bourbakis destructive influence on the mathematisation of economics

7/27/2019 Bourbakis Destructive Influence on the Mathematisation of Economics

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Economic & PoliticalWeekly EPW January 21, 2012 vol xlvii no 3 63

Bourbakis Destructive Influenceon the Mathematisation of Economics

K Vela Velupillai

I am using destructive as the antonym of constructive (in theBrouwer-Bishop senses rather than in its Russian sense). Of course Ishould have used non-constructive; but I am influenced in this choice bythe way Sokal, in his famous Hoax paper (1996; 2008) used profoundas the antonym of trivial, when he knew (ibid, # 98, p 36) he shouldhave used non-trivial!This is an ultra-condensed, summary, version of a more extensive paper,

with almost the same title, to appear as Appendix II, in my forthcomingbook:Algorithmic Foundations of Computation, Proof and Simulation inEconomics(Springer-Verlag 2012). Conversations with Adrian Mathias,whose lectures on the foundations of mathematics at Cambridge I at-tended, almost 40 years ago, during his visit to Trento a few months ago

were most helpful in the framing of some of the issues broached in thispaper as was his brilliant discussion of The Ignorance of Bourbaki(Mathias 1992; 2012). As always, discussions with my friends, StefanoZambell i, Selda Kao and V Ragupathy, on topics of relevance to thesubject matter tackled in this paper, were enlightening. Given thecontroversial stance I take, on Bourbaki in Economics, it is particularlyimportant to emphasise that none of the above four worthies are evenremotely responsible for any of the infelicit ies in it.

K Vela Velupillai ([email protected]) is at the department ofeconomics, Algorithmic Social Sciences Research Unit, University ofTrento, Italy.

The first appearance of a reference to a Bourbaki

mathematical result was the spoof by D D Kosambi,

published in the first volume of the Bulletin of the

Academy of Sciences of the United Provinces of Agra and

Oudh, 80 years ago, although it was not the first

reference to Bourbaki in a mathematical context. In

mathematical economics there seems to be anincreasing identification of Debreus mathematisation of

economics with Bourbakism, although the post-second

world war mathematics of general equilibrium theory

can be shown to be consistent also with the

contributions of the Polish School of Mathematics in the

interwar period. In this paper an attempt is made to

summarise the story of the emergence of Bourbakism,

originating in India, and its recent demise as well as how

it played a destructive role in mathematising economicsin one, uncompromisingly non-constructive, mode.

It is important to remember that Bourbaki almost never actuallyproved new theorems. Laurent Schwartz:A Mathematician Grapples with His Century, p 157.

1 An Indian Preamble on Bourbaki1

It is regrettable that it is not explicitly mentioned [by Kosambi] howmuch [in his paper] originates from [D] Bourbaki2and what the gen-eralisation consists of, since the mentioned work by the Russian is un-available to most scholars3 Shouten (1932).

Serendipitously, 2012 is the 80th anniversary of the publica-tion of Kosambis (1932) famous spoof on Bourbaki, mark-ing the first appearance of this name in a mathematical

publication. But it may not have been the last such Indiangrounded spoof! Halmos (1957: 89; italics added), in his elegantScientific Americanpiece on Nicolas Bourbaki pointed out:

At about the time Bourbaki was starting up, another group of wagsinventedE S Pondiczery, a purported member of the Royal Institute ofPoldavia.4The initials (E S P, R I P) were inspired by a projected butnever-written article on extrasensory perception. Pondiczerys main

work was on mathematical curiosa. His proudest accomplishment wasthe only known use of a second-degree pseudonym. Submitting a paper

on the mathematical theory of big-game hunting to The AmericanMathematical Monthly, Pondiczery asked in a covering letter that hebe allowed to sign it with a pseudonym, because of the obviously face-tious nature of the material. The editor agreed, and the paper appeared(in 1938) under the name H Ptard.5

Whether Pondiczery was a pun on Pondichery, or whether theauthor was an Indian (from Cambridge or Princeton), has neverbeen made clear. The Hilbert or Axiomatic Method, Axioms IandII, the Rule of Procedure and the Bolzano-Weierstrass Method, asdefined on the very first page of Ptard (1938), aka Pondiczery,cannot but be read as a dig at the destructive mode of the Hilbert-Bourbaki mathematical method.

The ultimate spoof was, of course, the more recent, spectacularone by Alan Sokal (1996; 2008), but to the best of my knowledgethere was no Indian angle to it, mathematical or otherwise.

Ever since the appearance of Andr Weils autobiography (inFrench in 1991; in English as Weil 1992), the Indian Bourbakiconnection has been well known to scholars and students inter-ested in the origins of this famous6pseudonym.

There is, therefore, no need for me to rehash known facts; butsome interpretation of the facts may well be in order, especiallysince I am less than convinced of Weils strong opinions on Indianand other personalities he met during his brief, two-year sojourn

in India in 1930-32, the workings of Indian institutions of higherlearning and even of Indian history and mythology.7In particu-lar, his caricature of the mercurial Syed Ross Masood,8whose


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January 21, 2012 vol xlvii no 3 EPW Economic & PoliticalWeekly64

personal generosity was instrumental in hiring him to come asprofessor of mathematics, at the tender age of 23, to Aligarh Mus-lim University (AMU). Syed Masood had assumed duties as its

vice chancellor, in 1928, with a brief to revamp its scholarlyunderpinnings with competent and prestigious appointments.

Apart from Weil, there was also the appointment of RudolfSamuel, an experimental physicist, to the physics professorship.

Of this appointment Weil is scathingly condescending, forgettingthat his own appointment to the mathematics professorship atAMUhad been on the basis of a recommendation by his Stras-bourg friend, Sylvain Lvi, the orientalist and Indologist, who

was also a friend of Syed Masood.9Moreover, the facts about C V Ramans visit toAMUdo not sub-

stantiate the assertions by Weil (ibid: 81) on the appointment ofSamuel who, after all did stay on atAMUtill 1936, trained somedistinguished Indians in experimental chemistry, particularlyspectroscopy, a field in which Indian science was not quite back-

ward, even at that time.10Moreover, Samuel also collaboratedproductively with the professor of chemistry, the Englishman,R F Hunter, before both of them leftAMUin 1936.11

Weils caricature of Babar Mirzas marriage to a German, andher alleged role in his support for Subhash Chandra Bose is,surely, an affront to the intelligence of this distinguished scien-tist, who servedAMUnobly and for long.

As far as I am concerned the true Indian Bourbaki connectionwas provided by the active political and humane personality ofthat outstandingly critical, sometime Bourbakist, LaurentSchwartz, especially as brilliantly and sympathetically por-trayed, in the review of A Mathematician Grappling with HisCentury(Schwartz 1997), by Hermann Weyls successor at ETH,

the famous Indian mathematician K Chandrasekharan12(1998).A point I shall make later, against economists who identify thework of a mathematician with and without his Bourbakist hat, isperceptively pointed out by Chandrasekharan, with respect toSchwartzs pioneering work on distributions (ibid: 1143-44;italics added):

It is an extraordinary instance of cerebral percolationthat, after such atrying period entirely taken up with problems of survival rather thanmathematics, Schwartz should have come up with his idea of distribu-tions[in 1944-45] Bourbakis influence on the process of percolationclearly is a moot point.

2 Bourbakis Methodology and Brief Reflections

The intuitionist school, of which the memory is no doubt destined to re-

main only as an historical curiosity, would at least have been of serviceby having forced its adversaries, that is to say definitely the immensemajority of mathematicians, to make their position precise and to takemore clearly notice of the reasons (the ones of a logical kind, theothers of a sentimental kind) for their confidence in mathematicsBourbaki (1984; 1994), p 38; italics added.

Robin Gandy (1959: 72; bold italics added), in his remarkablylucid and critical review of the first part of the original Thoriedes ensemblesby Bourbaki, reacted to the above premature Obitu-aryof the intuitionist school with much prescience:

[T]he logic that is developed is, right from the start, completely classi-cal. (Indeed, in the historical note [see the above quote]... it is assertedthat Intuitionism will subsist only as an historical curiosity...But it is

nowadays13clear that classical theory (as here formalised) is too am-biguous to give a definite answer to some problems of analysis (e g, thecontinuum problem). It is possible then that this book may itself

soon have only historical interest.

The possibility that Gandy envisaged has been realised; andthe intuitionist school is alive and well, with the new lease oflife provided by the exciting developments of providing founda-

tions for mathematics via category theory.14My aim in this brief section is simply to point out three direc-

tions of contemporary mathematical research, both in founda-tional studies and in mathematical practice, that make Bourbakianmethodology and epistemology to have only historical interest,especially for the mathematisation of economics: (a) the obsoles-cence of set theory or, to be more precise, Bourbakian structures as the sole provider of foundations for mathematics; (b) thepractical relevance of undecidability, even in economicallyapplicable dynamical systems theory; and (c) the resurgence ofintuitionistic logic and the relevance of varieties of constructivemathematics.

In the case of (a), it has been found, in the last 50 years, firstgradually, but now more clearly and increasingly, that categorytheory offers a more fruitful foundational framework formathematics than Bourbakis structures (Aubin 2008: 825).Here, more fruitful means, from the point of view of non-classi-cal, computationally oriented logic, an encapsulation of wholefields officially excluded from the Bourbakian algebraic pro-gramme (ibid; Schwartz 2001: 163).

In the case of (b), even as enlightened a Bourbakist as RogerGodement, in his superbly pedagogical Algebra15 (Godement1969), felt able to suggest that the reader is unlikely to meet

[a relation that is undecidable] in practice (ibid: 26; italicsadded). After the celebrated Paris-Harrington results (Paris andHarrington 1977) and the further results on Goodsteins sequence,even for applicable especially in economic dynamics naturalnumber constrained dynamical systems (Paris and Tavakol 1993),this same reader is, surely going to be routinised into meetingundecidable relations in practice. The added bonus of bringinginto the fold Ramsey Theory and questions on the valid mode ofinteraction between the finite and the infinite (cf Ramsey (1926;1978) and Ragupathy and Velupillai (2011) for a discussion fromthe point of view of economic theory and its formalisation).

Finally, in the case of (c), coupled to (a), there is the resurgence

of interest in the possibilities for new foundations for mathe-matical analysis16 in the sense of replacing the complacentreliance on set theory supplemented by ZFC(i e, Zermelo-Fraen-kel plus the axiom of choice) brought about by the developmentof category theory, in particular a category called a topos. Theunderpinning logic for a topos is entirely consistent with intui-tionistic logic, hence no appeal is made either to the tertium nondatur(the law of the excluded middle) or to the law of doublenegation in the proof procedures for topoi. This fact alone shouldsuggest that categories are themselves computational in the pre-cise sense of Brouwer-Bishop constructive analysis. As Martin

Hyland17

(1991: 282-83) emphasised:Quite generally, the concepts of classical set theory are inappropriate asorganising principles for much modern mathematics and dramatically so


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for computer science. The basic concepts of category theory are veryflexible and prove more satisfactory in many instances.

It is deficient scholarship at the deepest level of mathematics,and its frontier foundational developments, that allows aso-called mathematical economist to make senseless assertionslike the following (Ok 2007: 279)18:

If you want to learn about intuitionism in mathematics, I suggest read-ing in your spare time, please articles by Heyting and Brouwer...

No wonder economists have been, at best, spare timemathematicians.

3 Bourbakis Destructive

Mathematisation of Economics

The doctrine mentioned in the title19is the assumption, implicit or ex-plicit, that only formally defined notions and therefore only explana-tions in formal terms are precise. ...But, as is well known, mathemati-cal practice continues to use the rejected notions, and, more impor-tant, it makes no attempt to eliminate these notions even when this ispossible. In other words, there is a conflict between (mathematical)practice and (logical) theory on what is needed for precision (Kreisel1969: 2).

In the Preface to the Theory of Value (Debreu 1959: viii), weare informed that:

The theory of value is treated here with the standards of rigor of thecontemporary formalist school of mathematics. The effort towardsrigor substitutes correct reasonings and results for incorrect ones, butit offers other rewards too.

Clower (1995: 311), in his wisdom, surmised that Debreu wasreferring, presumably, to the school of Bourbaki. The caveatpresumably is crucial. Since nowhere in the rest of the book do

we find a precise definition of rigour, who or what consists ofthe contemporary formalist school of mathematics, or whatconsists of correct reasoning.20Yet many lesser economists thanClower have rushed into equate this reference to the contempo-rary formalist school of mathematics with the Bourbakists. Evena small, partial, list of such equators would include Boylanand OGorman (2009), Dppe (2010), Weintraub (2002) andWeintraub and Mirowski (1994).21

Consider, however, the following counterfactual scenarios:Imagine a mathematically competent economist, with su-

preme training in mathematics via the books of van der Waer-den (1930; 1970), Birkhoff and Mac Lane (1941) and Mac Lane

and Birkhoff (1967). Now imagine, also, another economist equal in mathematical competence to the first one trained ad-mirably in Kuratowski and Mostowski (1976)22and Kuratowski(1958; 1966). Add to this the innocuous, even fairly realistic,assumption that neither of these two mathematically compe-tent economists had ever heard of Bourbaki even in Kosambisspoofed version.23

Now suppose these two mathematically competent economistschance upon The Theory of Value. Why would they equateDebreus reference to the contemporary formalist school ofmathematics with the Bourbakists?24Economists who rush to

equate all kinds of axiomatic formalisations in economics withBourbakism, especially Debreus work, forget that there weremany formalist schools of mathematics, and almost all of them

had begun to evolve some, like the Polish School and theAlgebraists, quite independent of Hilbertian formalism and theHilbert Programme long before the Bourbakists were evenconceived, via spoofs or not.

Long before the Bourbakists had any influence in AmericanMathematics, especially in mathematical economics, MarshallStone, a doyen of the field at USuniversities, paid handsome trib-

ute to the Polish School of Mathematics of the interwar period(quoted in Kuzawa 1968: 15-16; second set of italics added):

The work of men who have founded and developedFundamenta Math-ematicaehas had a deep influence on the mathematical progress of thepast quarter-century. Starting with Jamiszewski and Sierpiski, therehas grown up a f ruitful movement with which American mathemati-cians have had intimate and effective relations. The work in Topologyand in abstract spaces is now recognised throughout mathematics asof fundamental character; the Polish Schoolunder such men as Ba-nach and Kuratowski constituted, before the present catastrophe(1939), one of the outstanding mathematical groups.

Stones tribute, concentrating on set theory, topology andfunctional analysis, does not go far enough: he has forgotten thepioneering contributions made by the Polish School, in that he-roic 20-year period of 1919-39, to recursion theory, recursiveanalysis and metamathematics.

If the economic theoretic crown jewels of orthodox mathemati-cal economics are the Arrow-Debreu equilibrium existence theo-rem and the fundamental theorems of welfare economics, thenthe mathematical crown jewels that underpin them are, surelyrelevant fixed point theorems (Brouwer, Knaster-Kuratwoski-Mazurkiewicz [KKM], Kakutani) and separating and supportinghyperplane theorems (especially the Hahn-Banach theorem). All

of these can be taught and learned in a completely rigorousaxiomatic framework by anyone trained only in Polish mathe-matics, without any help from the Bourbakists. Elementarytexts on equilibrium theory pay at least lip service to what Ihave come to call the Polish Fix Point Theorem in proving theexistence of an Applied Dynamic General Equilibrium, andthereby the crucial role it plays in the first fundamental theo-rem of welfare economics.

Reflections by the pioneers of mathematical economics (evenfor example by Debreu 1984: 268-69) emphasise the crucial roleplayed by the Hahn-Banach theorem in demonstrating the (non-constructive) validity of the second fundamental theorem of wel-

fare economics. Yet, it is von Neumann and Bourbaki who areconsidered the founding fathers of the mathematics and themathematical methods of orthodox mathematical economics andformal economic theory not the legions of interwar Polishmathematicians who framed, codified and formalised settheory, topology and functional analysis (to which the contri-butions of von Neumann and Bourbaki is a proverbial , at leastin my opinion).

Above all, the Polish School of mathematics did not neglecteither constructive or computable mathematics, as we know fromtheir pioneering contributions to both fields, all the way from

Emil Post s work in 1921, via the classics by Banach and Mazur inthe late 1930s and the threads that were taken up, on such foun-dations, to develop computable analysis by post-second world


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war Polish mathematicians. Unfortunately, the constructive andcomputable analytic trends that enriched the formalistic part ofthe Polish School of mathematics were not part of the Bourbakitradition. Thus, assuming pro tempore that the equators are

justified in identifying Debreus method with the Bourbakimethod, then it is easy to show its destructive content, bothmethodologically and epistemologically.

There are at least 31 propositions25in the Theory of Value,not including those in Chapter 1, titled Mathematics. TheBourbakian mathematical method comprises the triptych ofaxiomatisation,26 proof and rigour. I shall have to pass onaxiomatisation, purely for reasons of space (although thediscussion in Velupillai, 2012 will deal copiously with this sub-

ject, essentially based on Kreisel and K riv ine 1971 and Kleene1952). Rigour, for the purposes of my interpretation of Bour-bakian destructive economics, via the Theory of Value,is cou-pled to proof.

Thus, the destructive content of the Bourbakian mathemati-sation of economics can be narrowed down to methods ofproof and in this the Bourbakists are squarely adherents ofHilberts Dogma.27 They are destructive simply because themethods of proof used in the Theory of Valueare, all of them,non-constructive. Moreover, they are not of the destructive vari-ety that constructivisation of [them] is almost routine Bishop(1967: 359), referring to Brouwers fixed-point theorem as easyto constructivise routine and easy have to be interpretedin the appropriate context.

In particular, the two fixed-point theorems Brouwer andKakutani the separating hyperplane theorems and the Bolzano-Weierstrass theorems are not routinely constructifiable, i e,

irreversibly destroyed, by the reliance on Bourbakian mathemat-ical method. For example, in the Bolzano-Weierstrass theorem,in the Bourbakian version, adopted in the Theory of Value, thereare two destructive steps, one of which is routinely constructifia-ble, the other is not, without adopting constructive mathematics.For the separating hyperplane theorem(s), more constructivecallisthenics have to be indulged in, from the outset. For exam-

ple, by starting from separable (metric) spaces.28

4 Towards a Non-Destructive

Mathematisation of Economics

By the end of the 19th century... the two major figures were Poincarand Hilbert. Poincars thought was more in the spirit of geometry,topology, using those ideas as a fundamental insight. Hilbert wasmore a formalist; he wanted to axiomatise, formalise, and give rigor-ous, formal, presentations. They clearly belong to two traditions.Bourbaki tried to carry on the formal programme of Hilbert of axi-omatising and formalising mathematics to a remarkable extent, withsomesuccess (Atiyah 2001: 657-58).

Not even Bourbaki has succeeded in axiomatising and forma-lising Bishops constructive analysis. As Harvey Friedman (1977:1-2) explained very clearly:

In [Bishop] there is no systematic attempt to delineate what consti-tutes an admissible piece of constructive (analytic) reasoning, nor topresent and analyse an underlying conceptual framework necessaryfor such a delineation...Thus for Bishop, there is no need of a formal system to delineate theadmissible reasoning, for borderline cases do not naturally arise; andthere is no need for a philosophical analysis of a conceptual frame-

work. There is a body of mathematics, called constructive analysis,which has an existence of its own independent of any particularformal system or conceptual framework in which it is cast.

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Notes

1 Ramaswamy wrote me in July 2010, with an earlyversion of his tribute to Kosambi (fils); I wroteback (on 6 July 2010):

For a long time I have been campaigning againstthe Bourbakian influences in mathematical eco-nomics. One of my Cambridge mathematicsteachers, later a colleague at Peterhouse, AdrianMathias, contacted me after a silence of 30 years,a few months ago. He had written an elegant piecetitled: The Ignorance of Bourbaki(op cit),about adecade and a half ago, which I have been using

very regularly in my lectures. He is coming toTrento, from the island of Runion, where he nowlives and teaches, in late October, to give a talk onthe mysteries of Bourbakian rope tricks!

2 D Bourbaki was eventually baptised as NicolasBourbakiby veline Weil (Weil 1992: 101), the for-mer wife of another original Bourbakian, Ren dePossel (the author of one of the earliest, if not theearliest books, on game theory). The saga of ve-line de Possel becoming Mrs Weil, and the earlydeparture of Ren de Possel from the foundinggroup of Bourbakians, is poignantly outlined inSchappacher (2006).

3 My translation from the original German.Schoutens abstract seems to indicate that he tookKosambis spoof seriously. Schouten was pro-fessor of mathematics at Delft and he is relevantin this story for two personal from my point of

view reasons. His role in removing Brouwersinfluence on the journal Compositio Mathematica(cf, van Dalen and Remmert 2006 and Velupillai2010 and 2011 for some history of these shenani-gans against Brouwer by Hilbert and Schouten);and as the brilliant Dirk Struiks doctoral thesisadvisor.

4 Perhaps not unrelated to the fact that Nicolas Bour-Perhaps not unrelated to the fact that Nicolas Bour-bakis origins were inPoldevia(Weil, ibid, p 102)!

5 Ptard returned to the same scene, with the samepseudonym, with an equally facetious contribu-tion, almost 30 years later, and the earlier classicspoof was even the subject of an even more amus-ing spoof, also in AMM, in 1967 (Ptard 1938,1966 and Roselius 1967), by a Christian Roseliusof Tulane University. I leave it to the interestedreader, John Kelleys (1955: vi) elusive crea-

ture, to speculate on thispseudonym!6 Some would say infamous; a representativesample of the better researched and insider ac-counts can be found in Beaulieu (1999), Mashaal(2002) and Schwartz (1997), as well as variousarticles and interviews with the individual math-ematicians who made up the collective that was is Nicolas Bourbaki, in The Mathematical

Intelligencer(e g, Senechal 1998) in the 1990s andthe first decade of this century.

7 The general tenor of Schappal (op cit) on Weilscharacter, supports copiously, my scepticism ofthe reliability of Weils opinions on his Indianadventures. His justification for abscondingfrom serving in the second world war, invokingthe Gita and Dharma, is most amateurish, even ifthat I suspect, in fact, it is Karma he means, whenhe uses Dharma.

8 Incidentally,A Passage to Indiawas dedicated toSyed Ross Masood by E M Forster (mentioned alsoin Weil, op cit, p 59), who had tutored him at

Cambridge. Forster visited India, in 1912/3, atMasoods invitation.

9 Weils remarks on Rudolf Samuels appointment onthe basis of a recommendation letter from Einstein who was yet to make his decisive break withEurope is, to put it mildly, most condescending:

In physics, Masood thought it a triumph when hepushed through the appointment of a German whoseonly qualification was a letter of recommendationfrom Einstein, and whose merit in Einsteins eyescould only have been that he was an unemployedJew for he never displayed any other qualities.

Were Weils qualifications, after Cartans appoint-ment at Strasbourg, any better than Samuels,especial ly since in the former case the recommen-dation did not even come from a Jewish mathe-matician? Weil is only slightly less scathing aboutR F Hunters appointment to the chemistry profes-sorship at AMU. Hunter later became the head ofthe department of botanical studies at theScottish Hill Farming Research Organisation in Ed-inburgh.

10 Incidental ly, when Samuel, an ardent Zionist, re-Incidentally, when Samuel, an ardent Zionist, re-newed his efforts to emigrate to Palestine, Raman

wrote a warm and appreciative letter of recom-mendation on his behal f (in 1935).

11 An example would be The Transition of Covalencyto Electovalency, by R F Hunter and R Samuel,

Journal of the Society of Chemical Industry,pp 733-40, 25 September 1936.

12 In his email of 6 July 2010, Ram Ramaswamywrote me (italics added):

Chandrasekharan is alive and not very well inZurich. He has become a recluse, and has notallowed contact for several years now and heand Kosambi started the TIFR [Tata Institute forFundamental Research] mathematics school in the1940s and 1950s. I think they eventually fell outquite badly.

One of my earliest introductions to BrouwerianIntuitionism was through Chandrasekharans ele-gant pedagogical article, written in hisyouth andpublished in The Mathematics Student (Chan-drasekharan 1941). My introduction to this article

was via a reference in the rare classic by Rasiowaand Sikorski (1963). The distinguished Indianastrophysicist , Jayant Narlikar, who was a profes-sor at TIFR in Bombay, knew E M Forster very wellduring his Cambridge days at Kings College. It

was, by the way, Jayant Narlikar who introducedme to Luigi Pasinetti, during my first days as aPhD student at Kings College, in October 1973.

13 Gandys review appeared, recall, in 1959. Inciden- Gandys review appeared, recall, in 1959. Inciden-tally, Robin Gandy was Alan Turings only doctoralpupil and was Turings literary executor. AlanTuring, by the way, was conceived in India, exactlya century ago!

14 Itself developed by Eilenberg and MacLane(1945), the former while notwearing his Bourbakihat!

15 Which was my own textbook on the subject in mypostgraduate years.

16 Not that any serious student of Bishop (1967) as Ihave been for at least 30 years will feel the needfor anyfoundations for analysis, let alone newones! The brilliant review of this classic by Bishopon constructive analysis, contrasting it with the

claims and aims of classical analysis, by Stolzen-berg (1970), is still worth reading carefully, espe-cially in this post-Bourbaki era.

17 Indeed, it was Hylands early result on filter spac- Indeed, it was Hylands early result on filter spac-es that debunked Bourbakian beliefs that theirformalisation of continuous encapsulated com-pletely the intuitive notion of continuity(Gandy1995; Hyland 1979). Hyland had been a pupil of

Adrian Mathias, at Cambridge.18 The full quotation, in footnote 47 of p 279, is too

full of incorrect assertions that I constrain myselfto the least offensive part.

19 The Formalist-Positivist Doctrine of MathematicalPrecision in the Light of Experience (bold italicsadded).

20 Would Debreu claim that any example of rea-Would Debreu claim that any example of rea-soning in Bishop (1967) is incorrect because itdoes not adhere to the definition of rigour attri-buted to the contemporary formalist school ofmathematics? After all, Bishop explicitly dis-avows allegiance to any formalist school of math-ematics, contemporary or otherwise!

21 Although I endorse the philosophical message inBoylan and OGorman, the mathematical under-pinnings, particularly Bourbakian claims are lessthan satisfactory. Dppes claims are too absurdto warrant even a cursory discussion but I dodevote considerable space to debunking his ridic-ulous assertions in Velupillai (2012). One aspectof his mistake is the reliance on Giocoli (2003) for

his constructivist assertion on von Neumann.Giocoli, in turn, relies on the thoroughly incorrectdefinition of constructivity in Punzo (1991)

which also mars the otherwise meticulous Wein-traub (2002) discussion on Hilberts formalism,for which he, too, relies on the completely falseand unscholarly definition and discussion in Pun-zo (ibid). Dppes extraordinary phrase, Even if[Debreu] is now peremptorily rejected or belittledas outmoded , must, surely, make every Com-putable General Equilibrium theorist, Real Busi-ness Cycle theorists building on the foundationsof Recursive Competitive Equilibria and Stochas-tic Dynamic General Equilibrium theorists writhein intellectual pain! Finally, there is the followingassertion, about which I am at a loss to say any-thing sensible (ibid: 2):

Furthermore, although the entire effort of Debreuwas motivated by, and becomes intelligible onlybe reference to, his training in a specific school inmathematics Bourbaki there are but a feweconomists who have ever heard of the name, andeven less who have read it. Debreu did...and eve-rything he said about his own workcan befound almost word for word in Bourbaki.

There is no evidence that Dppe has even read De-breu (Uncertainty forms the subject matter ofChapter 7, not 6), let-alone anything serious byBourbaki! As for only a few economists who haveever heard of the name, Ill just leave it at that,pro tempore! Also, keeping close contact with Weilduring his Chicago days (ibid, footnote 1) is onething; to know, work with, and learn from Weil as aBourbakist is quite another thing, as tirelessly em-phasised by Schwartz, Cartier, Cartan, Diudonneand even Weil (see also footnote 14, above).

22 I remember with crystal clarity the following can-I remember with crystal clarity the following can-did signpost in this wonderful book following

The message, therefore, is the following: the path towards anon-destructive mathematisation of economics entails a completedestruction of the Bourbakian mode of delineating admissiblereasoning, appointing commissars to police valid modes of for-malisation and regimenting the notion of rigour to be utilised inproof procedures. It is to accept that rigour is undefinable, proofshould not be subject to Hilberts Dogma, nor axiomatisation a

desirable or necessary feature of a mathematised economics.

The exemplar of the kind of mathematisation of economics,emulating Bishops mode of doing constructive mathematics,that I have in mind is Sraffas classicProduct ion of Commoditiesby Means of Commodities (Sraffa 1960). Like Bishop, it hasresisted a complete axiomatised formalisation. Like Bishop,it employs a collection of admissible modes of reasoning,

without seeing any need to embed them in any particular

formal system.


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Principia Mathematicapractice (ibid: vi): In orderto illustrate the role of the axiom of choice wemarked by a small circle all theorems in whichthis axiom is used. Would that mathematicaleconomists did the same!

23 This is not an entirely fanciful counterfactual.Richard Goodwin told me that he once tookunpaid leave for one academic year, in the mid-1940s, and went away with Birkhoff-Mac Laneand Courant-Hilbert, and came back and wrote

The Multiplier as Matrix(Goodwin 1949) andmuch else. To his dying day he was blissful ly igno-rant of Bourbaki. He had, by the way, also attend-ed lectures by Marston Morse.

24 Let me to add that I was advised to train myself inthe methods of van der Waerden and Birkhoff-Mac Lane by my PhD supervisor, Richard Goodwin,and to read Mac Lane-Birkhoff by Lickorish, who

was lecturing on Algebra at Cambridge in theearly 1970s. I began reading Kuratowski andMostowski and, later, Kuratowski, also at aboutthe same time in any case, long before I knewanything about the Bourbakists.

25 Some, but not all, of them are called theorems.26 More precisely, formalisation by axiomatisation

but I shall return to these themes, in much greaterdetail, in Velupillai (2012).

27 Discussed in some detail in Velupillai (2011).28 I refer the interested reader to Bishop (1967),

Appendix A and B for hints on non-destructiveconstructions of proofs of Brouwers fixed-pointtheorem and the Hahn-Banach theorem andMandelkern (1988), particularly for elucidat ion ofthe two destructive aspects of the Bolzano-Weierstrass theorem.

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