made up of strictly indistinguishable elements which cannot be specified or told apart in terms of any particular features – such

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as those invoked by cultural relativists – that would constitute a unique, sui generis or privileged claim to attention. Thus ‘[i]t is pointless to search amongst differences for anything that might play a normative role, [since] if truths exist, they are certainly indifferent to differences’ (p. xii). Nor will this appear just a piece of ingenious paradox-mongering if one grasps the logic of Badiou’s consequent claim that ‘[c]ultural relativism cannot go beyond the trivial statement that different situations exist’, and for this very reason ‘does not tell us anything about what, among the differences, legitimately matters to subjects’ (p. xii). From which it follows, secondly, that truths are universal insofar as they con-cern every subject quite apart from his or her class, gender, social status, ethnic identity, occupation, cultural background and so forth, but also intrinsically subject-involving insofar as their dis-covery, further development and promulgation are the business of certain (no matter how few or how many) committed individuals. These are the exemplary ‘militants of truth’ whose company includes ‘the political militant working for the emancipation of humanity in its entirety’ along with ‘the artist-creator, the scien-tist who opens up a new theoretical field or the lover whose world is enchanted’ (xiii).

It is a safe bet that this latter set of claims would be received by most analytic philosophers with responses ranging from wry disdain to bewilderment or sheer incredulity. That is, they would most likely find something absurd in the very idea that a single project of thought – even (or especially) one grounded in a set-theoretically derived formal ontology – might have something significant to say about topics so diverse or downright ill-assorted as politics, science, art and love. However, that is precisely what this Preface sets out in summary style and what the book then proceeds to demonstrate by way of a highly structured, tightly reasoned and intricately cross-referenced process of argument sometimes reminiscent of the manner of reasoning more geomet-rico – ‘in the geometrical manner’ – that so appealed to rational-ist philosophers of the seventeenth century. Where his approach differs crucially from theirs is, first, in having far more powerful and refined mathematical, logical and conceptual resources at its disposal and, second, in coming from a philosophic standpoint deeply informed not only by those developments but also by cer-tain decisive advances (not merely shifts of paradigm, discourse

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or language-game) in each of its principal subject-domains. They constitute advances in a sense closely analogous to the sense in which mathematics can be shown to have made progress, that is to say, through periods of relative stability when the foundations of the subject seem secure and any problems look resolvable with-out major disruption, interspersed with (usually much shorter) periods when the problems turn out to shake the foundations and require some far-reaching overhaul. Such is most strikingly the case, he argues, with the advent of post-Cantorian set theory and the drastic transformation which this wrought in our ways of thinking about the infinite and its role – contra previous beliefs – in various kinds of operationally valid or formally speci-fiable procedure. However, it can also be seen to apply in those other fields of thought, from the natural sciences to politics, and art, where Badiou finds evidence of a similar pattern of punctu-ated equilibrium, or periodic irruptions of the radically new into states of relative epistemic calm.

So it is that certain breakthrough discoveries come to exercise the jointly attractive and repulsive force that ranges other think-ers – those who live in their immediate wake – most emphatically for or against the project of following through or drawing out their further consequences. Among these latter are the various kinds of truth (again with their model instance in the realm of mathematics) that have forced a radical re-thinking of what pre-viously counted as knowledge with regard to the natural, social and human sciences. Thus Badiou goes on to specify more pre-cisely how the relevant discoveries have come about and how they bear on his project. There are three aspects of its ‘current global state’ which he takes to be indicative of philosophy’s rela-tionship to those other disciplines of thought that require its help in order fully to recognize their own vocation yet which phi-losophy needs to bear constantly in mind if it is not to become narrowly self-occupied. These are, in Badiou’s succinct formula-tion, (1) that ‘Heidegger is the last universally recognizable phi-losopher’, (2) that the most crucial developments in mathematics, logic and the formal sciences must be credited to the line of think-ers that runs from the Vienna Circle of the 1920s to present-day analytic (largely Anglophone) philosophy and that has ‘suc-ceeded in conserving the figure of scientific rationality as a para-digm for thought’; and (3) that there is a new, ‘post-Cartesian’


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conception of the subject now emerging whose source must be traced to a certain range of ‘non-philosophical practices’ such as psychoanalysis and politics, and whose ‘regime of interpretation, marked by the names of Marx and Lenin, Freud and Lacan, is complicated by clinical or militant operations which go beyond transmissible discourse’ (p. 1).

Each proposition here would bear a great deal of commentary, not least No. 1 which will come as a surprise to those many ana-lytic philosophers who follow Rudolf Carnap and his Vienna Circle colleagues in regarding Heidegger as (hopefully) the last word in irrationalist bewitchment by language.6 In fact, as we shall see, Badiou is himself very far from simply or directly endorsing Heidegger’s depth-hermeneutic-ontological approach to what he (Badiou) continues to regard as distinctively philosophic ques-tions whereas Heidegger views the whole tradition of philoso-phy from Plato down as the perpetuation of that fateful error whose name is ‘Western metaphysics’.7 However, his strong res-ervations on this score – along with his equally strong misgivings with regard to Heidegger’s vatic elevation of poetry above all others modes of language or thought – are scarcely such as to placate his analytically minded opponents. For them it will be nothing short of unthinkable that proposition (2) might plausi-bly consort with propositions (1) and (3), or that a serious, philo-sophically informed engagement with issues in mathematics and logic might be compatible with a genuine respect for Heidegger’s work and, even more credibility-stretching, a belief that those issues might somehow relate to developments in psychoanalysis or the fortunes of Marxist political thought. Yet it is worth not-ing that some members of the Vienna Circle, Otto Neurath chief among them, were actively involved in the inter-war movement for progressive reform in their country and appear to have seen a very close connection between what they hoped to achieve polit-ically (a clear-eyed analysis of communal needs and the interests of social justice) and what they counted most important philo-sophically (that thought should aspire to the highest degree of conceptual or logico-semantic precision).8 Both aspects of their programme played a part in the antipathy towards Heidegger which often carried distinct political overtones (well before his public embrace of the National Socialist cause) along with that allergic response to Heidegger’s depth-hermeneutic or etymopoeic

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way with words that prompted Carnap’s famous essay title ‘The Elimination of Metaphysics through Logical Analysis of Language’.9

To be sure, Badiou goes nothing like so far in rejecting the Heideggerian turn towards poetry – and away from mathematics or the physical sciences – as the source of a privileged access to truths that have long been concealed, repressed or forgotten by the discourse of post-Hellenic philosophical reason. Indeed, the fact that he takes Heidegger seriously to the point of devot-ing some close and intensive commentary to various aspects of Heideggerian thought is one reason to count Badiou among the company of ‘continental’ philosophers, or at least to hesitate before claiming that he belongs much more to the ‘analytic’ line of descent. Thus, ‘along with Heidegger, it will be maintained that philosophy as such can only be re-assigned on the basis of the ontological question’ (p. 2). That is to say, Badiou differs sharply with the mainstream analytic view that Heidegger’s way of fram-ing this question – his distinction between Being and beings, or the ontological and the ontic – is just another species of ‘bewitch-ment by language’, a merely verbal confusion brought about by failing to apply certain nowadays basic logico-semantic distinc-tions. On the contrary, Badiou will have much to say (and much that is acutely perceptive) not only about Heidegger’s writings on poetry but also about those particular poets, Hölderlin chief among them, who stand out for Heidegger as exemplary ‘shep-herds of Being’ or conveyors of a truth inherently beyond reach of any logical, conceptual or plain-prose reasoning.10 Yet he will also have a good deal to say about the limits and the dangers of Heidegger’s thought, above all – here very much in accord with the analytic line of argument from Carnap down – as concerns its desire to elevate just those aspects or dimensions of poetic language that are taken as ‘ontologically’ prior to other modes of discourse, among them mathematics. Thus ‘along with analy-tic philosophy, it will be held that the mathematico-logical revo-lution of Frege-Cantor sets new orientations for thought’ (p. 2).

It is here that analytic philosophers are likely to find the great-est problems with making sense of Badiou’s (by their lights) inordinately large-scale, over-ambitious and hence philosophi-cally misconceived project. Most basic is the problem of grasping what he could possibly mean by invoking standards of objective


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truth and of ontological enquiry as constitutively aimed towards such truth while none the less adopting a criterion of ‘fidelity’ – of authentic dedication to that purpose – which seems to invite a subjective, even existentialist notion of good-faith commitment. After all, there is a near century-long tradition of analytic phi-losophers professing to find themselves puzzled, baffled or just plain appalled by the ‘continental’ tendency to mix up issues of factual truth or logical validity with issues of ethics or politics. However, as Badiou makes very clear, this response presupposes an idea of subjectivity that lags far behind what he takes to be the stage arrived at by current, psychoanalytically but also math-ematically and scientifically informed conceptions of how the subject stands with regard to matters of truth. Thus, on the one hand we are living through a ‘third epoch’ of science’ which has superseded both the ancient Greek inauguration of ‘demonstra-tive mathematics’ and the Galilean breakthrough that ‘mathe-matized the discourse of physics’. We are henceforth installed within a ‘split’ – a non-coincidence between what knowledge has attained and what thinking is able to conceive – which, accord-ing to Badiou, marks the point of transition whereby ‘the very nature of the base of mathematical reality reveals itself, as does the character of the decision of through which establishes it’ (p. 3). On the other hand, that ‘decision of thought’ must have reference to a drastically altered conception of the subject that is far removed from any existentialist or phenomenological con-ception of the human individual as a locus of purely autono-mous or self-willed choice between various, more or less live or mathematically viable options. ‘It is no longer’, he writes, ‘the founding subject, centered and reflexive, whose theme runs from Descartes to Hegel and which remains legible in Marx and Freud (in fact, in Husserl and Sartre)’. Rather, ‘[t]he contemporary Subject is void, cleaved, a-substantial, and irreflexive’, such that ‘one can only suppose its existence in the context of certain pro-cesses whose conditions are rigorous’ (p. 3).

Thus it is wrong to conclude that Badiou is out to undermine the objectivity of mathematics or, like anti-realists or intuition-ists in the ‘analytic’ camp, to bring mathematical truth within the compass of proof, ascertainment or human knowability-in-principle.11 Indeed nothing could be further from his aim since he devotes a good deal of hard-pressed argument to attacking

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any version of the claim that truth – whether in the formal, the physical or even certain branches of the social and human sci-ences – can be thought of as epistemically constrained, or as subject to the scope and limits of human knowledge. Thus, con-tra the anti-realists, Badiou looks to developments in set theory from Cantor down as a striking instance of the way in which thought may register what cannot as yet be discerned as an item of clear or distinct knowledge, and may therefore run ahead of any presently existing proof-procedure. This is how he is able to incorporate a range of otherwise diverse, heterogeneous and indeed (to an analytic way of thinking) strictly irreconcilable commitments, among them his realist yet subject-involving approach to mathematics and his high regard for Heidegger as a thinker who reopens long-neglected questions in the sphere of fundamental ontology along with his sharply limiting judgement on Heidegger’s over-promotion of language (especially poetic language) as a path to authentic truth. For on Badiou’s account – taking its lead from Plato, in this as in many other respects – it is to the discourse of mathematics, rather than poetry, that think-ing has been obliged periodically to turn in order to regain its ontological bearings and renew its creative-exploratory powers. In which case ‘philo sophy must designate, in so far as it is a mat-ter of being qua being, the genealogy of the discourse on being – and the reflection on its possible essence – in [the mathematicians] Cantor, Gödel, and Cohen rather than in [the poets] Hölderlin, Trakl, and Celan’ (p. 10).

Thus Heidegger is right, so far as Badiou is concerned, in naming ancient Greece as the inaugural site of that discourse on being and therefore as the source – and a constant point of refer-ence – for every subsequent, no matter how radical departure onto new tracks of thought. However, ‘it is not in the enigma and the poetic fragment that the origin may be interpreted’ but rather in ‘the mathematico-philosophical nexus . . . which defines, until Kant, the “classic” domain of its objects’ (BE, p. 10). Moreover, its domain has now been extended, with the advent of modern set theory, in directions strictly inconceivable to Kant and onto ground strictly inaccessible to those working within that ‘classic’ ontological paradigm.12 For what henceforth offers itself to thought is the prospect of an ontological enquiry that is restricted neither to the limits of unaided human intuition – limits already

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surpassed through the passage to Copernican–Galilean astro-nomy and decisively transcended through developments such as non-Euclidean geometry or Einsteinian Relativity – nor even to the bounds of what is knowable according to our current best methods of conceptual or empirical investigation. Such enquiry will take as its object-domain the most advanced researches of ‘working mathematicians’, that is, their most powerful and far-reaching discoveries with respect to the various orders of inclu-sion, exclusion, belonging, membership, excess, ‘inconsistent’ versus ‘consistent’ multiplicity and so forth. Thus, if philosophers wish to contribute usefully to this enterprise, then – as Plato required of all students signing up for his Academy – they will need to go to school with the mathematicians and not suppose (as philosophers have all too often been prone to do) that they are capable of excogitating such truths through some purely a priori process of thought. Among the benefits that this will bring (and here again Badiou has Plato’s example very much in mind) is the help it offers towards countering the claims of sophistry, rel-ativism or the ‘strong’ programme in sociology of knowledge.13

So philosophers will surely be self-deluded if they think that they can ‘do’ fundamental ontology, so to speak, right off the bat whether through the kinds of metaphysical system-building that rationalists such as Leibniz have typically attempted or else (like Heidegger) through a mode of poetico-philosophical brooding on themes of their choice. Indeed, Badiou even goes so far as to say that philosophy is ‘originally separated from ontology’, not – he hastens to add – because the object of ontological enquiry is merely chimerical or non-existent ‘as a vain “critical” knowledge would have us believe’ but rather because ‘it exists fully, to the degree that what is sayable – and said – of being qua being does not in any manner arise from the discourse of philosophy’ (p. 13). To this extent philosophy has no choice, or no intellectually rep-utable choice, but to take its lead from those signal developments in the most advanced quarters of mathematical thought that have opened up new possibilities in this regard. Yet Badiou is equally insistent that mathematicians are unwise to ignore what philosophers have to say when it comes to drawing out the wider implications of work in that specialist domain. For there is a cru-cial distinction – one that he shares with some analytic defenders of realism in philosophy of science – between knowledge and

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truth, or between whatever can be brought to the stage of explicit understanding, statement or articulate theory and whatever in the course of mathematical thought may open a way to such knowledge while itself neither requiring nor, perhaps, allowing such a measure of conscious or reflective awareness. Thus, ‘the new theses on being qua being are indeed nothing other than the new theories, and the new theorems to which working mathema-ticians – “ontologists without knowing so” – devote themselves; but this lack of knowledge is the key to their truth’ (p. 13). And again, if philosophers need reminding of the ontological prior-ity of truth over knowledge, and therefore (in this sense at least) of mathematics over philosophy, then mathematicians also need reminding that ‘the ontological dignity of heir research, despite being constrained to blindness with respect to itself, does not exclude, once unbound from the being of the working mathema-tician, their becoming interested in what is happening in meta-ontology, according to other rules, and to other ends’ (p. 14).

2. TRUTH, MATHEMATICS AND THE SUBJECT

No doubt there is a certain touchiness or prickliness about such remarks, as suggested by those insistent italics and by Badiou’s somewhat irritable comments elsewhere about the fact that ‘working mathematicians’ tend to dismiss what philosophers say on the assumption that they just don’t know enough about mathematics. As we shall see, this assumption is way off the mark in Badiou’s case – since he manifestly knows a great deal and communicates that knowledge to impressive effect – even if it has some force when applied to the sorts of mathematically humdrum set-piece example that typify a good deal of analytic work in this field.14 However, it is also important to grasp that the relationship between mathematics and philosophy as Badiou conceives it, although not one of equiprimordiality, does grant philosophy a strictly indispensable role in giving articulate form and expression to those wider, ‘meta-ontological’ truths that would otherwise go unspoken since mathematicians – naturally enough – tend to fight shy of pursuing them. Thus, it is up to philosophy to explain how and why all the major developments in set theory from Cantor to the present ‘resonate well beyond their technical validity, which has confined them up till now to the academic arena of the high specialists’ (p. 16). In the same


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way philosophy is best placed to show how a mathematically informed ontology can offer guidance in regions far removed from any such high specialism, that is, in the four chief areas of knowledge and experience – science, politics, art and love – which on the one hand provide philosophy with its strictly indispens-able ‘conditions’ of engaged or purposive thinking while on the other they depend on it precisely for this sense of their own standing vis-à-vis the dualism of being and event. For it is just Badiou’s point – one that will again strike many philosophers as involving a massive category-mistake or a wildly promiscuous conflation of disparate realms – that those epochal develop-ments in set theory must properly entail a radical revision of existing concepts not only with regard to mathematics, logic and the formal and physical sciences but also as concerns every aspect of our social, political and personal (or inter-personal) lives.

To the obvious question, ‘Why single out these four “condi-tions” in particular, given the great number of possible alternative candidates?’ Badiou would, I think, respond with three answers in ascending order of specificity or importance to his own project. They are privileged, firstly, because they are the central and defining realities of human existence; secondly, because they have already been subject to some highly perceptive and refined elaboration by thinkers whom Badiou holds in great esteem, whatever his (often profound) disagreements with them; and thirdly, because they each pose the antinomy of being and event in such a complex, distinctive and challenging way as to drive his principal thesis home with unmistakable force. Thus science, politics and art have this much in common; that they involve certain ‘generic procedures’ in a sense that derives from set the-ory, and more specifically from the work of Paul Cohen, one of Badiou’s most prominent mathematical sources.15 This is not for one moment to suggest that Badiou is in any way ‘ontologizing’ the social and human sciences, or – as might appear from what I have said so far – proposing that they henceforth be conceived on the model of a thoroughly mathematized natural science that would leave no room for choice, agency or ethical commitment. On the contrary, his whole point in making such a cardinal theme of the being/event dichotomy is to highlight the extent to which major developments in all of these spheres, the formal,

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physical, social and human sciences alike come about through a break with existing modes of thought and the advent of a new dispensation that could not previously have been known, recog-nized or brought within range of adequate conceptualization. Even so – as he is equally keen to stress, once again with primary reference to the set-theoretical domain – its emergence may later be discerned as having been prefigured (albeit indiscernibly so at the time) in the paradoxes, logical aporias, or unresolved dilem-mas that were kept from coming too clearly into view or from wreaking serious conceptual harm by the then-prevalent ideas of what counted as adequate proof or consistent reasoning.

A ‘generic procedure’ in mathematics, philosophy or one of its fourfold conditioning modes of knowledge or experience is that which enables thought to attain a presentiment or anticipatory awareness of whatever currently lies beyond reach of formal (demonstrative) proof or ascertainment on widely accredited grounds. Moreover, so Badiou contends, that procedure can be specified with adequate logical and existential precision only by way of a close and detailed reference to those set-theoretical concepts that occupy his main focus of attention in Being and Event. Hence the following statement, one that must appear excessively cryptic at this stage but which captures a number of essential points about Badiou’s project and which may perhaps resonate with aspects of my later, more detailed treatment of these themes.

What happens in art, in science, in true (rare) politics, and in love (if it exists), is the coming to light of an indiscernible of the times, which, as such, is neither a known or recognized multiple, nor an ineffable singularity, but that which detains in its multiple-being all the common traits of the collective in question: in this sense, it is the truth of the collective’s being. (p. 17)

The ‘collective’, here, must be understood to include not only those comparatively numerous multiples that constitute a social or political class, group, faction, movement or tendency but also the couple that typically forms the erotic or love relationship, and even – insofar as they must figure in any adequate account of these matters – the singular individual (whether scientist, artist,

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political activist or lover) whose very singularity, though not ‘ineffable’, presents a powerful challenge to received modes of thought, concepts of order, or collective representations. In this latter case the subject may be the locus – more aptly, the bearer or vector – of that which intrinsically eludes the grasp of any larger collective since its import or truth-content will emerge only at a later stage and through a more developed understand-ing of the symptomatic gaps, elisions or aporias that character-ize some presently existing discourse. In which case the subject has to be conceived as more active, self-willed, committed and (to this extent) autonomous than was ever allowed by those – the structuralists and post-structuralists – with whom Badiou is very often engaged in an overt or implicit running dialogue.16

Indeed this is one major point on which his thinking diverges sharply from the kind of linguistic-constructivist (and hence determinist) position implied by Lacanian psychoanalysis, despite – as we shall see – his frequent use of Lacanian concepts.17 After all, there is a clear conflict of aims between any project, like Badiou’s, which stakes its claim on the subject’s fidelity to the task of testing or bearing out certain scientific truth-claims, mathematical theorems, artistic practices, or political commit-ments and a conception, like Lacan’s, according to which the ‘autonomous’ subject is merely a figment of the imaginary, just as the conscious, reflective ego is merely a plaything of the Freudian unconscious to which it stands as the tip of a vast and largely submerged iceberg. There is a similar conflict between Badiou’s deep attachment to the idea that human beings can and do sometimes make their own history in ways that involve a substantial degree of active, engaged, transformative praxis on the part of collective agents and the Althusserian (or ‘structuralist-Marxist’) emphasis on the extent to which agents are always already constructed – or ‘interpellated’ – by existing socio-economic, cultural and ideological formations.18 Hence Badiou’s fidelity to Sartre (the later Sartre of the Critique of Dialectical Reason) very much against the fashionable tide on account of his con-serving this central role for human collective agency despite – what Sartre also acknowledges – the opposed, purpose-deflecting forces of ‘counter-finality’ and the ‘practico-inert’.19 Yet nobody who has read very far into Being and Event could suppose that Badiou subscribes to any version of the liberal-humanist creed

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that deploys a high-toned rhetoric of individual freedom, rights, authentic selfhood, autonomy and so forth very often as a means of effectively averting attention from exactly those antithetical forces that Sartre so tellingly describes. This is why the subject, for Badiou, is neither (as in post-structuralist theory) an empty place-holder or purely linguistic-discursive construct, nor again (as it is for liberals) the last guarantee of human freedom against such coercive or conformist pressures. Rather it is the locus of a strictly indispensable appeal to that which alone makes possible any advance beyond the confines of accredited knowledge or consensus beliefs but which cannot be conceived as somehow existing in a realm quite apart from those specific and exemplary modes of understanding that make up its enabling conditions.

Thus, according to Badiou, ‘[a] subject is manifested locally [and] is solely supported by a generic procedure. Therefore, stricto sensu, there is no subject save the artistic, amorous, scientific, or political’ (p. 17). What ultimately constitutes the subject is its involvement with a project in one of those domains such that the project in a certain sense ‘takes them over’ but also depends absolutely for its furtherance – its carrying-forward through var-ious kinds of experiment, proof-procedure, artistic practice or political activity – on the subject’s commitment but also on his or her inventiveness, intelligence, creativity or strength of politi-cal purpose. This is why he comes out strongly against both of those present-day intellectual orthodoxies that are often seen (and which see themselves) as politically and culturally poles apart but which in fact stand in a relationship of mutually sus-taining pseudo-opposition, with a typecast ‘liberal-humanist’ ideology serving as a handy target for post-structuralist, post-modernist or other such sceptical lines of attack while the latter fall just as handily into the former’s polemical sights. It explains his clear determination to break out of that false dilemma and develop an alternative conception of the subject that prevents it from getting a hold. That is to say, his approach will offer ade-quate scope for the exercise of human commitment, fidelity and resourcefulness in some given (e.g. scientific, political or artistic) cause and yet preclude any recourse to that notion of the subject – periodically resurgent among philosophers from Descartes to Husserl and beyond – as a purely self-sufficient or autonomous locus of thought and purposive agency.

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The link between the subject and truth is one that goes back, as Badiou duly notes, via Descartes to Plato but which assumes its characteristically modern form in the line of epistemo-critical thinking that begins with Descartes and receives its most elabo-rate treatment in Kant. However, it is one that has given rise to a great many problems with regard to epistemology and philoso-phy of science, as shown by the interminable disputes between rationalists and empiricists, realists and anti-realists or objectiv-ists and constructivists. At any rate there is prima facie much to be said for an approach, like Badiou’s, that aims to cut through this proliferating thicket of debates by grounding its case in ontology, rather than epistemology, and by grounding its ontol-ogy not (as with Heidegger) in language, hermeneutics or inter-pretation-theory but rather in mathematics, logic and the formal sciences. Hence the central thesis of Being and Event, namely that ‘mathematics is ontology’, a ‘meta-ontological or philosophical thesis’ which Badiou takes to be ‘necessitated by the current cumulative state of mathematics (after Cantor, Gödel and Cohen) and philosophy (after Heidegger)’(p. 15). I shall of course have more to say later on about the specific bearing on Badiou’s proj-ect of those developments in mathematical thought signalled by the names Cantor, Gödel and Cohen. For the moment, however, let me just make twoclarificatory points in brief. First is his iden-tification of ‘philo sophy’ with ‘meta-ontology’, that is to say, with a second-order discourse whose role is to elucidate those truths with regard to the structure of being discovered by mathe-matical enquiry. Second is the fact that Heidegger figures here – and throughout the book – as a thinker of great importance even though, in Badiou’s judgement, one who mislocated the nature and source of that depth-ontological renewal which he took to be the sole hope for authentic thinking in an age given over to rampant technology and the rule of instrumental reason.20 Indeed one reason for Badiou’s holding philosophy to this strictly ancillary role vis-à-vis mathematics – while also (and with equal emphasis) asserting its distinctive vocation and relative auton-omy – is his acute awareness (with Heidegger’s example vividly before him) of the way that philosophical thinking may be led in dangerous, even politically disastrous directions if it becomes too closely ‘sutured’ to any one of its fourfold conditions.21

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So there are large issues at stake when Badiou puts his case that philosophy’s engagement with those conditions should be under-taken always on the basis of a prior engagement with ontologi-cal questions, these latter conceived in mathematical terms since mathematics alone measures up to that demand of universal truth or validity that can act as a defence against the kinds of parochial (though quasi-universal) claim that Heidegger advanced on behalf of German culture, language and especially poetry. By insisting that philosophy not lose sight of this ‘meta-ontological’ role – that it defer to the achievements and expound the implications of set-theoretical enquiry – Badiou also seeks to ensure that philosophy should honour its responsibility to inter-ests transcending the merely local or partisan.

Thus he comes out strongly against any version of the argu-ment – at present almost de rigueur across wide swathes of ‘radical’ thinking in cultural theory – that justice can best be served or human welfare most effectively promoted through a maximal respect for the differences, rather than the commonali-ties, between people of various ethnic affiliation, cultural back-ground, social class membership, linguistic provenance, or sexual/gender orientation.22 On the contrary, he argues that this empha-sis on difference along with its sundry cognate terms (alterity, otherness, heterogeneity, incommensurability and so forth) very often betokens not so much a respect for the diversity of human values and beliefs but an absence of genuine, that is, reasoned and principled respect for any of them, one’s own included. In philosophical terms this cult of difference translates most directly into Wittgensteinian talk of multiple ‘language-games’ or cul-tural ‘forms of life’, Lyotard’s postmodernist celebration of ‘first-order natural pragmatic narratives’ as opposed to those obsolete ‘grand narratives’ of truth, progress, enlightenment and so on, Rorty’s debunking view of philosophy as just another more or less inventive or stylistically resourceful ‘kind of writ-ing’, and Levinas’s notion of a strict regard for the absolute oth-erness (or radical alterity) of the other person as that which constitutes the basis of any ethics meriting the name. Badiou rejects this whole line of thinking as just another minor update on age-old sophistical themes, that is, an inertly un-philosophical (rather than a bracingly anti-philosophical) doctrine according to


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which truth is either to be thought of as relative to some given cultural life form or else, after Levinas, not to be thought of unless on condition of the subject’s renouncing any claim to pur-sue it through the exercise of his or her own, merely ‘egological’ capacities of thought.23

What the Introduction thus makes clear is the fact that Badiou stands firmly apart from all those movements in recent French or French-influenced philosophy and critical theory that have shared the twofold aim of knocking truth off its ped-estal and demoting the subject – the Cartesian-Kantian ‘sub-ject-presumed-to-know’ – to a strictly subservient role in relation to that which exceeds its utmost powers of rational, epistemic or critical reflective grasp. On the other hand, Badiou is far from embracing any notion of the subject as privileged locus or source of truth, as shown by his requirement that philosophy acknow ledge the prior claim of ontological over epistemologi-cal (and, yet more emphatically, cultural-linguistic) concerns, along with his insistence that mathematics is the sole adequate basis or starting-point for any ontological enquiry. This is why he can claim, paradoxically enough, that his project in Being and Event ‘is organized around two affiliated and essentially new concepts, those of truth and subject’ (p. 15). ‘New’, that is to say, in the sense that they will here be radically redefined through a conception of the truth-event – the discovery of hith-erto unthought or unsuspected ontological resources – that finds its model in the methods and procedures of mathematical reasoning. More precisely, such significant stages of advance in this or other kindred fields can be shown to result from a far-reaching shift in the order of relationship between being and event, one that at first gives rise to a profoundly destabilizing effect and then brings about a new ontological dispensation or working grasp of that relationship. From which it follows – on Badiou’s account – that events of this momentous character will also produce a radical change in the domain of the subject, this latter construed not so much (or not at all) in phenomeno-logical or first-person experiential terms but rather as the locus wherein those events must be thought to occur and to exert their transformative power.

Thus the subject exists only in relation to certain epochal events whose import can itself be known only in retrospect – in light of

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developments that will bear out its truth through subsequent stages or more advanced procedures of investigation – and whose truth-content has to do either with mathematics, logic and the formal sciences or else with one of the four conditions that jointly compose its enabling element. To this extent Badiou belongs to the company of those French thinkers in the wake of phenomenology and existentialism who have cast a sceptical or dissident eye on any claims advanced on behalf of the autono-mous thinking, willing and acting subject of liberal-humanist ideology. Yet it is equally vital to Badiou’s entire project that the subject be conceived in its evental aspect as always potentially surpassing or exceeding the bounds of any in-place belief sys-tem or conceptual scheme. Hence what he takes to be philoso-phy’s chief and quintessentially motivating task: to conceive the possibility of thinking beyond the limits of whatever is pre-sented as the limiting condition of thought in some given disci-plinary domain. On his account this requires that we return via the path of philosophical reflection to a reengagement with the central issues of ontology (for which read: issues raised by the development of set-theoretical enquiry from Frege and Cantor to Cohen), and thence to a renovated concept of truth that would also and inseparably carry along with it a renovated concept of the subject.

It is here that Badiou first introduces another of Cohen’s set-theoretical concepts, namely that of ‘forcing’, or the process whereby certain paradoxes, contradictions, aporias or inconsis-tencies within some existing state of knowledge can later – from a more advanced stage of understanding – be seen to have marked precisely those symptomatic stress-points where know-ledge fell short of a truth beyond its best powers of discernment or conceptual grasp. Again, his chief reason for taking this expository route via such relatively technical areas of set theory and philosophy of mathematics is that it offers not just a sugges-tive analogy but a close and even (so Badiou would claim) a precise correspondence with the way that major advances are achieved in other fields of endeavour. So it is that ‘being can be supplemented’, or that ‘the existence of a truth’ can be thought of as ‘suspended from the occurrence of an event’, one whose implications will be far from clear at the outset since ‘the event is only decided as such in the retroaction of an intervention’ (p. 17).


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This is how Badiou can think of ontology in realist terms as involving the progressive exploration of territory that is ‘there’ to be explored – since by no means (as anti-realists would have it) created in or through that very process – and yet as always open to startling new discoveries which themselves require a great deal of thinking-through for a good while after their first, often somewhat tentative or stumbling formulation. Hence his descrip-tion of Being and Event as a book ‘designed to broadcast that an intellectual revolution took place at the beginning of the sixties, whose vector was mathematics, yet whose repercussions extend throughout the entirety of possible thought’ (p. 16). That revolu-tion he sees as having resulted from Cohen’s development of set theory to the point where it finally provided philosophy with the formal resources to explain how knowledge could be thought to fall short of truth – or truth to run ahead of knowledge – in such a way that this discrepancy supplied the very driving force or incentive required to motivate a long-term, intensive and sharply focused project of enquiry.

‘In the category of the generic’, Badiou writes, ‘I propose a contemporary thinking of these procedures which shows that they are simultaneously indeterminate and complete; because, in occupying the gaps of available encyclopedias, they manifest the common-being, the multiple-essence, of the place in which they proceed’ (p. 17). Of course this sentence would bear a good deal of analytical unpacking since it encapsulates much of the argu-ment that Badiou will go on to expound and refine over the fol-lowing large tract of closely reasoned, densely allusive, intricately cross-referenced text. For the moment let me note how it phrases his cardinal distinction between truth and knowledge as a mat-ter of those exploratory (e.g. mathematical, scientific, political or artistic) procedures that may be thought of – metaphorically but aptly enough – as tracking what is unknown even to the best, most informed and informative sources through those gaps in the current ‘encyclopedia’ of knowledge that reveal the existence of missing (‘indeterminate’) items precisely on account of its not living up to the promise of truly encyclopaedic com-pleteness. We should also bear in mind his identification of ‘common-being’ with ‘multiple-essence’, conveying as it does Badiou’s equally cardinal claim that the greatest innovation of

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set theory – and the yardstick of its various advances from Cantor to Cohen – is its formal demonstration of the fact that no ‘consistent multipli city’ (i.e. no dominant ordering or version of the ‘count-as-one’) can ever fully contain or comprehend the ‘inconsistent multiplicity’ that always pre-exists and to that extent exceeds or eludes such a count. For it is this central claim that enables and motivates his appeal to mathematics as the basis of ontology and to ontology, in turn, as the basis for what-ever we can justifiably assert concerning the truth-conditions for our various truth-apt statements, theories, hypotheses or conjectures.

As I have said, this puts him very much at odds not only with a swathe of linguistically or hermeneutically oriented move-ments within recent continental (i.e. mainland-European) thought but also with that strain of anti-realist thinking that is a prominent feature of mainstream analytic philosophy of lan-guage, whether in the Wittgensteinian or the more ‘technical’, logico-semantic mode.24 The issue of philosophy versus the soph-ists, rhetoricians and cultural relativists is one that had its first airing in the dialogues of Plato where it typically came up in conjunction with other main themes of Badiou’s work. Among them was the absolute pre-eminence of mathematics as an intel-lectual discipline, a training ground for philosophers, and (above all) a royal road to the discovery of truths that, once discovered, acquired a priori status or the character of seeming self-evident to reason yet which often had to be arrived at through a lengthy process of constructing hypotheses, devising proof-procedures, pursuing long sequences of hypothetico-deductive argument, or testing theorems and conjectures for any logical anomaly that might arise in the course of those procedures. Despite (as we shall see) his express reservations about being labelled a mathe-matical Platonist, Badiou nevertheless belongs very firmly on this side of the deep division that is commonly supposed to exist between thinkers of a Platonist and thinkers of an Aristotelian intellectual temperament.25 So we can now turn to Part I of Being and Event which takes the form of six ‘Meditations’ on mathematical/ontological themes, and which starts out from a sustained engagement with the dialectics of the one and the many as treated in Plato’s later writings.


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Discussion points

On the basis of your reading so far, what do you take to be Badiou’s most distinctive or far-reaching proposals with regard to philosophy’s role vis-à-vis mathematics and politics?

Why does Badiou take such a strong line against the recent ‘lin-guistic turn’ across various branches of philosophy and the social or human sciences?

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CHAPTER 3

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PART I. BEING: MULTIPLE AND VOID. PLATO/CANTOR

1. The one and the many: Plato to Deleuze

Plato’s dialogue Parmenides is one of those problematic works – ‘problematic’ at least for scholars and interpreters who wish to extract some fairly straightforward or unambiguous item of doctrine – where Platonism (or the collection of ideas generally given that name) finds itself exposed to some difficult questions which place it under considerable strain.1 For Badiou, conversely, this is a work whose ‘revolving doors . . . introduce us to the sin-gular joy of never seeing the moment of conclusion arrive’ (p. 23). That is to say, this is a thoroughly aporetic dialogue, one that fails to reach any decisive or definitive conclusion not through some weakness, dialectical wrong turn or argumentative path not taken but rather through inherent complications in its subject-matter that would have to wait more than two millennia before mathematicians and logicians came up with the conceptual means to handle them. Those complications have to do with the one and the many, or the issue – central to every aspect of Platonist metaphysics, ontology and epistemology – as to whether the one must be thought to have priority over the many or the many over the one.

The former position was adopted most strongly by Plato’s precursor Parmenides who taught that truth and reality must be thought of as timeless, unchanging and altogether beyond the realm of transient sensory experience which could not be grasped intellectually without giving rise to all manner of logical incon-sistency. Thus the famous paradoxes of time and motion devised by Parmenides’ disciple Zeno were designed to make the case for a purely monistic and rationalist conception of being in relation to thought that would escape those paradoxes by denying the rea-lity of time and change, these latter conceived as mere illusions


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brought about by our habitual over-reliance on the supposed ‘evidence’ of the senses. This ontological distinction was carried across into the Platonist dichotomy of epistεmε and doxa, or genuine knowledge with its grounding in logos (or the exercise of reason) as opposed to mere opinion or consensus belief, equated with the realm of sensory impressions on account of their shared impermanence or chronic liability to change. Many philosophers from Plato down have taken something akin to this position, either as a matter of explicit doctrinal adherence or else as an inbuilt assumption prerequisite to other main dimensions of their thought. Thus for Leibniz, as Badiou notes, ‘what is not a being is not a being’, since – ontologically speaking – the condition of existence for any given object is precisely that set of individuat-ing features or attributes that mark it out as that particular, self-same object and no other. Such is Leibniz’s cardinal precept concerning the ‘identity of indiscernibles’ and also its comple-mentary adjunct with regard to the ‘indiscernibility of identi-cals’. More recently it has found expression in W. V. Quine’s pithy saying ‘no entity without identity’, and it remains an item of shared (if often unspoken) belief across some otherwise large divergences of view in present-day philosophic debate.2

For Badiou, on the contrary, it is far from evident that this issue has been settled – philosophically resolved – in favour of the one over the many, or of that which can always be subsumed without remainder under some unifying concept as opposed to that which exceeds or eludes any such all-comprehending appli-cation of the dominant count-as-one. This is why he insists on reopening those questions from Plato that can now be seen, as a result of recent mathematical advances, to belong among the most potentially fruitful topics but also to the problems most frequently ignored, suppressed or finessed by thinkers in Plato’s wake. Thus Badiou follows Plato’s example in making mathe-matics the basis of ontology, ontology the starting point of all genuinely truth-seeking enquiry, and this Parmenidean/Platonic issue of the one and the many a privileged means of access to the questions thus posed for philosophical thought. Chief among them is the question whether being can be grasped in unitary terms as that which underlies and brings intelligible form to the passing sensory show or whether, on the contrary, this is just a fixed preconception that has typified much philosophical

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thinking from Plato down but which cannot be sustained against the various objections that rise against it.

Hence Badiou’s fascination with the game – the ingenious, inventive, conceptually resourceful but also consequential and serious game – played out between Plato and his interlocutors. What Plato glimpsed under pressure from the kinds of quandary, dilemma or conceptual impasse encountered in these late dia-logues was the impossibility of thinking such issues through to a conclusion on his own metaphysically favoured terms. The desired upshot – in line with the doctrine set out in Plato’s middle-period writings – was to vindicate his claim for the existence of supra-sensory forms or essences that would have their being above and beyond the transient flux of perceptual experience and thereby perform this imperative unifying function. However, the prob-lems emerged very clearly as soon as the obvious question arose as to how the one – the supposedly transcendent and all-embracing principle of unity – could manifest itself to human understanding except in and through the realm of ‘presentation’ which was also (inescapably) the realm where thought encountered multiplicity in all its forms. Hence the logical come-uppance delivered to that Parmenidean-Platonist hankering for the one through its encounter with inconsistent multiplicity, that is, with whatever intrinsically exceeds or disrupts its homogenizing drive. This rendition of Badiou’s central theme may strike some readers as excessively wire-drawn, or as just a piece of idle scholastic paradox-spinning. However, it not only goes to the heart (i.e. the mathematics-based formal core) of his entire philosophical proj-ect but also provides him with a highly effective point of entry to those various ‘conditioning’ fields of thought – from science to politics, art and love – which, as we have seen, compose a major part of that project. For it is Badiou’s claim that the issues first aired in these ancient Greek disputes are those which continued to exert a powerful hold on the thinking of philosophers over the next two millennia and have now – since the pioneering work in the foundations of set theory by thinkers such as Frege, Russell and (especially) Cantor – been brought to a stage where their true implications are at last coming into view.

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logical inconsistency of any attempt to prove the contrary case. Indeed this dialogue can be seen to mark the point at which Platonist thought turns against its own earlier, distinctly Parmeni-dean commitments and begins – in however tentative and self-resisting a fashion – to venture onto different, no doubt more vertiginous but also (as Badiou will proceed to demonstrate) philosophically more fertile ground. This is the stage to which thinking attains when it first proposes as a serious candidate for philosophical acceptance the statement that ‘being is not’, or (as will later become more clearly expressible with the advent of modern set theory) that the multiple is that which always and everywhere exceeds the grasp of any unifying function or any mode of knowledge premised on this or that application of the dominant count-as-one. What mathematical developments since Cantor have at last achieved is a genuine working grasp of those issues that proved so vexatious for Parmenides and Plato, and whose legacy Badiou will trace in Being and Event as it surfaces repeatedly to complicate the thinking of (among others) Aristotle, Descartes, Pascal, Leibniz, Spinoza and Hegel. Thus the one can now be treated as the product of a certain formal operation, that is to say, a procedure of counting or grouping that imposes some order on an otherwise inchoate since open-ended multiplicity but which is always – and for just that reason – exposed to the potentially disturbing effect of that which finds no place in the existing conceptual domain since it exists as a ‘supernumerary’ element excluded from the count-as-one.

Hence the two main theses which Badiou takes as ‘prerequi-sites for any possible ontology’: first that ‘[t]he multiple from which ontology makes up its situation is composed solely of multiplicities. There is no one. In other words, every multiple is a multiple of multiples’; and secondly, that ‘[t]he count-as-one is no more than the system of conditions through which the multiple can be recognized as multiple’ (p. 29). Another way of putting this is to draw a distinction between ‘consistent’ and ‘inconsis-tent’ multiplicity. The former is defined by Badiou as that which results from some preceding count or formal operation, whereas the latter must be thought of as itself pre-existing, surpassing and eluding the count-as-one yet also – since of course that operation must have something to operate on – as providing its necessary starting point or precondition. In which case, Badiou

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argues, we shall have to re-conceive ontology on the model of – or as identical with – certain specific situations, that is, certain specifiable stages in the process whereby inconsistent multipli-city is rendered consistent by this or that formal means, before once again turning out to possess elements or subsets that can-not be reduced to any such principle of order. Thus a ‘situation’, as Badiou understands it, is defined in essentially liminal or transitive terms as a ‘structured presentation’ which partakes of both consistent and inconsistent multiplicities. ‘This duality’, he writes, ‘is established in the distribution of the count-as-one; inconsistency before, and consistency afterwards’ (p. 25). And again, ‘[s]tructure is what obliges us to consider, via retroaction, that presentation is a multiple (inconsistent) and what authorizes us, via anticipation, to compose the terms of the presentation as units of a multiple (consistent)’ (p. 25). So it is – through this constantly evolving dialectic of containment and uncontainable excess – that thought is empowered to transgress and surpass the limits laid down by any regnant paradigm or merely de facto consensus of belief.

In its absence, thought will most likely yield to one or other of the two great opposing temptations that Badiou finds endemic across a whole range of present-day disciplines. On the one hand is the lure of that sophistical approach (post-structuralist, post-modernist, Wittgensteinian, hermeneutic, constructivist or neo-pragmatist) according to which it makes no sense to suppose the existence of truths beyond those that fall within range of our present epistemic, cognitive, conceptual or – what this is usually taken to entail – linguistic-expressive capacity. On the other, both provoking and reacting to such sceptical-relativist ideas, there is the older and more characteristic philosophical tempta-tion that consists in flatly asserting the claim for ontology as a discipline aimed towards matters of absolute, objective truth and just as flatly denying that it ought to be concerned with ‘situations’ or conjunctures of the kind that Badiou constantly invokes. If the greatest single problem for ontology is that thrown up in the course of Plato’s Parmenides and repeatedly in various guises thereafter then the greatest temptation is that which involves ‘removing the obstacle by posing that ontology is not actually a situation’ (p. 26). Badiou on the contrary describes it as the chief ‘wager’ of his book that ontological issues cannot be


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treated in all their specificity, complexity and depth except by taking adequate account of how they emerge in given situations and how they respond more or less inventively to various con-junctural problems or dilemmas. Moreover, this entails that there cannot be a concept of the multiple, at least if by ‘concept’ is meant a clearly defined – necessary and sufficient – set of condi-tions for what the term properly denotes. After all, ‘[o]ne would thus count it [the multiple] as one and being would be lost again’, insofar as being is equated by Badiou with whatever eludes our best-present powers of cognitive grasp precisely on account of its having no place in the conceptual order laid down by the currently prevailing count-as-one. This is why, for set-theoretical purposes, ‘the prescription is . . . totally implicit . . . it operates such that it is only ever a matter of pure multiples, yet there is no defined con-cept of the multiple to be encountered anywhere’ (p. 29).

All the same – what is perhaps the most difficult aspect of Badiou’s thinking for anyone new to his work – it is in conse-quence of just this ultimate elusiveness or resistance to concep-tual definition that set theory has displayed such a striking capacity for advancing through and beyond (and by means of) the various problems and paradoxes thrown up in the course of its development to date. Badiou will have a great deal more to say – and with closer reference to specific episodes in that deve-lopment – regarding this notable progress in its powers of con-ceptual-ontological grasp and the way that they spring from a repeated encounter with the limits of present understanding linked to an anticipatory sense of what as yet lies beyond reach of any adequate formal or conceptual statement. For the moment there are three main points that we should note by way of pre-liminary orientation. First is that ontology, as Badiou conceives it, has always to be treated by way of its relation to those various specific ‘situations’ or conjunctures of thought which determine the scope and limits of knowledge at any given time. Second – apparently in conflict with this – is the precept that ontology has to do with questions of truth, rather than knowledge, and hence that it is a gross confusion (albeit one endemic to many present-day movements of philosophic thought) to suppose that ontological issues could ever be settled or even usefully addressed through any kind of context-relative or historically indexed approach. Hence Badiou’s third and most crucial claim: that what accounts

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for the capacity of thought to transcend any current, however limited state of knowledge is the development of certain formal operations through which those limits show up symptomatically by acting as constraints on the quest for a more advanced, con-ceptually adequate mode of understanding.

Just how this occurs – or how it can possibly be thought to occur – is Badiou’s theme in later parts of Being and Event where he takes up Cohen’s set-theoretical ideas of forcing and the generic as between them offering a means to explain what would other-wise constitute a strictly unresolvable paradox.3 At this stage he makes the point by explaining how the axiomatic-deductive method in mathematics, logic and the formal sciences can indeed posit truths and anticipate the kinds of procedure required to establish those truths without as yet having actually arrived at any such procedure or laid it out as a valid (i.e. formally ade-quate) proof. His adherence to the axiomatic method – to a con-ception of truth as attainable solely by way of certain logically rigorous formal procedures – is a prominent feature of Badiou’s thinking with respect not only to mathematics but in each of those otherwise diverse fields where he takes the dialectic of being and event to constitute the driving force of scientific, political and cultural change. It is also what sets him implacably at odds with a great many present-day thinkers for whom such a claim on behalf of truth – let alone on behalf of mathematics as the royal road to truth – would betray nothing more than a quaint attachment to thoroughly antiquated ways of thought. Those thinkers would include the aforementioned company of post-modernists, post-structuralists, Wittgensteinians, neo-pragmatists and suchlike adepts of the ‘linguistic turn’ alongside other more highly esteemed antagonists, like Gilles Deleuze, with whom Badiou has taken issue on just these grounds.4

Deleuze stands out – and receives by far the most detailed and respectful of the many critiques that Badiou has devoted to philo-sophical sparring partners – by reason of his having emphati-cally espoused an ‘open’ rather than a ‘closed’ ontology, one that derives its chief inspiration from the differential calculus and ‘intensive’ (qualitative) rather than ‘extensive’ (discrete and quantitative) multiplicities, and which thus defines itself squarely against any formal or axiomatic conception of truth.5 Moreover, as becomes very clear in Badiou’s protracted engagement with


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Deleuze, this difference as regards relatively technical issues in the philosophy of mathematics has wider repercussions for their respective views concerning questions in the natural, social and human sciences and even – as Badiou would be quick to affirm – matters of an urgently political or ethical character. For it is his contention that in each case the interests of truth are best served by adherence to certain rigorously specified operative principles. These latter may be axioms, formal procedures, scientific hypoth-eses, research programmes, political initiatives or the kinds of constraint upon future conduct brought about by the demands of fidelity to some prior commitment undertaken with a view to its unpredictable yet none the less looked-for since partly antici-pated outcome. So there is a close connection between, on the one hand, Badiou’s disagreement with Deleuze over issues con-cerning mathematical ontology and the rival claims of extensive versus intensive (or axiomatic versus differential) modes of thought and, on the other, his disagreement concerning the need for concerted and organized action in the socio-political sphere.

This latter, well-publicized difference of views no doubt has its roots in their respective histories of political involvement before and after the watershed events of May 1968. However, it also has much to do with Deleuze’s almost visceral mistrust of large-scale ‘molar’ (as opposed to small-scale, ‘molecular’) forms of activist engagement and his broadly anarchistic idea of ‘desiring-production’ as the locus of those disruptive energies that alone have the potential to break down existing structures of power and control.6 Conversely, it is Badiou’s deep-laid con-viction that in politics as in other, more obviously formal disci-plines of thought there is no prospect of genuine advance or possibility of truly innovative thinking except by way of rigorous procedures that should always themselves be subject to likewise rigorous assessment at every stage. Closely allied to this is the willingness to venture far-reaching hypotheses that decide the whole course of one’s investigative work but for which as yet there exists no adequately formalized proof-procedure, or to stake large claims on some future event which thereby shapes one’s every act and decision even though that event is by no means certain – even (as things look at present) unlikely – to transpire. Moreover, it demands that one follows them through with the utmost fidelity and rigour, whether in terms of intellectual commitment to a

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detailed and demanding programme of research in the formal, physical or human sciences or in terms of a long-term political project entered into and sustained with unyielding commitment against whatever circumstantial odds. So it is not hard to see why Badiou should have singled out Deleuze as one of those exem-plary figures – along with (among others) Aristotle, St. Paul, Pascal, Leibniz, Spinoza and Heidegger – to whose thinking he is sharply, even passionately opposed in various respects while none the less finding in them just the kinds of adversary with whom he can most productively engage.

I have taken this slight detour from the path of strict exegeti-cal conduct so as to explain (for maybe sceptical readers) how issues in such technical or specialized fields as advanced set the-ory and philosophy of mathematics can bear something more than a suggestive, vaguely analogical relation to issues of politi-cal theory and practice. It comes out in what Badiou has to say about the need for an axiomatic-deductive approach to mathe-matics and the formal sciences in virtue of its singular capacity to pass beyond the limits of intuitive grasp or presently existing knowledge, devise proof-procedures for unproven theorems or conjectures, and thereby – when these work out according to that sense of anticipatory grasp – discover new stretches of mathe-matical or ontological terrain. The latter pair of terms Badiou would consider a pseudo-dichotomy – a distinction without a real difference – since for him mathematical discoveries are also and inseparably stages of advance into so-far unexplored onto-logical regions. Indeed it is one of philosophy’s primary tasks to expound and clarify these ontological advances and thus make up for the marked unwillingness, among most working mathe-maticians, to do so off their own bat. It is precisely the signal merit of such axiomatic-deductive thought that it allows enquiry to proceed along paths that are clearly marked out or rigorously formalized – where individual terms have a strictly specified ‘compositional’ role as defined by their place within the overall structure – while avoiding the kinds of premature appeal to some existing state of knowledge or stock of intuitions that would block any prospect of significant further advance. Thus again, ‘[i]t is clear that only an axiom system can structure a situation in which what is presented is presentation. It alone avoids having to make a one out of the multiple, leaving the latter as what is


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implicit in the regulated consequences through which it mani-fests itself as multiple’ (p. 30).

No doubt some explanation is required as to what Badiou means by the rather odd claim that ‘what is presented is presen-tation’. Here again he is referring to that basic set-theoretical precept according to which the one (i.e. first in the series of inte-gers or natural numbers) is in no sense a primitive term but, on the contrary, results from an certain operation brought to bear on the otherwise open multiplicity – the endless regress of multi-ples whose terms are themselves multiples whose terms are them-selves multiples, and so forth – that must be taken to precede and moreover to constitute the very condition of possibility for any such operation. In other words – Badiou’s own, from the useful glossary of technical terms appended to Being and Event – pre-sentation is itself the ‘primitive word of metaontology (or of philosophy)’ (p. 519). This is because it specifies only the bare fact of something’s having been presented, ‘something’ in this context taken to denominate the as-yet open or undefined con-tent of any presentation in general, rather than some determi-nate content as given by this or that specific application of the count-as-one. This is also to say that ‘[p]resentation is multiple-being such as it is effectively deployed’, and furthermore that it is ‘reciprocal with “inconsistent multiplicity”’ since the latter likewise exceeds and eludes any adequate specification. For ‘[t]he One is not presented, it results, thus making the multiple consist’ (p. 519). In other words, it is only at the stage of the count – in whatever onto-mathematically determined form – that inconsis-tent multiplicity (presentation as such) is confined or reduced to the realm of definite numerical, logical, propositional or other such formally specified content.

With all this in mind we are now better placed to see just how high are the stakes when Badiou returns to Plato in Medita-tion Two and stages a further dialectical encounter which in fact involves four disputants – himself, Plato, Parmenides and Socrates – although their voices are mostly unmarked. This sec-tion is headed by the Parmenidean epigraph – ‘If the one is not, nothing is’ – but goes on to turn that dictum around from its manifest sense (that everything must be one because otherwise all would be confused in a multiplicity of fleeting appearances) to what Badiou derives from it by way of set-theoretical reasoning

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(that since any such ontology founded on the one will break down through the logical contradictions exposed in Plato’s dia-logue therefore it is the nothing that must constitute the starting point of any viable ontology). For the sceptical reader still seek-ing evidence of Badiou’s philosophical acumen or power of sus-tained and concentrated argument there is no better evidence than that contained in this brief but pregnant chapter. As he notes, the Platonic dialogue ‘is consecrated to an “exercise” of pure thought proposed by the elderly Parmenides to the young Socrates’ (p. 31). It possesses a quality of intense and hard-pressed adversarial reasoning that is carried across with no dimi-nution to Badiou’s meticulously reasoned and attentive but also shrewdly transformative reading of the text. Thus his aim is to show – very much against the grain of Parmenides’ (and also, more ambiguously, Plato’s) intent – how any attempt to state a doctrine of being premised on the absolute priority of the one will end up despite and against its purpose by implicitly affirming a doctrine of the multiple as prior to the count-as-one. In short, ‘[w]hat Plato is endeavoring to think here, in a magnificent, dense text, is evidently inconsistent multiplicity, which is to say, pure presentation, anterior to any one-effect, or to any structure’ (p. 33). We shall soon see enough of its further consequences – not least with regard to those political issues of inclusion, exclu-sion and representation that are rarely absent even from Plato’s more arcane metaphysical debates – to recognize just how much is at stake when Badiou presses this ancient dialectic of the one and the many to the point of aporia and ultimate inversion of the Parmenidean doctrine.

What emerges from this reading, therefore, is a Plato who enunciates truths that contravene his own express metaphysical commitments and one such truth in particular: that ‘in the absence of any being of the one, the multiple in-consists in the presenta-tion of a multiple of multiples without any foundational stop-ping point’ (p. 33). ‘In-consists’ is one of those neologisms – in this case the making of an intransitive verb from the adjective ‘inconsistent’ – that may strike some readers as overly self-indulgent but which in fact serve very well to communicate a technical (in this case set-theoretical) concept. Thus its point is to capture not only the state of ‘inconsistent multiplicity’ which Badiou opposes to the order of ‘consistent’ multiples that result from


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some particular mode of presentation or product of the count-as-one but also the process or activity whereby any such consis-tent multiple is subject to the kind of potentially destabilizing force exerted by the various anomalies, discrepancies, exceptions and other problematical instances that find no place in the count. Here again there is a close and unmistakable connection between what Badiou has to say in this formal (set-theoretical) register and what he has to say on topics of an ethical, social and political nature. Indeed to put it like this is to understate and misrepresent his claim since, for Badiou, the structures and procedures involved in an adequately theorized social-political ontology are identical to those that emerge from an ontologically adequate account of devel-opments in set theory. As we shall see later on, it is precisely through the non-coincidence between belonging and inclusion – a disparity expressible with greatest precision in set-theoretical terms but one with far-reaching implications as regards those other subject-domains – that Badiou is able to stake this claim and, moreover, to unfold its implications witha high degree of formal rigour as well as extraordinary speculative sweep.

It is therefore natural that Being and Event should set out by engaging with Plato’s Parmenides as the text wherein – through its prior engagement with Parmenides himself – these questions receive their earliest and (from Badiou’s point of view) their ide-ally complex, dialectical and inconclusive or aporetic treatment. Not that Badiou is out to fault Plato on grounds of mere falla-cious reasoning or – as so often when analytic philosophers write about thinkers of the past – on grounds of his unfortunately not having had the benefit of those more advanced logical or con-ceptual resources enjoyed by anyone nowadays approaching the same sorts of problem. Nothing could be further from his prac-tice of critical commentary than the tone of somewhat pitying fondness that often creeps into such writings in the ‘rational-reconstructive’ mode.7 With Plato as with the many other think-ers discussed in Being and Event Badiou inclines more towards the ‘continental’ way of according those precursors serious, detailed and respectful attention on their own terms while also very clearly signalling the points where his thought diverges from theirs and the various respects in which later developments – most crucially, the advent of set theory – have pointed a way beyond the obstacles they once faced. After all, Plato is scarcely

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at fault if his failure (more like: his dogged refusal) to accept the priority of the many over the one, or of inconsistent over consis-tent multiplicity, is due not so much to some corrigible lapse of reasoning on his part but rather to the inherent difficulty – even the downright impossibility – of clearly and distinctly conceiv-ing any such thing.

All the same Badiou is very far from rejecting that conception of philosophy as wedded to the virtues of conceptual clarity and distinctness, one with deep roots in the French philosophi-cal tradition – going back to Descartes – and carried on by twen-tieth-century thinkers such as Gaston Bachelard and Georges Canguilhem.8 Its influence is everywhere apparent in Badiou’s thought, whether in his detailed reconstructions of the process by which conceptual advances come about in mathematics and the formal sciences or in his singling out just those thinkers – and just those aspects of their work – that most strikingly exemplify that claim. Hence his strong attachment to the seventeenth-century rationalists (despite many and various disagreements with them) insofar as they adhered to a conception of scientific, philosophical and social progress that placed chief emphasis on the power of reason to think its way through and beyond the limits of common sense, intuitive, uncritical, parochial, passively imbibed or other such customary habits of belief. Hence also his approach to Plato as a thinker more productively understood through a critical-diagnostic reading that treats him as at least partially accountable to later and, in certain respects, more advanced procedures of thought. If read in this way – on a quali-fied principle of charity that maximizes rational content while also (quite compatibly) making full allowance for explicable error – then Parmenides can be seen as a work that implicitly foregrounds its own dilemmas or moments of unresolved con-ceptual strain, and which thus looks far beyond its own tem-poral, intellectual and cultural horizons. That is to say, Plato’s statement, ‘If the one is not, nothing is’ can be left to stand wholly unaltered and yet, in consequence of such a reading, be construed as follows: ‘if the one is not, what occurs in the place of “the many” is the pure name of the void, insofar as it alone subsists as being’ (p. 35).

Thus the upshot of Plato’s exceptionally hard-pressed dialogue is to show Socrates – as seldom happens, though always (when it


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does) to highly revealing effect – in the role of a thinker caught very much on the hop by the unforeseen dialectical consequences of his own stated position. This ironic twist emerges in various ways, among them the change of sense that certain crucial words undergo as a result of their being caught up in a process of immanent self-examination or conceptual auto-critique that leads them beyond their established (and no doubt authorially intended) semantic range. They include the pair πλη′θος and πολλα′ , the former transliterating (or translating, since it has fully entered the English language) as ‘plethora’, whereas the latter signifies ‘the many’ and has achieved – aptly enough from Badiou’s viewpoint – a rather less ‘properly’ naturalized status through the phrase ‘hoi polloi’, ‘the common people’ or ‘the plebs’. His point (here summarizing rather brutally) is that Plato, for all his contrary intent, cannot help but demonstrate the con-ceptual impossibility of thinking πλη′θος as a fullness or pleni-tude of being, such as would underwrite the Platonist appeal to that realm of transcendent forms, ideas or essences which consti-tutes the sole guarantee of truth and knowledge against the ever-shifting, illusory character of sensory-perceptual experience. What Plato clearly wishes to maintain – and states as such – is the intrinsic and natural pre-eminenceof the one over the many, and hence the necessity (not merely the possibility) of gathering the plethoric multiplicity into some higher, ultimately truth-preserv-ing state of union. What the unfolding logic of the dialogue con-strains him to acknowledge – albeit despite himself – is the opposite consequence that flows from his failure to think through the required opposition between πλη′θος and πολλα′ with suffi-cient clarity or logical rigour to sustain his professed ontological commitment to the absolute priority of the one over the many.

2. Cantor: ‘theory of the pure multiple’

Badiou goes on to develop these themes from Plato in the context of modern set theory and what he proclaims as its radically trans-formative or innovative impact on our basic ontological concep-tions. Set theory has to do with relationships of membership, inclusion and exclusion among numbers or other entities that are taken as forming a unit of assessment for some given purpose. Thus sets are defined as products of the count-as-one, that is, the classifi-catory procedure that consists in grouping together a certain range

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of such entities and treating them as co-members of a single assem-blage whatever their otherwise diverse natures or properties. The latter point is crucial in mathematical terms but also for Badiou’s socio-political thinking since it allows the set theorist – or anyone who has truly absorbed its implications – to ignore any merely con-tingent or localized differences between such entities and accord them strictly equal status as regards their membership of any given set. Ironically enough, as Badiou notes, it was a point not fully taken by Cantor when he first enounced his ‘theory of the pure multiple’ and defined it as follows: ‘By set what is understood is the grouping into a totality of quite distinct objects of our intuition or of our thought’ (cited, BE, p. 38). ‘Without exaggeration’, Badiou responds,

Cantor assembles in this definition every single concept whose decomposition is brought about by set theory: the concept of totality, of the object, of distinction, and that of intuition. What makes up a set is not a totalization, nor are its elements objects, nor may distinctions be made in some infinite collec-tions of sets (without a special axiom), nor can one possess the slightest intuition of each supposed element of a mod-estly large set. (p. 38)

Of course this series of objections would also disqualify my own attempt at a brief definition as given above and along with it, truth to tell, the majority of such attempts to be found in refer-ence works and introductory texts. Indeed, if so minded, one can pick out a good few passages elsewhere in Being and Event where Badiou himself can be seen to indulge the same forbidden mode of talk. However, his purpose is to drive home the point that set theory has now progressed to a stage where it is (or should be) no longer necessary to fall back upon such notions, and more-over that the intervening post-Cantorian sequence of advances – which his book sets forth in some detail – were potentially contained within Cantor’s inaugural insight. And so it came about, Badiou writes, that ‘[a] great theory . . . was born, as is customary, in an extreme disparity between the solidity of its reasoning and the precariousness of its central concept’ (p. 38).

He reinforces this point by then describing the process of increasingly advanced and rigorous formalization whereby set


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theory was progressively uncoupled from all such naïve or restric-tive appeals to a domain of distinct objects and likewise distinct thoughts or intuitions concerning them. Above all, what Badiou seeks to dispel – not only for the benefit of relatively uninformed readers but also in riposte to some philosophers of mathematics who take a contrary view – is the idea of intuition as having any role to play in set-theoretical reasoning. Here he is in agreement with the majority of analytic philosophers who likewise adopt an extensionalist rather than intensionalist approach, that is, one that defines the conditions for membership solely and strictly with reference to the set of those entities (whatever their nature) that fall within the relevant domain, and not in terms of any qualifying attributes or distinctive features that mark them out as fit candidates according to this or that (e.g. intuitive) criterion. ‘What was thought of as an “intuition of objects” was recast such that it could only be thought of as the extension of a concept, or of a property, itself expressed in a partially (or indeed com-pletely, as in the work of Frege and Russell) formalized language’ (p. 39). If the latter project ran into problems with Russell’s dis-covery of certain paradoxes at the conceptual heart of set theory, then this was yet another indication that mathematical thinking, once launched on this investigative path, would continue to advance through repeatedly coming up against limits to its pres-ent (very often intuitive) and pointers to its future (more concep-tually adequate) state of understanding.

In effect, the very confidence initially displayed by Frege and Russell in the power of their logical language (technically speak-ing: that of the first-order quantified predicate calculus) to offer a complete and perfectly consistent formalization of the set-theoretical domain was itself a sure sign of the project’s being headed for just such an obstacle somewhere along the way. It is precisely this recurrent gesture of containment – this move to control and delimit the scope of enquiry through various tech-niques of always premature ‘totalization’ – that Badiou regards as having posed a chief obstacle to progress by evading the radical challenge that set theory presents to every existing ontological or, indeed, socio-political order. Hence his objection not only to the claim that intuition might yield valid insights or conceptual progress (since intuition is most often just the name applied to preconceived habits of belief) but also to that narrowly logicist

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idea that the exploratory scope of set theory might be circum-scribed by a purely formal programme whose terms are specifi-able in advance and which therefore pre-emptively restricts any future developments to what is conceivable at present. So it was, Badiou thinks, that the Frege-Russell project was predestined to run aground on those paradoxes that sprang to view as soon as it encountered its limit-point condition in that realm of the ‘pure multiple’ – or that formally unrestricted set-theoretical domain – which required that statements of the relevant class be open to the test of self-reflexive application.

I should offer some further detail at this stage since the epi-sode in question is among the most crucial for Badiou’s under-standing of set theory and of the complex relationship between genesis and structure that has characterized its history to date. The great promise of set-theory as envisaged by Cantor, Frege, Russell and its other early proponents was that of reducing mathematics to a purely logical or axiomatic-deductive structure of entailment relations that would leave no room for anomaly or paradox. That claim encountered its first major setback when Russell showed – by purely logical means – that set theory was intrinsically prone to generate just such problems, namely the kinds of self-reflexive, self-predicative or auto-referential para-dox that resulted from its dealing with formulas such as ‘the set of all sets that are not members of themselves’ or ‘he who shaves the barber in a town where the barber shaves every man except those who shave themselves’.9 Yet, as Badiou points out, despite their somewhat contrived appearance such paradoxes all derive from a basic formula (that of the set which is not a member of itself) which, so far from being forced or extraordinary, in fact turns up – and quite acceptably so – in each and every possible specification of a set. Thus ‘it is obvious that the set of whole numbers is not itself a whole number’, and so on for any range of similar instances (p. 40). To this extent it is an inbuilt feature of set-theoretical thought, one that arises whenever it is a ques-tion of asserting ‘the constitutive power of language over being-multiple’, and which therefore cannot be regarded as something pathological or (as Russell and Frege supposed) in need of surgi-cal excision. However, it does take on such a negative, subversive or system-threatening aspect when its implications are followed through in the context of an ultra-logicist programme which


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identifies truth with formal validity and validity, in turn, with the classical ideals of consistency and total closure under logical entailment. In that context the acceptable face of self-reference – its ubiquitous and therefore unobjectionable presence – under-goes a distinct change of expression and becomes, in effect, the un-doer of that whole optimistic logicist project.

Russell’s answer was to make it a stipulative rule that state-ments in formal languages such as those of mathematics or the logical calculus should not be self-referring in a way that gave rise to difficulties of this sort. Rather they could best be averted by a ‘Theory of Types’ which distinguished clearly between vari-ous orders or levels of statement, that is, those belonging to the first-order language of direct or material-mode assertion, those that referred to such first-order statements from a higher logical level, and so on up through successive stages of increasingly abstract formal (i.e. meta-linguistic) specification. Only thus, Russell thought, could set theory – as a crucial component of present-day developments in logic and mathematics – be kept on its path towards an ever more secure, since ever more precisely codified conception of validity or truth. Still his purported ‘solu-tion’ to these problems struck many, then and now, as objection-ably ad hoc and as having more to do with interests of pragmatic or methodological convenience than with principles self-evident to reason. Indeed, the set-theoretical paradoxes have remained a spur to philosophic thought and a potent source of speculative ideas both within mathematics and across a range of other disciplines ever since Russell first discovered them. Their impact was intensi-fied by various related developments, including – most notably – Go˝del’s undecidability-proof to the effect that any formal system sufficiently complex to generate the axioms of elementary arithme-tic or first-order logic would necessarily include or entail at least one statement the truth or validity of which could not be proved within the system itself.10 In other words, one could have either truth as matter of rigorous logical procedure or consistency (‘complete-ness’) as a matter of intra-systemic coherence but surely not both unless by some manoeuvre, like Russell’s, that looked suspiciously like a mere device for saving logico-mathematical appearances. Nevertheless set theory survived these and other challenges through the effort of various thinkers to provide some method of formal restatement in axio matic terms that would keep the paradoxes

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safely out of view or at least prevent them from doing real harm. During the past century it has become absolutely central to every branch of pure and applied mathematics, as well as to every math-ematics-based development in the physical and even (in certain contexts) the social and human sciences.

Badiou’s work is notable for not losing sight of the set-theoretical paradoxes – indeed, for placing them squarely at the centre of its philosophic interests – while regarding them more as an incen-tive to thought or a spur to renewed intellectual-creative activity than as an obstacle that has to be ignored or set aside if further progress is to be made. Thus although they ‘went on to weaken mathematical certainty and provoke a crisis which it would be wrong to imagine over [since] it involves the very essence of mathematics’, nevertheless – he asserts – the widespread accep-tance of Russell’s pseudo-solution meant that the problem with this logicist project ‘was pragmatically abandoned rather than victoriously resolved’ (p. 38). As for Cantor, Badiou sees an effort to ‘force a way through’ this looming impasse by resorting to quasi-mystical, even theologically inspired notions of absolute infinity as opposed to the realm of mathematically specifiable transfinite numbers which he himself had discovered, thereby opening up (in his famous phrase) a ‘mathematicians’ paradise’. Theology makes its re-entry to the otherwise radically de-theologized (since de-transcendentalized) realm of set theory as a result of Cantor’s retrograde tendency to equate absolute being ‘not with the (consistent) presentation of the multiple’, but rather with ‘the transcendence through which a divine infinity in-consists, as one, gathering together and numbering any multiple whatso-ever’ (p. 42). On the other hand Badiou is more than willing to credit Cantor with having grasped more vividly than any of his fellow pioneers what also drove him to seek refuge in such ‘onto-theological’ notions, namely the upshot of his own discovery when relieved of its inherited metaphysical baggage and pressed to its ultimate, strictly logical end point. Such was the incipient realization, already legible though not fully acknowledged in Cantor’s work, that any resultant (set-theoretically derived) concept of ‘being’ would resist or elude the best efforts of sys-tematic statement in terms compatible with that whole tradition of thought, whether in its mathematical, philosophic or (what effectively subtends both of these) its crypto-theological aspect.


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In this respect Cantor stands out as the most striking and, for Badiou’s purposes, the most intellectually heroic example of a thinking whose special virtue it is to confront the maximum challenge to its powers of rigorous development – a challenge as much internal or self-generated as brought to bear by external opposition – and thereby gain all the greater strength to overcome its own residual attachments and resistances. It is the same pat-tern that we have seen emerging in Badiou’s dialectical encounter with Plato, and that appears in varied forms whenever he engages with strong precursors such as Aristotle, Descartes, Leibniz, Spinoza and Hegel. Thus there are certain aspects of their think-ing that he finds problematic, unacceptable or retrograde while none the less considering their work to possess just this same self-resistant or – at their most impressive moments – self-transcending quality. It is well worth quoting another lengthy passage from his commentary on Cantor since it captures precisely what Badiou so values about those few select thinkers who, in his estimation, achieve that rank. ‘Cantor’s thought wavers’, he writes,

between onto-theology – for which the absolute is thought as a supreme infinite being, thus as trans-mathematical, in-numerable, as a form of the one so radical that no multiple can consist therein – and mathematical ontology, in which consistency provides a theory of inconsistency, in that what proves an obstacle to it (paradoxical multiplicity) is its point of impossibility, and thus, quite simply, is not. (p. 42)

What this passage displays most clearly – in his own way of thinking as well as in those aspects of Cantor’s thought that he finds exemplary despite their downright contradictory character – is the constant interplay of two terms, ‘consistent’ and ‘inconsis-tent’, as the main source of conceptual leverage or (at risk of sounding too Hegelian) the dialectical motor of Badiou’s entire project. He views the history of advances in mathematical knowledge as having most often come about through a process whereby various sorts of problem or paradox eventually gave rise to some new concept or agreed-upon way of proceeding which in turn – when its consequences became clear – could be seen to involve a further, deeper, and yet more thought-provoking chal-lenge to the enterprise in hand. If again this sounds decidedly

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Hegelian then the impression is not entirely wide of the mark although it does demand qualification in ways that will emerge later on. At any rate Badiou is absolutely firm in his belief that although knowledge must be held distinct from truth – since truth might always transcend the utmost limits of human know-ability – nevertheless knowledge is attainable, albeit with the strict fallibilist proviso that all and any present claims in that regard might conceivably be subject to future revision or out-right disconfirmation. Indeed, one of the philosophic traits that lifts his work well clear of post-structuralist, postmodernist and other recent Francophile movements of thought is Badiou’s unwavering commitment to the existence of language-independent or culture-transcendent truths and his equally strong rejection of the claim that this is in any sense a sign of dogmatism or entrenched doctrinal adherence. On the contrary, it is only by affirming that commitment and hence by conceding the possibil-ity of error in even our most deeply held theories, truth-claims or items of belief that we are saved from equating truth tout court with what counts as such for ourselves and fellow members of our own (whether specialized or culture-wide) community.11

Badiou makes this point most concisely in the context of describing those advances in the formal development (or axio-matization) of set theory that were carried through by post-Cantorian thinkers such as Zermelo and Fraenkel, the devisers of that particular version – the ZF system – that he adopts mainly on grounds of conceptual economy and ease of exposi-tory treatment. Nevertheless, as he is keen to impress upon the reader, where set theory is at issue ‘axiomatization is not an artifice of exposition, but an intrinsic necessity’ (p. 43). That is to say – contra a good many Wittgensteinians, neo-pragmatists, intuitionists, conventionalists or anti-realists – what is at stake in that process is not just a matter of finding some more convenient since compactly expressible means of formal presentation for concepts that might otherwise (and perhaps better) have been expressed in something less drastically divergent from the norms of natural or ‘ordinary’ language. Rather it is the very possibility of thinking beyond that tenacious since intuitively deep-laid and linguistically ingrained mindset which – if not subject to constant rectification and critique through just such axiomatic-deductive procedures of thought – will persist in presenting us


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with ‘common-sense’ ideas and pseudo-solutions to misconceived problems. Here again Badiou stands four-square with the ratio-nalist tradition of thought from Descartes, Leibniz and Spinoza to its latter-day progeny among both mainstream analytic phi-losophers who continue the basic Frege-Russell programme and also those French thinkers – Bachelard, Canguilhem, even the deeply Spinoza-influenced Marxist theoretician Louis Althusser – who stake their projects on the capacity of thought to transcend the deliverances of mere intuition or received (linguistically ensconced) doctrine.12 By the same token he stands just as squarely opposed to those latter-day ‘sophists’ – as distinct from the more intellectually reputable since dialectically challenging company of ‘anti-philosophers’ – who take refuge in just such sources of false assurance or just such appeals to the delusory idea of a wisdom vested in ‘ordinary language’ and its associated customary ‘forms-of-life’.

Nowhere is the fallacy of this way of thinking more clearly shown up, so Badiou maintains, than in the context of develop-ments in set theory and the problems faced by philosophy of mathematics to the extent that it strives to account for those developments in conceptually adequate terms. His preference for ZF over rival systems has to do precisely with its pressing as far as possible in this direction, avoiding all forms of premature conceptual (or ontological) commitment, and thereby faithfully pursuing what Badiou sees as the path of thought laid out for set-theoretical enquiry. This it does by allowing just one relation between terms – that of belonging, represented by the symbol ∈ – and excluding all reference to other properties that would bring such otiose commitments along with them. The purpose of adopting this austere approach is to avoid the constant tempta-tion (as witness Cantor’s ‘theological’ turn) of regressing to a more intuitively manageable concept of set theory which contin-ues, in the classical manner, to distinguish between objects, mul-tiples, multiples of multiples and so forth. Thus ‘[w]hen I write “α belongs to β, α ∈ β, the signs α and β are variables from the same list, and can be substituted for by specifically indistinguish-able terms’ (p. 44). That is to say, on the ZF system it is easier to conceive how thinking can dispense with the intuitively self-evident distinction between ‘individual’ objects and groups of objects, or particular (discrete) sets and assemblages composed

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of multiple sets under some higher level grouping principle. Moreover it leaves no room for what seems – on a more conser-vative or intuitive account – the self-evident truth that logically there must be a distinction between elements and the sets to which those elements belong or in terms of which they are speci-fied as elements. Indeed it is at just this point that set theory in its more developed forms departs from the ‘naïve’ or still intuit-able stage at which Cantor remained through his supposition that to think of sets was necessarily to think of them as entities that differed, ontologically and logically speaking, from the ele-ments that made them up. In other words, ‘[t]he sign ∈, unbeing of any one, determines, in a uniform manner, the presentation of “something” as indexed to the multiple’ (p. 44). What set theory most notably – and to some thinkers most disturbingly – con-jures up is the prospect of a bad infinity or a multiple that is not composed of so many fixed or definable units but must rather be thought of as a ‘uniformly pure multiplicity’ without any clearly specifiable constituent parts.

Here Badiou offers the suggestion that ‘[i]f one admits, with a grain of salt, Quine’s famous formula “to be is to be the value of a variable”, one can conclude that the ZF system postulates that there is only one type of presentation of being: the multiple’ (p. 44). This remark is best read as conveying a certain sympathy, on Badiou’s part, with the ‘austere desert landscapes’ that Quine famously preferred to the lush vegetation of more ample or profligate ontologies.13 However, the ‘grain of salt’ serves to indi-cate – distinctly in tension with that – a clear sense of just how restrictive is Quine’s echt-analytic desire to prohibit any reckless ontological ventures beyond the safe (‘scientifically’ validated) ground of a quantified first-order predicate logic coupled with a radically empiricist conception of epistemic warrant. That is to say, Badiou is by no means averse to the formal rigour or the extreme ontological austerity of Quine’s approach, accordant as it is with his own professed aims of giving logic precedence in all matters of ontological enquiry and moreover restricting such enquiry to what can be said – consistently maintained – on the basis of a disciplined investigation into the various set-theoretically thinkable modes of being. However he is sharply at odds with Quine in just about every other respect, including his commit-ment to a rationalist conception of ontology that could scarcely


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be further from Quine’s outlook of radical empiricism. Equally un-Quinean – and likewise reflecting his distinctly ‘continental’ angle of vision – is Badiou’s conception of progress in the formal (as well as in certain branches of the physical) sciences as typically powered by conflicts, anomalies or moments of productive fric-tion between the drive for consistency and that which will always elude or subvert any fully consistent methodology or set of results.

It is just this idea of a constant dialectical tension intrinsic to the very nature of thought – rather than of problems that crop up periodically and that have to be resolved before thinking can once again proceed along its appointed path – that Badiou finds most compellingly enacted in the sequence of set-theoretical advances from Cantor down. He makes the point with reference to Zermelo’s principle – a main component of the ZF system – that ‘a property only determines a multiple under the supposition that there is already a presented multiple’ (p. 45). That is to say, any imputed feature or attribute pertaining to some given mem-ber of some given set and on the basis of which their membership is taken to depend must itself suppose a pre-existent multiplicity sub-ject to no such selective constraint and therefore – by definition – more numerous or inclusive. Here again, in this idea of what is suppressed or marginalized by any determinate (e.g. ‘democratic’) instance of the count-as-one, we may glimpse some of the political or socio-critical implications that Badiou will go on to draw from his set-theoretical elaborations. In formal terms, ‘Zermelo’s axiom system subordinates the induction of a multiple by language to the existence, prior to that induction, of an initial multiple’ (p. 45). ‘Language’ here presumably includes not only those varieties of natural language to which, as we have seen, Badiou accords no authority in such matters but also those formal or regimented lan-guages – like that which he shares with Quine, that is, the language of the first-order quantified predicate calculus – whose very consis-tency is such as to ensure that they can serve only in a strictly heu-ristic, assistive or enabling (though also a strictly indispensable) role. For they will always result from the suppression of – and therefore, at certain crucial junctures, be subject to disruption by – an inconsis-tent multiplicity that cannot be fully grasped or encompassed but only more-or-less drastically reduced to order by any application of the count-as-one.

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What occurs at such moments is an especially forceful demonstra-tion of the truth that applies always and everywhere in matters of ontological import but which is mostly concealed – repressed or glossed over – by philosophic doctrines or ‘common-sense’ ideas premised on the plenitude or the positivity of being. It has to do with the essentially ‘subtractive’ character of ontological enquiry and the impossibility that thinking should ever fully coin-cide with the contents of thought as given by intuition, by language, or by any supposedly consistent apparatus of formal concepts that fails (or programmatically declines) to make allowance for that which might always elude its foregone systematic grasp. This is why such truths are visible only in the fissures, contradictions and aporias that mark the great majority of texts in the Western philosophical canon. The exceptions are those very few thinkers – Plato (if unwill-ingly) and certain set-theoreticians among them – who pressed the dialectic of being and non-being to its logical conclusion and also those philosophers such as Aristotle, Descartes, Leibniz, Spinoza and Hegel who – albeit in radically different ways – bore witness to the limits of a positive ontology through their failure to express or consistently articulate the conditions under which it might be achieved. Hence, according to Badiou,

the whole problem of the subtractive suture of set theory to being qua being. It is a problem that language cannot avoid, and to which it leads us by foundering upon its paradoxical dissolution, the result of its own excess. Language – which provides for separations and compositions – cannot, alone, institute the existence of the pure multiple; it cannot ensure that what the theory presents is indeed presentation. (p. 48)

This distinction between ‘what the theory presents’ and ‘presen-tation’ in another (ontologically prior) sense of the term takes us to the heart of Badiou’s political as well as his ‘purely’ mathe-matical and formal-philosophical thinking. ‘What the theory presents’ is what finds an accredited, duly acknowledged place in those various prevailing systems (mathematical, formal and natural-scientific but also – by more than suggestive analogy – political and socio-cultural) that decide what shall count as a member or constituent of some given set, group or class. What the term ‘presentation’ signifies, on the other hand, is the totality


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of those elements that offer themselves as potential candidates for membership, whether or not that potential is realized by their actually being so treated.

Hence Badiou’s central thesis in the formal (i.e. the ontological and set-theoretical) domain: that even though ‘inconsistency is not actually presented as such since all presentation is under the law of the count’, nevertheless ‘inconsistency as pure multiple is solely the presupposition that prior to the count the one is not’ (p. 52). Or rather, it is just because the first of these claims can be shown to hold – shown (that is) through Badiou’s elaborate work-ing-through of the set-theoretical paradoxes – that the second claim also goes through. His point, to repeat, is that the one is always the result of some such counting operation brought to bear in the act or through the process of transforming an incon-sistent into a consistent multiplicity, or deciding which elements shall count as members and which be consigned to the limbo of non-belonging. At the same time this central truth of ontology – the truth of its essentially subtractive character – is concealed from most enquirers simply through the fact that by very defini-tion those excluded elements cannot figure within the count-as-one or be perceived as integral or constituent parts of any existent situation. Thus ‘[n]othing is presentable in a situation otherwise than under the effect of structure, that is, under the form of the one and its composition in consistent multiplicities’ (p. 52). From which it follows that only within the discourse of mathematics and the formal sciences – that is, within the ambit of those disci-plines most readily amenable to set-theoretic formalization – can thinking resist the otherwise inevitable tendency to recognize only those elements that make up some known or acknowledged situa-tion and hence to ignore whatever eludes, escapes or exceeds the prevalent count-as-one. ‘Any situation, seized in its immanence, thus reverses the inaugural axiom of our entire procedure. It states that the one is and that the pure multiple – inconsistency – is not’ (p. 52). However – and this claim is absolutely central to what I shall describe as Badiou’s realist ontology, although he might well have certain misgivings about that description – the truth of such a situation is in no way dependent on what we may perceive, rec-ognize, believe or take ourselves to know concerning it.

This conception of truth as always potentially surpassing our best attainable state of knowledge – in the jargon, as ‘recognition-

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transcendent’ or ‘epistemically unconstrained’ – is one that unites Badiou with many realists in the analytic camp, whatever his differences with them in other regards.14 Moreover it is one in the absence of which his project would utterly founder since it would lack any means to explain how thought can advance through the process of discovering – rather than inventing – those anomalies and conflicts that previously passed unnoticed but which then at a certain point emerged clearly to view and set the conditions in place for re-thinking the issue at hand. What is also required in order for this to occur is a reversal, however short lived, of the imperative that governs most thinking at most times in most areas of thought, namely that such thinking be conducted very largely in terms of consistent multiplicity or structured situations so as to gain sufficient purchase on its vari-ous object-domains. Thus ‘[i]n a non-ontological (thus non-mathematical) situation, the multiple is possible only insofar as it is explicitly ordered by the law according to the one of the count’. And again, ‘[i]nside the situation there is no graspable inconsistency which would be subtracted from the count and thus a-structured’ (p. 52). However this restriction may be lifted, to some extent at least, insofar as thinkers in other disciplines acquire the conceptual resources made available by develop-ments in post-Cantorian set theory and thus come to grasp the basic point: that if the one is what results from some previous operation, then ‘of necessity “something” of the multiple does not absolutely coincide with the result’ (p. 53).

Indeed it is precisely in the need for such an operation – the inability of thought to achieve a proper sense of conceptual pur-chase except on condition of reducing inconsistent to consistent multiplicity – that the ‘something’ in question most strongly manifests itself as preceding and exceeding the count-as-one. The ‘wavering towards inconsistency’ is something that Badiou detects across a wide range of philosophical texts where the overriding drive for system and method – or (in Heidegger’s case) for access to a realm of ontologically authentic Being beyond the merely ontic or quotidian – is allowed to subdue any countervailing sense of that which would otherwise resist such appropriation. This applies especially to programmatic thinkers such as Spinoza and Leibniz whose ruling premise is that truth must be expressible in terms of a consistent, logically articulated


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system of propositions that admits of no internal gaps, discrep-ancies or other such faults and which thus stands proof against criticism or indeed – by implication – against any further progress beyond its self-achieved stage of advance. Nevertheless, as Badiou will endeavour to show, their projects encounter just the kind of resistance from internal anomalies – most often from unresolved conflicts between the large-scale (purported) logical structure of their argument and its detailed working-out – which is only to be expected given his claim concerning the ultimate prevalence of inconsistent over consistent multiplicity, and hence the ubiqui-tous (no matter how elusive) remainder or reminder of the ‘supernumerary’ element that haunts all systematic discourse.

3. Plato and the unpresentable

One striking example that Badiou adduces is Plato’s strange and in many ways untypical dialogue Timaeus where he advances a highly speculative piece of metaphysical argument – with distinct echoes of Anaximander and the pre-Socratic thinkers – concern-ing the emergence of order from chaos, of the many from the one, of phenomenal appearances from that which underlies and makes possible those appearances, and of sexual difference from a prior state of hermaphrodite dual gender.15 Badiou’s interest here is not so much in Plato’s ‘great cosmological construction’, although he is strongly drawn to works of this kind (grand efforts in the system-building vein like Spinoza’s Ethics or Leibniz’s Monadology) while looking out for all the tell-tale points at which their more ambitious or programmatic claims can be seen to break down on recalcitrant details and twists of argumenta-tion. Rather he is intrigued by what Plato – in the opening para-graph of Timaeus – refers to as the ‘errant cause’ and invokes as part of his attempt to explain how the cosmos originally came into being through a combination of chance, necessity and the working-out of a rational scheme or underlying principle of order that would somehow reconcile the warring claims of free-will and determinism. At this point Plato is led or obliged by the logic of his argument to introduce a term that seems to involve the familiar – if endlessly debated – notion of causation but under a guise (that of ‘errancy’) that marks it out as no normal or familiar case of the kind. In his Spinoza commentary, later on, Badiou will locate this ‘wandering’ and strictly ‘unpresentable’

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(since thoroughly anomalous) point of excess – or rather of lack, insofar as it marks what is superfluous and redundant but also, for that reason, inadequate to the system’s needs – in the Spinozist notion of ‘infinite modes’ which, like the Platonic ‘errant cause’, proves stubbornly resistant to a rational accounting on system-compatible terms.

What thought thus encounters in its dealing with these texts is the ‘void’ to which Badiou makes constant reference in his writ-ing on set-theoretical matters and also – via the series of closely argued connections that I have sketched out briefly above – his thinking in the socio-political sphere. Thus the void is that which cannot be conceived except in subtractive terms, or as a ‘non-one’ that eludes the count insofar as it both precedes the operation by which ‘inconsistent’ is converted to ‘consistent’ multiplicity and remains as a kind of internal exile or – in Badiou’s striking ana-logy – a (non-)member of the (non-)collectivity of sans-papiers whose absence from any official tally or electoral roll renders them publicly invisible. However, this is not to deny that they may yet exert a considerable power of resistance and disruption through their standing as blatant exceptions or counter-examples to the much-touted rhetoric of equality, freedom, participant democracy, social inclusion and universal human rights that – at least in the present-day French and similar contexts – forms the constant refrain of governments and political parties from main-stream left to mainstream right.16 So there is a political as well as mathematical question involved when Badiou puts the case for a strictly extensionalist approach that would take absolutely no stock of any attributes, properties or distinguishing (i.e. qualify-ing) features that might otherwise – on the opposite (intension-alist) account – play a crucial part in deciding issues of set-theoretical membership. All the more must this apply to the non-members or to those excluded from the count on whatever legal, ethnic, cultural or linguistic grounds since the main reason for their being so placed is their imputed lack of those (supposed) identifying features that are taken to mark out members in good legal, ethnic, or cultural-linguistic standing. Thus, ‘[i]f the void is thematized, it must be according to the presentation of its errancy, and not in regard to some singularity, necessarily full, which would distinguish it as one within a differentiating count’ (p. 57). For Badiou, there is nothing so politically and ethically


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retrograde – indeed so potentially lethal – as a version of the count that looks to such ‘singular’ (though type-indicative) fea-tures as a means of sorting those that belong from those that belong outside or elsewhere.

This is why he can say – at risk of being charged by analytic phi-losophers with flouting the fact/value distinction or claiming to derive an ‘ought’ from an ‘is’ – that ‘for the void to become localiz-able at the level of presentation, and thus for a certain type of intra-situational assumption of being qua being to occur, a dys-function of the count is required, which results from an excess-of-one’ (p. 56). The obverse of this is of course the implication that what is required – ethically or politically – in order to promote the interests of justice in an unjust society is a stronger, theoretically informed grasp of what is wrong with any thinking premised on ideas of identity or difference as intrinsic to the individuals or spe-cific relationships concerned. In which case those interests could be further advanced through an increased understanding, suchas he provides in Being and Event, of the various set-theoretical con-cepts (especially the opposed pairs of inclusion/belonging, part/member and inconsistent/consistent multiplicity) along with the relevant formal procedures whereby they might serve to demon-strate the disparity between any given situation and what that situ-ation holds in the way of unrecognized, unacknowledged or ‘uncounted’ parts. This implication Badiou takes to be already contained in Cantor’s Theorem – the starting point of all set the-ory – according to which the power set (i.e. the set of all subsets) of any given set has a greater cardinality (i.e. numerical value) than the set itself. Thus the latter will include not only all its members counted one-by-one but also the various combinations (subsets) into which those members may be placed, thus yielding an excess of subsets over set that grows exponentially with the size of the original set right up to the multiple orders of infinity where of course it surpasses the utmost powers of any finite enumerative reckoning.17 Paradoxically enough it is the knowledge of this shortfall in the present capacity of thought to encompass such truths or to bring them within the scope of attainable knowledge that Badiou sees as driving the constant quest for more powerful, refined and elaborate modes of set-theoretical investigation.

Nor indeed should this claim seem so very paradoxical if one considers the copious evidence from every branch of the formal,

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natural and human sciences that new discoveries are most often sparked by some increasingly acute sense of presumptive anom-aly. That is to say, they result from a grasp of possible or likely future development that cannot amount to knowledge – since it cannot so far be specified with adequate precision – yet the pros-pect of which is logically entailed by certain gaps, defects or dis-crepancies in the present state of understanding. Hence Badiou’s claim, in Being and Event, that the history of advances in set the-ory exemplifies this process to uniquely impressive and convinc-ing effect since it shows how such advances can come about through the process whereby thinking transcends those obstacles that are constantly placed in its path through its residual attach-ment to limiting conceptions of its own investigative scope. Among them – and most sharply limiting – is that idea of the relevant criteria for set-theoretical inclusion or exclusion which takes this to be crucially a matter of the possession or non-pos-session of certain distinctive or qualifying features. On the con-trary, the proven capacity of set-theoretical reasoning to overcome its various challenges or setbacks to date has resulted from its perfect indifference to any question of what makes up the multiples (and multiples of multiples) that fall within its formal domain. In short, ‘the attribute “to-be-a-multiple” tran-scends the particular multiples which are elements of a given multiple’ (p. 65). Which is also to say – with fairly obvious rele-vance to the social and political dimension of Badiou’s thought – that any issue concerning the status of elements will have nothing to do with their intrinsic natures, real or imputed, but solely with their set-theoretical relation to the multiples that include or exclude them as members and the multiples or elements which they in turn include or exclude.

At some risk of simplification one could say that this is the direct equivalent, in formal terms, of Badiou’s strongly univer-salist stance in matters of ethics and politics, and – by the same token – his firm rejection of those various strains of ‘radical’ difference-thinking (whether in epistemology, ethics, politics, post-colonial discourse or gender studies) that have lately cap-tured the high ground in critical and cultural theory.18 Indeed it is the main reason for his choice of mathematics, rather than language, as the basis on which to construct his critical ontology and also his account of those fourfold other subject-domains


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which can be conceptualized only in relation to that ontology even if their major occurrences belong to an order of ‘events’ that inherently eludes the grasp of any pre-existent ontological scheme. Where language – or the normative appeal to language – enjoins us to accept what is presently sayable, describable, or representable as fixing the ultimate scope and limits of truth or intelligibility mathematics on the contrary allows – indeed requires – that we surpass those limits and conceive what may extend that scope beyond anything yet achieved or remotely envisaged. This it does through the presence of those various anomalies, conflicts or unresolved dilemmas that will always exist so long as mathematics remains a live and intrinsically a problem-solving activity of thought. But there is also something peculiar to the nature of mathematical (more specifically of set-theoretical) thinking that reinforces the decisive advantage it enjoys over language as a source of ontological advances and a spur to new insights in those other subject-domains. This is its unique capacity for grasping the ‘subtractive’ dimension of being, that is, the ‘axiom of the void set’ according to which ‘there exists a set that has no elements’, one that is strictly ‘unpre-sentable’ in terms of any given ontology or particular instance of the dominant count-as-one. Moreover it is another basic axiom that the void (or null) set is necessarily a member – an integral or constitutive member – of every other set whatsoever. Thus it fig-ures to Badiou’s way of thinking – though also in set-theoretical discourse generally – as that which grounds the entire project (since all those other sets are built up successively by a process of recursive extrapolation from the null set) yet that which by very definition eludes the grasp of any definite, positive or clearly intuitable order of conceptualization.

Indeed, Badiou claims that this elusiveness can be seen to complicate a good many of the statements put forth by the pro-ponents of set theory, including the above-cited (pretty much canonical) formulation to the effect that ‘there exists a set which has no elements’. For of course any notion of ‘existence’ here is one that needs qualifying to the point of near-withdrawal since it runs flat up against the seeming contradiction, familiar from Plato’s Parmenides, involved in all forms of negative existential statement. However, as we have seen, Badiou is enough of a Hegelian – and ipso facto enough of an anti-Kantian – to take

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this not as a disabling dilemma or a sign that thought is over-stepping its appointed bounds but rather as a spur to thinking through and beyond what constitutes an impasse only on such a static conception of the scope and limits of reason. According to his alternative dialectical conception it is at just such points of seeming deadlock or aporia that thought typically turns that situation to advantage and achieves the leap to a terra incognita or whole new tract of hitherto unexplored but henceforth richly fertile terrain. Above all such advances have come about through the increasing centrality of that which Badiou designates the ‘subtractive’ dimension of mathematical thought, namely its encounter with the various crises periodically induced by what-ever is excluded from its zone of operations yet continues to exert an unsettling and hence a potentially transformative power. Such was the effect of those early developments in set theory that led via an ironic repetition of the paradoxes first aired in Plato’s Parmenides to a recognition of the null set as both the founding concept and, so to speak, the immanent driving force of all further advances, refinements and discoveries. It is at just this point, according to Badiou, that ‘the subtractive character of being causes the intuitive distinction between elements and sets to break down’ (p. 67). What we are thereby obliged to think – conceptually ‘forced’, as Badiou phrases it, deploying this term in the technically precise sense proposed by Cohen – is the abso-lute and principled impossibility of drawing that intuitively self-evident and yet, in this context, logically insupportable distinction. Which is also to say, in more positive terms, that what we are thereby enabled to conceive is the possibility that thinking can advance through a grasp of non-intuitive (even strongly counter-intuitive) truths that are manifest precisely in those conflicts, inconsistencies, conceptual tensions, or failures of logical coher-ence which signal the advent of a crisis in knowledge but also – and solely on that condition – the prospect of new discovery.

Badiou is well aware that such ideas can very easily be miscon-strued or tendentiously deployed in the service of quasi-mystical or negative-theological claims. The former have their chief source in the ancient Greek precedent of Pythagorean numerological doctrine – itself a strong influence on Platonist metaphysics and epistemology – whereas the latter, as we have seen, find expres-sion through Cantor as well as in the much-quoted statements


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of a good few other mathematicians and scientists. Badiou addresses this (to him) highly suspect tendency in one of the more cryptic paragraphs of Being and Event where he seeks to clarify the truly ‘remarkable conclusion’ arrived at by this stage: that ‘it is because the one is not that the void is unique’ (p. 69). Its uniqueness makes it a ‘proper name’ in the sense that, unlike all other set-theoretical entities, it can be picked out or specified – ‘marked’, in Badiou’s terminology – only by an act of pure des-ignation or ostension, rather than by some identifying list of its attributes, properties or status with regard to other sets (that is, all sets whatsoever) in which it is included as a member or in which it is included as a member. This gave rise, understandably, to a certain mystique of ineffability that brought it within the rarefied ambit of negative-theological thought. If those early mathematicians ‘had to shelter their own audacity behind the character of a forgotten language’ then this was in one sense an irrational regression to theological or mystical ideas that had no legitimate place in mathematical thought. Nevertheless their sense of the numinous and feelings of awe in this regard were justi-fied at least insofar as they recognized, however ‘dully’, the power of thought when it learns to proceed through the via negativa of a quest for truth that renounces any positive grounding in the sup-posed warrant of sensory-perceptual, commonsense-intuitive, or even (what passes for) conceptual or a priori self-evidence.

Badiou is a resolutely secular thinker though one who takes pretty much for granted the present-day obsolescence of the monotheistic religions – at least for the purposes of serious (truth-based or truth-oriented) philosophical, moral and cultural debate – rather than stoking the fires by adopting a high-profile atheist stance. All the same, he does find something exemplary for his own purposes in the figure of a religious ‘militant’ such as St. Paul, one whose crusading zeal (whatever its far from benign historical and cultural consequences) marked not only an irrup-tive event of the first magnitude but also – accordant with Badiou’s staunchly universalist precepts – a belief in the charac-ter of truth as transcending all merely national (e.g. Hebrew or Greek) ideas of cultural-linguistic belonging.19 In Pascal likewise, as we shall see, he discovers the paradigm case of an ‘anti-philosopher’ whose commitment to the precept that reason should always give way before the dictates of religious faith

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would appear to be drastically at odds with his own thinking (pp. 212–22). Yet, according to Badiou, Pascal’s work – in math-ematics and logic but also in its ethical and even its existential-theological aspect – is much better viewed in light of its power to challenge, provoke and ultimately vindicate philosophy’s universalist claims. For it is here, in the kinship between mathe-matics and a supra-cultural conception of political justice, that we should seek to make sense of Badiou’s above-cited statement to the effect that any rigorous reflection on the void as it figures in the progress of mathematical thought will reveal such thought as ‘liminal to language’ in ways that are analogous (though cru-cially not identical) to the pseudo-logic of negative theology. That the latter is indeed a pseudo-logic and shown up as such by contrast with the history of demonstrable gains or advances in the field of mathematical enquiry is explained, for Badiou, by the fact that negative theology.

Always involves a covert appeal to that which supposedly (if indescribably) sustains or underlies the various predicates that fail to capture its essence. Thus it represents a falling-back into that ‘presentifying’ mode of thought – that deep-laid set of ontological assumptions premised on the positive rather than the negative or subtractive nature of being – which had for so long acted as an obstacle to thought in the realm of mathematics and the formal sciences. What is required in order to remove that obstacle is a strictly axiomatic truth-procedure which ‘cannot propose anything in particular’, and can therefore ‘neither be a matter of the one, which is not, nor of the composed multiple, which is never anything but a result of the count, an effect of structure’ (p. 66).

4. Plenum and void: why Aristotle hated a vacuum

It is at this stage in the unfolding of his argument that Badiou turns to Aristotle as one of those thinkers with whom he is pro-foundly at odds in relation to some basic metaphysical and onto-logical commitments but whose thinking he typically treats as a challenge to be met so far as possible on shared argumentative ground rather than dismissed out of hand. The main point at issue is Aristotle’s denial that there could possibly exist any such thing as a void or vacuum in nature.20 This was a doctrine – on essentially metaphysical grounds – that continued to enjoy the


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authority of something like holy writ throughout the medieval and renaissance periods in European thought, and which began to lose ground only with the rise of early modern science and its rejection of scholastic dogma in favour of experimental methods that involved (in this case) the construction of increasingly effec-tive vacuum pumps. In recent years this received view has itself been subject to challenge from ‘strong’ sociologists of knowledge who reject the idea of scientific progress as a piece of self-serving mythology and who view the issue of experimental physics versus Aristotelian doctrine – or (in the classic encounter of this kind) of the seventeenth-century physicist Robert Boyle versus the philosopher Thomas Hobbes – as a case of two conflicting ideo-logies, rather than a conflict between scientific method and a legacy of outworn metaphysical baggage.21 Badiou’s aim is not so much to adjudicate in this quarrel (which has surely been set-tled on physical-scientific terms) but to shift the whole ground of debate and define what precisely Aristotle meant when he wrote of the non-existence of the void as a matter of metaphysi-cally ordained or a priori demonstrable truth. ‘For the Greek’, Badiou writes, ‘the void is not an experimental difference but rather an ontological category, a supposition relative to what naturally proliferates as figures of being’ (p. 71). That is to say, no merely ‘artificial’ creation of a void, such as Boyle was much later to achieve – albeit, on his own admission, to a very imper-fect or partial degree owing to the drastic limitations of available technology – could possibly count as refuting the Aristotelian doctrine, based as that doctrine was on a quite different order of purely ex hypothesi speculative reasoning that brooked no such empirical counter-evidence.

Badiou’s point here is not, of course, to uphold Aristotle’s claim against the massive, strictly unignorable weight of accu-mulated physical-scientific evidence nor indeed – as should be obvious by now – to defend it as a matter of prior metaphysical or ontological commitment. After all, nothing could be further from Badiou’s absolute and principled insistence on the essen-tially subtractive nature of ontology – the primacy of that which eludes specification in positive (‘presentifying’) terms – than the Aristotelian doctrine that ‘nature abhors a vacuum’, or that talk of the void must involve either scientific absurdity or logical self-contradiction. However this is to commit something like a

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basic category-mistake since, in Aristotle’s thinking as Badiou understands it, ‘the artificial [i.e. experimental] production of a void is not an adequate response to the question of whether nature allows, according to its own opening forth, “a place where nothing is” to occur, because such is the Aristotelian definition of the void’ (p. 71). He is drawing quite explicitly on Heidegger’s reading of Aristotle here, and this is indeed – so far as I recall – the most markedly Heideggerian passage of commentary anywhere in Badiou’s work. All the same it is not so purely echt-Heideggerian as to conjure a depth-hermeneutic realm of metaphysically unencumbered Being that would constitute both the primordial, long-forgotten source and the concealed ontological ground of all those merely ontical beings that science has taken for its object-domain. On the contrary, what most engages Badiou’s interest is Aristotle’s axiomatic mode of reasoning from first prin-ciples which, although sharply opposed to his own, can none the less be seen to have certain implications that are strikingly germane to much of what Badiou has to say on his three main topics of the void, infinity and the event.

Thus if Aristotle’s theses have not been refuted on their own metaphysical ground by subsequent advances in physical-scientific understanding then this is not because, as Heidegger would have it, the physical sciences (like philosophy) have long been mort-gaged to a technocratic will-to-power and an epochal oblivion of Being but rather because Aristotle’s reasoning places him in the company of those – Badiou included – ‘for whom the void is in truth the name of being, and so can neither be cast into doubt nor established via the effects of an experiment’ (p. 72). That is, any refutation of Aristotle’s doctrine at the level of ontological enquiry here in question – as distinct from the physical-scientific level where its claims can no longer stand up – will have to go by way of a critical engagement with the underlying logic or onto-logic of Aristotelian physics. It will therefore not be required to take sides on the kind of issue nowadays engaged between real-ists who suppose Boyle to have got it right – or at least to have been very much on the right scientific track – in asserting the pos-sibility of a vacuum and those on the opposite (strong-sociological or cultural-constructivist) wing who consider Boyle’s claim and Hobbes’s denial of it to be strictly on a par as regards their truth-content since each was the product of a certain ideological or


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socio-cultural mindset. For the record, one can state with a fair degree of assurance that Badiou would lean strongly in the for-mer (scientific-realist and rationalist) direction. However, this is not – or not primarily – where his interests lie. Rather he seeks to demonstrate that Aristotle’s staunch, metaphysically grounded opposition to any idea of the void as existing in nature or as having any legitimate place in the conceptual apparatus of the various physical or human sciences was in fact the result of his perceiving very clearly where such thinking led and the kinds of paradox to which it would surely give rise.

Thus in Aristotle’s case it is a consequence of his basic prem-ise that to entertain a notion of the void as anything other than a sheer impossibility or affront to rational thought is ipso facto to invite all the massive conceptual, metaphysical and ontolo-gical problems that come in its train. Chief among them are the problems of spatio-temporal indifference (since the void admits of no distinctions in this regard), un-measure (since it likewise prevents any meaningful comparison between different dimen-sions, velocities, or other such quantitative attributes), and above all infinity (by reason of its forcing thought to confront that intrinsically disturbing since – for Aristotle and even for Cantor – inherently excessive or paradox-inducing idea). This is why, as Badiou cautions, we should not be ‘led astray’ by physics ‘in the modern sense’, that is, by the otherwise spectacular self-evidence of past and presently continuing progress in the physical sciences. What Aristotle is asking us to think is something quite different: namely, that ‘every reference to the void produces an excess over the count-as-one, an irruption of inconsistency, which propa-gates – metaphysically – within the situation at infinite speed’ (p. 75). In which case – he concludes – ‘the void is incompatible with the slow order in which every situation re-ensures, in their place, the multiples that it presents’. It can only be conceived in terms of a potentially subversive or destabilizing threat to the entire onto-metaphysico-epistemological structure of thought that constitutes Aristotelian philosophy in its various specific regions of enquiry. When Aristotle says that ‘the void bears no ratio to the full, such that neither does movement [in the void]’, or again that ‘there is no ratio in which the void is exceeded by bodies, just as there is no ratio between the nothing and num-ber’, Badiou would scarcely take issue with the content of these

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statements – bearing out as they do his own central theses with regard to the subtractive nature of ontology – but only with Aristotle’s belief that they constitute a case by reductio ad absur-dum against the possibility of a void.

Thus he fully endorses the reasoning behind this Aristotelian doctrine – ‘to my mind, the ensemble of these remarks is entirely coherent’ (p. 74) – but takes it to demonstrate just the opposite of Aristotle’s intended point, namely that the impossibility in question is that of adequately conceiving the void in terms of a ‘presentifying’ metaphysics and ontology, rather than a concep-tual impossibility tout court. That we can grasp this distinction now where Aristotle couldn’t, despite being carried so far towards it by the rigour and consistency of his own logic, is mainly the result of our coming at the issue from a standpoint informed by set-theoretical concepts and techniques, above all with regard to infinity and transfinite mathematics. Moreover it enables us to read the works of earlier thinkers – among them (most impor-tantly for Badiou) Plato, Aristotle, Leibniz, Spinoza, Rousseau, Hegel and Heidegger – in such a way as to locate the various symptomatic tensions, conflicts and blind-spots of unexamined presupposition that mark their discourse where it touches on themes that involve some dealing with those problematic topoi and is thus forced up against its own conceptual limits.

I should make it clear – lest anything I have said give rise to the contrary impression – that Badiou is as far as possible from implying that his approach to these thinkers could fairly be described as a ‘psychoanalysis of philosophy’. Indeed it is one of his leading claims that philosophy should know its proper place not only in the sense of accepting a strictly ancillary role vis-à-vis the ground-breaking ontological work of mathematics but also in the more positive sense of maintaining its crucial measure of autonomy vis-à-vis those various enabling ‘conditions’ which might otherwise compromise its critical independence or intel-lectual integrity. All the same, as will emerge more explicitly in Part VIII of Being and Event when he turns to Freudian–Lacanian themes, Badiou’s practice of symptomatic reading is one that has much in common with the kinds of critical-philosophic dis-course that resulted from the structuralist-inspired rapproche-ment between Marxism and psychoanalysis from the mid-1960s on. That rapprochement came about very largely through the


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intensive theoretical labours of Louis Althusser, and it is thus worth noting that Badiou – despite his express opposition to the linguistic turn in its structuralist as well as its Wittgensteinian and other forms – is among the very few major present-day think-ers who have retained at least a qualified allegiance to the Althus-serian project.22 Indeed this is one striking example of what Badiou means by the fidelity to certain truth-procedures (whether in mathematics, science, politics or art) where the upshot – the prospect of their being carried through to a successful conclu-sion – may be highly uncertain, even subject at times to severe or calamitous setbacks, yet where the stakes are sufficiently high to warrant that kind of long-term commitment. This is why he is so drawn to thinkers, like St. Paul and Pascal, with whom he would appear to have little in common politically, ethically or philosophically. What they share is the distinctive coupling of a well-nigh existentialist conception of truth as a matter of authen-tic individual dedication to the project in hand with a strongly universalist claim to the effect that, should the project at last be vindicated, then its truth or validity conditions will apply across every kind of social, cultural, political, ethnic or other such restrictive boundary.

This conjunction of seemingly opposite doctrines will appear less strained or downright contradictory if one reflects on the way that mathematical discoveries – such as Andrew Wiles’s celebrated recent proof of Fermat’s Last Theorem after more than three centuries of intensive effort by numerous dedicated individuals – may well involve personal commitment on a quite heroic scale of mental and physical endurance and yet, once established, hold good without regard to any such (now) extra-neous facts about their psycho-biographical genesis.23 Not that Badiou would subscribe unreservedly to the distinction – com-monly advanced or assumed among analytic philosophers of science – between ‘context of discovery’ and ‘context of justifi-cation’, or the various circumstances under which some discov-ery may have come about and the various standards of empirical, logical, inductive, predictive or causal-explanatory justification that led to its widespread acceptance in the scientific community.24 His approach differs from theirs mainly in the much greater weight he attaches to questions of historical development – that is, the genealogy of set-theoretical concepts, techniques and

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proof-procedures – and also, consistent with that, his greater emphasis on how such discoveries can be shown to have occurred through certain highly specific procedures of thought on the part of certain mathematicians confronted with certain, likewise specific problems or obstacles to progress. On the one hand this is why he can cross disciplinary boundaries so as to stress the closely analogous relationship between cutting-edge work in mathematics, requiring as it does the highest degree of dedication to a given truth-procedure, and those other chief fields of human endeavour where the demands of rigorous and consequent thought go along with the demand for unswerving fidelity or single-minded commitment. On the other hand, it is why he comes out in such firm opposition to the idea that truth in any of those subject-areas might properly be thought of as ‘relative to’ or ‘constructed by’ the various languages, discourses, cultural communities, more-or-less specialized (e.g. mathematical) ‘forms of life’, and so forth, which supposedly constitute the ultimate horizon of intelligibility for those who inhabit them.

We are now better placed to take stock of his comments con-cerning Aristotelian physics (or metaphysics) and the mistake of supposing that Aristotle’s denial of the void has been – or could in principle be – refuted by modern science. This is not to place Badiou in the company of those above-mentioned ‘strong’ socio-logists who would insist that the principle of ‘parity of esteem’ be carried so far that we refrain from judgement as regards the (notional) truth of the issue between Boyle and Hobbes and instead seek an explanation for both of their conflicting views in the particular socio-cultural-political context wherein those views took rise. Rather – and contrary to any such flat-out rela-tivist approach – what Badiou wants us to see is that when Aristotle asserted the non-existence of the void with such (as it turned out) misplaced confidence and vigour he did so for rea-sons that had nothing to do with empirical or experimental proof and everything to do with his clear understanding that its exis-tence, or mere possibility, would utterly wreck his entire concep-tion of the cosmic and natural order of things. That Aristotle got it wrong in physical-scientific terms Badiou would not for one moment deny. That his denial of the void on metaphysical grounds was also an error – and one with negative repercussions for the history of thought right down to Hobbes and beyond – is


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likewise a conclusion that Badiou could scarcely wish to chal-lenge, given his own ontological commitments. However, this is just his point: that what Aristotle grasped was the range of (to him) deeply disturbing or downright absurd consequences that would follow necessarily from any proposition asserting the actual or possible existence of a void. That is to say, Aristotle’s logically precise and philosophically acute (though scientifically fallacious) arguments against the existence of a vacuum bear witness to his grasp of just how destructive were the implica-tions of the contrary thesis with regard to his own cosmological, metaphysical, natural-scientific and even – by close analogy – ethico-political precepts. What they threatened and what Aris-totle needed to keep very firmly at bay was also what Badiou sees as the liberating effect of set-theoretical thought not only in mathematics and the formal sciences but in every discipline, field of research, or project of discovery – politics included – where the issue can be stated with conceptual precision in terms of belonging, inclusion, membership and the count-as-one along with their negative (exclusionary) counterparts. In short, ‘[t]he void is in-numerable, hence the movement which is sup-posed therein does not have a thinkable nature, possessing no reason on the basis of which its comparison to other move-ments could be assured’ (p. 75). It is here that Aristotle’s think-ing encounters its own ‘point of impossibility’, the point at which its whole co-implicated structure of metaphysical, onto-logical and epistemological (not to mention ethical and politi-cal) assumptions comes up against the kind of aporetic challenge that would shake that structure to its very foundations if allowed to proliferate throughout its various regions in the way that Badiou describes.

So far our voyage of commentary through Being and Event has taken us from Plato to Aristotle, that is, from Badiou’s medi-tations on a thinker with whom he feels a strong (if qualified) intellectual kinship to his diagnostic reading of a thinker with whose central claims he is profoundly at odds yet whose reason-ing he finds both consequent and all the more revealing for its blind-spots of questionable presupposition. Having ‘placed’ his thought in relation to theirs and having used the resultant three-sided comparison as a means to draw out some of Badiou’s most important and distinctive themes we can now proceed to engage

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more directly with the intricate sequence of mathematical and other arguments that constitute Part II of Being and Event.

Discussion points

Badiou’s strong universalist stance in ethics and politics goes sharply against the present-day emphasis, among cultural and critical theorists, on the need to show maximum possible respect for differences of creed, tradition, cultural background, value priority or gender orientation. Are you persuaded by his argu-ments in support of this position

What exactly is the relationship between mathematics, onto logy and political thought as Badiou conceives them?

PART II. BEING: EXCESS, STATE OF THE SITUATION, ONE/MULTIPLE, WHOLE/PARTS OR ∈/⊂?’

1. Inclusion, belonging and the count-as-one

In the four Meditations that make up Part II Badiou sets out some basic set-theoretical concepts and procedures before explaining their wider (extra-mathematical) pertinence and then, typically, engaging with a past thinker – in this case Spinoza – whose con-trasting claims can be seen to throw his own into sharper relief. What none the less gives this part its strong sense of a developing and tightly structured argument is Badiou’s constant circling back to the relationship between being and event, or the domain of ontology (with its basis in mathematics) and the domain of events (taken as denoting whatever exceeds the bounds of any pre-existent ontology and establishes new terms for the conduct of future investigation). I shall therefore focus on the salient themes – in particular the set-theoretical concepts along with their emer-gent political implications – and also offer some background commentary on Badiou’s project in the wider context of present-day philosophical (including Anglo-American analytic) thought.

Meditation Seven is entitled ‘The Point of Excess’ and takes us directly to the heart of Badiou’s mathematically based concep-tion of ontology as applied to issues in the formal, physical, social and human sciences. In other words it carries forward the discussion that began with his intensely dialectical staging of the difference between Platonist and Aristotelian ontologies


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and that was all along rehearsed – though now more explicitly – in relation to developments in post-Cantorian set theory.1 Most fundamental here are the relations of belonging (symbolized ∈) and of inclusion (symbolized ⊂), the former indicating that ‘a multiple is counted as an element in the presentation of another multiple’, whereas the latter signifies ‘that a multiple is a sub-multiple of another multiple’ (p. 81). That is to say, as con-cerns the relation of belonging it is here a matter of some multi-ple α that forms an element of some other multiple β such that α is ‘presented’ by the count-as-one or the existing ‘situation’ as prescribed or dictated by β. Thus ∈ is the ‘unique foundational sign of set theory’ since it establishes the possibility of all those relations (among them inconsistent, anomalous, contradictory or suchlike problematic relations) that constitute both an obsta-cle to thought and the means by which thinking typically achieves its most decisive stages of advance. In the case of inclusion, con-versely, multiple α is taken to include all the subsets (i.e. constit-uent multiples) of β and β is thus defined as itself a subset of α and yet – as Cantor showed with respect to the different ‘sizes’ of infinity – a subset that must be thought of as equinumerous with α or as capable of having its members paired off one-for-one with the members of α.2 For clarity’s sake Badiou uses the term ‘element’ to signify belonging and ‘subset’ to signify inclusion, although these should not be taken to mark any further, that is, ontological distinction. His principal concern – here and through-out Being and Event – is to show how such seemingly abstract considerations in the realm of pure mathematics can have a direct (not merely suggestive, oblique or analogical) bearing on matters outside that realm.

So it is with the dualism of belonging and inclusion which, Badiou says, ‘directs, step by step, the entire thought of quantity and . . . the great orientations of thought, prescribed by being itself ’ (p. 82). And again, more specifically, ‘[i]n one case (the case ∈), the multiple falls under the count-as-one which is the other multiple. In the other case (the case ⊂), every element pre-sented by the first multiple is also presented by the second’ (p. 82). If the former (in Badiou’s clearly specified terminology) ‘pres-ents’ certain elements as ‘belonging’ to a given ‘situation’ while others are thereby excluded from it, the latter should be thought of as ‘representing’ all the subsets included in a given ‘state of

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the situation’, that is to say, as involving no such selective mem-bership conditions. It is precisely through the constant possibil-ity of a rift, a mismatch or breakdown of structural reciprocity between these two basic conditions of belonging and inclusion that there also emerges the potential for significant change – for revolutions of thought or theoretically informed practice – in the various spheres to which Badiou applies his dialectic of being and event. Above all, as we shall see shortly, it has to do with that breakdown in the order of commonsense-intuitive mathe-matical grasp signalled by the discovery of the power-set axiom, namely that ‘if a set α exists (is presented) then there also exists the set of all its subsets’, a set whose numerical value must clearly exceed that of set α by an order of magnitude that increases exponentially with the size of set α itself and which generates multiple orders of infinity as soon as mathematics goes trans-finite in the wake of Cantor’s revolution.

This means that the restrictive conditions on belonging which defined the membership of α must now be lifted or redefined so as to acknowledge the existence of β – the power-set of α – whose numerical value far exceeds anything admissible under those same restrictive conditions. What the power-set axiom requires us to think is the effect of that strictly ubiquitous ‘point of excess’ that will always signal the existence of a gap between belonging and inclusion, the situation and the state of the situation, or consis-tent multiplicity (as presented by the dominant count-as-one) and inconsistent multiplicity (as re-presented by all those supposedly non-belonging but none the less included subsets). This can also be phrased in terms of the rift between structure and meta-structure, or again (in Badiou’s chosen terminology) between the elements of a set and its numerically ‘larger’ multiple of subsets. That those quote-marks are required around the term ‘larger’ is one consequence of Cantor’s showing that seemingly different ‘sizes’ of infinite set – like ‘all the integers’ and ‘all the even integers’ – could be counted off one-for-one against each other without limit and could not, therefore, be thought of as larger or smaller in any such straightforward, self-evident or intuitive terms. However, it follows from the power-set axiom that in a different, mathematically definable sense the subsets of any given multiple will be larger (numerically greater) than the multiple itself and that when this axiom is extended to the


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realm of the infinite (or transfinite) it goes beyond the utmost scope of calculation.

Badiou terms this the ‘theorem of the point of excess’ and regards it as occupying a central place not only in the structure, history and genesis of set theory but also in his own elaborations of a set-theoretically based ontology with far-reaching philo-sophic, political and ethical consequences. ‘This is a crucial the-orem’, he writes, ‘which leads to a real impasse: it is literally impossible to assign a “measure” to this superiority in size. In other words, the “passage” to the set of subsets is an operation in absolute excess of the situation itself ’ (p. 84). Hence the need to distinguish ‘situation’ from ‘state of the situation’, the latter taken to include all those subsets whose number – by this theo-rem – absolutely exceeds that of the elements which belong to the situation according to the prevalent count-as-one. It is here – at this point where the resources of ontology are pressed to the limit and beyond – that philosophy finds itself equipped or com-pelled to conceive of the event as an ‘ultra-one’ or as a strictly ‘supernumerary’ item vis-à-vis the existing order of things, that is, an occurrence whose advent marks a decisive break with that order. Such would prototypically be instances of – in the proper as distinct from the debased or everyday usage of these terms – invention in science, creation in art, revolution in politics and passion in love.3 Each of these has its negative counterpart, according to Badiou: culture in place of art, management in place of politics, technique in place of science and sex in place of love. Moreover, it is chiefly on the strength of his set-theoretical elabo-rations – his formal rendering of the process whereby truth-events come about in excess of any prior reckoning, predictive capacity, or power of ontological grasp – that Badiou is able to draw these distinctions and to specify what counts as a genuine event in each of those subject-domains. On his account, ‘no mul-tiple is capable of forming-a-one out of everything it includes . . . [since] inclusion is in irremediable excess of belonging’ (p. 85). And again, ‘the included subset made up of all the ordinary ele-ments of a set constitutes a definitive point of excess over the set in question. It never belongs to the latter’ (ibid.).

So despite his extreme care to distinguish the evental and the ontological (since the former is here defined as that which can-not possibly be deduced, predicted or allowed for in accordance

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with some pre-existent state of knowledge or conceptual scheme) still there is a clear sense in which Badiou’s whole project rests on ontological foundations and indeed requires them just in order to make that same distinction. More precisely, what he sees as philosophy’s proper task is not that of making ontolo-gical discoveries or exploring new ontological regions on its own account – since this is a role best left to the mathematicians – but rather that of pursuing a ‘meta-ontological’ enquiry that expounds, clarifies and draws out the consequences (some of them decidedly extra-mathematical) of any results thus obtained. One can therefore see why he lays such stress on the claim, contra Wittgenstein and Heidegger, that ‘mathematics thinks’ insofar as it involves a creative, inventive and truth-disclosing activity of thought that cannot be reduced either (following Wittgenstein) to a mere assemblage of vacuous since purely self-confirming logical tautologies nor again (following Heidegger) to a mere expression of the techno-scientific-metaphysical will-to-power over nature and humankind alike.4 Indeed one fascinating aspect of Badiou’s work is the way he pursues a selective exegetical path among the many thinkers – from Parmenides, Plato and Aristotle, via Descartes and Pascal, to Frege, Russell, Heidegger, Wittgen-stein and others – against whose projects he measures his own with varying degrees of perceived affinity or (very often in the same thinker) perceived differences of view. In each case his basic argument is that we can and should read these thinkers differently in light of the epochal advance brought about by Cantor’s inaugural discoveries in set theory and their subsequent development by mathematicians and some (very few) academic or professional philosophers.

Most crucial here is the power-set axiom since it establishes the principle – the point of departure not only for Badiou’s mathe-matically based critical ontology but also for his thinking on mat-ters of political, scientific and ethical import – that no instance of the count-as-one, whatever its claim to universal inclusivity, could ever contain (or purport to represent) those endlessly proliferat-ing subsets of multiples revealed by a grasp of that axiom. Badiou follows his ‘technical’ rendition of the case as concerns set theory and its formal structure with a sentence that effectively re-states that case in a register whose normative moda lity seems to waver between laying down the necessary (non-negotiable) terms and


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conditions for any adequate address to these matters and pre-senting something like an ethical injunction to observe, respect, or act upon those terms and conditions. ‘Once this is admitted’, he writes, ‘one is required to think the gap between simple presen-tation and this species of re-presentation which is the count-as-one of subsets’ (p. 85). That requirement clearly derives its imperative force from the various formal demonstrations, from Cantor down, of how set theory achieves its most signal advances by a procedure of thinking through-and-beyond the various obstacles – mostly of a commonsense-intuitive kind, like that initially provoked by the power-set axiom – which have risen against it. To this extent it is normative in the sense that it pre-scribes what properly counts as a correct or valid application of the formal procedure concerned. That is to say, it belongs to the domain of mathematical truth where the issue of fidelity has nothing to do with ethics or the moral virtues and everything to do with domain-specific requirements such as consistency, rigour, demonstrative force, logical explicitness and so forth. However it also belongs to that other dimension where the term ‘fidelity’ does have a strong ethical toning and where issues of truth cannot be held entirely apart from matters of truthfulness or intellectual virtue.

Not that I would wish to place Badiou in the company of those who have argued for a virtue-based epistemology with its roots in the tradition of ethical thought descending from Aristotle and taken up lately by philosophers in quest of some alternative to the current range of often sharply conflicting (e.g. deontologi-cal versus consequentialist) views.5 In brief, this is an approach that envisages no possible solution to the ‘problem of knowledge’ in its various forms except by instancing the sorts of jointly moral and cognitive-investigative qualities, dispositions or intel-lectual traits that best, most reliably conduce to the advance-ment of human understanding. These would typically range from epistemic virtues like perceptual acuity or sensory-intuitive ‘feel’ for the physical items or properties under investigation to epistemologically salient aspects of intellectual character such as dedication, perseverance, open-mindedness, respect for the evidence, willingness to test even one’s most cherished or firmly held beliefs against that evidence, and a well-developed capacity for self-criticism. That Badiou has the utmost regard for those

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virtues is clear enough from his writing about certain exemplary figures – such as the mathematicians Jean Cavaillès and Albert Lautman, both of whom sacrificed their lives as members of the French Resistance – whose actions and work can be seen to have manifested them in the highest degree.6 All the same he is very careful not to confuse truth with truthfulness, or issues of the kind: ‘is statement x true or justified according to the strictest standards of evidence or of formal (e.g., set-theoretical) proce-dure?’ with issues of the kind: ‘has statement x been arrived at by someone (or some community of like-minded enquirers) possessed of all the relevant, truth-conducive or knowledge-promoting virtues?’. That these two conditions may often be satisfied by the same statement – since the latter is defined as one that should predictably give rise to the former – is everywhere implicit in Badiou’s account of those particular discoveries or stages of advance, in set theory and other domains, that have typically occurred through a combination of rigorous thinking with the courage to defend them against the weight of established doc-trine or commonsense-intuitive belief. Still they cannot be run together or this distinction effectively collapsed – as advised by some proponents of a virtue-based epistemology – without consequently opening the way to all manner of sceptical, social-constructivist or cultural-relativist ideas. For it is no great distance from this invocation of the intellectual virtues (however carefully or strongly specified) to the idea that any such appeal is reliant on the existence of a certain socially accepted or culture-specific conception of just what constitutes a virtuous epistemic practice.

The next pair of Meditations, Eight and Nine, bear the somewhat forbidding titles ‘The State, or Metastructure, and the Typology of Being (normality, singularity, excrescence)’ and ‘The State of the Historical-Social Situation’. All the same read-ers should not be tempted to skip since there is, as those titles suggest, substantive ethical and socio-political as well as ‘purely’ philosophic content in this formal demonstration of his central claim as concerns set theory and its implications for the various domains of applied ontological enquiry. Indeed they should if possible be read at a sitting since they run to just 18 pages in all and between them offer the clearest account in Being and Event of how the three main dimensions of Badiou’s work (crudely put: the mathematics, the ontology and the politics) should rather


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be seen as so many aspects of a single, strictly indivisible project. After all, Badiou is among the most committed – that is to say, the least ‘repentant’ or shiftily backsliding – Marxist intellec-tuals of our time, and it has always been a central Marxist thesis that any genuine project of political emancipation must achieve the overcoming of this false dualism through an active, that is, practically engaged yet also theoretically informed pursuit of specific political goals.7 His response to this vexed question of the relationship between theory and practice is remarkable chiefly for the fact that it pushes the dualism to what looks like a point of extreme – even irreconcilable – antinomy yet does so precisely in order to expose that falsehood and thereby make his case for the practical-political relevance (indeed the indispensabi-lity) of certain, on the face of it highly ‘abstract’ set-theoretical axioms and proof-procedures. Not that we should take this relevance-claim as adequately borne out just because there can be shown to exist an analogical relation between on the one hand set-theoretical terms such as ‘class’, ‘state’, ‘inclusion’, ‘belonging’, ‘member’, ‘part’, ‘subset’, or ‘count’ and on the other hand iden-tical or kindred terms that typically figure in the lexicon of polit-ical theory and the thinking of those whose primary aim is to transform or undermine existing structures of privilege and power. Badiou enters this cautionary note – albeit in a muted way – when he remarks during Meditation Eight that it is ‘due to a metaphorical affinity with politics’ that he will henceforth deploy the phrase state of the situation to signify ‘that by means of which the structure of a situation – of any structured presen-tation whatsoever – is counted as one, which is to say the one of the one-effect itself ’ (p. 95). This affinity will be explained, he promises, in Meditation Nine when the focus switches more explicitly to politics and when these so far ‘metaphoric’ connec-tions or suggestive cross-domain analogies acquire a more detailed working-out in conceptual-analytic terms.

The link is accomplished chiefly through that same technical coinage, ‘state of the situation’, which Badiou variously defines as ‘count-of-the-count’, ‘metastructure’, or ‘that by means of which the structure of a situation is, in turn, counted as one’. His point is that this involves a ‘second count’, a further operation – of the kind that has its place in all formal, logical or set-theoretical sys-tems – whereby the first count is subject to a duplicate reckoning

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so as to confirm its consistency and comprehensive grasp, or so as to ensure that nothing has gone uncounted by the prior opera-tion. To this extent, he claims, ‘concrete analysis converges with the philosophical theme’ since in both cases the thesis can be verified, whether through a process of formal (mathemati-cal or logical) analysis, a project of applied natural-scientific research, or a study of social and political structures along with their inherently problematic modes of self-legitimation. These latter can always be shown to involve the duplication of a first-order count that establishes the terms of belonging or member-ship by a second-order count – a re-presentation – which purports to secure the former’s status as a valid (since all-encompassing and non-exclusionary) compte rendu but which in fact, upon closer inspection, turns out to pose a strictly inescapable challenge to that very claim. Thus ‘the structure of the count is reduplicated in order to verify itself, to vouch that its effects, for the entire duration of its exercise, are complete, and to unceasingly bring the one into being within the un-encounterable danger of the void’ (p. 94). Such – one may fairly deduce – is the perceived or the generally dominant effect of this second-order count during periods of relative stability in various domains such as Kuhnian ‘normal’ (as opposed to ‘revolutionary’) science, or mathematics when it settles to working through the implications of some pre-vious major advance, or social existence when subject to no urgent or strongly disruptive forces of political unrest.8 It is also, in set-theoretical terms, what ensures the seeming predominance of ‘consistent’ over ‘inconsistent’ multiplicity, or of that which submits to the count-as-one over that which eludes or exceeds its calculative grasp. By this means it seeks to guarantee ‘that there is no possibility of that disaster of presentation ever occurring which would be the presentational occurrence, in torsion, of the structure’s own void’ (p. 94).

However, as should be evident from what has gone before, such stability – or the appearance thereof – is maintained only through the constant suppression, disavowal, or holding at bay of those anomalous instances that would otherwise mark the irruption of inconsistent multiplicity. For this shows (despite Russell’s purported ‘solution’ to the set-theoretical paradoxes) how precarious is any structure that resorts to some higher level, meta-structural concept in order to legitimize its claims or any


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situation whose state – as defined above – depends upon this doubling of the count so as to confirm its all-inclusive character or representative power. Any appearance of stability thereby achieved is one that perforce suppresses certain salient truths about its own production, among them the truth that ‘a struc-ture is precisely not a term of the situation’, that it cannot itself be counted, and that therefore ‘a structure exhausts itself in its effect, which is that there is oneness’ (p. 95). That is to say, the count-as-one may succeed in excluding or discounting ano-malous instances and thereby imposing what appears a perfectly consistent order wherein there exists a perfect, one-for-one correspondence between belonging and inclusion, members and parts, or presentation and representation. However it does so only at the cost of giving rise to inconsistencies elsewhere, as happens in mathematics and the formal sciences when some paradox achieves seeming resolution through stipulative fiat or pragmatic adjustment but then crops up at a later stage to yet more disruptive (but also, as Badiou is keen to stress, revelatory or knowledge-transformative) effect. It is through the series of encounters with this strictly unavoidable though often suppressed or ignored non-correspondence between the orders of belonging and inclusion – along with their various cognates – that thinking has typically found itself propelled into new and productive regions of enquiry. Moreover this has always involved the irruption of a seemingly negative factor – ‘the void, and the risk it represents for structure’ – which motivates both the reactive drive for order (for consistency, integrity or other such classical imperatives) and the countervailing drive to discover new ways of overcoming some present obstacle to thought.

2. State, subject, representation

So when we reach the end of Meditation Eight Badiou has placed before us a developed terminology and a set of clearly marked conceptual distinctions that promise a means of accomplishing the passage – the impossible passage, as some would have it – from issues within mathematics and the formal sciences to issues in the socio-political domain. He has done so, to repeat, by pur-suing the logic of those set-theoretical terms and operations that bring out the constant disruptive or destabilizing effect of incon-sistent upon consistent multiplicity, of that which is included in

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any given multiple upon that which is taken as belonging to it, and of all those uncounted parts of a certain situation upon the count-as-one that purports to represent its every last element. Thus, despite his above-mentioned caveat regarding its ‘meta-phoric’ character, Badiou’s usage of the term ‘state’ at this point of his exposition – in particular his claim (p. 101) that ‘in ontology, the state’s “anti-void” functions are not guaranteed’ – is sure to evoke its social-political as well as its up-to-now predominantly technical, i.e., set-theoretical sense. So likewise with Badiou’s sum-mary remark, again towards the close of Meditation Eight, that – according to the terms and conditions of membership laid down by an existing state – ‘inclusion must not arise on the basis of any other principle of counting than that of belonging’ (p. 101). While this still has primary reference to certain regulative precepts in the discourse of mathematics (in Kuhnian terms, those which charac-terize periods of ‘normal’, problem-solving or non-revolutionary work) it can scarcely be read without evoking a whole range of analogous political situations. Among them – most emphatically – is that of the displaced, immigrant, socially excluded and disenfran-chised minorities whose cause Badiou has taken up through his involvement with various extra-parliamentary activist groups.9

These themes are developed more fully in Meditation Nine, where in effect (although without declaring as much) he withdraws the cautionary note and allows that these points of cross-reference between mathematics and politics are indeed something more than merely metaphorical. Its purpose is to specify more exactly – by way of adequate conceptualization rather than suggestive analogy – the order of relationship involved. As so often it is Aristotle who provides Badiou with his opening example of a thinker whose basic orientation – whose entire apparatus of meta-physical, ontological, ethical, political and scientific concepts – stands squarely opposed to his own (since in various ways committed to denying the existence or possibility of the void) yet for just that reason throws the relevant issues into sharp relief. Badiou makes that case with respect to what he sees, along with other recent commentators, as the close affinity between Aristotle and Marx in certain crucial respects.10 Chief among them is Aristotle’s clear recognition that any dealing the state might have with those under its jurisdiction is not a matter of relating to individuals, to persons, or even to ‘subjects’ in a sense of that


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term connoting the possession of particular traits beyond their generic (structural) placement as ‘subject to’ the power and authority of the state.11 Rather – at its most fundamental – it has to be conceived in terms of precisely such an undifferentiating structural relationship maintained across and despite all distinc-tions between one and another individual.

This emphasis on the generic as opposed to the specific (i.e. experiential or existential) aspects of the relation – or quasi-rela-tion – between ‘subject’ and state is one that Badiou has taken from, or developed through critical engagement with, a wide range of philosophical precursors from Plato and Aristotle to Pascal, Spinoza, Rousseau, Hegel, Marx, Husserl and Heidegger. More directly, it shows his continued allegiance to certain aspects of Althusserian ‘structuralist’ Marxism, in particular Althusser’s conception of the subject as ‘interpellated’ by the ruling ideo-logy or as always, inescapably caught up in those structures of ‘imaginary’ misrecognition that grant it the illusory sense of individual subjecthood.12 Also, as we shall see later on, it reflects his likewise continued commitment to the central claims of Lacanian psychoanalysis. That is, Badiou subscribes to Lacan’s cardinal and again broadly structuralist idea that the ego as sup-posed locus of identity or conscious, reflective selfhood is in fact a mere epiphenomenon of discourse, a ‘plaything’ of the uncon-scious whose ubiquitous workings (like those of ideology in Althusser’s account) are there to be glimpsed – but never truly or fully comprehended – in the verbal exchange between analyst and analysand.13 However, what is most decisive here is the example of mathematics and its demands upon the conduct of disciplined enquiry both within and beyond the formal sciences. Above all it is the set-theoretical requirement that thought should be concerned with universal rather than particular, struc-tural or combinatorial rather than discrete, and – in the techni-cal sense of these terms current among logicians and philosophers of language – an extensional rather than intensional approach to issues of sense and reference. On this account the operative sense of a term is fixed entirely by the range of those objects, whether physical or abstract, to which it applies or extends and not by any-thing specific about those objects (their distinctive properties, qual-ities or attributes) that marks them out as rightfully falling under the term in question. That is to say, an extensionalist conception of

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sets and their membership conditions is one that rigorously excludes any thought of whatever might otherwise be taken to dis-tinguish potential or candidate members, and thus to determine which items qualify for inclusion.

There has been much debate among mathematicians as regards these rival approaches. Thus the advocates of extensionalism mostly espouse a realist principle according to which set-theoretic statements or theorems have their truth-value fixed by the way things stand in mathematical reality, while the advocates of intensionalism more often incline towards an anti-realist position whereby truth becomes a matter of epistemic warrant and there is no making sense of the claim that there exist such objective truth-makers beyond our furthest powers of proof or ascertain-ment.14 Badiou adopts an extensionalist, that is, a strictly non-differentiating ontology of sets, subsets, elements, parts, members and so forth, which leaves no room for any kind of qualitative distinction and which therefore conceives their various orders of relationship in purely numerical terms. It is on this basis that he is able to mount what most analytic philosophers would think an exorbitantly far-fetched claim, that is, to derive from certain axioms and theorems of set theory not only a generalized social ontology but also a conception of political justice grounded in the clear-cut distinction between two set-theoretical concepts, namely those of belonging and inclusion. Whence, as I have said, his mathematically derived and ethico-politically strengthened conviction that these interests can best be served – or the claims of justice best advanced – only by a radically egalitarian and universalist outlook opposed to any form of identity politics, or any notion of difference (whether ethnic, national, gender-based, or cultural-linguistic) as having a significant role to play in such matters.

Here it is worth noting a certain calculated ambiguity about Badiou’s usage of those two key terms ‘inclusion’ and ‘belong-ing’. On the one hand it is a thesis basic to his political-activist project – in the simplest terms, that of ‘counting those who aren’t counted’ – that the multiplicity of elements or parts included within a given multiple may far exceed the multiplicity of mem-bers taken as ‘properly’ belonging to it or accorded the rights (along with the concomitant responsibilities) of full participant status. In which case, it would seem, this excess of inclusion over


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belonging is the axiom of set theory most strongly conducive to the interests of a radical-emancipatory politics that would push so far as possible towards ensuring that the status of belonging was extended to all those included – though as yet not acknowl-edged to enjoy such status – within some given socio-political situation. On the other hand it is clear from Badiou’s formula-tions as cited above that inclusion is also the principle of reck-oning which the State applies in its false claim to treat everyone on equal terms, that is, in its claim to respect each and every person who figures in the dominant, supposedly comprehensive count-as-one – whatever their rank, class or socio-cultural posi-tion – as equally ‘belonging’ to the all-inclusive, democratically constituted body politic. Thus ‘[t]he “voter”, for example, is not the subject John Doe, it is rather the part that the separated structure of the State re-presents, according to its own one; that is, it is the set whose sole element is John Doe and not the mul-tiple whose immediate-one is “John Doe”’ (p. 107). There is a certain tension in his thinking here – even a sense of conceptual strain – between ‘inclusion’ as that which offers a merely formal or abstract (hence delusive) notion of equality and as that which affords a powerful means to conceptualize and thereby think a way beyond the drastic shortfall in every hitherto existent version of liberal or social democracy. Nor is this surprising, it might be said, given the notorious adaptability of mathematical (or pseudo-mathematical) techniques to all manner of politically or ideologi-cally loaded persuasive ends. However it is precisely against such abuses – so Badiou maintains – that one can muster the unique probative force of those set-theoretical concepts, axioms and procedures that he lays out for detailed inspection in Being and Event. That is, they can most aptly serve to demonstrate the kinds and degrees of democratic deficit – or political disenfranchise-ment – that typify the workings of ‘liberal democracy’, whether US-style or in various forms throughout Western Europe and the ex-Soviet bloc.

Thus his point about the State’s obsessive concern with inclu-sion is that this goes along not only with a systematic disregard for issues of belonging (or of how individuals relate to their con-ditions of socio-political existence) but also with a narrowed and bureaucratic sense – brought about by rigid application of the count-as-one – of what inclusion properly signifies.

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The individual is always – patiently or impatiently – subject to this elementary coercion, to this atom of constraint which constitutes the possibility of every other type of constraint, including inflicted death. . . . Any consistent subset is imme-diately counted and considered by the State, for better or worse, because it is matter for representation. On the other hand, despite the protestations and declarations to the con-trary, it is always evident that in the end, when it is a matter of people’s lives – which is to say, of the multiple whose one they have received – the State is not concerned. Such is the ulti-mate and ineluctable depth of its separation. (pp. 107–8)

It is characteristic of Badiou’s thought that the passage should convey such a clear sense of moral and political passion – of mixed anger and sadness at the cost in terms of human well being of such avowedly inclusive but in fact highly selective or exclu-sionary practices – while none the less making its point through a mode of argument deriving from certain, as it might seem, decidedly abstract set-theoretical procedures. However, this widely held idea (at least among arts-and-humanities types) of mathe-matical discourse as a realm of soulless abstractions devoid of human or creative content is one that Badiou has rejected with great energy and eloquence, most often by instancing the rich profusion of previously unthought-of entities that mathemati-cians have discovered by dint of their jointly conceptual and imaginative powers.15 At the same time he is keen to insist – against any kindred notion of mathematics as inherently remote from political concerns or the interests of social justice – that one very effective way to bring home the wrongs inflicted by unjust or oppressive (even if self-styled ‘democratic’) regimes is to deploy the conceptual resources of set theory as a means of clearly specified and logically precise articulation. That is, they offer the strongest purchase for an account of how presently existing forms of pseudo-democratic governance operate to ensure the belonging of all and only those members or subjects in good standing who qualify according to the count-as-one. What set theory makes possible is a formalized reckoning with just that excess of inclusion over acknowledged or recognized belonging – or of uncounted subsets over any count according to presently accepted rules and conventions – which constitutes


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the basis, in theory and principle, of a properly motivated chal-lenge to the socio-political status quo.

Nor should it be thought that this is an overly, even inhumanly ‘abstract’ approach to issues that can have so decisive a bearing on the welfare, the life-hopes and the very survival of so many people across such a range of real-world oppressive situations. After all, as Badiou pointedly remarks,

it is not for nothing that governments, when an emblem of their void wanders about – generally, an inconsistent or riot-ing crowd – prohibit ‘gatherings of more than three people’, which is to say they explicitly declare their non-tolerance of the one of such ‘parts’, thus proclaiming that the function of the State is to number inclusions such that consistent belongings be preserved. (p. 109)

What is striking here is Badiou’s ability to move straight across with such consummate ease – though without the least sense of tricksy semantics or conceptual legerdemain – between a regis-ter of set-theoretical ‘abstractions’ and a language of engage-ment with directly political or social-activist concerns. That he is able to do so against all manner of likely objections – including various analytic updates on the fact/value dichotomy or the sup-posed logical impossibility of deriving a normative-evaluative ‘ought’ from a purely constative ‘is’ – should be put down to Badiou’s highly focused and concentrated sequence of arguments up to this point, rather than to any evasive strategy or vaguely analogical habit of thinking on his part. More specifically, it has to do with his singular gift for locating just those erstwhile stress-points yet also (and for just that reason) points of subsequent radical advance within the history of set-theoretical thought which lend themselves to re-statement in political terms. If poli-tics is always, by very definition, ‘an assault against the State, whatever the mode of that assault might be, peaceful or violent’, then such a claim finds support in the proven capacity of formal reasoning to ‘mobilize the singular multiples against the normal multiples by arguing that excrescence is intolerable’ (p. 110).

An ‘excrescence’ is defined, in this context, as a term that is ‘represented by’ the state of some given situation while not ‘pre-sented in’ the situation itself, or again – what amounts to the same

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thing – as ‘included in’ that same situation while not recognizedly ‘belonging to’ it for social-political-administrative purposes. It is therefore a locus of maximal challenge to the structures of State-accredited authority and power that in turn derive their appearance of legitimate, that is, ‘democratic’ warrant from the count-as-one and its fallacious claim to represent a consistent multiplicity of free and equal subjects, rather than – as emerges through Badiou’s analysis – an inconsistent multiplicity that in truth subtends and belies that false appearance. What is uniquely capable of breaking through this ideological façade is the sudden irruption of those same ‘excrescences’ or stubbornly resistant, hence ‘intolerable’ instances of failure (or refusal) to be counted as belonging to some given self-image of the fully democratic since all-inclusive body politic. Thus ‘excrescence’ takes its place as another key term – alongside ‘void’, ‘event’ and ‘evental site’ – in the range of set-theoretically derived concepts whereby Badiou seeks to understand and explain how it is possible for something radically new to occur (whether a great advance in mathematics, discovery in science, or episode of major political change) despite and against all the odds of received belief or entrenched socio-political power. For this is the point at which his project hinges between its ontological orientation, that is, its concern to explicate the structures and modalities of being con-ceived not in static but dynamic or constantly self-transformative terms, and on the other hand its orientation towards an idea of the event as involving a more drastic (and wholly unpredictable) break with whatever may seem to have led up to it from a subse-quent or post-evental viewpoint.

Hence Badiou’s sceptical – some would say cynical – attitude concerning the prospects for genuine as opposed to merely cos-metic change through any ‘democratically’ sanctioned parliamen-tary means or any process that exists under present conditions in the United States or countries belonging to the (so-called) Euro-pean ‘community’. Hence also, conversely, his passionate convic-tion that alternative means can be found whereby to circumvent, outflank or break through what he cites as the phenomenon despairingly noted by Lenin and other revolutionary thinkers, namely the ‘obscene persistence’ of the State despite the best efforts of those ranged against it. All the same such despair of the State’s ever being brought to let go of its established powers

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and prerogatives should by no means be thought to entail a simi-lar despair with regard to the existence of other, extra-parlia-mentary or non-State-involving means by which to bring about that desirable upshot. ‘Rather than a warrior beneath the walls of the State, a political activist is a patient watchman of the void instructed by the event, for it is only when grappling with the event that the State blinds itself to its own mastery’ (p. 111). The power of the State is exercised – and perpetuated – only on condition of ignoring, discounting or (at the limit) forcibly suppressing those whose very marginality or social invis-ibility gives them the countervailing power which comes of their excluded, anomalous and hence potentially subversive or desta-bilizing presence. If granted recognition even in this outcast role as ‘excrescences’ or not-to-be-tolerated others they would be strongly placed to expose both the arbitrary, unjust nature of the State’s claim to power and the weakness of that claim when subject to challenge on ethico-juridical and indeed, according to Badiou’s central thesis, on formally specifiable grounds.

3. Spinoza: foreclosing the void

We can now proceed to Meditation Ten, ‘Spinoza’, where the argument takes another of its periodic excursions through an episode in the history of philosophy – like his previous engage-ments with Plato and Aristotle – with special relevance to Badiou’s project. In this case its relevance has to do with the relation – the highly problematical relation, as Badiou sees it – between three chief aspects of Spinoza’s thinking: his rationalist metaphysics, his monist ontology (i.e. his conception of mind and nature as simply two ‘attributes’ of the self-same substance) and, following directly from this, his outlook of strict determinism with regard to the springs of human thought and action. The problem is sharpened by Spinoza’s dedication to a range of (for his time) singularly radical proto-enlightenment projects – undertakings of an ethical, socio-political and speculative philosophic nature – to which Badiou is strongly drawn as a matter of shared intel-lectual and practical commitment, yet which he thinks wholly incompatible with those three primary components of the Spinozist worldview. Thus it offers a focus for the book’s central question: what role or what room can be found for the ‘event’ vis-à-vis the order of ‘being’, or for that which intervenes – in

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however unpredictable or unintended a manner – to disrupt and transform some existing situation?

Another shared precept that brings Badiou very much into Spinoza’s philosophic orbit is the conviction – flat contrary to much present-day thought – that language need not (and should not) be considered the primary concern or the condition sine qua non of philosophical understanding. Thus Badiou has no time for those varieties of pragmatist, hermeneutic, post-structuralist, postmodernist, anti-realist or ‘post-analytic’ approach whose chief common feature – despite their otherwise large divergences of view – is the claim that language in some sense goes all the way down, and therefore that truth (insofar as we can possibly know it) must be construed in language-dependent or language-relative terms.16 This is why he is fond of quoting Spinoza’s peremptory rationalist dictum ‘ideam enim veram habemus’ (‘For we have a true idea’), representing as it does a well-nigh scandalous affront not only to these schools of thought but also to thinkers of just about every epistemological persuasion from Kant to the present.17 It is Badiou’s principled and passionately held conviction that truth can always exceed or transcend our present-best powers of knowledge and must therefore be thought to set the standard for whatever we can rightly (or intelligibly) say about it. From which it follows – contra the above-mentioned schools of thought – that the criterion for what should count as an adequate, knowledge-conducive deployment of language is that it measure up to the requirement of truth, rather than the other way around. Like Spinoza, but unlike many philosophers nowadays, Badiou thinks of language – at any rate in certain disciplines such as philosophy – as properly aspiring to the highest degree of conceptual-semantic clarity and precision, and hence as subject to regulative norms that are not just those of custom-ary practice or communal warrant. Thus Badiou is one of the few present-day commentators who take seriously Spinoza’s attempt, in the Ethics, to lay out his arguments more geometrico, that is, in a Euclidean fashion that purports to arrive at its con-clusions through a process of rigorous deductive reasoning along with the full logico-mathematical apparatus of definitions, axi-oms, propositions, corollaries and scholia.18

In this respect – as in others – he takes a view sharply opposed to that of his erstwhile colleague and intellectual sparring-partner


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Gilles Deleuze.19 On Deleuze’s account all that creaky Spinozist scaffolding should best be ignored and the Ethics be read not for its (pseudo-)demonstrative logical structure but rather for the moments of passional intensity and highly charged personal reflection that erupt at various points of the text, especially in the scholia. Not that Badiou is in the least inclined to ignore this ‘other’ Spinoza, or to overlook the signs of a restless, unruly, desirous physical being that Deleuze places very much at stage-centre. On the contrary, his reading makes much of those stress-points, anomalous passages and other such crucial (though often disregarded) junctures in the Ethics where the supposedly seamless progression from stage to stage in its structure of argu-ment is interrupted by moments of a strikingly different, highly charged emotive or passionate character. Thus Badiou, no less than Deleuze, rejects any reading that would focus solely on its logical (or quasi-geometrico-deductive) structure at the cost of downplaying – or ignoring – that other, intensely affective dimension of Spinoza’s life and work. Indeed, it is crucial to his own thinking that Spinoza’s resolutely monist ontology – his conception of mind and body or thought and matter as two ‘attributes’ of the self-same substance – should be prey to just such uncontrollable intrusions not only from the realm of pas-sional experience but also from the world of contingent histori-cal and socio-political events. After all, Badiou’s entire philosophic project involves precisely this cardinal distinction between, on the one hand, the order of being as revealed or dis-covered through enquiry into the set-theoretical foundations of ontology and, on the other, the order of events as that which inherently eludes any such account and which sets new stan-dards – new fidelity conditions – for the exercise of thought in its other, for example, political, artistic and ethical spheres.

Spinoza famously broke off his work on the Ethics in order to write the Tractatus Theologico-Politicus and thereby intervene, to the best of his powers, in the crisis of conflicting religious as well as political allegiances which at that time threatened to overthrow the Dutch Free Republic.20 Badiou’s reading gains credence from this salient fact, along with the extent to which Spinoza’s passions, both positive and negative, were so often evoked by his intense participation in this struggle to preserve the hard-won freedoms of thought and speech. On the other

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hand – against Deleuze – he holds that we shall underrate the ethical and political as well as the philosophic force of Spinoza’s thought if we treat its geometrico-deductive mode of presenta-tion as just a handy formal device or a means of achieving maxi-mum rhetorical and argumentative effect. What Badiou finds so intriguing about Spinoza is precisely this unique combination of a mind fixed upon truths that are taken to hold sub specie aeternitatis, or as always potentially transcending the compass of time-bound human cognition, with an intelligence keenly and deeply aware of its temporal (e.g. cultural-historical and socio-political) involvements.

Spinoza thus stands as a test-case and notable precursor for the two major theses that between them motivate Badiou’s philo-sophical project. His thinking prefigures what Badiou has to say – with the advantage of set-theoretical, Marxist, psychoana-lytic and other kinds of informative hindsight – concerning mathematics as the basis of all ontology and the event as that which redefines our intellectual and ethico-political responsibil-ities vis-à-vis some thereafter strictly binding (since truth-pursu-ant) obligation. In this respect he manages to straddle the two major camps of recent French Spinoza interpretation. On the one side were those – like Althusser and the early Balibar – who recruited him to the cause of a ‘structuralist’ or critical-rational-ist Marxism conceived very much in the Spinozist manner of a quest for truth and knowledge ideally unclouded by the effects of false, deceptive, imaginary or ideological belief.21 On the other were those, including Deleuze, who reacted strongly against this idea (as much with regard to issues in present-day politics as to issues in Spinoza scholarship) and who swung right across to the opposite extreme of a reading that emphasized the philosophically exorbitant character of Spinoza’s thought and its affinity with such notions as desiring-production, libidinal economy or ‘deterritorialised’ energy-flows.22 As I have shown elsewhere, each of these drastically opposed readings is able to claim a good measure of exegetical warrant through the direct appeal to certain strongly supportive passages in Spinoza’s text.23 However, what is conspicuously missing from both inter-pretations – and what Badiou sets out to provide – is an ade-quate account of how the method of reasoning more geometrico relates to Spinoza’s treatment of the passions (positive and


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negative) along with his response to the pressures and prospects of historical-political life.

Thus Spinoza is a central figure in Badiou’s genealogy of modern thought since he, like Badiou, was above all concerned to understand the relations between truth and knowledge, the-ory and practice, reason as that which aspires to a timeless (pro-totypically mathematical) order of truth and reason as subject to practical constraints when required to adapt itself to chang-ing historical and socio-political conditions. However – and this is where Badiou parts company with Deleuze – we shall be in no position to appreciate the strength or intensity of Spinoza’s political passions and convictions unless we are willing to mea-sure them against the demonstrative force of his reasoning more geometrico and not treat the latter as a mere excrescence or a misconceived attempt to achieve scientific credibility for some otherwise highly dubious premises and conclusions. And again, we shall fail to grasp an important aspect of that reasoning – namely its role as both a critical check upon those passions and a motivating source for them – if we adopt a ultra-rationalist position, like Althusser’s, that very largely ignores both the affec-tive dimension and the circumstantial details of Spinoza’s con-joint life-and-thought. To be sure, there is a strong case to be made for viewing Spinoza as a thinker far ahead of his time and one who moreover managed to elaborate a proto-Marxist theory of truth, subjectivity and ideological misrecognition. This he achieved – so Althusser maintains – through his distinction between the ‘first’ and ‘second’ kinds of knowledge, that is, his account of how ‘confused’ or ‘imaginary’ ideas should properly give way to their ‘adequate’, clear and distinct (since rigorously theorized) replacements.24 Yet as Althusser’s critics have been quick to point out it is hard to extract any convincing account of political agency or motivation from his high-structuralist account of how subjects are passively interpellated by – or recruited to – this or that prevailing ideological formation.25 What is lacking in his general approach to these matters, as likewise in his reading of Spinoza, is what Badiou most importantly aims to provide: an account of how philosophy might reconcile the claims of conceptual rigour, clarity and precision with an openness to the contingency and unpredictability of real-world events, whether

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in the ethical, political, philosophical, artistic or personal (espe-cially amorous) spheres.

Hence the prominent position of Spinoza as an elective pre-cursor to Badiou’s philosophical project, despite the extreme contrast between Spinoza’s radical monism and Badiou’s com-mitment to an equally radical conception of inconsistent multi-plicity as that which precedes, subtends and surpasses any unity imposed upon it by various operations of the ‘count-as-one’. Hence also the lesson for those who might be tempted to dismiss Badiou’s writing on set-theoretical themes as at best a somewhat fanciful diversion and at worst a display of gratuitous expertise in a discipline utterly remote from his home-ground interests, that is, politics, ethics, aesthetics and psychoanalysis. They would be wrong about this for a number of reasons, among them – as I have said – the high sophistication and conceptual range of Badiou’s mathematical thought and the extent to which his ontological (i.e. set-theoretical) concerns intersect with his treat-ment of those other themes. For it is just Badiou’s point that they constitute the chief enabling ‘conditions’ for a project that would keep its sights firmly fixed on the standard of truth while none the less taking adequate account of those various kinds of event that can always intervene in such a way as radically to redefine what qualifies as thinking, acting or living in accor-dance with that same standard. This is why he is so critical, even contemptuous, of much that passes for philosophy of mathemat-ics in the recent analytic tradition, focused as it often is either on narrowly technical or on hyper-inflated issues – such as the seem-ingly endless debate around rule-following – that (in his view) merely trivialize the subject and deflect thinking from other, philosophically as well as mathematically more challenging paths.26 It is also why he rejects any version, no matter how qual-ified, of the Frege-Russell logicist programme that would seek to derive all the basic truths of mathematics from a handful of set-theoretical axioms and strictly deductive procedures of proof and demonstration.27

It is here that Badiou’s thinking comes closest to Spinoza even though he rejects the Spinozist idea of a single, undifferentiated order of being (interchangeably ‘God’ or ‘nature’) that contains or subsumes all its various ‘modes’ of objects, beings and events.28 So likewise with the two ‘attributes’ of mind and body which


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Spinoza conceives – in company with some present-day physical-ist or central-state-materialist philosophers of mind – as merely epiphenomenal, that is to say, as products of our humanly limited powers of apprehension.29 So it is not surprising that Badiou – whose ontology starts out from the notion of infinitely multiple infinities constrained by the stipulative count-as-one in its vari-ous forms – should make a point of staking his distance from Spinoza as the philosopher most committed to a radically moni-stic or anti-dualist, that is, anti-Cartesian metaphysics of mind and world. After all, it was Badiou who caused considerable upset among the followers of Deleuze by claiming that the latter – especially in his thinking about issues in mathematics – betrayed all the symptoms of a covert attachment to the ultimate univoc-ity of being and truth, despite his overt celebration of difference, heterogeneity or multiple and endlessly proliferating ‘lines of flight’.30 I shall not here attempt an adjudication of the issue between Badiou and Deleuze except to say that it reflects their very different views with respect to crucial topics in the history of philosophy from Plato down and – most crucially of all – the relationship between mathematics, philosophy and the various ‘conditions’ that constitute philosophy’s means of access to truth. If there is one point at least on which they agree it is the strict impossibility of thinking the multiple without reference to the count-as-one as that which seeks, albeit vainly, to comprehend the multiple and thereby get a purchase on what would other-wise exceed its utmost capacities of rational grasp.

Thus despite his rejection of Spinoza’s monist ontology Badiou can subscribe unreservedly to the Spinozist dictum ‘For we have a true idea’, and moreover to the Spinozist claim – on the face of it one with radically monist implications – that ‘the order of things’ and the ‘order of ideas’ are in fact one and the same order under different descriptions or aspects.31 This follows from his acceptance of mathematical Platonism, construed in a non-standard way according to which – contra the sceptics, anti-realists and conventionalists – there is simply no distinguishing the object-domain of mathematical entities and truths from the various procedures or acts of thought whereby they are brought within range of discovery or formal-demonstrative proof.32 What saves this conception from the much-touted Platonist ‘dilemma’ of objective truth versus humanly attainable knowledge is

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Badiou’s refusal (with good Platonist warrant) to allow any such gap to open up in the first place, along with his equally firm insistence on the way that mathematics – as our paradigm case of truth-oriented thought – typically achieves its most signal advances through a constant dialectic of problem-creating and problem-resolving initiatives. It is in the restless movement between these poles that mathematics exhibits both its own capacity for creative self-renewal and the extent to which its vari-ous formal procedures bear upon other fields of human experi-ence, knowledge and enquiry. This is why, as we have seen, Badiou takes his cue in matters ontological from those passages in Plato’s Sophist and Parmenides where Socrates most directly confronts the aporias of the one and the many, and where think-ing sets out on the long and tortuous path that will eventually lead to the paradoxes of classical set theory and the various attempts (by Russell and others) to resolve or at any rate defuse those paradoxes. Here as elsewhere – for instance in his com-mentaries on Spinoza and Hegel – Badiou’s is essentially a diag-nostic reading which aims to draw out those symptomatic moments of recalcitrant, resistant or non-assimilable sense that signal the presence of a counter-logic at odds with the thinker’s overt professions of intent.

Above all it is Spinoza who provokes this ambivalent response since Badiou is strongly drawn to certain aspects of Spinozist thought – chief among them his axiomatic-deductive style of rea-soning more geometrico – while none the less rejecting his radically monist conception of mind and nature as two ‘attributes’ of the self-same substance, itself manifested in the various ‘modes’ whose seeming multiplicity belies their true nature as so many aspects or phenomenal appearances thereof.33 For there is an obvious conflict between the central claim of Spinozist ethics, that is, that true free-dom lies in the acceptance of an all-encompassing (even if in large part humanly unknowable) order of necessity and Badiou’s great care to distinguish the realms of being and event. Thus ‘Spinoza represents the most radical attempt ever in ontology to identify structure and metastructure, to assign the one-effect directly to the state, and to in-distinguish between belonging and inclusion’ (p. 113). Such is the result of Spinoza’s radically monist conception when consistently applied to issues in the social, political and ethi-cal spheres as well as in realms – such as the formal and physical


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sciences – where it fails to account for the creative-transformative power of human intellect and also for the way that such advances are achieved against, despite, or (sometimes) as an unlooked-for result of contingent factors that find no place in the kindsof neatly tailored account that typify the ‘official’ histories of those disci-plines. In short, ‘it is clear that [Spinoza’s] is the philosophy par excellence which forecloses the void ’, and therefore that Badiou will need to demonstrate some flaw – some logical inconsistency or fail-ure to meet its own conceptual requirements – in the Spinozist sys-tem of thought. This he does by focusing on the notion of ‘infinite mode’ which Spinoza introduces as a means of bridging those var-ious dualisms (god and nature, infinite and finite, mind and body, essence and accident, freedom and necessity) that would otherwise lack any possible means of reconciliation.

His argument here is extremely compressed and resistant to any kind of summary account. Sufficient to say that he shows this idea of ‘infinite modes’ to bear the full weight of Spinoza’s problematical (indeed strictly unthinkable since logically self-contradictory) doctrine and reveal where that doctrine runs up against the need to acknowledge what it cannot accept in keeping with its own governing precepts. This involves an intensive critical engagement with the whole apparatus of Spinozist thought and, in particular, its strenuous though (as Badiou thinks) its inherently self-refuting attempt to encompass the transcendence of all those vexing antinomies through a radically monist conception – ‘god or nature’, deus sive natura – whereby freedom would achieve its true meaning as the knowledge or acceptance of necessity. Thus Spinoza’s strategy is doomed to fail and ‘the void, whose meta-structural or divine closure should ensure that it remains in-exis-tent and unthinkable, is well and truly named and placed by Spinoza under the concept of infinite mode’ (p. 113). That is to say, the definition of mode as that which pertains to a vast though finite variety of objects presented under either their physical (bodily) or mental (intellectual) attributes cannot possibly be rec-onciled with talk of the infinite, this latter having its legitimate place only in the Spinozist discourse on that which in essence – or by very definition – transcends or exceeds any such limiting con-ception. It is here that Spinoza’s monist doctrine (deus sive natura) can be seen to come up against its limit: that is, ‘the necessity of having to invoke a void term, whose name without a testifiable

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referent (‘infinite mode’) inscribes errancy in the deductive chain’ (p. 120). Or again, the ‘great lesson’ of Spinoza’s thought can be expressed in the following terms: ‘even if, via the position of a supreme count-as-one which fuses the state of a situation and the situation (i.e. metastructure and structure, or inclusion and belonging), you attempt to annul excess and reduce it to a unity of the presentative axis, you will not be able to avoid the errancy of the void’ (p. 120).

Thus it is hard to conceive how Spinoza’s purported deriva-tion of ethics from ontology – entailing his idea of ‘freedom’ as in truth nothing other than the wise acceptance of an all-embrac-ing necessity – could possibly be reconciled with Badiou’s demand for a clear and principled distinction between the realms of being and event. The former (i.e. ontological) domain is that which Spinoza went some way towards charting through his insistence on the discipline of argument more geometrico and hence his historically precocious understanding – one that would not be fully borne out until the advent of modern set theory – that mathematics provides the sole adequate basis for a critical ontology able to account for our knowledge of the growth of knowledge across the formal, physical and social-scientific domains. However what Spinoza’s system crucially lacked was any allowance – on its own terms, any means of making allowance – for the evental dimension wherein human purposes, values and commitments come up against the sheerly contingent character of real-world circumstance. Here it is worth noting that Badiou reads Spinoza very much on Spinozist terms, that is, not in the fideist mode of commentary that would simply report and endorse his ideas but rather through the kind of ‘symptomatic’ approach that Spinoza himself (much to the horror of his piously orthodox contemporaries) applied in his reading of scriptural texts.34 From this standpoint it is possible to see ‘that the infinite mode is where Spinoza designates, despite himself – and thus with the highest unconscious awareness of his task – the point (excluded everywhere by him) at which one can no longer avoid the supposition of a subject’ (p. 113). A ‘subject’, that is, in the precise sense specified by Badiou: one whose existence is predi-cated on or synonymous with the project that commands their utmost fidelity and which thus defines their very condition of subjecthood.


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Spinoza is here being read through a present-day critical optic that incorporates the Freudian–Lacanian concept of ‘imaginary’ misrecognition, Althusser’s structural-Marxist understanding of ideological interpellation, and Marx’s well-known claim to the effect that human beings can indeed make their own history, but not under conditions or in circumstances of their ownchoosing. Yet, far from emerging in any way diminished, Spino-za’s thought is thereby revealed as remarkably prescient to the extent that he can now be seen to have discovered – often (if not always) at a high level of conscious reflection – so much that has gone into those critical protocols now brought to bear upon his work.

Discussion points

What do you find most striking or revealing about Badiou’s con-ception of the human subject and its role in relation to issues of truth and knowledge?

How do you understand Badiou’s closely connected claims with regard to the state, the situation, the void, inclusion and belonging?

PART III. BEING: NATURE AND INFINITY. HEIDEGGER/GALILEO

1. The Greek inauguration: poetry or mathematics?

Part III of Being and Event contains Meditations Eleven to Fifteen and covers a large, not to say giddying amount of philosophic, cultural-his-torical and – increasingly from this point on – mathematical ground. It begins with a section entitled ‘Nature: Poem or Matheme?’ in which Badiou reflects at greater length on a topic first raised in his Introduc-tion, namely the issue between poetry and mathematics as claimants to the title of ‘first philosophy’. That is, he sets out to contest Heidegger’s claim for the role of poetry (rather than mathematics or the physical sci-ences) as having marked the inaugural moment – indeed the very condi-tion of emergence – for ancient Greek thought and all that followed in its cultural-historical wake. This claim was the basis of Heidegger’s late ‘turn’ towards an outlook that placed less emphasis on the idea of reso-lutely facing-up to the call of Being and counselled in stead a mood of Gelassenheit, of receptive openness to truths vouchsafed by poets whose language (especially the German language through its supposed

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deep affinity with the ancient Greek) was still capable of bearing wit-ness to that otherwise long-forgotten source.1

Hence the depth-hermeneutic approach whereby Heidegger claims to think his way back through and beyond the various accretions of ‘Western metaphysics’ from Plato and Aristotle down to Husserl.2 Chief among them is the drastically reduced, denatured and inadequate conception of truth that starts out with Plato’s idealist doctrine of transcendent forms or ideas, that finds its next major statement in Aristotle’s theory of truth as homoiosis or correspondence, and thereafter undergoes a suc-cession of modifications and refinements which on Heidegger’s account do nothing to redeem those defects.3 What they share is a failure to allow for that further, deeper, more authentically revealing since primordial dimension of truth that the Greeks – especially certain of the pre-Socratics – invoked in the name of aletheia, a term that translates literally as ‘unforgetting’, or retriev-ing from oblivion truths that had once been known to the soul but then overlaid (as the story goes in its more mystical Pythago-rean, neo-Platonic or proto-Wordsworthian guise) by false or distracting appearances. For Heidegger, this failure is redeemable only through a deconstruction (Aubbau or ‘unbuilding’) of the various inherited concepts that have held philosophy in thrall, along with an ear and mind sufficiently attuned to the ways in which poetic language may communicate beyond any level of meaning that might be accessed by conceptual analysis or plain-prose paraphrase. Moreover – and it is here that Badiou most emphatically takes leave of Heidegger – such thinking would go furthest towards restoring the kind of receptive or responsive awareness that might yet save us from the depredations of a technocratic reason with its joint source in the genealogy of Western post-Socratic philosophic rationalism and post-Galilean, that is, mathematics-based physical science.4

This diagnosis can only strike Badiou as a gross distortion of intellectual history brought about partly by an under-valuation of the disciplined yet none the less creative activity of thought manifest in mathematics and partly by a basic misconception of poetry’s past and continuing role in that history. For Badiou, one may conjecture, it is just another version of the old logical-positivist doctrine that accepted as genuinely meaningful only that class of empirically vacuous statements (or tautologies)


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whose truth was self-evident purely in virtue of their logical form and that other class of statements whose truth or untruth could be established by observation or empirical warrant.5 Thus Heidegger’s disdain for mathematics and the physical sciences – no matter how disguised by his talk of their predestined emer-gence from the epoch of Western metaphysics – can be seen as just the mirror-image of that positivist disdain for any kind of discourse (including poetry) which failed to meet either require-ment. Badiou is just as far from sharing Heidegger’s hostile atti-tude towards the sciences as he is from sharing this positivist idea of the sciences, physical or formal, as philosophy’s sole source of guidance and therefore of poetry (along with other such ‘emotive’ or merely ‘metaphysical’ modes of expression) as lack-ing any properly assignable truth-value.6 Above all he is deter-mined to show – as against the former mistaken supposition – that mathematics has creative resources in plenty as well as the conceptual or demonstrative rigour whereby to bear out Plato’s claim for its status as first philosophy.

In this respect Badiou stands firmly opposed to those philos-ophers, among them Heidegger and Wittgenstein, who go so far as to deny that mathematics involves ‘thinking’ in any proper sense of the term, that is, any sense that would not confuse think-ing with mere calculation, mechanical reasoning or rule-gov-erned formal procedure.7 Thus he rejects the echt-Heideggerian idea that those poetic resonances, once strongly conveyed and still distantly audible in the Greek word φυ′ στς, ‘nature’, were effectively occluded in its translation to the Latin natura and there after – increasingly so – in the history of usage through which that primordial theme was subject to treatment in con-ceptual or mathematico-scientific terms. For Heidegger, ‘the word “nature”, especially in the aftermath of the Galilean rup-ture, is commensurate with a complete forgetting with regard to what is detained in the Greek word φυ′ στς’ (p. 123). For Badiou, conversely, the Greek inauguration of philosophy – of a think-ing turned towards its future potential for truth rather than turned back towards a deeply conservative mystique of origins – is one that has to do first and foremost with mathematics and not with poetry, at least on Heidegger’s idiosyncratic under-standing of just what an adequate response to poetry involves or requires. If Plato’s dereliction (as Heidegger views it) consists

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in his having chosen wrongly ‘at the ambivalent frontiers of the Greek destiny of being’ and having proposed ‘an interpretation of φυ′ στς as ι′δε′α′ ’ – that is, of the depth-ontological and poetic as the merely abstract or conceptual – then Badiou reads this episode and its subsequent (post-Galilean) history as, on the contrary, a sequence of intellectually liberating events that became possible only through that mathematically inspired break with a language given over to metaphor, image and poetic reverie. To repeat, it is not at all Badiou’s intention to devalue poetry vis-à-vis mathematics, a false impression that might be left by these remarks about Heidegger but which can scarcely survive a reading of his commentaries on Hölderlin and Mal-larmé later in Being and Event. Rather it is to give mathematics its due as the matrix or generative source of any ontology that would seek to conceptualize the precise order of relationship between knowledge and truth, or human understanding at some given (historically defined) stage of advance and whatever, by eluding its grasp at that time, acts asa spur to subsequent progress.

Thus, contra Heidegger, Badiou rejects the depth-hermeneutical idea of language – or our well-attuned harkening to language – as the locus of authentic or primordial truths that were later concealed by the fateful turn towards technology, science, mathe-matics and other such products of the Western drive for concep-tual mastery. As we shall see, poetry (especially that of Mallarmé) has pride of place within Badiou’s conception of art as one of those enabling ‘conditions’ that allow us to grasp the history of truth as progressively revealed through a sequence of world-transformative events which set new terms for artistic, political, scientific or ethical fidelity. On the other hand he takes issue with the later-Heideggerian notion of poetic language as the sole or uniquely privileged vehicle of truth.8 In so far as that role is occupied by any one discipline it is mathematics, rather than poetry, that Badiou conceives as pointing back to the ancient Greek instauration of a thinking that decisively breaks with the doxa of received opinion or commonsense belief and thereby launches both its own and other kindred since truth-oriented projects of enquiry. For Badiou, what is truly unique about that ancient Greek event – what constitutes its properly evental status – is the breaking-through to an order of thought that renounces


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the desire for some long-lost plenitude of being or ‘originary link between being and appearance’, such as might lead thinking back to the condition it once enjoyed before the emergence of all those vexing antinomies (subject/object, mind/nature, reason/intuition and so forth) that took firm hold on the discourse of Western post-Hellenic culture.

Hence the two chief orientations which, according to Badiou, can thereafter be seen to have competed for dominance within that line of descent, whether directly – as in recurrent bouts of the perennial ‘two cultures’ controversy – or in various displaced or surrogate forms. Among the latter would be counted those differences of interest, priority or emphasis (for instance, with regard to the authority of ‘ordinary language’ as opposed to the claims of conceptual analysis) that divide Anglophone philoso-phers along clearly marked lines but which must appear some-what parochial or intra-mural if viewed in this larger perspective. By going back to that inaugural moment when the two great ‘orientations’ first emerged we can perceive more clearly how the ‘[o]ne, based on nature in its original Greek sense, welcomes – in poetry – appearing as the coming-to-presence of being’, whereas ‘[t]he other, based on the Idea in its Platonic sense, submits the lack, the subtraction of all presence, to the matheme, and thus disjoins being from appearing, essence from existence’ (p. 125). What then becomes possible – through recognition of the void that opens up (much to Heidegger’s regret) between being and appearance – is a sharpened since mathematically grounded sense of the kindred distinction between knowledge and truth, or the scope and limits of present understanding as revealed by the proleptic power of thought to perceive not only certain unresolved problems in its present state but also the prospect, by means as yet unspecified, of eventually overcoming them. It is this aspect of its ‘mathematico-ideal’ as distinct from its ‘poetico-natural’ dimension that Badiou considers to have marked the great leap forward in ancient Greek thinking. And if that phrase ‘great leap forward’ might be felt to have irrele-vant or downright unwelcome political, that is, Maoist connota-tions then one needs to think again about what he has to say concerning the relation between mathematics and politics. For it is just Badiou’s point – one with particular resonance and force in the context of this engagement with Heidegger – that from

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Parmenides and Plato, via Leibniz and Spinoza, to develop-ments in the wake of Cantorian set theory there has always been a close if often occluded or disavowed relationship between mathematics (along with the various philosophic doctrines espoused in its name) and issues of political power, justice, equality and representation.

This is why Badiou places such great emphasis on the sheer originality and cultural uniqueness of that Greek mathematical breakthrough, as compared with the poetic inauguration which – as he says – occurred across a much wider (e.g. Indian, Chinese and Egyptian) range of national-cultural contexts. Of course it may well be objected that these cultures also contributed their share to the development of mathematics, along with some notable advances in the field achieved by thinkers from the Arab world or the tradition of Islamic scholarship. However it is Badiou’s more particular thesis that the Greeks played a crucial role in discovering those two most essential components of modern mathematical thought, namely the axiomatic-deductive mode of reasoning and the willingness – as shown in their very different ways by Parmenides, Plato and Aristotle – to reckon with the kinds of paradoxical challenge thrown up by any rigor-ous or sustained address to the question of non-being. Thus ‘[t]he Greeks did not invent the poem. Rather, they interrupted the poem with the matheme. In doing so, in the exercise of deduc-tion, which is fidelity to being such as named by the void, the Greeks opened up the infinite possibility of an ontological text’ (p. 126). Or again, they severed the thought of being from its ‘poetic enchainment to natural appearing’, and thereby replaced the idea of presence – that which ‘demands an initiatory return’ – with an idea of ‘the subtractive, the void-multiple, which com-mands a transmissible thinking’ (p. 126). ‘Transmissible’, that is, in the sense of having founded and opened up a potentially end-less sequence of discoveries, each of them building on what went before while also – as Badiou makes clear – pointing forward to further such signal advances through a sense of problems that have yet to be confronted or conceptual anomalies yet to be resolved. As he more evocatively (even poetically) puts it: ‘[t]he poem entrusts itself nostalgically to nature solely because it was once interrupted by the matheme, and the “being” whose pres-ence it pursues is solely the impossible filling in of the void, such


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that, amidst the arcana of the pure multiple, mathematics indefi-nitely discerns therein what can, in truth, be subtractively pro-nounced of being itself ’ (pp. 126–7).

What is at stake here beyond the issue of priority between these rival ideas of the ancient Greek inception is the claim – central to Badiou’s entire project – that from Parmenides to Cantor and thence to more recent developments there has always existed this power of mathematics, whether zealously pursued or strongly resisted, to explore new realms of ontological possibi-lity and hence to raise questions of a far-reaching kind with respect to other regions of thought, among them that of politics. Thus it is chiefly a matter of those various set-theoretically speci-fied concepts – of membership, inclusion, belonging, the void, the situation, the state of the situation, the evental site, inconsis-tent versus consistent multiplicity and so forth – that permit the elaboration of a social and political ontology with resources to explain both the status quo ante of any existent order and the stress points within that same order where there is the greatest likelihood of its coming under pressure from elements excluded by the count-as-one. That is to say, the possibility that first emerged in those ancient Greek debates and which came to frui-tion with Cantor was that of working out a full-scale, formalized ‘typology of the multiple’ that allowed for the valid, that is, the precise, non-abusive and more than merely metaphoric extra-polation from pure mathematics to the socio-political domain. What emerges very clearly at this point is the contrast that Badiou perceives between a Heideggerian ‘poetico-natural’ ontology, a conception of authentic being and truth turned back towards some long-lost primordial source, and on the other hand a ‘mathematico-ideal’ conception turned towards the possibility of future advance, whether in terms of mathematical discovery or socio-political progress. Where the latter stands to gain from Badiou’s set-theoretical approach is through the purchase this offers for a formalized, that is, conceptually precise and logically articulated treatment of the relationship between actually exist-ing forms of (so-called) social democracy and the form it might take should certain discrepancies – those shown up to striking effect by analysis along such lines – be subject to the kind of recti-fication envisaged by its more progressive or radical-egalitarian reformers.9

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Thus, here as in the case of mathematics, it is the sense of such hitherto contained or unrecognized but now clearly emergent problems that provides both the chief incentive for further prog-ress and also – paradoxical as this might seem – the means of grasping in advance that a solution lies within range of possible achievement even though (necessarily) without as yet the means to encompass or achieve it. As I have mentioned already and will explain in more detail later on, Badiou’s main line of response to this (as he considers it) false dilemma or pseudo-paradox is to invoke the set-theoretical concepts of ‘forcing’ and the ‘generic’, both developed by the mathematician Paul Cohen and both hav-ing to do with this question of how – by what conceivable stretch of intellect or imagination – thought can leap so far ahead of itself as somehow to grasp what it cannot yet know, prove or ascertain.10 That the claim should make sense and should indeed be shown to constitute the sole basis on which to account for the advancement of mathematical knowledge to date is absolutely essential to Badiou’s case not only with respect to intra-mathe-matical issues of truth, knowledge and warrant but also as regards the pertinence of those issues to matters of a larger (at any rate wider) human socio-political concern. Moreover, it is a claim that has lately been disputed – treated as strictly unintelli-gible, even nonsensical – by anti-realist philosophers of mathe-matics and logic who maintain that it involves the absurd (since self-contradictory) idea of our somehow knowing things unknown, or asserting the existence of truths for which we con-fess to having no sufficient evidential, demonstrative or proba-tive grounds.11 In which case, so these thinkers conclude, we had better replace objectivist truth-talk with more sensibly modest or moderate talk of ‘warranted assertibility’, and thereby accept the anti-realist thesis that truth (on whichever preferred under-standing) is always ‘epistemically constrained’, that is to say, restricted to the scope and limits of what we are able to know. Thus realism should be seen not only as a lost cause, philosophi-cally speaking, but also as having created a great many needless and intractable problems since by conceiving truth in objectivist or mind-independent terms it effectively denies (like sceptics down the ages) that we could ever get to know anything.

The most frequent realist rejoinder to arguments of this sort is to point out the crucial difference between claiming to know


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that there are things we don’t know – the contrary of which (that we now know absolutely everything) is after all pretty implausi-ble – and claiming to know just what it is we don’t know, that is, the specific nature and content of those truths that presently elude our epistemic grasp.12 This in turn means respecting the distinction – and (as realists would have it) the order of priority – between ontological and epistemological issues, or questions of the type ‘What exists? What are its objective properties, attri-butes, causal dispositions, etc.?’ and questions of the type ‘What can we know concerning such matters and how do we come to know it?’. As should hardly need saying by now, this distinction is basic to Badiou’s thought with its squarely ontological focus on the various (e.g. physical and abstract) modes of being and the equally various ways and means by which human enquiry is – sometimes, not always – able to cognize or comprehend them. Only thus, so he argues, can we hope to explain the process or indeed the very possibility of advancement in the powers of human cognitive, conceptual or explanatory grasp in and beyond the realm of mathematics and the physical sciences. This goes along with Badiou’s countervailing (yet by no means contradic-tory) insistence that we also take account of that evental domain – the irreducibly contingent situation or context wherein such discoveries come about – which intrinsically resists assimilation to any pre-established or consistent ontology. It is here that he most emphatically takes leave of Heidegger’s depth-ontological hermeneutic broodings, devoted as they are to a ‘poetico-natu-ral’ conception of thought as having once existed – and poten-tially as once again coming to exist – in a state of union or harmony with nature.

Such is the echt-Heideggerian demand that thinking aspire to that ‘self-homogeneous self-presentation’ that fully guarantees – which cannot but guarantee – the unimpeded communion of subject and object, mind and nature, discursive thought and sen-suous intuition. However this wished-for state (this illusion, according to Badiou) must be seen as dearly bought if it entails the abandonment of any claim that thinking might transcend or surpass the limits of whatever presently constitutes its ‘normal’ or ‘natural’ condition. For if ‘nature never internally contradicts itself ’ – if it is the name for that which just cannot give rise to discrepancies, anomalies, paradoxes or other such temporarily

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arresting yet ultimately progress-inducing problems – then the kind of ‘normality’ that goes along with any such naturalizing ground of appeal is one that can also be seen to foreclose any prospect of intellectual, ethical or socio-political advance. ‘Schematically’ Badiou writes,

if N is the situation in question, every element of N is also a sub-multiple of N. In ontology, this will be written as: when one has n ∈ N (belonging), one also has n ⊂ N (inclusion). In turn, the multiple n is also a natural situation, in that if n’ ∈ n, then equally n’ ⊂ n. We can see that a natural multiple counts as one normal multiples, which themselves count as one normal multiples. This normal stability ensures the homo-geneity of natural multiples. (p. 128)

That is to say, the upshot of any such attempt to naturalize (or normalize) the multiplicity of being is a failure – or (in political terms) a programmatic refusal – to recognize those various internal contradictions that emerge most clearly at ‘evental sites’ where normality is exposed to maximum strain and which thus prefigure some imminent crisis or breakdown in the presently existing order. Moreover it is at just these critical junctures that thought may discern the possibility of moving beyond its cur-rent state of impasse by application of new-found conceptual resources discovered or devised precisely in response to that same predicament.

2. Hölderlin: poetry, nature, history

This is why there is such a marked difference of emphasis, tone and level of engagement between Badiou’s two main passages of literary commentary in Being and Event, namely the sections (pp. 191–8 and 255–61) on Mallarmé and Hölderlin. I trust that it will not be too disruptive if I jump ahead briefly at this point since the passages in question are, I think, best treated in the present context of discussion. In the latter case he is writing about a poet whose dominant themes – of homeland, nature, the genius loci, the spiritual affinity of latter-day German with ancient Greek culture, and language as a kind of dwelling-place or native habitat – are such as understandably exerted a power-ful philosophic and imaginative hold upon Heidegger but which


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strike nothing like so resonant a chord with Badiou.13 Thus his reading takes the form of a sometimes strongly evocative yet also, for the most part, a critically detached and even diagnostic meditation on the political perils as well as the poetic allure of a creative vision so deeply grounded in this heady confluence of ‘poetico-natural’ values and beliefs.14 With Mallarmé, on the other hand, Badiou is very much in his element since here the poetry is itself preoccupied – both thematically and by formal implication – with that whole intricate complex of ideas around chance and necessity, being and event or the emergence of the radically new from pre-existent modes of thought, whether these take the form of discoveries in mathematics or truly significant ‘breakthrough’ developments in the creative arts. What sets these poets apart – and what accounts for Badiou’s equally intense but very differently angled engagement with both – is the contrast between, on the one hand, a poetics deeply wedded to the notion of a privileged (i.e. geographically specified yet quasi-universal or world-historical) relation between ancient Greek and modern German culture and, on the other, a poetics that explores regions of thought more often considered the domain of mathematicians, logicians or analytically minded philosophers of language.

With regard to Hölderlin Badiou starts out by acknowledging that ‘any exegesis [of this poet] is henceforth dependent on that of Heidegger’ (p. 255).15 However he then proceeds to qualify and even implicitly revoke that declaration through a reading more alert to the dangers of cultural-aesthetic nationalism than anything to be found in Heidegger’s depth-ontological but – in certain crucial and symptomatic ways – insufficiently critical commentaries. ‘On occasion’, Badiou writes, ‘Hölderlin is quite close to a prophetic conception of this bond, and thereby exposed to the danger of imagining that Germany fulfils the Greek promise’ (p. 259). His poetry is saved or at any rate pro-tected from this danger by its constant feel for the distance – the gulf of historical-cultural ethos – between what he perceives as the ‘Asiatic’ (i.e. what Nietzsche would later identify as the vio-lent, irrational, frenzied, Dionysian) ethos of Greek tragic drama and its German reception under different, highly responsive yet radically transformed conditions of interpretative uptake.16 I shall not here offer a detailed commentary on Badiou’s complex, the-matically rich and – in itself – intensely poetic meditation on

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Hölderlin’s treatment of these powerfully evocative (as well as politically ambivalent) themes. Sufficient to say that his writing on Hölderlin stands out, in contrast to the dominant approach in Being and Event, for its dwelling almost exclusively by way of extended paraphrase on matters of thematic content and paying little or no regard to formal or structural considerations.17 The reason, I think, is that Badiou is here trying to navigate a diffi-cult path between the opposed temptations of a Heideggerian giving-way to the mystique of origins that identifies ‘authentic’ thought with the return to some ultimate source or cultural-linguistically privileged means of access to being, and on the other hand a critique of such notions that would go so far in a sceptical direction as to leave poetry largely bereft of its expres-sive or evocative power.

This tension emerges most strikingly in the last few sentences of his Hölderlin commentary where Badiou can be seen steering away from the Heideggerian hermeneutic depths towards a re-statement of the poet’s main themes in a more ‘philosophical’ or conceptually mediated key. On this interpretation the poet is envisaged as an ‘intervener’, one who essays a ‘second fidelity’ by abandoning himself to ‘the present of the storm’, though strictly on condition of ‘abolishing himself in the void’ and naming the wished-for cataclysmic event – the ‘return of the gods’ – only at a certain protective distance of historical and finely judged poetic reserve. ‘Such’, Badiou writes, is ‘the intervener’:

one who knows that he is required to be faithful: able to fre-quent the site, to share the fruits of the earth; but also, held by fidelity to the other event, able to discern fractures, singulari-ties, the on-the-edge-of-the-void, which makes the vacillation of the law possible, its dysfunction, its crookedness; but also, protected against the prophetic temptation, against the canonical arrogance; but also, confident in the event, in the name that he bestows upon it. (p. 261)

This is a remarkable passage by any philosophic or literary-critical standard but especially in view of the sensitive issues that Badiou is here handling. Among them – not least – is the long-running controversy as to just what kind or depth of elective affinity might exist between Hölderlin’s invocation of the gods in


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the context of German national destiny and later assertions of that same destiny in an altogether darker, more brutally assertive key.18 What Badiou is here doing with extraordinary subtlety and tact is deploying a mixture of sensitive paraphrase, thematic commentary and deft allusion to his set-theoretically derived idea of the event (or ‘singularity’) as that which issues from a void in the order of being, and which thus marks the locus – the ‘evental site’ – where such a strictly unpredictable occurrence may none the less with hindsight appear to have been, so to speak, very much on the cards. This in turn goes along with a marked shift of emphasis in relation to Hölderlin’s more overtly expressed and, again with hindsight, more ideologically trouble-some nationalist themes. Thus Badiou’s reading is one that lays maximum stress on their tentative, subjunctively qualified char-acter and their remoteness from any political programme that would carry such thinking – such a powerful combination of ‘poetico-natural’ and ‘national-aestheticist’ themes – right through to its ultimate, potentially disastrous political conclusion. After all, it was Heidegger who travelled precisely that route and whose wholesale endorsement of the Nazi cause during the 1930s – as well as his failure (or refusal) thereafter to issue anything like an express repudiation or apology – offers the single most striking cautionary lesson in this regard.19

So one can best read Badiou’s exceptionally subtle and fine-tuned Hölderlin commentary not as a revisionist interpretation designed to head off any idea of the poet’s having offered cul-tural sustenance to the ideology of National Socialism but rather as a reading altogether more responsive to matters of nuance and thematic implication that pass unnoticed through Heidegger’s desire to enlist the poetry in service of his own phil-osophico-political themes and imperatives. This is, to repeat, one of the examples that Badiou has most vividly in mind when he writes about the risks of serious compromise that philosophy always runs when it allows itself to become too closely ‘sutured’ to one or other of those four conditions – in Heidegger’s case politics – towards which it should properly exhibit a certain degree of critical reserve. The equivalent of this in terms of lit-erary criticism (more exactly: in terms of philosophy’s engage-ment with poetry through its own distinctive mode of textual exegesis) is a step back from the direct endorsement of certain

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presumptive natural bonds, like those between poetry and the genius loci or language and its native ground, which might oth-erwise exert just such a powerfully distorting effect. Hence Badiou’s claim with regard to the intense but also the deeply ambivalent ethos of Hölderlin’s poetry: that ‘whilst Greece accomplishes its being in the excellence of form because its native site is Asiatic and furious, Germany will accomplish its being in a second fidelity, founded upon the storm, because its site is that of the golden fields, of the restrained Occident’ (p. 260). So it is, on his account, that ‘[t]his evental unbinding . . . prohib-its any frequentation of the site in the assurance of a straight path’. What saves Hölderlin’s poetry from being drawn into a deeper, more ideologically laden complicity with any potent mystique of origins – any thinking that asserts some kind of rooted natural affinity between national locale, native language and poetic authenticity – is also what enables it (when read with sufficient attentiveness and tact) to hold out against Heidegger’s mode of pre-emptive thematic exegesis.20

With Mallarmé it is clear that Badiou is much more on home ground, philosophically speaking, since this is a poet whose characteristic themes and preoccupations – as well as the various formal devices through which they are typically expressed or conveyed – are strikingly close to his own. It is not surprising, therefore, that his commentary focuses on ‘Un coup de dés’ (‘A cast of dice . . . ’), Mallarmé’s most ‘philosophical’ poem and one quite explicitly concerned with just those themes and con-cerns that so preoccupy Badiou in Being and Event.21 Thus he begins by citing a pointedly relevant line – ‘or was the event brought about in view of every null result?’ – and goes on to offer a technically impressive but also a powerfully evocative account of how the poem stages or enacts (rather than describes or represents) a series of strictly evental, that is, unpredictable and unprepared-for occurrences that none the less take their place in what appears (retroactively, no doubt) an unfolding ‘solar drama’ of determinate roles, locations, trajectories, tem-poral conjunctures and so forth. Not that Badiou merely uses the poem as a springboard for developing his thoughts on these topics or a handy source of metaphors, images and suchlike ana-logical (‘poetic’) devices whereby to put across some otherwise demandingly complex and abstract ideas. On the contrary, it is one


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of his leading precepts that ‘poetry thinks’, if not in the same way or with the same degree of formal-conceptual rigour as in mathematics, then at least to the extent that it can offer a working example (as opposed to a merely suggestive illustration) of how thinking fares at the point of encounter with that which exceeds any power of straightforward intuitive grasp. Thus Mallarmé figures as the writer who, more than any other, refuses to be categorized according to the terms – ‘Nature: poem or math-eme?’ – put forward by Badiou in the earlier section bearing that title.

As we have seen, the fact that this appears to pose a choice between mutually exclusive or sharply polarized modes of thought can be explained in large part by Heidegger’s insistence – calling Hölderlin to witness, along with Rilke and other German poets – on poetry’s pre-eminent claim to attention as the chief source of that ancient-Greek ‘inauguration’ that is still, if fleetingly, our sole means of access to truths long since covered up by the accretions of ‘Western metaphysics’. However it soon becomes clear from his reading of Mallarmé that this is a mistaken under-standing of Badiou’s ‘either-or’ question since it ignores the fact that he phrases that question in pointedly interrogative form. Thus he is as far as could be from endorsing the sorts of argu-ment put forward by detractors of poetry from Plato to the logi-cal positivists, that is, the idea that poetry trades in fictions (or falsehoods), imaginary (or just plain make-believe) worlds, and heightened (hence seductive and dangerous) states of emotion, for all of which reasons it should either be banned – Plato’s pre-ferred option – or at least put firmly in its place.22 What Badiou seeks to emphasize, rather, is the standing possibility of a different, essentially non-Heideggerian or non-hermeneutically oriented way of reading poetry. This is a reading that resists the allure exerted on the one hand by an over-regard for those supposedly primor-dial truths-of-Being beyond the grasp of mere analytic reason and on the other by a corresponding under-valuation of any approach that would seriously challenge the received idea of that ‘ancient quarrel’ between philosophy and poetry that Plato was already treating as a cultural truism. On this view the very nature of poetry – or of the mind-set most readily responsive to poetry – is such as to place it intrinsically at odds with every form of genuine truth-seeking enterprise, including philosophy

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and (above all) mathematics. For Badiou, such thinking results from a twofold misapprehension whereby it is supposed that poetry doesn’t think – at least in any sense of ‘thinking’ that would count with those (like mathematicians and philosophers) for whom that activity defines their vocational raison d’être – and moreover that values such as ‘creativity’ and ‘inventive-ness’ have only a limited role to play in the discourse of the formal or physical sciences.

Hence Badiou’s likewise twofold polemic: against literary crit-ics or theorists who fail to appreciate the richness, diversity and truly creative potential of mathematics and also against those in the analytic camp who exhibit a kindred blindness with respect to the kind of conceptual rigour and logical precision that poetry is able to exhibit. It is vital, he maintains, that the difference between them not be collapsed in some vaguely postmodernist mélange of sundry ‘discourses’ from which truth has been sum-marily expelled in the name of an all-encompassing, all-homog-enizing ‘textuality’ but rather that they each be respected as raising its own distinctive kinds of truth-claim. Above all, what this involves – and here Badiou takes Mallarmé as a prime exem-plar – is a break with the idea of poetic intuition as a source of creative or imaginative truths surpassing those of plain-prose reason, and hence a recognition that poetic thinking may well achieve an order of formal or logical complexity raised to the highest power. Thus he takes it that thinking is something quite distinct not only from intuition – since advances in thought, especially in the physical sciences, so often involve the exposure of errors brought about by the presumed self-evidence of sense-certainty – but also from consciousness insofar as it is supposed to be the element within which thinking proceeds or the precon-dition of any thought whatsoever. In this respect Badiou belongs squarely to the company of recent French thinkers in various fields – among them philosophy of science, epistemology, anthropology, linguistics and psychoanalysis – who have rejected any version of Descartes’ appeal to the cogito – the conscious and self-conscious subject – as an anchor-point of certain know-ledge and a citadel against sceptical doubt.23

These developments have ranged from Husserl’s phenomeno-logical re-working of Cartesian themes, via Heidegger’s wholesale ‘depth-ontological’ critique of Western post-Hellenic (especially


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Cartesian) epistemology, to Sartre’s existentialist notion of con-sciousness as ‘for-itself ’ rather than ‘in-itself ’, as that which always already ‘transcends’ any presently ascribable state or con-dition of its own being.24 They were followed and pressed yet fur-ther by the structuralist and post-structuralist turn towards forms of theoretical anti-humanism – such as Althusserian structural Marxism and Lacanian psychoanalysis – which found (or at any rate purported to find) absolutely no room for what they saw as the residual elements of Cartesian subject-centred epistemology in those earlier movements of thought.25 It is fair to say that Badiou takes something of crucial significance for his own proj-ect from each of the above-mentioned sources, from Lacan in particular but also from Sartre and Althusser – improbably enough, given the deep-laid differences of philosophic outlook between these two – and even from those with whom he is him-self sharply at odds in sundry ways, among them Husserl and Heidegger. However what comes across most strikingly in the section on Mallarmé is Badiou’s way of distancing his project from theirs with varying degrees of emphasis even while implic-itly acknowledging those debts. Above all, as I have said, it is his rejection of the basic phenomenological, that is, Husserlian-Heideggerian claim that any thinking directed towards issues of ontology (or questions with regard to the nature, structure and properties of being) must go by way of a reflection on human consciousness, one that for Heidegger in turn takes its bearings from a depth-hermeneutic meditation on the life-world or expe-riential contexts wherein consciousness discovers its primordial sources of meaning and truth.26

3. Mallarmé: poem as event

In Mallarmé Badiou finds a poetry that treats of ‘topics’ or ‘themes’ such as chance, necessity, structure, event, the multiple and the count-as-one – clearly the kinds of subject-matter clos-est to his own interests – while avoiding any kind of commen-tary (as in Heidegger’s exegeses of Hölderlin) that would make them the focus of an existential brooding on the forms of awareness most deeply bound up with our situated modes of being-in-the-world. That is to say, Mallarmé’s preoccupations do have a topical-thematic character insofar as they are explic-itly named, mentioned or referred to in the poem and, beyond

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that, insofar as they provide a structural – even narrative – dimension to the otherwise obscurely related sequence of epi-sodes that constitute Un coup de dés. Yet the poem must also be thought of in performative terms as enacting or directly present-ing that sequence and as doing so, moreover, through a subtle registration of the way that various imagined events impact upon the very process of thought – not (to repeat) the state of consciousness – that it seemingly describes, represents or nar-rates. If ‘[a] poem by Mallarmé always fixes the place of an alea-tory event’, and if that event must necessarily ‘be interpreted on the basis of the traces it leaves behind’, then this is because ‘the meaning (univocal) of the text depends on what is declared to have happened there’ (p. 191). The meaning is ‘univocal’ not – I take it – in the sense that a reading along these lines must prop-erly close off any possibility of figural (e.g. metaphoric, sym-bolic or likewise analogical) readings but in the sense that they will have to start out from an awareness of what is actually, momentarily going on at the level of those textual events – including the titular ‘cast of dice’ – that punctuate the poet’s and the reader’s trains of thought, rather than their ‘stream of consciousness’. Thus it might be said that Badiou reads the poem allegorically, as an imaginative mise-en-scène of precisely those ideas about being and event – or ontology and that which intrinsically exceeds any power of ontological specification – that stand very much at the philosophic heart of his own work. However this claim would need to be qualified by a due recogni-tion that Mallarmé’s poem both anticipates much of what Badiou will have to say concerning set-theoretical matters and also, more importantly, bears out his contention that poetry thinks not so much in the Heideggerian depth-hermeneutic mode, but rather in a mode that invites comparison with certain distinctly mathematical procedures.

At this point the notion of a straightforward contest between ‘Poem or Matheme?’ as rival claimants to truth gives way to the proposal that we conceive their relationship in terms that are less sharply polarized though still – as Badiou is careful to maintain – sufficiently distinct for the purpose of continued, dialectically pro-ductive exchange. The extraordinary way that his reading of Mallarmè combines literal citation with deft paraphrase, textual exegesis, thematic analysis and philosophic commentary is perhaps


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best captured by the following passage where Badiou picks up cer-tain salient motifs from the poem and allows them to resonate with aspects of his set-theoretically based ontology. ‘Allows them’, that is, rather than anything more programmatic or pre-emptive: what is involved here is a reading practice that follows this poem through the various imagined but none the less decisive episodes that peri-odically (yet unpredictably) intervene to change the course of its thinking on the relationship between being and event. Thus his commentary moves from the famous opening line of the poem – ‘A cast of the dice will never abolish chance’ – to a reflection on what this should be taken to imply for our prospects of advance-ment across the whole range of ethical, social and political prac-tices as well as (more clearly) in the formal and physical sciences. ‘If the event is erratic’, he writes,

and if, from the standpoint of situations, one cannot decide whether it exists or not, it is given to us to bet; that is, to legislate without law in respect to this existence. Given that undecid-ability is a rational attribute of the event, and the salvatory guarantee of its non-being, there is no other vigilance than that of becoming, as much through the anxiety of hesitation as through the courage of the outside-place, both the feather, which ‘hovers above the gulf’, and the star, ‘up high perhaps’. (BE, p. 198)

That is to say, there is no logically or rationally guaranteed passage from the realm of ontological (ultimately set-theoretical) think-ing where events can be conceived only as anomalous, marginal or problematic instances that occur at certain localized stress points of the dominant count-as-one to the evental realm prop-erly so-called where that count no longer succeeds in exerting such a dominant role. So it is – in Badiou’s submission – that the most significant breakthrough moments in the history of math-ematical, natural-scientific, political, ethical and artistic thought can be shown to have resulted from just such a rupture with established or accredited modes of conceptual grasp. For there is simply no explaining significant advances in any of these fields by reference to in-place theories, paradigms, frameworks, belief-systems or conceptual schemes that pre-existed the discovery in question.

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However, this should not be taken for some kind of downright decisionist creed to the effect that any genuinely self- or world-transformative commitment – whether in the realms of ethics, politics or scientific enquiry – can be undertaken only on the strength of a radically (and rationally) under-motivated choice between strictly incommensurable options. To be sure, one can see why some commentators have drawn this conclusion from Badiou’s account of the way that commitments of that order come about through just the kind of sudden, dramatic and unprepared-for conversion experience that would appear to preclude anything like a judgement arrived at on adequate evidential grounds or on the basis of a rational decision-procedure. Such, after all, is his view of St. Paul as a ‘militant for truth’ whose revelation on the road to Damascus set the terms and conditions for what henceforth counted as a way of life and a teaching faithful to just that revelatory event.27 This idea – that the Pauline experi-ence is in some sense a paradigm case of what occurs in episodes of radical theory-change across the whole gamut of natural-scientific and ‘humanities’ disciplines – is one that can easily give rise to suspicions that Badiou is espousing a decisionist, fideist or (for those who have registered his firmly secular and atheist standpoint) Sartrean-existentialist standpoint. Nevertheless such a reading would perforce ignore his strong countervailing empha-sis on the need for conceptual rigour in grasping what he reads in Mallarmé’s poem, that is, the aporetic yet strictly required (since logico-mathematically based) conjuncture of necessity and chance and also the fact that choices or decisions are likewise subject to rigorous constraints as regards their fidelity, or lack thereof, to the inaugural event in whose name they are under-taken. Were it not for this distinctly non-decisionist aspect of his work – the room it allows for a post-evental analysis of reasons, motives, interests and consequences – it would of course be open to the damaging charge that ‘fidelity’ amounted to no more than a variant of Heideggerian ‘authenticity’, or a standard of truth to one’s innermost convictions that treated them as simply not accountable beyond that self-enclosed and hence self-justifying, self-exonerating sphere. In which case there could be no means of distinguishing, say, St. Paul’s or Pascal’s exemplary fidelity to what is all the same, so far as Badiou is concerned, an instance of false revelation or a strong yet delusory assurance of truth


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from other cases – whether in mathematics, science, ethics, poli-tics or art – where fidelity is defined not only in terms of authentic dedication to the task in hand but also in terms of its validity or truth as gauged by other (likewise regional or discipline-specific) standards.

So it is important not to mistake Badiou’s stress on the sheerly unpredictable character of events and their transformative impact on the lives and beliefs of those whom they most power-fully affect for a decisionist outlook that would leave no room for the exercise of rational, deliberative thought even if – in the nature of some such events – this would scarcely be a matter of conscious, reflective or present-to-mind cogitation. After all, as we have seen, it is just his point against the whole tradition of Cartesian philosophizing down to Husserl that thinking and consciousness are so far from synonymous that certain kinds of thought (among them some of the most advanced, creative or intellectually far-reaching) may in fact exclude the very possibility of conscious grasp in the process of their first elaboration. This is how Badiou is able to square his claim that such acts, decisions or commitments involve the readiness to stake one’s life on fidelity to that which so far as conscious-ness goes surpasses any presently available evidence or justifi-catory grounds with his belief that such criteria crucially apply not just as a matter of subsequent (post-evental) understand-ing but also as concerns the processes of thought, at whatever preconscious level, on which they are based or from which they result. It is also what emerges in his reading of Mallar-mé’s poem through the interplay of various elements – the-matic, structural, semantic, syntactic and (as Badiou is at pains to bring out) quasi-mathematical in character – which between them capture precisely the way that such thinking tends to elude the best powers of phenomenological (i.e. self-focused and supposedly lucid) reflection. In ‘Un coup de dés’, he writes,

given that the essence of the event is to be undecidable with regard to its belonging to the situation, an event whose con-tent is the eventness of the event (and this is clearly the cast of dice thrown ‘in eternal circumstances’) cannot, in turn, have any other form than that of indecision. Since the master

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must produce the absolute event (the one, Mallarmé says, which will abolish chance, being the active, effective, concept of the ‘there is’), he must suspend this production from a hes-itation which is itself absolute, and which indicates that the event is that multiple in respect to which we can neither know nor observe whether it belongs to the situation of its site. (BE, p. 193)

Here we see the connection between Badiou’s post- (or anti-) phe-nomenological stance and his set-theoretically inspired thinking about issues of necessity and chance, being and event, or ontology and that which intrinsically exceeds any power of ontological spec-ification. Just as thought is always capable of venturing beyond the bounds of what is presently accessible to conscious or self-con-scious reflection, so the event can always exceed or transcend any limit fixed by present conceptions of the count-as-one, whether on the home-ground terrain of mathematics or in those various kin-dred realms – including philosophy and its fourfold ‘conditions’ – where the movement of transcendence can likewise be discerned.

It is by allowing his commentary to replicate this movement and ply back and forth between thematic exposition and state-ments of a highly abstract, formal and ultimately logico-mathe-matical character that Badiou is able to enlist Mallarmé for his own philosophic purposes. I hesitate to put it this way because the complaint has been raised by some otherwise well-disposed literary theorists that Badiou is overly prone to look sharp for motifs that lend themselves readily to such an illustrative role. As a result (so it is said) he tends to read through the text for its philosophic yield and thereby to ignore – or at any rate down-play – those formal, stylistic or thematically ‘redundant’ aspects that inherently resist such treatment.28 There is an element of truth in this charge if his readings of Hölderlin and Mallarmé are measured against the high-formalist (or structuralist) demand that nothing be said with regard to matters of ‘content’ or ‘theme’ until the reader has taken full account of everything that should properly engage her interest at the level of linguistic or formal device where literature stakes its chief claim to our attention.29 No doubt they are motivated largely by a range of concerns – chief among them mathematics and politics – which tend to exert a strong influence in that direction, that is, to ensure that Badiou’s

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interest will be focused on just those thematic (though sometimes also formal or structural) aspects of the text in hand that most readily conduce to such a purpose. However this is really no more than to say that he is a reader of exceptional insight, acuity and powers of conceptual grasp whose approach to poetry (and, besides that, whose choice of these particular poets for detailed analysis) reflects his commitment to a certain view of how phi-losophy can best, most productively engage with literary texts.

4. Versions of infinity: Galileo to Cantor

We can now go on – or rather back a little way, since Hölderlin and Mallarmé involved us in a fast-forward episode – and take a close look at Meditations Twelve to Fourteen where Badiou comments further on the revolution in thinking across a great range of disciplines which he sees as having been brought about by the advent of post-Cantorian set theory. More specifically, these sections have to do with the impact on thought of a set-theoretical conception of the infinite, that is to say, the infinite multiplicity of multiples (or subsets within sets) that will always at some point give rise to paradox no matter how adroitly the problem is managed by formal ‘solutions’ (such as Russell’s The-ory of Types) that in truth come down to a pragmatic doctrine of convenience. Meditation Twelve takes up at the point where we left our orderly manner of proceeding – with the question ‘Nature: poem or matheme?’ – and presses it further by taking as its theme ‘The Ontological Schema of Natural Multiples and the Non-Existence of Nature’. Since it is here that Being and Event takes another more sustained, intensive and demanding turn towards mathematics as its basis for a generalized formal, physi-cal and social ontology it is time to lay out the set-theoretical sources and structure of Badiou’s thinking in a somewhat more detailed (though still necessarily simplified) form. His most insistent point throughout this portion of Being and Event is that ‘[t]here is no infra-mathematical concept of infinity, only vague images of the “very large”’ (p. 145). Thus if thinkers (whether mathematicians or philosophers) evoked the idea of infinity before the advent of Cantor’s set-theoretical revolution then they did so either with a view to denouncing that idea as the source of insoluble and best-avoided paradoxes or – more approvingly but no less confusedly – by way of gesturing towards

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some ineffable realm beyond the furthest bounds of rational thought.30 Moreover, so he maintains, the question ‘What is an infinite multiplicity?’ is one that ‘has not yet been entirely dealt with today’, despite the whole range of impressive advances – by thinkers from Cantor to Cohen – that have made it possible at least to formulate that question in mathematically intelligible terms. Badiou’s point here is that there is and will always remain something distinctly counter-intuitive about the notion that an infinite set (such as that of the natural or counting numbers) might include a ‘smaller’ subset (such as that of the even num-bers) and yet the two of them somehow be capable of having their elements paired off one-by-one. Or again – rather like visual illusions where we ‘know’ that our senses are being deceived yet continue to be taken in at a sensory-perceptual level – there is resistance to accepting the idea that the power-set of any given infinite set, that is, the sum-total of all those subsets included within it must likewise allow for this procedure even though intuition tells us that it must be reckoned far (indeed infinitely) greater than the set itself.

Such is Badiou’s main point at this stage: that there are ‘infinite multiples which can be differentiated from each other to infinity’, and therefore that ‘[t]he ontologization of infinity, besides abolishing the one-infinite, also abolishes the unicity of infinity’, since ‘what it proposes is the vertigo of an infinity of infinities indistinguishable within their common opposition to the finite’ (pp. 145–6). It is only with the advent of Cantor’s revo-lution that it becomes possible to think in such terms without giving way to that sense of vertigo, or to conceive how those var-ious orders or ‘sizes’ of infinity must none the less be counted numerically equal as regards their single most salient defining characteristic. There is the same periodically recurrent sense of strain whenever set theory achieves some further significant stage of advance despite and against sizeable odds of conceptual or intuitive resistance. Thus if thinkers all the way from Aristotle to Hegel denied the very possibility of a ‘completed’ or ‘positive’ infinite – if they deemed it an outright affront to reason insofar as it engendered all manner of paradox or (as they supposed) unresolvable aporia – then undoubtedly they had good com-monsense-intuitive warrant, even though they were wrong as a


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matter of objective or mathematically demonstrable truth. Badiou’s point here is that the claims of intuitive self-evidence have very often gone along with the appeal to nature as a joint guarantee of those underlying laws – or at any rate constant con-junctions – that are taken to supply the needful link between processes of thought and physical events. That is to say, there is a perfectly natural (though not for that reason valid or rationally justified) assumption that when reason is exercised to optimal, that is, most reliably knowledge-promoting or truth-conducive effect then this can only be a matter of such natural convergence on that which lends itself, naturally enough, to human cognitive or intellectual grasp. It is just this kind of emptily circular or self-confirmatory mindset that Badiou sets out to challenge by bringing its claims – its presumptive self-evidence – flat up against the range of conceptual resources available to thinkers who have followed the development and absorbed the implica-tions of modern set theory.

Hence his leading theme in this section: that wherever the appeal to nature is pressed hardest or assumed to carry greatest intuitive weight one is likely to find a deep-laid resistance to pre-cisely the kind of challenge represented by a thinking beyond the furthest limits of currently accredited truth. This is also why he claims – again as it might seem absurdly – that ‘nature does not exist’, at least in any sense of ‘exist’ that would measure up remotely to the set-theoretical conception of being in its infinitely multiple (and multiply infinite) modes of existence. Mathematically speaking, it is in the realm of the ordinals – the ‘placing’ numbers: first, second, third and so on – that ideas of nature and the natural can be felt to exert their strongest and thus their most actively misleading force. This follows from the basic principle (of which Badiou offers a brief formal proof ‘just for fun’) that every multiple belonging to an ordinal will itself also be an ordinal, in which case there is a strictly transitive relation between them and ‘everything which belongs to it is also included in it’ (p. 133). Since ‘an ordinal is thus a multiple of multiples which are themselves ordinals’, we are entitled to deduce that ‘[t]his concept literally provides the backbone of all ontology, because it is the very concept of Nature’ (p. 133). In this respect it involves the collapse of that crucial distinction between belonging and inclusion which, for Badiou, marks the

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crisis-point where established structures of whatever kind – from the mathematical and natural-scientific to the socio-political and cultural – come most visibly under strain. Thus the nature of the ordinals, the fact that they exist in a strictly homogeneous order of relationship one with another, is such as to give them a unique degree of ontological purchase on certain aspects, prop-erties or features of the natural world. However it is also such as to preclude their having any role in the transformative process whereby some existing, ontologically determinate state of affairs comes up against that which exceeds its grasp and can hence reg-ister only as anomalous, discrepant or unaccountable.

In other words, it is the presupposed ‘homogeneity of nature’ that lends the ordinals their character of absolute, invariant conform-ability to type and also, conversely, that aspect of their character which renders nature sufficiently consistent – or homogeneous – for the purposes of rational thought. What Badiou thus asserts against the massed schools of formalist, conventio nalist, fictionalist, con-structivist or intuitionist thought is the thesis that certain eminent features of the mathematical domain – such as the transitivity of ordinals – provide the only adequate basis or starting-point for any ontological enquiry.31 To this extent, but decidedly no further, it is the means whereby thinking can legitimately claim to compre-hend those regions of natural (including natural-scientific) real-ity that would otherwise have to be considered, in Kantian fashion, as lying forever beyond our utmost powers of cogni-tive grasp. Where the limiting condition comes in is once again with Badiou’s insistence on distinguishing sharply between the realms of being and event, or ontology and that which eludes or transcends any mode of ontological specification. Thus he utterly repudiates Kant’s famous turn – his self-ascribed ‘Copernican revolution’ – from ontology to epistemology, along with the phenomenal/noumenal dualism and all the subsequent (as Badiou thinks them) confused and misbegot-ten philosophical endeavours to which that dualism has given rise.32 Rather, for Badiou, the distinction falls between ontol-ogy (which thereby regains its status as first philosophy) and the realm of events where the writ of previous ontological com-mitments ceases to run even though – or just because – it is here alone that such commitments undergo their most decisive chal-lenges and transformations.


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That such events can never be predicted or allowed for in advance – that they always arrive with the shock of the new – is precisely what places them beyond the reach of any going onto-logical scheme. It is also what sets them in a realm quite apart from that of the ‘natural multiples’ or those which, according to Badiou, find their prototypical manifestation in the sequence of ordinals conceived as corresponding to the structure of ‘natural’ reality. ‘If you say that a multiple is an ordinal – a transitive set of transitive sets – this is an absolute determination, indifferent to the situation in which the multiple is presented’ (p. 133). That is, it excludes the very possibility that some event might inter-vene to disrupt and transform the existing situation, or some strictly supernumerary element arrive to exert its force at just the critical point – or evental site – where this state comes up against the void in the form of that which finds no proper or legitimate place in the dominant count-as-one. Nature, or the natural, can hence be defined as that which imposes limits on the range and scope of what is counted fit for presentation (as opposed to what presents itself without being recognized as such) under this or that prevalent conceptual, juridical or socio-political order. ‘Natural consistency – to speak like Heidegger – is the “holding sway”, throughout the entirety of natural multiples, of the origi-nal Idea of multiple-presentation that is belonging’ (p. 134). For if one takes it that ‘Nature belongs to itself ’ – if the conception of nature is precisely that of a realm wherein consistency rules to the effect of excluding or discounting whatever fails so to belong – then one will also take his point that this idea of the natural is prone to certain kinds of abusive extrapolation. Among them is that which more-or-less surreptitiously derives a notion of cultural, civic, socio-political or ethnic community from a notion of the properly or naturally belonging-together with its ultimate source, so Badiou maintains, in a refusal of thought to stretch itself around the kinds of challenge presented by whatever exceeds or transcends its naturalized (intuitive) powers of conceptual grasp.

So there are, on his account, some large and politically as well as philosophically crucial issues bound up with the question of how thought is able to negotiate this fraught passage from finite to infinite orders of reckoning. One is the issue that Galileo bequeathed to the modern physical sciences when he famously

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declared that the book of nature was ‘written in the language of mathematics’. As should hardly need saying by now, Badiou is very far from adopting the fashionable line – whether anti-realist, constructivist, postmodernist, Wittgensteinian or strong-sociological – that would reverse the terms of this statement and assert that mathematical and scientific truths must themselves be conceived as dependent upon, or as intelligible only in the context of, some given natural ‘language’ and its associated range of background beliefs, conventions or ‘forms of life’.33 On the contrary, he is absolutely firm in maintaining that the lan-guage of mathematics, so far from being in any sense beholden to natural language, is the single-most effective and powerful means of detecting, opposing and potentially transcending the kinds of commonsense-intuitive illusion fostered by any such bottom-line linguistic ground of appeal. For it is just this natu-ralizing tendency of ‘natural’ language that Badiou regards as having always exerted – nowadays (alas) with the encouragement and blessing of large sections of the intellectual community – a conformist or downright soporific influence whose source is the idea that thought cannot possibly (intelligibly) claim to break with the informing values and beliefs of its own cultural community.

Thus it is clear that when Badiou cites Galileo’s dictum it is neither with a view to denying the pre-eminence of mathematics as a means of intellectual advancement in and beyond its own specialist sphere nor with the aim of turning it around so as to maintain that such advances are restricted to the scope and limits of what is linguistically expressible (hence conceptually achievable) at this or that time. Such notions are altogether alien to Badiou’s thought, involving as it does an unqualified commit-ment to the twin precepts (1) that truth always might and often does transcend the limits of present knowledge, and (2) that knowledge – especially mathematical knowledge – can itself go beyond what is capable of being expressed in any natural or even any formally refined or regimented language. So it will scarcely be supposed that, by closing Meditation Twelve with a section entitled ‘Nature Does Not Exist’ and by asserting in the course of it that ‘nature has no sayable being’, Badiou means to say that modern (post-Galilean) science has all along been subject to a false or delusory idea of its own truth-telling, truth-conducive


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or at any rate truth-approximative power. Still his reference to Galileo on the ‘book of nature’ does carry a certain force of limiting judgement insofar as it points to the specific respect in which modern (i.e. set-theoretical) developments have moved decisively beyond what Galileo conceived as the natural order of relationship between mathematics and the physical world. Thus, if ‘the set of all the ordinals . . . defines, in the framework of the Ideas of the multiple, the ontological substructure of nature’, then that basic Galilean assumption cannot outlive the classical paradigm where the infinite figured only in a virtual or limit-point role and not – as it would after Cantor – in that of a well-defined and fully operational if still, with every new advance, paradoxi-cal or counter-intuitive concept.

At this stage, Badiou writes, ‘a new theorem of ontology declares that such a set [that of the ordinals] is not compatible with the axioms of the multiple, and could not be admitted as existent within the frame of ontology . . . . There are only some natural beings’ (p. 140). That is to say, where thinking goes beyond the idea of a limit-point ordinal to conceive the existence not only of a positive (genuine) infinity but of multiple orders, ‘sizes’ or cardinalities thereof it finds itself on radically new conceptual ground. Henceforth it will have to make do without certain (as they once seemed) solid and reliable props, such as the ‘natural’ order of relationship between number and the structures of physical reality, while at the same time – and for just that reason – enjoying a massively extended scope of ontological-investigative thought. What most clearly marks the border between these domains – the point of transition to a set-theoretical (hence operative) concept of infinity – is a drastic change in the presup-positions governing the conduct of enquiry not only in logic, mathematics and the formal sciences but also in other disciplines of thought ranging from physics to theology. With regard to the latter, Badiou remarks that Christianity has always been closely tied up with ideas of the infinite, though an infinite conceived in terms that precluded – and came under powerful challenge from – the other, ontologically farther-reaching since thoroughly secularized mode of set-theoretic thought. It is in this sense, he writes, that ‘Christian monotheism, despite its designation of God as infinite, does not immediately and radically rupture with Greek finitism’ (p. 142). And again, if ‘the infinite God of medieval

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Christianity is, qua being, essentially finite’ then surely this is why ‘there is no unbridgeable abyss between Him and created Nature, since the reasoned observation of the latter furnishes us with proof of His existence’ (p. 143).

Indeed one could say – although Badiou doesn’t put it in quite these terms – that there exists a precisely inverse relation between the process of conceptual boundary-marking by which theology has sought to tame the otherwise exorbitant resources of infinity and the opening-up of new, ever more expansive ontological terrain that has come about through the advent of set theory. On the one hand, he suggests, ‘we should note that when [Descartes] was on the point of recognizing the infinity of created Nature itself, under the effect of the Galileo event, [he] also had to change proofs as to the existence of God’ (p. 143). That is, Descartes had to abandon the idea of God as ‘immobile supreme mover’ whose being was defined in contradistinction to the changeable, time-bound, finite realm of merely natural or physical existence and in stead resorted to a mode of argument that placed God in a transcendental realm altogether outside and beyond such modes of analogical thinking. Even so, this religiously inspired concep-tion of the infinite remained tied to a restrictive ontology that still closed off any prospect of advance towards the modern, post-Cantorian conception. This is why, as Badiou says, ‘[t]he effective infinity of being cannot be recognized according to the unique metaphysical punctuality of the substantial infinity of a supreme being’ (p. 143). It is also why the set-theoretical revolu-tion in thinking about these issues could take place – or be car-ried through to a stage beyond Cantor’s residual craving for some kind of crypto-theological warrant – only with the advent of a thoroughly secularized conception of infinity. When this conception finally enters the scene it has two decisive effects: to decouple the infinite (i.e. the infinitely multiple orders of infinity) once and for all from any such metaphysical or supra-mundane grounding, and by the same token to redeploy all these new-found ontological resources in a project of thought that has chiefly to do with mathematics and the natural sciences. Thus ‘[t]he thesis of the infinity of being is necessarily post-Christian, or, if you like, post-Galilean’ (p. 143). And moreover, ‘the radi-cality of any thesis on the infinite does not – paradoxically – concern God but rather Nature’ (p. 143).


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These two propositions fly so directly in the face of widespread modern philosophical and scientific-historical belief that they clearly require further comment here. After all, there is a strong supposition (which Badiou would himself scarcely wish to deny) that Galileo stood up with considerable courage for the helio-centric hypothesis – whatever his short-term accommodating tactics when confronted with the threat of religious persecution – and that his stated refusal to have any truck with talk of the infinite was perfectly understandable, given both its dubious theological associations and the insoluble (or so it seemed at that time) mathematical and philosophic problems to which it gave rise.34 Also there is the prominent strain in French philosophy and history of science that takes a lead from Alexander Koyré’s widely influential account of Galileo as the thinker who, more than anyone, opened the way to a conception of the universe as – in a certain sense – infinite or unbounded.35 However Badiou’s point is neither to deny Galileo’s centrality to the history of early modern science nor to reject that particular claim when properly understood. Thus: ‘[i]f the infinity of Nature solely desi-gnates the infinity of the world or the “infinite universe” in which Koyré saw the modern rupture, then it is still possible to conceive this universe as an accomplishment of the being-existent of the one’ (p. 144). So far from devaluing the Galilean achievement this also allows us to see that he recognized, albeit obscurely, how much more subversive of established beliefs – scientific as well as theological – would be any thought of the infinite that went beyond that metaphysically governed conception to raise the possibility of multiple infinities or an infinite multiplicity. If it is true to say – as Badiou does in this Galilean context – that the question ‘has not yet been entirely dealt with today’, then it has none the less worked strongly upon thinkers from Plato down who have had only partial success in avoiding its powerful, seductive as well as threatening force.

Indeed Badiou makes just this point later on (in Meditation Twenty-six) when he remarks that Galileo, like Pascal, was acutely aware of such logic-wrenching paradoxes as the fact that ‘adding’ some finite quantity to an infinite quantity makes no numerical or quantitative difference to the latter. Or again, since the series of nat-ural numbers was infinite and the series of even number likewise infinite – since they could be placed in a one-for-one correspondence

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ad infinitum – then it would seem (clean against the witness not only of commonsense-intuitive but also of a strongly held logical convic-tion) that the two series must be equinumerous. When confronted with that dilemma, Badiou acknowledges, Galileo ‘quite wisely concluded . . . that the notions of “more” and “less” were not pertinent to infinity, or that infinite totalities were not quantities’ (p. 266). This verdict would be countermanded only with Cantor’s realization that the paradox could be turned into a concept, or that the scandal of the infinite – of a part that must somehow be conceived as equal to the whole – could in fact serve as its very definition or distinguishing mark. However Badiou’s point here is to stress the dialectical process of thought by which the idea of infinity, once it takes hold, has an inbuilt tendency to break through the limits imposed upon it by established, meta-physically constrained habits of belief and thereby assume more radical forms than anything consciously or willingly envisaged by its various proponents.

We have seen already how Cantor exhibited a reluctance to press right through with the implications of his own discovery and a proneness to fall back into more secure, in his case crypto-theological modes of thought so as to keep those implications at least temporarily at bay. Thus the history of set-theoretical enquiry is one that carries on that same dialectic of co-implicated insight and blindness which had marked earlier attempts, from Aristotle down, to grapple with the paradoxes of infinity. At every stage – so Badiou maintains – there is a double and contradictory movement whereby the emergence of new, more radically challeng-ing conceptions goes along with the inertial force of received ideas, thus requiring that such ‘evental’ innovations occur very often despite or against their discoverers’ intent. What transpires at such moments is something that cannot be accounted for by any appeal to rules, methods, research-programmes or established procedures since it consists precisely in the movement beyond what had hitherto specified a valid, consistent, knowledge-conducive or scientifically acceptable way of conducting enquiry. In the case of the infinite multiple – involving as it does just such a movement-beyond – ‘the rule will not present this multiple, since it is by failing to completely traverse it that the rule qualifies it as infinite’, from which follows the necessity ‘that it be presented “elsewhere”, as the place of the rule’s impotence’ (p. 147). Since the infinite multiple stands by definition


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on the far side of that domain traversed by any rule with application to numbers or quantities up to and including the limit-ordinal – that is to say, everything short of infinity – therefore, again by definition, such a rule could neither produce nor encompass such a multiple. This ‘other’ of finite reckoning is thus, according to Badiou, ‘the multiple subtracted from the rule, and it is also what, if reached by the rule, would interrupt its exercise. It is clearly in the position of limit for the rule’ (p. 147).

5. Rule-following and the axiom of choice

Clearly there is a sense which Badiou’s way of raising this issue brings his thought within reach of the Wittgenstein-inspired debate around rule-following that has generated a vast bulk of commentary among analytic philosophers over the past three decades.36 However his approach contrasts sharply with theirs insofar as analytic philosophy of mathematics, logic and the formal sciences tends to focus on a handful of topics – like the rule-following ‘paradox’ or the question whether truth in mathe-matics can be thought of as potentially transcending the limits of human epistemic grasp – which are likelier to engage the interest of philosophers than that of working mathematicians.37 Moreover this difference is reflected in the sharp contrast between Badiou’s choice of complex, challenging and often problematical cases for discussion and the analytic fondness for five-finger-exercise instances (such as that of elementary addition or con-tinuing the sequence ‘n + 2’) which may be thought to throw up interesting issues about truth, knowledge and sceptical doubt but which hardly require any great stretch of mathematical intel-ligence or (even less) intellectual creativity. Indeed, one aspect of Badiou’s work that has probably antagonized those few ana-lytical philosophers willing to give it a glance is his somewhat disdainful attitude towards those ‘minor’, unadventurous, onto-logically modest modes of thought which eschew the risks but also the rewards of engaging such issues at the limits of current mathematical or mathematico-philosophical grasp.

Hence his very different approach to the rule-following issue, one that stands in marked contrast to the set-piece agenda and the narrowly professionalized ethos of mainstream analytic philoso-phy, just as the mathematicians he most admires stand out from the rest on account of their intellectual creativity and speculative

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range. As he sees it, that issue has to do with a ‘dialectic of the “already” and the “still-more”’, a dialectic that has taken in all the most significant episodes in Western thinking about the one and the many from Plato down. In essence it is the question ‘What are the means of thought for rendering effective the thesis “there exists an infinity of presentation”?’, where ‘[b]y “means” we understand methods via which infinity would occur within the thinkable with-out the mediation of the one’ (p. 146). To which his response – borne out by the detailed working-through of some crucial axiomatic-deductive chains of argument and proof-procedures – is that this requires a choice or decision on the thinker’s part, a commitment to explore possibilities of thought beyond those restricted, limit-point notions of the infinite. That is to say, it involves a willingness to leave the safe ground of calculable method or reliable procedure and venture into regions of conceptual terra incognita that offer scant purchase for well-established forms of valid reasoning, let alone for intuitive grounds of the sort that have long been rendered suspect by the progress of mathematical thought.38

All the same this stress on the element of choice – on the fact that even the most rigorous and consequent formal procedures will at some point require an optative rather than a strictly req-uisite formal operation – should not for one moment be con-fused with any kind of irrationalist or (in the usual sense of that term) decisionist approach to these matters. On the contrary, as Badiou is at pains to show, it is only by means of some highly complex and demanding formal-demonstrative procedures that mathematicians arrived at the point of formulating the ‘axiom of choice’ in such a way as to specify this ultimate limit to the compass of deductive or strict probative warrant. What the axiom entails, in its basic form, is that for any given set α there exists another set β which includes a single representative from each element or subset of α. It plays a crucial role in Badiou’s thought for two main reasons: that it is ‘illegal’ in the sense of not being provable by any rigorous formal procedure, yet is none the less strictly indispensable since it provides set theory with its basic working concept of well-orderedness. Thus it captures pre-cisely what he seeks to reveal as the radical dichotomy of being and event, or ontology and that which exceeds the grasp of any consistent ontological account. ‘This axiom’, Badiou writes, ‘is the ontological scheme of intervention (+) but without the event


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(+); it is the being of in tervention which is at stake, not its act’ (p. 500). In keeping with his subtractive conception it marks a point of conflict, conceptual tension or inconsistency within the presently existing ontological schema, and hence perhaps the site of some imminent transformative event though only as a matter of so-far unactualized and therefore as yet unspecific or conceptually indeterminate possibility.39 Badiou remarks that the axiom was – and remains – unacceptable or highly proble-matic to some on account of its seeming to admit an element of irrationality at the heart of mathematical thought. However, this is more a sign of their conservative (if quite understandable) desire to preserve the claims of objectivity and truth, classically conceived, than of their actually applying or maintaining such superior standards in the face of some threatened collapse. For it is just Badiou’s point in working through these issues that the axiom of choice emerged by way of response to foundational problems in the edifice of modern mathematics and logic which imposed themselves with the strictest necessity and not (as regards the requirement to decide one way or the other) as itself a matter of choice. This is mainly because there turned out to be certain basic assumptions – chief among them that which supposed the real numbers to be well-ordered – that were taken as axioms only on the strength of their seeming self-evidence to reason and their absolute indispensability to other, likewise (though derivatively) taken-for-granted truths. Thus the axiom of choice is an exem-plary case of what Badiou puts forward as the single most decisive criterion of truth in mathematics and elsewhere. To repeat, it is the procedure by which an initial venturing beyond the domain of presently attainable knowledge – in this sense only, a ‘leap of faith’ – is later borne out or its validity confirmed by the extent of its truth-conferring potential or knowledge-conducive power.

No doubt it will be said – with some justice – that Badiou’s emphasis on the axiom of choice and its implications for our think-ing about the nature of mathematical truth has much to do with other leading themes of his work. Chief among them is his focus on the epochal event, whether in mathematics, politics, the physical sciences or the arts as that which likewise eludes any means of prior specification or any possible technique for predicting its future occurrence. Along with this goes Badiou’s distinctive idea of fidelity as an attribute manifest in the conduct of those – whether

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mathematicians, political activists, creative artists, philosophers or others – who qualify as ‘militants for truth’ in virtue of their will-ingness to stake all on a commitment to that which exceeds their own or anyone’s else present-best powers of proof or ascertain-ment. On both counts he might be expected to show a special inter-est in and attraction to the idea that elements of choice or decision have a strictly ineliminable role to play in even the most rigorously formalized areas of mathematical thought. Indeed it provides an answer – albeit a mathematically as well as philosophically hetero-dox answer – to this necessity of finding some means of articula-tion between the otherwise totally disjunct realms of being and event, or that which belongs to the order of received (ontologically established) truths and that which can only figure as anomalous or strictly off-limits by any such standard. Hence the sharpness of Badiou’s focus on the movement or decisive moment of thought that led from a variety of vague, ill-defined or approximative notions of infinity to one that for the first time – in Cantor’s work – managed to produce a specification of the infinite that allowed for its taking-up into the history of operative (i.e. truth-apt) rather than merely suggestive mathematical concepts.

Meditation Fourteen continues with the process of demonstrat-ing how that concept underwent a succession of refinements, elab-orations and radical re-statements which brought about its passage from the thesis ‘God is infinite’, to ‘nature is infinite’ and thence to the thesis ‘there exist infinitely multiple orders of infinity’. This involves a fair amount of detailed exegetical treatment aimed towards establishing the sequence of increasingly venturesome hypotheses leading on to increasingly rigorous proof-procedures by means of which that passage was accomplished. Thus the sequence moves – in brief – from the notion of infinity as an end-less series of successor ordinals to the idea of a limit-ordinal mark-ing the inception of whatever lies beyond that series, and from there to a full-fledged (set-theoretical) grasp of the infinite as that to which the finite stands as a special or limiting case, rather than the other way around. The first of these moves occurs with the recog-nition of a ‘qualitative discontinuity in the homogeneous substruc-ture of natural multiples’, that is, the conceptual rupture that occurs when thought comes up against the classical paradoxes concerning the one and the many discovered by philosophers and mathematicians from ancient Greek times to the present. At this


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stage it becomes possible to stake a ‘wager of infinity’, a speculative venture that exceeds any presently available method of proof but which none the less sets terms in advance for what will in due course come to offer the basis for just such a proof, largely or wholly in con-sequence of just such a wager. Where the discontinuity appears is through the fact that successor ordinals are ‘determined on the basis of the unique ordinal which they succeed’, whereas limit ordinals, ‘being the very place of succession, can only be indicated beyond a “finished” sequence – though unfinishable according to the rule – of ordinals previously passed through’ (p. 155).

I don’t have space for an adequate account of the reasoning (much of it symbolically notated for the sake of formal conci-sion) by which Badiou leads us up to this point, let alone for a sizable digression – rewarding though that might be – on how his address to the rule-following issue might let some fresh intel-lectual air into current analytic debate.40 Sufficient to say that these densely argued pages of his book proceed through a kind of via negativa, or by gradually homing in on the post-Cantorian concept of infinity by working through those earlier ideas that can each be shown to have encountered its limit at a certain stage in the ongoing process of elaboration and critique. Thus the statement ‘there exists a limit ordinal’ is the second most basic axiom (or existential assertion) that can be shown to constitute a point of departure for all further mathematical thought. The first and most fundamental – to repeat – is that which asserts the exis-tence of the void as the non-one that inhabits every count-as-one, or the point of ‘subtraction’ from any such count that marks the ‘wandering’ elements presented but not represented within it, and hence its failure to secure a correspondence between belong-ing and inclusion. The concept of the limit ordinal follows through a simple procedure of recursive reasoning whereby thought envisages the possibility of an ‘endless repetition’, or a ‘still-one-more’, which none the less leads – as the next stage in this dialectical progress – to ‘the recognition of a place which is also a beyond’. Here again Badiou invokes Mallarmé as the poet who most strikingly managed to convey this intersection of the finite and that which thinking is driven to conceive as surpassing the finite without, in the process, annulling or negating its limits.

So the concept of the limit ordinal is one of those signal stages in the development of mathematical thought that can be seen to fall

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short of its ultimate aim as revealed by the subsequent course of discovery – that is, with the advent of a set-theoretically defined concept of infinity – and yet to have performed (and still to per-form, for anyone thinking their way through these issues) a decisive role in that development. For it is precisely through bumping up against the problems, puzzles and perplexities engendered by some partial state of knowledge – some limited advance which goes just so far as to expose them – that thinking is impelled to press further along the dialectical path thereby opened up. Thus at this stage ‘we have not yet defined infinity’, at least in any sense remotely ade-quate to its present-day set-theoretical usage. Indeed it is just that sense of a shortfall or state of impasse in our present-best powers of conceptual grasp which itself gives rise to a movement of thought onto new and as yet unmapped mathematical terrain.

A limit ordinal exists; that much is given. Even so, we cannot make the concept of infinity and that of a limit ordinal coincide; consequently, nor can we identify the concept of finitude with that of a successor ordinal. If α is a limit ordinal, then S(α), its successor, is ‘larger’ than it, since α ∈ S(α). This finite successor – if we pose the equation successor = finite – would therefore be larger than its infinite predecessor – if we pose that limit = infi-nite – however, this is unacceptable for thought, and it suppresses the irreversibility of the ‘passage to infinity’. (p. 157)

From here on the course of Badiou’s exposition takes him through the various stages by which the old supposition (that infinity should figure as a deeply problematic and suspect adden-dum to the normal resources of a finite mathematics) gave way to its opposite (that the finite was in truth just a vanishingly small sub-region of the worlds upon worlds of conceptual space opened up to thought by an operative grasp of the infinite).

6. Hegel: how not to think about the infinite

At this point Badiou takes another strategically placed excur-sion through the work of a thinker whose ideas have enough in common with his own but are also sufficiently at variance with them to create a rewarding dialectical encounter. With Hegel it is a question of philosophic reasoning pressed up against the limit of its own conceptual resources in the attempt to think its way


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through and beyond the aporias engendered by reflection on the infinite. What adds to the interest, even (for Badiou) the peculiar piquancy of Hegel’s case is the fact that this thinker went out of his way to devalue or disparage mathematics as a discipline capable of finding out only such trivial since merely formal and content-less truths as belonged to its own, narrowly technical sphere.41 Thus mathematics was, he thought, intrinsically devoid of that world-disclosive and humanly revealing dimension that philosophy was able to provide through its grand-scale pheno-menological project of thinking-back into the whole vast suc-cession of world-historical creeds, cultures, political systems, intellectual developments and artistic forms or genres. On Hegel’s account this was owing to the abstract or purely ratiocinative nature of mathematical thought, that is to say, its taking place in a realm at the greatest possible remove from those vivid actualities of human experience – all the way from ‘primitive sense-certainty’ to the highest forms of artistic expression or philosophical articulation – that made up the well-stocked nar-rative vista of his Phenomenology of Mind.42

In short, Hegel offers a striking case-in-point of that deep-laid prejudice that Badiou is always quick to denounce: the idea of mathematics as a ‘dry’ subject or a discipline inherently deficient in the kinds of creative or inventive resourcefulness that charac-terize the arts or humanities. Indeed, some of Badiou’s most elo-quent passages have to do with the sheer proliferation of axioms, theorems, hypotheses, conjectures and suchlike ventures into new-found land – along with the equally innovative proof-proce-dures that were later devised by way of securing the conceptual terrain thereby opened up – that constitutes the history of math-ematics, especially in Cantor’s wake. So one can see well enough why Badiou should be intrigued by the chapter of Hegel’s Science of Logic devoted to the theme of ‘quantitative infinity’, and especially by the passage – taking the form of a ‘gigantic “remark”’ – wherein Hegel ‘proposes to establish that mathemat-ics, in comparison to the concept, represents a state of thought which is “defective in and for-itself”, and that its “procedure is non-scientific”’ (p. 161). The term ‘concept’ – in Hegel’s philo-sophic parlance – should be taken to signify not just (as per its commonplace usage) an intellectual, mental or ideational com-ponent with its duly assigned place in some system of intelligible

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thoughts or representations but rather a momentarily captured phase in that complex, dynamic, constantly evolving relation-ship of subject and object whose progress to date is his central theme in the Phenomenology and elsewhere. Moreover it is in and through the agency of the concept, thus understood, that Hegel claims to demonstrate the various dialectically interlinked stages that consciousness must be supposed to have achieved en route to its final destination of Absolute Knowledge.

Thus by excluding mathematics – especially where it touches on the theme of infinite quantity – from the realm of the concept Hegel would appear to have two purposes in mind. One is to keep its more threatening, that is, conceptually non-subsumable aspects at bay while the other is to ensure that philosophy is kept safely ensconced within a sphere of constant dialectical self-overcoming where the history of Mind can be written from the standpoint of an all-encompassing retrospect.43 This requires that it be passed in synoptic review as having moved through successively identifiable stages of primitive sense-certainty (or un-self-consciousness), dawning self-knowledge, deepening self-consciousness, emergent self-division and prospective reconcilia-tion between subject and object or mind and reality when the self will at last make good its claim to transcend all such vexing antinomies. Hence Badiou’s assertion that Hegel ‘does not cease to inscribe in-difference toward the Other’, and that the annul-ment of difference – for which read also ‘multiplicity’ and ‘posi-tive infinity’ – within Hegel’s system is always achieved by making sure in advance that consciousness will have the dialecti-cal resources to amortize any residue of that which resists such recuperative treatment. ‘All of Hegel’, he writes, ‘can be found in the following: the “still-more” is immanent to the “already”: everything that is, is already “still-more”’ (p. 162). And again, ‘Hegel, with a special genius, set out to co-engender the finite and the infinite on the basis of the point of being alone. Infinity becomes an internal reason of the finite itself, a simple attribute of experience in general, because it is a consequence of the regime of the one . . . . Being has to be infinite’ (pp. 163–4).

Thus a chief consequence of Hegel’s refusal to accord mathe-matics the dignity of conceptual thought – that is, his confinement of it to the realm of mere formal or mechanical technique – is the way that his system is able to deflect or simply not register anything


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that might place an obstacle in the way of its preordained dialec-tical progress. More specifically, the outcome is to make it incon-ceivable, in terms of that system, that infinity could ever confront thinking with a real (as opposed to merely notional or tempo-rary) challenge to its sovereign re-integrative power. To be sure, Hegel himself seems in quest of a more radical, less self-assured or restrictive conception when he famously refers to the ‘bad infinity’ that thought engenders when subject to a ‘law of repeti-tion’ which merely involves the reiterated instancing of ‘n + 1’ or some kindred formula applied in the absence of any determinate or specified halting-point.44 Such a ‘boring’ conception of infin-ity is just what Hegel purports to overcome – or dialectically transcend – through the calling of thought to a ‘higher duty’, a requirement that ‘the passing-beyond be passed beyond’, or that ‘the law of repetition be globally affirmed; in short, that the Other come forth’ (p. 164). However, so Badiou argues, this claim is undermined by the fact that Hegel’s notion of the ‘good’ infinity not only bears a curiously close resemblance to the ‘bad’, but also counts as ‘good’ on his own submission for precisely that reason, that it remains within the compass of a dialectical schema that is always guaranteed (by its own lights) to restore the sovereignty of reason. Thus in truth there is a sense – one that Hegel’s text does its best to suppress but which emerges to view through a symptomatic reading of the kind here essayed by Badiou – in which the ‘bad infinity’ can be seen to subvert the claims advanced on behalf of its ‘good’ counterpart. That is to say, if the ‘bad infinity is bad due to the very same thing which makes it good in Hegelian terms’, namely that ‘it does not break the ontological immanence of the one’, then surely one has to conclude that the good infinity likewise suffers from that ‘limited or finite’ character which, in the case of the bad infinity, ‘origi-nates in its being solely defined locally, by the still-more of this already that is determinateness’ (p. 165). In other words the supposedly decisive passage from bad to good, local to global, quantity to quality or finite to (properly) infinite is a passage that must rather be seen to ‘impose a disjunctive decision in which the being of the one will falter’.

From a certain point of view – that expressly taken by Hegel and his fideist interpreters – it is fair to say that ‘[t]he Hegelian artifice is at its apogee here’, in the process of self-assured dialectical

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sublation by which that passage from lower to higher is purport-edly achieved. From another, more sceptical or critical viewpoint it is at just this point that the Hegelian system breaks down on its failure to contain, subsume or comprehend that which belongs to the realm of sheer multiplicity, that is, of the infinite properly con-ceived as by the very definition exceeding every limit imposed by the dominant count-as-one. What Hegel attempts is a form of dialectical Aufhebung – a process of conflict-resolution com-bined with the movement of transcendence to a higher, more advanced stage of understanding – that would finally transform the bad into the good, or the quantitative into the qualitative mode of infinity. However, according to Badiou, Hegel ‘fails to intervene on number’ insofar as his conception of the good (qualitative) infinite can count as such, that is, as an infinity in any meaningful or contentful sense of the term only on condition of its making a surreptitious appeal to that ‘still-more’ or ‘n + 1’ conception that typifies the bad infinite. No matter how ‘heroic’ the attempt, Badiou concludes, ‘it is interrupted de facto by the exteriority itself of the pure multiple’ (p. 169). Hegel’s highly spiritualized or sublimated idea of transcendence achieved through the qualitative overcoming of merely quantitative notions is itself overcome – shown up as a false or delusory idea – by the power of a rigorous conception of the infinite to break all the bounds imposed by this or that version (no matter how dialecti-cally elastic) of the count-as-one. ‘In wishing to maintain the con-tinuity of the dialectic right through the very chicanes of the pure multiple, and to make the entirety proceed from the point of being alone, Hegel cannot rejoin infinity’ (p. 170).

Thus Hegel is a prime example of the need, as Badiou sees it, to read the cardinal works of philosophy in such a way as to bal-ance respect for their argumentative purpose with a critical focus on those symptomatic passages that reveal the presence of con-ceptual strains unacknowledged by the text as likewise by its more fideist or orthodox commentators. Hegel is constrained by the logic of his own argument – as soon as it touches on the deeply problematic and paradox-engendering topic of infinity – to bear involuntary witness to certain truths that find no place in his express philosophical creed. Among them is the truth that mathematics takes priority in matters of ontological enquiry and, following from this, that where such enquiry is pursued beyond


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the stage of naïve or intuitive self-evidence then it cannot but engage with issues (like that of the infinite in its non-dialectically-subsumable form) that challenge its very foundations. ‘In qua-lity, what is repeated is that the other be that interior which has to cross its limit. In quantity, what is repeated is that the same be that exterior which has to proliferate’ (p. 168). It is between the poles of this double aporia – this condition of a thinking brought up against the limit of its own conceptual resources – that Badiou locates the ‘real’ of Hegel’s philosophy, that is, the point of encounter with that which resists its utmost powers of dialectical assimilation.

Badiou’s can thus be seen as the most recent in a series of French engagements with Hegel over the past six decades or so that have all purported, in various ways, to present a strong-revisionist or heterodox reading of Hegel’s work. These interpre-ters – often taking a lead from Georges Bataille – have focused chiefly on the Phenomenology of Mind with a view to subverting its more systematic (or authoritarian) claims and stressing those recal citrant details, implications or narrative events which can-not be straightforwardly subsumed under Hegel’s grand dialecti-cal schema.45 Such readings typically reject any notion that history in its manifold aspects – political, cultural, artistic, religious, philosophical and so forth – could ever be subject to the kind of large-scale teleological or ‘totalizing’ grasp that would interpret its every last episode in terms of some quasi-providential unfold-ing of the Idea in its various world-historical manifestations. As against that notion Hegel’s postmodern exegetes stress the way that history mocks any claim to reveal its ultimate intelligibility by throwing up a constant succession of random, contingent, unpredictable events which tax the best efforts of conceptual, dialectical or synthesizing grasp to the limit and beyond.

Badiou may be said to belong to their company up to a point insofar as he likewise reads against the grain of Hegel’s explicit or professed intent, and does so moreover with the purpose of resisting some of Hegel’s more coercive claims. However it should also be said that he is far from endorsing any version of the facile postmodernist creed that offers nothing in place of those lost certitudes except a doleful or jubilant (according to taste) dec-laration that history has come to an end or no longer makes any kind of sense, least of all from a ‘meta-narrative’ perspective of

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the sort exemplified by Hegel.46 Where Badiou takes leave of these thinkers is chiefly by insisting that any attempt to go beyond Hegel must first have gone through the dialectical stages of his thought with the degree of tenacity and critical rigour required to understand where its limits occur and why they emerge at those particular points in its unfolding. Moreover, the aspect of Hegel’s thinking that alone gives adequate conceptual purchase for a critique of this kind is the opening onto issues in the realm of mathematics – that is to say, issues concerning the ‘good’ and ‘bad’ infinite – rather than the opening onto ques-tions of language (of hermeneutics, narrative or representation) such as tend to preoccupy Hegel’s postmodernist or all-out revi-sionist interpreters. What the approach via mathematics achieves is an especial clarity of focus on just those crucial argumentative turns and resultant stress-points in Hegel’s system – like his notion of the mind’s ‘interiorizing’ power as a means of warding off any threat from the realm of unbridled numerical prolifera-tion – that are merely ‘reinterpreted’ and rendered innocuous by the language-first account.

Thus ‘[o]n the basis of the very same premises as Hegel, one must recognize that the repetition of the One in number cannot arise from the interiority of the negative’ (p. 169). Only by engag-ing Hegel through his treatment of such highly sensitive since always potentially system-disruptive topics can one hope to reveal those instructive stress-points through a ‘symptomatic’ reading of the sort that Badiou no doubt learned from an earlier generation of anti-Hegelian Marxist theoreticians, among them Louis Althusser and Pierre Macherey.47 Along with this – and from the same, broadly structuralist intellectual stable – goes that concept of the ‘real’ for which Badiou is mainly indebted to the psychoanalyst Jacques Lacan, and which he puts to work (in his reading of Hegel and elsewhere) as a shorthand reference to whatever exceeds, eludes or obstructs a given project of thought while also in some sense representing its sole object or motivat-ing interest.48 What Badiou retains from the structuralist ‘moment’ in French thought is its adherence to certain clearly formulated precepts of what Althusser termed (with a Marxist inflection) the domain of ‘theoretical practice’ together with its basically rationalist conviction – in sharp contract to current postmodernist ideas – that such practice cannot do without the


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virtues of conceptual rigour and a due regard for the claims of system and method. Thus despite its appeal to the Saussurean paradigm of language (la langue), synchronically conceived, as a model for the conduct of enquiry in manifold disciplines besides that of linguistics itself there is still a clear sense in which structuralism held out against the subsequent slide into a post-structuralist notion of language, discourse or textuality as going ‘all the way down’.

That is to say, it is a crucial tenet of Badiou’s undertaking – one which reveals the formative influence of certain structuralist as well as deeper-laid rationalist precepts – that thought is not passively or unresistingly subject to language as its ultimate horizon of intelligibility. On the contrary, its main task is to question and criticize false or unwarranted beliefs, even (or espe-cially) where these result from certain presuppositions ingrained in our habitual thinking through the suggestive, analogical or suasive power of language. Here again, one can see why Badiou comes out so emphatically against that strain of Wittgensteinian ‘ordinary-language’ philosophy that purports to release us from the various puzzles and perplexities induced through our prone-ness to ‘bewitchment by language’, but tends rather to strengthen their hold through its own failure – or resolute refusal – to adopt a more critical or analytic stance. This opposition emerges to striking effect in his commentary on Hegel where Badiou notes how the ‘disjoining decision’ – Hegel’s wish to ‘maintain the con-tinuity of the dialectic right through the very chicanes of the pure multiple’, and his decision to do so by ‘mak[ing] the entirety proceed from the point of being alone’ – can be seen to ‘make its return in the text itself ’ as a symptomatic tension that no amount of conceptual finessing can resolve or assuage. Where it registers most pointedly is in the ‘split between two dialectics, quality and quantity, so similar that the only thing which frees us from hav-ing to fathom the abyss of their twinhood, and thus discover the paradox of their non-kindred nature, is that fragile verbal foot-bridge thrown from one side to the other: “infinity”’ (p. 170).

This passage makes clear the distance between Badiou’s critical-rationalist conception of philosophy’s legitimate role and any version of the doctrine – whether Wittgensteinian, post-structuralist or Heideggerian-hermeneutic – which would con-fine that role to the provision of constant reminders à propos the

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ubiquity of language, along with the error of supposing that thought might exert a corrective or emancipatory effect despite and against the powers of linguistic bewitchment. It brings us to a stage in the unfolding argument of Being and Event where Badiou has laid out his main philosophical and (most importantly) his chief mathematical terms of reference, and can now move on to a consideration of how these bear on issues in the wider historical and socio-political domain. All the same, and despite Part IV bearing the title ‘The Event: History and Ultra-One’, there is still some way to go in the exposition of set-theoretical themes before that bearing is made sufficiently clear for Badiou’s purpose.

Discussion points

Why do you think Badiou makes such a cardinal point of con-trasting the two ancient Greek ‘inaugurations’, those of mathe-matics and poetry? Is he justified in drawing this sharp distinction or does it result largely from his own preconceived set of philo-sophic and historical-cultural-political priorities?

What is the relationship, as Badiou sees it, between the way that various thinkers (from Aristotle to Hegel) have shied away from any conception of ‘positive infinity’ in the realm of mathematics and their attachment to a more-or-less conservative view of the social and political order?

PART IV. THE EVENT: HISTORY AND ULTRA-ONE

1. History, politics, evental site

Part IV starts out by offering a precise specification of the terms for what many philosophers, analytic philosophers especially, would regard not only as a forced relationship but as the sign of an egregious category-mistake or failure to respect the proper distinction between utterly disparate realms. That is to say, he purports to move by a logical (not merely analogical) procedure of thought from the realm of mathematics to that of the vari-ous social and political structures that involve similarly diverse – though here always partial – modes of inclusion, belonging and membership. His starting point, as in earlier passages, is that cardinal distinction between nature and the non-natural or


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anti-natural which Badiou takes as the definitive mark of a thinking addressed to decisive questions in both domains. Indeed they are questions that are bound to arise in any domain – here following Hegel – where it is a matter of breaking with intuitive sense-certainty and exploring the capacity of reason to discover as-yet unknown and hitherto unknowable regions of conceptual space.1

The dualism of the natural and anti-natural finds a number of parallels at this point of Badiou’s argument, among them being those between normal and abnormal, member and part, repre-sentation and presentation, belonging and inclusion, or element and subset. Although these distinctions don’t map precisely or perfectly onto one another they do capture his general point: that a set-theoretically based ontology taking full advantage of recent (post-Cantorian) progress in the field has a fair claim to articulate the logic and the salient structural features of those other, that is, socio-political states, situations, or – in Badiou’s pointedly innovative phrase – ‘states of the situation’. In answer to the question ‘What is the abnormal?’ Badiou writes, ‘what are initially opposed to normal multiplicities (which are presented and represented) are singular multiplicities, which are presented but not represented. They are multiples which belong to the situ-ation without being included in the latter; they are elements but not subsets’ (p. 174). From which he derives some highly conten-tious but, for his purposes, absolutely crucial and (as I have here sought to demonstrate) logically consequent series of proposals. It is on this basis – by proposing a rigorous distinction between the domain of strictly ‘singular’ (i.e. evental) occurrences and whatever can be annexed without remainder to the naturalized or normalized domain of the prevalent count-as-one – that Badiou mounts his case for a radical re-thinking of history and politics. Thus he goes on to explain, albeit briefly at this stage, how research into the causes and course of such events will typi-cally show them to have taken rise within a site – a complex or overdetermined conjuncture of social and political circumstance – whereby they are placed ‘on the edge of the void’ and can be seen to constitute a multiple ‘made up exclusively of non-presented multiples’ (p. 175). At this stage, as in the early phases of the French or Soviet Revolutions, there develops an exceptional – indeed wholly unique – state of affairs which has the effect of

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reversing or inverting the normal relation between centre and margin, included and excluded, those who ‘count’ in socio-political terms and those whose presence (or whose bare exis-tence) counts for nothing.

What then occurs, on Badiou’s account, is a process of inten-sive and collective mobilization brought about by the emergent sense of common purpose among such hitherto excluded ele-ments, together with a corresponding sense of breakdown, of looming catastrophe or legitimation-crisis among those who up to now formed the dominant class-fraction or ideological bloc. His thinking here has a good deal in common with Sartre’s more detailed description, in the Critique of Dialectical Reason, of how certain extreme pressures of circumstance (i.e. of social injustice and political exclusion) will sometimes – through the impact of some crucial triggering event – create the requisite con-ditions of emergence for a pre-revolutionary ‘group-in-fusion’.2 This group then succeeds, at least for a while, in overcoming the usual state of human social relations, which Sartre takes to be marked chiefly by mutual hostility or reciprocal mistrust, relieved only by periods of downright indifference to other people’s needs or desires. It is during those exceptional (since up to now regret-tably short-lived) periods of group agency inspired and directed by a shared sense of moral-political purpose that there also emerges the distinctive possibility of true revolutionary praxis, that is, of political acts and interventions likewise undertaken with a clear-eyed collective grasp of their aims and likely conse-quences. Badiou has less to say on this topic in Being and Event than in some of his other, more directly political writings.3 All the same it is a constant if muted refrain to his many remarks – especially in this present sequence of Meditations – about the emergence of sin-gularities, ‘excrescences’, or evental sites and their power of resis-tance to established (de facto ‘legitimate’) socio-political structures and modes of representation. Above all, what captures his attention is the way that such localized stress-points can intensify to the point where they take on a globally disruptive or full-scale system-sub-verting character, whetherin the context of political situations, in formal disciplines such as mathematics and logic, or any field which typically moves through protracted phases of ‘normal’ develop-ment interspersed by shorter periods of pre-revolutionary crisis and consequent paradigm-change.

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However there are three main features that distinguish Badiou’s account of such episodes from the widely influential treatment to be found in Thomas Kuhn’s book The Structure of Scientific Revolutions.4 One is of course that Badiou seeks to theorize the conditions of possibility for the occurrence of such transforma-tive events through a set-theoretical investigation of the points at which some existing (whether social or scientific) ontology comes under paradigm-breaking strain from unexpected devel-opments within or outside its home territory. The second is the far greater stress he lays on those exemplary figures (or ‘mili-tants’) whose life histories, commitments, discoveries, conver-sion experiences, holdings-out against political pressure or orthodox scientific belief, and so forth, he sees as providing a nec-essary focus or rallying point for the various dedicated groups – of whatever kind – that form in their historical wake. The third, somewhat in contrast to these, is Badiou’s clear-eyed recognition of the forces that have always been ranged against any such threatening mobilization of the socially excluded or politically oppressed. Here again he is in agreement with Sartre, in particu-lar with those passages of Sartre’s Critique which analyse the way that praxis comes up against the thwarting and distorting effects of the ‘practico-inert’, that is, the often obstructive force of mate-rial circumstance and also the cumulative after-effects of past actions that have been deflected from and turned back against the agents’ original aim. This diagnosis goes along with Sartre’s description of that effect of ‘counter-finality’ that likewise tends to thwart even the best-laid revolutionary plans and which again most typically takes the form of an ironic, often brutal reverse.

In Badiou, the process receives a less dramatic but equally sober description according to which it is ‘one of the profound charac-teristics of singularities’ that ‘they can always be normalized’, or of inaugural events – such as those that spark a revolution – that they can always be drawn back into the ambit of non-singular (i.e. historically assimilated and politically no longer threaten-ing) occurrences. After all, ‘as is shown . . . by socio-political his-tory, any evental site can, in the end, undergo a state normalization’ (p. 176). However this is no more than a due recognition that events – and the sites from which they take rise – may undergo a great variety of subsequent fortunes, some of which conserve and carry forward their progressive or liberating impulse while

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others bring about a reversion to the status quo ante or a restrain-ing effect that cancels or negates that impulse. Indeed, it is just this aspect of contingency with respect to both their conditions of emergence and their likewise contingent (or non-preordained) legacy of positive or negative effects that most emphatically marks out genuine events from other episodes that might be counted ‘world-historical’ on a more conventional estimate. Thus ‘[i]t is solely in the point of history, the representative precari-ousness of evental sites, that it will be revealed, via the chance of a supplement, that being-multiple inconsists’ (p. 177). ‘Incon-sists’, that is, in the set-theoretically derived sense that Badiou assigns to this crucially loading-bearing neologism. Thus such an instance or condition of being-multiple may be said to ‘exist’ owing solely to the absence or determinate lack of some element whose presence would so radically subvert or transfigure the existing situation as to re-define the very terms of inclusion, membership and belonging.

This is why Badiou is so intent on maintaining the categorical distinction between on the one hand such basically normalizing concepts as nature, consistency, representation and the ‘state’ conceived in onto-mathematical or onto-political terms and, on the other, such intrinsically resistant or inassimilable terms as event, presentation and singularity, taken as defining the realms of history and politics. It is what he means by saying that ‘histo-ricity is presentation at the punctual limits of its being’ and that, contra Heidegger, being ‘comes forth’ only ‘by way of historical localization . . . because something is subtracted from represen-tation, or from the state’ (p. 177). Moreover, it is through the habit of reverting to ‘natural’ (i.e. socially normalized) concepts and categories that thought is deprived of its critical edge and rendered compliant with the dictates of received political wisdom. Thus the appeal to some ideological surrogate or substitute for nature often serves as a means of deflecting, containing or (at the limit) forcibly suppressing whatever might challenge the estab-lished order of things. ‘Compact excess of presence and the count, nature buries inconsistency and turns away from the void. Nature is too global, too normal, to open up the evental convocation of its being’ (p. 177). Here again, one can see the close affinity between Badiou’s thought and that phase of classic high struc-turalism – typified by Roland Barthes’ Mythologies – which

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aimed to contest the ideological self-images of the age by reveal-ing how purely cultural constructions of meaning, value and belief were passed off as altogether natural or as belonging to an order of timeless, culture-transcendent truths.5

It is for this reason chiefly that Badiou defines ‘historical’ situ-ations as those in which ‘at least one evental site occurs’, thereby placing them in direct contrast to ‘the intrinsic stability of natural situations’ (p. 177). It is also why he insists that ‘historicity is a local criterion’ rather than claiming any large-scale, encompass-ing or teleological significance such as Hegel notoriously took it to possess and more recent thinkers – Sartre among them – have mostly denied but very often been tempted to smuggle back under a range of more modest-sounding substitute terms. Thus Sartre’s idea, in the Critique, of ‘totalization without a totalizer’ can best be seen in the light of this desire (as it were) to eat one’s dialectical cake but not swallow it whole, or hang onto the useful sense-yield-ing kernel in Hegel’s thought while rejecting its more grandiose or universalist claims. Indeed, as Badiou would be quick to point out, that same ambivalence can be seen in yet more flagrantly self-compromising form when postmodernist gurus such as Jean-François Lyotard pronounce an end to the epoch of modernist ‘grand narratives – including that of Hegel – while delivering themselves of just such a narrative (and one with very definite designs on the reader) concerning that supposed epochal event.6

Badiou’s work stands in marked contrast to this and other reactive trends, among them the post-structuralist revolt against structuralist ideas of system and method in the name of a hence-forth infinitized play of subversive ‘signifying practice’. If this notion already has its place reserved in the grand panorama of Hegel’s Phenomenology – a place somewhere in the regions of ‘unhappy consciousness’ and ‘romantic irony’ – then it is also very firmly located by Badiou in the zone of those regressive movements of thought that typically emerge at times of political setback or perceived defeat.7 Such times he labels ‘Thermidorean’ with reference to that month in the French Revolutionary Calen-dar when power and initiative slipped from the hands of the radical vanguard and events started to take a regressive or increasingly counter-revolutionary turn. Since then it has been applied in various similar contexts, as when Trotsky lambasted the ‘Thermidorean’ disaster of Stalin’s ascent to power in the

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Soviet Union, or when Badiou puts it to use in describing the retreat from genuine, that is, grass-roots as well as intellectual radicalism in post-1968 French society and culture. Thus in reproaching the Nouveaux Philosophes and other such fashion-able, media-savvy intellectuals for having betrayed their calling he sees them very much as willing apologists for a Thermidorean political culture that has sold its revolutionary birthright for a mess of state-sponsored liberal-democratic pottage. The same goes for modish schools of thought like post-structuralism and postmodernism which likewise bear witness to a widespread sense of disillusion with any form of philosophic or theoretical commitment that would lay claim to real – as opposed to merely notional – world-transformative power.8

Where Badiou’s thought departs most sharply from theirs is in its steadfast rejection of the linguistic turn in whatever guise and, as the positive counterpart to that, in the privileged role it accords to mathematics as a means to establish both the groundwork of an ontology for the formal, physical and social sciences and the point of demarcation where events can be thought to exceed any limit hitherto established with respect to those ontological domains. Thus if historicity is defined only by ‘local’ or context-specific criteria, then this can most aptly be expressed in mathe-matical terms to the effect that ‘one (at least) of the multiples that the situation counts and presents is a site, which is to say it is such that none of its proper elements (the multiples from which it forms a one-multiple) are presented in the situation’ (p. 177). From which it follows also – as against those of a postmodern-Thermidorean or suchlike politically jaded persuasion – that contingency is so much an integral, indeed a defining aspect of historical events that any verdicts purportedly derived from those events and couched in the ‘end-of-ideology’ mode are not merely premature but grossly misconceived.9 They result from a failure to grasp the fact that the political significance of certain events cannot be evaluated simply as a function of their short-term or even their long-term impact on the course of history to date. For those events may turn out to possess a hitherto unrec-ognized potential, an exemplary character that has so far counted for nothing with the arbiters of political ‘realism’ but which might yet serve as a motivating force or a uniquely power-ful source of inspiration to those embarked upon later projects

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of revolutionary change. And conversely, there are certain ‘great’ events according to the standard (i.e. orthodox-historical or mainstream-political) account that may well turn out to have been nothing like so significant in the light of as-yet unpredict-able changes in the course of future history.

All the same – and this is where the distance opens up between Badiou’s mathematically based approach and any version of the hermeneutic, linguistic or wholesale narrative turn – there is absolutely no question of his lending support to the idea that interpretation goes ‘all the way down’.10 Still less would he sub-scribe to the strong-revisionist/constructivist claim that past events are always and inevitably products of the interests, pri-orities or kinds of ideological parti pris brought to them by present-day historians.11 Rather it is a question of those highly specific but multiply overdetermined varieties of possible out-come that follow in the wake of any major (or potentially major) historical event and whose significance, like that of the event itself, is always in some respects yet to be decided. Thus Badiou has no truck with the claim of those postmodern histo-riographers – sceptics, relativists, constructivists, ‘strong’ textu-alists or anti-realists – who press too far (or in the wrong direction) with the idea that past events are endlessly open to the winds of ideolo gical change. In Meditations Seventeen and Eighteen, ‘The Matheme of the Event’ and ‘Being’s Prohibition of the Event’, he returns to this theme of how mathematics as the basis of a critical, that is, non-dogmatic yet none the less rig-orously formalized ontology can offer invaluable guidance here. That is,it suggests how thinking might combine a due respect for the requirements of objective historical truth with a due sense of how historical events can, so to speak, flash up from the van-tagepoint of later situations and acquire an altogether new sig-nificance. After all, ‘[i]t is always possible that no event actually occur’ since ‘[s]trictly speaking, a site is only “evental” insofar as it is retroactively qualified as such by the occurrence of an event’ (p. 179).

Indeed it is at this point, Badiou remarks, that ‘I touch upon the bedrock of my entire edifice’. For if the character of events is precisely such as to elude the cognizance of those who in some sense inhabit or live through them – if the question of an event’s ‘belonging to the situation of its site is undecidable from the

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standpoint of the situation itself ’ (p. 181) – then clearly the phi-losopher of Being and Event will have his work cut out trying to explain how such events can none the less require and elicit fidelity. Or again, if ‘the signifier of an event is necessarily super-numerary to its site’, then just as clearly the situation of which that site forms a problematic (though perhaps as yet inconspicu-ous) part will be far from yielding its political significance or its latent revolutionary potential to even the shrewdest, most acutely perceptive observers of the scene. Thus there seems to be a strictly unresolvable – and so, one would think, a thoroughly disabling – paradox about Badiou’s idea of the event as that which transpires altogether beyond the domain of specifiable being and which hence disqualifies any contemporaneous attempt to state the conditions for truly judging some candidate occurrence to pass the test for properly evental as distinct from pseudo-evental status. On the one hand, ‘if the event belongs to the situation – if it is presented therein – it is not, itself, on the edge of the void’ (p. 182). Such would be the case either of a pseudo-event that in truth had little or no valid claim to such momentous import or of a potentially genuine event that had been deprived of the capacity to motivate further such events by its subjection to a mode of ‘normalizing’ uptake or interpretation. On the other hand, if one accepts the contrary hypothesis – that ‘the event does not belong to the situation’ – then it follows that ‘nothing has taken place except the place’, that the void (or abso-lute negation of determinate being) is all that remains after this terminal demise of ontology, and hence that ‘no presentable multiple responds to the call of such a name’ (p. 182).

It should be clear that this is no merely abstract or worked-up artificial dilemma but, on the contrary, a problem that touches the deepest and most persistently intractable of all philosophic problems, those of free-will versus determinism or rational autonomy versus the claim of an all-embracing necessitarian creed. For Badiou, these issues pose themselves chiefly in the form of a genuine (on its own terms inescapable) dilemma between acknowledging that which calls for recognition as hav-ing occurred – and which thus exerts a claim on our respect for truth in historical matters – and acknowledging how far such past events, if they really merit that title, remain always open to revaluation with the advent of new political conjunctures or

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unexpected turns in their own after-history of outcomes, results or repercussions. Thus, in one of his more tortuous formulations, ‘[e]ither the event is in the situation, and it ruptures the site’s being ‘on-the-edge-of-the-void’ by interposing itself . . . ; or, it is not in the situation, and its power of nomination is solely addressed, if it is addressed to anything, to the void itself’ (p. 182). If the first con-ception tends always to normalize the event – to bring it safely back within the ambit of familiar or ideologically received modes of thought – then the second risks treating the event as a singular-ity, or ‘ultra-one’, that would somehow exist so far beyond the space of judgement, historical comparison or conceptual repre-sentation as to lose all substantive political content.

2. Representing history: truth and event

As I have said, Badiou is very much alive to the worrisome possibility that something like this – a fully fledged nominalist conception of linguistic reference allied to a fully fledged post-modernist outlook in matters political and ethical – might be read into his own idea of the event as existing on the very edge of the void and hence as inherently resistant to any kind of clear-cut specification. It is the danger, he remarks, that ‘if you start posing that the “French Revolution” is purely and simply a word, you will have no difficulty in demonstrating, given the infinity of presented and non-presented facts, that nothing of the sort ever took place’ (p. 182). And of course this danger is all the more acute for someone of Badiou’s political persuasion at a time when historical revisionism of a mostly right-wing character has gained considerable ground among French historians, not least with respect to the French Revolution.12 Thus it no doubt strikes him as yet another instance of that widespread and ideologically suspect linguistic-discursive-narrative-textualist-hermeneutic turn that has lately exerted such an untoward effect on thinking in various disciplines. At any rate Badiou insists on the need to make judgements concerning both the truth of historical claims and the relative significance, politically speaking, of the events concerning which those claims are made. Nor is he at all averse to making such judgements, sometimes of a highly heterodox sort. For instance, he maintains that certain episodes commonly thought of as major, even epochal events – such as the collapse of Soviet-type communism in 1989–91 or the 2001 attack on the

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Twin Towers in New York – should instead be viewed as the dra-matic yet in many ways perfectly intelligible (if not predictable) outcome of developments already in train.13

That is to say, there is a sharp and necessary distinction which historians, politicians and ideologues ignore at their peril between the sorts of response that ‘naturally’ or instinctively follow in the wake of certain (on the face of it) world-transformative events and the sorts of response that may later turn out to be justified in light of a deeper, more historically and politically informed understanding of how those events transpired. This is not just a matter of what is, after all, the sometimes perfectly reasonable claim to benefit from the wisdom of hindsight or to learn from past errors of judgement. More specifically, it has to do with Badiou’s anti-Cartesian outlook and his consequent belief that know ledge and the consciousness of possessing that knowledge – let alone the self-conscious awareness of being thus conscious of possessing it – are quite distinct states, conditions or capaci-ties which need not (and very often do not) go together. In this regard he is fully in agreement with analytic philosophers like Timothy Williamson who have likewise sought to resolve some of the problems bequeathed by Cartesian epistemology by denying that knowledge either need or typically could be a matter of lucid, perspicuous, apodictically self-evident mental grasp.14 Rather it is subject to manifold restrictions on the capacity of thought to attain such a state, whether through the influence of causal fac-tors in the acquisition of knowledge that elude conscious grasp, through the inability to grasp all the logical implications of our various firmly held convictions, or through the numerous items of ‘standing’ as distinct from ‘occurrent’ belief that we are simply not aware of unless the issue happens to arise.

Such is the lesson driven home with particular force by the fact that epistemology from Descartes down – including the lengthy and complicated chapter that runs from Kant to debates within analytic philosophy – can be seen to have run aground over and again on the same old problems thrown up by the mistaken Cartesian quest for an order of knowledge present to mind in the form of ‘clear and distinct ideas’. Hence Badiou’s emphatic rejection of that whole approach and his insistence that the only way to put philosophy back on its rational, constructive and (above all) its scientifically and politically progressive feet is to

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address those basic ontological issues that have been demoted from this eminent role first by the epistemological ‘way of ideas’ in its varied empiricist or rationalist showings and then by the equally multiform ‘linguistic turn’.15 What then becomes appar-ent is the foredoomed character of any quest to vindicate the claims of knowledge, reason or humanly attainable truth that goes by way of an appeal either to the lucid realm of self-conscious ideas, concepts or representations or else to the tribunal of lan-guage (and our supposedly perspicuous grasp of language) as a substitute means of assurance. For Badiou, this need to draw sharp limits to the scope of conscious and especially of reflective or self-consciously accessible thought is further reinforced by the lessons of psychoanalysis as relayed via Jacques Lacan’s broadly structuralist account of the central Freudian concepts.16 Hence, as we shall see later on, the importance he attaches to Lacan’s claim that the ego is so far from being master in its own house that in truth it is little more than a plaything or dupe of the ubiq-uitous unconscious.

This demotion (or radical relocation) of epistemic privilege applies just as much to political and ethical judgements as to judgements of truth or validity with respect to statements, theo-ries or hypotheses in the formal and the physical sciences. In both cases – albeit in different ways – one has to recognize not only the errant or misleading nature of many commonsense-intuitive beliefs but also the extent to which knowledge may surpass the deliverances of conscious (no matter how lucid or perspicuous) thought. Thus agents living and making commit-ments under pressure of complex and turbulent events – like scientific thinkers assessing rival theories at times of pre-revolu-tionary crisis in the state of received knowledge – may very well act or reason in accordance with likewise immensely complex processes of thought which they would find it hard, maybe impossible to formulate clearly or bring to the level of conscious reflection. To this extent, as regards the scientific issue, Badiou is very much in accord with that prominent strain in present-day epistemology which takes external or causal factors to play a large role in reference-fixing and hence in deciding the truth-value of our various candidate (truth-apt) statements. Naturally enough these approaches also tend to play down – even to deny – the need for any specification of ‘internal’, subjective or conscious

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states whose presence or absence decides what should count as knowledge in this or that particular case.17 At any rate there is wide agreement nowadays that purebred internalism just doesn’t work – that it gives rise to strictly insoluble problems – and that the kinds of discovery procedure that typify scientific thought at its most resourceful and creative are such as to require a compli-cated process of inference to the best, most adequate explana-tion which cannot be apodictically accessible at the time to those directly involved.18

To be sure, Badiou comes at these issues from a different, pre-dominantly rationalist rather than empiricist philosophic tradi-tion and therefore treats them like his great rationalist precursors, especially Leibniz and Spinoza, as finding their paradigm or ultimate test-case in mathematics and the formal sciences. What is remarkable about his project – and sets it yet further apart from that other tradition – is its way of combining this formal orientation with an emphasis on matters of historical, social, ethical and practical-political concern that would surely strike most analytic philosophers as involving a gross conflation of disparate domains. However this is just his point; that once we shift from epistemology and philosophy of language as our two main foci of attention to ontology as that which should right-fully hold such a privileged position then it will no longer seem mistaken, eccentric or downright perverse to conjoin mathemat-ical with political concerns in this way. For on Badiou’s account, it is only by pressing as far as possible with a rigorous deduction of ontological concepts and categories from the resources made available by modern set theory that we are able to specify with due precision both the limits of ontology as a discipline of thought and the character of that which exceeds those limits, namely the event as an ‘ultra-one’ that by very definition cannot be brought under any such concepts or categories.

Thus his point is not to rule against offering historico-political verdicts of any sort, nor to claim in decisionist fashion that they can only be avowed or maintained in good faith through a ratio-nally under-motivated choice to endorse the subsequently bind-ing character of this or that signal event. Rather it is to make the case that what goes on at such decisive moments – whether in the sciences (formal and physical) or in the ethico-political sphere – is very often subject to a complex, largely tacit but none the less


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rational process of evaluative judgement that may issue either in the form of revised (or radically changed) beliefs and convic-tions or in the form of demonstrative acts. These latter are the kind that Aristotle counted as the proper outcome of ‘practical syllogisms’, that is, modes of reasoning whereby certain statements (minor premises) about some given situation, along with a state-ment of principle (major premise) relevant to that same situation, should most fittingly be taken to conclude not in a further state-ment but in a suitable, appropriate or rationally deducible action. Badiou’s thinking is far closer to Aristotle in this regard than to the whole post-Cartesian tradition of epistemologically oriented philosophy that gave rise, among many other dead-end dilemmas, to Hume’s flat statement of the impossibility of logically deriving an ethical ‘ought’ from a factual or constative ‘is’.

Thus despite their significant points of difference, as noted above, what none the less unites Badiou with Aristotle and sets them both very firmly against that tradition is the priority that both thinkers assign to ontological rather than epistemological questions, and hence their refusal to draw such an ethically and politically disabling conclusion. So Badiou is very far from adopting any version of the sceptical, relativist, ‘strong’-con-structivist or generic postmodernist stance according to which the ‘truth’ of historical events is always a product of interpreta-tion, and hence nothing more than the import they acquire in response to present-day interests and priorities. Indeed he could hardly be more emphatic in his rejection of that whole epistemo-logical-linguistic-narrative turn and the various fashionable ‘end-of-history’ or ‘end-of-ideology’ slogans to which it has given rise. For Badiou, an event – in the highly specific sense that he ascribes to the term – has a claim on the ethico-political allegiance as well as the truth-seeking motivation of those to whom it figures as something more than an interpretative Rorschach blot. On the other hand – or perhaps for that very reason – it is always possi-ble for certain historical events to gain a special salience (or equally to lose it) in consequence of later developments or epi-sodes which cast a new light on its longer term implications.

Clearly it is a very fine line that Badiou attempts to draw between this idea of fidelity or truth to the witness of historical events as a condition of their acquiring, retrieving or retaining such salience and that other, typically postmodernist idea of history as always a

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‘history of the present’ wherein events are viewed as mere figments of our own narrative or fictive contriving.19 After all, as he remarks (p. 180), even in the case of so momentous an occurrence as the French Revolution one can patiently list its social origins, background condi-tions, triggering episodes, stages of development, landmark advances, signal reverses, symptoms of decline and so forth – whatever stands out in that immediate context as most significant or representative – and yet say nothing of decisive import as regards its true historical import. What is required at this stage in the argument is a further and tighter specification of the means by which events might be singled out as just that class (more precisely: that unclassifiable assemblage) of his-torical episodes that can be seen to have belonged to no existing space of social-political possibility and thus to have required an exceptional degree of ‘militant’ fidelity on the part of their early protagonists. That is to say, if Badiou is to make good his claim to repudiate postmodern-sceptical relativist ideas about history while establishing this (for him) vitally important link between truth and truthfulness then he has some more detailed explaining to do. And so to Part V of Being and Event where he undertakes to meet that demand through a series of six Medi-tations under the title ‘The Event: Intervention and Fidelity’.

Discussion points

Badiou writes: ‘If the void is thematized, it must be according to the presentation of its errancy, and not in regard to some singu-larity, necessarily full, which would distinguish it as one within a differentiating count.’ How would you set about explaining or interpreting this cryptic statement on the basis of what you have read so far?

How should we understand Badiou’s anti-Cartesian assertion that thinking can proceed – and does very often proceed at its moments of greatest creativity or most decisive intellectual advance – with-out going by way of the conscious or self-conscious mind?

PART V. THE EVENT: INTERVENTION AND FIDELITY. PASCAL/CHOICE; HÖLDERLIN/DEDUCTION

1. Pascal: mathematics, miracles and ‘infinite thought’

Here Badiou follows his usual practice of developing some cen-tral philosophic themes – event, situation, inclusion, belonging, intervention and the supernumerary – while also devoting some

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detailed analytic commentary to an earlier thinker whose work may be shown to have prefigured certain aspects of his own. In this case the figure concerned is Blaise Pascal, the seventeenth-century mathematician, moralist, theologian, aphorist and spec-ulative thinker whose extraordinary range of interests has created considerable problems for – and divisions among – his commen-tators.1 That he figures so importantly in Being and Event is a striking example of the way that Badiou is able to discover revealing affinities or terms for comparison even in thinkers with whom one would expect him to be sharply at odds on ethical, political, philosophic and (not least) religious grounds.

In this context it is worth recalling the distinction Badiou draws between, on the one hand, latter-day ‘sophists’ such as Wittgenstein and Richard Rorty whose attitude to philosophy is mainly one of indifference, rejection or a wish to have done with such pointless concerns and, on the other, ‘anti-philoso-phers’ who brace themselves against the philosophic challenge and against whom philosophy is likewise compelled to test its own claims as a putative discourse of reason and truth.2 Badiou makes the point by contrasting Descartes and Pascal, the one a thinker for whom (purportedly at least) everything proceeded from the application of rational methods and deci-sion-procedures, the other a believer for whom his own achieve-ments in mathematics, logic and natural science were as nothing compared with the leap of faith – the supposed aban-donment of all such rational criteria – that opened the way to authentic religious belief.3 His intention is not for one moment to endorse Pascal’s doctrinal stance or the claim that reason should know its proper limits and thereby make room for that existential foray into the realm of supra-rational paradox and inward, revealed or spiritual truth. Rather it is to emphasize his point with regard to an age-old, conflictual yet productive relationship – that between reason and faith – which finds one of its most striking expressions in St. Paul’s (albeit for the most part mutually baffling) exchanges with the Greek philos-ophers and which has since then re-surfaced in manifold forms wherever there is a question of reason encountering some real or presumptive limit to its proper scope.4 Thus the ‘anti-phi-losopher’ – unlike the sophist – is perpetually engaged in a pro-cess of essaying that limit, provoking the philosopher who will

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typically resist any such claim, but also – most importantly as Badiou sees it – showing how the powers of reason may them-selves be refined and extended precisely through the challenge from a sharply opposed though at times strangely intimate quarter.

Indeed this conflict may sometimes exist within the work of a single thinker, as for instance with Cantor, who (as we have seen) continued to hold out for a ‘virtual’ conception of the infinite that would render it compatible with Christian belief – insofar as it surpassed the finite character of human intellectual or cal-culative grasp – even while he was laying the foundations for a thoroughly secularized transfinite conception that would render the existence of God an otiose hypothesis.5 What Badiou finds exemplary about Pascal is not so much the doctrinal content of his Christian faith but rather his having staked everything on that hypothesis as one that could be verified only through some future, as yet inconceivable event that would retroactively confer a determinate truth-value on those hitherto strictly undecidable conjectures. And so we come back to the question posed at the end of my last chapter: what is it that marks out the genuine event as a singular and – in its own time and place – unclassifi-able occurrence which none the less exerts (or is capable of exert-ing) a likewise singular demand for allegiance among those who are responsive to it? Badiou cites Lacan’s cryptic remark to the effect that ‘if no religion were true, Christianity, nevertheless, was the religion that came closest to the question of truth’ (p. 212). He goes on to gloss this saying very much in his own terms, and in a way that shows why Pascal figures so importantly here. Thus, ‘in Christianity and in it alone it is said that the essence of truth supposes the evental ultra-one, and that relating to truth is not a matter of contemplation – or immobile knowledge – but of intervention’ (p. 212).

Of course there is a risk of serious misunderstanding at this point, given Badiou’s clear attraction to just those elements in Pascal’s thought that will probably strike a non-believer as most open to question on moral as well as on philosophic grounds. Thus it might seem that he is adopting something like the doc-trine of ‘eschatological verificationism’ advanced by some theo-logians as a counter to the logical-positivist claim that the only meaningful statements were those that were either verifiable/


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falsifiable through methods of empirical (e.g. scientific) testing or else self-evidently true (hence tautologous and empirically vacuous) in virtue of their logical form.6 To this the theologians sometimes respond that the postulates of Christian faith are such as will eventually be verified or falsified although under evidential conditions that at present cannot be clearly envisaged or specified with great accuracy.7 However there is all the dif-ference in the world – so to speak – between, on the one hand, a realist ontology (Badiou’s) that locates the truth-makers for truth-apt but as-yet unverified conjectures or hypotheses in a realm of future discovery that is strictly intra-mundane even if it extends to abstract entities such as numbers, sets and classes and, on the other, a theological position that goes so far beyond anything that counts (on empirically or logically adequate grounds) as proof, knowledge or evidence. That is to say, there is nothing in the least eschatological about Badiou’s conception of truth – be it in mathematics, the natural sciences or politics – as requiring an attitude of future-oriented openness to that which may always turn out to have surpassed our best current means of proof or ascertainment. On the contrary, what distinguishes the genuine (epochal) event from the run of more-or-less significant occurrences or happenings is the fact of its standing in a certain retroactively transformative relationship to previous episodes by which it was obscurely pre-figured, and also – as follows nec-essarily from this – in a proleptic relationship to later events whereby its truth-content will be further revealed or progres-sively unfolded. For that content has everything to do with real developments, whether of a natural-scientific, socio-political, or formal-conceptual kind, and nothing whatsoever to do with hypotheses that by their very nature – pace the above-mentioned theologians – lie beyond the utmost reach of verification.

It is important to be clear about this since it bears on one objection that is sometimes raised to Badiou’s closely related ideas of the event as that which disrupts any given situation and of the subject as existing – indeed as quite literally brought into being – through his or her ‘militant’ fidelity to the event. Again there is a risk, not least on account of his taking St. Paul as an exemplary character in this regard, that Badiou will be interpreted as some kind of crypto-theological or (perhaps more to the point) crypto-Kierkegaardian thinker whose professions of religious

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unbelief are not to be taken at face value.8 Thus he might be understood as endorsing something very like Pascal’s famous wager, that is, his purported proof on probabilistic grounds that we had better place our faith in an omniscient, omnipotent and omni-benevolent deity since even if the chances of his actually existing are close to zero still we are better off believing in him than not since the prospect of eternal salvation is infinitely bet-ter than the prospect of eternal damnation.9 It strikes me that nobody who has read very far into Badiou’s work could suppose him to have any sympathy with this line of argument, at least as regards its moral, religious and (not least) its socio-political implications. After all, it goes clean against two main precepts of Badiou’s work, namely his commitment to a thoroughly secular-ized, materialist ontology – one that most emphatically leaves no room for the Christian God or any other deity – and also his insistence on the absolute necessity of thinking things through with the maximum degree of conceptual and logical rigour.

From this point of view he would doubtless be in sympathy with atheists-on-principle like Mill and Russell who have offered the best, intellectually and morally most decisive answer to Pascal: that there is a plain obligation not to acquiesce in any holy para-dox that requires belief in an executive god whose supposed attributes (as listed above) cannot be reconciled one with another or jointly with the facts of human experience.10 On the other hand Badiou is enough of a ‘continental’ rationalist to judge that there is much to be learned from a thinker like Pascal if one takes due stock of his great mathematical advances and translates his other (theological) statements into ethico-political terms.11 What then emerges is a way of conceiving Christianity as some-thing like a figural or even allegorical substitute for that which could not be expressed more directly owing to the pressures of doctrinal adherence or social-political prudence. Thus, in Pascal’s conception, ‘[a]ll the parameters of the doctrine of the event are disposed within Christianity: amidst, however, the remains of an ontology of presence – with respect to which I have shown, in particular, that it diminishes the concept of infinity’ (BE, p. 212). This latter qualification is crucial since it sets Badiou’s argument firmly apart from any strain of theological thought, however ‘negative’, that would conserve some remnant of divine being beyond all the multiplied denials that God could ever be defined


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or conceptualized in humanly available terms. It also serves to make his point, once again, that the operative concept of infinity that entered the discourse of mathematics with Cantor had noth-ing in common with that earlier ‘romantic’, mystical, religious and (purportedly) supra-rational idea of infinity that still left its mark on Cantor’s more backward-looking pronouncements.12 Where Pascal stands apart from that strain of thought – despite his seeming dedication to it as a matter of overt faith – is in vir-tue of his having pressed further than anyone up to that time in striving to think the paradoxical relationship between the exer-cise of reason conceived as subject to the limits of human under-standing and the requirements of faith conceived as inherently surpassing or transcending those limits.

Thus, on Badiou’s account, the rational content of Pascal’s avowedly anti-rationalist (Christian-fideist) thought is the idea that any decisive advance toward truth will be ‘decisive’ not only in the sense that it marks a clear stage of progress in the growth of human knowledge but also in the sense that it bears witness to a choice and thereafter to a deep-laid intellectual and/or ethi-cal commitment on the part of its early advocates and subse-quent upholders. This is why Badiou finds something altogether exemplary in Pascal’s famous ‘leap of faith’, that is, his convic-tion that ‘the heart had reasons of which reason knew nothing’, or that logic was a poor thing in comparison to the promise of redemption – no matter how groundless in rational or probabi-listic terms – held out by Christian belief. Such was Pascal’s motive in declaring the futility of all endeavours to prove the existence or define the attributes of God through scholastic, logic-based, metaphysical or onto-theological modes of thought. For it was only through faith – and moreover through faith in the occurrence of miracles as that which stretched rational cre-dence to the limit and beyond – that Christian belief could pos-sibly be justified, witnessed or defended against the encroachments of a new mathematico-natural-scientific rationality to which, ironically, Pascal himself was a leading contributor. So if his thought still retains its ‘disconcerting’ or ‘provocative’ power even today then it has to do with the vexing question of ‘why does this open-minded scientist, this entirely modern mind, abso-lutely insist upon justifying Christianity by what would appear to be its weakest point for post-Galilean rationality, that is, the

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doctrine of miracles?’ (p. 215). Moreover, what ‘madness’ – or perverse desire to maximize the odds against his own persuasive success – could have led Pascal to make faith in miracles the basis of his efforts to convert a fictive partner in dialogue, a self-professed ‘libertine’ and materialist disciple of Epicurus, Lucre-tius and Gassendi, to embrace the truths of revealed religion?

Badiou’s answer is quite straightforward: that ‘miracle’ for Pascal, like ‘chance’ for Mallarmé, is the name of whatever tran-scends or eludes definition according to the norms of some exist-ing situation, state of knowledge, conceptual scheme or received ontological framework. In this sense it is ‘the emblem of the pure event as resource of truth’, one whose very character is always ‘to be in excess of proof’ and which therefore serves – to Pascal’s fideist way of thinking – as a failsafe guarantee of God’s ‘not being reducible to this pure object of knowledge with which the deist satisfies himself ’ (p. 216). However, once again, it would be wrong to conclude either that Badiou’s reading of Pascal itself has certain fideist or crypto-theological undertones, or indeed that Pascal’s treatment of these issues – the radical dis-junction he proposes between reason and faith, or mere ‘correct-ness’ and truth – is capable of valid interpretation solely on its self-avowed, Christian-theological terms. For it is Badiou’s main purpose in this section to demonstrate that, on the contrary, Pascal’s professed aim of subjugating reason to the claims of faith is placed in doubt – despite and against his intent – by a precocious grasp of certain mathematical truths that went far beyond what had yet been discovered, proved or even conjec-tured at the time. If his thoughts about the infinite are known to most people mainly through his famous confession of terror when confronting the ‘eternal silence of the infinite spaces’, it is not so much this aspect of existential dread that Badiou has in mind but rather Pascal’s having been the first thinker to explore what might be the consequences (mathematical and scientific as well as theological) of adopting a perspective whereby the finite became a special case or limiting instance of the infinite, rather than the other way around. ‘In a spectacular reversal of the ori-entation of antiquity, he clearly states that it is the finite which results – an imaginary cut-out in which man reassures himself – and that it is the infinite which structures presentation’ (p. 220). And again, ‘Pascal thus simultaneously thinks natural infinity,

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the “unfixable” relativity of the finite, and the multiple-hierar-chy of orders of infinity’ (p. 220).

This he achieves, moreover, through a mode of thought that combines the utmost degree of formal and conceptual precision with a willingness – indeed, a religiously motivated compulsion – to pursue the antinomies of speculative reason (those inherent liabilities of human thought that Kant would later place under strict rules of confinement) to the point where they acquire an altogether new creative-productive power.13 That is to say, the rigour of his thinking about issues of mathematics and logic should not be conceived as opposed to, incompatible with or compromised by his insistence on the prior claim of faith over reason. Rather it depends crucially on that claim – on the idea that truth might always transcend the utmost capacities of pres-ent-best knowledge, proof or ascertainment – in order to conceive how such formal procedures can sometimes produce decisive advances and, more rarely, full-scale revolutions in thought. This in turn has to do with what Badiou defines as the axiological dimension of Pascal’s project, that is, his ‘formal doctrine of inter-vention’ whereby the event – whether Christ’s birth and death, a revolution in moral or political thought, a mathematical or sci-entific breakthrough or even a subjectively life-transformative episode – acquires the kind of paradigmatic force that thereafter compels the choice between fidelity and rejection. Thus ‘beyond Christianity, what is at stake here is the militant apparatus of truth: the assurance that it is in the interpretative intervention that it finds its support, that its origin is found in the event; and the will to draw out its dialectic and to propose to humans that they consecrate the best of themselves to the essential’ (p. 222).

Here again it needs stressing – as against the objection that is likely to be raised by analytic philosophers – that Badiou’s use of a voluntarist language (‘assurance’, ‘will’, ‘consecrate the best of themselves’) is not just an instance of gross confusion between ‘context of discovery’ and ‘context of justification’.14 This would be the error – currently widespread among constructivists’, descriptivists, cultural relativists and ‘strong’ sociologists of knowledge – which in effect makes a full-scale programme of refusing to distinguish whatever belongs to the subjective sphere of motivational psychology and whatever pertains to the norma-tive dimension wherein scientific and other truth-claims are held

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accountable to the strictest standards of rational accountability.15

Instead it is a question of acknowledging how difficult it is to disentangle the distinct yet closely imbricated concepts of truth-fulness and truth, or how the latter becomes void of substantive (genuinely rational) content if decoupled from any reconstruc-tive account of the processes of thought by which those who have pursued various kinds of truth have typically set about bringing them to light and defending them against sceptical attack.16 This is another way of saying that Badiou is one of the few philosophers not merely to face both ways across the analytic/continental divide as need or inclination dictate but rather to situate his thinking on alternative ground beyond that stereo-typical binary conception. Such, to repeat, is the central thesis of Being and Event: that while a mathematically informed (i.e. set-theoretically based) ontology offers the closest approach to truth within some existing, currently most advanced state of know-ledge still there is always an appeal open to speculative claims whose truth-content can be fixed – or whose truth-value can be borne out – only through the work of thinkers or ‘militants’ committed to just that presently unfinished task.

Thus Badiou shares with mainstream analytic philosophers the belief that there is a vital distinction to be drawn between truth and truthfulness, or that which pertains to the correctness of statements, propositions, theorems, conjectures and so on, and that which pertains to the motivating interests, the strength of purpose or depth of commitment exhibited by those for whom it is a matter of overriding concern. On the other hand, he parts company with them in allowing far greater weight – in scientific-philosophical as well as psycho-biographical or socio-cultural-historical terms – to the way that certain decisive interventions (those that have demonstrably changed the course of history) can be shown to have involved a fidelity to previous events which provided both an example or paradigm of how such thinking should proceed and, through their legacy of unresolved problems, a spur to more advanced investigative work. Meditations Twenty-Two to Twenty-Four can best be read as a sequence of reflec-tions on the nature of this complex relationship between truth as that which in some sense precedes and awaits discovery, truth as what transpires through the occurrence of a signal or epochal event, and truth as the product of a human intervention with its


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source in some historically specific project of thought. They begin with his treatment of the Axiom of Choice (see discussion above), taken by Badiou as the single most striking instance of a truth about the nature of mathematical reasoning that requires not so much a decision-procedure (of the kind typically involved, say, in rational-choice theory) but rather a decision in the radical sense of staking one’s entire project on the validity of certain other axioms or theorems without which that project would collapse even though they themselves lack any adequate proof.17

2. Fidelity, choice, ‘State of the Situation’

Hence Badiou’s central hypothesis in this connection, that ‘within ontology, the axiom of choice formalizes the predicates of intervention’ (p. 227). Here we should recall his insistence that the axiom is a strictly formal requirement – a conceptual neces-sity if various well-entrenched truths of mathematics and logic are to stay in place – and is therefore not to be confused with other meanings of the word ‘choice’ that carry more subjective, more ethically charged or existentially laden overtones. All the same those other meanings are not altogether ruled off-bounds, as can be seen from the role that this axiom plays in Badiou’s way of accomplishing the passage from issues in set-theory and the formal sciences to issues of a chiefly political import. ‘In its final form’, he writes,

it posits that given a multiple of multiples there exists a mul-tiple composed of a ‘representative’ of each non-void multiple whose presentation is assured by the first multiple. In other words, one can ‘choose’ an element from each of the multiples which make up a multiple, and one can ‘gather together’ these chosen elements: the multiple obtained in such a manner is consistent, which is to say it exists. (p. 224)

Thus the axiom has to be presupposed – has itself to be ‘chosen’ or decided upon as a matter of basic commitment – if mathema-ticians wish to maintain the integrity, rationality or logical coherence of the set-theoretical project. To this extent it might just as well be said that in truth they have no choice in this matter since refusing to ‘decide’ for acceptance of the axiom and

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thereby renouncing the various other, mathematically indispens-able axioms and theorems that depend on it would amount to a vote of no confidence in their own enterprise. However, for Badiou, this is a false way of posing the issue since it is precisely where formal reasoning achieves its greatest measure of consis-tency – the maximal degree of coherence between what is con-sistently thinkable and what is taken to exist – that thought is confronted with a further choice, that is, between halting at this stage of advance in its ontological enquiries and going on to dis-cover further possibilities along with as-yet unrecognized prob-lems, dilemmas and obstacles. After all, it is a leading precept of Badiou’s thought that inconsistent multiplicity is logically, math-ematically and ontologically prior to consistent multiplicity and that the latter maintains its seeming pre-eminence – its role as self-evident source of rationality and order – only by means of the count-as-one and its various surrogate functions. So what the axiom of choice brings out most sharply is the non-finality of any such order and the fact that thinking always has the option of locating certain emergent stress points in its presently accepted ontological scheme or range of commitments. For these can then be seen to mark the location of evental sites where an existing order first comes up against ‘supernumerary’ elements that resist assimilation to the count-as-one and thereby reveal a disparity between inclusion and belonging, parts and members, or presentation and representation.

Badiou thinks this to be the main reason why the axiom of choice generated such a great deal of controversy among math-ematicians during the early years of the twentieth century. What made it threatening to received views of good mathematical practice at the time was the fact that it found room for those other indispensable axioms but ‘only at the price of endangering the one . . . thereby naming, by default, the non-one of the inter-vention’ (p. 228). We have seen already that he considers it a basic, even definitional truth that ontology cannot have any-thing to do with the event since the event has to do with whatever eludes, exceeds or transcends some existing (ontologically defined) state of affairs. However this cannot be the case with regard to that other transformative occurrence, the intervention, since without it – without the power of thought to intervene and bring about decisive advances in the scope of human ontological


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grasp – there could simply be no possibility of change and we should find ourselves once again confronting those paradoxes of the One and the Many that Badiou explored in his opening sections on Parmenides and Plato. Thus ‘intervention’ is the name for whatever marks the entry of a power of thought which goes beyond the limits of present-best knowledge or currently available proof procedures and which postulates truth-apt theo-rems or conjectures whose truth-conditions have yet to be for-mulated. According to Badiou, ‘[t]here could be no better indication of the discernment in which all the zeal of fidelity is realized: the discernment of the effects of the supernumerary multiple whose belonging to the situation has been decided by an intervention’ (p. 228). So it is only in this way – through that which intervenes to disrupt, unsettle, and thereby transform an existing situation – that ontological enquiry can ever achieve real progress as distinct from merely throwing up minor innova-tions within the limits of existing thought.

It is here that Badiou makes some of his strongest claims con-cerning what I think is best described as the precise structural homology (not just the vaguely analogical grounds for compari-son) between conceptual revolutions in the formal or physical sciences and revolutions in the socio-political sphere. Above all, it is a matter of that which provokes and challenges the forces typically massed against any such truly revolutionary idea, that is, any notion that thought might be up against an obstacle that cannot be resolved by recourse to various local adjustments, small-scale revisions or suchlike accommodating pragmatist tactics but which requires that it face the challenge head-on and thus undergo some much farther-reaching transformative process. So it is that the axiom of choice, though retaining its character as a product of formal necessity rather than some kind of existential dilemma, none the less comes to figure as an issue within the discourse of mathematics which itself demands a choice – a decisive commitment – one way or the other. Thus ‘[t]he conflict between mathematicians at the beginning of the [twentieth] century was clearly – in the wider sense – a political conflict, because its stakes were those of admitting a being of intervention; something that no known procedure or intuition justified’ (p. 228). For it is just Badiou’s reiterated point that such standing guarantees of correctness – whether intuitive or

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rule-governed – are always and of their very nature restricted to that which conforms with existing (whether commonsense or formally prescribed) criteria. To be sure, there were various attempts to defuse this crisis by presenting the axiom as a sensi-ble since crisis-averting response to problems that could then be contained without inflicting any damage beyond the recourse to a precept which in truth involved nothing more than a due allow-ance for the role of human judgement at certain points in the process of mathematical reasoning.18 However he concludes that ‘this ethics cannot dissimulate the abruptness of the interven-tion on intervention that is formalized by the existence of a function of choice’ (p. 229). And again he states that ‘the most profound lesson delivered by the axiom of choice is therefore that it is on the basis of the couple of the undecidable event and the interventional decision that time and historical novelty result’ (p. 229). That is to say, the decisive intervention comes about through fidelity to some previous event which in turn enables those who come after – who inherit the privilege and responsibility of thinking its consequences through – to fulfil what had so far remained within the realm of prefigured yet unactualized possibility.

Again Badiou is anxious to avoid any risk that the term ‘fidel-ity’, in this context, might be understood as referring to ‘a capac-ity, a subjective quality, or a virtue’ (p. 233). To construe it like this would of course be to play straight into the hands of those who police the boundary between ‘context of discovery’ and ‘context of justification’, or what properly belongs to the subor-dinate realm of psycho-biographico-historico-cultural enquiry and what properly belongs to the other (rational-normative) dimension of truth and knowledge. Hence his care to explain – and demonstrate – that the axiom of choice followed by the strictest logical or formal necessity from the need to conserve certain highly specific and deeply entrenched set-theoretical pre-cepts that would otherwise have to be abandoned. All the same his desire to head off any imputation of conceptual slackness in this regard goes along with a countervailing (though by no means contradictory) stress on the extent to which issues of choice and commitment in mathematics, logic and the formal sciences find a more than vaguely suggestive analogue in the realms of ethics and politics. So it is wrong – getting things

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exactly back-to-front – if one concludes that the main effect of Badiou’s determination to establish this structural link between such disparate domains is to give us the worst of both worlds, that is, to undermine any semblance of formal or logical rigour in his work on mathematics while effectively reducing his pro-fessions of ethico-political commitment to a level of merely abstract, formal or procedural (pseudo-)choice.

Indeed, as I have said, this is the issue at the heart of his long-running quarrel with Deleuze over what he sees as the lat-ter’s retreat into an inchoate philosophy of flux, infinitized dif-ference, intensive as opposed to extensive multiplicities and so forth.19 According to Badiou this results from Deleuze’s failure – or unfortunate refusal – to grasp the demanding yet liberating insights afforded by a set-theoretical approach that respects the requirements of conceptual rigour even (or especially) when venturing into speculative regions of thought beyond the scope of present-best knowledge or formal provability. In both con-texts – mathematics and politics – he offers more than enough in the way of formal-demonstrative reasoning or historico-political documentation to rebut any claim that his assertion of the strong con ceptual relation between them is just the upshot of promiscu-ous discipline-jumping or vaguely analogical thinking. Thus it is not at all a case of Badiou’s work in mathematics and philosophy of the formal sciences being compromised or rendered intellectu-ally suspect by association with ‘softer’ disciplines like history, politics or (God forbid!) cultural studies. Rather it is a question of the latter being spurred to more careful, precise and ade-quately theorized efforts of self-critical reflection by exposure to the kinds of reasoning developed in those other disciplinary domains.

So when Badiou invites us (p. 234) to think of rival or conflict-ing versions of ‘fidelity’ in terms of Stalin versus Trotsky on the one hand or of mathematical intuitionists versus axiomatic-deductive practitioners of set theory on the other he is not going out on a wildly speculative limb. What gives that claim philoso-phical substance and credibility is that he then goes on to specify the terms – the set-theoretical terms – in which this comparison can be worked through and applied so as to distinguish varying (dogmatic or critical, orthodox or heterodox, ‘statist’ or dissident, repressive or liberating) modes of fidelity to some given event.

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‘That a multiple, α, is counted by the state essentially signifies that every multiple β which belongs to it is, itself, presented in the situation, and that as such α is a part of the situation: it is included in the latter’ (p. 236). Situations like this – whether in mathematics or in politics – are situations that effectively debar any challenge to the status quo ante by ruling that, quite simply, there is no possible disparity between the orders of belonging and inclusion, since to be included just is to be a member in good standing accordant with the terms and conditions laid down by this or that dominant count-as-one. Hence the very pointed con-trast that Badiou draws with situations wherein there does exist a definite potential – or structural opening – for the emergence of just such a disparity between what is presented and what is represented in or by them at any given time. It is here, at the ‘evental site’ marked out by its extreme marginality or down-right exclusion from the realm of acknowledged (whether math-ematical, scientific, political or artistic) possibility, that one is likeliest to find those irruptions of the radically new or unfore-seen to which Badiou applies the terms ‘event’ or ‘ultra-one’. In which case, he reasons, the surest sign of fidelity to such decisive or transformative episodes is the fact that it truly ‘discerns the connection of presented multiples to a particular multiple, the event, which is circulated within the situation via its illegal name’ (p. 236). The name is ‘illegal’ just insofar as it is de facto deprived of legitimate status by its counting for nothing under the preva-lent membership conditions that would close (or, more accu-rately, render invisible) the gap between belonging and inclusion, or presentation and representation.

In order to formalize this relation of fidelity Badiou assigns to it the operator , mostly used in the context of modal logic – the branch of logic concerned with matters of necessity and possi-bility – to indicate that ‘it is necessary that’ or ‘from which it follows necessarily’, as distinct from the operator ◊, signifying ‘it is possible that’ or ‘from which it possibly follows’.20 For ana-lytically minded modal logicians the main use of this distinction is to clarify issues in metaphysics and ontology, among them issues with regard to causal explanation where it serves to con-struct counterfactual possible-world scenarios and provide a relevant contrast-class by which to identify the operative causal factors.21 The argument is typically framed as a question in

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subjunctive-conditional terms: would some particular real-world event actually have occurred had some specific antecedent event (or conjunction of such events) not occurred and thereby set in train the relevant sequence of concatenated causes and effects? It is by means of such thought-experimental excursions into alternative (i.e. non-actual) possible worlds – way things might conceivably have been without any infraction of physical laws or other such domain-specific constraints – that scientists, histori-ans and thinkers across a wide range of disciplines attempt to specify what must have been the one or more salient causal fac-tors in any given case.

Thus the kinds of reasoning that find their formalized expres-sion in the terms, structure and symbolic apparatus of modal logic are also those that play a prominent role in our everyday and more specialized modes of causal-explanatory thought. Where Badiou’s deployment of the necessity-operator departs from this analytically standard usage is in assigning it to con-texts where the necessity in question has to do not only with the objective truth or validity of some given statement, hypothesis, theory, conjecture, and so on, but also with the question of just how far its adherents persevered in asserting or bearing witness to its truth against whatever odds of entrenched opposition or apparent contra-indication. Fidelity of this sort is apt to mani-fest itself in the form of intransigent resistance to prevailing norms of what counts as a legitimate claim, that is, what pur-ports to minimize (or annul) the ontological gap between belong-ing and inclusion. In Badiou’s words

[a] typology of fidelities would be attached to precisely such proximity. Its rule would be the following: the closer a fidelity comes, via its operator , to the ontological connections – belonging and inclusion, presentation and representation, ∈ and ⊂ – the more statist it is. It is quite certain that positing that a multiple is only connected to an event if it belongs to it is the height of statist redundancy. For in all strictness the event is the sole presented multiple which belongs to the event within the situation: ex = ex. If the connection of fidelity, , is identical to belonging, ∈, what follows is that the unique result of the fidelity is that part of the situation which is the singleton of the event. (p. 237)

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So there is another, altogether less strenuous or ethically and intellectually demanding mode of ‘fidelity’ which consists in just a slavish endorsement of things as they are, whether this takes the form of a Hegelian assertion that ‘the rational is the real’, a Leibnizian assertion that necessarily we live in the ‘best of all possible worlds’, a Wittgensteinian notion of ‘language-games’ or cultural ‘forms of life’ as the bottom line of justification, or a pragmatist claim that truth just is what’s currently and contin-gently ‘good in the way of belief ’. Despite its appearance of tolerant pluralism this would be a ‘dogmatic fidelity’, one that ‘separates nothing’ and ‘admits no negative atoms’ (p. 237). Thus it blocks any move to question its legitimate status by adverting to the gap between belonging and inclusion that marks the existence of anomalies or unresolved problems in the formal and physical sciences, or likewise of inequities, exclusions, class-based or ethnic divisions, and other such sources of injustice or oppression in the ethico-political sphere. However it is always possible that the tension thus created will itself give rise to a resistant or strongly countervailing force that leaves no room for any such thoroughly ‘statist’ conception of the body politic. In that case fidelity amounts to a ‘counter-state’, one that takes up the task of ‘organiz[ing], within the situation, another legitimacy of inclusion’, or of ‘build[ing], according to the infinite becom-ing of the finite and provisional results, a kind of other situation’ (p. 238). This would be the upshot of an altogether different mode of allegiance, a commitment to keep faith not with some currently prevailing idea of what counts (or doesn’t count) in a given situation – whether a state of knowledge or state of socio-political affairs – but rather with the event as that which inter-venes to challenge and transform any such received mode of thought.

In Meditation Twenty-Four (‘Deduction as Operator of Onto-logical Fidelity’) Badiou makes a similar point with regard to the role of subjectivity vis-à-vis truth, but here by remarking on the singular richness, creativity and scope of mathematical thought as compared with the notably restricted range of those logical ‘rules’ that constitute its basic principles or framework of opera-tive concepts. Indeed he sets them out in a half-page of summary statements (p. 242) covering the logical signs for negation (∼), implication (→) and the universal quantifier (∀), along with the


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three most elementary principles, modus ponens (if p → q, and p, then necessarily q), modus tollens (if p → q, and ∼q, then neces-sarily ∼p) and generalization (if A is true with respect to any particular instance of variable α then this follows of necessity from the fact that A is true for any and every instance of α).22 Of course modern, post-Fregean symbolic logic has vastly more notational and conceptual resources than this at its disposal, even if confined – as by lovers of ‘austere desert landscapes’ like Quine – to the apparatus of the first-order predicate calculus.23 However Badiou’s point is to emphasize the remarkable con-trast between this extreme poverty (or so it would appear) at the level of basic terms, structures or modes of canonically valid logical argument and the extraordinary range or creativity of which mathematics is somehow capable while none the less respecting the rules and constraints of that same logical regi-men. Thus ‘it is, after all, in conformity with the ontological essence of this universe that the difficulty of fidelity lies in its exercise and not in its criterion’ (p. 243; Badiou’s italics). That is to say, he rejects the Wittgensteinian (and nowadays more widely held ‘analytic’) idea that truth consists in a manifest con-formity to certain agreed-upon criteria of correct or commu-nally sanctioned reasoning whose terms and conditions are effectively settled in advance since they belong to a practice out-side of which there could be no such normative standards.24 Rather, on Badiou’s account, although ‘tactics . . . are rigid and almost skeletal’, nevertheless ‘[t]he entire art lies in the general organization, in demonstrative strategy’ (p. 243). It is for this reason that ‘great mathematicians often step right over the detail and – visionaries of the event – head straight for the general conceptual apparatus, leaving the calculations to the disciples’ (p. 243).

As I have said, it is among the most impressive features of Being and Event that the work very pointedly offers in itself what it seeks to convey about the character of modern mathematics, that is, the combination of a rare capacity for far-reaching spec-ulative thought with a keen attentiveness to matters of formal or demonstrative reasoning. Indeed it is precisely this conjunction of attributes – often taken as mutually exclusive and as marking the respective proprietary domains of ‘continental’ and ‘analytic’ philosophy – that Badiou, on the contrary, takes to constitute

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the hallmark of all genuinely ground-breaking achievements in mathematics and the formal sciences. Thus, ‘[j]ust as the strict writing of ontology, founded on the sign of belonging alone, is merely the law in which a forgetful fecundity takes flight, so logi-cal formalism and its two operators of faithful connection – modus ponens and generalization – rapidly make way for procedures of identification and inference whose range and consequences are vast’ (p. 244). Among the chief means or most prolific sources of this exponential growth in mathematical knowledge is the method of reasoning by reductio ad absurdum which takes a proposition, a theorem or hypothesis and derives from it a logi-cally entailed consequence whose patent falsehood (or absurdity) is sufficient to constitute a downright refutation.25 Badiou makes a strong and principled case for the absolute indispensability of such reasoning, and it is a case that merits further attention since it goes clean against one prominent trend in recent analytic phi-losophy of mathematics. This is anti-realism of the ‘technical’ sort whose most noted exponent is Michael Dummett and which holds, in brief, that truth cannot possibly exceed the limits of proof, ascertainment or demonstrative warrant since to assert otherwise – to claim (in typically realist fashion) that we know that there exist unknown or even for us unknowable truths – is to fall into manifest self-contradiction or plain nonsense.26 In which case, so anti-realists conclude, we should accept that warranted assertibility (not truth) is what mathematicians, along with enquirers in every other field, should properly and, so to speak, realistically set out to achieve. Since Badiou goes on at this stage to rehearse some of his most important mathematical precepts in the context of debates around precisely that issue I shall here do likewise, albeit necessarily in somewhat simplified and summary form.

3. Logic and truth: against anti-realism

Fundamental to these debates is the question whether truth-values are objective – ‘verification-transcendent’, in the jargon – or whether they are ‘epistemically constrained’ and thus have no existence beyond the scope and limits of our present-best (or maybe our best-attainable) knowledge. If this latter is the case, as anti-realists believe, then it follows that there must be a great range (what Dummett calls the ‘disputed class’) of well-formed


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and seemingly truth-apt yet in fact undecidable statements or propositions whose truth-value we cannot assign since they lie beyond reach of proof or verification, and which must therefore be reckoned as neither-true-nor-false or as belonging to a third (indeterminate) category where the true/false dichotomy has no purchase. That is to say, anti-realism of this kind entails a rejec-tion of the precept of classical logic known as ‘excluded middle’ and often expressed by the Latin phrase tertium non datur, that is, ‘there is no third or midway option between the twin poles of truth and falsehood’. This principle has to be abandoned since there is no longer any place reserved for those objective truth-values that the realist regards as having nothing to do with our present or even our future-best-possible state of knowledge, but rather as making our statements objectively true or false quite apart from any such humanly indexed epistemological issue. However – as Badiou is quick to note – there are other very size-able costs to be paid for this act of renunciation, among them that it undercuts a mode of reasoning much practised by mathemati-cians and logicians, as well as by thinkers across a wide range of disciplines where (on the classical or realist view) the truth of some given statement can be deduced by showing that its contrary is false. This often takes the form of a reductio ad absurdum, a demonstration that the falsehood concerned has one or more pat-ent absurdities among its logical entailments or among the range of further statements to which one is committed (whether know-ingly or not) by asserting its truth. Anti-realism has to make do without this resource since its rejection of excluded middle has the extra – according to Badiou, the logically disastrous – conse-quence that it must also renounce the principle of double-nega-tion-elimination, that which holds a double negative to cancel out and thus constitute a positive.27 For this is the principle on which all reductio-type arguments are based, and therefore one that forms a crucial resource in the realist’s conceptual armoury.

Among mathematicians – as likewise for a philosopher of mathe-matics like Dummett – the label more usually affixed to approaches that renounce bivalent truth/falsehood and reject the rule of double-negation-elimination is ‘intuitionist’, rather than ‘anti-realist’. However they can both be seen to figure in much the same role and to generate much the same range of arguments and counter-arguments. Badiou offers the following succinct account of the main issue at stake.

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If, from the supplementary hypothesis ∼A [not-A], I deduce a proposition which is incoherent with regard to some other proposition that has already been established, then the nega-tion of the negation of A is deducible. To conclude in the deducibility of A, a little extra is necessary – for example, the implication ∼∼A → A – which the intuitionists refuse without fail. For them, reasoning via the absurd does not permit one to conclude beyond the truth of ∼∼A, which is a proposition of the situation quite distinct from A. Here two regimes of fidelity bifurcate: in itself, this is compatible with the abstract theory of fidelity; it is not guaranteed that the event prescribes the criterion of connection. In classical logic, the substitution of the proposition A for the proposition ∼∼A is absolutely legitimate; for an intuitionist it is not. (p. 249)

Despite its appearance of even-handedness, one has only to place this passage in the context of Badiou’s approach to issues of truth and knowledge as conceived through the being/event dichotomy in order to see how strongly he sides with the realist (and, one is tempted to say, not just the anti-anti-realist) position. Thus he takes it that the ‘strict equivalence of A and ∼∼A’ is a principle ‘linked to what is at stake in mathematics, being-qua-being’, although one that is so markedly at odds with so much of our everyday experience as well as more special-ized modes of investigative thought that ‘ontology is simultaneously vulnerable in this point to the empiricist and to the speculative critique’ (p. 248). Its dependence on excluded middle and the various concep-tual resources bound up with the maintenanceof that principle renders ontology a prime target for anti-realist thinkers – whether of a logico-semantic (Dummettian), neo-pragmatist, social-constructivist, Witt-gensteinian or other such persuasion – who can point to what they take as its inordinate (indeed, its strictly nonsensical) demand that truth should exceed the bounds of knowability, that is, the utmost limits of formal proof or epistemic grasp. However this is no more than tosay that any really decisive intervention in the realm of ontological enquiry is sure to come up against strong resistance fromthe commonsense-intuitive or orthodox, conventionally right-thinking quarter.

So it was – by venturing theorems or hypotheses beyond current norms of intelligibility – that pioneering thinkers from Galileo to Cantor and Cohen have succeeded in ‘turning a paradox into a concept’, or using obstacles as springboards for inventive


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and transformative thought. So it was likewise that Cantor ‘had the brilliant idea of treating positively the remarks of Galileo and Pascal . . . in which these authors had concluded in the impossibil-ity of infinite number’ (p. 267). Hence the decisive set-theoretical advance whereby it became possible to think of the one not as the foundation or precondition of all mathematical reasoning but rather as a product of the count-as-one, and moreover, to think of inconsistent (rather than consistent) multiplicity as that which underlies and may always turn out to exceed or disrupt any unify-ing order thereby imposed. Thus

because the set theory doctrine of the multiple does not define the multiple it does not have to run the gauntlet of the intuition of the whole and its parts. . . . We will allow, without blinking an eye, that given that it is a matter of infinite multiples, it is possi-ble for what is included (like square numbers in whole numbers) to be as ‘numerous’ as that in which it is included. . . . There is a subversion herein of the old intuition of quantity, that subsumed by the couple whole/parts: this subversion completes the innova-tion of thought, and the ruin of that intuition. (p. 267)

So Badiou is very firm – as against many adversaries, present and past – in maintaining the absolute and principled indepen-dence of mathematical thought from any conditions having to do with the scope and limits of linguistic-symbolic expression or the verdicts of intuitive judgement. Where the former places him squarely at odds with Wittgenstein the latter flags up both his distance from Kant or any notion of synthetic a priori mathe-matical truths and also his rejection of intuitionist (or anti-realist) approaches to issues in philosophy of mathematics.

This is why Badiou takes a strong line against the intuitionists’ refusal to endorse the logical axioms of bivalence or excluded middle, that is, their claim that for certain mathematical state-ments – those belonging to Dummett’s ‘disputed class’ of formally unproven theorems, hypotheses or conjectures – these classical axioms simply don’t apply.28 This follows directly from their pre-cept that truth cannot possibly be thought to exceed the scope of warranted assertibility, in which case any statement of this type that we venture to assert cannot be rendered true or false by the way things stand in mathematical reality quite aside from the issue

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as to whether we are now – or shall ever be – suitably placed to decide either way. Thus it is merely self-evident, or so they would claim, that our access to that truth is always and ineluctably con-strained by our various perceptual, cognitive or conceptual capaci-ties and cannot be thought of (in objectivist or realist terms) as always potentially transcending or eluding our present-best state of knowledge.29 As we have seen, when combined with their rejec-tion of another classical axiom – that of double-negation-elimination – this leads to the anti-realist verdict that unproven conjectures are neither true nor false to the best of our attainable knowledge and hence neither true nor false sans phrase. Moreover, consistently with this, they have to deny the validity of any argu-ment which purports to derive its demonstrative force from pre-cisely that axiom, that is, to establish the truth of some given hypothesis by establishing the falsehood (or absurdity) of its nega-tion. For Badiou, conversely, such principles of deductive reason-ing are a sine qua non of mathematical thought and cannot be abandoned without, in the process, renouncing any claim to be concerned with discovering truths of mathe matics rather than con-venient working fictions or handy techniques for avoiding trouble.

Badiou is quite clear that the axiom – and along with it the force of reductio-type or apagogic arguments – can readily be made to seem artificial and unconvincing by the standards of everyday commonsense thought. Thus it might seem wide open to the kinds of objection mounted not only by empiricists like Hume but also by a speculative thinker like Hegel who sought to overcome the charges of tautology, vacuity, vicious regress, cir-cularity and so forth, through a different, that is, dialectical logic with substantive rather than abstract or merely formal content.30 However Badiou comes back most emphatically against that whole range of arguments – as likewise against the intuitionist refusal to countenance those classical axioms – by claiming that mathematics simply cannot do without such resources unless at the cost of triviality or (contra Hegel) giving up any claim to engage with matters of real-world ontological concern. Indeed, ‘it is on the basis of the ontological vocation of mathematics that one can infer the legitimacy of the equivalence between affirmation and double negation . . . and, by consequence, the conclusiveness of reasoning via the absurd’ (p. 250). As regards the reference to Hegel here – more specifically, to Hegel’s grand


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dialectical system and its claim to subsume or transcend all such instances of determinate negation – I must refer readers back to Meditation Fifteen and my all-too-brief commentary on it. What chiefly needs stressing in the present context is Badiou’s absolute conviction that fidelity to the truth-procedure set in train by some particular transformative event in the history of thought is what separates the genuine quest for discoveries beyond the present most advanced state of knowledge from the routine pursuit of near-at-hand solutions to familiar types of problem. Thus, contrary to much present-day thinking in phi-losophy of mathematics, Badiou takes his stand fairly and squarely on the capacity of reason to postulate truths that exceed any currently available means of formal-demonstrative proof yet which none the less lie within range of its speculative grasp and its power to devise more refined, elaborate and resourceful truth-procedures that will at length bring such a method to light.

Among the most fruitful modes of such deduction is the pro-cedure of reductio ad absurdum which – as we have seen – requires the principle of double-negation elimination. Insofar as intu-itionist or anti-realist conceptions deprive mathematical reason-ing of this vital resource – deny it the power of deriving truths from the demonstrable falsehood of their contraries – these con-ceptions fail to offer any remotely adequate account of how mathematics has managed to achieve such impressive (and surely undeniable) advances. Not that this constitutes a failsafe method or problem-resolution technique that can always be deployed through some rule-governed usage of reductio-type arguments. On the contrary, Badiou declares that ‘[t]he goal of the exercise is indistinct, and it is quite possible that you will have to search blindly, for a long time, before a contradiction turns up from which the truth of the proposition A can be inferred’ (p 251). It is precisely the unpredictable or non-formalizable character of these discoveries – at least until they take their appointed, quasi-providential place in the record of past triumphs – that makes mathematics such a paradigm case of the always evolving but never resolved dialectic of being and event. The impossibility of closure, Badiou writes, ‘is due less to its [mathematics’] employ-ment of double negation than to its strategic quality, which con-sists, on the one hand, of an assurance and a prudence internal to order, and, on the other hand, of an adventurous peregrination

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through disorder’ (p. 251). Otherwise of course the history of mathematical thought would have exhibited a far more steady, predictable and (in Badiou’s sense) non-eventful sequence of cumulative progress throughout its development to date. In con-trast, what is seen to mark that history when treated from a standpoint (like Badiou’s) conversant with its problems and com-plexities is a constant alternation – more precisely, a joint and inseparable movement of thought – between the strength of com-mitment that typifies a faithful adherence to certain imperative truth-procedures and the openness to new (perhaps highly dis-concerting) challenges along the way.

We now move to Part VI of Being and Event (Meditations Twenty-Six to Thirty) where Badiou continues to elaborate his account of being as pure multiple, along with what he takes to be its far-reach-ing implications for every discipline of thought where the foremost concern is (or ought to be) that of the relationship between being and event. More specifically, it will take us into regions of set-theo-retical enquiry that are the home ground for mathematicians and logicians but terra incognita for most of those who come to Badiou with their interest primed by an involvement in cultural studies, lit-erary theory or some other branch of the human or social sciences.

Discussion points

What connection do you see between Badiou’s development of set-theoretical themes and his idea of ‘fidelity’ as applied to vari-ous contexts of intellectual, political and artistic endeavour?

Why does Badiou come out so forcefully against intuitionist or anti-realist approaches to mathematics and philosophy of mathematics?

PART VI. QUANTITY AND KNOWLEDGE. THE DISCERNIBLE (OR CONSTRUCTIBLE): LEIBNIZ/GÖDEL

1. Reckoning with the infinite: on the limits of naturalism

We have seen that Badiou introduces a number of related set-theoretical concepts, all of which bear directly on the issue of just how advances may be thought to come about in the formal (as well as the physical) sciences. They hold out the prospect of explaining how truth can be conceived as running ahead of some


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current state of knowledge while not losing touch so completely with the powers and capacities of human reason as to conjure up the threat of epistemological scepticism or – as a putative escape-route from it – the fallback appeal to various kinds of anti-realist, intuitionist or constructivist doctrine. Among them are the con-cepts of the void, the generic, the indiscernible, the ultra-one, the evental site and the supernumerary, which between them can be seen as laying the groundwork of Badiou’s ontology and his account of what exceeds any power of ontological reckoning. I should mention, lest readers note the anomaly, that I propose to hold over my discussion of Gödel until Chapter Nine where it fits more readily with other central themes, despite his conspicu-ous appearance in the title above.

So (to recapitulate): the void is that which is included in every situation or every set-theoretical ensemble yet the presence (or determinate absence) of which can be felt to exert a destabilizing pressure only at moments of challenge or threat to the dominant count-as-one. It achieves its maximum impact or power to trans-form the status quo ante – whether in the formal, the natural, the social or the human sciences – at just those times when the reign of ‘consistent multiplicity’ gives way to the irruption of an ‘inconsistent’ (i.e. anomalous and crisis-inducing) multiplicity. The generic (from Cohen) is that which derives from some ano-malous even though, to begin with, indiscernible element in this or that given situation – one that doesn’t figure or qualify for membership according to the count-as-one – yet which turns out to possess just that kind of long-range transformative or paradigm-shifting power. What enables those elements to acquire such power is their existing as supernumerary parts of a situation whose total state – including its various constituent sub-sets – must always be reckoned as greatly in excess of whatever is rec-ognized as belonging to it by current membership criteria.

It is here – with the emergence of anomalies, excrescences or ‘supernumerary’ elements – that the situation comes under strain or confronts an as yet scarcely discernible challenge from that which it contains yet cannot properly accommodate. It is here also, in the wider context of Badiou’s work, that the discourse of mathematics – that is to say, of fundamental ontology as he con-ceives it – provides the meeting point for all those concerns and involvements that have occupied his thinking over the past three

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decades. Thus the entire conceptual ensemble of count-as-one, event, void, evental site, the generic, consistent versus inconsistent multiplicity, situation versus state-of-the-situation and so forth enables him to make the crucial link between issues in mathe-matics and issues of political power, agency and representation. Moreover, this allows for the extension of those same heuristic concepts to a range of other fields – including certain aspects of the arts and the natural sciences – where he shows them to apply with equal pertinence and force. Here again it is a question of doing justice on the one hand to truth-conditions or validity claims that transcend any merely de facto state of knowledge, understanding or best judgement and on the other to those vari-ous procedures whereby the dedicated subject affirms his or her commitment to the truth in question. Badiou is absolutely insis-tent on the point that we are not then faced with any version of the false dilemma so eagerly touted by some occupants of the ‘nothing works’ camp in philosophy of mathematics, or the idea that we can either have objective, recognition-transcendent truth or humanly attainable knowledge but – on pain of manifest self-contradiction – surely not both.1 Anti-realism in this form is plainly untenable, he thinks, since there is just too much evidence that genuine discoveries have been made (like those of Cantor concerning the infinite) which cannot be explained or rendered intelligible unless on the assumption that they have to do with an expanding knowledge of that which may always potentially elude or transcend our best powers of epistemic grasp. Hence what I think will be the growing conviction – for anyone who reads Being and Event with sufficient attentiveness – that in this field especially sceptical doubt of the kind professed by some philosophers can only be the product of a failure to engage with the activity of real mathematical thought as distinct from reflect-ing on the nature of that activity from a disengaged standpoint.

What makes his project altogether unique in contemporary terms is its cleaving to truth as an indispensable standard across these diverse regions of enquiry and yet – consistently with that – its stress on the irruptive or unpredictable nature of those epochal events that impose new demands of fidelity on subjects who have known or experienced their impact. To recapitulate briefly: this came of their seeing (like Galileo) no sense to the claim that one infinite quantity was greater or larger than another, while none


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the less holding that infinites were (in some albeit anomalous sense) to be counted as quantities and that quantities must, on any workable definition, allow for comparison in (what else?) quantitative terms. Faced with such a dilemma it was only ‘sensi-ble’, Badiou concedes, for Galileo to adopt the logical-conceptual line of least resistance and declare ‘that the notions of “more” and “less” were not pertinent to infinity, or that infinite totalities were not quantities’ (p. 266). All that remained – though it took three centuries for mathematics to arrive at this point – was for Cantor to take the (as it now seems) inevitable step of reversing this dic-tum, ‘turning a paradox into a concept’, and maintaining, contrary to all the evidence of intuitive judgement, that ‘it is possible for what is included (like square numbers in whole numbers) to be “as numerous” as that in which it is included’ (p. 267). Indeed it was Galileo’s decision not to take this further step – his choice of remaining true to certain very strong and (at that time) scientifi-cally productive intuitions rather than launching into strange and knowledge-threatening seas of thought – that can now be seen to have provoked Cantor’s decisive break with that entire antecedent tradition. All it took for the break to be accomplished was Cantor’s (again as it now seems) predictable realization that there existed a one-for-one correspondence between every whole number n and its square n2, and so on for all the relations or ratios of inclusion which turned out to require that thinking go drastically against what appeared self-evident by commonsense-intuitive standards.

Thus the ground has been prepared for Badiou’s conception of mathematics as the key to ontological enquiry and of ontology as pointing the way – through a grasp of its operative scope and limits – to a sense of how truth can coherently be thought of (contra the intuitionists and anti-realists) as always potentially exceeding the compass of present-best knowledge. ‘This concept of term for term “correspondence” between a multiple, be it infi-nite, and another multiple provides the key to a procedure of comparison: two multiples will be said to be as numerous (or, a Cantorian convention, of the same power as each other) if there exists such a correspondence’ (p. 268). Moreover those italics for the word ‘exists’ are clearly intended to carry the force of an assertion to the realist effect that what is at stake here, in this talk of ‘correspondence’, is precisely the question that anti-realists consider ill-formed or beyond hope of any adequate answer. It is

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the question as to how we should best conceive the relationship between on the one hand mathematical entities, structures, theo-rems, statements or hypotheses and on the other hand that to which they are taken – by the realist at any rate – as properly referring or applying. Hence Badiou’s pointed reminder that ‘the concept of quantity is thus referred to that of existence, as is appropriate given the ontological vocation of set theory’ (p. 268). In which case the question arises as to just which modes, dimen-sions or domains of ontological enquiry might be involved, or in just what sense Badiou is here invoking a term like ‘existence’ which notoriously lends itself to all manner of logico-semantic confusion or cross-purpose argument. Thus it brings us up against the issue that has divided philosophers from the pre-Socratics down, namely that of where the emphasis should fall between the claims of objective, timeless, absolute or verification-transcendent truth and the claims of human knowledge as a time-bound, uneven and culture-specific even though truth-oriented process of investigation. In other words it is the issue that Badiou takes up from a great range of philosophical perspectives in Being and Event, and which indeed gives that book both its title and its single most prominent theme.

At Meditation Twenty-Six (‘The Concept of Quantity and the Impasse of Ontology’) he commences a series of tightly reasoned and mathematically based reflections on the impact of Cantor’s discovery that there exist different ‘sizes’ (cardinalities) of the infinite that are capable of being ordered, compared and vari-ously reckoned with even though – or just because – they stand on a scale that intrinsically transcends the kinds of reckoning involved in calculations of a finite or ordinal-based character. An ordinal, recall, is defined as a set whose elements are all transitive and which therefore exhibits a consistency, that is, a property of well-orderedness or perfect correspondence between belonging and inclusion that is taken to require – or to permit – no intervention of disruptive, anomalous, uncounted or likewise disordering elements. That is to say – and here we reach the heart of Badiou’s concerns in this section – it can also be defined as the schema that embraces all natural multiples, where ‘natural’ denominates that which is taken to exist (or subsist) in a realm of plenary being quite apart from the vicissitudes of history, time and change. If there is a sense in which ‘nature measures


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being’ (p. 271), then it is not – most emphatically – because every last dimension of being (or existence) is or would be exhausted by a thorough compte rendu of that which pertains to the realm of natural objects, processes or causes and effects. Such, as we have seen, is the strongly determinist aspect of Spinoza’s think-ing that Badiou is keen to disavow, the aspect that makes such a sharp contrast with the Spinozist stress – which he is just as keen to endorse – on promoting the ‘positive’, affirmative emotions against their ‘negative’, reactive counterparts.

Hence his pointed reminder at this stage that ‘not every pre-sentation is natural’ and that ‘historical multiples exist’, even if ‘every multiple can be referred to natural presentation, in partic-ular with respect to its number or quantity’ (pp. 269–70). How-ever, as he quickly goes on to remark, this is no more than a straightforward consequence of the theorem – ‘one of ontology’s crucial statements’ – that ‘every multiple has the same power as at least one ordinal’. From which it follows both that ‘the “class” formed out of those multiples which have the same quantity will always contain at least one ordinal’, and that ‘[t]here is no “size” which is such that one cannot find an example of it amongst the natural multiples’ (p. 270). However, this should not be taken to entail a concession to what analytic philosophers often call ‘bald naturalism’, that is, the kind of thinking which takes it as self-evident – even if at present and perhaps for all time beyond human demonstrative reach – that the question ‘What exists?’ can always be re-phrased ‘What exists as a matter of natural or phys-ical fact?’ without the least change to its sense, purport or range of pertinent answers.2 For despite his firm and explicit commit-ment to a materialist ontology for the physical and social sci-ences as against all varieties of idealist, quasi-spiritualist or – to steal a nice term from Richard Harland – ‘superstructuralist’ thought Badiou is at one with those analytic philosophers who diagnose a chronic normative deficit or lack of adequate ratio-nal resources in any such hard-line naturalist creed.3 More than that (and here Badiou remains true to Sartrean as well as Cantorian precedent): it forecloses every possible mode of tran-scendence whereby the subject – the knowing, thinking, willing, desiring or acting subject – is always so placed as potentially to break with its previous (or present) state of existence. Thus natu-ralism of that sort cannot find room for the exercise of cognitive,

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intellectual, ratiocinative, ethical, creative or socio-political pow-ers which he takes to characterize the history of every significant human enterprise and hence to form an essential part of any project, like his own, seeking to render that history intelligible.

This is partly a matter of events which arrive, so to speak, from outside the existing state of knowledge or prevailing state of affairs – ‘singularities’ that typically first show up at some ‘evental site’ whose marginal location excludes it from the domi-nant count-as-one – and thereafter exert an inescapable claim on the fidelity of those who have registered the force of their rup-ture with previous modes of thought. Yet it is also a matter of that same internal dialectic or process of advancement through self-interrogation and immanent critique which Badiou finds perpetually at work in mathematics as likewise in any project of enquiry or activist commitment that requires such a faithful yet critical following-through of its inaugural challenge. This other mode of transcendence finds its model, as we have seen, in the process whereby mathematical thought came upon those special kinds of ordinal that marked the critical point of transition to a realm beyond anything conceivable in finite, that is, perfectly consistent or commensurable terms. Such an ordinal, Badiou reminds us, will render it impossible ‘for there to exist a one-to-one correspondence between it and an ordinal smaller than it’ since it will ‘mark the frontier at which a new order of intrinsic size commences’ (p. 270). All the same this is no vague gesture towards some utterly untravelled or unknowable domain since, as Cantor was the first to demonstrate, that new order is capable of formal treatment and hence of reckonings with ‘intrinsic size’ which are just as precise – albeit in accordance with a different, more elusive standard of precision – as those concerning the normal or commonplace range of ordinals. After all, ‘[t]hese ordinals can be perfectly defined: they possess the property of tolerating no one-to-one correspondence with any of the ordi-nals which precede them’, which is why ‘[a]s frontiers of power, they will be termed cardinals’ (p. 270). This property (that of being a cardinal) can itself be formalized as follows: ‘Card (α) ↔ ‘α is an ordinal and there is no one-to-one correspondence between α and an ordinal β such that β ∈ α’. What it amounts to in ontological terms is the statement of a major set-theoretic precept concerning infinite quantities, that is, that this provides


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a scale of measurement – a class of limit-point markers – for comparison between their relative ‘sizes’ or cardinalities. Cantor’s great advance was to establish the existence of just such differ-ently sized infinite quantities and, moreover, the fact of their existing at every point on a scale that runs continuously all the way ‘from the empty multiple (which numbers nothing) to an unlimited series of infinite cardinals, which number quantita-tively distinct infinite multiples’ (p. 273).

Hence the theorem that bears his name: that which holds the cardinality of any given set to be exceeded by the cardinality of its various subsets or component parts. Hence also Cantor’s development of the formal procedure known as ‘diagonaliza-tion’, that is, the technique of logically proving this to be the case through a demonstration that there is not and could not be a one-to-one correspondence between α and p(α) – the set and its parts or subsets – since the latter will include at least one element β that does not belong to f(β) where f is defined as that (supposed but in fact impossible) condition of one-to-one correspondence. For what results from applying the ‘diagonal’ technique – as opposed to a straightforwardly ‘vertical’ pairing-off between set and subsets – is proof that there also exists an element δ that is neither internal nor external to f. But if such an element can be shown to exist then of course this entails the non-existence of f, or the conclusion that it is strictly impossible to establish that putative one-to-one correlation between a set and its subsets or parts. In which case, clearly, the argument goes through: that ele-ment β may likewise not be included in the set which supposedly includes all those elements, multiples, subsets and so on, that fall under the superordinate count-as-one. Badiou presents the for-mal proof (top of p. 275) as a pair of reductio-type arguments which demonstrate first the contradictory outcome of asserting that δ is f-internal and second the contradictory outcome of asserting that δ is f-external.

From which it follows, to repeat, that β might always turn out to be just that anomalous or uncounted element that is either undecidably placed in this regard or capable of being shown to exist outside and beyond the utmost powers of reckoning or ‘legal’ inclusion exerted by the prevalent count. In this way the method of ‘diagonal’ reasoning invented by Cantor is able to reveal ‘a “one-more” (or a remainder) of a procedure which is

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supposedly exhaustive, thus ruining the latter’s pretension’ (p. 274). Moreover, this mode of proof by drawing out the logical contra-dictions produced if one adopts the opposite premise is itself ‘typical of everything in ontology which is related to the prob-lem of excess, of “not-being-according-to-such-an-instance-of-the-one”’ (p. 274). In other words it is a form of reasoning that banks heavily on the power of thought to achieve genuine dis-coveries or arrive at knowledge-expanding as well as logically valid conclusions on the basis of certain formal arguments that none the less have decisive implications beyond the purely formal domain. So one can see why he took such pains in Meditation Twenty-Four (‘Deduction as Operator of Ontological Fidelity’) to repudiate the constructivist, intuitionist or anti-realist idea that the domain of mathematical truth is co-extensive with that of mathematical knowledge or provability. For, as we have seen, this often goes along with the claim that proofs by reductio cannot work since they require the validity of a rule in classical logic – that of double-negation elimination – which in turn rests on the principle of bivalence, that all well-formed or truth-apt state-ments are either true or false.

In that case Badiou’s above-summarized argument with regard to the excess of parts or subsets over any given set would have to be counted invalid since it purports to do just what those thinkers rule out as an abusive overextension of our cognitive, rational or epistemic powers. Thus it would claim to produce substantive results by a via negativa that entails the existence of objective truth-values for certain statements – those belonging to Dummett’s ‘disputed class’ – which are taken to lack such a value (or not to be candidates for truth or falsehood) since they cannot be proved or ascertained by any method acceptable on constructivist or intuitionist terms. Indeed Badiou pauses at this stage to reflect on just how substantive or potentially far-reach-ing those results may be. What they show is not only the excess of parts over that which supposedly contains, comprehends or exhaustively counts those parts but also the extent to which any given ‘situation’ is numerically exceeded by the ‘state of the situ-ation’, that is, by the greater multiplicity of multiples that turns out to exist within it when subject to a second-order analysis in set-theoretical terms. All the more so with infinite quantities that offer the prospect of a likewise infinite increase in the size of that


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disparity which opens up between the situation and the state of the situation. Thus ‘however exact the quantitative knowledge of a situation may be, one cannot, other than by an arbitrary decision, estimate by “how much” its state exceeds it’ (p. 278). Rather one has to recognize the fact that in such circumstances action is not so much a hostage to fortune as a wager – at any rate in no sense a pure calculation – and that ‘of this wager it is known . . . that all it can do is oscillate between overestimation and underestimation’ (p. 278).

Thus Badiou concludes that it is only by way of a decisive and unforeseeable intervention that the excess is temporarily held within check and the discrepancy annulled or at least prevented from coming too plainly into view. Yet it is also the case that this suppressed anomaly – this structural imbalance or lack of com-mon measure between ‘situation’ and ‘state of the situation’ – will exist as a potential destabilizing force wherever the interests of maintaining the status quo ante come into conflict with those of a dissident, heterodox or critical mindset that refuses to endorse such received views through its sharper perception of what they habitually ignore or conceal. Not that Badiou is locat-ing the anomaly in the mindset, or making the conflict out to be a product of ideological tensions that can be thought to exist only insofar as they register with suitably attuned – that is to say, conflictually motivated – subjects. Nor does it bring him out in agreement with those constructivists, intuitionists or anti-realists who see no possible room for truth in the discourse of mathe-matics except on condition that it is so re-defined as to equate with provability, epistemic warrant or the scope and limits of humanly attainable knowledge.4 On the contrary, Badiou leaves the reader in no doubt that when he makes this claim concerning the excess of the state of the situation over the situation – and when he says that this ‘problem of excess’ marks every branch of ontology whether in the formal, natural, social or the human sciences – then he intends it as a matter of objective truth that pertains to the way things veridically stand, rather than the way they are thought or believed to stand according to our present-best or future-best-possible state of knowledge. However he is also very clear that such anomalies can be brought to light only through an act of discovery that requires the utmost dedication and strength of purpose on the part of its inaugural thinker.

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Moreover, it demands a comparable measure of commitment among those later ‘militants of truth’ whose fidelity becomes the sole guarantee that its further implications will continue to be worked through, even – or especially – where these involve an encounter with obstacles that tax its resources to the limit and beyond.

To Badiou, this represents an object-lesson in the soundness of that basically Marxist conception that maintains the dialec-tical inter-involvement of subject and object in the process of knowledge-acquisition, and hence the inadequacy of any approach – like that of mathematical anti-realism – which treats them as two quite distinct ontological domains.5 For on the latter conception there appears such a sharp and immoveable wedge between objective truth and attainable knowledge that the issue becomes a downright dilemma and relinquishing truth comes to seem by far the lesser of two privations.6 Badiou puts his case concerning the theorem of the point of excess – ‘Easton’s Theorem’, named for the mathematician W. B. Easton – in terms that emphasize its strongly realist (truth-based) character as concerns both those abstract entities that figure in the discourse of mathematics, logic and the formal sciences and also those other subject-areas where he takes the theorem to possess a com-parable power and scope of application. This is why there will always be a deficit or shortfall of knowledge concerning the extent and the precise location of those points where the count-as-one – the ‘situation’ insofar as it is recognized in accordance with the dominant mode of reckoning – comes up against that which surpasses its powers of computation or pre-assured con-ceptual grasp. Thus it may well be possible to know for sure (per-haps by reading Badiou on set theory and its ontological import) that in any situation there is bound to be some such point of excess, and also an evental site where its effects are likeliest to show up, and yet quite impossible to predict when or where that event will occur.

To this extent ‘[a]ction receives a warning from ontology: that it endeavours in vain when it attempts precisely to calculate the state of the situation in which its resources are disposed’ (p. 278). This is not to say (far from it) that to act or decide on some particular course of action must always be a matter of passing beyond any maxims, principles, reasons or kindred justificatory


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grounds and staking one’s claim to right conduct on a leap of faith – or an ethical commitment – that wholly transcends any such rule-based and to that extent strictly non-ethical mode of reasoning. It is a tricky point for any commentator on Badiou since he does, after all, attach great importance to the way that major advances in mathematics and other disciplines must always involve a decisive break with received modes of thought. Such a break can come about only through belief in some as-yet unproven conjecture which thereafter – until proved or admitted to the range of well-founded theorems – depends for its continued upholding on the ‘militant’ or fideist commitment of those for whom its truth and its pursuit into hitherto unexplored regions of thought are likewise matters of intense personal dedication. However, as should be clear by now, there is one crucial differ-ence between Badiou’s conception of fidelity as an intellectual as well as ethical virtue and the kind of decisionist approach that would place any genuinely ethical choice or commitment beyond the furthest bounds of rational thought or argued justification. The difference lies in his showing that the limit of ontology – the point where it gives way to recognition of the event as that which potentially subverts all received or established modes of onto-logical reasoning – is one that can be reached only by way of just such a rigorous, that is, axiomatic-deductive procedure.

At this limit, Badiou suggests, there is no contradiction or flat conflict but rather a highly productive tension between what per-tains to the formal, logical or scientific ‘context of justification’ and what pertains to the ‘context of discovery’, that is, the sphere of subjective commitment or will to push through with the relevant proof procedures and the testing of further derivative hypotheses. What he seeks to maintain – against any quasi-Hegelian overcoming of the object/subject divide, as likewise against that Deleuzean strain of radical difference-thinking that purports to dissolve such ‘molar’ oppositions into ‘molecular’ or ‘deterritorialized’ energy-flows – is the absolute necessity that thinking preserve the sharpest, most decisive and rigorous aware-ness of how that dualism structures the relationship between truth and the various processes of human knowledge-acquisition.7 For we shall otherwise have no conceptual or critical purchase on those specific instances where the excess of parts over wholes, sub-sets over sets, inclusion over belonging, the state of the situation

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over the situation and so forth, is such as to require the most meticulous degree of analytical precision. Hence – to repeat – Badiou’s espousal of set theory with its well-defined terms, structures, logical relations and range of operative concepts as opposed to Deleuze’s express preference for those regions of mathematical thought – like the differential calculus – that may be held to resist such (as Deleuze would claim) strictly regi-mented and hence inherently coercive, restrictive or ‘re-territori-alizing’ modes of thought.8

2. Orientations: mathematics, ontology and multiple-excess

Meditations Twenty-Seven and Twenty-Eight – ‘Ontological Destiny of Orientation in Thought’ and ‘Constructivist Thought and the Knowledge of Being’ – go on to explore the implications of this for mathematics and all those other regions of enquiry to which mathematics stands as a privileged source of insights or conceptual advancement. What they chiefly seek to establish is the claim that ‘it is for ever that this provocation to the concept, this un-relation between presentation and representation, will be open in being’ (p. 281). The argument is pursued at various levels and in various contexts of investigation, from mathematics and logic to issues in the ethical, socio-political and artistic spheres. These latter are tied in closely with his set-theoretical enquiries in a passage that links the Greek inauguration of certain still very active concerns in the formal sciences to the likewise Greek inception of themes – those at the heart of tragic drama in its classical mode – that still find a resonance and even a direct structural analogue in present-day politics and social ethics. The passage in question is worth citing at length since it brings together so many of Badiou’s leading preoccupations in a con-cise and perspicuous way.

Dissatisfaction, the historical law of thought whose cause resides in a point at which being is no longer exactly sayable, arises in each of three great endeavors to remedy this excess which the Greek tragedians quite rightly made into the major determinant of what happens to the human creature. Aeschylus . . . proposed its subjective channeling via the imme-diately political discourse to a new symbolic order of justice. For it is definitely, in the desire that is thought, a question of


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the innumerable injustices of the State: moreover, that one must respond to the challenge of being by politics is another Greek inspiration which still reigns over us. The joint inven-tion of mathematics and the ‘deliberative form’ of the State leads, amidst this astonishing people, to the observation that the saying of Being would hardly make any sense if one did not immediately draw from the affairs of the City and histori-cal events whatever is necessary to provide also for the needs of ‘that-which-is-not-being’. (p. 282)

I should emphasize that Badiou is not merely drawing certain suggestive analogies between those fourfold ‘conditions’ (science, politics, art and love) which he takes jointly to constitute the realm of practical engagement whereby philosophy is enabled to pursue its primary vocation, that is, exploring the scope and limits of ontology vis-à-vis those conditions. Rather this passage, through its reference to the ‘desire that is thought’, gives the first clear indication in Being and Event of how the fourth element might stand in relation to the other three which have so far received much more detailed attention. Thus its pointed gram-matical ambiguity (to paraphrase: ‘that specific kind of desire that takes the form of rational-discursive thought’ or ‘that kind of desire that has been subject to thought and thus achieved greater self-knowledge’) may be seen as reinforcing his case with regard to the close relationship between love – whether eros or agape – and philosophical reflection from Plato’s Symposium to Freudian and Lacanian psychoanalysis.

The ‘three great endeavors’ referred to in the above-quoted passage all have their source in that inaugural perception of a mismatch, discrepancy, lack of proportion or strictly unpredict-able (‘errant’) excess that haunted the discourses of mathematics, ontology, politics, ethics and – in its distinctively ancient-Greek form – tragic drama. Thereafter, according to Badiou, thinkers of the first persuasion have sought (like Leibniz with his law of the identity of indiscernibles) to contain or prevent the emer-gence of any such excess by decreeing that all differences are marked, that what cannot be distinguished in and through such a mark cannot ipso facto be different, and hence that language – or the discourse that articulates all known terms or available means of distinction – must, quite literally, have the last word in

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matters of ontology. In mathematics this doctrine, if carried through consistently, would amount to a requirement that think-ing remain within the limits set by received modes of reasoning or established proof-procedures, and hence – in effect – to a veto on the sorts of strictly unpredictable (since ground-breaking) advance singled out by Badiou. In politics likewise it would ulti-mately signal a failure or refusal to conceive of possibilities beyond those that found expression in some pre-existent dis-course of values and beliefs, or again – as Wittgensteinians and suchlike latter-day ‘sophists’ would have it – some communally sanctioned language-game or cultural ‘form of life’. For these thinkers, ‘[l]anguage assumes the role of a law of being insofar as it will hold as identical whatever it cannot distinguish’ (p. 283). This in turn serves to ward off recognition of whatever might threaten to subvert the existing status quo, whether as concerns the nature and scope of sound mathematical reasoning, the currently prevailing paradigm of scientific method, or accepted ideas of just what constitutes a proper (legitimate or socially warranted) mode of political activity. ‘Thereby reduced to count-ing only those parts that are commonly nameable, the state, one hopes, will become adequate to the situation again’ (p. 283).

The second ‘great endeavor’ would appear to be that which has found voice in those thinkers of the flux – from Heraclitus to Deleuze – who have argued or polemicized against what they consider the tyranny of concepts and the life-denying force exerted by abstract, logical or suchlike regimented thought. As might be expected Badiou sees small hope of progress in this direction, whether as regards mathematics and the natural sci-ences (where it offers only the notional prospect of a wholesale ‘liberation’ from all those standards of rational and purposive thought that would make such a freedom worth having) or as regards politics, ethics and the arts (where it would likewise amount to a merely gestural since wholly indiscriminate break with everything that constitutes a challenge to human creative, responsive and adaptive powers). To this way of thinking ‘the excess of the state is only unthinkable because the discernment of parts is required’ (p. 283). When faced with the coercive and impla-cable demand that we should constantly sift, sort, classify and draw ever more sharply categorical yet also minutely detailed dis-tinctions the only adequate response – so these thinkers maintain


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– is one that refuses to go along with it and in stead elects to cele-brate the endlessly proliferating play of differences, intensities and ‘de-territorializing’ flows. ‘What is proposed this time, via a doc-trine of indiscernibles, is a demonstration that it is the latter which make up the essential of the field in which the state oper-ates, and that any authentic thought must first forge for itself the means to apprehend the indeterminate, the undifferentiated and the multiply-similar’ (p. 283). Such thinking may very well task itself to analyse, describe and even to ‘interrogate’ the forms and structures of representation but only ‘on the side of what it num-bers without ever discerning: parts without borders, random conglomerates’.

Here we see, briefly stated, the main gist of Badiou’s case against Deleuze and his reason for insisting – contra the vogue of difference-thinking in its multiform modes and guises – that the precondition for any genuinely radical or transformative project of thought is that it exercise a maximum power of discernment with respect to those details, aspects or features of some given situation that would otherwise (and normally) escape notice or fail to register according to the prevalent count-as-one. He is thus squarely at odds with any approach whose chief effort is to ‘dispose of a matheme of the indiscernible, which brings forth in thought the manifold parts that cannot be named as separate from the crowd of those which – in the myopic eyes of language – are absolutely identical to them’ (p. 283). The result of such thinking is to multiply differences to a merely notional point of infinity where they blur into something like Hegel’s famous night in which all cows are black, a realm of absolute, endless, undif-ferentiated difference which offers no kind of conceptual pur-chase and hence no possibility of change through a process of rational evaluation and critique. Rather it serves only to obscure the sources of error and falsehood with respect to mathematical and scientific knowledge, or the sources of oppression, injustice, inequality and exclusion with respect to the social and political spheres. ‘Within this orientation’, Badiou remarks, ‘the mystery of excess will not be reduced but rejoined’ (p. 283). Where the crucial difference lies between this and his own, set-theoretically based approach is in the latter’s holding out for the values – ethico-socio-political as well as logico-mathematico-scientific – of analytical precision and conceptual clarity, as against the

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strong yet spurious appeal of a rhetoric of difference which finally collapses into just the sort of thinking that it claims and strives to avoid.

The third ‘great endeavor’ is basically that which we have seen most clearly exemplified in Badiou’s treatment of the stage at which mathematical thought makes the passage from limit-ordinals to large cardinals, and thence to an operative rather than merely notional conception of infinity. What this princi-pally involves is the attempt ‘to fix a stopping point to errancy by the thought of a multiple whose extension is such that it orga-nizes everything which precedes it, and therefore sets the repre-sentative multiple in its place, the state bound to a situation’ (p. 283). That is, the procedure here takes the form of positing so ‘gigantic’ an infinity that thought would be enabled to exceed or transcend and thereby (supposedly) contain any such ‘errant’ quantity as might otherwise elude its best efforts of conceptual grasp. This it would seek to achieve not, like the first endeavour, by ‘reinforcing rules and prohibiting the indiscernible’, but rather ‘directly from above’ by asserting the existence of ‘transcendent multiplicities’ that can then serve as limit-point notions whereby to ‘unveil the very law of multiple-excess’ and hence offer the prospect (paradoxically enough) of a ‘vertiginous closure to thought’ (p. 284). While fully alert to those manifold kinds and degrees of ‘unmeasure’ that mark all received (scientifically or socio-politically dominant) modes of being this conception stakes its claim on the standing possibility that thought might achieve such closure through an idea of that which by definition transcends any limit assignable to it by existing powers of ratio-nal, procedural or calculative reckoning. That is to say, ‘the grand cardinals approximate the virtual being required by theologies’, or play their role in attempts to place a limit on the otherwise truly ‘vertiginous’ spawning of infinite quantities by reasoning – like Cantor – in terms that overtly or covertly invoke some sub-stitute for the God whose being is definable only by negation.

I should make it clear that Badiou’s account of these three endeavours (or ‘orientations’) is not designed to represent them as a sequence of increasingly adequate, complex, analytically precise or conceptually powerful resources in the quest for clarity concerning ontological issues. Rather, he is at pains to insist that each corresponds to a certain aspect or dimension of onto logy


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since each ‘implies that a certain type of being is intelligible’ (p. 284). Moreover none of them is, or could be, rendered invalid through the confrontation with some past, present or future development in the sphere of set theory since mathematics plays the role of provider with regard to their various conceptual requirements and not the role of judge or arbiter concerning their respective claims to truth. Thus ‘[m]athematical ontology does not constitute, by itself, any orientation in thought, but it must be compatible with all of them: it must propose and dis-cern the multiple-being which they have need of’ (p. 284). So far is he, indeed, from the suggestion of any such ranking order in point of ontological grasp that he considers the second orien-tation – despite what I have here described as its distinctly Deleuzean character – as also including among its most notable advances Cohen’s concept of the generic, a development abso-lutely central to every aspect of Badiou’s own endeavour. Still these three modes of thought altogether represent what he con-siders a threshold point in the process of conceptual and onto-logical advancement whereby set theory – having itself passed through and beyond them – can now bid fair to locate their most salient stages of transformation to date. And if this sounds rather like Hegel writing from the standpoint of Absolute Knowledge then the imputation is valid insofar as Badiou is undoubtedly entering some large claims for the truth-content of the viewpoint thus attained but false insofar as he situates that content in a realm where certain strictly formal operations are unpredictably subject to the kind of disruptive, potentially trans-formative event that resists any form of dialectical subsumption or phenomenological uptake. What Badiou is here determined to show is the element of strictly uncontainable excess that afflicts any effort to conserve thought within a smoothly functioning economy where difference can either be annulled by absorption back into the realm of identity-thinking or extended to the point of a purely notional ‘infinity’ where in effect this becomes just a substitute term for the God of negative theology.

Hence the fourth and final ‘orientation’, one that is able to take thinking beyond that point and to offer a working idea, if not (in the nature of the case) a fully adequate concept, of that which had hitherto eluded its furthest powers of rational grasp. According to Badiou this path of enquiry has been ‘discernible

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from Marx onwards’ and was ‘grasped from another perspective in Freud’ (p. 284). It is best conceived as ‘transversal to the three others’ since, somewhat like the set-theoretical technique of diagonalization invented by Cantor, it allows thought to achieve a decisive advance by showing how certain formal procedures generate results that fail to square with some foregone (often intu-itive or commonsensically ‘self-evident’) supposition, and which thus force a radical re-thinking of that supposition along with a grasp of hitherto unglimpsed possibilities. Thus the fourth orien-tation is that which locates the ‘truth of the ontological impasse’ beyond the domain of ontology itself and beyond any ‘meta-ontological’, that is, purely analytic or formalized approach – here, one assumes, with Russell chiefly in mind – that would seek to maintain the logical consistency of set theory (and to hold any looming paradoxes firmly at bay) by decreeing a strict separation of realms between language, meta-language, meta-meta-language and so forth. For of course it is Badiou’s leading claim that such consistency is in truth not to be had, unless at the cost of producing a closed and immobile ontology (like that of Parmenides and, at least on one interpretation, Spinoza) which excludes any possibility of change. What therefore sets this orientation apart from the others – and at odds with them in certain crucial respects – is its challenge not only to such extreme versions of the monist ontological creed but also to ontology itself insofar as that project has always, by its nature, tended towards a conception of being that would leave no room for the disruptive impact of whatever lies beyond its (supposedly) con-sistent and all-encompassing grasp.

It is here – in this exposition of the ‘fourth way’ – that Badiou offers one of his clearest statements concerning the need for ontology, that is, for a mathematically based exploration of being in its various modes but also the need for ontology to recognize that which by very definition exceeds its conceptual grasp. Thus ‘[i]ts hypothesis consists in saying that one can only render justice to injustice from the angle of the event and intervention’, in which case ‘[t]here is no need to be horrified by an un-binding of being, because it is in the undecidable occurrence of a supernumerary non-being that every truth-procedure originates’ (p. 285). This is where Badiou most decisively parts company with other writers on the theme of general ontology, among them Dale Jacquette in


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a recent book that in certain respects steers a similar course to his own.9 For both of them Heidegger stands out as a thinker of great originality and insight, one who did much to revive the for-tunes and shape the development of modern ontological enquiry, yet whose choice of a phenomenological starting-point – an appeal to the subject, albeit redefined in (supposedly) post-humanist or non-anthropomorphic terms – was a wrong turn and, in effect, an abandonment of anything that warrants that description.10 If the term ‘ontology’ is to have any proper or distinctive meaning then it has to signify a mode of investigation that aims to estab-lish the objective, mind-independent structures of reality (whether physical or abstract), rather than – as Heidegger would have it – their ultimate non-objective source in the moda lities of human perception, experience and (above all) temporal awareness. As Jacquette pointedly puts it, ‘[w]here can so much information about being qua being as Heidegger discovers possibly come from if it has not been surreptitiously smuggled into the assump-tion that phenomenology is the only method of ontology?’.11 Badiou and Jacquette are likewise agreed in drawing a sharp dis-tinction between general ontology (that which seeks to limn the structures of reality aside from all distinctions of particular domain) and those local or regional ontologies that have to respect such distinctions if they are not to lose touch with their own specific standards of truth or objectivity.

However they are at odds on two major points that can be seen to set Badiou’s project decidedly apart from most conceptions of the aim and scope of ontological enquiry. One is his opting, unlike Jacquette, for a strictly extensionalist rather than inten-sionalist approach to set-theoretical issues, an approach that defines its operative terms, concepts or functions in a purely numerical or quantitative way – as entities referred to without any kind of qualitative distinction between them – and which thus programmatically avoids the introduction of preconceived evaluative judgements or priority rankings. This choice is moti-vated not only by considerations of a relatively technical, that is, formal, mathematical or logical nature but also – as we have seen – by Badiou’s conviction that it has a crucial bearing on issues of political, social, sexual and ethnic equality. The other main point where his thinking diverges from Jacquette’s is with regard to the latter’s guiding assumption that general ontology should or must

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have to do with a maximally consistent combination of objects, properties and attributes which between them constitute the actual world, as distinct from those other possible worlds that may exhibit such consistency in some degree but which fall short of it in certain decisive respects. Their non-actuality is precisely a function of their failing to cohere – to yield an exhaustive range of true, valid and logically consistent statements – in such a way as must of necessity be the case with respect to the actual world, since after all that world must itself be thought to cohere as a total ensemble of how things stand and moreover to contain everything that fixes the truth-value of those statements. Thus Jacquette suggests that ‘the word “world” [should be] restricted to the unique maximally consistent proposition set associated with the actual world’, while as regards ‘submaximally consis-tent proposition sets’ we should do better to speak of ‘near worlds’ or ‘world-like structures’.12 Such a usage would have the wholly beneficial effect – he believes – of maintaining a firm ontological line between reality (i.e. the sum-total of objects, events and properties both abstract and physical, whether past, present or future, throughout the entirety of this as opposed to some alternative conceivable universe) and the various near-by or remote quasi-‘worlds’ that can be shown to be submaximally consistent and hence – ex hypothesi – defective in point of actuality.

This goes along with Jacquette’s programmatic commitment (again placing him at odds with Badiou) to the thesis that ‘combinations’, rather than sets, are the best linguistic-conceptual currency in which to conduct ontological discussion since they are strictly neutral as concerns content or subject matter and hence ideally suited to the purposes of a general ontology which then provides the basis and starting point for more specialized regional enquiries. On this account, ‘[t]he concept of a combina-tion is schematic for reference to any logically possible selection of logically possible properties, or of objects defined in terms of properties combined with properties’.13 For Badiou, on the other hand, there is nothing to commend such topic-neutrality or lack of ontic specification and indeed every reason to suppose that, in keeping with the choice of an intensionalist rather than exten-sionalist ontology, it serves mainly as a pretext for deflecting attention from substantive issues in the formal as well as the


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physical and social sciences. That suspicion would no doubt be reinforced by Jacquette’s taking it as further good reason to substitute talk of ‘combinations’ for set-theoretical talk that the former has to do only with entities – that is, with existent (whether physical or abstract) objects and properties thereof – rather than hobnobbing with the kinds of non-entity for which set theory not only finds room but reserves a privileged since system-grounding or edifice-supporting place. Thus, in Jacquette’s words, ‘[u]nlike sets, which include the possibility of a null set contain-ing no members whatsoever, there can be no null combinations’.14 Whereas, for him, the exclusion of this possibility speaks strongly in favour of a combination-based rather than set-theoretical approach, for Badiou it would betray a shying away from the transformative potential – the openness to unpredictable events or encounters with that which eludes its calculative grasp – that thinking gains by adopting the latter rather than the former. It is in consequence of just such disruptive encounters with the limits of self-assured reckoning – limits that have often shown up through a failure to accommodate those crisis-inducing irrup-tions of the void – that mathematics has periodically been jolted into new and intensely productive tracks of thought.

3. Leibniz: logic, language and the plenitude of being

With Meditation Thirty Leibniz makes his entry as one of those strong precursors who exemplify a particular, clearly marked and crucial stage in this perpetual dialectic of containment and excess, or that which preserves and that which periodically unset-tles and disrupts established conceptions of truth. What emerges here is the difference between Badiou’s and Leibniz’s way of conceiving the being/event dichotomy, that is, the question – cen-tral to both their projects – of how and why contingency should have any place in what would seem (from the ontological view-point) an exceptionless order of necessity governed by causal, metaphysical or logical laws.15 For Badiou this is not – as it is for Leibniz – primarily a matter of human ignorance vis-à-vis the total chain of concatenated causes and effects or a product of our limited powers of rational grasp as compared with those of an omniscient knower for whom ex hypothesi nothing would be contingent since everything would occupy its rightful place in the pre-established order of things. Rather, it is a question of

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necessities that come into being only as a consequence of certain events – certain epochal changes or discoveries – whose extraor-dinary nature is precisely a matter of their having been prepared or their advent guaranteed by no such providential scheme. After all, Badiou’s entire project rests on the distinction between math-ematics as fundamental ontology (and therefore as philosophy’s primary source of guidance in such matters) and those various kinds of historically located thought, knowledge and experience that provide the essential enabling ‘conditions’ for philosophic thought. Where they come into contact is not through some Leibnizian God’s-eye perspective wherein that distinction would fall away and the notion of contingent matters of fact – as opposed to necessary truths of reason – at last be revealed as merely a product of our time-bound perceptual, cognitive, epis-temic or other such creaturely limits. On the contrary, according to Badiou, it is through the always unfolding and strictly open-ended dialectic between those two dimensions that philosophy discovers its true vocation as a creative as well as a rigorous and disciplined mode of enquiry.

All the same what most attracts him to Leibniz – as likewise to Spinoza – is the intriguing combination of a systematic doc-trine that he firmly rejects with a speculative brilliance that he finds both congenial and productive of numerous otherwise unachievable insights. Of course the doctrine must strike Badiou as downright unacceptable on account of its commit-ment to a thoroughgoing metaphysical-determinist creed which leaves no room for events sub specie aeternitatis except insofar as their apparent contingency results from our ignorance con-cerning their causal antecedents or place in the overall rational scheme of things. However, as he then proceeds to argue, this goes along with a remarkable, well-nigh baroque profusion of ideas, images and metaphors that lead into unexplored regions of thought far beyond the system’s programmatic remit. Hence Badiou’s major thesis: that ‘Leibniz is able to demonstrate the most implacable inventive freedom once he has guaranteed the surest and most controlled ontological foundation – the one which completely accomplishes, down to the last detail, the constructivist orientation’ (p. 315). ‘Constructivist’, that is, in the sense that Leibniz’s four best-known rationalist principles – those of non-contradiction, sufficient reason, the identity of


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indiscernibles and the indiscernibility of identicals – between them constitute the logical-metaphysical framework of a monist ontology that would indeed, if everything went per-fectly to plan, have the effect of excluding any possible depar-ture from its immanent laws of structure, causality, or logical consequence.16

That these latter are so closely intertwined or conflated in Leibnizian metaphysics is one main reason why his system offers such a consummate example of constructivist thought in its drive to preclude – to render strictly unthinkable – whatever might otherwise exceed its conceptual power. The second main reason is that this comes down to a matter of nomination insofar as Leibniz’s two identity-laws can be seen to rest on a jointly ontolo-gical and linguistic-representational thesis whereby it is assumed that all relevant distinctions (i.e. all those that latch onto some genuine, objectively existent, truly discernible difference between entities) will find adequate expression in language. No doubt this claim could be made good only in the case of a formal, that is, logical, symbolic or algebraic language – such as Leibniz (like Frege and others after him) hoped to construct – that would leave no room for vagueness, semantic imprecision, metaphorical licence, syntactic ambiguity or other such ills to which natural languages are (naturally) prone.17 Not that Badiou is by any means predisposed against an idea – that of the ‘ideal and trans-parent language that Leibniz worked on from the age of twenty’ – which after all has much in common with his own allegiance to a formal (mathematics-based) ontology and his principled rejection of the ‘linguistic turn’ in its various Wittgensteinian, her-meneutic, post-structuralist or neo-pragmatist modes. However what crucially distinguishes his from Leibniz’s formalist concep-tion is the fact that Leibniz adopts this approach by way of securing – or appearing to secure – an absolute dominion for the dictates of logic over the realm of ontology and for the rule of ontology, in turn, over everything that has its properly assignable place in the great concatenated chain of logical-rational-causal necessity. Thus ‘[w]hat Leibniz absolutely rejects is chance . . .if it means an event whose sense would have to be wagered’ (p. 316). Here one can see why Badiou is so drawn to Pascal despite sharing nothing of the latter’s theological or existentialist-fideist convictions, and is so fundamentally at odds with Leibniz

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despite his (on the face of it) far greater depth of intellectual kinship with Leibniz’s rationalist metaphysics and his idea of mathematics and logic as limning the ultimate structures of real-ity. For, as Leibniz conceives it, ‘[a] multiple, and the multiple infinity of multiples from which it is composed, can be circum-scribed and thought in the absolute constructed legitimacy of their being’ (p. 316).

This follows directly from the cardinal Leibnizian precept according to which the domain of possible beings or potential existents is entirely determined by those four basic principles of logic mentioned above, chief among them that of non-contradic-tion. For if indeed reality contains all and only those objects, properties, conjunctures, causal relations and so forth that don’t imply some logical contradiction – even if way out among their furthest and to us unknown entailments – then ‘being-possible is subordinate to pure logic’. In which case, Leibniz maintains, it is only through the sharply restricted nature of human knowledge and understanding that we count some truths (e.g. those of mathematics and logic) as holding good by necessity and others (e.g. those of a historical or circumstantial character) as belong-ing to the realm of contingent or might-have-been-otherwise factual record. Besides, this conviction has its ultimate ground-ing in the God of Leibnizian metaphysics and ontology whose office it is to stand guarantor that this is the ‘best of all possible worlds’ since it is just that unique and particular world among the myriad that God might have made which supposedly exhibits the greatest plenitude of being and the highest attainable perfec-tion of properties or attributes.18 Moreover it is precisely on account of his constructivist approach – his presumption that this ultimate, logically articulated structure of being can and must find expression in a likewise perfect (since constructed-to-order) language – that Leibniz is able to project his entire meta-physical worldview, along with his theodicy and also his doctrine of reality as composed of innumerable windowless monads whose lack of direct communication one with another is made up for by God’s having set them in a state of preordained perfect harmony.

So one can see well enough why Badiou should feel compelled to take Leibniz on as one of those leading adversaries whose very closeness to his own way of thinking in certain respects is such


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as to make their divergences all the more provocative and, as this Meditation shows, intellectually productive. What divides them at root is Leibniz’s espousal of a totally integrated worldview – a jointly ontological, metaphysical, cosmological and theological outlook – wherein all things have their appointed place and where nothing could possibly be otherwise because every last object, property or event is subject to the principle of sufficient reason, that is, the necessity of its being just so in order to satisfy the rationalist demand that it not be put down to mere contin-gency or chance. Thus a great deal depends on those other two principles – the identity of indiscernibles and the indiscernibility of identicals – since they articulate Leibniz’s grounds for main-taining a plenist ontology or metaphysical schema that excludes any possible irruption of just those continuity-shattering events that play so central a role in Badiou’s thinking. If the principles hold good then there is simply no room for any localized emer-gence of the void or for any incursion of that which signals a gap – precisely an indiscernible element according to the domi-nant count-as-one – in what purports to be a seamless continuum of causal and logical necessity. Granted that two indiscernible yet somehow separate or numerically non-identical beings existed, this would mean that language – even (or especially) the perfected formal language of Leibniz’s dream – was unable to attach dif-ferent names to them, and hence that thought was utterly unable to tell them apart. In which case, he reasons, ‘this pure “two” would introduce nothingness into being. . . . It would be super-numerary with regard to the axioms, effective contingency, “super-fluous” in the sense of Sartre’s Nausea’ (p. 318). Neither God nor nature (insofar as they are distinct) could possibly countenance any such senseless and irrational duplication since it would then require that the two entities be treated differently in some respect, a difference that – in view of their strict indiscernibility – must lack rational justification.

Thus it is clear why Leibniz should have invested so much intel-lectual energy and ingenuity in constructing a logico-metaphysi-cal scheme that sought to close off any points of entry for whatever carried the disruptive force of the void, the singularity or the ultra-one and was hence perceived as the greatest threat to this otherwise ideal plenitude of causal-logical space. Which is also to say that this plenum is so devised – or this continuum so

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constructed – as to ensure in advance that no event can possibly occur within it, at least no ‘event’ in the specific or qualitative sense that Badiou has now defined at considerable length in mathematical (as well as, more sketchily, in socio-political and ethical) terms. Were it not for that absolute plenitude of being then there would always be the chance of some ‘hiatus’ occur-ring in the order of things, some break in the chain of concate-nated cause and effect or ubiquitous logical interconnection. The consequence, Leibniz cautions, would be to ‘overturn the great principle of sufficient reason, and . . . oblige us to have recourse to miracles or to pure chance in the explanation of phe-nomena’ (Leibniz, cited by Badiou, p. 319). Such would be the double catastrophe for thought should the void turn out to exist (or ‘inexist’) and the formal language thereby be revealed as per-force unable to discern the indiscernible and hence as inadequate to its proper task of clear and distinct nomination. Rather than risk the occurrence of that catastrophe Leibniz sets about build-ing up the ontological, metaphysical and conceptual-linguistic resources that would fend it off by securing their own exception-less and indissoluble unity. This he does by way of mathematics but with the emphasis on those branches of it that privilege the gradual, the continuous or the smoothly differential over the discrete, the sharply defined or clearly quantified.

As we have seen, it is primarily for similar reasons that Badiou takes issue with Deleuze’s appeal to those branches of mathematics – such as the differential calculus – that allow him (albeit with some benefit of metaphoric licence) to privilege his own favoured ontology of fluxes, intensities, ‘desiring-produc-tion’, ‘deterritorialized’ energy-flows and so forth.19 For Badiou, conversely, it is the greatest virtue of set theory taken as the leading-edge development in modern mathematical thought that it permits an otherwise unattainable degree of conceptual precision in the marking-off of quantities one from another, and hence in the locating of precisely those points – as arrived at by Cantor and subsequent thinkers – where ‘paradox turns into concept’ and thereby opens up entire new regions of exploration and discovery. Moreover it is Badiou’s contention that if one essaysa non-fideist, that is, a resistant or critical-symptomatic reading of Leibniz’s work then there emerges a sub-text of logical implications markedly at odds with its overt


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content or express doctrinal creed. This reading cuts across the grain of his project to construct an ontology of pre-harmo-nized but non-inter-communicating monads that would each contain or reflect that entire system within itself and hence perfectly instantiate the order of God’s maximally rational, consistent and self-sufficient creation. What brings the internal tension out to most telling effect is the presence, in Leibniz’s monadology, of a constantly implied yet occluded role for the subject – that is, the subject as putative bearer of truth – who is summoned into being by and through the very act of assuming that project as its own. To this extent the Leibnizian system is no more exempt than any other from dependence on those who either manage to initiate a project through their creative or inventive genius, or else succeed in carrying that project for-ward through their commitment and fidelity to it. However, this role is here reduced to an absolute minimum – indeed, to near-total passivity – by Leibniz’s desire to construct a system wherein the overriding requirements of logical consistency and all-encompassing (if largely notional) causal-explanatory scope are such as to render the subject little more than a ghost in the machine.

Badiou doesn’t hesitate to link this with ‘what is weak and con-ciliatory in Leibniz’s political and moral conclusions’, and even with those aspects of his personal life and public (diplomatic) career that may be seen to evince the same tendency. All the same – as elsewhere when he makes connections between work and life in the case of other thinkers such as Pascal, Spinoza, Hegel and Heidegger – this is no mere recourse to anecdotal tidbits as a substitute for genuine critical engagement. Rather it results from a clear-eyed recognition of how such character-traits or motivating interests play a role in the biographical-cultural-historical-political ‘context of discovery’ even though they have no legitimate place in the intellectual-scientific-philosophical ‘context of justification’. With Leibniz, it is that very attempt to minimize the subject’s role to vanishing-point that betrays both the ideologically driven nature of his project and – since it cannot be carried through without remainder – the fact that any such attempt will at some point founder on the subject’s re-emergence, albeit (as here) in a reduced or attenuated form. ‘What we should see in this is the instance of the subject such

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that constructivist thought meets its limit in being unable to exceed it . . . a subject required by the absence of the event, by the impossibility of intervention’ (p. 323). Such is the formal impasse of Leibnizian ontology and metaphysics as Badiou understands it, and such – by implication – the impasse faced by any politics that might (improbably enough) take lessons from Leibniz’s doctrines, or perhaps (more plausibly) have ideological interests or commitments of its own that lead to a similar conclusion. Thus ‘[i]t is difficult to not recognize therein the singleton, such as is summoned, for example – failing the verita-ble subject – in parliamentary elections: the singleton, of which we know that it is not the presented multiple, but its representa-tion by the state’ (p. 323).

Such is the standard by which, in Badiou’s assessment, Leibniz must be judged to have erred with respect to the social and polit-ical as well as the ‘purely’ philosophical dimension of his think-ing. It is therefore impossible to admire Leibniz for the ‘audacity and far-sightedness of his mathematical and speculative intel-lect’ while discounting those aspects that cannot but conduce to a sharply diminished sense of the potential for radical change brought about in large part by the active intervention of engaged and committed subjects. What prevented Leibniz from acknow-ledging this possibility was his no doubt variously motivated need to make good the case for a plenist metaphysics, a maxi-malist ontology and a wholly consistent rationalist or logicist account of how things hung together from a God’s-eye (omni-scient) viewpoint. Thus ‘[w]hatever genius may be manifested in unfolding the constructible figure of an order, even if this order be of being itself, the subject whose concept is pro-posed in the end is not the subject, elusive and split, which is capable of wagering on the truth’ (p. 323). Here again one notes the clear echo of Badiou’s commentary on Pascal and his insistence that truth cannot be discovered, devel-oped or conserved without the involvement of ‘militant’ subjects devoted to precisely that end. Such is always the case even though he is equally at pains to insist – as against any purely fideist or echt-Pascalian interpretation – that truth may always in principle surpass or transcend the grasp of any subject, no matter how intense or single-minded their devotion.


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Thus, according to Badiou, there is indeed a truth to be deci-phered in Leibniz’s texts, one that runs directly counter to his overt or express argumentative design and is therefore legible only through the kind of symptomatic reading that allows for the emergence of just such thematic, conceptual and logical tensions. As should be clear by now, it consists in his strongly denied or unacknowledged – one might say even repressed – need to fall back upon a certain notion of the indiscernible even while striving to assert its non-existence or rational unthinkability. ‘One of Leibniz’s great strengths’, Badiou writes, ‘is to have anchored his constructivist orientation in what is actually the origin of any orientation of thought: the problem of the continuum’ (p. 320). That is to say, he was led by his extraordinary gift for inven-tive or creative-anticipatory thought to the point of grasping that same central issue in set-theoretical ontology that so vexed Cantor two centuries later, only to shy away from its full (and to him like-wise deeply troubling) implications through the recourse to a notion of language – of an adequately formalized and disambiguated lan-guage – as providing the necessary means whereby to avert any threat to his system and its all-embracing metaphysical claims. It is at this stage of Being and Event – having taken Leibniz as his single most striking instance of a thinker impressively ahead of his time with respect to such issues – that Badiou turns his full attention to developments in the wake of Cantor’s inaugural discovery.

Discussion points

What exactly is meant by the ‘state of the situation’, as Badiou deploys that phrase? How does it relate on the one hand to his utilization of set-theoretical concepts and on the other to his activist political concerns?

Why does Badiou dedicate such a deal of close analytical atten-tion to philosophers, among them Leibniz, with whose leading metaphysical, ontological or political commitments he is often profoundly at odds?

‘Action receives a warning from ontology: that it endeavours in vain when it attempts precisely to calculate the state of the situation in which its resources are disposed.’ What is the bearing of this claim on Badiou’s philosophico-political project in Being and Event?

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PART VII. THE GENERIC: INDISCERNIBLE AND TRUTH – THE EVENT: P. J. COHEN

1. Forcing, the generic and subtractive ontology

We have now seen several instances, Leibniz most notable among them, of a thinking that is so far ahead of its time as to encounter problems that as yet elude any clear or overt formulation, while also sufficiently of its own time as to resist even acknowledging their presence. What chiefly occupies Badiou’s interest in Part VII is the question why this should be the case and then, in more positive terms, why very often these are just the problems that subsequently prove most productive of fresh insight and major advance. It is by way of getting a preliminary hold on this question that he introduces the distinction between truth and knowledge as he proposes to deploy it throughout the remainder of Being and Event. Since ‘everything is at stake’, according to Badiou, ‘in the thought of the truth/knowledge couple’ it is worth citing him on this topic as a prelude to further commentary. ‘What this amounts to’ he writes,

[i]s thinking the relation – which is rather a non-relation – between, on the one hand, a post-evental fidelity, and on the other hand, a fixed state of knowledge, or what I term below the encyclopaedia of the situation. The key to the problem is the mode in which the procedure of fidelity traverses existent knowledge, starting at the supernumerary point which is the name of the event. (p. 327)

This idea of a faithfully enacted truth-procedure ‘traversing’ the body of knowledge – or what passes for knowledge at some given time – would seem to derive partly from Cantor’s ‘diagonal’ proof technique as applied to set-theoretical conjectures. However it also appears to owe something to the less well-defined but none the less crucial realist idea of discoveries in mathematics and the formal sciences as involving an exploration of unknown concep-tual terrain and discovery of salient landmarks that pre-existed the explorers’ arrival and are in no way dependent for their being or location on the explorers’ having come across them.1 Both analogies go strongly against the constructivist/anti-realist conception of mathematical ‘truth’ as co-extensive with – since


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entirely created by – those methods, techniques, proof-procedures and so forth, that constitute the means and (very largely) the sub-stance of this or that state of accepted mathematical knowledge.2 Where this idea lacks credibility, so the realist will argue, is in its failure to explain how advances in knowledge could ever occur were it not for the capacity of thought to conceive truth as always potentially distinct from – or in advance of – what presently counts as such. However the constructivist will then come back and demand how it could possibly make sense to claim that one knows, or has some adequate ground for asserting, the existence of truths that supposedly surpass one’s utmost range of epis-temic, cognitive or rational-demonstrative grasp.3

Hence Badiou’s invocation of recent developments in set theory which offer an account of just how it is that problems engen-dered by a present-best state of knowledge can often pass unno-ticed and yet be seen later to have given rise to tensions which then proved a spur to some signal advance in the powers of mathe-matical thought. It is through the process of constantly con-fronting discrepancies, exclusions, anomalies or whatever doesn’t ‘count’ according to the dominant count-as-one that mathemat-ics – like politics in this regard – most often finds itself forced to acknowledge some previously unrecognized problem and, in consequence, some new and hitherto strictly inconceivable advance. This is where Badiou locates the productive tension – sometimes the outright conflict of aims – between a thinking that remains within certain pre-defined limits and therefore has to repress or ignore the existence of such problems in its own constitution and a thinking that ventures beyond those limits through fidelity to a new project whose exact nature, structure and conceptual implications are as yet unknown. On the one hand there is a ‘language of the situation’, that is to say, a state of knowledge that finds perfectly adequate expression in and through that language, and which consists in ‘the capacity to discern multiples within the situation which possess this or that property’ (p. 328). What chiefly characterizes such thinking is a perfect correspondence of the ‘rule of knowledge’ with ‘a crite-rion of exact nomination’. This latter is conceived as enabling the knower to exercise an infallible (by its own lights) power of ‘discernment’ which picks out all and only those properties – those (again by its own lights) most conspicuous and salient

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properties – that determine which multiples should be taken to possess some ‘nameable characteristic’ in common and therefore to exhibit ‘the connection between language and presented or presentable realities’ (p. 328). In short, it is an ‘encyclopaedic’ conception which assimilates truth to knowledge and knowledge in turn to that which can be fully, comprehensively, and (as sup-posed by the projectors of a perfectly logical language) trans-parently represented. On the other hand there is that which intrinsically escapes or eludes such classification since it has to do with those singular events that find no place within the range of currently accredited mathematical, scientific, political, ethical or other discourses which claim to articulate the truth of their respective domains. Indeed it is just this fact of non-belonging – of exceeding and thereby (potentially) transforming the situation wherein it has so far gone unrecognized – that sets such occur-rences apart from the run of ‘events’ in the commonplace, con-ventional or (as Badiou considers it) falsely levelling sense of that term.

This is why he devotes some of the most intricately argued sections of Being and Event to the question of how it might be possible for thought to run ahead of its present-best capacities of proof, knowledge or demonstrative reasoning and raise issues that would not be posed in an adequate or fully articulate con-ceptual form until the occurrence of just such a future advance. Most important here are his closely related concepts of forcing, the generic and indiscernibility, all of them derived from the work of Paul Cohen and each of them explained with great care and precision in this part of the book.4 Moreover, as Badiou stresses, they are closely related to the point of being inter-definable, that is, of near-synonymy. ‘The term “generic” positively designates that what does not allow itself to be discerned is in reality the general truth of a situation, the truth of its being, considered as the foundation of all knowledge to come’ (p. 327). It is precisely the indiscernible element – that which eludes the count-as-one in any given situation – that will always mark the critical point at which knowledge encounters its limit and where thinking confronts the possibility of moving decisively beyond it. This is why Badiou adopts a negative or privative terminology by way of describing how progress is achieved not so much through a patient Baconian accumulation of knowledge, nor again


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(following Kuhn) through drastic but rationally under-motivated switches of allegiance, but rather through the singular capacity of thought to grasp what is absent or lacking in a situation and thereby most effectively orient and motivate its own future projects. Thus he puts the case for a ‘subtractive’ ontology as that which alone makes room for the occurrence of genuine events, and maintains that such events typically result from the power of thought to ‘indiscern’ (i.e. to perceive as lacking or non-existent) the completeness or consistency of a given situation.5 ‘“Indiscernible” implies a negation, which nevertheless retains this essential point: a truth is always that which makes a hole in knowledge’ (p. 327). It is through the exercise of this subtractive power that thought becomes open to a sharpened sense of yet-to-be-resolved problems or aporias in the currently prevailing state of knowledge, and hence subject to the ‘forcing’ effect of that which eludes its comprehension at present yet none the less exerts a transformative pressure on its current methods, tech-niques and procedures.

It is here that Badiou introduces his Cohen-derived concep-tion of the ‘generic’ as that which distinguishes authentic from pseudo-events, or those that involve some genuine advance in the resources of mathematical, scientific, political or creative-artistic thought from those which assume that epochal aspect only in a short-term, parochial or ideologically driven perspective. Thus the term ‘generic’ applies to certain conceptually recalcitrant yet ultimately truth-conducive theorems or conjectures whose effect is to stimulate enquiries or open up paths of thought that would otherwise – according to alternative (e.g. constructivist, intuitionist or instrumentalist) philosophies – have absolutely no place in mathematical or other kinds of rigorous and disciplined thinking. It is on this basis that Badiou develops a number of important distinctions, among them those between the veridical and the true and – as mentioned above – the discernible and the indiscernible. Thus, ‘[t]he discernible is veridical. But the indis-cernible alone is true. There is no truth apart from the generic, because only a faithful generic procedure aims at the one of situ-ational being’ (p. 339). That is to say, what sets the ‘veridical’ apart from the ‘true’ is also what distinguishes the as yet indis-cernible elements of truth in some given situation from every-thing that would, on a more advanced reckoning, be assignable

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to ignorance, error or the limits of presently attainable knowl-edge. This is why Badiou takes so strongly against any approach to mathematics – or, for that matter, to the natural sciences, poli-tics, psychoanalysis or any other discipline of thought – that endorses the anti-realist idea of truth as epistemically con-strained, that is to say, as ineluctably subject to those same human, all-too-human cognitive limits. It is also why he comes out firmly opposed to the ‘linguistic turn’ in its manifold forms and guises except where – as in the line of analytical descent from thinkers like Frege and Russell – it holds language accountable to stan-dards of logical consistency and truth that may well involve (con-tra Wittgenstein and other exponents of the ‘language-first’ approach) a willingness to claim that everyday usage sometimes stands in need of corrective analysis or clarification.6

Thus Badiou operates a point-for-point reversal of the argu-ment by which philosophers of an anti-realist or constructivist persuasion have started out by conceiving truth as co-extensive with the scope and limits of attainable knowledge, and then moved on – with Wittgenstein’s blessing – to conceive knowledge itself as co-extensive with the scope and limits of linguistic representation.7 This is why he is so implacably opposed to any notion of truth as subject to constraints of a linguistic, commu-nal or epistemic nature that would leave us at a loss to explain how it might at once transcend those restrictions and yet lie within the bounds of conceivability. Hence his insistence on the subtractive dimension of truth, or its distinctive property of taking away from that range of accepted, received or conventionally sanctioned items of knowledge that falsely lay claim to a positive character grounded in the plenitude of being, the ideal con-sistency of rational thought and the notion of an ultimate (if asymptotic) convergence between knower and known. This is the reason, he suggests, ‘why the world is full’, and also why ‘it can rightfully be held to be linguistically familiar’, namely that ‘a finite set of presented multiples can always be enumerated’, in which case ‘[t]he totality of these discernments constitutes an encyclopaedic determinant’ (p. 331). With regard to mathemat-ics, the natural sciences, politics and art what distinguishes the authentic event is its capacity to point beyond any presently established evidential or probative grounds and to signal the truth of that which will – at a later, more developed stage of


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understanding – turn out to have provided sufficient warrant for certain truth-apt theorems, conjectures or hypotheses whose truth-value could not be known at the earlier time. So it is that Badiou introduces Cohen’s concept of ‘forcing’ as a means to explain how such as-yet unknown (even presently unknowable) truths may none the less play a crucial transformative role in some existing state of knowledge through the fact of their marking a gap – a def-inite lack or a falling-short of adequate demonstrative grasp – with respect to that current stage of epistemic advance.

For Badiou this truth finds its classic exemplification in the power-set axiom whereby, to repeat, it is conclusively shown that the sub-sets of any given set will always exceed the cardinality of the set itself, and moreover that the disproportion will exceed any calculable limit where it is a question of infinite or transfinite sets. This is why, as he says, ‘the true only has a chance of being distinguished from the veridical when it is infinite’, and again, ‘[a] truth (if it exists) must be an infinite part of the situation, because for every finite part one can always say that it has already been discerned and classified by knowledge’ (p. 333). Thus the concept of ‘forcing’, as Badiou understands it, takes rise from some of the most basic conditions of all set-theoretical enquiry. It involves the triple premise – derived (as we have seen) from ancient Greek as well as from modern mathematical thought – that the One is a product of conceptual imposition, that the multiple is ontologically prior, and hence that any product of the count-as-one will always potentially run up against this kind of internal, self-generated challenge to its power of numerical containment or comprehension. What marks off the true from the (merely) veridical is precisely its resulting from a procedure of thought which confronts that challenge and strives to trans-form it into a means of surpassing its own present limits, that is, a technique – as so often in the history of set-theoretical enquiry – whereby what once presented itself as an insoluble dilemma or paradox can henceforth be turned into a fully operative concept.

However the transformation cannot be achieved by a purely formal, quasi-mechanical or algorithmic procedure that would somehow function of its own accord quite apart from the com-mitment of those dedicated subjects whose willingness to break with received (i.e. accredited or ‘encyclopaedic’) ideas of veridi-cal knowledge is the precondition for any such advance. This

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should not be taken as in any sense opening the door to some kind of unbridled subjectivist, psychologistic or – least of all – intuitionist/constructivist approach. Badiou stresses the point that what constitutes ‘fidelity’ in such matters, or what sets a faithful procedure apart from the numerous alternatives at any given time, is the fact of its engaging with a project that involves some one or more truth-apt conjectures or hypotheses, and which moreover turns out to have been in accordance with a valid (i.e. truth-tracking) procedure. It is therefore strongly opposed to all those constructivist or anti-realist schools of thought according to which truth must be thought of as always in some sense epis-temically constrained, whether by the limits of language, logic, rational inference, present-best knowledge or available proof-procedures.8 Above all it has to be distinguished from the veridi-cal, since the latter – etymologically, that which is ‘true-to-say’ – belongs to a realm of knowledge in Badiou’s heterodox (non-factive) sense of that term, the domain of received, accredited or taken-for-granted belief. ‘What we are looking for’, he writes, ‘is an ontological differentiation between the true and the veridical, that is, between truth and knowledge. . . . The requirement will thus be that the one-multiple of a truth – the result of true judge-ments – must be indiscernible and unclassifiable for the encyclo-paedia. This condition founds the difference between the true and the veridical in being’ (p. 333). In which case, he deduces, it will have to be an infinite truth in the sense first specified by Cantor and thereafter developed to progressively higher levels of formal power, precision and refinement by the various set theo-rists whose work Badiou passes in review, Cohen chief among them. In other words, it will involve the kind of radical transfor-mation – the leap of thought that typically converts paradox into concept, or some standing impasse into the source of some star-tling new advance – which Cantor achieved when he defined an infinite set as one whose members could be paired off with the members of one of its own sub-sets (e.g. the integers with the even numbers).

In many ways – as regards its detailed exposition of develop-ments in the field from Cantor down – Badiou’s is an orthodox, even text-book account of the relevant intra-mathematical issues. Where he does break ranks with the majority of mathematicians – as likewise with mainstream philosophic thinking – is in two main


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respects, one of them basically formal or procedural (though with a range of wider implications), the other basically ‘applied’ or practical (though depending directly on that same formal com-mitment). The first has to do with Badiou’s emphatically ‘sub-tractive’ ontology, that is, his stress on the capacity of thought to ‘indiscern’ whatever it is about some given state of knowledge, situation or dominant count-as-one that marks the potential point of entry (or the probable locus of irruption) for that which is at present excluded, unacknowledged or denied recognition on account of its failing to meet the membership criteria laid down by the existing ‘encyclopaedic’ state of knowledge. The second – closely connected with this – has to do with his very different take on the question as to how such seemingly ‘abstract’ concerns can possibly claim any kind of real-world descriptive or explanatory purchase, not only with respect to the natural sciences but also as concerns those other (i.e. social, political, historical and even artistic) orders of event that Badiou sees as no less susceptible to treatment in these terms. Here also it is a matter of grasping both the scope and the limits of a set-theoretically based onto-logy, that is to say, its strictly indispensable character as a means of grasping how thought proceeds in the exploration of new conceptual terrain but also its inherently restricted nature as that which is able to account for such discoveries only with rational benefit of hindsight or (very literally) after the event. That is to say, it belongs very much on the side of accredited knowledge – of what has already entered the repertoire of items known, methods familiarized, theorems proven, conjectures verified (or falsified) and so forth – rather than on the side of those truly evental or breakthrough advances which up until their very moment of occurrence could not have been predicted by even the best-informed and most expert practitioners.

Yet despite this drastic disjunction, as Badiou conceives it, between the ontological and evental domains there is absolutely no question of events being shunted off into some realm of ulti-mate mystery where logic and reason should properly fear to tread. On the contrary, he offers a detailed account of how events – whether mathematical, natural-scientific, historico-political or cultural-artistic – typically transpire at an ‘evental site’ which occupies a marginal space vis-à-vis the main body of established truths at some given time. Its precise location in that liminal

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domain is decided by the localized presence of certain especially sharp and pressing anomalies or, to be more exact, by the absence of certain results, methods or proof procedures that would coun-terfactually serve to resolve those anomalies. It is here that Cohen’s concept of the ‘generic’ comes in as a means of explaining how this might transpire. In brief, the generic is that which figures nowhere in the count-as-one, which finds no place in the current ‘encyclopaedia’, which resists being named or ‘discerned’ by the language of any given situation, and which is therefore ‘sub-tracted from knowledge’ or present only as ‘indiscernible of the situation’ (338). Moreover – and crucially – it is distinguished from other parts (or multiples) by the fact that it uniquely ‘pos-sesses the “properties” of any part whatsoever’, that is to say, has no distinctive properties of its own that would place it in any way apart but rather just the one generic property, that of being, which necessarily pertains to each and every part of each and every situation. ‘It is rightfully called generic’, Badiou remarks, ‘because, if one wishes to qualify it, all one can say is that its ele-ments are’ (p. 339).

However Badiou’s main point – and the nub of his case for the politically as well as mathematically crucial role of the generic part – is that this universally shared because minimal property of ‘mere’ being is just what provides the basic orientation for any project of thought that would seek to go beyond received ideas of legal membership or proper belonging. Such, we should recall, was his main reason for adopting a strictly extensional rather than intensional approach to set-theoretical semantics and ontology: that this made it possible to focus on their purely mathematical and logical aspects (i.e. relations of inclusion, belonging, cardinality, transitivity, consistent versus inconsistent multiplicity and so forth) quite apart from – and indeed in stead-fast opposition to – any assignment of differential content to the various parts and multiples concerned. For this latter (inten-sional) approach would expose the procedures of set-theoretical reasoning to all manner of ‘encyclopaedic’ meanings, values, associations or elements of knowledge imported from outside the formal-procedural domain. On the other hand ‘[t]he fact that the procedure is generic entails the non-coincidence of this part with anything classified by an encyclopaedic determinant’ (p. 338). That is, his specification of its generic character along with his


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strictly extensional treatment of set-theoretical terms is the means whereby Badiou seeks to ensure that no extraneous consider-ations – or elements of foregone (presumptive) knowledge deriv-ing from that same encyclopaedia – can exert a restrictive or distorting influence on the truth-procedure concerned. Some readers may find it remarkable (not to say incredible) that Badiou should claim any kind of socio-historico-political relevance for a doctrine so strongly wedded to the virtues of formal rigour and conceptual precision. However they might recall Roland Barthes’ on the face of it perversely paradoxical claim that ‘a little formal-ism turns one away from history, but a lot brings one back to it’.9 Such is also the great central wager of Badiou’s project in Being and Event: the conjecture – staked on his commitment to provide a full-scale working vindication of its truth across the widest possible range of issues and precursor texts – that the formal resources of set theory are the best, indeed sole adequate means of accounting not only for the possibility of progress in the for-mal sciences but also for any prospect of advance in the socio-political sphere.

2. Subject, situation and post-evental truth

The obverse can just as clearly be seen in Badiou’s diagnosis of what gave rise to certain errors, distortions, deformations or (as he would surely not hesitate to say) certain downright betray-als of that same prospect through the failure to pursue such inaugural insights or discoveries with sufficient dedication and conceptual rigour. In the case of ‘vulgar’ Marxist and likewise of ‘vulgar’ Freudian thought the error was one of premature identification or reified reference, that is to say, of a false equa-tion between certain privileged terms in their own lexicon and certain presupposed groups, classes, social fractions, distinctive psychological traits and so forth. ‘Vulgar’ Marxism claimed (reasonably enough) that ‘truth was historically deployed on the basis of revolutionary events by the working class’, but went on (mistakenly) to count ‘the working class’ as ‘the class of work-ers’, this latter defined in a reductively sociological or economic-determinist fashion. To Badiou’s way of thinking – informed as it is by his set-theoretically derived concepts as well as by a radi-cally egalitarian politics – it is clear that ‘“the workers”, in terms of pure multiples, formed an infinite class’, so that ‘it was not the

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sum total of empirical workers that was at stake’ (p. 334). Thus Marxism ran into error – politically, historically and strategically as well as in ‘purely’ philosophic terms – by failing to grasp that vital point about the ‘infinite class’ that might always turn out to be signified by any such term, or (what amounts to the same thing) the impossibility of assigning it any determinate ‘empiri-cal’ content without thereby placing a restrictive and unjustified quota on those who would count as belonging to that class. The result was to leave Marxism (i.e. this doctrinaire, reductive and conceptually hidebound version of Marxism) devoid of resources when it came to the test – for Badiou, one supposes, the acid test of May 1968 – and its limited perspective showed up as a failure to grasp that specific evental conjuncture.

In the case of psychoanalysis likewise it was the cooption of Freud’s inaugural project by a socially conformist school of thought – American ego-psychology – that blocked its capacity to open up a range of liberating insights far beyond even the founder’s powers of speculative anticipation. Thus, if ‘today this Freudianism looks like a state corpse’, then that outcome has directly and chiefly to do with the movement’s ‘claim[ing] to form a section of psychological knowledge’, and thereby ‘assign-ing truth to everything which was connected to a stable class, the “adult genital complex”’ (p. 334). It is at this point – by way of exemplary contrast – that Badiou invokes Lacanian psychoanal-ysis as a radically different and (as he construes it) more authen-tic understanding of the Freudian discovery, one that holds out to the limit against such perversions of its true import. Lateron – in his concluding Meditation Thirty-Seven – Badiou will enter certain reservations regarding Lacan’s much-heralded turn to language (the Saussurean or structuralist conception of lan-guage) as a putative means of bringing greater precision not only to the reading of Freud’s texts but also to the ‘talking cure’ itself, that is, the elusive passages and detours of the signifierin the interaction between analyst and analysand.10 Still there can be no doubt concerning the extent of Badiou’s intellectual indebt-edness to Lacan and, moreover, the depth and intensity of alle-giance that marks his engagement – however critical at times – with the legacy of Lacan’s thought. Indeed this offers a particularly striking case-in-point of the way in which a ‘truth-procedure’ initially takes rise from some decisive intervention or inaugural


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event and is then maintained, conserved, refined, developed, extended and subject to various (on occasion strongly dissenting or heterodox) further interventions on the part of its faithful – yet by no means dogmatic or overly devoted – legatees.

Above all what he finds manifested, albeit unevenly, in the theory and practice of Freudian–Lacanian psychoanalysis is that process whereby a subject is literally brought into being through some such inaugural event and thus enabled to discern what Badiou takes as the founding distinction of every authentic revolution in thought or in practice. This is – to repeat – the dichotomy between knowledge and truth, or that which belongs to the ‘encyclopaedia’ of presently accepted lore and that which can be made out only through a critical-diagnostic or ‘sympto-matic’ mode of engagement which seeks to locate the gaps, aporias, contradictions, non-sequiturs, unrecognized dilemmas, suppressed premises, logical equivocations and so forth, which bear witness to whatever eludes or exceeds that state of knowledge. Yet the question remains: how could anyone possibly get the least degree of intuitive, empirical, epistemic, conceptual, ratio-nal or other such cognitive purchase on truths that ex hypothesi transcend the utmost bounds of present cognizability? Badiou sets out to answer this question in the remainder of Part VII through a sequence of closely argued and, at times, intensely demanding sections devoted for the most part to what he takes as Cohen’s truly epochal discoveries within and beyond the strictly mathematical realm. Crucial here is the hypothesis that ‘a truth groups together all the terms of the situation that are positively connected to the event’, where the fact of that positive connection is symbolized by x(+) while its contradictory – the absence or the failing-to-hold of any such connection – appears, naturally enough, as x(−). Of the latter one can say with full assurance that it ‘does no more than repeat the pre-evental situ-ation’, that it ‘has no relation whatsoever with the name of the event’, and therefore that ‘it is in no way “concerned” by that event’ (p. 335). Moreover, one can also project forward and adopt the future-anterior (or ‘will-have-been’) perspective which plays such a central role in Badiou’s thinking, as likewise in Cohen’s conception of how mathematical advances come about through a truth-procedure involving the generic, the indiscern-ible and ‘forcing’. In brief, this involves the capacity of thought

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to exceed the limits of presently attainable knowledge or proof through a grasp of those operative truth-conditions that will or necessarily would have been satisfied if this or that statement, theorem, or conjecture is eventually to count as proven or valid. Thus, in the case of an x(-) term, ‘[i]t will not enter into the new-multiple that is a post-evental truth, since, with regard to the fidelity, it turns out to have no connection to the supernumerary name’ (p. 335). For an x(+) term, conversely, the fact that it has maintained such fidelity – in however critical or dissident a fash-ion – ensures its link with any future advance, discovery, or breakthrough that can be shown to have resulted from precisely that sequence of previous developments. Where ‘[t]he x(−) terms remain indifferent, and solely mark the repetition of the pre-evental order of the situation’, the x(+) terms must be thought of as belonging by very definition to a valid procedure that started out from the occurrence of a genuine (truth-conducive or truth-oriented) event and the end-point of which will be something in the order of a decisive formal proof, a momentous advance in the scope and reach of that procedure, or (very often) a further such event whose effect is to pose some yet more complex and demanding challenge.

What is so difficult to grasp about all this, as anti-realists are fond of pointing out, is the sense in which truth can possibly (non-contradictorily) be conceived as somehow latent within a given situation or state of knowledge and yet as surpassing the utmost powers of any available proof-procedure or investigative method. And again, how is it that the probative upshot of some given hypothesis, conjecture or prediction can turn out to vali-date not only the first (perhaps highly tentative) formulation of it but also the subsequent labours of those who strove – with great tenacity but without success – to discover or devise a fully adequate proof? For Badiou, this becomes the question as to how his capital distinction between truth and knowledge can be upheld despite its apparently confronting just such powerful objections. Thus ‘[o]ur problem is finally the following: on what condition can one be sure that the set of terms of the situation which are positively connected to the event is in no manner already classified within the encyclopaedia of the situation?’ (p. 336). This amounts to a reverse-order statement of the same problem that anti-realists routinely adduce against realists, that


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is, the supposedly knock-down argument that counts it non-sensical to posit the existence of objective (hence recognition-transcendent, hence strictly unknowable) truths. Where Badiou markedly raises the stakes is in maintaining not only, with the realist, that such truths exist and, moreover, that if they didn’t then there could be no making sense of mathematical know-ledge, discovery or progress but also – the single-most distinctive thesis of Being and Event – that they do indeed possess that deci-sive power of retroactive intervention with regard to what can now be seen as having constituted a faithful or truth-oriented mode of enquiry. Here again his claim rests crucially on Cohen’s set-theoretical advance and on what Badiou takes (in this respect breaking with the more cautious or restrained approach of Cohen and the majority of mathematicians) to be its wholly jus-tified extrapolation to the realm of speculative ontology.

Thus any rendition of Cohen’s thesis must be couched in the future-anterior tense and also in the conditional or subjunctive mode since it has to do with what cannot yet be formally proved or verified while none the less following by the strictest necessity from certain other propositions which, if true, will be recognized as lending decisive support to the given hypothesis or theorem. Furthermore any gaps, inconsistencies or contradictions in some given state of knowledge may force the invention (in a somewhat archaic but aptly ambiguous sense of that word: the creative devising but also the discovery) of a new working hypothesis. This latter is then subject to further testing, refinement and elaboration until the stage when its validity – if and when con-firmed – will retroactively endorse whatever led up to it in the way of conjectural (though none the less rigorous and faithful) truth-procedures. Cohen’s thesis therefore offers Badiou a means of sharpening his own distinction between being and event, or explaining how events which exceed the limits of conceptual specification can none the less effect a decisive change in the state of mathematical knowledge. Above all it is the idea of ‘forcing’ that played an indispensable role in enabling Badiou to think his way through and beyond the dilemma (or the pseudo-dilemma, as he came to think it) that is commonly supposed to result from affirming the existence of recognition-transcendent truths. At this stage we shall have to run ahead of ourselves and glean some relevant passages from Meditation Thirty-Six – ‘Forcing: From

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the Indiscernible to the Undecidable’ – since despite Badiou’s deferring this discussion until later in the book it is, as I have said, a concept most readily understood in conjunction with his likewise Cohen-derived notions of the generic’ and the ‘indis-cernible’. Thus, ‘[f]oreclosed from ontology, the event returns in the mode according to which the undecidable can only be decided therein by forcing veracity from the standpoint of the indiscern-ible’ (p. 429). It is precisely here that the subject makes its entry since, on Badiou’s account, the subject can best be defined as ‘that which decides an undecidable from the standpoint of an indiscernible . . . or that which forces a veracity, according to the suspense of a truth’ (p. 407). That is, the subject is here con-ceived as the locus of certain commitments, attachments, priori-ties, hypotheses, research agendas or projects and so forth, any one of which may properly be said – in some particular, well-defined context – to constitute their very identity or mode of existence.

That he is able to formulate such claims without manifest self-contradiction – or without pressing paradox beyond the furthest stretch of credibility – is largely owing to Badiou’s heterodox conception of the subject and his likewise singular deployment of set-theoretical topoi such as the generic, the indiscernible and ‘forcing’. What this conjunction of terms makes possible is a radical break with all previous (e.g. Cartesian, Kantian, Hegelian or Husserlian) modes of thought that equate subjectivity, true to the word’s etymological origins, with that which somehow underlies, supports, upholds, maintains and – transcendentally speaking – forms the condition of possibility for all human thought, knowledge and experience. In its place Badiou posits his theory of the subject not – I should repeat – in a place-holder role or as a mere by-product of some given post-evental truth-procedure but rather as summoned, convoked, inspired, borne along and sustained by the commitment to a truth that has already received its initial (hypothetical or tentative) statement but has not yet been brought to the final stage of proof or formal validation. Thus the subject, for Badiou, is always defined rela-tive to a certain specific project of enquiry, or as discovering its purpose, its vocation and indeed its very sense of identity in and through that same definitive project. Moreover – and it is here that his thought has such far-reaching implications for the realist/


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anti-realist debate – those involved in it may not (in fact very often cannot possibly) have knowledge of the truth or its means of ascertainment despite their significant, perhaps indispensable role in forwarding the process of investigation that will eventu-ally bring it to light. This is what Badiou calls the ‘capital result’ of Cohen’s mathematical researches, along with his own more speculative (ontologically-oriented) understanding of them: namely, that ‘although an inhabitant of the situation does not know any-thing of the indiscernible, and so of the extension, she is capable of thinking that the belonging of such a condition to a generic description is equivalent to the veracity of such a statement within that extension’ (p. 411). In other words, truth may indeed run ahead of achieved or presently achievable knowledge not only in the purely objectivist (Platonist) sense that creates such epistemological problems – since it opens up that seemingly unbridgeable gulf that anti-realists are wont to exploit – but also insofar as the truth-knowledge gap may itself be inscribed as a symptomatic absence or determinate lack in the present state of understanding.

No doubt this requires that a subject – a thinker with some quite specific range of motives, interests, formative influences, intellectual incentives and so forth – should register that lack at some level of consciousness and thereby be enabled to think their way through to a further, more advanced stage on the path to truth. However, so Badiou cautions, we should not be misled into supposing that the ‘subject’ here invoked in the context of mathematics is the same ‘subject’ that routinely figures in the discourses of the human and social sciences, including most (i.e. non-Lacanian) versions of psychoanalysis. Thus the phrase ‘inhabitant of the situation’ in the above-quoted passage should not be taken at face value but rather as ‘a metaphor’, one that ‘does not correspond to any mathematical concept’, nor indeed to any concept in good ontological standing since ‘ontology thinks the law of the subject, not the subject itself ’ (p. 411). That is to say, the subject ‘inhabits’ a situation just to the extent that she registers what Badiou defines as the discrepancy between it and the ‘state of the situation’, or belonging and inclusion, or – again – between the dominant count-as-one and those multiples which (as in the power-set axiom) exceed, surpass or intrinsically elude its grasp. From which it follows that the ‘subject of truth’,

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as distinct from the subject in the commonplace acceptance of that term, should be thought of always according to its role in the furtherance of just those projects or enquiries that most strongly define its ‘militant’ vocation. On the other hand this somewhat rarefied conception doesn’t go so far as to render the subject a merely nominal entity or a product of purely formal definition with no power of active intervention beyond what is programmed in advance throughout the stages of some given truth-procedure. ‘Just as it cannot support the concept of truth (for lack of the event), nor can ontology formalize the concept of the subject’ (p. 410). Still it is able to investigate the scope of its own conceptual resources and, in the process, indicate those points where ontology encounters the event and the subject as absolute limits on its power of comprehension.

So on Badiou’s account there is room – indeed a strictly indis-pensable role – for the subject as involved in the production, dis-covery or progressive emergence of truth though not in any sense that could possibly set it apart from or over against those proce-dures that constitute the methodology of the formal sciences. Such is the ‘capital’ philosophic breakthrough that Badiou attri-butes to Cohen, albeit – as he fully concedes – without the latter’s having grasped or been willing to acknowledge its full implications. What Cohen’s work shows is that the ‘existence of the subject is compatible with ontology’, thus in effect ‘ruin[ing] any pretension on the part of the subject to declare itself “contradictory” to the general regime of being’ (p. 410). On the other hand it also shows that a ‘subtractive’ ontology is the only one capable – for reasons we have seen – of doing justice to the scope and reach of mathe-matical thought, the potential for future discovery latent in existing (however limited) states of knowledge, and the way in which, con-tra anti-realists, truth can and must be conceived as transcending or surpassing those same limits. Thus if the subject cannot pretend to ‘contradict’ the ontology (or prevalent ‘regime of being’) that defines the conditions under which it exists and exerts its powers of understanding, nevertheless it can discern – more precisely, ‘indis-cern’ – those gaps, fissures, stress-points, aporias, intrusions of the void or irruptions of inconsistent multiplicity that signal the encounter between ontology and that which exceeds its grasp.

It is here that Cohen’s concept of ‘forcing’ goes furthest towards resolving certain crucial issues not only in mathematics,


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epistemology and philosophy of the natural sciences but also in those social-science and humanities disciplines where there has lately developed a deep-laid suspicion of any appeal to the human subject as locus, source or guarantor of truth. What he offers is (on the face of it) a purely formal and procedural state-ment of how ‘it is possible, in a quasi-complete fundamental situation, to determine under what conditions such or such a statement is veridical in the generic extension obtained by the addition of an indiscernible part of the situation’ (p. 410). Yet it is just Badiou’s point – and his reason for devising that odd but very apposite negative verb ‘to indiscern’ – that there must be something more, some ratiocinative power or mode of active intel-lectual engagement that is able to register those gaps, absences or problematic loci in the body of current ‘encyclopaedic’ know-ledge, and moreover to deploy them as a means of advancing beyond that stage in the continuing quest for truth. So it is that the subject, on Badiou’s understanding, can be said to have its founding or inaugural moment in and through its recruitment to a truth-procedure which itself started out from some preceding notable event and which thus sets the terms for whatever he or she is able to achieve in the way of new discovery. This is also why his two leading concepts of ‘subject’ and ‘event’ are so closely bound up one with another, not only in thematic or semantic-associative terms but also as a matter of logical necessity if his argument is to have any force. Their relationship is absolutely central to Badiou’s case for set theory as a guide to ontology and for ontology as the discipline of thought that limns both the structure of abstract or physical reality and the investigative methods, resources and procedures which bring that structure to light. In so far as the event is intrinsically ‘foreclosed from ontol-ogy’, its occurrence must be thought of ‘in the mode according to which the undecidable can only be decided therein by forcing veracity from the standpoint of the indiscernible’ (p. 429). Yet this forcing must itself be conceived as requiring the involvement of a subject whose office it is – quite aside from all the usual, for example, social, psychological or personal accoutrements of subjectivity – to press beyond a currently existing state of knowl-edge through maximal commitment to the truth of that which at present can be specified only in a tentative, conjectural or strictly hypothetical form. Thus ‘[e]verything of the Subject which is its

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being – but a Subject is not its being – can be identified in its trace at the jointure of the indiscernible and the undecidable; a jointure that, without a doubt, the mathematicians were thor-oughly inspired to blindly circumscribe under the name of forc-ing’ (p. 429).

This is not, despite appearances, a backhanded compliment to mathematicians such as Cantor and Cohen. Rather, it is a com-ment perfectly in line with Badiou’s aforementioned belief that while thinkers like these at the leading edge of mathematical research are engaged in a procedure with decisive implications for ontology and other extra-mathematical branches of enquiry they cannot – or should not – be expected to draw out those implica-tions in an overt, methodical or conceptually explicit way. Rather that task is left to the philosopher since she is the one whose voca-tion it is – in virtue of her wider involvement with disciplines or areas of knowledge and experience beyond that specialized sphere – to articulate the precise order of relationship between being, event and the subject as a site where those otherwise exclusive domains can be seen to intersect. We should pause a while over that parenthetical clause in the above-cited passage which cau-tions that ‘a Subject is not its being’, before going on to define its role in relation to forcing, the event and the indiscernible. Its sense is somewhat clarified when Badiou further remarks that ‘despite it depending on the event, which belongs to “that-which-is-not-being-qua-being”, the Subject must be capable of being’ (p. 429). What I take this to mean is that it, like the event, cannot be assimi-lated without remainder to any accomplished or even provision-ally stabilized ontology since the latter is precisely that which would deny any room for such (on its own terms) random, unac-countable or strictly aleatory occurrences. Subject and event are likewise marginalized vis-à-vis the project of a classical ontology based on the premise that being should be thought of as maxi-mally consistent or as lending itself to the kinds of all-embracing descriptive-explanatory treatment attempted in their different ways by Aristotle, Leibniz and Spinoza. Moreover it is just on account of their both inhabiting this marginal zone beyond reach of ontological specification that event and subject can each exert a pressure at the stress-points that emerge in received ‘encyclopae-dic’ modes of knowledge so as to expose what is missing from – or occluded by – any such prevalent consensus.


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Having briefly swerved from the straight and narrow by jump-ing forward to a later section of Being and Event we can now revert to Part VII (‘The Generic: Indiscernible and Truth) with the advantage, hopefully, of pre-acquaintance with Badiou’s closely related thoughts about ‘Forcing: Truth and the Subject’. What that earlier section is principally concerned to explain is just how a purely formal truth-procedure can somehow be thought of as surpassing or transcending the current most advanced state of knowledge in some given discipline or context of enquiry. While the focus in Part VIII is on the subject as the agent whereby such breakthroughs are achieved – albeit the ‘subject’ in a some-what formal or attenuated sense remote from its commonplace usage – Part VII approaches the issue from a standpoint more concerned to articulate those strictly formal procedures which, according to Badiou, offer the subject its sole possibility of entering that intimate relationship to truth which constitutes its primary vocation. Here again he looks to Cantor for a model instance of how thinking should proceed in such matters. Both claims depend entirely for their credibility on Badiou’s having convincingly shown how ‘the subject’, in any philosophically accountable sense of that term, is always directly related to and even – he would argue – engendered by some specific project of thought, enquiry, artistic creativity or politically motivated action. Just as ontology, taking its lead from mathematics, requires a ‘subtractive’ conception of being if it is ever to achieve any prog-ress beyond the naïve self-evidence of commonsense-intuitive ‘knowledge’, so ‘love, art, science and politics generate – infi-nitely – truths concerning situations; truths subtracted from knowledge which are only counted by the state in the anonymity of their being’ (p. 340). That they need philosophy, or at any rate a philosophically informed view of their own proceedings, is not (Badiou stresses) a signal of its standing in some privileged position as dispenser of truth or arbiter of knowledge across the whole range of human investigative practices. Rather it is a mark – at best a willing acknowledgment – of philosophy’s strictly ancillary role vis-à-vis mathematics or any mathemati-cally based project of ontological enquiry, along with the need for philosophy as a point of reference for any other discipline that seeks to orient itself in relation to the interests of truth. Thus it is very much a matter of gains and losses, or even – one

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might somewhat cynically say – of a trade-off between philoso-phy’s demotion at the hands of mathematics where truth is directly in question and its promotion to a higher ranking discourse when-ever it is the case that mathematical truth has need of philosophy’s mediating offices in order to achieve adequate formulation in onto-logical and, beyond that, in political, scientific and artistic terms. Indeed Badiou makes a point of remarking, here and elsewhere, that even the most gifted and ground-breaking mathematicians have very often either been prone (like Cantor) to derive ill-founded theological or quasi-mystical implications from their work, or else (like Cohen) deeply resistant to any idea that their discoveries might have wider ontological or extra-mathematical bearings.

Thus although ‘the question of the being of truth has only been resolved at a de jure level quite recently (in 1963, Cohen’s discovery)’, nevertheless that signal advance has been achieved ‘without the mathematicians – absorbed as they are by the for-getting of the destiny of their discipline due to the technical necessity of its deployment – knowing how to name what was happening there’ (p. 341). And if philosophy has to adopt this double-edged relation to mathematics then it is similarly placed with regard to those other disciplines or topic-domains that make up its enabling ‘conditions’ of existence. That is to say, it is obliged to combine a due respect for their claims to scientific, political, artistic or ethical autonomy with a due sense of its own capacity to speak the kinds of truth that define their very being yet for which they cannot find any adequate means of conceptual-theoretical articulation without philosophy’s assistance. This helps to explain the puzzling conjunction in Being and Event of an extreme (some might think arrogant) confidence with regard to that work’s philosophical significance with a modest, even self-effacing attitude with regard to the achievements of mathe-maticians and – albeit somewhat less emphatically – thinkers, practitioners, creators and activists in other fields. What draws these otherwise diverse projects together is their shared capacity for participation in certain types of generic procedure, those that make room for a fidelity whose mark is the setting of its sights on an ‘infinite’ truth in Badiou’s carefully defined sense of that term. This is a truth that would not be confined to the limiting terms of some particular given situation, but would rather induce its seekers to locate those symptomatic points of conflict, excess,


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anomaly, aporia or (at present) irresolvable tension that indicate precisely where thought can get a purchase for further, hitherto unlooked-for and potentially ground-breaking advances.

What most often brings this about is the act or process of ‘indiscernment’ whereby thought is enabled – pace anti-realists and intuitionists – to exceed the present-best capacities of know-ledge or formal-demonstrative proof and to do so, moreover, through a grasp (albeit a not fully conscious or deliberative grasp) of the particular gap, deficiency, lack, conceptual short-fall or other such impediment that has hitherto stood in the way of any such advance. And it is not in mathematics, logic and the formal sciences alone that this effect can be observed (even if they provide some of the most striking examples), but also in those strictly autonomous yet strongly analogous realms of intellectual, creative and political-activist endeavour that are likewise central to Badiou’s project. Thus ‘art, science and poli-tics do change the world, not by what they discern, but by what they indiscern therein’ (p. 343). To this extent they share the essentially subtractive character of mathematical thought, the way in which ‘a faithful generic procedure renders the indiscern-ible immanent’, or the process whereby the most significant advances and momentous discoveries result from a suddenly sharpened perception of whatever is lacking in some given situa-tion or marked in its absence by various symptoms of logical or conceptual tension. In the case of poetry, music and the visual arts this finds a close equivalent in that sense of formal or expres-sive strain – of an existing language, style, technique, genre or mode of representation forced up against and striving to pass beyond its inherited limits – which, for Badiou, can be seen to mark the works of (among others) Mallarmé, Schoenberg and Malevich. However, it is in the socio-political sphere above all that he explores the implications of a mode of thought that would adopt this subtractive conception of truth and deploy it as the basis for a series of kindred ideas – notably those of the event, the indiscernible, the generic procedure and the passage to infinity – that between them constitute a veritable programme for re-conceiving the always problematical relationship between agency and structure, or – in Marx’s phrase – what human beings make of their history and those conditions ‘not of their own choosing’ under which they confront that task.

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3. Rousseau’s Social Contract: subject, citizen, general will

It is with just these issues in mind that Badiou now turns to Jean-Jacques Rousseau as the thinker in whose work he finds them strikingly prefigured and, moreover, treated in a way that poses them with striking urgency and force, as well as a high degree of theoretical refinement. What makes Rousseau’s Social Contract stand out in this way and merit its place alongside Plato, Aristotle, Spinoza, Leibniz, Hegel, Marx and the others whom Badiou selects for extended commentary is the fact that he not only thinks about politics from this or that standpoint (philosophical, ethical, legal, constitutional and so forth) but purports to think the very nature of politics as a matter of its basic ontological standing vis-à-vis the various possible modes and structures of human being-in-the-world.11 If this description suggests that Badiou is here slipping into a more Heideggerian mindset then the impression is mistaken since it is just his point that Rousseau’s way of raising these questions – a way that has often provoked deep suspicion as well as fascination and puzzlement among his commentators – is at the furthest remove from Hei-degger’s intensely conservative hearkening-back to a time when mind and world, subject and object, or the thought-of-being and being itself had not yet been rudely put asunder by the history of Western metaphysics from Socrates down.12 On the contrary, when Rousseau puts his central question concerning the putative origins of the social contract – how was it that people first consented to surrender their individual needs, desires and interests to a General Will that would henceforth be taken at once to include and tran-scend them? – he does so not with any heavy investment in the his-torical (factual) truth of that hypothesis but rather by way of a thought experiment or, as Badiou understands it, a procedure that shows the necessity of thinking this to have been the case even if there exists no empirical or other evidence for it. Thus ‘[Rousseau’s] method is to set aside all the facts and thereby establish a founda-tion for the operations of thought’ (p. 344). And again, ‘[t]he social pact is not a historically provable fact, and Rousseau’s references to Greece or Rome merely form the classical ornament of that temporal absence’ (p. 345).

Of course the same point has often been made – usually by way of criticism or downright rejection – with regard to Hobbes, Spinoza, Locke and other advocates of the contractualist


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approach to issues of social and political justice right down to modern theorists such as John Rawls or Thomas Scanlon.13 However it is Badiou’s claim that Rousseau presses this mode of reasoning very much further, indeed (although he doesn’t quite say so) to the point where it becomes something more like a formal demonstration or an axiomatic-deductive procedure designed to establish the truth of some candidate theorem. Here the reader might well think that there is a gross inconsistency between Badiou’s apparent admiration for Rousseau on this account and his express opposition to other thinkers – Leibniz and Spinoza among them – on account of their having con-structed a system of thought that presumed to accommodate every last item of knowledge within its vastly overweening metaphysical-rationalist purview. However there is one decisive respect in which Rousseau stands apart from that way of think-ing, namely in his stress on the evental (i.e. the strictly contin-gent, hence unpredictable and non-deducible) character of that which brings about the inception of a new political order and, along with it, the General Will as embodying – in principle at least – the collective wishes of a people united in their readiness to see individual interests give way to those of the henceforth sovereign community. It is this aspect of Rousseau’s thought that has provoked the most serious misgivings among his com-mentators since it strikes some of them as coming dangerously close to a conception of politics based on the idea of ‘the people’ – that is, some particular people defined in ethnic or cultural-linguistic terms – as privileged bearers of a national destiny vouchsafed or entrusted to them and them alone.

Badiou does little to allay those misgivings when he writes that ‘Rousseau’s goal is to examine the conceptual prerequisites of politics, to think the being of politics’ insofar as ‘[t]he truth of that being resides in “the act by which a people is a people”’ (p. 34). All the same one should note – lest this create a thor-oughly false impression – how any ominous, even Heideggerian echoes in the last couple of phrases here are markedly offset by what he says with regard to Rousseau’s project of examining ‘the conceptual prerequisites of politics’. In effect this stakes a claim for Rousseau as having exercised the kind of critical vigilance or power of conceptual-analytic thought that saves him from fall-ing into any such perilous, strongly nationalist and potentially

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totalitarian way of thinking. Hence Badiou’s insistence on the point that Rousseau is as far as possible from thinking of poli-tics as a ‘natural’ expression or manifestation of some particular, geographically and historically located life-form, culture, lan-guage or distinctive mode of being-in-the-world that would most likely take the shape of a nation-state or equivalent (however notional) entity. Here more than anywhere one can see how Badiou’s counterposed themes of being and event relate to his re-thinking of political theory as involving a perpetual critique of just this tendency – most clearly and disastrously visible in Heidegger’s ringing endorsement of National Socialism – to conceive politics and history in terms that are borrowed (at whatever metaphorical stretch) from natural processes of organic growth, development, or decline. Thus, contra Aristotle, ‘[m]an is not a political animal: the chance of politics is a supernatural event’ (p. 345). ‘Supernatural’, of course, not in any sense that would invoke divine or transcendent intervention but rather in the sense that it springs from a choice, a wager, or a willing-ness to stake everything on the faithful adherence to a truth-procedure – a course of action along with all its strictly incalculable implications and consequences – whose outcome is by no means guaranteed in advance, least of all by its predestined arrival as the end-point of some quasi-natural evolution. ‘If Rousseau for ever establishes the modern concept of politics, it is because he posits, in the most radical fashion, that politics is a procedure which originates in an event, and not in a structure supported within being’ (p. 345). That is, the Social Contract is too intelli-gent and self-critical a text to have its ultimate sense and logic dictated by that nexus of themes and preoccupations – among them the notion of an idealized primitive (or ‘natural’) commu-nity as yet untouched by all the bad accoutrements of ‘civilized’ social life – that are prominent elsewhere in Rousseau’s writing.14

What accounts for this acuity, so Badiou maintains, is the fact that here ‘[he] clearly designates the necessity, for any true poli-tics, to articulate itself around a generic (indiscernible) subset of the collective body’ (p. 346). The effect is to prevent any possible foreclosure of the body politic around some privileged marker of belonging or membership-condition that would constitute at best the basis of an unjust, divided and exclusive social order, and at worst the notional justification for racist or ethnically


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driven campaigns of violence. What Rousseau envisages, on the contrary, is a ‘split’ within the person or the (so-called) ‘individ-ual’ such that – in consequence of the ‘event-contract’ – they will each take on a double role as ‘subject’ and ‘citizen’, the former defined primarily in terms of ‘his or her subjection to the laws of the state’, while the latter signifies ‘his or her participation in the sovereignty of general will’ (p. 346). This ensures that there will always exist a disparity or non-coincidence between, on the one hand, the body politic as represented by some given, pre-sumptively legitimate (since communally sanctioned) structure of laws, social institutions, and political procedures and, on the other, the body politic insofar as it includes what doesn’t belong, or – to spell out once again the political import of this set-theoretical precept – insofar as it contains parts that don’t legally qualify as members since excluded by the count-as-one. So it is that the general will (to this extent a near-synonym of the ‘generic’) must be thought of as intrinsically ‘tied to the indiscernible’. This is because, by very definition, it leaves room for the exis-tence of that which cannot be discerned or duly acknowledged within the currently prevailing situation, yet which none the less stakes its claim to recognition – to acceptance as a member in good standing – on the bare fact of its inclusion quite aside from all qualifying attributes or predicates.

This, we should recall, was Badiou’s reason for adopting an extensionalist rather than intensionalist construal of set-theoretical discourse: that it served to ensure the equality (i.e. the non-discriminatory treatment) of each and every term within the various formal operations concerned, as opposed to assigning different meanings or values and thus contravening the generic demand for a principle of strict equality. So likewise the ‘[g]eneral will never considers an individual nor a particular action . . . . [t]his evidently results in the general will being intrinsically egali-tarian, since it cannot take persons or goods into consideration’ (p. 347). Moreover, it is just this egalitarian priority that leads Badiou to hail Rousseau’s thinking – despite and against its sup-posedly ‘totalitarian’ character – as having marked not only a decisive break with all previous (e.g. Hobbesian and Lockean) forms of contractualist theory but also the equally decisive turn towards a social ontology open to that which inherently eludes its grasp, namely the event as ‘ultra-one’ or as inhabiting

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a zone of the excessive or anomalous beyond ontological speci-fication.15 Hence Badiou’s claim that Rousseau here achieves, albeit imperfectly, a conception of politics that brings together the idea of a democracy more radically egalitarian than any-thing hitherto achieved with the conception of a politics that would find its model of progress in the formal rather than the social or human sciences.

This is why – in flat contrast to most interpretations of the Social Contract, whether admiring or otherwise – Badiou sees fit to praise Rousseau as a thinker who has managed to ‘formalize’ the discourse of political theory to the point where it becomes capable of statement in a mode that derives much (not all) of its conceptual apparatus from the resources of set theory. What cannot in principle be so derived – since it belongs by definition to the realm of that which exceeds or eludes the grasp of any ontology – is also (as we have seen) what constitutes the limit of achieved mathematical knowledge and at the same time points beyond it to those presently uncharted or unchartable regions that confront knowledge with the gaps or lacunae in its own cur-rent state of advance. Thus,

[t]he most remarkable thing about the Social Contract is that it establishes an intimate connection between politics and equality by an articulated recourse to an evental foundation and a procedure of the indiscernible. It is because general will indiscerns its object and excludes it from the encyclopaedias of knowledge that it is ordained to equality. As for this indis-cernible, it refers back to the evental character of political creation. (p. 347)

It is owing to just this aspect of his thought – an aspect (so Badiou asserts) that has been ignored, obscured or unwittingly passed over by most of his commentators – that Rousseau stands as a truly revolutionary thinker, albeit one whose texts have been grievously misread on both hands of the left-right divide as con-ventionally (party-politically) understood. ‘It is not a matter’, Badiou rather tetchily remarks, ‘of knowing whether a statement originates from good or bad politics, from the left or the right, but of whether it is or is not political’ (p. 349). On his account politics – in any valid or meaningful sense of that word – takes


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rise from a disruption to the normal functioning of all those dis-courses, representations, ‘democratic’ procedures, systems of knowledge, or modes of ideological containment that typically masquerade as ‘politics’ under presently existing conditions. In order to grasp how this can come about despite such conditions one has to read the Social Contract with an eye to that so-far neglected dimension of Rousseau’s argument that lends itself to formal re-statement in terms (notably those of the generic and the indiscernible) that bring with them a much higher degree of conceptual clarity and rigour.

Still Badiou needs to press his objection to the ‘totalitarian’ reading of Rousseau, whether from left or right, that would charge him with having made of the general will a conceptual instru-ment whereby to theorize – and, worse still, to recommend – a collective tyranny that enslaves each and every ‘willing’ subject in the name of a communal good supposedly transcending their individual rights and interests. That this is not, in fact, an adequate or accurate reading of the Social Contract is a case that Badiou argues with great emphasis and which involves some of his own most crucial, politically as well as philosophically load-bearing concepts and commitments. What makes all the difference is Rousseau’s concept of the will not as an expression or implemen-tation of power – even a power that we are to think of as some-how vested in the people by common assent or through popular identification with it – but rather as that which eludes and resists its otherwise ubiquitous workings. The will is enabled to bring this about through the space that it perpetually opens up for something other and more (i.e. more disturbing to the presently existent status quo) than could ever be reckoned with on terms deriving from the count-as-one or the dominant system of representation. ‘[Rousseau’s] distinction between power (trans-missible) and will (unrepresentable) is very profound. It frees politics from the state. As a procedure faithful to the event-contract, politics cannot tolerate delegation or representation. It resides entirely in the “collective being” of its citizen-militants.’ (p. 347). This is what Badiou has in mind when he equates Rousseau’s political radicalism with his having pushed so far towards a clear-headed treatment of these issues that could itself be treated – no doubt with benefit of political and social-scientific as well as set-theoretical hindsight – in formal (or at any rate

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quasi-formal and conceptually rigorous) terms. It is also what he sees as marking the scope and limits of social ontology, or the furthest extent to which thought can legitimately go with such a project of formal enquiry. For in this context, as in mathematics and the natural sciences, there often comes a point where knowl-edge runs up against an obstacle to progress or a check to its powers of conceptual grasp. That check is encountered not only by those in the original ‘context of discovery’ when some deci-sive or epochal event was in the offing but also by those looking back upon it from a ‘post-evental’ standpoint and seeking to account for its occurrence in an adequate or rationally compre-hensible way.

Thus there is always a stage of potential ‘impasse’ – a moment of suspense, indetermination, aporia, dilemma, unresolved par-adox or formal undecidability – when the sole possible outcome for thought is a commitment to one or other of the those options which (to adopt William James’s criteria) present themselves as ‘live’, ‘forced’ and ‘momentous’.16 Such is the stage, in Badiou’s reading of Rousseau, where a cleft opens up – both in logical-conceptual and factual-historical terms – between the dominant structures of power (‘popular’ power included) at any given time and the general will insofar as it expresses or embodies a collective interest at odds with those structures. The disparity in question exists to some degree in all societies at all times but on occasion assumes so acute a form as to constitute a breakdown in the normal modes of ideological containment. It reaches the critical point when there erupts a no-longer-negotiable conflict between those various components of the current political conjuncture (belonging and inclusion, sets and sub-sets, members and parts, situation and state of the situation, the count-as-one and the excess of what goes uncounted) that Badiou sets out in set-theoretical terms but which always – through an order of neces-sity announced by the axiom of choice – point beyond any such purely formal reckoning. This is why critics of Rousseau get him wrong when they take the Social Contract to propose an extreme and especially dangerous form of what John Stuart Mill would later call the ‘tyranny of the majority’.17 What they miss is the crucial distinction – one that Badiou claims to capture more pre-cisely in set-theoretical terms – between arbitrary decrees and justifiable laws, or enactments given force by some ‘particular’


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will and enactments of the general will that derive their legiti-mate (non-coercive) authority from their implicitly referring to that which transcends the dictates of brute power. It is the same with critics of Badiou – as he is quick to point out – who commit a similar fallacy when they think to convict him of importing mathematical (or quasi-mathematical) modes of thought into a realm, that of politics or political theory, where such analogies are at best misleading and at worst apt to furnish an excuse or rationale for ‘totalitarian’ thinking. Those critics have simply ignored his insistence on the gap that exists – more at certain times than others, but always to some degree – between what counts and what could or should count as a constituent part of the presumptive totality in question.

It is this decisive shift to a different register of thought – one marked by an openness to multiplicities or modes of inclusion ruled out by earlier concepts of well-orderedness – which enables political theory to accomplish something closely analogous to the advance that occurred in mathematics with Cantor’s revolution. It is why Badiou very often deploys the term ‘politics’ in a dis-tinctive or qualitative sense that sets it off from whatever per-tains to the merely ‘governmental’, or to the exercise of power through channels of state-sanctioned and legally administered authority. What is altogether lacking from such pseudo-political procedures is the standing possibility of that impasse that is always apt to emerge when the general will comes into conflict with the social-institutional status quo. ‘On the other hand’, Badiou writes, ‘the impasse remains in its entirety when politics is at stake; that is, when it is a question of decisions which relate the people to itself, and which engage the generic nature of the procedure, its subtraction from any encyclopaedic determinant’ (p. 352). It is in this sense that the general will is ‘qualified by indiscernibility’, since its truth lies beyond any state of know-ledge attainable at the time when it first showed up as a potential for change far out on the horizon of thought and will only become knowable as such from a retrospective standpoint achieved through fidelity to the truth-event in question. ‘If it was determined by an explicit statement of the situation, politics would have a canonical form’ (p. 353). In mathematics and the natural sciences likewise, if truth were restricted – as anti-realists and intuitionists would have it – to the well-defined compass of

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knowable results, provable theorems, verifiable (or falsifiable) conjectures and so on, then quite simply there would be no possi-bility of progress since no making sense of the basic claim that thought has advanced and may yet advance further precisely on account of the gap that might always turn out to exist between present-best knowledge and objective truth.

That gap is the space within which – according to Badiou – there can be seen to have transpired all the most decisive, that is, scientifically crucial or politically transformative episodes of change since the idea of progress first took hold and came to define what should count as genuine (not merely random or rationally under-motivated) change. Yet as regards the context of discov-ery it is still his leading claim that the methods and procedures of logical thought can only have gone so far, since beyond that limit any instance of progress which now appears rationally intelligible must at the time have required a great investment of fidelity to various strictly hypothetical or speculative theses. Thus ‘[g]eneric truth suspended from an event, it is part of the situation which is subtracted from established language, and its form is aleatoric, for it is solely an index of existence and not a knowledgeable nomination’ (p. 353). However it would be alto-gether wrong to suppose that the term ‘aleatoric’ should here be taken as referring to a realm of purely chance occurrences or as collapsing any remnant of the standard analytical distinction between context of discovery and context of justification. To be sure, the generic truth-procedure ‘is supported uniquely by the zeal of citizen-militants, whose fidelity generates an infinite truth that no form, constitutional or organizational, can adequately express’ (p. 353). Yet Badiou is quite clear that this has to do with the motivating energies, passions and commitment of those whose unswerving dedication to the task is required in order to carry such projects through, and not with the truth-conditions that decide whether or not any such project will at length turn out to have been on the right track. In other words – to repeat – he is by no means rejecting the ‘two contexts’ principle, even if the context of discovery acquires a more prominent role through Badiou’s stress on the under-determination of rational theory choice or the extent to which political decisions and commit-ments are likewise undertaken in the absence of fully adequate justificatory grounds.


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It is mainly on this account that he credits Rousseau as the thinker whose peculiar ‘genius’ it was ‘to have abstractly circum-scribed the nature of politics as generic procedure’ (p. 353). Rousseau had the sagacity to grasp the formal as well as the urgently political necessity of respecting those crucial distinc-tions between power and right, decree and law, the general and the particular wills, or again – in Badiou’s terminology – between the state with its power (and de facto right) to determine what belongs or doesn’t belong according to the authorized count-as-one and whatever in the present ‘state of the situation’ indiscern-ibly yet none the less decisively marks the disparity between inclusion and belonging. What set certain limits to this notable advance – so Badiou concedes – was Rousseau’s continuing engagement with a ‘classical approach’ that took the issue of sovereignty as a central theme, and which thereby foreclosed any further thinking-through of those radical implications that would otherwise have followed (or been seen to follow) by the strictest logical necessity. So it was that Rousseau retreated to the posture of ‘consider[ing] – albeit with paradoxical precau-tions – that the majority of suffrages was ultimately the empiri-cal form of this legitimacy’ (p. 353). In so doing he effectively left room for that whole range of more-or-less restrictive compro-mise formations, from king-in-parliament to the various current modes of ‘social-democratic’ thinking, that have always placed firm limits on the scope for any genuine, that is, properly inclu-sive exercise of popular will. Thus Rousseau’s legacy to modern thought, as Badiou understands it, is the question: ‘what is it that distinguishes, on the presentable surface of the situation, the political procedure?’ (p. 353).

This Meditation closes with the outline of an answer to that question, namely, that ‘[p]olitics is, for itself, its own proper end; in the mode of what is being produced as true statements – though forever un-known – by the capacity of a collective will’ (p. 354). But his answer clearly raises further issues concerning both the means by which thought might be expected to tran-scend that limit to the radicalism of Rousseau’s Social Contract and the claim, so central to Badiou’s entire project, that this might be accomplished through a procedure analogous to (even identical with) those practised in mathematics, logic and the for-mal sciences. Making good this claim will be Badiou’s enterprise

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inthe two remaining Meditations of Part VII, ‘The Matheme of the Indiscernible: P. J. Cohen’s strategy’ and ‘The Existence of the Indiscernible: the power of names’. His main purpose here is to explain in formal but practically relevant terms how it might (indeed must) be possible for truth to surpass the limits of knowledge, or – what amounts to almost (not quite) the same thing – for thought to surpass the limits of consciousness. This in turn requires that Badiou demonstrate how the indiscernible can exist within some given situation and exert a singularly powerful, disturbing and thought-provocative force upon those of a sufficiently open-minded or receptive disposition even though their residual attachment to prevailing norms of epis-temic warrant prevents that awareness from attaining adequate expression as a matter of conscious or articulate grasp. Thus the task of these two Meditations is to gather up the themes, mathe-matical and political, that have been the main focus of his inter-est so far and give them a more theoretically precise or rigorously formalized treatment.

4. Truth beyond knowledge: indiscernibility, generic extension

I trust – on the basis of my commentary up to this point – that the reader will have some idea of what is involved in this formal procedure. Briefly put, it is a logical proof of the claim that there exist certain ‘quasi-complete situations’ that are marked by cer-tain anomalous features – gaps, absences, conceptual defects, lacunae, failures of representation, paradoxes of self-reference and the like – which are presently unknown or unrecognized yet none the less real and potentially open to discovery through some future advance that is already implicit, latent or prefigured in the current ‘state of the situation’. That is, we are here work-ing once again with a ‘subtractive’ ontology, one that delimits the realm of substantive or plenary being in order to emphasize the strictly evental (hence ontologically fugitive) character of truth. What Badiou wishes to formalize in these sections of Being and Event is the precise sense in which ‘[t]he generic and indiscernible multiple is in situation; it is presented despite being subtracted from knowledge’ (p. 355). Cohen’s set-theoretical advance made this possible, Badiou proclaims, since it had the effect of ‘authoriz[ing] the existence of the result-multiple of the


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generic procedure suspended from the event, despite it being indiscernible within the situation in which it is inscribed’. He then goes on to explain at greater length – in 30 pages of densely argued and highly demanding exposition – how thought can make room, via these concepts of the generic and indiscernible, for the advent of truths that as yet lie beyond the compass of achieved (or achievable) knowledge yet can even so be felt to exert a power of orientation that draws its directive capacity from those very defects in our present state of understanding.In order to bring this about, Badiou writes, we shall need to ‘install our-selves in a multiple which is very rich in properties (it “reflects” a significant part of general ontology) yet very poor in quantity (it is denumerable’) (p. 356). In other words, this choice is such as to maximize the content or the ontological range and grasp of any such speculative venture onto thought-experimental ground while at the same time keeping that venture within conceptually man-ageable bounds by restricting its order of multiplicity to the min-imum required for investigation of the properties concerned.

‘Inside this fundamental situation’, Badiou continues, ‘we will define a procedure for the approximation of a supposed indiscern-ible multiple’, one that at this stage intrinsically eludes nomination or specification (since of course its properties are unknown) and therefore has to be baptised by a ‘supplementary letter’ which as yet has no assignable referent. In the standard, Cohen-derived notation this multiple is signified by G (for generic), whereas in Badiou’s preferred symbolism it is rendered – ‘due to a predilec-tion whose origin I will leave the reader to determine’ – by the female marker . His point is that this ‘extra signifier’ stands in for whatever is unknown or unknowable – whatever is not pre-sented in the current situation – although its absence is such as to register by way of the disturbances, tensions, and aporias which it induces rather than existing merely in a privative or negative mode. That is to say, it is the ‘ontological transcription of the supernumerary nomination of the event’, or – to adopt a phrase from Lacan and thus look forward to Badiou’s Part VIII Medi-tations on ‘Truth and the Subject’ – the ‘hole in being’ through which there may emerge, albeit on rare occasions, the first elusive sign of that which exceeds the cognitive grasp of any pre-established or existent ontology. So this project quite liter-ally has its work cut out in advance: that of exploring what

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will necessarily have been the case with regard to the truth-proce-dure in question granted the generic (indiscernible) status of the multiple taken to provide that procedure with its as-yet unknown warrant or justification. What should then emerge is that the multiple (‘very rich in properties’ but ‘very poor in quantity’) from which this speculative venture started out will itself supply the elements that make up the ‘substance-multiple of the indis-cernible’, since the latter will constitute a part of the former, that is, a constituent veritably included within it even though not veri-fiably belonging to it by any proof procedure or mode of knowl-edge currently at our disposal.

Badiou is quite aware that this will strike many readers – especially those of an anti-realist or intuitionist persuasion – as implausible to the point of downright perversity. After all, they will say, how could we be in a position to assert the existence of truths that generically transcend the utmost bounds of present knowledge or formal provability? Nor will these objectors be much impressed by his claim that such truths are somehow latent or implicit in the very gaps that structure the relation between knowledge and ignorance in our present state of understanding. For that claim rests in turn on Badiou’s central thesis – one that goes clean against the anti-realist grain – concerning not only the recognition-transcendent character of certain truths in the formal sciences but also the capacity of thought to register such truths, so to speak, in absentia through their disturbing or anomaly-inducing effect on the discourse of present knowledge. This second line of argument is likely to perplex even those in the mainstream analytic community who would count them-selves realists according to the usual definition, that is, subscri-bers to the first (objectivist or, in this particular context of debate, avowedly Platonist) line. However it is Badiou’s conten-tion that one cannot have the one without the other since, quite simply, there is no making sense of any version of the Platonist-objectivist case that doesn’t find room for the capacity of mind to outrun its existent cognitive limits in grasping the possibility of truths that themselves require just such a stretch in its powers of exploratory-investigative thought. After all, it is a mainstay of anti-realist arguments that objectivist realism inher-its all the problems of a purebred Platonist metaphysics insofar as it places truth in a realm of absolute ideal objectivity to which


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per definiens we mere human knowers cannot possibly have access by way of any epistemic or cognitive route at our crea-turely disposal. What Badiou is thus seeking to explain – through a process of argument none the less rigorous for its partly specu-lative character – is how this much-exploited gap between truth and knowledge can be seen as a source of dialectical tension and thereby a means of otherwise unattainable advances in mathe-matics and other disciplines.

Hence his invocation, at this point, of Gödel’s famous incompleteness-theorem to the effect that any formal system rich enough to generate the axioms of (say) elementary arithmetic or the first-order predicate calculus in logic will of necessity contain at least one axiom that is not provable in terms of the system itself.18 Gödel himself took this as a strong vindication of mathematical Platonism since – as he argued – how else could one account for the mind’s capacity to follow the proof of that theorem (i.e. to grasp its truth or demonstrative warrant) if its purport was precisely to deny that any formal proof could be carried through to decisive or probative effect?19 Only by suppos-ing the existence of a faculty able to transcend the commonplace limits of such reasoning could thought lay claim to a grasp of truths – including paradoxical truths like that of Gödel’s the-orem – that yielded to no available means of formal or purely axiomatic-deductive reasoning.20 Thus there is clearly a sense in which Badiou’s strong claims for the truth-tracking power of mathematical thought (i.e. for its capacity to outrun the scope of present-best knowledge through a grasp of those symptomatic gaps or aporias that mark its formal incompleteness) can be seen as closely analogous to Gödel’s case for mathematical Platonism. All the same it would be highly misleading – for reasons that Badiou is careful to explain – if one pushed this analogy too far and presumed to capture his ‘philosophy’ of mathematics (a phrase that he likewise very forcefully abjures) in Gödelian or purebred Platonist terms. For that reading would ignore not only his insistence on the intra-worldly character of even the most (apparently) abstruse or abstract mathematical researches but also his commitment to giving an account of mathematical disco veries, advances or stages of progressive knowledge-acquisition in such a way as to respect both their genesis or historically specific condi-tions of emergence and their structure as that which intrinsically

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transcends any specification in such terms. Moreover, it would ignore his repeated point that advances of the kind exemplified in the work of set theorists from Cantor to Cohen can only be explained through a grasp of those particular thought-processes whereby a range of truths that had hitherto eluded the best efforts of enquiry were at length brought within the compass of attain-able knowledge or formal proof.

This in turn demands an active, that is, mathematically engaged understanding of how those thinkers typically responded when they first became aware of certain problems or anomalies which as yet lacked any adequate means of formal articulation, let alone any adequate treatment that would ‘turn paradox into concept’. To this extent at least Badiou agrees with those philo-sophers who take mathematical knowledge to entail something more – to require a far greater depth of jointly conceptual-analytic and creative-exploratory involvement – than could ever be accounted for in the quasi-behaviourist terms that have characterized so much discussion of this topic in the wake of Wittgenstein’s provocative but highly misleading remarks about ‘following a rule’.21 On the other hand, as I have said, he is just as firmly opposed to any line of thought within philosophy of mathematics that would adopt a phenomenological (Husserlian) perspective according to which mathematical truths are conceived as ‘absolute ideal objectivities’ that must ultimately depend – for their discovery, transmission and subsequent repeated ‘reactiva-tion’ – on those human (albeit transcendental) structures of thought that alone give access to them.22 Still less would he sub-scribe to any version of the psychologistic or subjectivist approach that Husserl was so anxious to disavow as a result of Frege’s having mistakenly laid that charge at his door.23 What is most distinctive about Badiou’s approach is its managing to find sufficient room for the subject as actively and indispensably involved in every such valid truth-procedure, while at the same time managing to steer well clear of subjectivism in any, no matter how conceptually refined or phenomenologically sophis-ticated a form. And what enables him to situate his thinking squarely on ground that has not been trodden into ruts by adher-ents of either party is Badiou’s way of redefining the subject such that its very conditions of existence – those under which it embarks upon its own sui generis projects of enquiry, discovery


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or creative exploration – are specifiable precisely in terms drawn from those various disciplinary contexts.

That is to say, we have to do here with a radically heterodox conception of the subject which conserves its powers of ratio-nality, inventiveness, agency, decision-making and ethical com-mitment yet treats them as attributes of this or that specific undertaking which in turn provides the subject with his or her sense of purposive selfhood or identity. So it is not so much a question – as in many postmodernist or post-structuralist debates on this topic – of ‘what is left of the subject?’ but rather a more exactly framed question concerning the subject’s role in taking thought beyond some limit-point of paradox or impasse to a more advanced stage where those obstacles at last fall away and open up hitherto unforeseen prospects along with further, as yet unpredictable challenges. Badiou goes on, in Meditation Thirty-Four, to give a detailed and meticulously reasoned formal account of how such advances in knowledge – or abridgements of the gap between knowledge and truth – may best be understood to have come about. What it involves is the thought-experimental ploy of ‘adding [i.e., the indiscernible element] to the fundamental sit-uation S’, as a result of which ‘we would have a new situation to which would belong’ (p. 375). This is a situation that can then be referred to as the ‘generic extension of S’ [symbolized S( )] since it has to be thought of as somehow implicit in the earlier, less advanced situation and hence as potentially intelligible to the denizens of that situation even though beyond their powers of fully achieved (i.e. conscious) epistemic or cognitive grasp. Of course there is something deeply problematical about any such claim, namely that it posits a capacity of thought to transcend or surpass the scope of its own currently most advanced state of knowledge. Thus ‘[t]he extreme difficulty of the question lies in this “addition” having to be made with the resources of S; other-wise it would be unintelligible for an inhabitant of S’. So much would appear self-evident in commonplace epistemological as well as metaphysical and even, it might be thought, straightfor-wardly logical terms. Yet, as Badiou continues, on his own reck-oning ‘~( ∈ S) [i.e., it is not the case that the indiscernible belongs to the multiple S]’, in which case ‘[h]ow can any sense be made of this extension of S via a production that brings forth the belonging of the indiscernible which S includes?’ (p. 375).

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In short, Badiou’s chief concern here is to think his way through and beyond the dilemma of objective truth versus humanly attain-able knowledge that has hobbled so many branches of philoso-phy – especially philosophy of mathematics and science – during the past century and more. ‘The solution’, he writes, ‘consists in modifying and enriching not the situation itself, but its language, so as to be able to name in S the hypothetical elements of its extension by the indiscernible, thus anticipating – without pre-supposition of existence – the properties of the extension’ (p. 375). What this amounts to, simply paraphrased, is the idea that a reasoner situated in S and possessing only the resources avail-able within that same situation will none the less be able to assert with good warrant: ‘If there exists a generic extension, then this name, which exists in S, designates such a thing within it’ (p. 375). Since from her point of view the generic extension is purely ‘void’, that is, indexed to a truth-procedure that has yet to be assigned any definite content this yields no self-contradictory entailment of the type instanced above, that is, no claim (such as would immediately draw the fire of anti-realists) that we can somehow get to know truths that exceed the compass of our best knowledge. That she possesses the concept of genericity, even if in this maximally abstract or non-content-specific form, is suffi-cient to ensure that she will grasp not only the in-principle pos-sibility of such truths but also – what Badiou is here seeking to demonstrate – their existence within situation S as the indiscern-ible yet none the less active source of those various anomalies, dilemmas, paradoxical entailments, and so forth, that do fall within the scope of present knowability. Thus the argument offers a strengthened because more detailed, explicit and for-mally developed version of the familiar realist case: that only by supposing the existence of objective, verification-transcendent truths can we account for advancements in knowledge or pro-duce a viable concept of progress with respect to any field of enquiry.

However there is plainly a limit to the reach of this hypotheti-cal (i.e. only-later-if-at-all-to-be-verified) claim, proposed as it is from the epistemic standpoint of one who commands, as yet, only such an indirect or purely conjectural access to truth. This shortfall is made up, on Badiou’s submission, by introduc-ing a second standpoint, that of the ‘ontologist’, who is conceived


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as looking back upon situation S from the perspective of a more advanced state of knowledge wherein those erstwhile indis-cernible truths will have come clearly to view, along with their role in producing the various symptomatic strains and ten-sions that bore oblique witness to what once lay beyond the bounds of attainable knowledge. ‘From the outside’, Badiou suggests,

the ontologist will realize the hypothesis, because he knows that a generic set exists. For him, the referents of the names, which are solely articles of faith for an inhabitant of S, will be real terms. The logic of the development will be the same for whoever inhabits S and for us, but the ontological status of these inferences will be entirely different: faith in transcen-dence for one (because is ‘outside the world’), position of being for the other. (pp. 375–6)

This fiction (more precisely: this thought-experiment or exploratory hypothesis) concerning the ‘ontologist’ blessed with benefit of hindsight is one that may well strike the reader as somewhat fanciful or far-fetched. However it goes no fur-ther out on a speculative limb than those analytic thinkers who deploy (for instance) the idea of an ‘omniscient inter-preter’ by way of addressing various issues in epistemology and philosophy of language.24 What Badiou claims to show by a different though closely related procedure is how progress may plausibly be thought to come about, in mathematics and other fields, through an anticipatory and hence conjectural yet none the less truth-conducive grasp of that which eludes any currently available method of proof or demonstration. More-over, this indiscernible element can still leave its mark on situ-ation S (or the present context of enquiry) in ways that may decisively affect the given situation even if their true import will not become clear until the occurrence of that very advance whose prospect – whether imminent or more remote – they can then be seen to have signalled. At which point we proceed to the final part of Being and Event where we shall find Badiou’s most sustained and focused engagement with the question of the subject in relation to issues of knowledge and truth.

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Discussion points

What do you take Badiou to mean by his (to say the least) heterodox definition of the subject as ‘that which decides an undecidable from the standpoint of the indiscernible’?

What are Badiou’s main reasons for holding not only that truth might always exceed the compass of present-best know ledge but also – paradoxically – that our grasp of truth can sometimes out-run the limits of conscious or reflective understanding? Do you find those reasons cogent or persuasive?

PART VIII. FORCING: TRUTH AND THE SUBJECT. BEYOND LACAN

1. Truth, indiscernment and the impasse of being

It is at this late stage, having pressed ontology to the point of exposing its inherent limits, that Badiou is now able to focus attention on the topic that has really been his guiding thread and motivating interest all along, namely the ‘theory of the subject’ as that which alone offers the means to understand how thought is on occasion able to surpass and redefine those limits. The upshot – again with Gödel’s results in mind, as well as every-thing that Badiou has shown with regard to the chronically para-dox-prone yet also paradox-driven development of set theory – is his claim that even mathematics, logic and the formal sciences have need of such a theory if they are seeking to account for their own history of advances to date and, more specifically, what this reveals concerning the complex dialectical relationship between structure and genesis. Meditation Thirty-Four brought us up against a further set-theoretically induced (and Gödel-rein-forced) aporia whereby ‘[t]he coherency of ontology – the virtue of its deductive fidelity – is in excess of what can be demon-strated by ontology’ (p. 360). Yet it is by way of truth-procedures whose characterizing mark is that of unswerving fidelity to the project in hand that thought can turn this deficit to advantage by acknowledging that which leads enquiry beyond any outlook of settled adherence to received, orthodox or quasi-foundational (ontologically secure) modes of understanding. Which is also to say that ‘what is at stake here is a torsion which is constitutive of the subject: the law of a fidelity is not faithfully discernible’ (p. 360). Since truth must be conceived as always potentially


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surpassing the limits of any given state of knowledge, so likewise the subject – as seeker, discoverer, conveyor or faithful ‘militant’ of truth – must always be accorded a central role in the process by which knowledge encounters the limits of whatever it is pres-ently able to discern, and hence the need to envisage the exis-tence of truths that lie beyond its epistemic grasp.

So it is that Badiou embarks upon Part VIII with the forthright statement: ‘I term subject any local configuration of a generic procedure from which a truth is supported’ (p. 391). What should have become clear by now is that ‘subject’, in Badiou’s usage, has little in common with any understanding of the term bequeathed by thinkers in any of the main philosophic lines of descent whether rationalists, empiricists, phenomenologists, depth-ontologists, hermeneutic adepts or even those (Foucauld-ians and post-structuralists) who purport to have theorized their way far beyond such delusory humanist notions.1 A subject is neither a substance (since ‘if the word substance has any mean-ing it is that of designating a multiple counted as one in a situa-tion’, and Badiou claims to have established ‘that the part of a situation constituted by the true-assemblage of a generic procedure does not fall under the law of the count of the situation’) nor a void point (since ‘the proper name of being, the void, is inhuman, and a-subjective’) (p. 391). Rather it is the name – and Badiou is a reso-lute nominalist in this respect at least – for whatever goes beyond those conceptions of subjectivity that equate it with the deliver-ances of sensory experience, or the a priori self-evidence of tran-scendental subjecthood, or the likewise self-evident though more worldly (i.e. more perceptually or physically rooted) subject evoked by the procedures of phenomenological reflection. Above all Badiou is concerned to decouple his thinking from any notion of the subject defined, in broadly Kantian terms, as the site or locus of a union achieved between the manifold of sensory experience and the synthesizing power of a mind that somehow bridges the stark dichotomy of intuition and concept. For ‘if the word “experi-ence” has any meaning, it is that of designating presentation as such’, whereas – on Badiou’s account – ‘a generic procedure, which stems from an evental ultra-one qualified by a supernumerary name, does not coincide in any way with presentation’ (p. 391).

Thus his usage of ‘subject’ marks Badiou’s distance not only from those who have sought to uphold that concept, in whatever

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philosophic guise, but also from the presently large company of sceptics who have sought to reveal it as in truth nothing more than an ideological fig-leaf or alibi.2 This latter line of thought is one that could only strike him as yet another variant of that same ‘linguistic turn’ across the social, human and even certain branches of the natural sciences to which he has been firmly opposed on account of its strong if covert idealist leanings as well as the strain of linguistic determinism implicit in much of this thinking. For him, such ideas – whether of a structuralist, post-structuralist, Wittgensteinian, Foucauldian or Heideggerian depth-hermeneutic provenance – are deficient to the point of self-refutation. What they cannot explain is how thought could ever accomplish the passage from a state of knowledge defined by its partial character, that is, by its failure to recognize the exis-tence or latency within it of one or more as-yet indiscernible (since knowledge-transcendent) truths to a more advanced state whereby those truths would enter the domain of fully achieved or operative concepts. If language is taken as the basis, framework, horizon or ground of subjectivity – according to some particular version of the thesis – then it leaves no room for those truth-pro-cedures which lend themselves to formal specification after that advance has come about but which involve something more in the original ‘context of discovery’ than could ever be captured by any such strictly ex post facto reckoning. Thus ‘subjectivization is that through which a truth is possible, [since] it turns the event towards the truth of the situation for which the event is an event’ (p. 393). It is this irreducibly subjective dimension of truth – one that is oriented more towards the ethical idea of ‘truthfulness to’ – which Badiou takes as having an essential part to play in any adequate account of our know ledge of the growth of knowledge. When coupled with his other chief items of conceptual innova-tion it opens up a space of inventive possibility within which thinking is able to conceive how truths may emerge through just such a process of transforming the indiscernible, via the event of its first (unrecognized) occurrence, into that which at length acquires the status of proven or accredited knowledge.

This is not to say that the subject is capable of knowing, com-prehending or accessing truth in such a way as to attain what Spinoza – in classical rationalist style – defined as ‘adequate ideas’, that is, concepts that exhibited a perfect grasp of the


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object (physical or abstract) to which they referred.3 For there is always, according to Badiou, a disparity or lack of adequation between subject and truth, or what the subject is able to know from his or her particular, situated standpoint and what they would (counterfactually) know from that mythical god’s-eye perspective wherein the truth/knowledge distinction would be finally annulled. ‘Because the subject is a local configuration of the procedure, it is clear that the truth is indiscernible “for him” – the truth is global’ (p. 396). And again, ‘a subject, which realizes a truth, is nevertheless incommensurable with the latter, because the subject is finite, and the truth is infinite’. By now we should be under no illusion that Badiou is here speaking loosely or ges-turing towards some ineffable notion of the infinite. Rather he is deploying that term in just the sense and with just the range of specific (if paradoxically mind-stretching) conceptual resources that it first acquired through Cantor’s set-theoretical revolution and which thereafter proved intensely productive of advances in mathematics, logic and other fields. Indeed it is precisely on this account – through this tendency of thought, when engaging with the infinite, to find itself repeatedly brought up against aporias which force it to confront and surpass its pre-existent limits – that Badiou is best able to articulate his case for the knowledge-transcendent character of truth and hence the need to reformulate our notion of the subject in keeping with this principle.

Thus ‘the subject, being internal to the situation, can only know, or rather encounter, terms or multiples presented (counted as one) in that situation’, while conversely ‘a truth is an un-presented part of the situation’ (p. 396). In which case there is a clear structural homology between the surplus of truth over knowledge, the excess of the subject’s capacity for thought over that which presently gives itself to being known, and the way in which set-theoretical conceptions of the infinite from Cantor to Cohen have typically shown that capacity at work to the utmost paradoxical or mind-stretching effect. This also goes to reinforce Badiou’s claim about the gross insufficiency of language – or any doctrine that purports to maintain the priority of language vis-à-vis thought – when it comes to offering an adequate, that is, a philosophically cogent, historically informed and properly for-malized account of how such advances could have come about. For if truth is indeed, as Badiou asserts, an ‘un-presented part of

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the situation’, then ‘the subject cannot make a language out of anything except combinations of the supernumerary name of the event and the language of the situation’, and it is therefore ‘in no way guaranteed that this language will suffice for the discernment of a truth [that is] indiscernible for the resources of the language of the situation alone’ (p. 396). What this amounts to, in fact, is a formal refutation not only of the various linguistic-constructivist doctrines cited above but also of any philosophical approach to the formal sciences which takes the anti-realist/intuitionist line of equating truth tout court with epistemic warrant, or which counts the idea of verification-transcendent truth a world well lost for the sake of attainable knowledge. Badiou sees the two fallacies as closely, even inseparably bound up together, with the over-valua-tion of language serving to lend greater credence or plausibility to the overemphasis on human knowledge as the arbiter of truth for all practical (humanly relevant) purposes.

On the other hand he is very far from dismissing that whole range of developments in philosophy, political theory, psycho-analysis and other branches of the social and human sciences that arose very largely from the French ‘re-discovery’ of Saussurean semio-linguistics in the 1960s and 1970s.4 As we have seen, his work bears the deep imprint of two of those nominally ‘struc-turalist’ thinkers – the Marxist theoretician Louis Althusser and the psychoanalyst Jacques Lacan – both of whom placed lan-guage (or a certain theoretically elaborated model of language) at the heart of their respective projects and each of whom did much to propagate the claim that thought was ultimately subject to – rather than merely mediated by – certain deep-laid linguistic or discursive structures of representation.5 Thus where Lacan draws on Saussurean concepts and terms for his basic topologi-cal account of the unconscious as very literally structured ‘like a language’, Althusser similarly draws on Saussure – and on Lacan’s psychoanalytic re-working of Saussure – for his own account of how subjects are ‘interpellated’ by some dominant ideology and thus recruited to its service through a structural logic analogous to that which assigns them their proper, linguis-tically defined role within social discourse. All of which might seem to place them squarely within the constructivist camp and hence squarely at odds with everything that Badiou urges against that whole school of thought. However, what sets these projects


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decisively apart from the wider turn towards language or dis-course as the bottom-line of enquiry is the commitment, in Althusser and Lacan alike, to the existence of that which stub-bornly resists, eludes or exceeds the structures of imaginary or ideological misrecognition whereby the subject lives out his/her relation to an always fugitive, never fully accessible reality. No doubt it is hard – as many commentators will attest – to assign any definite locus or conceptual import to Lacan’s in itself highly elusive notion of ‘the real’, and just as hard to see how Althuss-er’s immensely complex (some would say scholastically over-wrought) ‘materialist’ theory could be said to escape the oft-repeated charge of conceptual-linguistic idealism.6 But the crucial point remains: that both Lacan and Althusser explicitly (indeed emphatically) make allowance for an ontological domain of the real, albeit very differently defined in each case, which must be thought of as lying outside and beyond the realm of subjective representation.

So there is nothing inconsistent about Badiou’s having recourse to certain elements of structuralist thinking – more specifically, the structuralist theory of the subject in relation to issues of ideo-logy, knowledge and truth – despite his principled rejection of the language-first doctrine in its wholesale form. He can take them on board without compromise since for him, as likewise for Althusser and Lacan, it is an error (and the source of much confusion in various fields of enquiry) to suppose that truth or reality are in any way dependent on the comprehension, uptake or epistemic grasp of the knowing subject. Thus ‘[b]eing the local moment of the truth, the subject falls short of supporting the latter’s global sum’ (pp. 396–7). Moreover, and especially (though not exclusively) with reference to the formal sciences, ‘[e]very truth is transcendent to the subject, precisely because the latter’s entire being resides in supporting the realization of truth. . . . The subject is neither consciousness nor unconscious-ness of the true’. With these last two denials Badiou can be seen to situate his thinking on the often fiercely contested ground between philosophy and psychoanalysis, ground to which Freud and Lacan in particular were anxious to stake their own territorial claims.7 By denying that the subject has conscious know ledge of truth he might seem to be siding with that strain of anti-philosophic (more specifically anti-Cartesian) thought

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that is so prominent a feature of Lacan’s work. Yet by following up with his denial of the claim that the subject has ‘unconscious’ knowledge of the true Badiou effectively puts psychoanalysis in its place as a relevant, useful, in some respects indispensable but not – as Lacan would doubtless claim – uniquely privileged discourse. For it is crucial to his project that truth should be conceived as always potentially eluding the compass of con-sciously accessible or self-aware knowledge but also as requiring an active and intensely focused activity of thought which cannot plausibly be located at the level of any unconscious (or even pre-conscious) process. Rather it involves a post-Cartesian concept of the subject – such as a great many French thinkers during the past century have sought to provide – which conserves the pre-eminence of critical reason against its manifold present-day detractors but works to detach it from the notion of transparent first-person epistemic access that has proved a chronic liability from Descartes down.8 If Lacan drew attention to its problems in an especially striking way – as with his famously riddling re-formulation of the cogito: ‘where I think “I think, therefore I am”, that is just where I am not’ – still Badiou holds out against any too complete, dogmatic or reductive assimilation of the sub-ject to a notion of the Freudian unconscious that, if pressed to its conclusion, would leave no room for rational thought or rationally-motivated agency.9

It is at this stage that Badiou undertakes his most detailed commentary on Cohen’s development of set-theoretical discourse through the closely interlinked concepts of ‘forcing’, the ‘generic’ and the ‘indiscernible’ which between them achieve – on his sub-mission – a notable advance not only in the field of mathematics but also (inseparably from that) in our thinking about the sub-ject. Meditation Thirty-Six (‘Forcing: From the Indiscernible to the Undecidable’) starts out from the central thesis of Being and Event, and one that Badiou takes as sufficiently established by this stage: that ‘[j]ust as it cannot support the concept of truth (for lack of the event), nor can ontology formalize the concept of the subject’ (p. 410). If there is something odd, to received ways of thought, about this notion of ‘formalizing’ the subject – producing a theory that would account precisely for its mode of intervention in this or that specific truth-procedure – we should none the less be willing at least to entertain it as a means of entry


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to Badiou’s otherwise thoroughly alien conceptual domain. For it is just his point with respect to Cohen that the concept of forcing captures exactly that paradoxical juncture of truth as what obtains quite independently of human understanding or epistemic grasp and truth as what transpires through a certain, historically dated event which in turn comes about only on con-dition of its being the result of some humanly instituted practice of enquiry. What Cohen demonstrates, according to Badiou, is that ‘the existence of a subject is compatible with ontology’, thus ‘ruin[ing] any pretension on the part of the subject to declare itself “contradictory” to the general regime of being’ (p. 410). However this is clearly not to suggest that the subject be con-ceived as existing within the ontological domain in such a way that its various thoughts, actions, commitments, truth-procedures, investigative projects, and so forth, necessarily conform to the scope and limits of that pre-existent domain. Nothing could be further from Badiou’s insistence on the radical disjunction between being and event, along with his thesis – the chief claim advanced in this final part of the book – that it is solely through the subject’s capacity to think through and beyond those limits that an event (in the sense of that term here in question) can occur with what seems, at the time, a wholly unpredictable and hence revelatory force. Thus the subject may indeed be ‘compat-ible with’ or not ‘contradictory to’ the ‘general regime of being’, yet must none the less be thought of – if Badiou’s central claim is to make any sense – as existing always in a state of potential conflict or tension relative to that regime. After all, his main pur-pose in Being and Event is precisely to explain (or, as he would say, so far as possible to ‘formalize’) the process by which this inherent disparity on occasion gives rise to some epochal event in the history of mathematical, scientific, political or artistic-creative thought.

What enables this to occur is the existence of certain validity conditions that are known to our imaginary friend the External Ontologist whose facilitating role I described in my commentary on Part VII. Those conditions form the link – recognizable only in the wake of some such crucial event – between the earlier situ-ation wherein they are taken to ‘control’ what should count among the range of yet-to-be-validated theorems, hypotheses or candidates for truth and the later situation – albeit unknowable

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in advance to anyone bar the Ontologist – where they count among those whose truth has been established by procedures belonging to the generic extension. (To repeat, this device of appealing to a kind of super-knowledgeable party or ‘omniscient interpreter’ is one that, although regarded with suspicion by some philosophers, has all the same been made to do useful work by others, Donald Davidson among them.10) Such reasoning finds its clearest exemplification in Cohen’s book Set Theory and the Continuum Hypothesis where he puts it to work in treating one of the great unresolved problems that dogged Cantor during his later years.11 This was the question – in brief – as to whether there existed an infinite set with cardinality (or ‘size’) between that of the integers, that is, the whole or counting numbers and that of the ‘larger’ infinity of real numbers which intuitively marked the next distinct range of numerical values or discrete order of magnitude. Cantor hypothesized, without being able to prove, that there could not be any such intermediate set, to which Gödel then added a formal proof that Cantor’s hypothesis could be accommodated without producing any contradictions within the currently accepted (Zermelo-Fraenkel) axioms of set theory.12 In the standard notation, if ℵo (Aleph-null) signifies the ‘small-est’ order of infinity, that of the integers, and c signifies the con-tinuum of real numbers then according to Cantor’s hypothesis ℵ1 = c (i.e. ‘nothing in between’). Cohen’s contribution – which, as we have seen, Badiou makes absolutely central to his own project of thought – was to offer a formal demonstration that the continuum hypothesis could be rejected without giving rise to any recalcitrant or logically unacceptable result.

Thus the joint upshot of Gödel’s and Cohen’s proofs was to show that the validity or otherwise of Cantor’s hypothesis could not be decided by any self-sufficient or formally adequate proof-procedure but was crucially dependent on the system of axioms (e.g. Zermelo-Fraenkel) deployed in any given case. Nevertheless mathematicians are mostly convinced, as was Gödel, that the hypothesis is false and that there can (indeed must) exist one or more such classes of infinite number between the order of inte-gers and the order of reals.13 Hence what Badiou calls the ‘abso-lutely capital’ result of Cohen’s discovery for mathematics, logic and – not least – for his own work on the scope and limits of a formalized ontology which, across a range of subject-domains,


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reveals the essentially disjunct relation or lack of any ultimate common measure between the two realms of being and event. Its significance has to do mainly with its showing to powerful effect how a formal (axiomatic-deductive) truth-procedure may none the less leave room for that crucial element of choice that distin-guishes a faithful following-through of those further, as-yet unknowable consequences that may at length turn out to furnish its probative warrant. Badiou’s point is that the Cohen result will be seen to possess just such exemplary force if taken together with Cantor’s and Gödel’s arguments and with various later, post-Cohen interventions that have kept the continuum hypoth-esis in play as one that can be known to possess an objective truth-value (true or false) even though the issue cannot be resolved either way for lack of any clinching, formally adequate proof or disproof.14 This is why it is often discussed in conjunc-tion with the Axiom of Choice (quick reminder: ‘given a set, there exists a set composed exactly of a representative of each of the [non-void] elements of the initial set’; p. 499). As we have seen, this is another main component of set-theoretical thought whose validity is unproven – or its truth-value strictly unknown – even though it has been subject to a great deal of formal elaboration and is taken as an axiom in good standing by most (not all) present-day mathematicians and philosophers. For Badiou it figures, along with the continuum hypothesis, as a paradigm case of how thinking may be drawn into new and productive regions of enquiry through its orientation towards certain truths that elude its present-best or even, at the limit, its utmost achiev-able powers of conscious or deliberative grasp.

Here it is worth noting that of the three main options available to set theorists regarding the Axiom of Choice – unqualified acceptance, faute de mieux acceptance and outright rejection – it is the anti-realists, constructivists or intuitionists who most often adopt option three since their outlook precludes any notion that the axiom might be objectively true or false quite aside from our happening not to possess the requisite means of proof.15 To a realist (or Platonist) way of thinking, conversely, the axiom is both well-formed and consistent with such a large body of estab-lished set-theoretical lore as to render it a fit candidate for objec-tive truth or falsity whatever the limits of our knowledge in that regard. Once again, and despite his express reservations on the

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point, Badiou can be seen to align himself squarely with the real-ist camp insofar as he locates the chief spur to mathematical cre-ativity, inventiveness and progress in precisely that ever-present possibility of a gap between knowledge and truth. Thus the next portion of Being and Event (pp. 412–16) is devoted to a detailed and formalized account of those conditions under which the sub-ject – again in Badiou’s highly distinctive sense of that term – quite literally comes into being with the advent of a previously indiscern-ible truth that now exists within the bounds of conceptual possibil-ity and, beyond that, of potential proof. At such a moment, she finds herself taken up by the demand of fidelity to a truth-event whose further exploration or progressive unfolding it is thenceforth her task to pursue. This involves procedures that begin from the stage of its ‘indiscerned’ (i.e. its unrecognized yet somehow latent and subliminally marked) existence in the earlier situation and which then lead on, via the generic extension, to the point of estab-lishing its formal validity or its claim to have passed from the realm of conjecture to that of veridical statement. ‘A Subject alone pos-sesses the power of indiscernment’, since the truth-event can only occur through the ‘impasse of being’ brought about by this capital-ized Subject whose role it is to intervene decisively at some crucial juncture, reveal those hitherto unperceived signs of conceptual strain, and thereby become the agent of a forced but none the less subjectively willed since rationally motivated transformation.

That is to say, there is no validity – according to Badiou – in that deeply entrenched system of binary oppositions that draws a categorical line between objective and subjective orders of truth-claim, or on the one hand those whose conditions can be specified in purely formal or empirical terms, and on the other those that involve an irreducible reference to some particular knower in some particular, more or less knowledge-conducive context. Not that he is for one moment recommending the kind of post-disciplinary utopia envisaged by a thinker like Richard Rorty, one for whom the highest philosophical as well as more broadly cultural good would be a full-scale collapse or abandon-ment of all such hopelessly outmoded and irksome distinctions.16 On the contrary, as must be clear to any reader who has come this far with Being and Event, Badiou is a veritable sticker for the strictest, most demanding standards of specialist competence in those fields (whether mathematics and the formal sciences or


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the exegesis of challenging poetic texts) where such expertise is called for. Nor, as we have seen, is he in any way attracted to those currently widespread strains of linguistic-constructivist, depth-hermeneutic or strong-descriptivist thinking that would count such notions just a throwback to old, academically hide-bound habits of thought. For Badiou, this whole debate is just a ping-pong match between two equally misconceived notions of where (if anywhere) the requirements of formal or epistemic rigour leave off and those of subjective fidelity or hermeneutic tact should be thought to begin. Thus he rejects the very terms in which this question has typically been posed since the nine-teenth-century parcelling-out of territorial claims between the Geisteswissenschaften and Naturwissenschaften, or the human and natural sciences. However he does so not, like Rorty and other proponents of the full-strength hermeneutic turn, with a view to treating the latter on terms laid down by the former, but – quite the opposite – in order to insist that conceptual rigour has its necessary place on both sides of that notional divide.

2. Descartes and Lacan: the subject at stake

That his approach to issues in the philosophy of mathematics and logic lives up to this self-imposed requirement will I hope have become evident enough in the course of my commentary so far. But I should also want to claim that it is borne out, albeit more controversially, by those sections of Being and Event where Badiou extends these formal concerns into regions of enquiry (chief among them politics, art and love) that he considers no less demanding of treatment with the utmost conceptual precision. The first two categories (politics and art) have here received a fair amount of detailed attention through my account of what Badiou has to say about thinkers and poets – a distinction that he would surely reject out of hand – such as Aristotle, Hegel, Heidegger, Hölderlin, Mallarmé, Marx (not the subject of a named section but a constant point of reference) and Rousseau. The third subject, love, is one that he broaches mainly by way of psychoanalysis and, more specifically, through Lacan’s often riddling and opaque meditations on amorous, erotic, mathemat-ical, hermeneutic and structural-linguistic themes.17 As readers may have noticed – perhaps after having their interest piqued by some early passing mentions – it has figured only peripherally so

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far but comes very much into its own in Meditation Thirty-Seven, ‘Descartes/Lacan’, which concludes our (in every sense) eventful odyssey through this remarkable work.

One point on which he seeks to rectify a commonplace error concerning Lacan is the latter’s supposed out-and-out repudia-tion of Descartes, that is to say, his putative rejection of any idea that the cogito (first-person-singular thinking subject) might have some real, other than delusory role to play in our under-standing of what properly constitutes a truth-procedure. To be sure, the effect of Lacan’s prose with its constant veering between the labyrinthine and the cryptic – as well as his intensely (even obsessively) close-focused reading of certain passages in Freud – is often to suggest that he is out to create such thickets of oblique or devious signification that the rational, autonomous, self-con-scious subject of Descartes’ wishful imagining is played com-pletely off the field. Nor has there been any shortage of opponents convinced that all this is merely a charade, and that when Lacan speaks – in response to such claims – of the ‘tyranny’ exerted by an academic discourse of clear and distinct ideas he is really just putting on a charlatan’s display for the dubious benefit of credulous readers. For his disciples, conversely, Lacan has indeed shown that the subject (at any rate the subject con-ceived in Cartesian, Kantian or Husserlian terms) has undergone a radical ‘decentering’ or displacement in the wake of Freud’s great discovery, and can henceforth no longer be thought of as master in its own, rationally administered and consciously open-to-view house. To them there is no question but that he is right in interpreting Freud’s remark ‘wo es war, soll ich werden’ (‘where it/the id was, shall I/the ego be’) not as an injunction that the conscious mind should reclaim territory once occupied by the unconscious but rather as stating that the unconscious will always turn out to hold undisputed sway in those regions where the conscious mind self-deludingly thinks to impose its rule.18 Thus Lacan evokes passionate responses pro and contra to a degree unusual even by the standards of the strife-ridden ‘com-munity’ of psychoanalysis. Indeed his claims have been fiercely contested both within that community and among representa-tives of other disciplines (philosophy included) by whom they have often been denounced as nothing more than a species of rhetorical imposture.19


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It will be clear from what I have said so far that Badiou would repudiate any such charge with a forceful affirmation of his own belief that Lacan is most decidedly one of those thinkers who have left so deep and consequential a mark on the present-day human sciences as to make this situation wholly understandable. All the same Badiou makes sure to place a large distance between his own faithful yet critical approach to Lacan’s thought and the kinds of unquestioning doctrinal adherence – oddly coupled with far-out relativist, constructivist or anti-realist postures – that have characterized the uptake of Lacanian psychoanalysis among post-structuralists and others for whom it has often served as a handy means of mounting their assault on the claims and prerogatives of reason.20 Thus he opens this Meditation with a pointed reminder that ‘the Lacanian directive of a return to Freud was originally doubled: he says . . . “the directive of a return to Descartes would not be superfluous”’ (p. 431). For if Lacan can be read – and not without strong textual warrant – as recommending (indeed mandating) a break with that Cartesian idea of the autonomous, transparent, self-knowing subject that has caused such a deal of trouble for subsequent epistemology and philosophy of mind then it should none the less be clear how deep is the kinship – albeit the fraught and antagonistic kinship – between Lacanian psychoanalysis and that which it so strenuously seeks to repudiate. ‘What localizes the subject’, Badiou writes, ‘is the point at which Freud can only be under-stood within the heritage of the Cartesian gesture, and at which he subverts, via dislocation, the latter’s pure coincidence with self, its reflexive transparency’ (p. 431).

This is not merely to say that any idea stands to benefit from having some contrary idea to rub up against, or that progress comes about – in Hegelian fashion – through a constantly evolv-ing and self-transformative dialectic of opposed concepts, beliefs or ideologies. Rather it is to make the more substantive claim that reason, or the exercise of rational thought, is not an option but an inescapable necessity even – or especially – for those, like Lacan, who aim to reveal how much of the unconscious ‘pri-mary process’ is lost or disfigured in the course of its re-working into consciously accessible/expressible guise. In fact it is on just this latter point, that is, the priority granted to issues of language, expression or representation that Badiou takes issue not only with

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the line most often adopted by Lacan’s post-structuralist followers but also with certain prominent aspects of Lacan’s own work. After all, Badiou could scarcely rest content with the three main Lacanian proposals: (1) that reason is no more than a helpless ‘plaything’ of the unconscious, (2) that the unconscious is struc-tured as a language, and (3) that this language can best be con-ceived after Saussure as a system of differences or purely contrastive relations ‘without positive terms’. Taken together they go flat against everything that Badiou most emphatically maintains con-cerning the precedence of thought over language, or – more pre-cisely – the shortfall of any ‘language-first’ approach when it comes to explaining how thought can accomplish the kinds of inventive or creative as well as formal or conceptual advance that require a decisive break with existing (linguistically or culturally entrenched) modes of thought. This argument is likeliest to gain assent when applied to the domain of mathematics, logic and the formal sciences where it chimes with the widely held view – borne out by various items of expert testimony – that some of the most notable achievements have come about through processes far removed from any verbal or quasi-verbal train of reasoning. Thus it also consorts naturally enough with Badiou’s strong leaning in that direction and his focus on developments in set theory as a test-case example of the kind, albeit one that would be fiercely disputed by anti-realists, constructivists, some intuitionists and certainly all those who have gone along with the linguistic turn whether in its post-structuralist or its Wittgensteinian form.

It is therefore quite understandable that Badiou’s reading of Descartes should resist the somewhat facile pronouncements of those on both sides of the ‘analytic’/‘continental’ split who have announced an end to the Cartesian regime of ‘clear and distinct ideas’, along with its subject-centred epistemology and its failure to acknowledge the prior claims of language, commu-nity and culture. One point that needs stressing here is that the Descartes whom Badiou wishes to defend – or at least partially to vindicate – against those adversary legions is not so much the Descartes of the solitary, self-grounding cogito (a notion that he, like them, finds problematic to the point of nonsensicality) but Descartes the logician, mathematician, scientist and author of the Discourse on Method and Rules for the Direction of the Mind rather than the much better-known Meditations.21 This is


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a thinker who, despite his deep investment in that first-person privileged epistemology, none the less makes room for Badiou’s cardinal distinction between consciousness and thought. That is, Descartes himself offers the grounds for rejecting that delusive appeal to the cogito as ultimate bulwark against sceptical doubt and replacing it with the kinds of axiomatic truth-procedure that involve no such appeal since their validity conditions are entirely independent of recognition, acceptance or ratification by the knowing or self-conscious subject.

So one can see why he goes to such lengths, in this closing Medita-tion, to mount a kind of double rescue-operation, on the one hand in order to save Descartes from the obloquy heaped upon him by those who ignore that crucial distinction between consciousness and thought, and on the other so as to save Lacan from those who (most of them) would heartily endorse his critique of the Cartesian ‘subject-presumed-to-know’ but regard him as having made the case in a thor-oughly obscurantist and ineffectual way. To be sure, the operation is by no means straightforward since Lacan’s texts are hardly clear and distinct on the point, and hardly lacking for passages that tend to support the opposite (i.e. downright anti-Cartesian) interpretation. Badiou puts it with untypical caginess when he writes that ‘one is not obliged to read into this a complete rupture with Descartes’, because ‘Lacan signals that he “does not misrecognize” that the conscious certitude of existence, at the centre of the cogito, is not immanent, but rather transcendent’ (p. 432). If we read Descartes aright, this sug-gests, then we shall see that he already anticipates Kant’s well-known criticisms of him – in the section of the First Critique entitled ‘Paralo-gisms of Pure Reason’ – for having supposedly confused the tran-scendental and experiential dimensions of subjectivity.22 Thus Descartes effectively pre-empts and deflects the charge of having thought to derive a substantive, psychologically specified conception of the subject from that which in truth affords no more than an abstract (hence empirically vacuous) account of the transcendentally deduced conditions of possibility for subject-hood.

Where Badiou characteristically claims to outgo or overleap the Kantian position is by giving a more radical, indeed distinctly Sartrean-existentialist import to the term ‘transcendent’, defining it in direct opposition to the ‘immanent’ and taking it to entail that ‘the subject cannot coincide with the line of identification proposed to it by this [i.e. cogito-based, phenomenological or

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consciousness-indexed] certitude’ (p. 432).23 In short, it is impor-tant for Badiou at this late stage of Being and Event to cla rify not only his stance vis-à-vis Descartes and the non-coincidence or cleft between consciousness and thought but also what he sees – in closely related terms – as the best, most productive way to inhabit that contested zone (more so than ever in the wake of Lacan’s provocative intervention) between philosophy and psy-choanalysis. ‘Taking a short cut through what can be inferred as common to Descartes, to Lacan, and to what I am proposing here – which ultimately concerns the status of truth as generic hole in knowledge – I would say that the debate bears upon the localiza-tion of the void’ (p. 432). In reading Descartes we have to do with a conception of mind that is less dependent than traditionally supposed on the myth of first-person epistemic access, and more akin to those modes of thought – whether in mathematics or in certain (especially Lacanian) discourses of psychoanalytic theory – which raise significant questions with regard to the cogito as a privileged means of such access. What Badiou is here proposing is a radical shift away from the sorts of Descartes-inspired debate in epistemology that have focused on the problem of knowledge as a matter of somehow outflanking the demon of sceptical doubt by discovering indubitable grounds of certainty within the con-sciously (as well as self-consciously) thinking subject.

The result of that shift would be to redirect attention from what philosophers have found so endlessly intriguing and prob-lematical about Descartes’ epistemological project towards the relationship – likewise problematic in various ways but not unre-solvably or hopelessly so – between his logical, procedural and scientifically oriented methods of enquiry. It would thus offer a more positive outcome to the impasse proclaimed by those, like Rorty, who find nothing more than a sad delusion and a squan-dering of intellectual energies in the post-Cartesian idea of knowledge as standing in need of epistemic, cognitive or justifi-catory warrant. Rorty would have us simply drop that entire misbegotten enterprise and henceforth take the linguistic turn towards a view of philosophy as just another, strictly non-privileged voice in the ongoing cultural conversation.24 Badiou proposes a dif ferent, more principled change of course which likewise starts out from the assumed obsolescence of a certain way of philosophizing – one with the conscious and knowing


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subject at the heart of its epistemological concerns – but which takes this as a reason to examine and reformulate those concerns rather than as a pretext for giving them up altogether. This is why Badiou stakes his claim for mathematics, not linguistics or philosophy of language, as the renovating source for a discipline of thought that would sustain the project of truth-seeking enquiry that he traces from its Greek origins to the present while none the less acknowledging the various objections that have risen against it and seeking to provide an alternative approach that squarely addresses those objections.

His purpose in this final Meditation – one that in a sense gath-ers up all the main lines of argument in Being and Event – is to make the case that we misread Descartes if we fail to see that he is thinking primarily in a mathematical (indeed in what might be called a proto-set-theoretical) mode and only secondarily in terms of the cogito with all its otiose metaphysical baggage. ‘What still attaches Lacan . . . to the Cartesian epoch of science is the thought that the subject must be maintained in the pure void of its subtraction if one wishes to save truth. Only such a subject allows itself to be sutured within the logical, wholly transmissible, form of science’ (p. 432). Where Lacan takes his conceptual bearings from a structural-linguistic (i.e. Saussurean) model Badiou takes his from mathematics, with the crucial dif-ference that Badiou’s case goes by way of a more demanding and rigorous process of demonstrative argument, and thus avoids the sorts of facile misinterpretation to which Lacan has been subject by post-structuralists and others. This means that Badiou can retain a much stronger, more adequate conception of the subject – the enquiring, willing, self-committing or ‘faith-ful’ subject – than could ever be available to those who adopt the idea of subjectivity as just a product (some would say delusory figment) of language or discourse. Thus he not only leaves room for a substantive or more-than-nominal concept of the subject, but also gives it a primary and strictly indispensable role in the discov-ery of mathematical and scientific truths, the production of genu-inely original or creative-exploratory artworks, and the pursuit of radically progressive or transformative political aims beyond any-thing acknowledged by current consensual, that is, liberal-demo-cratic norms. What stands clear to view as a result of Badiou’s critique is the close connection – not to call it ‘complicity’ – between

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the linguistic turn and this resort to a conception of truth and jus-tice based on currently prevailing ideas of the relevant standards or criteria. It would be hard to over-state the extent and signifi-cance of this decisive break with a linguistically based (whether post-structuralist or Wittgensteinian) conception of thought, rea-son and intelligibility. It provides both a rehabilitation of truth – a concept that is sharply defined and developed in context-specific terms across a range of disciplines or topic-areas – and a rehabili-tation of the subject as locus of a truth that might always surpass both its own conscious grasp and the limits of currently available knowledge.

Hence Badiou’s distinction between ‘veracity’ and ‘truth’, where the former (but emphatically not the latter) involves and requires some reference to language or to communally shared means of expression. Such is also the relationship, as he conceives it, between language, consciousness, judgement and that ‘encyclopaedic’ dimension wherein can be found whatever belongs to the realm of received or epistemically secure since well-established veridical opinion. Thus Badiou sides firmly with Plato in opposition to the Protagorean-relativist doctrine that ‘man is the measure’, or the idea (taken up by these latter-day schools of thought) that truth must always conform to the scope and limits of human knowledge. This is also his reason for rejecting that entire chapter of developments opened up by Kant’s self-proclaimed ‘Copernican revolution’, that is, the idea that philosophy could be rescued from its age-old sceptical dilemmas only through a radical switch of focus from ontologi-cal to epistemological issues. Such a change would have the wholly beneficial outcome, Kant believed, of diverting philoso-phers’ attention from strictly unanswerable questions about the nature of objective, mind-independent reality to questions that inherently lay within the bounds of human cognizance since they concerned what we could establish with respect to the mind’s a priori capacities and powers of understanding, reason and judgement.25 For Badiou, conversely, there is nothing to com-mend this Kantian recourse to the subject conceived as a shifting alliance of diverse ‘faculties’ which somehow co-exist under the terms laid down by the conditions of possibility for its own unknowable (transcendentally deduced but purely noumenal hence empirically/psychologically vacuous) selfhood.26


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Least of all can he accept the role that Kant assigns to ‘judge-ment’ as the crucial mediating power between sensuous intuitions and concepts of understanding, and therefore as that which enables every act or instance of empirical knowledge, as well as providing – under a different dispensation of the faculties – the basis of practical reason (ethics) and aesthetic appreciation. To Badiou’s way of thinking this is merely the altogether bad result of Kant’s retreat from ontology to epistemology, and thence to the idea (with untoward consequences for so many present-day thinkers) that truth must be conceived in terms of humanly attainable knowledge and knowledge in terms of those various types or modalities of judgement that between them establish its scope and limits. His response, as we have seen, is to reassert the claims of ontology as first philosophy and also to insist that the subject not be thought of in this Kantian fashion as the locus of various disparate yet somehow mysteriously unified and cooperative powers. Rather we should seek – with greater formal precision – to define its role and its strictly sui generis effects as that which eludes any form of ontological specification since it intervenes to create, discover, invent, devise, faithfully sustain or further develop a truth-procedure whose beginning cannot know its end but whose end – from what he here terms a ‘future ante-rior’ perspective – will constitute its ultimate justification.

Thus Badiou’s proposed new inauguration is one that would restore ontology to its rightful place and thereby mark a signal advance in philosophy’s conceptual resources as applied to areas of investigation beyond the strict remit of mathematics, logic and the formal sciences. Moreover it would have the same transfor-mative effect with regard to philosophy’s longstanding and fre-quently baffled attempt to arrive at some conception of the human subject that is neither abstract to the point of irrelevance nor ‘subjective’ in that philosophically pejorative sense of the term which ranges from ‘personal’ to ‘ill-disciplined’ or ‘downright whimsical’. What Badiou sets out to achieve in Being and Event is just such a working conception, and one that he develops with direct reference to the various specific truth-procedures that arise from specific events – advances or discoveries – in likewise specific contexts of enquiry. Moreover he is equally insistent that subjects come into existence only through some such domain-specific event and its transformative effect both on their own

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lives (insofar as those lives are thereafter dedicated to following out its as-yet unperceived consequences) and on the future deve-lopment of the discipline concerned. Thus there is little in com-mon between Badiou’s usage of ‘subject’ and the way this word is deployed not only in a relatively specialized sense by philo sophers from Kant to Husserl, but also as a matter of everyday parlance where it signifies something more broadly encompassing in per-sonal, psychological and experiential terms. In effect Badiou is pointing back to a stage before there occurred certain complex semantic shifts whereby its meanings separated out into (1) ‘sub-ject = person, individual, human being, first-person locus of iden-tity’ and so on, and (2) ‘subject = theme, topic, discipline, or focus of investigation’.27 What this enables him to claim – without any Heideggerian straining after obscure or downright fake etymolo-gies – is that subjects (sense 1) can best be thought of as acquiring their salient profiles or chief individuating features from the sub-jects (sense 2) wherein they have made some decisive intervention.

In fact – and here we touch the paradoxical heart of his argu-ment – one would have to go back beyond Kant to Descartes in order to find a formulation that poses the relevant issues (as Badiou sees them) with comparable clarity and force. Although Badiou is at odds with Descartes – emphatically so – as regards the latter’s notion of the subject as able to achieve apodictic knowledge through transparent or privileged first-person epis-temic access he is none the less keen to reclaim what he considers its kernel of truth or validity from those who have rejected it out of hand. That is to say, his main reason for ending the book with this seemingly improbable conjunction of Descartes and Lacan is that both thinkers have been exposed to a widespread and tenacious misreading. In the one case this has tended to obscure how Descartes, in his writings other than the Meditations, goes far towards conceiving the subject (sense 1) as intrinsically bound up with those procedures, methods or protocols of thought that constitute its chief points of engagement with the subject (sense 2) to which it dedicates the best of its truth-seeking endeavours. In the other case it has likewise had the effect of disguising how close – almost symbiotic – is the relation of mutual dependence between Lacan’s psychoanalytically inspired challenge to that bugbear entity, the Cartesian subject-presumed-to-know, and the version of Descartes that has come down (from his own time


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to the present) as exactly what is needed in order for these con-trary doctrines to establish their claim.

So Badiou’s purpose in yoking them together is not to deny the very real problems that Lacan poses for any orthodox Cartesian (privileged-access epistemic) approach but rather to demon-strate how a reading of Descartes via Lacan – and of Lacan via Descartes – can bring out the extent of this reciprocal inter-involvement. It shows how Lacan’s drive to revoke the privilege of reason by placing it under the sway of the ubiquitous Freudian–Lacanian unconscious comes up against that strictly inescapable need for the recourse to rational procedures of thought. At the same time it reveals how Descartes’ appeal to the autonomous cogito as presumptive anchor-point of mind and world encounters its limit in that which eludes the supposedly transparent and self-grounding character of conscious reflection. What emerges from this joint demonstration is the insufficiency of any approach that either treats knowledge (and consciousness thereof) as the criterion of reason and truth or else goes so far in an echt-Lacanian direction as to make some show of rejecting reason and truth along with the traditionally privileged claims of knowledge and consciousness. That this can only be a show – that it cannot consistently avow such a goal and yet profess to argue its case in a logical, coherent, rationally cogent way – should not be taken as a flat repudiation of Lacan’s (seemingly) anti-Carte-sian stance. Rather it should serve as a reminder first of that ‘other’ Descartes whose rationalist precepts are not so closely tied to the cogito and secondly of Lacan’s own highly formalized and far from ‘irrationalist’ understanding of the unconscious and its structural – even systematic –workings.

Thus the juxtaposition of two such, on the face of it, starkly antithetical thinkers offers a means whereby Badiou can re-affirm the central thesis of Being and Event. This concerns the remark-able yet well-documented capacity of reason to transcend the limits of conscious, reflective or epistemically accessible thought while yet remaining subject to the dictates of a truth-procedure that acts as both a stimulus and check to its ventures beyond the confines of received knowledge or accepted method. Where Badiou departs most markedly from Lacan is in attaching no special importance to language and in supposing thought at its most rigorous, resourceful or inventive to be sustained very largely

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without dependence on the structures of linguistic meaning or representation. This is why Badiou sees Lacan as belonging very much to the Cartesian line of descent, despite all his vigorous protestations to contrary effect and despite the drastically dis-ruptive or destabilizing effect that his thinking brings to bear on all conceptions of the sovereign or autonomous subject, from Descartes down to Kant, Husserl and the proponents of psycho-analysis in its corrupted (especially American ego-psychological) guise. What kept him tied to a tradition of thought that his work had otherwise so powerfully challenged was its subscription to a variant of the language-first thesis with its source in Saussurean linguistics. That thesis can be seen to have continued, in a differ-ent register, the kind of representationalist doctrine exemplified by the earlier ‘way of ideas’ that had formed such a largely unquestioned staple of rationalist and empiricist philosophies alike.28

It is this long-established but fallacious paradigm that Badiou considers to have been instrumental in bringing about the two-fold inversion of priorities whereby epistemology displaced ontology as the prime focus of enquiry and was itself then dis-placed by the notion of language as marking the ultimate hori-zon of knowledge and truth. So Lacanian psychoanalysis has both a backward and a forward-looking aspect, on the one hand remaining partially in thrall to a modern, ‘linguistified’ variant of the old Cartesian way of ideas while on the other – through its formal, often mathematical modes of articulation – marking the imminent point of transition to a new, altogether more ade-quate paradigm. ‘What is at stake’, Badiou writes, ‘is an opening on to a history of truth which is at last completely disconnected from what Lacan, with genius, termed exactitude or adequation, but which his gesture, overly soldered to language alone, allowed to subsist as the inverse of the true’ (p. 433). Thus the formalism, the recourse to symbolic logic, and the intermittent quest for conceptual rigour – however suspect to his many antagonists – were sufficient to offset the strain of linguistic idealism that charac-terized Lacan’s work though not to enable a decisive break with that whole structuralist or nascent post-structuralist way of thinking. What might have pointed a way forward from this impasse in his thinking was a more sustained engagement with just those develop-ments in recent mathematics which offered the sole adequate means


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of understanding how thought could surpass the limits of in-place (epistemically accredited or linguistically transmissible) knowledge.

Hence Lacan’s place of honour as the last of those highly select maîtres à penser who have figured at various crucial junc-tures throughout Being and Event. This is chiefly on account of his standing, like John the Baptist, at the threshold of that epochal shift in the configuration of thought that Badiou identi-fies with the advent of a set-theoretically inspired return to onto-logy – rather than epistemology or philosophy of language – as the discipline best (or solely) equipped to underwrite and com-prehend such advances. One result of adopting the ‘ontology-first’ standpoint is to lend renewed credence to talk of advances or progress and not require that it always come surrounded by a host of visible or invisible quote-marks. For this entails the two main realist/objectivist precepts (1) that truth may always sur-pass the limits of attainable knowledge, and (2) that knowledge may always transcend whatever finds adequate expression in this or that currently accepted communal language. Moreover, as concerns the formal sciences, it follows from Badiou’s set-theoretical account (3) that the defects – the fallings-short of truth – in the state of knowledge at any given time are such as might conceivably be brought to light through a more advanced grasp of the latent tensions, contradictions or aporias that were hitherto more-or-less successfully concealed from view.

So it is, on the last page of this final Meditation, that Badiou looks back to the book’s Introduction after a long and immensely complex process of argument and claims to have offered demon-strative proof (nothing less) for what was there set forth in pro-spective or promissory terms. Thus he hopes to have shown how ‘[i]t is possible to re-interrogate the entire history of philosophy, from its Greek origins on, according to the hypothesis of a math-ematical regulation of the ontological question’, in which case ‘[o]ne would then see a continuity and a periodicity quite differ-ent from that deployed by Heidegger’ (p. 435). To which one might add: a conception of truth-seeking enquiry across every main area of human intellectual, scientific, political and artistic endeavour quite different from – and venturesome beyond – any-thing essayed in a single work of philosophy since Hegel’s Phenom-enology of Mind. I would hope, in the course of this detailed commentary, to have offered good reason for counting Badiou fully

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justified in his statements of vaulting ambition at the outset and his equally elevated sense of achievement at the close. In coming years it is likely that Being and Event will be taken up by an increasing number of well-prepared readers with a deepened grasp of its philo-sophic challenge, its mathematical complexities and its force of political commitment allied to an ethical standpoint none the less demanding for his overt suspicion of ethics as a discourse most often enlisted in the service of dubious or downright repugnant political ends.29 This will help to overcome the resistance engendered by its singular power to disturb and unsettle orthodox habits of thought, whether those of an expressly conservative character or those – more typically, given Badiou’s ‘natural’ readership – that combine a radical-seeming post-structuralist or ‘strong’ linguistic-constructivist rhetoric with a failure to think through its less than radical (indeed its markedly conservative) implications. By that time Being and Event may well have acquired, and deservedly so, the evental status of a major and transformative episode in the history of thought that Badiou accords to those select few figures whose decisive interventions he can fairly claim to have carried forward through his own mathematically informed re-thinking of issues cen-tral to every discipline where truth is fundamentally in question.

Discussion points

How do you understand the complex relationship of Badiou’s thought to Lacanian psychoanalysis and, via Lacan, to Descartes’ subject-centred epistemology or ‘project of pure enquiry’?

Given the centrality of mathematics to Badiou’s project and, more generally, the formal-abstract cast of his thinking one might expect him to share the suspicion or downright hostility that many analytic philosophers have expressed towards the claims of psychoanalysis. Why is this so strikingly not the case?

Would you say that Badiou has now fulfilled his promise, at the outset of Being and Event, that this would be not only a significant work but a transformative event in the history of modern thought? If so, what exactly is the nature of that trans-formation and which are the disciplines, subject-domains or areas of wider (e.g. social and political) concern where you would expect its impact to be most keenly and deeply felt?

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CHAPTER 4

FURTHER READING

Badiou has been a notably productive writer, more than ever in recent years, so this will be a highly selective bibliography and very much tailored to the likely needs of those who are approach-ing his work for the first time by way of this introductory guide. It contains items chosen on the following grounds:

1. that they bear importantly on themes and ideas developed in Being and Event;

2. that they are central to his project as a whole;3. that they throw a revealing light on some aspect of his work

that I have not been able to treat adequately or discuss at all owing to restrictions of length.

Fortunately there are two online bibliographies that between them offer an impressively detailed survey of Badiou’s work and commentaries on it up to the end of 2006 and 2007, respectively. The first is by Alan Ashton and appeared in the journal Cosmos and History [http://www.cosmosandhistory.org/index.php/journal/article/viewFile/124/73]. This has helped me greatly in hunting out some otherwise fugitive details, and I am grateful to author, editor(s) and publisher for providing such an excellent resource. The second is available from a website dedicated mainly to debates around the work of Jacques Lacan, and therefore has a strongly psychoanalytic slant in its coverage [http://www.lacan.com/bibliographyb.htm]. The ready availability of both has enabled me to focus here on just those items that will be of most use to non-spe-cialists. Works of a more general character (e.g. on topics in philosophy of mathematics, logic, epistemology, onto logy, political theory, psychoanalysis and so on) are covered where appropriate in the chapter endnotes, so I have excluded any such references so as to avoid duplication. All in all readers should find sufficient guidance to the best and most accessible sources for pursuing their engage-ment with Badiou’s thought to a further, more ambitious level.

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1. OTHER BOOKS BY BADIOU AVAILABLE IN ENGLISH TRANSLATIONManifesto for Philosophy (trans.) Norman Madarasz (Albany, NY: State

University of New York Press, 1999).Deleuze: The Clamor of Being (trans.) Louise Burchill (Minneapolis:

University of Minnesota Press, 2000).Ethics: An Essay on the Understanding of Evil (trans.) Peter Hallward

(London: Verso, 2001).Infinite Thought: Truth and the Return to Philosophy (trans. and ed.)

Oliver Feltham and Justin Clemens (London: Continuum, 2003).Saint Paul: The Foundation of Universalism (trans.) Ray Brassier (Stan-

ford, CA: Stanford University Press, 2003). Theoretical Writings (trans.) Ray Brassier (London: Continuum,

2004).Handbook of Inaesthetics (trans.) Alberto Toscano (Stanford U.P.,

2005).Metapolitics (trans.) Jason Barker (London: Verso, 2005).Briefings on Existence: A Short Treatise on Transitory Ontology (trans.)

Norman Madarasz (New York: State University of New York Press, 2006).

The Century (trans.) Alberto Toscano (London: Polity Press, 2007).The Concept of Model: An Introduction to the Materialist Epistemology

of Mathematics (trans.) Zachary Luke Fraser and Tzuchien Tho (Victoria: re.press, 2007).

Polemics (trans.) Steven Corcoran (London: Verso, 2007).The Meaning of Sarkozy (trans.) David Fernbach (London: Verso,

2008).Number and Numbers (trans.) Robin MacKay (London: Polity Press,

2008).Logics of Worlds (trans.) Alberto Toscano (London: Continuum, forth-

coming 2009).Theory of the Subject (trans.) Bruno Bosteels (forthcoming, London:

Continuum, 2009).

2. ARTICLES BY BADIOU IN ENGLISH TRANSLATION‘On a Finally Objectless Subject’, (trans.) Bruce Fink, in Eduardo

Cadava, Peter Connor and Jean- Luc Nancy (eds), Who Comes After the Subject? (London: Routledge, 1991), pp. 24–32.

‘Gilles Deleuze, The Fold: Leibniz and the Baroque’, (trans.) Thelma Sowley, in Constantin Boundas and Dorothea Olkowski (eds), Deleuze and the Theatre of Philosophy (New York: Columbia Univer-sity Press, 1994), pp. 51–69.

‘Philosophy and Politics’, (trans.) Thelma Sowley, Radical Philosophy (July/August1999), pp. 29–32.

‘Metaphysics and the Critique of Metaphysics’, (trans.) Alberto Toscano, Pli: Warwick Journal of Philosophy, No. 10 (2000), pp. 174–90.


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‘On a Contemporary Usage of Frege’, (trans.) Sam Gillespie and Justin Clemens, Umbr(a) 2000, pp. 99–115.

‘The Ethic of Truths: Construction and Potency’, (trans.) Selma Sow-ley, Pli: Warwick Journal of Philosophy, No. 12 (2001), pp. 245–55.

‘Who is Nietzsche?’, (trans.) Alberto Toscano, Pli: Warwick Journal of Philosophy, No. 11 (2001), pp. 1–10.

‘Logic of the Site’, (trans.) Steve Corcoran and Bruno Bosteels, Dia-critics, Vol. 33, No. 3 (2003), pp. 141–50.

‘Some Replies to a Demanding Friend’, (trans.) Peter Hallward, in Think Again: Alain Badiou and the Future of Philosophy (London: Continuum, 2004), pp. 232–7.

‘The Adventure of French Philosophy’, New Left Review, No. 35 (2005), pp. 67–77.

‘Democratic Materialism and the Materialist Dialectic’, Radical Philos-ophy, No. 130 (2005), pp. 20–4.

‘Lacan and the pre-Socratics’, (ed.) Slavoj Zizek, in Lacan: The Silent Partners (London: Verso, 2006), pp. 7–16.

3. ENGLISH-LANGUAGE BOOKS AND ARTICLES ABOUT BADIOUAshton, Paul, A. J. Bartlett and Justin Clemens (eds), The Praxis of

Alain Badiou (Victoria: Re.Press, 2006).Balibar, Etienne, ‘The History of Truth: Alain Badiou in French

Philosophy’, in Hallward (ed.) (2004), pp. 21–38.Barker, Jason, Alain Badiou: A Critical Introduction (London: Pluto

Press, 2002).Bensaid, Daniel, ‘Alain Badiou and the Miracle of the Event’, in

Hallward (ed.) (2004), pp. 94–105.Bosteels, Bruno, ‘Alain Badiou’s Theory of the Subject: The Recommence-

ment of Dialectical Materialism? (Parts I and 2)’, Pli: Warwick Journal of Philosophy, No. 12 (2001), pp. 200–29 and No. 13 (2002), pp. 173–208.

Bosteels, Bruno, ‘On the Subject of the Dialectic’, in Hallward (ed.), (2004), pp. 150–64.

Bosteels, Bruno, ‘Can Change Be Thought? A Dialogue with Alain Badiou’, in Riera (ed.) (2005), pp. 237–61.

Brassier, Ray, ‘Nihil Unbound: Remarks on Subtractive Ontology and Thinking Capitalism’, in Hallward (ed.) (2004), pp. 50–8.

Brassier, Ray, ‘Badiou’s Materialist Epistemology of Mathematics’, Angelaki: Journal of the Theoretical Humanities, Vol. 10, No. 2 (2005), pp. 135–50.

Clemens, Justin, ‘Platonic Meditations’, Pli: Warwick Journal of Phi-losophy, No. 11 (2001), pp. 200–29.

Clemens, Justin, The Romanticism of Contemporary Theory: Institution, Aesthetics, Nihilism (Aldershot: Ashgate, 2003).

Copjec, Joan, ‘Gai Savoir Sera: The Science of Love and the Insolence of Chance’, in Riera (ed.) (2005), pp. 119–35.

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Critchley, Simon, ‘Demanding Approval: On the Ethics of Alain Badiou’, Radical Philosophy, Vol. 100 (2000), pp. 16–27.

Critchley, Simon, ‘On the Ethics of Alain Badiou’, in Riera (ed.) (2005), pp. 215–35.

de Beistegui, Miguel, ‘The Ontological Dispute: Badiou, Heidegger, and Deleuze’, in Riera (ed.) (2005), pp. 45–58.

Desanti, Jean Toussaint, ‘Some Remarks on the Intrinsic Ontology of Alain Badiou’, in Hallward (ed.) (2004), pp. 59–66.

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NOTES

1. CONTEXT

1 Alain Badiou, Being and Event (trans.) Oliver Feltham (London: Continuum, 2005). All further references throughout this book will be given parenthetically by BE and page number in the text.

2 For further discussion, see Graham Bird (ed.), A Companion to Kant (Oxford: Blackwell, 2006); Ruth F. Chadwick and Clive Cazeaux (eds), Immanuel Kant: Critical Assessments 4 vols (London: Rout-ledge, 1992); Paul Guyer (ed.), The Cambridge Companion to Kant (Cambridge: Cambridge University Press, 1992).

3 See especially Richard Rorty, Contingency, Irony and Solidarity (Cambridge: Cambridge University Press, 1989); also Christopher Norris, What’s Wrong with Postmodernism? (Brighton: Harvester, 1990) and The Truth About Postmodernism (Oxford: Blackwell, 1993).

4 See for instance Michael Devitt, Realism and Truth, 2nd edn (Oxford: Blackwell, 1986) and Norris, Philosophy of Language and the Chal-lenge to Scientific Realism (London: Routledge, 2004).

5 See especially Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1951).

6 See for instance Paul Benacerraf and Hilary Putnam (eds), The Phi-losophy of Mathematics: Selected Essays, 2nd edn (Cambridge: Cambridge University Press, 1983); also W. D. Hart (ed.), The Phi-losophy of Mathematics (Oxford: Oxford University Press, 1996). On the vastly overworked topic of rule-following see Ludwig Wittgen-stein, Philosophical Investigations (trans.) G. E. M. Anscombe (op. cit.), Sections 201–92 passim; Saul Kripke, Wittgenstein on Rules and Private Language: An Elementary Exposition (Oxford: Blackwell, 1982); Alexander Miller and Crispin Wright (eds), Rule-following and Meaning (Chesham: Acumen, 2002). For Badiou’s decidedly adverse view of much of this work, see ‘Ontology is Mathematics’, in Theoretical Writings. All subsequent references to works of Badiou other than Being and Event will be given by title only since full publi-cation details may be found in Chapter 4, ‘Further Reading’.

7 For some highly illuminating discussion, see Badiou, Metapolitics, Polemics, and Century.

8 For a clear and highly relevant survey, see Michael Potter, Set Theory and its Philosophy: A Critical Introduction (Oxford: Oxford University Press, 2004).

9 Bertrand Russell, Introduction to Mathematical Philosophy (London: Allen & Unwin, 1930).

10 Paul J. Cohen, Set Theory and the Continuum Hypothesis (New York: W. A. Benjamin, 1966).

NOTES

290

11 For an account of these developments, see Norris, Against Relativism: Deconstruction, Philosophy of Science and Critical Theory (Oxford: Blackwell, 1997) and Philosophy of Language and the Challenge to Scientific Realism (London: Routledge, 2004).

12 See especially the various remarks to this effect in Badiou, ‘Onto-logy is Mathematics’ (Note 6).

13 For further argument in support of this claim, see Norris, ‘Change, Conservation and Crisis-Management in the Discourse of Analytic Philosophy’, in Language, Logic and Epistemology: A Modal-realist Approach (Basingstoke: Palgrave-Macmillan, 2004), pp. 227–66.

2. OVERVIEW OF THEMES

1 See especially Immanuel Kant, Political Writings (ed.) Hans Reiss (Cambridge: Cambridge University Press, 1970) and Kant on His-tory (ed.) L.W. Beck (Indianapolis: Bobbs-Merrill, 1963).

2 For further discussion of these various developments, see Christo-pher Norris, Against Relativism: Philosophy of Science, Deconstruc-tion and Critical Theory (Oxford: Blackwell, 1997) and Philosophy of Language and the Challenge to Scientific Realism (London: Routledge, 2004).

3 For a useful sampling, see Richard Rorty (ed.), The Linguistic Turn (Chicago, IL: University of Chicago Press, 1967).

4 See especially Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1953) and OnCertainty (ed. and trans.) Anscombe and G. H. von Wright (Blackwell, 1969).

5 See Rorty (ed.), The Linguistic Turn (op. cit.). 6 See especially Rudolf Carnap, ‘The Elimination of Metaphysics

through Logical Analysis of Language’, in A. J. Ayer (ed.), Logical Positivism (New York: Free Press, 1959), pp. 60–81; also – for some fascinating background history – Michael Friedman, A Parting of the Ways: Carnap, Cassirer and Heidegger (Chicago: Open Court, 2000).

7 Martin Heidegger, Being and Time (trans.) John Mcquarrie and Edward Robinson (Oxford: Blackwell, 1980).

8 Nancy Cartwright, Jordi Kat, Lola Fleck and Thomas Uebel, Otto Neurath: Science Between Philosophy and Politics (Cambridge: Cambridge University Press, 1996).

9 See Note 6. 10 Heidegger, Poetry, Language and Thought (trans.) A. Hofstadter

(New York: Harper & Row, 1971). 11 See especially Michael Dummett, Elements of Intuitionism (Oxford:

Oxford University Press, 1977) and Truth and Other Enigmas (London: Duckworth, 1978); also – for a critical review of these devel-opments – Norris, Truth Matters: Realism, Anti-realism, and Response-dependence (Edinburgh: Edinburgh University Press, 2002).

NOTES

291

12 For a detailed study, see J. Alberto Coffa, The Semantic Tradition from Kant to Carnap: To the Finland Station (Cambridge: Cambridge University Press, 1991).

13 These issues receive a good critical airing in Ian Hacking, The Social Construction of What? (Cambridge, MA: Harvard Univer-sity Press, 2000). See also, from a range of perspectives, David Bloor, Know ledge and Social Imagery, 2nd edn (Chicago: Univer-sity of Chicago Press, 1991); Harry Collins and Robert Evans, Re-thinking Expertise (Chicago, U.P., 2007); Kenneth J. Gergen, Realities and Relationships: Soundings in Social Construction (Har-vard, U.P., 1994); Norris, Against Relativism (op. cit.) and Philosophy of Language and the Challenge to Scientific Realism (London: Routledge, 2004).

14 See for instance entries for Chapter 1, ‘Context’, Note 6. 15 Paul J. Cohen, Set Theory and the Continuum Hypothesis (New York:

W. A. Benjamin, 1966). 16 For some informative accounts, see Catherine Belsey, Critical Prac-

tice (London: Methuen, 1981); Richard Harland, Superstructuralism: The Philosophy of Structuralism and Post-structuralism (London: Routledge, 1991); Mark Poster, The Mode of Information: Post-structuralism and Social Context (Chicago, IL: University of Chicago Press, 1990); Robert Young (ed.), Untying the Text: A Post-structuralist Reader (London: Routledge & Kegan Paul, 1981).

17 See especially Jacques Lacan, Ecrits: A Selection (trans.) A. Sheri-dan-Smith’ (London: Tavistock, 1977); also (for an excellent short introduction) Malcolm Bowie, Lacan (London: Fontana, 1991).

18 See for instance Louis Althusser, For Marx (trans.) Ben Brewster (London: New Left Books, 1969); also Gregory Elliott, Althusser: The Detour of Theory (London: Verso, 1987).

19 Jean-Paul Sartre, A Critique of Dialectical Reason, Vol. 1, Theory of Practical Ensembles (trans.) A. Sheridan-Smith (London: New Left Books, 1976) and Vol. 2 (trans.) Quintin Hoare (London: Verso, 2006).

20 See Notes 7 and 10; also Heidegger, The Question Concerning Tech-nology, and Other Essays (trans.) W. Lovitt (New York: Harper & Row, 1977).

21 For further discussion see Richard Wolin (ed.), The Heidegger Controversy (New York: Columbia University Press, 1991).

22 See for instance Iris Marion Young, Justice and the Politics of Differ-ence (Princeton, NJ: Princeton University Press, 1990); also – for a critical review of such thinking across various disciplines – Norris, Truth and the Ethics of Criticism (Manchester: Manchester University Press, 1994).

23 Immanuel Levinas, Totality and Infinity (trans.) A. Lingis (Pittsburgh, PA: Duquesne University Press, 1969).

24 See Note 11; also Rorty (ed.), The Linguistic Turn (op. cit.). 25 For a strikingly heterodox view, see Lloyd P. Gerson, Aristotle and

Other Platonists (Ithaca, NY: Cornell University Press, 2005).

NOTES

292

3. READING THE TEXT

Part I. Being: multiple and void. Plato/Cantor 1 Plato, Parmenides (trans.) Mary L. Gill and Paul Ryan (Indianapo-

lis: Hackett, 1996). See also F. M. Cornford, Plato and Parmenides: Parmenides’ Way of Truth and Plato’s Parmenides (trans. and intro-duction) Cornford (London: Routledge & Kegan Paul, 1939) and John A. Palmer, Plato’s Reception of Parmenides (Oxford: Clarendon Press, 2002).

2 W. V. Quine, Ontological Relativity and Other Essays (New York: Columbia University Press, 1969), p. 23.

3 Paul J. Cohen, Set Theory and the Continuum Hypothesis (New York: W.A. Benjamin, 1966). See Abraham A. Fraenkel, Elements of Set Theory, rev. edn (Amsterdam: North-Holland, 1973) for a main source of Badiou’s thinking about issues of set-theoretical ontology; also Michael Potter, Set Theory and its Philosophy: A Critical Intro-duction (Oxford: Oxford University Press, 2004).

4 Badiou, Deleuze: The Clamor of Being. 5 See Gilles Deleuze, The Logic of Sense (trans.) Mark Lester (London:

Athlone Press, 1990) and Difference and Repetition (trans.) Paul Patton (Athlone, 1994).

6 Gilles Deleuze and Félix Guattari, Anti-Oedipus: Capitalism and Schizophrenia (trans.) Robert Hurley, Mark Seem and Helen Lane (London: Continuum, 2004).

7 See especially Bertrand Russell, A History of Western Philosophy (London: Allen & Unwin, 1961); also – for some highly illuminating commentary on these issues of interpretation – Jonathan Bennett, Learning from Six Philosophers: Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume, Vols 1 and 2 (Oxford: Clarendon Press, 2001).

8 See Gaston Bachelard, La formation de l’esprit scientifique (Paris: Vrin, 1938); The Philosophy of No (trans.) G. C. Waterston (New York: Orion Press, 1969); The New Scientific Spirit (trans.) Arthur Goldhammer (Boston: Beacon Press, 1984). See also Mary Tiles, Bachelard: Science and Objectivity (Cambridge: Cambridge University Press, 1984).

9 See Note 3; also Bertrand Russell, Introduction to Mathematical Philosophy (London: Allen & Unwin, 1930).

10 Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems (trans.) B. Meltzer (New York: Basic Books, 1962).

11 For further arguments to this effect, see Stathis Psillos, Scientific Realism: How Science Tracks Truth (London: Routledge, 1999); J. Aronson, R. Harré and E. Way, Realism Rescued: How Scientific Progress is Possible (London: Duckworth, 1994); Michael Devitt, Realism and Truth, 2nd edn (Oxford: Blackwell, 1986); Jarrett Leplin (ed.), Scientific Realism (Berkeley & Los Angeles: University of California Press, 1984); Christopher Norris, Language, Logic and Epistemology: A Modal-Realist Approach (Manchester: Manchester University Press, 2004).

NOTES

293

12 See Note 8; also Norris, Spinoza and the Origins of Modern Critical Theory (Oxford: Blackwell, 1991).

13 See Quine, ‘On What There Is’, in From a Logical Point of View, 2nd edn (Cambridge, MA: Harvard University Press, 1961), pp. 1–19.

14 See Note 11. 15 Plato, Timaeus and Critias (ed. and trans.) Desmond E. Lee (Har-

mondsworth: Penguin, 1974); also Reason and Necessity: Essays on Plato’s Timaeus (ed.) M. R. Wright (London: Duckworth, 2000).

16 See especially Badiou, Metapolitics, Polemics, and Century.17 For further discussion, see A. W. Moore, The Infinite (London:

Routledge, 2000) and Potter, Set Theory and its Philosophy (op. cit.). 18 See for instance Iris Marion Young, Justice and the Politics of Dif-

ference (Princeton, NJ: Princeton University Press, 1990); also Recognition and Difference: Politics, Identity, Multiculture (eds) Scott Lash and Mike Featherstone (London: Sage, 2002); Chris Weedon, Feminism, Theory and the Politics of Difference (Oxford: Blackwell, 1999).

19 Badiou, Saint Paul: The Foundation of Universalism. 20 Aristotle, Physics (trans.) Robin Waterfield (Oxford: Oxford Uni-

versity Press, 1996); also Articles on Aristotle, Vol. 1, Science (eds) Jonathan Barnes, Malcolm Schofield and Richard Sorabji (London: Duckworth, 1975).

21 Steven Shapin and Simon Schaffer, Leviathan and the Air-Pump: Hobbes, Boyle and the Experimental Life (Princeton, NJ: Princeton University Press, 1985); also Norris, ‘Why Strong Sociologists Abhor a Vacuum: Shapin and Schaffer on the Boyle/Hobbes controversy’, in Against Relativism: Philosophy of Science, Deconstruction and Critical Theory (Oxford: Blackwell, 1997), pp. 265–94.

22 See Althusser, For Marx (op. cit.) and Essays in Self-criticism (trans.) Ben Brewster (London: New Left Books: 1976); also Althusser and Etienne Balibar, Reading Capital (trans.) Brewster (New Left Books, 1970).

23 Simon Singh, Fermat’s Last Theorem: The Story of a Riddle that Confounded the World’s Greatest Minds for 358 Years (London: Fourth Estate, 1997).

24 For the classic exposition of this ‘two contexts’ approach, see Hans Reichenbach, Science and Prediction (Chicago: University of Chicago Press, 1938).

Part II. Being: excess, state of the situation, one/multiple, whole/parts or ∈/⊂?

1 See A. A. Fraenkel and R. Bar-Hillel, Foundations of Set Theory (Amsterdam: North-Holland, 1958); Azriel Levy, Basic Set Theory (New York: Dover, 2002); Patrick Suppes, Axiomatic Set Theory (New York: Courier Dover, 1972).

NOTES

294

2 See also Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, NJ: Princeton University Press, 1990).

3 See various texts and passages to this effect in Badiou, Infinite Thought, Handbook of Inaesthetics, Metapolitics, and Century.

4 See especially Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1953) and On Cer-tainty, (ed. and trans.) Anscombe and G. H. von Wright (Oxford: Blackwell, 1969); also Martin Heidegger, ‘The Question Concerning Technology’ and Other Essays (trans.) William Lovitt (New York: Harper & Row, 1977). For Badiou’s most explicit and powerful critique of such ideas, see the essays collected in ‘Ontology is Math-ematics’, Section One of Theoretical Writings, pp. 3–93.

5 For relevant discussion, see Lorraine Code, Epistemic Responsibility (Hanover, NH: University Press of New England, 1987); M. DePaul and L. Zagzebski (eds), Intellectual Virtue: Perspectives from Ethics and Epistemology (Oxford: Oxford University Press, 2002); M. Steup (ed.) Knowledge, Truth, and Duty: Essays On Epistemic Justification, Responsibility, and Virtue (Oxford University Press, 2002).

6 See Badiou, ‘Ontology is Mathematics’ (op. cit.). 7 See especially Badiou, Metapolitics and Century; also Polemics. 8 Thomas S. Kuhn, The Structure of Scientific Revolutions, 2nd edn

(Chicago: University of Chicago Press, 1970). 9 See entries under Note 7. 10 See for instance C. B. MacPherson, The Political Theory of Posses-

sive Individualism (Oxford: Oxford University Press, 1962). 11 For a closely though not precisely analogous argument, see Louis

Althusser, ‘Ideology and Ideological State Apparatuses: Notes Toward an Investigation’, in ‘Lenin and Philosophy’ and Other Essays (trans.) Ben Brewster (London: New Left Books, 1971), pp. 127–86.

12 Ibid; see also Althusser, For Marx (trans.) Ben Brewster (London: Allen Lane, 1969).

13 Jacques Lacan, Ecrits: a selection (trans.) Alan Sheridan-Smith (London: Tavistock, 1977).

14 For further discussion, see Dale Jacquette (ed.), Philosophy of Logic: An Anthology (Oxford: Blackwell, 2002).

15 See especially Badiou, ‘Ontology is Mathematics’ (op. cit.). 16 See for instance Kurt Mueller-Vollmer (ed.), The Hermeneutics

Reader (Oxford: Blackwell, 1988); Richard Rorty (ed.), The Linguistic Turn (Chicago: University of Chicago Press, 1967); Robert Stainton (ed.), Perspectives in the Philosophy of Language: A Concise Anthol-ogy (New York: Broadview Press, 2000).

17 For some useful discussion of this focal shift from the ‘way of ideas’ to the ‘way of words’ (or from epistemology to philosophy of lan-guage), see Ian Hacking, Why Does Language Matter to Philosophy? (Cambridge: Cambridge University Press, 1975).

18 Spinoza, Ethics, in The Collected Writings of Spinoza (trans.) Edwin Curley (Princeton, NJ: Princeton University Press, 1985).

NOTES

295

19 See Badiou, Deleuze: The Clamor of Being; also Gilles Deleuze, Spinoza: Practical Philosophy (trans.) Robert Hurley (San Francisco: City Lights Books, 1988) and Expressionism in Philosophy: Spinoza (trans.) Martin Joughin (New York: Zone Books, 1992), along with the various references to Spinoza in Deleuze and Félix Guattari, A Thousand Plateaus (Capitalism and Schizophrenia, Vol. 2) (trans.) Brian Massumi (Minneapolis: University of Minnesota Press, 1987).

20 See Etienne Balibar, Spinoza and Politics (London: Verso, 1998); Jonathan Israel, Radical Enlightenment: Philosophy and The Making of Modernity, 1650–1750 (Oxford: Oxford University Press, 2002); Yirmiyahu Yovel, Spinoza and Other Heretics, Vol. 1, The Marrano of Reason, and Vol. 2, The Adventures of Immanence (Princeton, NY: Princeton University Press, 1989).

21 See for instance Louis Althusser, For Marx (trans.) Ben Brewster (op. cit.) and ‘Elements of Self-criticism’, in Essays in Self-criticism (London: New Left Books, 1976), pp. 101–61; Althusser and Etienne Balibar, Reading Capital (trans.) Brewster (New Left Books, 1970); Balibar, Spinoza and Politics (London: Verso, 1988); Pierre Mach-erey, In a Materialist Way: Selected Essays (ed.) Warren Montag, (trans.) Ted Stolze (Verso, 1998).

22 See Note 19.23 Christopher Norris, Spinoza and the Origins of Modern Critical

Theory (Oxford: Blackwell, 1991).24 Spinoza, Ethics (op. cit.).25 See for instance Ted Benton, The Rise and Fall of Structural Marx-

ism (London: New Left Books, 1984) and Gregory Elliott, Althusser: The Detour of Theory (London: Verso, 1987).

26 See Note 15.27 See especially Badiou, ‘Ontology is Mathematics’ (op. cit.). 28 Spinoza, Ethics (op. cit.).29 For a classic statement of the hard-line case, see D. M. Armstrong,

A Materialist Theory of the Mind (London: Routledge & Kegan Paul, 1968); also Paul M. Churchland, Scientific Realism and the Plasticity of Mind (Cambridge: Cambridge University Press, 1979).

30 See Note 19.31 Spinoza, Ethics (op. cit.).32 Badiou, ‘Ontology is Mathematics’ (op. cit.).33 Spinoza, Ethics (op. cit.); Badiou, ‘Spinoza’s Closed Ontology’, in

Theoretical Writings, pp. 81–93; also Norris, Spinoza and the Origins of Modern Critical Theory (op. cit.).

34 See Note 20.

Part III. Being: nature and infinity. Heidegger/Galileo 1 See especially Martin Heidegger, Poetry, Language and Thought (trans.)

Albert Hofstadter (New York: Harper & Row, 1971); also Early Greek Thinking (trans.) David Krell and Frank Capuzzi (Harper & Row, 1975) and Heidegger: Basic Writings (ed.) Krell (Harper & Row, 1977).

NOTES

296

2 Martin Heidegger, Being and Time (trans.) John Mcquarrie and Edward Robinson (Oxford: Blackwell, 1980).

3 For a useful historical-comparative account of these various con-ceptions, see Barry Allen, Truth in Philosophy (Cambridge, MA: Harvard University Press, 1993).

4 See for instance Heidegger, ‘The Question Concerning Technology’ and Other Essays (trans.) William Lovitt (New York: Harper & Row, 1977) and ‘The End of Philosophy and the Task of Thinking’, in Time and Being (trans.) Joan Stambaugh (Harper & Row, 1972).

5 For further discussion, see A. J. Ayer (ed.), Logical Positivism (New York: Free Press, 1958).

6 Some literary critics, among them I. A. Richards, tried to make the best of this situation by embracing the idea of poetry as an ‘emo-tive’, that is, strictly non-cognitive or non-truth-apt mode of discourse. See Richards, Principles of Literary Criticism (London: Kegan, Paul, Trench & Trubner, 1924).

7 See especially Ludwig Wittgenstein, Remarks on the Foundations of Mathematics (ed.) G. H. von Wright, R. Rees and G. E. M. Ans-combe (Oxford: Blackwell, 1956); Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1954); Crispin Wright, Wittgenstein on the Foundations of Mathematics (London: Duckworth, 1980); Heidegger, ‘The End of Philosophy and the Task of Thinking’ and ‘The Question Concerning Technology’ (see Note 4).

8 See Note 1. 9 See especially Badiou, Theoretical Writings, Polemics, and The

Century; also De quoi Sarkozy est-il le nom? 10 Paul Cohen, Set Theory and the Continuum Hypothesis (New York:

Addison-Wesley, 1966).11 Michael Dummett, Truth and Other Enigmas (London: Duckworth,

1978); Neil Tennant, Anti-realism and Logic (Oxford: Clarendon Press, 1987); also – for a strongly opposed line of argument – Christopher Norris, Truth Matters: Realism, Anti-realism and Response-depen-dence (Edinburgh: Edinburgh University Press, 2002).

12 For further discussion of these issues, see Michael Devitt, Realism and Truth, 2nd edn (Oxford: Blackwell, 1986); also Norris, Episte-mology: Key Concepts in Philosophy (London: Continuum, 2005).

13 See Notes 1 and 2; also Christopher Norris, ‘Settling Accounts: Heidegger, de Man and the Ends of Philosophy’, in What’s Wrong With Postmodernism: Critical Theory and the Ends of Philosophy (Hemel Hempstead: Harvester-Wheatsheaf, 1990), pp. 222–83.

14 See also Jean-Luc Nancy and Philippe Lacoue-Labarthe, The Literary Absolute: The Theory of Literature in German Romanticism (trans.) Philip Barnard and Cheryl Lester (Albany, NY: State University of New York Press, 1988).

15 Heidegger, Poetry, Language and Thought (op. cit.). 16 Friedrich Nietzsche, The Birth of Tragedy out of the Spirit of Music

(trans.) Shaun Whiteside, (ed.) Michael Tanner (Harmondsworth: Penguin, 1993).

NOTES

297

17 See for instance Jean-Jacques Leçercle, ‘Badiou’s Poetics’, in Peter Hallward (ed.), Think Again: Alan Badiou and the Future of Philosophy (London: Continuum, 2004), pp. 25–47.

18 See Notes 13 and 14. 19 Tom Rockmore and Joseph Margolis (eds), The Heidegger Case: On

Philosophy and Politics (Philadelphia, PA: Temple University Press, 2008); also Rockmore, On Heidegger’s Nazism and Philosophy (Berkeley & Los Angeles: University of California Press, 1992).

20 For an argument to somewhat similar effect, see Paul de Man, ‘Heidegger’s Exegeses of Hölderlin’, in Blindness and Insight: Essays in the Rhetoric of Contemporary Criticism, 2nd edn (London: Methuen, 1983), pp. 246–66.

21 By far the most convenient source is the online, freely downloadable English translation of Mallarmé’s ‘Un coup de dés’ by A. S. Kline which includes an introduction, the poet’s original preface, and a complete French text. This may be accessed at [http://www.tonykline.co.uk/PITBR/French/MallarmeUnCoupdeDes.htm]. See also Sté-phane Mallarmé, Selected Poetry and Prose (ed.) M. A. Caws (New York: New Directions, 1982). For further detailed commen-tary, see Malcolm Bowie, Mallarmé and the Art of Being Difficult, 2nd edn (Cambridge: Cambridge University Press, 2008).

22 For a wide-ranging and highly perceptive survey of the field, see Graham Dunstan Martin, Language, Truth and Poetry: Some Notes Toward a Philosophy of Literature (Edinburgh: Edinburgh Univer-sity Press, 1975).

23 See for instance Gaston Bachelard, Le rationalisme appliqué (Paris: Presses Universitaires de France, 1949); The New Scientific Spirit (Boston: Beacon Press, 1984); Georges Canguilhem, Ideology and Rationality in the History of the Life Sciences (trans.) A. Goldham-mer (Cambridge, MA: MIT Press, 1988); also Mary Tiles, Bachelard: Science and Objectivity (Cambridge: Cambridge University Press, 1984).

24 For a useful survey of these developments in a broad intellectual-historical context, see Peter Sedgwick, Descartes to Derrida: An Introduction to European Philosophy (Oxford: Blackwell, 2001).

25 See especially Louis Althusser, For Marx (trans.) Ben Brewster (London: Allen Lane, 1969) and Jacques Lacan, Ecrits: a selection (trans.) Alan Sheridan-Smith (London: Tavistock, 1977).

26 For a comparable critique of Heidegger’s depth-hermeneutic and phenomenologically grounded approach to these questions, see Dale Jacquette, Ontology (Montreal: McGill-Queen’s University Press, 2003).

27 Badiou, Saint Paul: The Foundation of Universalism. 28 See Note 17. 29 For the most extreme statement of this high-formalist view, see

Veronica Forrest-Thompson, Poetic Artifice: A Theory of Twenti-eth-century Poetry (Manchester: Manchester University Press, 1978).

NOTES

298

30 For an exceptionally acute and wide-ranging commentary on the history of thinking about the infinite, see A. W. Moore, The Infinite (London: Routledge, 2001).

31 For a first-rate online survey of these various schools of thought together with a useful selective bibliography, see Leon Horsten, ‘Phi-losophy of Mathematics’, The Stanford Encyclopaedia of Philosophy [http://plato.stanford.edu/entries/philosophy-mathematics/].

32 Immanuel Kant, Critique of Pure Reason (trans.) N. Kemp Smith (London: Macmillan, 1964).

33 See especially Badiou, Manifesto for Philosophy and Theoretical Essays.

34 See Moore, The Infinite (op. cit.). 35 See Alexandre Koyré, From the Closed World to the Infinite Universe

(New York: Harper & Row, 1958). 36 On the rule following issue, see Ludwig Wittgenstein, Philosophical

Investigations (op. cit.), Sections 201–92 passim; Saul Kripke, Witt-genstein on Rules and Private Language: An Elementary Exposition (Oxford: Blackwell, 1982); Alexander Miller and Crispin Wright (eds), Rule-following and Meaning (Chesham: Acumen, 2002).

37 Badiou makes this point with great argumentative as well as polemi-cal force in the sequence of essays entitled ‘Ontology is Mathematics’, Section One of Theoretical Writings, pp. 3–93.

38 For a detailed survey of this history of thought, see J. Alberto Coffa, The Semantic Tradition from Kant to Carnap: To the Vienna Station (Cambridge: Cambridge University Press, 1991).

39 For a detailed discussion, see Gregory H. Moore, Zermelo’s Axiom of Choice: Its Origins, Development and Influence (Berlin & New York: Springer Verlag, 1982).

40 See Note 36. 41 G. W. F. Hegel, Science of Logic (trans.) A. V. Miller (London: Allen

& Unwin, 1959). 42 Hegel, Phenomenology of Mind (trans.) A. V. Miller (Oxford:

Clarendon Press, 1987). 43 Ibid. 44 For more detailed discussion see Moore, Infinity (op. cit.). 45 Stuart Barnett (ed.), Hegel After Derrida (London: Routledge,

1998); Christopher M. Gemerchak, The Sunday of the Negative: Reading Bataille Reading Hegel (Albany, NY: State University of New York Press, 2003); Alexandre Kojève, Introduction to the Read-ing of Hegel (New York: Basic Books, 1969).

46 For the best-known and most widely influential instance, see Jean-François Lyotard, The Postmodern Condition: A Report on Knowledge (trans.) Geoff Bennington and Brian Massumi (Manchester: Manchester University Press, 1984).

47 See Note 25; also Pierre Macherey, In a Materialist Way: Selected Essays (ed.) Waren Montag, (trans.) Ted Stolze (London: Verso, 1998).

48 See Note 25.

NOTES

299

Part IV. The event: history and ultra-one

1 G. W. F. Hegel, Phenomenology of Mind (trans.) A. V. Miller (Oxford: Clarendon Press, 1987).

2 Jean-Paul Sartre, A Critique of Dialectical Reason, Vol. 1, Theory of Practical Ensembles (trans.) A. Sheridan-Smith (London: New Left Books, 1976) and Vol. 2 (trans.) Quintin Hoare (London: Verso, 2006).

3 See especially Badiou, Metapolitics and Polemics; also ‘Philosophy and Politics’ and ‘Ontology and Politics: An Interview with Alain Badiou’, in Infinite Thought, pp. 69–78 and 169–94.

4 Thomas S. Kuhn, The Structure of Scientific Revolutions (Chicago: University of Chicago Press, 1970).

5 Roland Barthes, Mythologies (trans.) Annette Lavers (London: Paladin, 1972).

6 Jean-François Lyotard, The Postmodern Condition: A Report on Knowledge (trans.) Geoff Bennington and Brian Massumi (Man-chester: Manchester University Press, 1984).

7 See for instance Badiou, Metapolitics and The Century; also De quoi Sarkozy est-il le nom?

8 Badiou, Manifesto for Philosophy. 9 For an early example of the kind, see Daniel Bell, The End of Ideol-

ogy: On the Exhaustion of Political Ideas in the Fifties (Glencoe, IL: Free Press, 1960); also – more recently – Francis Fukuyama, The End of History and the Last Man (London: Hamish Hamilton, 1992).

10 See Kurt Mueller-Vollmer (ed.), The Hermeneutics Reader (Oxford: Blackwell, 1988); Richard Rorty (ed.), The Linguistic Turn (Chicago: University of Chicago Press, 1967); Robert Stainton (ed.), Perspec-tives in the Philosophy of Language: A Concise Anthology (New York: Broadview Press, 2000).

11 See for instance – for a range of views – Keith Jenkins (ed.), The Postmodern History Reader (London: Routledge, 1997).

12 See for instance François Furet, Interpreting the French Revolution (trans.) Elborg Forster (Cambridge: Cambridge University Press, 1968); Furet and Mona Ozouf (eds), A Critical Dictionary of the French Revolution (trans.) Arthur Goldhammer (Cambridge, MA: Harvard University Press, 1989).

13 Badiou, Infinite Thought, pp. 46–7. 14 Timothy Williamson, Knowledge and its Limits (Oxford: Oxford Uni-

versity Press, 2000). I should say that Williamson arrives at these questions – in particular the issue between realists and anti-realists con-cerning the (non-)existence or (in)conceivability of unknown/unknowable truths – from a standpoint notably akin to Badiou’s, despite their other-wise large differences of philosophic idiom and orientation. Thus Williamson similarly sets out to show, contra anti-realists like Dummett, how we can indeed make sense – logically and epistemologi-cally speaking – of the claim that such truths may be known to exist, and moreover that certain conditions may be specified for the kinds of

NOTES

300

advance in our powers of epistemic or cognitive grasp that would (even if at an impossible stretch for our present best state of understanding) bring them within reach. ‘Once we acknowledge that the domain [that of unknowability] is non-empty, we can explore more effectively its extent. In order to be able to set a limit to knowledge, we do not have to find both sides of the limit knowable. Although, trivially, we cannot know that which we cannot know, we can know that we cannot know something’ (pp. 300–01). And conversely, though by the same token, we can know or excogitate a great many things that we are not consciously aware of in the Cartesian sense of having them transparently, directly, reflectively or occurrently present to mind. Williamson is fully in agree-ment with Badiou on these two major points, as likewise in adopting a ‘classical’ (i.e. bivalent or non-intuitionist) approach to mathematics, logic and the formal sciences.

15 Note 10; also Ian Hacking, Why Does Language Matter to Philo-sophy? (Cambridge: Cambridge University Press, 1975).

16 Jacques Lacan, Ecrits: A Selection (trans.) Alan Sheridan-Smith (London: Tavistock, 1977).

17 Hilary Kornblith (ed.), Epistemology: Internalism and Externalism (Oxford: Blackwell, 2001).

18 See in particular Alvin Goldman, Pathways to Knowledge: Private and Public (Oxford: Oxford University Press, 2002).

19 See Note 11; also Frank Ankersmit and Hans Kellner (eds), A New Philosophy of History (London: Reaktion, 1995).

Part V. The event: intervention and fidelity. Pascal/choice; Hölderlin/deduction

1 For a representative selection, see Blaise Pascal, Pensées and Other Writings (ed.) A. and H. Levi (Oxford: World Classics, 1999).

2 Badiou, Manifesto for Philosophy. 3 Pascal, Pensées (op. cit.). 4 See also Badiou, Manifesto for Philosophy. 5 See Badiou, ‘Philosophy and Mathematics: Infinity and the End of

Romanticism’, in Theoretical Writings, pp. 21–38. 6 For a sampling of views, see A. J. Ayer (ed.), Logical Positivism

(New York: Free Press, 1958). 7 See for instance John Hick, Faith and Knowledge (London: Macmillan,

1967). 8 Badiou, Saint Paul: The Foundation of Universalism. 9 See Alan Hajek’s excellent discussion of Pascal’s Wager, download-

able from the Stanford Encyclopedia of Philosophy at [http://plato.stanford.edu/entries/pascal-wager/].

10 See for instance J. S. Mill, An Examination of Sir William Hamilton’s Philosophy, and of the Principal Philosophical Questions Discussed in his Writing (London: Longmans, Green & Dyer, 1878) and Bertrand Russell, Why I am Not a Christian, and Other Essays on Religion and

NOTES

301

Related Subjects (ed.) Paul Edwards (London: Allen & Unwin, 1957).

11 See Nicholas Hammond (ed.), The Cambridge Companion to Pascal (Cambridge: Cambridge University Press, 2003) and Robert J. Nelson, Pascal: Adversary and Advocate (Cambridge, MA: Harvard University Press, 1982).

12 For some illuminating commentary, see A. W. Moore, The Infinite (London: Routledge, 2001).

13 Immanuel Kant, Critique of Pure Reason (trans.) N. Kemp Smith (London: Macmillan, 1964).

14 This distinction was first formulated by Hans Reichenbach in his book Experience and Prediction (Chicago, IL: University of Chicago Press, 1938).

15 For further argument to this effect, see Christopher Norris, Against Relativism: Philosophy of Science, Deconstruction and Critical The-ory (Oxford: Blackwell, 1997) and Philosophy of Language and the Challenge to Scientific Realism (London: Routledge, 2004).

16 On this topic see especially Bernard Williams, Truth and Truthful-ness (Princeton, NJ: Princeton University Press, 2004).

17 G. H. Moore, Zerlemo’s Axiom of Choice: Its Origins, Development and Influence (New York: Springer Verlag, 1982).

18 For further discussion, see E. Mendelson, Introduction to Mathe-matical Logic, 4th edn (London: Chapman & Hall, 1997).

19 Badiou, Deleuze: The Clamor of Being. 20 For a lucid survey, see G. Hughes and M. Cresswell, A New Intro-

duction to Modal Logic (London: Routledge, 1996).21 See Saul Kripke, Naming and Necessity (Oxford: Blackwell, 1981);

Hilary Putnam, Mind, Language and Reality (Cambridge: Cambridge University Press, 1975); Stephen P. Schwartz (ed.), Naming, Necessity and Natural Kinds (Ithaca, NY: Cornell University Press, 1975).

22 For a thorough treatment of these topics, see especially Irving M. Copi and Carl Cohen, Introduction to Logic (New York: Prentice-Hall, 2005); also – for a more concise exposition – W. H. Newton-Smith, Logic: An Introductory Course (London: Routledge, 1985).

23 W. V. Quine, Philosophy of Logic, 2nd edn (Cambridge, MA: Har-vard University Press, 1986).

24 Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1953).

25 For a useful discussion, see Anthony Weston, A Handbook for Argu-ments (Indianapolis, IN: Hackett, 2001), p. 48ff.

26 Michael Dummett, Truth and Other Enigmas (London: Duckworth, 1978) and Elements of Intuitionism, 2nd edn (Oxford: Oxford University Press, 2000).

27 Deleuze: The Clamor of Being28 See Note 26; also – for a dissenting view – Norris, Truth Matters:

Realism, Anti-Realism, and Response-Dependence (Edinburgh: Edin-burgh University Press, 2002).

NOTES

302

29 For further discussion, see Crispin Wright, Realism, Meaning and Truth (Oxford: Blackwell, 1987) and Truth and Objectivity (Cam-bridge, MA: Harvard University Press, 1992).

30 G. W. F. Hegel, Hegel’s Logic (trans.) William Wallace (Oxford: Clarendon Press, 1975).

Part VI. Quantity and knowledge. The discernible (or constructible): Leibniz/Gödel

1 See especially Paul Benacerraf, ‘What Numbers Could Not Be’, in Benacerraf and Hilary Putnam (eds), The Philosophy of Mathematics: Selected Essays, 2nd edn (Cambridge: Cambridge University Press, 1983), pp. 272–94; also W. D. Hart (ed.), The Philosophy of Mathe-matics (Oxford: Oxford University Press, 1996) and Hilary Putnam, Mathematics, Matter and Method (Cambridge University Press, 1975).

2 For more on ‘bald naturalism’, see John McDowell, Mind and World (Cambridge, MA: Harvard University Press, 1994).

3 Richard Harland, Superstructuralism: The Philosophy of Structural-ism and Post-structuralism (London: Methuen, 1987).

4 See Note 1; also Michael Dummett, Elements of Intuitionism, 2nd edn (Oxford: Clarendon Press, 2000) and Truth and Other Enigmas (London: Duckworth, 1978).

5 See especially the essays collected in Badiou, ‘Ontology is Mathe-matics’, Section One of Theoretical Writings, pp. 3–93.

6 See Notes 1 and 4. 7 See Gilles Deleuze, Difference and Repetition (trans.) Paul Patton

(London: Continuum, 2004); Deleuze and Félix Guattari, The Anti-Oedipus: Capitalism and Schizophrenia, Vol. 1 (trans.) Brian Massumi (Minneapolis: University of Minnesota Press, 1987); Badiou, Deleuze: The Clamor of Being; also Todd May, ‘Badiou and Deleuze on the One and the Many’, in Peter Hallward (ed.), Think Again: Alain Badiou and the Future of Philosophy (London: Continuum, 2004), pp. 67–76.

8 Deleuze and Guattari, Anti-Oedipus (op. cit.). 9 Dale Jacquette, Ontology (Montreal: McGill-Queen’s University

Press, 2002).10 Martin Heidegger, Being and Time (trans.) J. Mcquarrie and

E. Robinson (Oxford: Blackwell, 1962).11 Jacquette, Ontology (op. cit.), p. 29.12 Ibid, p. 79.13 Ibid, p. 42.14 Ibid, p. 45.15 See G. W. Leibniz, New Essays on Human Understanding (trans. and

ed.) Peter. Remnant and Jonathan Bennett (Cambridge: Cambridge University Press, 1996); The Monadology and Other Philosophical Writings (trans.) Robert Latta (Oxford: Oxford University Press,

NOTES

303

1925); Philosophical Essays (trans.) Roger Ariew and Daniel Garber (Indianapolis, IN: Hackett, 1989).

16 See Note 15.17 See Gottlob Frege, Conceptual Notation and Related Articles (trans.

and ed.) T. W. Bynum (Oxford: Clarendon Press, 1972). 18 See Leibniz, Monadology (op. cit.). 19 See Note 7.

Part VII. The generic: indiscernible and truth – the event: P. J. Cohen

1 See especially Jerrold J. Katz, Realistic Rationalism (Cambridge, MA: M.I.T. Press, 1998).

2 See Michael Dummett, Truth and Other Enigmas (London: Duck-worth, 1978); also – from a range of more-or-less qualified anti-realist standpoints – Crispin Wright, Truth and Objectivity (Cambridge, MA: Harvard University Press, 1992); Paul Benacerraf, ‘What Num-bers Could Not Be’, in Benacerraf and Hilary Putnam (eds), The Philosophy of Mathematics: Selected Essays, 2nd edn (Cambridge: Cambridge University Press, 1983), pp. 272–94; Neil Tennant, The Taming of the True (Oxford: Clarendon Press, 2002).

3 See for instance Michael Devitt, Realism and Truth, 2nd edn (Oxford: Blackwell, 1986); Norris, Truth Matters: Realism, Anti-realism and Response-dependence (Edinburgh: Edinburgh University Press, 2002); Stathis Psillos, Scientific Realism: How Science Tracks Truth (London: Routledge, 1999).

4 Paul J. Cohen, Set Theory and the Continuum Hypothesis (New York: W.A. Benjamin, 1966).

5 Badiou, ‘The Subtraction of Truth’, in Theoretical Writings, pp. 97–160.

6 See for instance Richard Rorty (ed.), The Linguistic Turn: Essays in Contemporary Philosophical Method (Chicago: University of Chicago Press, 1967).

7 See Note 2; also Dummett, The Logical Basis of Metaphysics (London: Duckworth, 1991) and The Seas of Language (Oxford: Clarendon Press, 1993).

8 See Notes 2 and 7. 9 Roland Barthes, Mythologies (trans.) Annette Lavers (London:

Jonathan Cape, 1972), 112. 10 Jacques Lacan, Écrits: A Selection (trans.) Alan Sheridan-Smith

(London: Tavistock, 1977). 11 Jean-Jacques Rousseau, The Social Contract (trans.) J. H. Tozer

(London: Wordsworth Classics, 1998).12 Martin Heidegger, Being and Time (trans.) John Mcquarrie and

Edward Robinson (Oxford: Blackwell, 1962).13 For further discussion see David Boucher and Paul Kelly (eds), The

Social Contract from Hobbes to Rawls (London: Routledge, 1994).

NOTES

304

14 For a reading of Rousseau that is comparable in certain respects, see Jacques Derrida, Of Grammatology (trans.) G. C. Spivak (Baltimore, MD: Johns Hopkins University Press, 1976).

15 See Note 13; also Gerald F. Gaus, Justificatory Liberalism: An Essay on Epistemology and Political Theory (New York: Oxford University Press, 1996).

16 William James, The Will to Believe and Other Essays in Popular Philosophy (New York: Dover, 1956).

17 John Stuart Mill, On Liberty (ed.) Gertrude Himmelfarb (Har-mondsworth: Penguin, 1985).

18 Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems (trans.) B. Meltzer (New York: Basic Books, 1962).

19 See especially Kurt Gödel, ‘What Is Cantor’s Continuum Problem?’, in Benacerraf and Putnam (eds), Philosophy of Mathematics (op. cit.), pp. 470–85; also Jerrold J. Katz, Realistic Rationalism (op. cit.).

20 See also Roger Penrose, The Emperor’s New Mind (Oxford: Oxford University Press, 1990).

21 On this hugely overworked topic, see Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1953), Sections 201–92 passim; Saul Kripke, Witt-genstein on Rules and Private Language: An Elementary Exposition (Oxford: Blackwell, 1982); Alexander Miller and Crispin Wright (eds), Rule-following and Meaning (Chesham: Acumen, 2002).

22 See especially Edmund Husserl, Logical Investigations, 2 vols (trans.) J. N. Finlay (New York: Humanities Press, 1970).

23 Gottlob Frege, ‘Review of Dr. E. Husserl’s Philosophy of Arith-metic’, (trans.) E. W. Kluge, Mind, Vol. 81 (July 1972), pp. 321–37.

24 See for instance Donald Davidson, ‘The Method of Truth in Meta-physics’, in Inquiries into Truth and Interpretation (Oxford: Clarendon Press, 1984), pp. 199–214.

Part VIII. Forcing: truth and the subject. Beyond Lacan

1 See for instance Anthony J. Cascardi, The Subject of Modernity (Cambridge: Cambridge University Press, 1992); E. Cadava, P. Connor and J.-L. Nancy (eds), What Comes After the Subject? (London: Routledge, 1991); Paul Smith, Discerning the Subject (Minneapolis: University of Minnesota Press, 1988).

2 See Note 1; also – for a vigorous literary-theoretical take on this topic – Sean Burke, The Death and Return of the Author (Edinburgh: Edinburgh University Press, 1992).

3 Baruch Spinoza, Ethics (trans.) Edwin Curley (London: Penguin, 1996); also Christopher Norris, Spinoza and the Origins of Modern Critical Theory (Oxford: Blackwell, 1991).

NOTES

305

4 For a commanding overview, see François Dosse, History of Struc-turalism, 2 vols (trans.) Deborah Glassman (Minneapolis: University of Minnesota Press, 1997).

5 See especially Jacques Lacan, Ecrits (trans.) Alan Sheridan (London: Tavistock, 1977) and Louis Althusser, For Marx (trans.) Ben Brewster (Harmondsworth: Penguin, 1969).

6 See especially E. P. Thompson, The Poverty of Theory, 2nd edn (London: Merlin, 2004).

7 See Michel Henry, The Genealogy of Psychoanalysis (trans.) Douglas Brick (Stanford, CA: Stanford University Press, 1993) and David Macey, Lacan in Contexts (London: Verso, 1988).

8 For a good synoptic treatment, see Peter Sedgwick, Descartes to Derrida: An Introduction to European Philosophy (Oxford: Blackwell, 2001); also Gary Gutting, French Philosophy in the Twentieth Century (Cambridge: Cambridge University Press, 2001).

9 Lacan, ‘The Insistence of the Letter in the Unconscious’, (trans.) Jacques Ehrmann, Yale French Studies, Nos. 36/7 (1966), pp. 112–47.

10 Donald Davidson, ‘The Method of Truth in Metaphysics’, in Inqui-ries into Truth and Interpretation (Oxford: Clarendon Press, 1984), pp. 199–214.

11 Paul Cohen, Set Theory and the Continuum Hypothesis (New York: W.A. Benjamin, 1966).

12 Kurt Gödel, The Consistency of the Continuum-Hypothesis (Prince-ton, NJ: Princeton University Press, 1940).

13 For further discussion see J. Ferreiros, ‘The Notion of Cardinality and the Continuum Hypothesis’, in Labyrinth of Thought: The History of Set Theory and its Role in Modern Mathematics (Basel: Birkhäuser, 1999), pp. 171–214.

14 See J. R. Lucas, The Conceptual Roots of Mathematics (London: Routledge, 2000) and Penelope Maddy, ‘Believing the Axioms, I’, Journal of Symbolic Logic, Vol. 53, No. 2 (1988), pp. 481–511.

15 For further discussion see Thomas J. Jech, The Axiom of Choice (New York: Dover, 1973).

16 See especially Richard Rorty, Objectivity, Relativism, and Truth (Cambridge: Cambridge University Press, 1991).

17 See Notes 5 and 9; also Lacan, The Language of the Self: The Function of Language in Psychoanalysis (trans.) Anthony Wilden (Baltimore, MD: Johns Hopkins Press, Baltimore, 1968); The Seminar XI, The Four Fundamental Concepts of Psychoanalysis (ed.) Jacques-Alain Miller, (trans.) Alan Sheridan (New York: W.W. Norton, 1977); The Seminar XX, Encore: On Feminine Sexuality, the Limits of Love and Knowledge (ed.) Miller, (trans.) Bruce Fink (New York: Norton, 1998).

18 Lacan, ‘The Insistence of the Letter’ (op. cit.).19 See for instance Raymond Tallis, Not Saussure: A Critique of Post-

Saussurean Literary Theory (Basingstoke: Macmillan, 1988). 20 On post-structuralism and its relativist/constructivist/anti-realist

excesses, see Christopher Norris, Uncritical Theory: Postmodernism,

NOTES

306

Intellectuals and The Gulf War (London: Lawrence and Wishart, 1992) and Reclaiming Truth: Contribution to a Critique of Cultural Relativism (Lawrence and Wishart, 1996).

21 See René Descartes, ‘Rules for the Direction of the Mind’ in Descartes: Philosophical Writings (trans.) Peter Geach and G. E. M. Anscombe (London: Thomas Nelson, 1954), pp. 153–80.

22 Immanuel Kant, Critique of Pure Reason (trans.) N. Kemp Smith (London: Macmillan, 1964). See Chapter I, Book II, Division II of Part II, ‘Transcendental Doctrine of Elements’.

23 See especially Jean-Paul Sartre, The Transcendence of the Ego (trans.) Forrest Williams and Robert Kirkpatrick (New York: Noonday Press, 1948).

24 See Rorty, Philosophy and the Mirror of Nature (Oxford: Blackwell, 1980).

25 Kant, Critique of Pure Reason (op. cit.).26 For a perceptive commentary on this system of ‘rotating chairman-

ship’ between the various Kantian faculties, see Gilles Deleuze, Kant’s Critical Philosophy: The Doctrine of the Faculties (trans.) Hugh Tomlinson and Barbara Habberjam (London: Athlone Press, 1984).

27 For more on this semantic history, see Raymond Williams, Keywords: A Vocabulary of Culture and Society, rev. edn (Oxford: Oxford Uni-versity Press, 1985).

28 Rorty, Philosophy and the Mirror of Nature (op. cit.); also Ian Hacking, Why Does Language Matter to Philosophy? (Cambridge: Cambridge University Press, 1975).

29 See especially Badiou, Ethics.

307

absolute infinity 55absolute knowledge (Hegel) 145,

206activist engagement 44agency 28–30Althusser, Louis 58, 90, 99, 100,

149, 263–4Althusserian structural

Marxism 121–2analytic philosophy 12, 15, 21–2,

161, 182Anaximander 64ancient Greece 23

culture of 115–16drama 201–2philosophy 106–15

anomalies 190, 198–9anti-Cartesian metaphysics 102anti-humanism 121–2anti-philosophers 58, 166–7anti-realism 22–3, 34, 35, 91,

113, 183–90, 223, 225, 231Aristotle 61, 71–9, 83–4, 89–90,

106, 107, 129, 164, 243arts 2, 19, 26, 27, 109, 191,

202, 203astronomy 24atheism 169Austin, J. L. 3, 16authenticity 125autonomy 29axiomatic-deductive

reasoning 43, 45, 111, 200, 254

axiomatic truth-procedure 71axiom of choice 138–43, 174–83,

268

Bachelard, Gaston 49, 58bad infinity 146, 149Barthes, Roland 155–6Bataille, Georges 148being 8–9, 11, 37–9, 61, 63, 64,

73, 266–7vs. beings 21concept of 55impasse of 269subtractive character 68

being/event dichotomy 26–7, 32–3, 79–80, 185, 210

belonging 48, 58, 78, 79–88, 92, 103, 132, 155, 180, 181

bivalence 186Boyle, Robert 72, 73, 77breakthrough discoveries 19–20

Canguilhem, Georges 49, 58Cantor, Georg 5, 39, 112, 128–9,

136–7, 167, 186, 192, 193, 195–7, 219, 225, 237–8, 267–8

Cantor’s theorem 50–64, 66cardinality 196, 267cardinals 195, 196, 205Carnap, Rudolf 20–1causal explanations 179causation 64Cavaillès, Jean 84change 37–9chaos, order from 64choice, axiom of 138–43, 174–83,

268Christianity 134, 167–72citizenship 244cogito 271, 273–4, 276, 280

INDEX

INDEX

308

Cohen, Paul 6, 26, 33, 113, 221, 225, 230–6, 251–2, 265–8

Cold War propaganda 14–15collectives and collectivity 27–8combinations 210commonalities 31common-being 34–5consciousness 126, 145, 161, 275conscious subject 121conscious thought 162consistency 155consistent multiplicity 40–1, 47–8,

62–5, 81, 88, 175, 190, 227constructivism 41, 222, 223, 225containment 52context of discovery 76, 172,

177, 200, 216, 261context of justification 76, 172,

177, 200, 216continental philosophy 12, 15,

21, 22, 35, 48, 182contingency 210–11continuum hypothesis 268Copernican-Galilean

astronomy 24Copernican revolution 131, 277cosmos, creation of 64count-as-one 40–2, 46, 47, 50,

60–2, 68, 78, 79–89, 92–3, 94, 101–2, 104, 112, 147, 175, 224–6

counterintuitive truths 69 counter-state 181count-of-the-count 86critical theory 32, 67Critique of Dialectical Reason

(Sartre) 153, 154, 156crypto-theology 135cultural-aesthetic

nationalism 116cultural construction 155–6cultural left 17cultural relativism 15, 16–17, 18

cultural theory 31, 67culture-transcendent truths 57

Davidson, Donald 267deductive reasoning 187Deleuze, Gilles 43–5, 97–8,

99–100, 102, 178, 200–1, 203depth-hermeneutic approach

20–1, 109Descartes, René 30, 49, 58, 61,

121, 135, 166, 270–83determinism 28, 96, 159, 211diagonalization (Cantor) 196–7,

207, 219difference 18, 31, 51, 66, 91differential calculus 43, 201, 215discernibles 222discoveries 19–20, 67, 69,

112–14, 211discovery, context of 76–7, 173–4,

177, 199–200, 216, 261disenfranchisement and social

injustice 7–8, 89disputed class (Dummett) 197double-negation-elimination 184,

187doxa 38drama 116, 201, 202dualisms 86, 104

false 1–2natural/non-natural 152phenomenal/noumenal 131

Dummett, Michael 183, 186, 197Dutch Free Republic 98

Easton, W. B. 199Easton’s Theorem 199ego 162ego-psychology 229, 281Einsteinian Relativity 24empiricism 60end-of-history thesis 157, 164end-of-ideology thesis 157, 164

INDEX

309

epistεmε 38epistemically constrained

truth-values 183–4epistemology 10, 30, 84–5,

161–4, 258epochal events 156, 160, 191, 211errancy 64eschatological

verificationism 167ethical judgements 162ethics 67, 103, 177, 203Ethics (Spinoza) 97–8evental site 118, 179, 190, 195, 226event(s) 9, 11, 96, 155, 158–9

being and 32–3, 79–80, 266epochal 156, 160, 191, 211historical 164past 158–9poem as 122–7situations and 159, 168truth and 160–5unpredictability of 125–6

excess, theorem of the point of 81

excluded middle 184, 185, 187exclusion 50, 67excrescences 94–6, 153, 190existence 38, 68, 193existentialism 33extensionalist approach

(set theory) 52, 65, 90–1

fact/value distinction 66, 94faith 166–8, 169–70falsehood 184–5Fermat’s Last Theorem 76fidelity 84, 109, 125, 140, 159,

164, 177–82, 199, 200, 225finitism 134first-order statements 54forcing 33–4, 43, 113, 219–24,

232–5, 236–8, 265–6formal languages 60–1

forms of life (Wittgenstein) 31, 58, 181, 203

Fraenkel, Abraham 57freedom 29, 104–5free will 159‘free-world’ liberalism 14–15Frege, Gottlob 16, 39, 52, 53, 101French philosophy 32, 49, 136, 149French Resistance 84French Revolution 152, 156, 160,

165Freud, Sigmund 20, 271, 272Freudianism 229

Galileo 132, 133, 135, 136, 192general ontology 208–9general will 241, 242generic, the (Cohen) 43, 113,

190, 206, 221–7, 233, 265generic extension 256–7, 269generic procedure 27, 29German culture 116God 134, 135, 141, 167, 169–71,

206, 213–14Gödel, Kurt 54–5, 254, 259,

267–8grand narratives 156Greek drama 116, 201, 202Greek philosophy 106–15group-in-fusion (Sartre) 153

Harland, Richard 194Hegel, George Wilhelm

Friedrich 61, 129, 143–52, 156, 181, 187

Heidegger, Martin 19–21, 23, 30–1, 73, 106–8, 114, 116, 118–19, 121–2, 208, 242–3

heliocentric hypothesis 135hermeneutics 3, 41, 270historical judgements 163–4historical revisionism 160historical situations 156

INDEX

310

history 151–60of the present 164representing 160–5

Hobbes, Thomas 72, 77–8, 241Hölderlin, Friedrich 21, 115–21homogeneity of nature 131human agency 28–30human rights 15Hume, David 187Husserl, Edmund (and

phenomenology) 106–7, 121–2, 255

hypotheses, far-reaching 44

identicals, indiscernibility of 212, 214

identity of indiscernibles 38identity politics 91imaginary misrecognition 105impasse of being 269inclusion 48, 50, 66, 78, 79–89,

91, 92, 103, 155, 180–1, 192incompleteness theorem

(Gödel) 254inconsistency 47–8, 62, 87–8inconsistent multiplicity 39,

40–1, 46–9, 60, 62–4, 81, 87, 175, 190, 227

indiscernibility 219–27, 251–9indiscernibles 38, 190, 212, 214,

233–4, 265indiscernment 269individual 92infinite/infinity 5, 55, 66, 74, 192,

204, 205–6absolute 55axiom of choice and 138–43‘bad’ 146, 149concept of 128–37, 142Hegel on 143–51of nature 135–7paradoxes of 136positive 145

quantitative 144religious belief and 167,

169–70‘sizes’ of 193supra-rational idea of 170

infinite modes 104–5infinite sets 81, 129–30, 267

multiple orders of 5subsets of 5

internalism 162–3intrinsic size 195intuition 51, 52, 121intuitionists 22–3intuition of objects 52

Jacquette, Dale 208–9, 210justification, context of 76

Kant, Immanuel 1, 15, 23, 30, 131, 172, 277–8

knowledge 161–2absolute (Hegel) 145, 206acquisition of 199–200,

251–9advances in 56–7, 112–14,

202consciousness and 161generic procedure for 27humanly attainable 278limits of 23mathematical 56–7, 113, 133,

183, 197, 220, 232–3, 245, 255

present-best state of 187, 192, 198, 220–8

problem of 84transcendence of current 43truth and 8–9, 23, 24–5, 34,

57, 62, 99, 102, 183–9, 192, 219–28, 230–2, 251–8, 261–2

Koyré, Alexander 136Kuhn, Thomas 154

INDEX

311

Lacan, Jacques 20, 149, 162, 229, 263–5, 270–83

Lacanian psychoanalysis 28, 90, 122, 229–30, 272, 281

language 10, 23, 35, 67, 68, 149–50, 203, 262–3

depth-hermeneutical idea of 109

formal 60–1logical 210–12of mathematics 132–3natural 57–8, 60, 132–3ordinary 3, 58, 110, 150philosophy of 163, 258poetic 21, 23, 109of the situation 220truth and 97

language-first approach 3–4, 223, 273

language-games (Wittgenstein) 15–16, 31, 181, 203

language-independent truths 57Lautman, Albert 84leap of faith 170, 200Leibniz, Gottfried 24, 38, 58, 61,

63, 64, 163, 181, 210–18Lenin, Vladimir 20, 95les évènements 15Levinas, Emmanuel 31liberal democracy 7, 92liberal-humanism 29liberalism, ‘free-world’ 14–15liberal pluralism 16–17life-forms 15–16, 31, 58, 181, 203limit ordinal 142–3linguistic constructivism 28linguistic philosophy 3–4linguistic turn 3–4, 15–16, 43,

157, 162, 212, 223, 261literary criticism 118Locke, John 241logic 212

anti-realism and 183–9modal 179–80

logical language 52, 210–12logical rules 181logico-semantic distinctions 21logos 38love 2, 26, 27, 202, 270Lyotard, Jean-François 31, 156

Macherey, Pierre 149Mallarmé, Stéphane 109, 115–16,

119–27, 142many, priority of over the

one 49, 50Marx, Karl 15, 20, 89Marxism 75, 85, 90, 99, 100, 122,

199, 228–9mathematical knowledge 56–7,

113, 133, 183, 197, 220, 232–3, 245, 255

mathematical thought, subtractive dimension of 69

mathematical truth 22–3, 219–20mathematics 4–5, 9, 67, 68, 203

advances in 19, 56–7, 182, 200, 230

axiomatic-deductive approach to 45

as basis of ontological inquiry 10, 30, 32, 38–49, 67–8, 98–9, 187, 192, 211

discourse of 23, 190–1Hegel on 144–51history of 144language of 132–3philosophy and 24–6, 30–1, 101poetry and 106–15, 120politics and 7–8, 89, 94, 110, 112pre-eminence of 35

membership (set theory’) 50–1, 52, 60, 78, 155

meta-narrative 148meta-ontological truths 25, 83

INDEX

312

meta-ontology 25, 30, 31, 46, 83metaphysical determinism 211metaphysics 20, 96, 106, 107, 120metastructure 81, 86militants of truth 10, 18, 125,

140, 154, 168, 199, 217, 260Mill, John Stuart 169, 247mobilization 153, 154modal logic 179–80monist ontology 96, 98, 103,

104, 212monotheism 134multiple(s) 67–8

comparison of 192–3concept of the 42indiscernibility of 251–2multiplicity of 128–37, 197, 213natural 131–2, 193, 194priority of over the one

39–40, 49theory of the pure 50–64typology of the 112

multiplicity 35, 39, 40, 46consistent vs. inconsistent 39,

40–1, 46–9, 60, 62–4, 81, 87, 175, 190, 227

infinite, of multiples 128–37of multiples 197, 213

nationalism 116–18National Socialism 118, 243naturalism 189–201natural language 57–8, 60, 132–3natural multiples 131–2, 193, 194natural numbers 136natural sciences 19, 191, 203natural situations 156nature 108, 114, 115, 132, 155

homogeneity of 131infinity of 134–5, 136, 141vs. non-natural 151–2

Nazis 243Nazism 118, 243

necessitarianism 159necessity-operator (modal

logic) 180negative theology 71, 206neopragmatism 41Neurath, Otto 20Nietzsche, Friedrich 116non-being 61, 111, 208non-belonging 221non-contradiction 212non-Euclidean geometry 24non-intuitive truths 69non-natural 151–2normality 114, 115Nouveaux Philosophes 157null set 68

objective truth 21–2, 102, 257objectivist realism 253–4object/subject dialectic 199,

200, 269ontological advances 45ontological inquiry 23–4

mathematics as basis of 10, 30, 32, 38–49, 67–8, 98–9, 187, 192, 211

subtractive character of 61, 62ontology 2, 8–11, 17, 30, 33–4,

41–3, 79–80, 163, 182, 192–3, 206–7

general 208–9limit of 200meta-ontology 25, 30, 31,

46, 83monist 96, 98, 103, 104, 212open 43philosophy and 24–5prerequisites for 40realist 62social-political 48subtractive 72, 222–8, 251

oppressed minorities 7–8order, from chaos 64

INDEX

313

ordinals 130–1, 131, 141–2, 195ordinary language 3, 58, 110, 150orientations 201–10

paradox into concept 255Parmenides (Plato) 37–41, 46–9,

68, 69Pascal, Blaise 70–1, 76, 165–74,

217past events 158, 159phenomenal/noumenal

dualism 131phenomenology 33Phenomenology of Mind

(Hegel) 144, 145, 148philosophy

Badiou’s view of 2–5, 11–12, 24–5, 33

continental 2, 12, 15, 21, 35, 48, 182

first 106–15of language 163, 258linguistic 3–4mathematics and 24–6, 30–1,

101meta-ontological role of 30–1ontology and 24–5psychoanalysis of 75relationship to other

disciplines 19–20task of 33

physics 74, 77, 134Plato 23, 24, 30, 35, 56, 61, 102,

106–7, 120, 277Parmenides 37–41, 46–9, 68–9Timaeus 64

Platonism 37, 102, 254pluralism 181poetic intuition 121poetry 20, 21, 23

of Hölderlin 115–22of Mallarmé 115, 122–7mathematics and 106–15, 120

point of impossibility 78political disenfranchisement 92political emancipation 86political judgements 162, 163political justice 71politically oppressed,

mobilization of 154political philosophy

Kant’s 15set theory and 7–8

political realism 157–8political representation 7political theory 45politics 2, 7, 19, 26, 27, 44, 67,

202, 203mathematics and 7–8, 89, 94,

110, 112rethinking of 151–60Rousseau on 241–51

positive infinity 145post-Cantorian set theory 19–20,

128post-evental truth 228–40postmodernism 3, 41post-structuralism 3, 28, 41, 121,

156, 157power set 66, 81, 83, 84, 129practical syllogisms 164presentation 45–6, 47, 61, 138,

155, 201primitive terms 46pseudo-dichotomy 45psychoanalysis 20, 28, 121, 122,

162, 229–30, 264–5, 271, 272, 281

psychoanalysis of philosophy 75

pure multiple 53Pythagorean numerology 69

quantitative infinity 144quantity 193Quine, W. V. 38, 59–60

INDEX

314

radical empiricism 60radicalism 157rational autonomy 159rationalist metaphysics 96rationalists 49, 58rational reconstruction

(philosophy/history of ideas) 48

Rawls, John 242realist ontology 62reason 49, 166–7, 211, 214, 215reasoning

axiomatic-deductive 43, 45, 111, 200, 254

deductive 187more geometrico 97, 99, 100,

103–5recognition-transcendent

truth 62–3reductio ad absurdum 183, 184, 188reference 90regime of interpretation 20relativism 17, 24religion 70, 166–70religious belief 166–70representation 88, 93, 155, 201revolutionary praxis 153revolutions 153–4, 156–9, 176rights 29Rorty, Richard 3–4, 31, 166, 269,

270, 275Rousseau, Jean-Jacques 241–51rule-following

(Wittgenstein) 138–43rules 137, 181Russell, Bertrand 6, 16, 39, 52,

53, 54, 55, 87, 101, 128, 169Russian Revolution 152–3

sans-papiers 7, 65Sartre, Jean-Paul 28, 121, 122,

153, 154, 156, 214Saussure, Ferdinand de 263

Scanlon, Thomas 242science 2, 26, 27, 178, 202

advances in 72, 183–9history of 136

self-conscious subject 121, 145, 274, 275

self-consciousness 161selfhood 90self-referring expressions 6, 54semio-linguistics 263sense 90sense-certainty 144, 145senses, evidence of the 38September 11, 2001 160sets

vs. combinations 210elements of 81infinite 5, 81, 129, 267membership in 50, 51, 52,

60, 65null 68power set 81, 83, 129subsets of 128, 129transitive sets 131–2void 68without elements 68

set theory 5–7, 23, 26, 42, 186, 201, 206, 207, 259

advances in 57, 67advent of 9, 48challenge of 52–3impact of 83modern 50–2, 105paradoxes in 52–5philosophy and 12political philosophy and 7–8post-Cantorian 19, 128ZF system 57–60, 267

singularities 8, 117, 153, 154, 195, 214

situations 8, 41, 42, 80, 81, 228–40

events and 159, 168

INDEX

315

Social Contract (Rousseau) 241–51

social democracy 112social justice 7, 66, 93, 242social ontology 7, 8, 48societal change 8socio-political sphere 65Socrates 49–50, 103sophistry 24sophists 58, 166, 203Soviet Union 156, 160Spinoza, Baruch 58, 61, 63, 64,

79, 96–106, 163, 194, 211, 241–2, 261

stability 87–8Stalin, Joseph 156, 178state 88, 89–90, 92, 95–6, 155state of the situation 8, 80, 81,

86, 197–201, 234, 251St. Paul 70, 76, 125, 166, 168strong programme (sociology of

knowledge) 15, 24, 72, 77structuralism 28, 90, 99, 121–2,

149–50, 156, 264structure 81Structure of Scientific Revolutions

(Kuhn) 154concept of 260, 278conscious 121link between truth and 30–1relationship between event

and 33role of 244theory of the 259–70truth and 234–5, 261–2

subjectivity 22, 181–2subject/object dialectic 199,

200, 269subject of truth 234–5subjects 89subsets 66, 128, 129subtractive ontology 72, 222–8,

251

sufficient reason, principle of (Leibniz) 212, 214

supernumerary elements 40, 175, 190–1

‘superstructuralism’ 194supra-sensory forms (Plato) 39

‘talking cure’ (psychoanalysis) 229

theology 55, 134–5negative 71, 206

theorem of the point of excess 81–2

theory of the pure multiple 50–64

theory of the subject 259–70Theory of Types’ (Russell) 54, 128Thermidorean political

culture 157Timaeus (Plato) 64time 37–8totalization 51Tractatus Theologico-Politicus

(Spinoza) 98transcendence 194–5, 205transfinite numbers 5, 55transformative thought 201–10transitive sets 131–2transmissible thinking 111Trotsky, Leon 156, 178truth

absolute 193attainment of 43axiomatic conception of 43concept of 107culture-transcendent 57epistemically constrained 113,

183–4event and 160–5falsehood and 184–5knowledge and 8–9, 23, 24–5,

34, 57, 62, 99, 102, 183–9, 192, 219–28, 230–2, 251–8, 261–2

INDEX

316

truth (Cont’d)language and 97language-first approach to 3–4language-independent 57link between subject and 30–1mathematical 22–3, 219–20meta-ontological 25militants of 10, 18, 125, 140,

154, 168, 199, 217, 260non-intuitive 69objective 21–2, 102, 257ontological enquiry and 10–11ontology and 42–3post-evental 228–40subject and 234–5, 261–2subjectivity and 180–1vs. truthfulness 173universal 18, 31, 70verification-transcendent 183

193truth-event 32, 269truthfulness 84–5, 173, 261truth-procedures 11, 76, 188–9,

232, 259, 261, 265, 268, 271, 278–81

‘two cultures’ controversy 110tyranny of the majority 247

ultra-one 160, 163, 167, 179, 190, 214

unconscious 162, 263, 265, 271, 272

‘Un coup de dés’ (Mallarmé) 119, 122–126

undecidability-proof (Gödel) 54

United States 95unity 39–40universalism 67, 70, 91universal truths 18, 31universe, infinite 136unknown, the 113un-measure 74

vacuum, impossibility of (Aristotle) 71–9

validity 54, 266veracity 277veridical, concept of 222, 224verification-transcendent

truth-values 183 193Vienna Circle 19, 20virtue-based epistemology 84, 85void 65–6, 71–9, 88, 110, 190,

214–15void set 68

Western metaphysics 20, 106, 107, 120

Wiles, Andrew 76Williamson, Timothy 161Wittgenstein, Ludwig 3, 4,

15–16, 31, 41, 108, 150–1, 166, 181, 182, 223

working mathematicians 24, 25world-transformative events 161

Zeno 38Zermelo, Ernst 57Zermelo’s principle 60ZF system 57–60, 267

chi stop her norris badiou

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