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Differentiation and Distinction on the problem of individuation from Scotus to Deleuze with an appendix on the history of calculus1 gil morejn, 5.26.2015

Introduction This paper takes up the philosophical problem of individuation, or differentiation. We will argue that there are important resources to be found in Duns Scotus and Gilles Deleuze that allow us to articulate a crucial insight: that what exists prior to differentiation or individuation, which is figured variously as the common nature and the virtual, is not at all undifferentiated. This is an important insight because it forces a re-evaluation of the idea of possibility, which was classically understood as the negation of actuality.2 Hegels famous critique, that Schellings philosophy of Indifferenz amounted to the night where all cows are black, exposes the paucity of such a conception, through which we ultimately are left incapable of accounting for the reality of actual individuals or distinguishing between empirical instances.3 However, ironically, we would claim that the Hegelian hypothesis of an ontologically absolute process of self-negation is similarly inadequate, insofar as it posits the relationship between possibility and actuality as one of an abstract, that is, insufficiently dialectical negation, in which the passage from the chaos of the latter to the distinctness of the former remains inexplicable or mysterious. While we will not pursue this critique here, what we will suggest is that the positive notion of common nature or virtuality, as something both completely determined and undifferentiated, far exceeds in its explanatory capacities a negative notion of possibility as purely indifferent.4 But in

1 Page 18. 2 In particular here we have in mind Aristotles Metaphysics, in which he argues for the temporal and logical priority of actuality to potentiality. Cf., for instance, Metaphysics Z.8 3 Hegel, Phenomenology of Spirit, Preface, 16. 4 Similar notions can be found in Gilbert Simondon, who with his concept of metastable states as determinate but preindividual systems suffused with potentials clearly influenced Deleuze; and Manuel DeLanda, whose Deleuzian-inspired philosophy of science invokes phase spaces and possibility spaces. Miguel De Beistigui, John Protevi, and Alberto Toscano have all contributed substantially to

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order to think this pre-individual nature, to pose philosophically the problem of genesis, as Scotus already demonstrates and as we will see, it is necessary to abandon hylomorphism.

Why do we say individuation or differentiation? We mean this not in the sense of an alternative but in the sense of a Spinozist Deus sive Natura that is, as a polemical formulation indicating an inclusive disjunction.5 To the extent that one thinks the individual under the aspect of identity, one might conclude that individuation and differentiation are opposed. However, the individual is precisely distinct insofar as it is not identical to other individuals, although the positive account of in what this nonidentity consists may remain obscure. For this reason, as we will see, Scotus is already able to develop a concept of singularity whose fundamental trait is its irreducibility or uniqueness with regard to what is common, which is opposed to a concept of individuality whose fundamental character is its indifferent participation with regard to a universal. Moreover, this peculiar coincidence of irreducibility and commonness in actual singularity cannot, as we will see, be understood in terms of negation, but only by reference to the positivity of a difference. In other words, the process by which differentiation unfolds is precisely the genesis of individuals.

In this paper, we will first assess Scotus critique of the conceptual inadequacy of Aristotelian hylomorphism to the issue of individuation: neither form, matter, nor their relationship can provide a convincing account of what constitutes an individual. Scotuss exhaustive negative-critical analysis yields the positive result that some minimal immanent criteria are ascertained regarding the problem of individuation and the concept of common nature. Subsequently, we will turn to Deleuze, and in particular to his interpretation of differential calculus, since it is here that he uncovers substantial resources for thinking the process of differentiation, which he articulates in terms of the actualization of a problematic Idea (his theory of dialectics). Thus we will ask: why does Deleuze turn to the calculus, and what is at stake in his interpretation of it? For although Deleuzes reference to the calculus does not commit him to a mathematical ontology la Badiou, he nevertheless clearly insists that neither is it to be understood as metaphoric. As we will argue, the metaphysics of differential calculus

research into Deleuzes relationship to the philosophy and history of the problem of individuation. References for all of these works can be found in our bibliography at the end of this essay. 5 Cf., for example, Spinoza, Ethics, IV, Preface: "the eternal and infinite being, whom we call God, or nature, acts by the same necessity whereby it exists." For a thoughtful discussion of this rhetorical tactic in Spinoza and contemporary philosophers following in his footsteps, see Montag, Althusser and His Contemporaries, 116.


provides a model for grasping how, according to Deleuze and following the insights of Scotus, pre-individual reality is completely determined in a way which is obscure (rather than clear) and distinct.

I. Scotus immanent critique of individuation Duns Scotus asks: how can we account for individuality? To reformulate the question in such a way that elucidates the intimate connection between individuation and differentiation, we might say: How is a specific difference the difference that distinguishes the essence of one species from that of another within the same genus or generic difference itself differentiated into individuals of that species? To answer this question, the Aristotelian framework that formed the conceptual and metaphysical basis for much of the scholastic tradition would prove inadequate. Scotus exhaustive method shows that none of the received categories can possibly account for this differentiation of specific differences, and thus that this problematic of individuation requires the development and deployment of new categories entirely: no longer the universal, but the common; no longer the individual, but the singular. Let us follow his path and join in the exhaustion, seeking the point at which there are no more possibilities; from this vantage point, significant characteristics of the common nature will become apparent.6

In Ordinatio II, Distinction 3, Part 1, Scotus concerns himself with the issue of individuation, rejecting in turn the following suggestions for what could count as a principium individuationis:

1. that things are individual by virtue of their nature (question 1); 2. that things are individual by virtue of a double negation (q. 2); 3. that things are individual by virtue of their act of existence (q. 3); 4. that things are individual by virtue of their accidents, especially quantity (q. 4); 5. that things are individual by virtue of their matter (q. 5).

Numbers 1 and 2 establish the coordinates within which the rest of his investigation proceeds: they make it clear that our principle must be something positive which is distinct from the individuals specific difference. The subsequent questions, all answered in the negative, all serve to demonstrate the inadequacy of the accounts of his predecessors and interlocutors, insofar as they seek to locate the principle of

6 Cf. Deleuze, The Exhausted.


individuation in the already existing individual (3), some one of its accidental qualities (4), or in the inadequate hylomorphic concept of matter (5). 1. The first possibility, that a thing is individual by virtue of its very nature, or from itself, is the position Scotus attributes to Roger Marston.7 This would mean that the very nature of a thing, or its specific difference, is that by virtue of which there are individuals of that species. However, his leads to the problematic contention that the nature of stoneness is what accounts for the individual differences between this and that stone, whereas it seems clear that what two things have in common cannot be that by which they are distinguished from each other.8 Scotus argues that there is some real unity in the thing that is less than numerical unity or the proper unity of the singular. This unity belongs to the nature in itself.9 In other words, that which is held in common is real, but its unity is less than numerical insofar as it is insufficient on its own terms to constitute any this it does not individuate on its own.

This common nature, as less than numerical, is nevertheless not incompatible with universality, or else we would not be able to talk about what it is to be a stone, nor with individuality, since there are indeed individual stones.10 Consider that when we say: these two are both stones, we mean at once that these two are both stones, or really have something in common, and at the same time that these two are both stones, that is, they are nevertheless distinct from one another, or they are individuals. Universality and individuality, therefore, are both compatible with the nature insofar as it is common, but insofar as it is less-than-numerical it is neither universal nor singular on its own: although [the nature] never really exists without some of these [features], nevertheless of itself it is not any of them. Rather, [the nature] is naturally prior to all 7 Here and in what follows, we are not concerned with whether Scotus representations of his interlocutors Roger Marston, Henry of Ghent, and so on are accurate or fair; even if he misrepresents them, we are concerned with the logic of his arguments. 8 This is the same argument that Spinoza makes in Ethics I, Proposition 5, according to which there cannot be more than one substance with a given attribute. For if an attribute is what the intellect perceives as constituting the essence of a thing, and we say that two things are distinct insofar as they have the same attribute, then we are claiming that we distinguish them from one another by virtue of the essence they have in common, which is absurd. (Cf. Spinoza, Ethics, I P5 Dem.) 9 Scotus, Ordinatio II, Distinction 3, Part 1, Question 1, 30. Every reference to Scotus in what follows will be from this First Part of Distinction 3 of Book II of the Ordinatio, commonly referred to as the Treatise on the Common Nature and its Individuation; thus we will cite this hereafter as Treatise with Question and Paragraph number. 10 It would be interesting here to compare the logic of this objection to those raised by Malebranche's fictional interlocutor Aristes in his Dialogues on Metaphysics and Religion, regarding the conditions for the possibility of generalizing concepts, in Dialogues I and II. These powerful objections were very influential in the development of Hume's empiricism.


of these [features].11 The common nature is neither universal nor particular, neither singular nor plural; rather, it is prior to and compatible with receiving any of these determinations.

Scotus argues that we must be careful not to think that the nature is common insofar as we understand it, and singular insofar as it really is out there in the world, which would simply repeat the nominalist position. This is inadequate because common natures are not abstractions:

Commonness is suitable to the nature outside the intellect, and likewise singularity: commonness is suitable to the nature from itself, while singularity is suitable to the nature through something in the thing that contracts [the nature]. But universality is not suitable to the thing from itself. And so I grant that a cause of universality should be looked for, [as maintained in n.6]. Nevertheless, no cause of commonness other than the nature itself need be looked for. Once commonness is postulated in the nature itself according to its proper beingness and unity, one must necessarily look for the cause of singularity, which adds something more to the nature to which it belongs.12

Scotus recognizes, on the one hand, that insofar as we agree that something is held in common between individuals, it must be something real. But, on the other hand, and against the (Platonic) realists, he holds that it is in fact singulars which are ultimately actual, since only they, and not abstract universals, have numeric unity; this numeric unity is the something more they add to the common nature. Hence when we say that the common nature has less than numeric unity, as Paolo Virno argues, this amounts to a realism of the Common and a nominalism of the Universal.13 Now, on the basis of this formulation of the common nature, we will need to ask the question of individuation anew: what is it that, as he puts it, contracts the nature to individuality? 2. Scotus moves immediately to argue that this contracting factor must be something positive, contra Henry of Ghent, for whom individuation is the result of a twofold negation. According to the latter, an individual is not its species, and it is not any other individual of its species. On this account, since singularity or individuality only mean a double negation, it is not necessary to seek something positive as its cause, for the negation is sufficient.14 However, Scotus asks, what do we

11 Scotus, Treatise, Q1 32 12 Scotus, Treatise, Q1 42 13 Virno, Angels and the General Intellect, 62 14 Scotus, Treatise, Q2 44


mean when we say that something is an individual? We mean, above all else, that it cannot be divided into subjective parts: [there is something] to which being divided into many parts of which any given one is that thing is formally incompatible.15 While the species human may be divided into members of that species, an individual human cannot be divided without being destroyed as such. As Timothy B. Noone writes, for Scotus instantiability is contradictory with the notion of individuality as such.16 The question is, by virtue of what is an individual incompatible with being divided? And as Scotus argues, the answer cannot be a negation or privation, but must be something positive: nothing is simply incompatible with some being through a privation in it alone, but rather through something positive in it.17 Or again: No imperfection is formally incompatible with something except through some perfection.18 Individuality, or incompatibility with being divided, is in fact a positive feature of things, and thus cannot be explained by means of negation alone.

But even more directly, Scotus argues that this attempt in fact begs the question: I ask whence a negation is a this, since it is of the same account in this [singular] and in that one.19 If the question is: by virtue of what is something a this? then to answer: this negation is obviously insufficient and circular. Clearly, it is true that individuals are not their species and not other individuals of that species, but these negative relationships are secondary effects of that which positively individuates them.20 3. In Question 3, Scotus very quickly dismisses the idea that it is the act of existing that individuates or contracts the common nature. The suggestion is that whatever distinguishes something cannot be its potential being, but its actuality; but existing is the ultimate act of individuals, compared to which everything else about them is 15 Scotus, Treatise, Q2 48 16 Noone, Universals and Individuation, 114 17 Scotus, Treatise, Q2 49 18 Scotus, Treatise, Q2 52. Again the parallels with Spinoza are striking. For Spinoza as for Scotus, negation is insufficient to account for the positive qualities of any given thing. See, for instance, Ethics I, Axiom 3; I P4; I P26-28; III P4-8, etc. For this reason, we would argue that Hegel's influential interpretation of Spinozism in The Science of Logic and the Lectures on the History of Philosophy turns on a problematic misreading of "omnis determinatio est negatio." On this point we follow Macherey, Hegel or Spinoza; see in particular Chapters 1 and 4. 19 Scotus, Treatise, Q2 56 20 Scotus, Treatise, Q2 58. Deleuze says almost the same exact thing in criticizing the Hegelian theory of the primacy of the Negative: It is of the essence of affirmation to be in itself multiple and to affirm difference. As for the negative, this is only the shadow cast upon the affirmations produced by a problem: negation appears alongside affirmation like a powerless double, albeit one which testifies to the existence of another power, that of the effective and persistent problem. (Deleuze, Difference and Repetition, 267)


merely possible; and thus this very act of existing, esse existentiae, is that which distinguishes or individuates the common nature.21 Scotus objects to this, first, that whatever distinguishes or determines individuals must itself be distinct and determinate, but the act of existing is itself neither of these things; in fact, 'that it exists' seems to be about as indeterminate a claim as can be made about something.22 And second, he repeats the argument from 56, since it can be asked through what the existence is a this.23 Simply put, this existence is again circular and insufficient as an answer to the question of what makes something a this. 4. Scotus turns then to the position that it is matter, and material quantity, that individuates. This seems minimally plausible as an account of individuation given a hylomorphic model in which an individuals form is distinct from its matter: since ostensibly the formal cause is identical in the case of two individuals for example, universal humanity would be the same for any two humans then perhaps it is the material cause, or the relationship between the form and its matter, which provides the principle of individual difference. Scotus quotes Boethius: The variety of accidents produces difference in number, for three men differ neither in species nor in genus, but in their accidents.24 We can therefore also formulate this suggestion as follows: two members of a species are the same in essence; but they are not identical insofar as they are individuals with different accidental qualities; thus it must be their accidental qualities are what individuates them. But Scotus argues that this attempt also fails, for the simple reason that an individual substance is prior by nature and by definition to its accidents, and it is absurd to argue that anything can cause what precedes it: no accident can per se be an account through which material substance is individuated.25 Thus, the general lesson is that the principium individuationis cannot be found among any individuals accidental characteristics, and in particular the quantity of material that an individual possesses falls into this category.

Moreover, if we understand by a substance something that is logically prior to its accidents, then we must similarly reject the idea that it is the composite of substance and its accidents which individuates the former, since any such aggregate would itself have to be accidental.26 Simply on the basis of the logical priority of material

21 Scotus, Treatise, Q3 60 22 Scotus, Treatise, Q3 61-2 23 Scotus, Treatise, Q3 64 24 Scotus, Treatise, Q4 67 25 Scotus, Treatise, Q4 76 26 Noone, Universals and Individuation, 115


substantial individuality to its accidents, Scotus concludes that it is impossible for substance to be individual through some accident.27 We can also put this in the now-familiar terms of the charge of circularity: we presuppose its being a this when we say that this quantity of material can be that through which something is individuated.28

Even the apparently more sophisticated suggestion, that the individuating difference is the relationship between a things form and matter, fails for the same reason. Given that the form, ex hypothesi, is identical in the case of two members of a species, the difference in the relationships between their forms and the matter that together constitute them will turn out to be nothing but the difference in their matter. For there are only two terms in this relationship, the form and the matter; and so if, when we compare two such composites, one of the terms is identical in both, then the difference between these two composites will be entirely dependent on the second term, in this case matter.29 5. But there is another way in which this hylomorphic account cannot account for individuation by material difference. In the third question, Scotus did not object to the idea that individuals must be distinguished in their actuality rather than their potentiality; he objected rather to the suggestion that the simple act of existence sufficed to account for individuation. Indeed of two things that are actually identical but which may both be different, we must conclude that they are in fact not distinct. But on the hylomorphic account, matter is taken to be the principle of pure potentiality or dynamis; how, then, would it be possible for a differential distribution of potentiality to account for actual individuation? As Noone puts it, positing matter as the principle of individuation seems to entail locating the source of the greatest unity (unitas

27 Scotus, Treatise, Q4 111 28 Scotus, Treatise, QQ5-6 138 29 We will return to this later, but we note in passing that Scotus beautiful critique of this relational position amounts to demonstrating that it is not relational at all, because one of the terms of the relation is completely static, and so what was supposed to have been a dynamism is anything but. Following the same logic, we will come to see, according to Deleuze, that it is not enough to relate dx to x, or dy to y; in these relations they are completely undetermined, since x and y are algebraic symbols representing values that are always actually only indifferently determinate; that is, x and y are, in principle, constants. It is only when we relate the two dynamic variables, dx and dy, that we find that they are reciprocally determinable (dydx), and this is completely determined when its power is conditioned by a distribution of singular points (dydx=f'(x)). (Deleuze, D&R, 172-6)

It is not going too far to suggest that Scotus is among the first to recognize that individuation or differentiation remains inexplicable in principle so long as we remain within the ambit of a relation between terms that may appear variable but which are actually just indifferent or indeterminate constants.


maxima) and actuality in a principle that is ordinarily the source of multiplicity and potentiality.30 For this reason, matter is not capable of furnishing us with a principle of individuation: the pure capacity to receive a form logically cannot be a positive factor that contracts a common nature into singularity.

This distinguishing or contracting positive factor cannot be an essential, specific, or quidditative distinction, since this would entail a difference at the level of the common nature and not at the level of the individual: the singular is not definable by some definition other than the definition of the species. The [singular] is nevertheless per se a being, adding some beingness to the specific beingness. However, the per se beingness that [the singular] adds [to the specific beingness] is not quidditative beingness.31 Thus there is no science of the singular. If we ask after the quidditative being or essence of an individual, we must conclude that it is its specific difference; however, it is also the case that the individual adds something to the common nature that it contracts. As Virno argues, this yields the surprising, even paradoxical conclusion that the individual is both more and less than the species: the individual adds something to the common nature and, in contracting it, gives it the mode of ultimate existence or actuality; however, the common nature is never exhausted by the singular into which it is contracted.32 And one immediate consequence of this analysis, as we suggested at the outset, is that the common nature is not thinkable as indifferent, even though it is logically pre-individual. That is, the positive contracting factor is not the only positivity at play in differentiation: the common nature, too, is positive and determinate, although on its own it is neither singular nor universal. This pre-individual determinacy is what Deleuze, in Difference & Repetition, calls the virtual, and it is to his investigation that we will now turn.

II. Deleuze: the distinctness of the virtual The virtual occupies a vexing position in Deleuze's philosophy and in studies on his thought. On the one hand, it is quite obvious, at least in Difference & Repetition, that it plays a central role in his metaphysics; on the other hand, any survey of the secondary literature is more likely to confuse than clarify what precisely this role is. For instance, iek, in his monograph on Deleuze, writes that we must be careful not to conflate it with virtual reality, before attempting to reformulate it using his familiar Lacanian

30 Noone, Universals and Individuation, 117 31 Scotus, Treatise, QQ5-6 192 32 Virno, Angels and the General Intellect


vocabulary, in which he somewhat confusingly claims both that it as such is the Real, and that it is also the Symbolic as such.33 Moreover, he suggests that we understand by it a kind of quantum oscillation,34 and that the virtual names an ineliminable excess of becoming over being.35 On the other hand, Badiou, in his own monograph Deleuze: The Clamor of Being, argues that the virtual is the name of Being, the very Being of beings, ultimately suggesting that the virtual is an idealist Platonic hangover, the postulate of a transcendent One-All that results from Deleuzes inadequate reckoning with post-Cantorian mathematics.36 Even more sympathetic readings of Deleuze are often misleading and contradictory. Brian Massumi claims that the virtual names the real but abstract incorporeality of the body,37 arguing that certain fashionable forms of social critique are not abstract enough; he tends to characterize it as a kind of total space of incompossible and unlivable coexistences whose affective relation to the actual is apparently the object of quantum mechanical inquiry.38 It seems difficult to reconcile this characterization of the virtual as real but abstract, given that Deleuze cites Proust as having articulated the formula of virtuality: it is real without being actual, ideal without being abstract.39 For my part, I want to suggest that it is within the context of the problem of individuation, which we have been exploring in Duns Scotus, that the concept of the virtual gains its sense and becomes intelligible. The theory of the virtual, as I will try to show, is Deleuzes response to the question of how and in what way pre-individual 33 iek, Organs Without Bodies, 3-4 34 iek, Organs Without Bodies. Here as elsewhere iek follows the so-called Copenhagen interpretation of quantum physics, which is by no means the only one possible, even if it is currently the dominant one. But even in terms of this approach, his presentation is somewhat problematic; for instance, he suggests that Werner Heisenbergs uncertainty principle is what Niels Bohrs complementarity is all about, when these two were actually competing, incompatible interpretations, and Heisenberg ultimately retracted his position, ceding to Bohr. See iek, The Indivisible Remainder, Chapter 3. For a recent critical analysis of ieks interpretation and usage of quantum physics in his Lacanian-Hegelian ontology, see Johnston, Adventures in Transcendental Materialism, Chapter 7. For helpful introductions to quantum physics and its history, see Feynman, QED; Omns, Quantum Philosophy; Barad, Meeting the Universe Halfway; and Prigogine, e.g. From Being to Becoming. For a rigorously informed polemic against the hegemony of the Copenhagen indeterminist interpretation in favor of a Bohmian determinist one, see Drr et al., Quantum Physics Without Quantum Philosophy. For a helpful discussion of the role in Deleuzes philosophy of quantum field theory, see Plotinsky, Chaosmologies. 35 iek, Organs Without Bodies, 18-20 36 Badiou, Deleuze: The Clamor of Being, 43, 48 37 Massumi, Parables for the Virtual, 19-21 38 Massumi, Parables for the Virtual, 36-8. See note 33 above for some observations about the surprisingly common references, usually confusing when not confused, to quantum physics in this context. 39 Deleuze, Difference & Repetition, 208


reality, like Scotus common nature, is anything but indistinct. Rather than the abyss of undifferentiated (in)difference, the virtual is completely determined, in a way that is distinct and obscure.

Just as Scotus reached the postulate of a common nature essentially prior to and compatible with both individuality and universality, Deleuze argues that we can grasp the virtual only on condition that we think it as pure past: The virtual object is not a former present, which is the thought of a relative past, a past thought only in relation to the contracted present; rather, it is the past as contemporaneous with its own present, as pre-existing the passing present and as that which causes the present to pass. Virtual objects are shreds of pure past.40 The virtual is thus what is behind the actual present in two senses: it is behind logically and temporally, as that which has been actualized; but moreover it is what is behind the contraction of the present as what causes it. Indeed, the contemporaneity of the virtual past with regard to the actual present is a consequence of the former being the cause of the latter. Here already at the level of language we can begin to see the connection between the problem of individuation in both Scotus and Deleuze: the virtual is that by virtue of which the present is contracted into actual states populated by individuals.

It is in the fourth chapter of Difference & Repetition, Ideal Synthesis of Difference,41 that Deleuze attempts to work out the logic of this virtual contraction. Claiming to find in Kant a theory of Ideas as being essentially problematic, and in this way objective, he suggests that they present three moments: undetermined with regard to their objects, determinable with regard to objects of experience, and having the ideal of an infinite determination with regard to concepts of the understanding.42 The plausibility of his theory of Ideas will turn on whether or not he is able to demonstrate intrinsic and necessary connections between these three modalities of determination, which are the key moments of a theory of differentiation: the undetermined, the determinable, and the completely or infinitely determined. In order to do this, he turns immediately to the differential calculus and proposes a novel interpretation.43

40 Deleuze, Difference & Repetition, 101 41 Translation modified. Patton has Ideas and the Synthesis of Difference, while Deleuze writes Synthse idelle de la diffrence. 42 Deleuze, Difference & Repetition, 169 43 Deleuze presupposes a fairly robust understanding of the calculus and its history as the backdrop against which the novelty of his interpretation would be apparent. Such an understanding is only presupposed at the risk of being totally inaccessible to many readers, while on the other hand an


The inadequacy of transcendental philosophy, from Deleuzes perspective, is most acutely articulated in the early critiques of Salomon Mamon: it is necessary to pass from a theory of extrinsic conditions to one of intrinsic genesis. From the point of view of transcendental conditions alone, the relationship between the determinable and the determined remains extrinsic: if the condition for the possibility of any representation are the pure forms of intuition, this tells us nothing about this representation not only does it tell us nothing about what it is in its particularity, but we have no way of grasping why we have this one and not another. Deleuze considers the introduction of the schematism to be evidence of this inadequacy, through which difference remains external and as such empirical and impure, suspended outside the construction between the determinable intuition and the determinant concept.44 Hence if we are to develop a theory of individuation, it cannot be grounded upon a system of extrinsic conditioning. The theory of conditions is necessarily formal, but the genetic problem of individuation, as we saw with Scotus, is one that cannot be resolved by formal or extrinsic accounts; surely it is necessary that individual substances have matter (angels notwithstanding), but in any event the fact of having matter is not sufficient for their individuation.

For Deleuze, every individual is the actual solution or resolution of a virtual problem. As Dan Smith argues, it is false, or at least inadequate, to consider Deleuze an anti-dialectical thinker; rather, he proposes a new concept of dialectics.45 Deleuze writes: Problems are always dialectical: the dialectic has no other sense, nor do problems have any other sense [] it must be said that there are mathematical, physical, biological, psychical and sociological problems, even though every problem is dialectical and there are no non-dialectical problems.46 Dialectics is the art of problems, or of problematic Ideas; we say art, on condition that it is understood that art no less than science is compelled in its movement by a strict necessity in fact, by a

explanatory digression at this juncture risks losing the thread of our philosophical discussion. As such, the reader can find at the end of this essay an extensive Appendix, which attempts to provide an accessible introduction to the motivating problems and stakes of the differential calculus, as well as a sketch of its scientific development and the history of its philosophical interpretation. In what follows we will attempt to discuss the philosophical problem at hand while presupposing as little technical mathematical understanding as possible, and will seek to explain the relevant mathematical concepts and terms when their introduction is unavoidable. 44 Deleuze, Difference & Repetition, 173-4 45 Smith, Axiomatics and Problematics, 147; see also Smith, Deleuze, Hegel, and the Post-Kantian Tradition, and Deleuze, Kant, and the Theory of Immanent Ideas. 46 Deleuze, Difference & Repetition, 179


principle of sufficient reason. In his lectures on Leibniz, Deleuze argues that the difference between cause and reason comes down again to the distinction between condition and genesis, extrinsic and intrinsic relations of determination: Cause is never sufficient. One must say that the principle of causality poses a necessary cause, but never a sufficient one. We must distinguish between necessary cause and sufficient reason [] sufficient reason expresses the relation of a thing with its own notion, whereas cause expresses the relations of the thing with something else.47 The problematic, which Deleuze defines as the ensemble of the problem and its conditions,48 is a virtual multiplicity in which the relations between the undetermined, determinable, and completely determined particular to it are intrinsic, thus constituting a principle of sufficient reason.

Differential calculus, according to Deleuze, provides a model for such a problematic or dialectical Idea, without constituting an archetype or even being the exclusive formulation of the mathematical Idea. It is not a matter of opposing the infinitely small (Leibniz) to the infinity large (Hegel), but of exhibiting the moments of the process of differentiation which constitute its sufficient reason:

The symbol appears as simultaneously undetermined, determinable and determination. Three principles which together form a sufficient reason correspond to these three aspects: a principle of determinability corresponds to the undetermined as such (,); a principle of reciprocal determination corresponds to the reciprocally determinable (!"!"); a principle of complete determination corresponds to the effectively determined (values of !"!"). In short, is the Idea the Platonic, Leibnizian or Kantian Idea, the problem and its being.49

For a given function (), represents the rate of change of the value. In relation to alone, is completely undetermined nothing about the value indicates anything about the rate at which the value changes, and at any given point literally does not exist; and the same holds for in relation solely to . However, they are perfectly determinable in relation to one another. For this reason, a principle of determinability corresponds to the undetermined as such.50 As Hegel recognized, in

47 Deleuze, Lecture on Leibniz, April 15 1980 48 Deleuze, Difference & Repetition, 177 49 Deleuze, Difference & Repetition, 171 50 Deleuze, Difference & Repetition, 172


the differential !"!" the two terms are only moments of a process, and have sense as determinable only in relation to one another.51 Deleuze argues therefore that the principle of reciprocal determinability is not extrinsic to, but entailed by, the very nature of the undetermined.

Moreover, in this passage from the undetermined to reciprocal determination, Deleuze argues, we have also passed from quantitability to qualitability. Leibnizs method for finding the derivative invoked infinitesimals, infinitely small quantities, which traverse an ideal distance and at the limit vanish entirely, such that, based on his principle of the identity of indiscernibles, the secant becomes identical to the tangent.52 But until the middle of the nineteenth century, much of the controversy about the legitimacy of these methods came down to a concern over the validity of mathematical concepts derived from geometric intuition, and in particular about the geometric intuitions of continuous spatial magnitudes and their infinite divisibility.53 Now Deleuze is similarly skeptical about geometric intuition, but unlike the mathematicians who sought to eliminate any trace of these concepts (Cauchy, Weierstrass), he seeks to retain the concept of continuousness as the pure element of quantitability, which must be distinguished both from the fixed quantities of intuition [quantum] and from variable quantities in the form of concepts of the understanding [quantitas].54 Reversing the classical order, he argues that this continuity is grounded by, rather than grounding, the limit as a genuine cut [coupre], a border between the changeable and the unchangeable within the function itself.55

When we find the derivative of a function (), we obtain a new function: !"!" () or !(). But between these two functions there is a difference which is no longer simply quantitative, but rather

51 See Hegel, Science of Logic, 253; his interpretation of the calculus is discussed in Section II of the Appendix below. 52 See Section II of our Appendix. 53 See Section III of our Appendix. 54 Deleuze, Difference & Repetition, 171 55 Deleuze, Differnece & Repetition, 172

Figure 1


qualitative, insofar as the functions necessarily have different powers.56 That is, for example, there is a qualitative difference between a parabolic curve and the straight line of its derivative [figure 1]. This is the sense in which Deleuze writes that the differential relation is determinable in a qualitative form. But in the example we have given, as with any such example, in so far as it expresses another quality, the differential relation remains tied to the individual values or to the quantitative variations corresponding to that quality.57 However, if we attend to the universal logic of the differential determination, rather than the particular qualitative difference, we discern in this transformation a pure element of qualitability which derives from the principle of reciprocal determination.58

The qualitative difference here is, as we said, one of powers; and it is in this way, Deleuze argues, that we move from the second principle to the third: the differential relation presents a third element, that of pure potentiality. Power is the form of reciprocal determination according to which variable magnitudes are taken to be functions of one another [] A principle of complete determination corresponds to this element of potentiality.59 For every power, there

corresponds a distribution of singular points, which are completely determined. Simon Duffy offers an illuminating diagram illustrating what is meant by singular point [figure 2]. Singular points are points of articulation where the nature of the curve changes or the function alters its behavior [] The differential relation characterises or qualifies not only the distinctive points which it determines, but also the nature of the regular points in the immediate neighbourhood of these points.60 If we were to take the derivative of the function graphically represented in figure 2, we would obtain a new

56 Hegel too argued that the reciprocal relation of dydx has an essentially qualitative, rather than simply quantitative, character. See Section II of our Appendix. 57 Deleuze, Difference & Repetition, 172 58 Deleuze, Difference & Repetition, 173 59 Deleuze, Difference & Repetition, 174-5 60 Duffy, The mathematics of Deleuzes differential logic and metaphysics, 128

Figure 2


function with a lower power61 describing a parabola, in which the stationary points would become the points at which the curve passes through the x-axis (!() = 0), and the point of inflection would become the parabolas nadir (!!() = 0).

The singular points on their own do not yet constitute the function they are not the principium individuationis, although they provide the coordinates within which the functions individuation occurs. It is the distribution of these singular points that is the object of the principle of complete determination.62 The differential establishes not only the singular points, but also the way in which they establish neighborhoods within which the ideal infinite continuity is quantitatively determined. In other words, each completely determined pre-individual singular point ranges over an infinity of ordinary points within its area of influence: a series of powers with numerical coefficients surround one singular point, and only one at a time.63 As Dan Smith writes, in this way we obtain a definition of the individual: one can say of any determination in general (any thing) that it is a combination of the singular and the ordinary: that is, it is a multiplicity constituted by its singular and ordinary points.64

It is here, finally, that we return directly to our question about individuation: the completely determined distribution of such singular points, as fully real but pre-individual and only actualized through infinite variations at the level of individual differences, is precisely the way in which the virtual is not undifferentiated: they constitute the distinctness of Ideas.65 For this reason Deleuze develops the notion of different/ciation:66 We call determination of the virtual content of an Idea differentiation; we call the actualization of that virtuality into species and distinguished parts differenciation.67 As with Scotus investigation, all the difficulty lies in evading the Scylla of affirming actuality (Platonic realism), and the Charybdis of denying reality (nominalism), when it comes to the virtual or the common nature. For

61 depotentialization conditions pure potentiality by allowing an evolution of the function of a variable in a series constituted by the powers of i (undetermined quantity) and the coefficients of these powers (new functions of x), in such a way that the evolution function of that variable be comparable to that of others. (Deleuze, Difference & Repetition, 175) 62 Deleuze, Difference & Repetition, 175 63 Deleuze, Difference & Repetition, 176 64 Smith, Deleuze on Leibniz, 56 65 Deleuze, Difference & Repetition, 176 66 If not intentionally significant, it is perhaps serendipitous for our purposes that, unlike in the English publication, the original French publication of Difference & Repetition utilized mathematical fractional notation to mark this distinction: diffren!!iation. 67 Deleuze, Difference & Repetition, 207


in both cases, we are indeed considering a reality: The reality of the virtual consists of the differential elements and relations along with the singular poins which correspond to them [] far from being undetermined, the virtual is completely determined.68 This is the sense in which, as Deleuze argues, the undetermined of the problematic Idea is nevertheless objective, and non-knowledge is transformed not into an absence but a positivity of what is to be learned.

Conclusion More than a shared affirmation of the univocity of being,69 it is the discovery of and rigorous insistence upon the objectivity and positivity of the undetermined, of the pre-individual, that constitutes the thread uniting both Deleuze and Scotus in their efforts to think through the problem of individuation. As Deleuze writes, the virtual or the Idea is that which must be understood as both distinct, as the completely determined distribution of singular points (differentiated) but also as obscure, as not yet actualized or distinguishable as individuals (differenciated). In this sense pre-individual singularities and their distribution possess, as Scotus would say, a less-than-numeric unity. In describing distinctness-obscurity, Deleuze refers to Leibnizs notion of petites perceptions: the sound of a crashing wave that I hear is actually constituted by an infinite multiplicity of minute perceptions, each drop of water contributing to the cacophony, of which I am unconscious and which I am entirely incapable of distinguishing.70 The astonishing thing is that these indistinguishable infinitesimals might be contracted into an individual, that somehow less-than-numeric unity or pre-individual reality is actualized. With this insight, that of the determinacy of problematics, the very questions of metaphysics change: no longer what is this? but why this one? and even why this now?71 And it is on the basis of this insight and these transformations that philosophical inquiry can become adequate to the politically exigent tasks of intervening not at the level of actual distributions, but that of virtual structures and pre-individual determinations.

68 Deleuze, Difference & Repetition, 209 69 Deleuze, Difference & Repetition, 35-40 70 Deleuze, Difference & Repetition, 213; see also the Lecture on Leibniz, April 15 1980; and Leibniz, New Essays, 54-7. This is also why Dan Smith suggests a Leibnizian inspiration underlying Anti-Oedipus, in which the unconscious is differential and genetic rather than representative. (Smith, Deleuze on Leibniz, 55) 71 See Lampert, Deleuze and Guattaris Philosophy of History, especially chapters 1 and 7-9.


Appendix

A Brief History of the Calculus and its Metaphysics72

I. Introduction: what is differential calculus? Differential calculus attempts to establish the rate of change, also called the derivative, !"!" (), for any given function (). On a Cartesian coordinate plane, the value of a point on the x-axis, or ordinate, is that which enters into the function (), yielding a new value that is plotted on the y-axis, or abscissa. Graphed this way on a plane, many functions can be geometrically represented as curves, in which the rate at which the value changes relative to rate of change of the value is not constant: the value may change at a greater or lesser rate given the same change in . Differential calculus seeks to establish this variable rate. Now, given any two points on a curve, (!,!) and (!,!), the rate at which the value changes relative to the value of between these two points, !"!", can therefore be calculated as !!!!!!!!!!.

For linear equations, which can be expressed in the form = () = + , where and are both constants, the derivative or differential is nothing other than the slope of the line or the value of , and the derivation is therefore entirely trivial, since the rate of change of y is the same at every x. This is not the case for polynomial functions that involve variables with powers greater than 1, such as the function we will take as our example throughout this appendix: () = ! 2 + 3. Here the function can be geometrically represented as a parabolic curve, and therefore the rate at which changes relative to is different depending on the value of . The rate of change of a curve is equal to the gradient or slope of the line tangential to the curve at a point, and that tangent is different at different points. Differential calculus thus tries to determine the function that expresses this difference in the rate of change of relative to the change in .

But at a given point on a curve, (!,!), how do we calculate the rate at which y is changing relative to x? The problem is that our method for determining the rate of change requires calculating the difference between the rates at which y changes relative

72 Please let it be understood that what follows is necessarily cursory and, at times, surely oversimplifies the complexities of the work discussed. As the author is not a trained mathematician, it is also possible that some aspects of the exposition are incorrect or misleading. Any objections, clarifications, suggestions, or complaints will be received with great appreciation at [email protected].


to x of two different points, !!!!!!!!!!; this actually determines the gradient of a secant of the curve, not a tangent. But, by hypothesis, at a single point on a curve there is no second point, (!,!), with which we could make this calculation. How then can we determine the differential of a curve at a single point on that curve? Attempting to answer this question is what motivated Leibnizs approach, which employed infinitesimals or infinitely small quantities.

II. From Leibniz to Hegel on mathematical infinities73 Leibniz and Newton are generally credited with having discovered the calculus independently, although to claim that either of them is its discoverer is problematic and reductive given the long history of contributions to mathematical thought.74 Let us look at Leibnizs method for determining the derivative, which relies on the notion of infinitesimals, or infinitely small quantities. We have our function, () = ! 2 + 3. We want to determine !"!", the rate at which changes relative to the change of . We will calculate !!!!!!!!!! on the hypothesis that the difference beween ! and !, or , and that between ! and !, or , is infinitely small that it infinitely approaches and becomes functionally equivalent to zero.

First we note that = ! !. In our function, is the independent variable, which means that is a function of ; so, ! is a function of ! and ! is a function of !. Thus we can now articulate strictly in terms of :

1. = ! ! = (!! 2! + 3) (!! 2! + 3), and simplify: 2. = !! 2! !! + 2!

But we also know that = ! ! or, equivalently, ! = + !. So now: 3. = ( + !)! 2( + !)!! + 2!

Now we expand our terms using the binomial theorem and the distributive property: 4. = ! + 2! + !! 2 2! !! + 2!, and simplify: 5. = ! + 2! 2

Dividing both sides by gives us what we were after, a formula for !"!": 6. !"!" = + 2! 2

Now, however, we seem to have a problem, for our equation literally begs the question: we are defining the rate at which changes relative to the change of in terms of the 73 In these sections, I draw substantially from Gerdes 1985 and Duffy 2006. 74 Cf. Boyer, The History of the Calculus,177-8; consider also Anne McClintocks reflections on the logic of narratives of male birthing in Imperial Leather, e.g. in Chapter 6.


rate of change of ; that is, appears both on the left and right sides of the equation. Leibnizs solution is, at this point, to suppress or eliminate any terms on the right side of the equation that still include . Why is he able to do this? Because, as per our hypothesis, is infinitely small, or approximately and thus functionally equivalent to zero, and so can be ignored. As Leibniz writes in justification of this procedure: anyone who is not satisfied with this can be shown in the manner of Archimedes that the error is less than any assignable quantity and cannot be given by any construction.75 So, we are left with the general formula that expresses the rate at which changes relative to the change of for our function:

7. !"!" = 2 2 And indeed, for any point on the curve described by () = ! 2 + 3, the slope or gradient of the line tangential to it at that point is given by our derivative function () = 2 2, as verification by experimental application makes apparent.

Leibniz provides a diagram in his justification for this infinitesimal method [figure 3].76 We have two straight lines, and , which meet at point . Lines and are perpendicular to the line . Let be the length of line , be the length of , be the length of , and be the length of . Now, and form two similar right triangles, meaning that the only difference between them is one of scale i.e., their angles map onto one another. Due to this similarity, we note that !!!! = !!. If now we continuously slide the line rightward while maintaining its gradient (as in the dotted line), the lengths and triangle will shrink, while the ratio of to will remain constant. As he writes:

Now assume the case when the straight line passes through itself; it is obvious that the points and will fall on , that the straight lines and , or and , will vanish, and that the proportion or equation !!!! = !! will become !! = !!. Then in the present case, assuming that it falls under the general rule, = . Yet and [which now both = 0] will not be absolutely nothing, since they still preserve the ratio of to [...] and still have an algebraic relation to each other. And so they are treated as infinitesimals, exactly as are the elements which our differential calculus recognizes in the ordinates of curves for momentary increments and decrements.77

75 Leibniz, Justification of the Infinitesimal Calculus by That of Ordinary Algebra, 546 76 Leibniz, "Justification", 545 77 Leibniz, "Justification", 545


The central insight here is that, although these quantities approach zero, they still maintain a nonzero relationship with respect to one another even as they vanish. Or in other words, even if and both themselves = 0 at the terminal point of the continuous movement of the line, the ratio !"!" is not therefore nothing; rather, paradoxically, we obtain a !! which expresses a specific algebraic relationship: the function of the tangent of a curve at a given point. More precisely, we can say that at this limit the gradient of secant of the curve and that of its tangent at a single point become identical.

However, some serious issues trouble this method. First, in practically working through our calculations, we must at first and for the most part treat and as if they were not equal to zero, for if we immediately eliminate or suppress them we end up with nothing at all. If in step 3 above when we substituted + ! for !, we took to be functionally equivalent to zero, we would get:

8. = (0+ !)! 2(0+ !)!! + 2! 9. = !! 2! !! + 2! 10. = 0

Which obviously doesnt get us anywhere; as Marx emphatically notes, this leads literally to nothing.78 So what justifies holding off on treating and as zero until the very end, in step 7 above? That the derivation works, in that our new function () = 2 2 in fact yields the slope of the line tangent to the curve described by () = ! 2 + 3, is in no way a proof that the method of derivation is accurate or scientific. Marx refers to this Leibnizian approach as the mystical variant of differential calculus. Following Hegel, he argues that its procedures, and in particular the suppression of and at the very end, were justified not by rigorous mathematics but precisely because the experimental results were known in advance.79

78 Marx, On the Concept of the Derived Function, 3 79 Marx, On the History of Differential Calculus, 91-4; Hegel, Science of Logic, 273

Figure 3


Leibniz was well acquainted with these charges, and struggled to demonstrate the scientificity of this procedure.80 As Deleuze and Guattari write in A Thousand Plateaus, for a long time, [differential calculus] had only parascientific status and was labeled a Gothic hypothesis; royal science only accorded it the value of a convenient convention or a well-founded fiction,81 since its reliance on the ambiguous notion of the infinitesimal was considered unacceptable. As infinitely small quantities, and are in this method sometimes treated as very small but nonzero finite quantities (steps 1-6) and at other times treated as zero (step 7). This in fact was the thrust of Berkeleys influential response to Leibnizian and Newtonian calculus, The Analyst: Or a Discourse Addressed to an Infidel Mathematician.82

Many of the subsequent attempts at providing a rigorous ground for differential calculus, such as those of DAlembert, Euler, and Lagrange, sought to eliminate the need to make use of these infinitely small quantities, which were seen as ad hoc solutions at best, and metaphysical absurdities compromising the rigor of mathematics at worst. However, it was not until Bolzano, Cauchy, and Weierstrass that these fundamental issues were considered to be resolved.

Before turning to them, however, let us consider briefly Hegels reflection on the differential calculus as he found it in the early nineteenth century, in the first Remark on quantiative infinity in the Science of Logic. Hegels remarks are interesting in that, while he agreed that the justifications of the infinitesimal calculus were inadequate and its methods ultimately inconsistent, he nevertheless considered the calculus to have effected a genuine breakthrough in the problem of the representation of the mathematical infinite. He tries to show that the true mathematical infinite erupts, as though irrepressible, in the work of Euler and Lagrange, in spite of their efforts to do away with the infinitesimals via limits and series. This tension was expressed in the fact that while the methods employed in the calculus at this stage were not consistent, its outcomes were anything but faulty:

The procedure of the infinitesimal calculus shows itself burdened with a seeming inexactitude, namely, having increased finite magnitudes by an infinitely small quantity, this quantity is in the subsequent operation in part retained and in part

80 The same can be said of Newton, whose method relied on fluxions which were, despite his protests to the contrary, functionally the same as infinitesimals, except that they represented infinitely small increases rather than vanishingly small quantities. See Boyer, The History of the Calculus, 190-6 81 Deleuze and Guattari, A Thousand Plateaus, 363 82 Cf. Boyer, The History of the Calculus, 224-9


ignored. The peculiarity of this procedure is that in spite of the admitted inexactitude, a result is obtained which is not merely fairly close and such that the difference can be ignored, but is perfectly exact. In the operation itself, however, which precedes the result, one cannot dispense with the conception that a quantity is not equal to nothing, yet is so inconsiderable that it can be left out of the account.83

Hegels distinction between spurious and true mathematical infinities turns on the way in which the mathematical infinite is represented. Consider the equation: !! = 2.58714 On the right side, we have the bad representation: an infinite series of numbers, which exhibits the contradiction of representing that which is a relation possessing a qualitative nature, as devoid of relation, as a mere quantum, as an amount.84 As compensation for this reduction of qualitative relation to quantitative amount, the infinite series is always incomplete the representation always terminates with an ellipsis; in trying to represent the determinate finite quantity in this way, we must therefore always add another number, and so the infinite is represented as an essential incompleteness or the positing of a beyond. On the left side, we find the other side of the contradiction: here there is no beyond, as its expression lacks nothing, but at the same time this captures the infinite series by means of a relationship between finite quantities: in the finite expression which is a ratio, the negative is immanent as the reciprocal determining of the sides of the ratio and this is an accomplished return-into-itself, a self-related unity as a negation of the negation (both sides of the ratio are only moments), and consequently has within it the determination of infinity.85 Paradoxically, then, it is in fact the endless series which is the deficient finite expression of the infinite.

With the calculus, new modes of mathematical representation were developed, particularly in the infinitesimals and , and it here that Hegels real interest lies. In what he calls power-relations, such as the function of a parabola, !!! = , the quantitative relationship between and is not constant but variable; he argues that this indicates that the relationship of a magnitude to a power is in fact fundamentally qualitative. Moreover, he suggests that this should have prompted the development of distinct symbols expressing the difference between a variable understood as a genuinely variable magnitude, on the one hand, and one understood as an unknown 83 Hegel, Science of Logic, 242 84 Hegel, Science of Logic, 247 85 Hegel, Science of Logic, 248


quantity which is completely determined on the other (we will return to this).86 But whatever the nature of this relationship, and still signify quanta here. This is not the case in the equations of the calculus:

, are no longer quanta, nor are they supposed to signify quanta; it is solely in their relation to each other that they have any meaning, a meaning merely as moments. They are no longer something (something taken as a quantum), not finite differences; but neither are they nothing; not empty nullities. Apart from their relation they are pure nullities, but they are intended to be taken only as moments of the relation, as

determinations of the differential co-efficient !"!" . In this concept of the infinite, the quantum is genuinely completed into a qualitative reality; it is posited as actually infinite; it is sublated not merely as this or that quantum but as quantum generally.87

In other words, differentials really do not express finite magnitudes; they are completely undetermined in themselves, but reciprocally determining in relation to one another. Hegel argues that the vanishing of the magnitudes and , when they stand in this reciprocally determining relationship, testifies to the qualitative nature of what is quantitative, of a moment of a ratio as such;88 and that what remains is, as with Leibniz, not absolutely nothing but indeed their quantitative relation solely as qualitatively determined.89 Hegel thus reproaches the mathematicians, not for their making use of mathematical concepts of the infinitely small, but for their failing to draw out the radical conclusions of this sense of the mathematical infinite; in their work, the genuine Notion of the infinite is, in fact, implied in them, but [] the specific nature of that Notion has not been brought to notice and grasped.90 Indeed the persistence of infinitesimals, infinite series, and approaches toward limits remained something of a scandal, and mathematicians sought to eliminate the need to refer to them entirely. It was not until Abraham Robinson provided rigorous but controversial foundations for the concept of the infinitesimal with the development of the axioms of non-standard

86 Hegel, Science of Logic, 252-3 87 Hegel, Science of Logic, 253 88 Hegel, Science of Logic, 267 89 Hegel, Science of Logic, 269 90 Hegel, Science of Logic, 260


analysis in the late twentieth century that the Leibnizian approach found itself vindicated rather than repudiated.91 But now let us turn to the other trajectory, which culminated in Weierstrass fully arithmetized approach to the problems of calculus.

III. Bolzano, Cauchy and Weierstrass: towards non-geometric foundations In his landmark The History of the Calculus and its Conceptual Development, Carl Boyer astutely notes that the contradictions haunting the formulations of the calculus from Leibniz and Newton until the nineteenth century were in the last analysis equivalent to those which Zeno had raised over two thousand years previously and were based on questions of infinity and continuity.92 The apparently insoluble paradoxes of the infinitely small in the calculus stem from a confusion between geometric and arithmetic concepts, which do not have the same foundation or status. Arithmetic concepts deal with the properties of numbers, and the relations between them. Geometric concepts, on the other hand, deal with the properties of spatial figures. It became apparent, by the beginning of the nineteenth century, that many of the concepts essential for the method of calculus, in particular function and limit, were based on geometric intuition. As we have seen, such intuitions are metaphysically suspect insofar as they seem necessarily to involve speculation as to whether or not continuous spatial magnitudes, such as any given section of a curve, are infinitely divisible.93

Another aspect of the problem, however, was quite practical rather than metaphysical, and lay in the fact that geometric intuition proved to be frequently unreliable when it came to problems of differentiation. As Dan Smith writes, geometric intuition refers not to empirical perception but rather to the ideal geometrical notion of continuous movement and space.94 The trouble began with Bernard Bolzano's construction, in 1837, of a function which, although continuous everywhere, could not

91 Duffy, The mathematics of Deleuzes differential logic and metaphysics, 125 92 Boyer, The History of the Calculus, 267 93 To provide yet another example of the kind of metaphysical speculation into which geometrical intuitions can lead: when Descartes, in his Principles of Philosophy, argues for the infinite divisibility of matter, or against the possibility of indivisible atoms, he does so on the basis of a geometric intuition of the nature of extension, rather than on the basis of numerical properties of matter: "it is impossible that there should be atoms, that is, pieces of matter that are by their nature indivisible. For if there were any atoms, then no matter how small we imagined them to be, they would necessarily have to be extended; and hence we could in our thought divide each of them into two or more smaller parts, and hence recognize their divisibility." (Descartes, Principles of Philosophy, II 20) 94 Smith, Axiomatics and Problematics, 153


be differentiated at any point; this is to say that he proved that a function might have no tangents at all even if it is entirely continuous a claim unlikely to be intuitively accepted.95 Even more strikingly, it was later shown that there are continuous curves defined by motion through space that have no tangents, which again is incommensurable with geometric intuition.96 Hence in order to avoid the metaphysically problematic consequences that otherwise threatened to condemn the calculus to the status of mere unscientific conjecture, and in order to ground rigorously a method which was not liable to be misled in being guided by a geometric intuition which had proven itself unreliable, mathematicians by the early nineteenth century sought to provide arithmetic, non-geometrical foundations and definitions for all the concepts involved in the methods of differentiation and integration.

Let us take the concept of limit. In its geometrical articulation, which we saw above in Leibniz, a limit is intuitively intelligible as a kind of threshold point at which differences tend to vanish. A classical example of the geometric concept of limit is that of a polygon inscribed within a circle: the more sides the polygon has, the closer it seems to come to being identical with the circle in which it is inscribed. At the limit, if it had infinitely many sides, would the polygon become actually identical with the circle? Boyer cites a late nineteenth century mathematician who advances such a geometric conception: whether one calls the circle the limit of a polygon as the sides are indefinitely decreased, or whether one looks upon it as a polygon with an infinite number of infinitesimal sides, is immaterial, inasmuch as in either case in the end 'the specific difference' between the polygon and the circle is destroyed.97 It is clear how this notion of the dynamic disappearance of differences was central to the infinitesimal calculus: the real differences between (!,!) and (!,!) at the limit, where the distance between them become infinitely small, are taken to actually vanish, and we obtain the derivative by their elimination.

An arithmetic definition of the concept of limit was proposed by Augustin-Louis Cauchy in the early nineteenth century: When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.98 As Boyer writes, Cauchy's definition appealed to the notions of number, variable, and function, rather than to 95 Boyer, The History of the Calculus, 269-70 96 Cf. Neikirk, A Class of Continuous Curves 97 Boyer, The History of the Calculus, 272 (citing Giulio Vivanti, "Il concetto d'infinitesimo e la sua applicazione alla matematica." Mantova, 1894.) 98 Cauchy, uvres (2), III, 19 (cited in Boyer, The History of the Calculus, 272; his translation)


intuitions of geometry and dynamics.99 That the variable can differ from the limit as little as one wishes suggests that it is always possible to determine a finite, nonzero quantity which is closer to the limit than any other given quantity. We might say that Cauchy reversed the order of definition: rather than defining the limit as that into which infinitesimal differences vanish, he defined it in terms of the indefinite diminution of a variable without ever vanishing; and on the basis of this new arithmetic definition of limit, he redefined the infinitesimal arithmetically as a variable quantity whose indefinite diminution converges toward zero.100

With this arithmetic conception of limit, Cauchy was able to articulate a non-geometric method for differentiation. Still, Cauchy's formulations indefinite approach, as little as one likes, and so on risk suggesting precisely the geometric intuitions, the dynamics of infinitely vanishing movements across continuous distances, whose avoidance was his aim. As Alfred North Whitehead wrote, as long as we retain anything like tending to , as a fundamental idea, we are really in the clutches of the infinitely small; for we imply the notion of being infinitely near to . This is just what we want to get rid of.101 It was Karl Weierstrass whose formulations in the middle of the nineteenth century finally removed all remnants of geometric intuition and dynamism from the foundations of the calculus.

Weierstrass developed what is often referred to today as the epsilon-delta definition of the limit. Boyer expresses this definition with clarity: The number is the limit of the function () for = ! if, given any arbitrarily small number [epsilon], another number [delta] can be found such that for all values of differing from ! by less than , the value of () will differ from by less than .102 While this is basically a reiteration of Cauchy's definition of limit, it notably no longer has any reference to the apparent dynamism of variable values approaching it. This in fact involves a redefinition of the concept of variable, which does not represent a progressive passage through all the values of an interval, but the disjunctive assumption of any one of the values in the interval.103 That is, there is no longer anything in the definition that might suggest that either or () are mobile; they do not move at all, but rather simply stand in for one value within a fixed range of values. This, it is worth recalling, is precisely the distinction that Hegel suggested we make 99 Boyer, The History of the Calculus, 273 100 Boyer, The History of the Calculus, 273-84 101 Whitehead, An Introduction to Mathematics, 229 102 Boyer, The History of the Calculus, 287 103 Boyer, The History of the Calculus, 288


between variable magnitudes and unknown but completely determined quanta; on Weierstrass account, however, only the latter are legitimate.

For example, if we consider the difference between () where = 1 and again where = 3, we are quite clearly considering two different cases which are perfectly static, where the function yields two distinct outputs; nothing about this difference indicates that somehow has moved to 3 from 1, or that (1) has moved to (3). We need not make any recourse to infinitely small or vanishing quantities to establish the limit of a function () where = . It suffices to show that, for any given value between = and = + , some finite value ! can always be determined such that (!) quantitatively differs from () less than both ( ) and ( + ).

With this clarified concept of limit, we can now turn back to Cauchys method for finding the derivative of a function, which was essentially the same as that given earlier by Bolzano,104 without fear of sliding back into conceiving of approaching a limit as the continuous movement of a variable. For a given function () = , we want to know the rate at which varies relative to the rate at which does, !"!", at a given point, . Let = , that is, an incremental difference in magnitude added to . = ( + ) ()( + ) Simplifying the denominator, we obtain the general form of the derivative of a function as a ratio: = ( + ) () Now, we establish the limit of this ratio as converges on zero: !! ( + ) () The solution to which establishes the rate of change of relative to the rate of change of , or the gradient of the tangent to the curve described by () at a given point. This arithmetic definition of the derivative of a function is entirely non-geometric, relying only on the concepts of function, as a particular relationship between sets of ordered pairs; and limit, as defined above.

As we have said, the insight in this reformulation of the method for finding the derivative is that it does not rely on an understanding of the variables , !, and so on 104 Boyer, The History of the Calculus, 275-9


as traversing the distance of an ideally continuous space. Rather, they represent fixed and discrete magnitudes, which are in the last analysis each completely determined and static.

IV. Concluding remarks: on scientific images of thought

While the approach championed by Cauchy and Weierstrass has been generally accepted as successfully having provided at long last a rigorous foundation for the calculus, a few final comments are in order. First, not only did proceeding via infinitesimals persist as a practical approach for deriving functions long after Weierstrass formulation, but also, as we mentioned, these methods also found rigorous formulation in the work of Abraham Robinson, who wrote in 1966: Even now, there are many classical results in differential geometry which have never been established in any oher way [than through the use of infinitesimals], the assumption being that somehow the rigorous but less intuitive , method would lead to the same result.105 The crucial philosophical questions here, as in the case of the contemporary victory of the Copenhagen interpretation of quantum mechanics,106 are: what are the conditions under which a particular scientific interpretation gains legitimacy and acceptance? On the basis of what, and by whom, is this legitimacy delivered? These are emphatically political questions.

It must be said that the program of arithmetization and discretization advanced by Weierstrass and his students, along with the related post-Cantorian impulse to reduce all mathematical objects to set-theoretic constructions, has become something of an orthodoxy in contemporary mathematics and its spontaneous philosophy. It is the implicit and explicit presuppositions of such a scientific image of thought (e.g., that only arithmetic is rigorous107) which stand in need of critique. Moreover, there are socio-political and historical reasons to be suspicious of these axiomatic analytic approaches, as for instance Nicolas Bourbaki and more recently Alex Galloway have argued.108 Deleuze and Guattari, in A Thousand Plateaus and elsewhere, argue for a metascientific distinction between the nomad or minor sciences that develop problematics, and the major sciences that axiomatically pursue their formalization

105 Robinson, Non-Standard Analysis, 83; cited in Smith, Axiomatics and Problematics, 155 106 See note 33, above. 107 Smith, Axiomatics and Problematics, 154 108 Cf. for example Bourbaki, The Architecture of Mathematics, 31; Galloway, The Poverty of Philosophy


(and often their neutralization) under conditions determined by state interests.109 The philosopher of mathematics Fernando Zalamea has recently made compelling arguments against the hegemony of axiomatic set-theory, suggesting that the diverse advances in twentieth-century mathematics can furnish conceptual frameworks adequate to articulating transitory ontologies of irreducibly complex metastable objects.110 All of this is simply to suggest that what is meant by rigor is not always obvious; and that philosophical histories of science, if they are to avoid relapsing into rehearsals of state-sanctioned ideologies of inimitable progress, require a sensitivity to the political stakes of how rigor and legitimacy are conceived and established that is, they must take seriously the historical character of scientific practices. As Althusser said of the spontaneous philosophy of scientists, which is to say their spontaneous ideology: not only is it inseparable from scientific practice, but it is spontaneous because it is not.111

109 Cf. Deleuze and Guattari, A Thousand Plateaus, Chapter 12 and 13; see also the critique of logicism developed in Chapter 6 of What is Philosophy? On this distinction see Smith, Axiomatics and Problematics and Mathematics and the Theory of Multiplicities 110 Cf. Zalamea, Synthetic Philosophy of Contemporary Mathematics, especially Part Three 111 Althusser, Philosophy and the Spontaneous Philosophy of the Scientists, 88


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