BADIOU_ .....1IiIt .. original phtloeophcr wo~killg in. Swl 'RI~ ..~ of postmodrm orthO antnct &om fonhi:omiDg work. on th of appcaranCf' and

01' oriel', ph ted heft' in mMK't' of its Frendl publication .

....' & ft' -r . oi.uch aDOIricaI figures as Spinoza. Kant and Hegel_;::~~ ... ~.~ . ;and radical politics, TMord kQJ

_ tMb..t lr" "cdwa to oar oI lbr great thinkers of our time .

Also available from Continuum:

Beillg and Iioenr, Alain BadiouInfinite Though t : T ruth and the Return 0/ Ph ilosophy , Alain Badiou

Think Again: Atain Badiou and she Future of Philosophy ,edited by Peter Hallward

TheoreticalWritings

Alain Badiou

Edited and t ranslated by Ray Brassier and AlbertoToscano

All right s reserved . No part of thi s publication may be reprodu ced or trnnsmiucd in anyfo rm or by any mean s, elect ronic or mechanical, including phoeccopying, r=rding or anyinformation sto rage or retrieval system, with out prio r pe rmission in wr-itmg from thepublishers.

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For Sam Gillespi e ( 1970--2003), whose pioneeri ng work and ten a-cious, passiona te intellect remain an abid ing inspiration to bo th of us.

R.B and A .T.

Bri li sh Library Calaloguing- in-I~ublicali on Dala1\ cata logue rCC(lrd for this book is available ftOm the Brtrish Library .ISBN: HB : 0-826-1-6145-X

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Contents

List of Sources viiiEd itors' Note IXAu t hor's Preface xiii

Section I . O ntology is MathematicsL M athematics and Philosoph y: The G rand Sty le and the L itt le Style 32. Ph ilosoph y and M ath ematics; Infin ity and the End of Roman ticism 2 13. T he Question of Being T oday 394. Platonism and Ma thematical O ntology 495. The Being of Number 596. One , Multiple, Mul tiplici ties 677. Spiooee 's Closed Onto logy 8 1

Section II . The S ubtraction of T ruth8. T he Event as Trans-Being 979. On Subtraction 103

10. T ru th : Forcing and th e Unnameab le 119I I. Kant 's Subtractive Ontology 13512. E ight Theses on the Universal 14313. Politics as Truth Procedur e 153

Section III. Logics of Appearance14. Being and Appearance 16315. Not es T oward a T hinkin g of Appearance 17716. T he Transcendental 18917. Hegel and the Whole 22 118. Lan guage, Thou gh t, Poet ry 233

Notes 243Index of Co ncepts 253Index of Nam es 255

List ofSources

'l\h thc matics and Philosoph y: T he Grand Style and the Linfc Style' istranslated from an unpu blished manuscript; ' Philosophy an d M athematics:In finity and the End of Roman ticism ' originally appeared as ' Phi losophic ermarhemariquc' in Conditions (Paris: Scuil , 1992), pp. 157- 78; 'T he Questionof Being Today' originally appeared as ' La question de l'etrc uujc urd'hui' inCour t sraite d'omologie transitaire (Paris: SCUll, 1998), pp. 25-38; ' Platonismand Mathemati cal Ontology' origina lly appeared in Cour t traite d'ontoiogietra nsimire, pp . 95-119; 'T he Being of Number ' or iginally appeared in Courttrait e d'ama lcgie transitoir e, pp. 141-5 1; 'One, Mul tip le, Mu ltiplicities' ori-gina lly appeared as ' U n, mu ltipl e, multiplici ters), in multitudes 1 (2000) , pp .195-2 11; 'Spinoza's Closed Ontology' ori ginally appeared as ' L'on tologiefcrmec de Spi noza ' in Court trait e d'ontoiogie transitoire, pp. 73-93; 'T heEvent as T rans-Being' is a revised and expanded version of ' L'cycnemen tcomme trans-eire' in Court traitc d'omologie transitoire, pp. 55-9; 'On Sub-tra ction ' originally appea red as ' Co nference sur la sousrracuoo' in Condif iolls,pp. 179-95; 'T ruth: Forcing and the Unnameable ' originally ap peared as'V eri te: forcagc ct inuomablc' in Conditions, pp. 196-2 12; ' K an t's Su btractiveOntology' orig inally appeared as ' L'ontologie soustracttve de Kan t' in Courttraite d'cntoiogie transitoire, pp . 153-64; ' Eight Theses on the Universal' o ri-g inally appeared as ' Huit theses sur I'univcrscl ' in Uniuersei, singulier, sujet,ed. I clica Sumic (Paris: K ime, 2000 ), pp . 11- 20; ' Politics as a Trut h Pr oce-du re' origi na lly ap peared in A brigi de metapolitiqnc (Paris: SCUll, 1998), pp.155-67; ' Being and Appearan ce' origina lly appeared as ' L'etrc cr l'apparaitrc'in Court trai ts d'ClI/tolllg;e transiroire, pp. 179-200 ; ' Notes Toward a T hinkingof Appearance' is trans lated from an unpublished manuscript; 'The T rans-cendental' and ' Hegel and th e Wh ole ' are translated from a draft man uscr iptof Logiql/t'S des ntondes (Paris: Seuil , fo rthcoming); ' La ng uage, T hought ,Pocu-y' is tra nslated from the aut hor's manuscript , a Portuguese lang uageversion has been pu blished in Para 11111

extracts p reviewing The Login of Worlds. In spite of the heterogen eity of th esources, and the constrain ts these inevi tab ly imposed , we have deliberatelyassembled the material in such a way as to articu late and exhi bi t the funda-mental structu re of Bad iou's system. Accordi ngly, Th eoretical tl'lri lillKs isdivided into th ree distinct sections, each sect ion anchored in the preced ingouc. Th us the book is exp licitly designed to be read in seq uential orde r . Eachsection unfolds the content and ramifications of a core component of Badicu ' sdoctrine. Sec tion t, OmofoCJ' is Mathematics, introduces the read er to thegroundi ng ges tu re behind Badio u's philo sophical p roject , th e ident ification ofontology with math emat ics. Section II , The Subtraction 0/ Truth , putsfo rward the link be tween th e fundamental concepts of event. tru th andsubject as they are articu lated onto th e on tological doctrine ou tlined inSection L Sect ion III , Logics of A ppearance, outlines the recen t developmentin Badiou of a theory of appearan ce that seeks to localize the truth -even twithin the speci fic cons istency, or transcendental logic. of what he calls awo rld. In conform ity with the arch itectonic JUSt outl ined , each section beginswith direct treatments of the relevant featu re of Bad iou 's system (onrolcgyand the axiom; subjectivity, subtraction and th e even t; appearan ce, logic,world), before going on to elaborate on these: features th rough ( I) targetedengagemen ts with key ph ilosophica l interlocu tors and/or rivals (D eleuze onth e Status of the multiple; Spinoza on axiomatic onlOlogy ; Kant on subtrac-tio n and subjectivity; Hegel on totality and appearance). and (2) brief exem-plifications of phi losophy's engagement with its extra-philosophicalconditions (emancipation and un iversality; the num erical schematization ofpo lit ics; the relation between language and poetry).

S ince we cons ide r Bad iou 's original materi al and our ar rangement thereof10 ren der any further prefatory remarks a h ind rance to th e reader's engage-ment with th e work itse lf', we have chosen to confine our ow n remarks to apostface, which will IT)' to gauge th e conseq uences and explicate the stakes ofBadiou 's p roject vis-a-vis the wider phi losophica l land scape. Were th e readerto encounter int ractab le d ifficu lt ies in navigating Badiou 's co nceptual ap pa-ratus, we strongly recomm end that he or she refers [Q what will un doub ted lyremain the 'canonical' com mentary on Badtou 's thought, Peter Hallward 'sBad/Oil: A S llbjr!ct to T ruth (M inneapolis: Minnesota Un iversity Pr ess, 2003),comp h::mclll ing it if need s be with writings from the burgeo ning secondaryliterature.

We have tried 10 keep ed itorial inrcrvenrions to a strict min imum, pr ovid -ing biblicgruphica! references or cla rifications wherever we deemed it neces-sary. All no tes in sq uare bra ckets arc ours.

T he cd ltors would like to thank Trista n Palmer , who first comm issionedthis project , Hywcl Evans, Veronica Mi ller and Sarah Do uglas at Con-

x Theoret ical Wnt!ngs Editor s' Note

tinuum, and Keith Ansell Pearson for pr ovidi ng us with th e init ial contact,We wou ld also like to express o ur grat itude to those friends who have con-tribu ted , in one way or ano the r, to the concep tion and producti on of th isvolume, whether th rou gh ongo ing debate or ed itorial interven tions: JasonBarker. Lorenzo Chiesa. John Co llins, Oliv er Feltham, Peter Hallward, Ni naPower and Damian Vea l. M ost of all, our th anks go 10 Alain Bad iou, whoseuns tin ting gene rosity and con tinuo us suppo rt for this ven tu re over th e pastth ree yea rs have proved vital.

R.B. , AT .Londo n. Novcrnber 2003

Author's Preface

Philosophical wor ks come in a pecu liar variety of forms. Uhimarely, however,they all seem to fall somewhe re between two funda mental but opposing ten-de ncies . At one ext rem e, we find the comp lete absence of writing and theespousal of oral tra nsmi ssion and critical debate. T his is th e path chosen bySocrates, the venerable incepto r. At the o the r extreme, we find the single'great work', perpetually reworked in solitu de. This is basically the case withSchope nha ucr and his end lessly rev ised World as W ill ami Representa tion.Between these two ex tremes, we find the classical alternation between pre-cisely focused essays and vast synoptic trea tises. T his is the case with Kant,Descartes and many othe rs. But we also encounter the aphoristic approach,much used by Nietzsche , or the carefully orc hestrated success ion of worksdealing with problems in a clearly d iscerni ble seq uence, as in Bergson . Alter-natively, we have an amassing of brief bu t very dense texts, without an yatt empt at systematic overview, as is the case with Leibniz; or a d ispa rateseries of long , qua si-novelistic works (sometimes involving pseudon yms), likethose produced by Kic rkcgaard and also to a certain extent by JacquesDcrrtda. We should also no te the significant number of wor ks that haveacqu ired a mythical status precisely because the y were announced but neverfinished: for example, Plato' s dialogue, The Philosopher; Pascal's Pensees, thethird volume of Marx's Capita l, pa rt two of Heidegger's Sein und Zeit, or Sar-Ire's boo k on morality. It is also important to note how many ' books' of philo-sophy are in fact lecture not es, eithe r kept by the lecturer him self lindsubsequent ly published (th is is th e case for a major po rt ion of Heidegger 'swork, but also fo r figures like Jules Lagncau , Merleau-P onty and othe rs), ortaken by students (this is the case fo r almos t all the works by Aristo tle thathave been handed down to us, but also for important parts of Hegel's work,such as his aestheti cs an d his history of philosophy). Let's round off this briefsketch by remark ing that the phi losophical corpus seems to encom pass everyconceivable style of presen tation: dramatic d ialogue (Plato, M alcb ranchc,Sche lling ... ); novelist ic narrative (Rousseau, Hold erlin , N ietzsch e ); mat h-ematical treat ises in the Euclidean manne r (Descartes, Spinoza ); auto-

Theoret ical Wrltlngs

biography (St. Au gu stine, Kie rkegaard .. .); expans ive treatises for me pur-poses of wh ich the author has forged a new concep tual vocabulary (K an t,F ichte, Hegel ...); poems (Pannenides, Lucretius .. .); as well as many others-ba sica lly, anything whatsoever thai can be classified as 'writing '.

In other words, it is impossible to prov ide a clear-cut cri terion for whatcou nts as a book of philosoph y. Co nsider the n the case of these TheoreticalIf/ ri l i tlgJ; in what sense can th is presen t book really be said to be one of mybooks? Specificall y, one of my books of phi losop hy? Is it not rather a book bymy friends Ray Brassier and Alberto T oscano? After all , they gathered andselected Ihe texts from several d ifferent books, which for th e most part werenor stri ctly spea king ' works' but rather collections of essays. They decidedthai these texts merited the adjective 'theoretica l'. And they translated theminto English. so th at the end resu lt can be said not to have existed anywhe repri or to this publication .

Basically, I wou ld like abo ve all to than k these two fr iend s, as wel l asT ristan Palm er fro m Co ntin uum, who agree d to publish all this work. Iwould like to thank them because mel' have provided me, along with otherread ers, wit h th e oppo rtun ity of reading a new, p reviously unpublished book,appa rently autho red by someone called 'Alain Badiou ' - who is repu ted to benone othe r than myself.

What is the principal interest of th is new book? It is, I think, that it p ro-vides a new fonnulation of what can be cons idered to be the fundamentalcore of my philosophical doctrine - o r ' theory', 10 adopt the term used in thetitle of this book . Rather than linger over exa mples, de tails, tangentialhypoth eses, the ed itors have co-ordinated th e sequence of fundamental con-cepts in such a way as to con struct a fram ework for th eir articulation . Theytry to show how, start ing from an onto logy whose paradigm is mathemat ical,I am able to propo se a new vision of wha t a trlllh is, along with a new vis iono f what it is to be the Jubject of such a tru th.

This pairing of subject and truth goes back a long way. It is one o f th e o ldes tpairings in th e entire history o f ph iloso phy. Mo reover, the idea that the roo t o fthi s pairing lies in a thi nking of pu re being , or being qua bei ng , is not exac tlynew either . But this is the whole po in t: Ra y Brassier and Albe rto T oscano areconv inced tha t th e way in which I propose to link the three term s being, tr uth ,and subject, is novel and pe rsuasive; perhaps because th ere arc rigorouslyexac ting conditions for th is lin king . In order for bei ng to be thi nkable, it has tobe considered on the basis o f th e mathematical theory of multipliciti es. Ino rde r for a tru th to come forth, a hazardous su pple me nting o f being isrequired , a situa ted but incalculab le event . La stl y, in o rder for a subject 10 beconstitu ted, wha t must be deployed in the situation o f this sub ject is a multi-p licity tha t is ano nymous and ega lita rian, which is to say, gen er ic.

Author's Preface

Wh at these essay s, which my two fr iends have gathered and basically rein-ven ted here, sho w - at lC2St in my eyes - is that in o rder for the theor eticaltriad o f being, truth, and su bject to hold , it is necessary to think the triadtha t follows from it - which is 10 say the triad of the mu lrip le (along with thevoid), the event (along with its site) and the generic (along with th e newforms of knowled ge which it allows us 10 force).

In other word s, what we have here is the theo re tical core of my phi loso-phy , becau se th is book exhibits, non-deductively, new technical concept s th atallow us to transcribe the classical problematic (being, truth, su bject) into aconcep tual assemblage that is not on ly modem, but perhaps even 'more-than-modem ' (given that th e adjective ' pos trnod ern' has been evacuated o f allcon ten t). These concepts arc: mathem atical multiplicity, th e plurality of infi-n ities, the void as p roper nam e of being, th e even t as trans-be ing, fidelity , thesu bject of enq uiries, the generic and forci ng. These concep ts provide us withthe rad ically new term s req uired for a reformulation of H eid eggcr's funda -me n tal qu estion: 'What is it 10 think ?'

But one of the aims of my translator friends is also 10 explain why my con-cep tion of philosophy - and hence my answe r to the qu estion about th ink ing- requires that phi losophy remain under the combined guard of the mathe-matica l cond ition as well as th e poetic cond ition . Generally, the con-tempo rary phi losop hics that place them selves un der th e auspices o f the poem(e.g. in the wake of Heidegger ) diffe r essentially fro m those that place th em -selves under the auspices of th e ma thcme (e.g. th e various b ranches of anal y-tical phi losophy). One of the pecu liar characteristics of my ow n projec t is th atit requires bo th the reference to poe try and a basis in mathem atics. It doesso, moreover, th rou gh a combined critique o f the way in which Heideggeruses poe try and the way ana lyti ca l phi losop hers use mathemati cal logic. Ibe lieve th at th is double req uiremen t follows fro m th e fact th at at th e core ofmy Ih ink ing lies a rat ional denial of finitude, and th e convict ion that think-ing, ou r th inking, is essentially tied to th e infinite. But the in finite as form ofbeing is mathemat ica l, while th e in fin ite as resource for th e po wer of lan-guage is poe tic .

For a long time, Ray Brassier and Alberto Toscano hoped th e titl e of thisbook wou ld be The S ulfur M uthet1le. Perhap s thi s is tOO esote r ic an expres-sion . 8 Ul it encapsu lates what is essen tial 10 my th ink ing . T hought is a'math eme' insofar as th e pure mu h iple is on ly th inka ble th ro ugh math cma-tical insc riptio n. But rhough r is a 's te llar mathem c' in so far as, like th esym bo l of the srar in the poe try of M allarmc, it constit utes, beyond its ownempiricallimits, a reserve of etern ity .

A .BParis, Sp ring 2003

SECTION I

Ontology is Mathemat ics

CHAPTER I

Mathematics and PhilosophyThe Grand Style and the LittleStyle

In order to address th e relati on between mathematics and philosophy, wemust first distinguish between the grand style and the little style.

The litt le style painstakingly construc ts mathemati cs as an objul for ph ilo-sophical scrutiny. I call it ' the litt le style' because it ass igns mathematics asubservient role, as somethi ng whose only function seems to consist inhelpi ng to perpe tua te a well -defined area of ph ilosophical specialization, Thisarea of specia lization goes by the name ' philosop hy of mathematics", whe reth e genitive 'or is objective. The philosophy of mathematics can in tu m beinsc ribed within an area of specialization that gOC1 by th e nam e 'epistemologyand history of science'; an area pos sessing its own special ized bu reaucracy inthose acade mic comminees and bod ies whose role it is to manage a personnelcomprising teac hers and researchers.

But in philosophy, specialization invariably gives rise to th e littl e style. InLacanian terms, we could say that it collapses the discou rse of the Master -which is rooted in the master-sign ifier, the 5 1 that gives rise to a signifyingchain - onto the d iscourse of th e University, that perpetual commentarywhich is well represen ted by the second moment of all speech, the 52 whichexists by making the M aster disappear thro ugh the usurpation of commen-tary.

The Hnl c sty le, which is characteristic of the ph ilosophy and epi stemologyof math ematics, str ives to d issolve the ontological sovereign ty of mathe-matics, its aris tocratic self-sufficiency, its unrivalled mastery, by confining itsdr amat ic, almost baffl ing existence 10 a stale com partment of academ icspecialization.

The most tellin g feature of th e littl e style is th e manner in whi ch il exertsits gr ip upo n its objec t th rough historicization and classification . We couldcharacterize th is object as a neutered mathematics, one which is the exclusivepreserve of the little sty le precisely beca use it has bee n created by ir.

When the goal is to eliminate a frighten ing master-sign ifier , classificationand hisrc ricizaricn are the ha llmarks of a very liul e style.

4 Theore t ical Wri t ings t-tatremaucs and Philosophy 5

LeI me straigh taway provide a genuinely worthy ins tance o f the lill ie sty le;in o the r wo rd s, a great example of the lill ie style. I refer to the 'phi losophicalremarks' thai conclu de a tru ly remarkable work entitled Foundations of Set-Theory, whose second ed itio n, from wh ich I am q uoting here, dares from1973 . I ca ll ir grea t beca use , amo ng othe r things, it was written by th ree firs t-rate logicians and mathematicians : Abraham F raenkcl , Yehoshu a Bar -H illeland Azricl Le vy. This book's concl ud ing philosophica l paragraph bald lystates that:

Ou r first problem regards the ontological starus of SClS - not of this or th eothe r set, bu t se ts in general. S ince sets, as o rdina rily und ers tood , arc wha tph ilosop he rs call ll1liw rsals, o ur presen t prob lem is pari of the well-knownand amply discussed problem of th e ontological status 0/ ulliversals.1

Let us immed iately note th ree fea tures of th is bri ef parag raph, with whichan y adept of th e little style would unhesitatingly concur .

Firstl y, what is at stake is not what mathematics might enta il for ontology ,bu t ra ther th e specific ontology of mathem ati cs. In othe r wo rds, mathematicshere simply represent s a particular instance of a ready- made phi losophicalqu estion, ra ther tha n some th ing capable of cha llenging o r underm in ing thatqu estion, an d still less some th ing capab le o f providing a paradoxical o rdramatic solutio n for it.

Second ly, what is th is read y-made ph ilosophical q ues tion? It is act ually aquestion conce rn ing logic, or the capaci ties of lan guage. In short, thequestion of universals. Only by way of a p reliminary redu ct ion of ma them a-tica l problems to logical and linguistic problem s docs one become able toshoehorn mathematics into the realm of philosophical quest ioning and tran s-fo rm it into a specialized objective region su bsumed by ph ilosoph y. T h ispa rticu lar move is a fundamen tal hallmark o f the lillie style.

Third ly, th e philosoph ical problem is in no sense spa rked or provoked byth e ma th ematical prob lem; it has an independent history and, as the authorsremind us, featu red prominentl y in ' th e scho lastic debates of the mid dleages' . It is a cla ssical problem , with regar d to which mathema tics represen tsan opport unity for an updated , regional ad justm ent.

Th is becom es ap parent whe n we conside r th e classi ficatory zeal exhibi tedby the autho rs when they come to outl ine the possibl e responses to thep rob lem:

The th ree main tradi tional answe rs 10 the p roblem of un iversals, stemmingfrom medieval di scussions, are known as realism, nominalism, and concet nu-atism, \'(' e shall no t dea l here wit h these lines of though t in th eir tradi tional

vers ion bu t only with th eir modem counte rpa rts known as Platonism, neo-nominalism, and " eo-conceptu alism ( though we shall mostly omit th e prefix'noo-' since we sha ll have no opportun ity to deal with the o lder versions).In addition , w 1.' shall d eal wit h a fou rth attitude wh ich rega rd s the who leprob lem of th e onto logica l status of un iversa ls in general an d of se ts inparticular as a metaph ysical pseud o-problem.I

Clearly, the phi losophical incorporation of mathem atics carried out by thelittle style amount s to a neo-classical ope ra tion pur e and simple. It assumesthat ma thema tics can be trea ted as a particular area of ph ilosophi cal conce rn ;tha t thi s treatm ent necessaril y p roceeds through a cons ide ration of logic andlangu age; that it is entirely compatible with ready-made philosophical cate-gories; and th at it lead s to a class ification of doctr ines in terms of p ropernames.

T here is an old tech nical term in phi losop hy for th is kind of neo-classicistap proach: scho lastici sm.

Where ma th ematics is concerned, the little style amounts to a regionalscholasticism .

We find a perfect example of thi s regional scho lasticism in an interventionby Pasca l Engel. P rofessor at the Sorbonne, in a boo k called M athematicalObject ivity.J In th e course of a grammatical eXCUr1US concerni ng th e sta tus ofstatemen ts. Engel manages to use no less th an twen ty-five cla ssificatorysyn tagm s. These are, in thei r order of ap pea rance in th is littl e jewel of scho-lasticism: Platonism , on tological realism , nom inalism, ph enomenalism ,red uctionism, fictionalism , ins trumen talism, on tological antireali sm , sem anticrealism, seman tic an tirealism, intuitionism , idealism, verificationism, form-alism , cons tructivism, agnos ticism, onrologjcal reduct ion ism, on tologicalinflatio n ism. semantic atom ism, holi sm, logicism , ontological neutralism.conceptualism, empirical rea lism and conce ptual Platoni sm. M oreo ver,remarkable thou gh it is, Engel 's compulsive labelling in no way exha ust s thepossible categorial perm uta tions. These are p roba bly infinite, which is whyscholasticism is assu red of a bu sy fu tu re, even if, in conform ity with th escholastic injunction to inte llectual 'ser iousness', its wor k is invari ab lycarried ou t in team s.

N evertheless, it is po ssible to sketch a b rief survey of modem scholasticismin the company of F racnkcl, Bar-Hillel and Levy. Fi rst , they propose defin i-lions for each of the fundament al ap proac hes. T hen they cautiously poi nt ou tthat, as we have already seen with Engel, there are all sorts o f int ermediarypositio ns. F inally, they designate the pur est standard -bearers fo r eac h o f thefou r positions.

Lei 'S take a closer look.

6 The oretical Writ ings t-tatr emaucs and Philosophy 7

First, the definitions. In the following passage, the word 'set ' is to beunderstood as designating an y math emati cal configura tion that can bedefined in rigo rous language:

A Platonist is convinced that correspondi ng to each well-defined (monadic)cond ition there exis ts, in gene ral, a set, or class, wh ich comprises all andon ly those en tities th at fu lfil th is condi tio n and which is an enti ty in itsown right of an on tologica l status sim ilar to that of its membe rs.A neo-naminalist declares himself un able (0 un derstan d wh at oth er peo plemean when they arc talki ng about se ts un less he is ab le to interp ret theirtal k as a fa;otl de parler. The only language he professes himself 10unders tan d is a calculus o f ind ivid uals, cons tructed as a firs t-orderth eory.There are authors wh o are att racted neith er by the luscious jung le flo raof Platon ism nor by the ascetic desert landscape of neo- nominalism .They prefer to live in the well-d esigned and perspicuous orcha rds o f neo-conceptualism. They claim [0 unders tand what sets are, though th emetaph or they p refe r is that o f constructing (or invemi"g) rather than thatof singling out (or disCQtJed"g), which is th e one cheris hed by the Plato-nis ts ... [Tlhey are not ready to accept axioms or theorems that wouldforce them to admi t the existence of se ts which are not constru ctivelycharacterizable ."

T hu s th e Platonist admits the existence of entities that are ind ifferen t tothe lim its o f language and transcend human const ructive capacities; thenomina list on ly adm its the existence of verifiab le: indiv id ual s fu lfilling atranspa ren t syntactic form ; and the conceptua list dem ands that all existencebe subo rd inated to an effec tive construct ion , wh ich is itself dependent uponthe existence of enti ties that are either already eviden t o r cons tructed.

Church o r Godel can be invoked as un comprom ising Platon ists; Hilbert o rBrouwer as uneq u ivocal conce ptua lists; and Goodman as a rabid nominalist.

We have yet to ment ion the app roach whic h remains rad ically agnostic, theone tha t always comes in fourth place. Follow ing thesis I (' Sets have a realexi stence as ideal en tities independent of the mind '), thesis 2 (' Sets exist on lyas ind ividu al enti ties valida ting lingu istic expressions'), and th esis 3 (' Setsexist as men tal constructio ns'), comes th esis 4, the supern ume rary thesis:'T he quest ion abo ut the way in which sets exist has no mean ing Outside agiven rhcorcuca t con text ':

Thc prevalent o pin ions [i.e. Platon ism, nom inalism and conceptu alism ) arccaused by a fusion of, and confusion between , tWO different qu estions: the

one whether certain existen tial sentences can be proved, or d isproved , orshown to be und ecidable, withi" a giw II lheory, th e other wheth er thi stheo ry as a whole sho uld be accepted,"

Cam ap, the theoretician most rep resen tati ve of this clarificatory ap proach ,suggests that th e first prob lem , which depe nds on the reso urces of the theoryin q uestion , is a pu rely technical one , and that the second problem boilsdown to a practica l issue thai can on ly be decided according to variouscriteria. which Fraenkel et a!. summa rize as :

[L )ikelihood of being cons istent , ease of maneuverability, effectiveness inderiving classical analysis, teachabiliry, perhaps possession o f standardmod els, etc.6

It is by failing to disti nguish between th ese: two questions that one ends upform ulating mean ingless metaphysical problem s such as; Are there non-den ume rab le in finite sets?' - II question that can onl y lead to irresolvab le andul tima tely steri le controvers ies because it mi stakenly invokes existence in anabsolute rather than merely th eory -relati ve sen se.

Clearly then. the littl e style encompasses all four of lh CSC' optio ns, andholds sway wheth er one adopts a realist, lingu istic, constructivist or purelyrelativist stance vis-a-v is th e existence of mathematical en tities.

But this is beca use one has already presu pposed that philoso ph y relates tomathem atics through II critical examination of its objects , that it is th e mod eof existence of these objects that has to be interroga ted , and that there areultimate ly four ways of conceiving of that existence: as intri ns ic; as nothingbut th e correlate o f a na me; as a mental construction; o r as a variable prag-matic correlate.

T he grand style is entirely d ifferen t. It stipu lates th at mathematicsprovi des a d irect illum ina tion o f ph ilosop hy, rather tha n th e op posite , andthat th is illu mination is carried ou t thro ugh a forced or even violent interven-tion at the core of these issues.

I will now run th rough five majest ic examples of th e grand style:Desca rte s, Spinosa , Kant , Hegel and Leutreamont .

Firs t exam ple; Descartes, Regulae ad directianem iI/gel/ii, ' Ru les for theDirection of the M ind ', Rule II :

T h is furni shes us with an evide nt explanatio n of th e grea t su perio rity incert itude of Arithmet ic and Geome try to other sciences. The forme r alonedeal with an object so pu re and uncomplicated , that the y need make noassumptions at all which experience ren ders uncer tain , but wh olly consist

8 r-tathemancsand Philosophy 9

in the rat ional deduction o f conseq uences. T hey arc on that account mu chthe easiest and dearest of all , and possess an object such as we require, forin them it is scarce humanly possib le for an yone to crr except by inadve r-ten ce....

But one conclus ion now eme rges ou t o f th ese cons idera tions, viz, no tindeed , th at Arithmetic an d G eometry are the sole sciences to be stud ied,but onl y th at in o ur search for th e di rec t road towards truth we shou ldbu sy ou rselves with no object abo ut whic h we cannot att ain a cert itudeequal to that of the demonstrations of Arithmetic and Geometry."

For Descartes, math ematics d early provides the pa rad igm for ph ilosophy,a paradigm of certainty . But it is important not 10 confuse th e latter with alogical pa radigm . It is not p roo f that lies behind th e paradigm atic value o fma thematics for th e phi losoph er . Rather, it is the absolu te simplicity andcla rity of the ma thematical object.

Second example: Spinoza, appendix to Book On e of the Ethics, a text dearto Lo uis Alth usser :

So they maintained it as certa in that the judgm ent s of the god s far surp assman 's gra sp . This alone , of course, would have caused th e truth to behidden from the human race to ete rn ity, if mathem atics , wh ich iscon cerned not with ends, but on ly with th e essences and properties offigures, had not shown men ano ther standard of truth ....That is why we have such sayings as: ' So man y heads, so man y att itudes','everyone finds his own jud gment more than eno ugh', and 'th ere are asmany differences of b rains as of pa lates'. These proverbs show sufficien tlythat men judge thi ngs accordi ng to the d ispos it ion of th eir brain. andimagine, ra th er than unders tand them. Fo r if men had underst ood them ,th e th ings would at least convince them all, even if they d id not attractthem all, as the exam ple of mathem atics shows. '

It wou ld be no exagge ration to say th at , for Spmoza, mathematics governsthe his torlal destiny of knowledge, and hen ce the econo my of freedom, o rbeatitude . Withou t mathemat ics, humanity langu ishes in the night of su per-stitio n, which can be summarized by the ma xim : there is some th ing wecan not think. T o whic h it is necessary to add that mathematics also teachesus some th ing essential: that whatever is thought truly is immed iate ly shared.Mathematics shows that wh ate ver is un der stood is radically un d ivided . T oknow is to be absolutely and un iversally conv ince d .

T h ird example : Kam , Critique of Pure Reason, Preface to th e secondedi tion :

In the earliest times to whi ch the h istory of human reason extends, mathe-mat ics, among that wonderful peop le, the G reeks, had already enteredupon the su re path of scie nce . But it must not be sup posed that it \\'35 aseasy fo r math ematics as it was for logic - in which reaso n has to deal withitse lf alone - to ligh t upon, or ra ther construct for itself, th at ro yal road.On the contrary , I bel ieve th at it long remained , especially among th eEgyp tians, in the groping stage, and that th e tra nsformation must ha vebeen due 10 a revolution broug h t about by th e hap py th ough t of a singlema n, the experiments which he devised marking out the path upon whichthe science must en ter, and by following wh ich , secu re prog ressthro ughout all time and in end less expans ion is in fallib ly secu red . ..A new light Hashed upon the mind of the firs t man (be he Thales or someother) who demo nstrated the p roperti es of th e isosceles tri angle. The truemethod, so he foun d , was not to inspect what he discerned either in th efigu re, o r in the bare concept of it, and from this, as it were, to read off itsp roperties; but to b ring ou t what was necessari ly implied in the conceptsthat he has himse lf formed a prio ri and had pu t into the figu re in theconstruction by wh ich he p resen ted it to himse lf.9

T h us K an t thinks, firstl y, that mathem atics secured for itse lf from its veryorigin th e sure path of a science . Second ly, that th e cre ation of mathematicsis tantamount to an absolute his torical singu larity , a ' revolution ' - so muehso that its eme rgence deserves to be singu larized : it was du e to the felicito usthought of a single man . Nothing could be further fro m a h istoricist orcul ruralist exp lanation. Thirdly, Kant thinks th at , once ope ned u p, the pathis in finite, in time as we ll as in space. This un iversalism is a con crete un ivers-alism becau se it is the un iversalism of a tra jectory of though t that can alwaysbe retraced , irrespective of th e tim e o r the place. And fou rth ly, Kant sees inmathematics some thing that marks the perpetual rediscovery of its paradig-matic funct ion , the ina ugu ral conception of a type of knowledge tha t isneither empi rica l ( it is not what can be di scerned in the figu re), no r formal (itdocs nor consis t in the pu re, static, identi fiable propert ies o f the concept).T hus mat hematics pave s the way for the critical representation of th inking,which consists in seeing knowledge as an instance o f non-em pirical p rodu c-lion or construction, a sensib le const ruc tion that is adequate to the cons ti-tu ting a p riori. In other words, 'T hales' is th e pu tat ive name for a revolu tionthat extends to the en rircry of philosophy - whieh is to say that Klint 'scritica l project amounts to an exa mina tion of the condi tions of possib ilitythat un der lie Thales' con structio n.

Fourth example: Hegel , S cience of Logic, the leng thy Rem ark th at followsthe explica tio n of the in fin ity o f th e q uantum :

10 Theoretcal Writings Mathemat ICSand Pt'III0s0phy"

[I)n a philosophical respect th e mathematical infini te is im portant becauseunder lying ir, in fact . is th e notion of the genuine in fini te and it is farsuperio r to the o rdi nary so-called metaphYfital j"/illite on which arc basedthe objections to the mathematica l infinite. ...It is worthwhile: considering more close ly the ma them atica l conce pt o f theinfinite together with the most noteworthy o f the attempts aimed at just i-fying its usc: and eliminat ing the di fficu lty with which the method feelsitself burdened. T he considerat ion of these justifications and characteristicsof the mathematical infinite which I shall undertake at some length in th isRemark will at the same lim e th row the best ligh t on the natu re of the trueNotion itself and show how th is latt er was vaguely present as a basis forthose procedures. 10

The decisive poi nt here is th at, for Hegel , ma th ematics and philosoph icalspeculation sha re a fundamental concept: th e conce pt of the infinite. M orepan icularl y, th e des ti tu tion of the metaphysical conce p t of infinity - in otherwords, the d estitution of classica l theology - is ini tia lly undertaken th roughth e determination of the math emati ca l concep t o f the in finite. H egelobv iously has in mind the creation of the d ifferential and in tegra l ca lcu lusduring the seve nteenth and eigh teen th cen tur ies. He wants to show how thetrue (i.c. d ialect ica l) concep tion of the infinite makes its historica l appearanceund er the auspices of mathematics. His method is rem arkable: it consists inexami ning the contradic to ry labour o f the Not ion in so far as the lat ter canbe seen to be at wo rk within the mathematica l tex t itself. The Notion is bo thacti ve and manifest, it ru ins th e transce nden t theological concept o f th ein finite, but it is not yet th e consc ious know led ge o f its own acti vity . Unli kethe metaphysical infinite, the math em atica l infinite is the same as the goodinfinite of the d ialect ic. But it is lhe same on ly acco rding to th e d ifferencewhereby it does not yet kn ow itsel f as the same . In this ins tance , as in Platoor in my own work, ph ilosop hy's role cons ists in infonning math ematics ofits ow n specu lative grandeu r. In Hegel , th is takes the fonn of a deta iledexam ination of wha t he refers to as th e 'j ustifications and characteristics ' ofthe mathem atical concept of the infin ite; an examination which , fo r h im,consists in carryi ng ou t a meticulous analysis of the ideas of Euler andLagrange. T hro ugh th is analysis, on e sees how the math ematical concep tionof the infini te, whic h for Hegel is st ill ham pe red by 'the dl fficulry wit h whichthe method feels itse lf bu rdened ' , harbours with in itself the affirmativereso urce of a gen uinely absolu te concep tion o f quanti ty.

It seems fitti ng that we shou ld conclude th is su rvey of the grand style witha figu re who stradd les the margin be tween philosophy and the poem: IsidoreD ucasse, aka th e Co mte de Lauireamon r. Li ke Rimbaud an d Nietzsche,

Lau rreamom, using th e post - Romantic name ' Maldo ror ', wants to bringabout a denatu ring of man , a transmigration of his essence, a positiveb

12 Theoret ical Wnt lngs r- te tbemaucs and Philosophy 13

J ust as Nietzsche wished 10 surpass Ch rist and announce the adven t ofD ion ysus by having Zarathustra speak in the language of the Gospels (' intruth ' , ' I say unto you ', CIC. ) , Laurreamon r, by coupling Masonic csotcricismwith Old T estamen t language, wants to delineate the monstrous beco mi ng towhic h an exhau sted , defiled mankind is dest ined . In thi s regard , mathe-matics, wh ich is divided in ro algebra, arithmetic and geo me try - i.e. ' laconicequations", 'cabbalistic ciphers' and 'sc u lpted lines' - renders an indispc n-sab le service: it imposes on us a kind of implacable ete rnity whic h d irectl ychallenges the humanist conception of man . Mathematics is, in effect, 'o lderthan the sun' and will remain intact 'on the ruins of time' . Malhemalics isthe d iscipl ine and th e severity, the immutabili ty and th e ima ge of ' th atsu p reme truth ' . This is only a short Step away fro m saying that mathemat icsinscribes being as such; a Step which. as you know. I have take n. But forLautreamom, ma thematics is some thing even better : it is what furn ishes theinte llect with 'alien qu ali ties ", This is an essen tial po int: there is no intrinsicharmon y between math ematics and the human intellec t. The exercise o fmathematics. th e lessons - 'sweeter than honey' - that it teaches, is th eexe rcise of an alteration, an estrangem ent o f intelligence. And it is first andforemost th rough this resource of strangeness that mathematica l eternitysu bve rts ord inary thi nk ing. Here we have the profound reason why, witho utmathematics, without the infection of conventiona l thinking by math ematics.Maldoror wo uld not have p revailed in his fundamental struggle againsthumanist man. in his struggle to bring forth th e free monster beyondhuman ity of which man is capable .

On all these points. from glacial ami-humanis m to thc trans-hu man advento f truths. I th ink I may welt be Is idore Ducessc's one and on ly genuined isciple. Why then do I call my self a Platonist rath er than a Ducassean o r ason of Maldoror?

Because P lato says exactly th e same thing.L ike Is idore Ducasse, Plato claims that ma thematics und oes doxa and

defea ts the soph ist. Withom mathematics there cou ld ne ver arise, beyondexisting h umanity. those philosopher-kings who represent th e overm an' s alle -gorical name in the conceptual city erected by Plato . If there is to be anychance of seeing these ph ilosoph er-kings appear. the young mu st be taughtarithmetic, plan e geome try. so lid geo me try and astronomy for at least tenyears. For Plato , what is admirable about mathem atics is not JUSt that, as iswell know n. it lieu its sights on pu re essenc es, on the idea as such. but alsotha t its utility can be exp licated in ter ms of the only pragma t ics of any worthfo r a man who has risen beyond man , which is 10 say , in term s of war.Conside r lo r exam ple this passage from The Republic, Book 7, 525c (which Ihave taken Ihe libert)' to retran slate):

Soc ra tes: So o ur o verm an mu st be bo th ph ilosopher and sold icr?G laucon: Of course .Socrates: Then a law must be passed - immed iately.Glaucon: A law? Why a law. in God's nam e? W hat law?Soc ra tes: A law stipulating th e teaching of higher arit hme tic, you dullard .Hut we'll have trouble.Glaucon: Trouble? Why?Soc ra tes : T ake a young fellow who want s to beco me admi ral of the fleet. ormi nis te r, or president . or some th ing of tha t ilk. A youn g ho tshot straightout of th e L SE o r Yale . Do you imagine he'll be ru shing to en rol at th eins ti tute o f h igh er arithmetic? We'll have some serious convi ncing to do ,let me tell you.Glaucon: I can 't imagine wh at we're goi ng to tell h im .Socrates: The tru th . Some thi ng ha rsh. For exam ple: ' My dear fellow. if youwant to become mini ster or admiral, firs t you have to srop being such anagreea ble young man , a common yu ppie. Take numbers, for ins tance, doyou know what numbe rs are? I'm no t ta lking about what you need to knowto carry out your petty little business transactions, or count whatever it isyou' re flogging on th e market ! I 'm l3lking about number in so far as youcontem plate it in its eternal essence through the shee r power of you r yuppieintellect, which I p romise to dc-yupp ify! Number such as it exists in war. inthe terrible reckoning o f weapons and corpses . But abo ve all , number aswha t brings about a complete upheaval in thinking. as what erases approxi-mat ion and beco mi ng to make way fo r be ing as suc h, as well as its truth .'Glaucon: After hearing your lillie speech , I th ink ou r yuppie friend willru n like he ll. sca red om of his wits.

T his is what I mean by th e grand style: arit hmetic as an ins tance of ste llarand warlike inhumanity !

It should come as no surp rise. then , tha t today we see mathematics beingattacked sys tematica lly from all sides. J ust as po litics is being systematicallyatt acked in the nam e of economic and state manageme nt ; or art sys tematicallyatt acked in the nam e of cu ltural re lativity; or love systematically attacked intin' name of a pragmatics of sex. The littl e style of epis temo logical specializa-lion is merely an unw illing pawn in th is anack. So we have no cho ice: if wearc to defend ourselves - 'we' who speak on behalf of philosoph y itsel f and ofthe sup plementary step it can and must take - we have to find the new termsrcquircd for the grand style.

Hut let us fir st rccapirulutc the teaching of our admirab le p redecessors.It is obvious that for each of the m. the confrontation with ma thematic s is

an absolu tely indispe nsab le condition for philosophy as such; a conditio n that

14 Tbecreucat Wri tings r-ietreroeucs and Philosophy 15

is at once descriptively external and presc rip tively immanent fo r philosop h y.This holds even where mere are eno rmous d ivergen ces as 10 what const itu testhe fundamental proj ect of phi losophy. For Plato, it consists in creating anew conception of politics. For Descartes. in enla rging the scope of absolutecertainty to encompass the essent ial ques tions of life. For Spinoza, inattaining the intellec tual love of God. For Kant, in knowing exa ctly whe re tod raw th e line between faith and know led ge. For Hegel , in showing thebecoming-subject of the absolute. For Lautrearnom, in d isfigu ring and ove r-coming human ist man . But in each case, it is a q uestion of giving thanks 10' rigorous math em atics ' , II doesn 't matter whether phi losophy is conceived ofas a rationalism tiro to transcendence, as it is fro m Descart es to Lacan ; as avita list immanentism, as it is from Spinoza to Delcuee, as pious criticism. asit is from Kan t to Ricoeur; as a d ialect ic of the absolute, as it is fro m Hegelto M ao Zedong; or an aestheticist crea tionis m, as it is from La u trea mo n r toN ietzsche. For the founders of each of th ese lineages, it still remains th e casethat the cold radicali ry of ma thema tics is th e necessary exercise th rou ghwhic h is forged a th inking subject adeq uate to the tra nsformations he will beforced to undergo .

ExactJy the same holds in m y case. I have assigned ph iloso ph y th e tas k ofcons tructing thought's embrace of its own time, of refracting newborn tru thsth ro ugh the unique pri sm of concepts. Ph ilosoph y must intensify and gath ertogether. under th e aegis o f systematic thinking, not just what its timeimagines itself 10 be, but what irs time is - albeit unknowing ly - capable of.And in order to d o th is, I too had to labo riously set down my own lengthy' thank you' to rigorous mathem atics.

Let me pu t it as bluntly as possible: if there is no grand sty le in th e wayphilosop hy relates to mathematics , the n th ere is no grand style in phi losoph yfu ll stop.

In 1973. Lacen, using a 'we' that , for all its imperiousness, included bo thpsychoanalys ts and psychoan alysis, decla red: 'M athematical formalization isour goal , our ide aL, 16 U sing the same rh etoric, and a ' we' that now incl ud esbo th ph ilosophe rs and ph ilosoph y, I say; 'M athematics is our o bligation, oural teration .'

...

N one of the parti san s of the grand style ever be lieved that th e phi losophicalidentification o f mathema tics had 10 proceed by way of a logicizing orlingui st ic red uction . Suffice it to 53)' that for Descartes , it is the intui tiveclarity of ideas that foun ds th e mathematical parad igm , nOI th e automaticcharacte r of th e deducti ve p roce ss, which is merely the uninterest ing , scho-

13stic aSIXCt of mathematics. Similarly, for Kant . the hisrorial dest iny ofmathem atics as constru ct ion of th e concep t in in tuition cons titutes a revol u-lion that is entirely independen t of the destiny of logic, whic h is al read ycomplete and has sim ply been treading wate r since the time of its founder,Aristotle. Hegel examines the foundation of a conce pt , that of the infini te,and dis regards the apparel of proof. And altho ugh Lautreamont certainly,lpp reciates th e iron nece ssity of the deductive p rocess and the cohe rence ofligu res, what is most important for h im in ma th ema tics is its icy d isciplineand power o f ete rnal survival. As for Spinoza , he sees sa lvation as resid ing inthe onto logy that und erlies ma thematics , whic h is to say, in a conception ofbeing sho rn of every appeal to meaning or pu rpose. and prizing only thecohesiveness of consequences .

There is nOI a single mention o f language in all thi s.LeI us be blum and remark in passing that. in thi s regard, Wi ugenstci n,

despi te the cunn ing of his sterilized loquacity and despite the un den iablefor mal beau ty o f the Tracuu ut - without doub t one of the masterpieces ofant i-p h ilosophy - must be counted amo ng th e architects of the lit tle style,whose p rincip le he sets ou t with his customary b ru ta lity. T hus, in proposi-tion 6.2 1 of the Traclatus, he declares: 'A proposi tion of mathematics doesnot exp ress a thought.' J7 O r worse sti ll, in h is Remarks on the Foundations ofM athematics. we find this son of tri te pragm atism , whic h is very fashionablenowadays:

I should like to ask somethi ng like: 'Does every calc ula tion lead you tosomething useful? In that case, you have avo ided con trad iction. And if itdocs ne t lead you to an yth ing usefu l then wha t di fference does it make ifyou ru n in to a con u adiction?d 8

We can fo rgive Wittgenstei n. Bu t not those wh o shelter behind h isaesthetic cunning (whose entire impetus is eth ica l. i.e . religious) the be tte r 10ado pt the littl e style once and for all and (vain ly) try to throw to th e mod emlions of ind ifference those de te rmi ned to rem ain faithfu l to the grand st yle .

In any case , our maxim is: philosophy must .mter into logic v ia mathematics,'10 / inca mathematics v ia logic.

In my wo rk this translates into: mathem atics is th e science of bei ng quabeing. Lo gic perta ins to the coherence of ap pea rance . And if the study ofappearance also mobilizes certai n areas of mathem atics , th is is simplybeca use, followi ng an insigh t formalized by Hegel but which actually goesback to Plato , it is of th e essence of being to appea r. This is wh at maintain sthe form of all ap pea ring within a mathem anzable transcendental o rder. Bu the re, once again , transcendental logic, whi ch is a part of mathema tics tied to

contem porary sheaf theo ry, holds sway over formal or linguis tic Jogic, whichis ultimately no more than a superficial trans lation of the former .

Reiterating the ' we' I used earlier, I will say: M ath ematics leaches us abo utwhat must be said concern ing what is; not about wha t it is permiJfib/~ to sayconcern ing wha t we rhi" k there is.

M atbemattcs provides philosop hy with a weapon, a fearsome machine ofthough t, a catapult aimed at the bastio ns of ignorance, supers tition andmen tal servitude. It is not a docile grammat ical reg ion. For Plato, mathe-matics is what allows us to break free from the sophist ical dictatorship oflinguistic immed iacy. For La utreamont, it is what releases us from themoribund figu re of the human . Fo r Spino za, it is what breaks with supe rs ti-tion . But you have read their texts. Some today would have us believe th atmathematics itself is relative, prejudiced an d inconsistent, need lessly aristo-cratic, or alternately, subservient to technology. You should be awa re thatth is p ropagand a is trying to undermine what has always been most implac-abl y opposed to spiritualist approximation and gaud y scepticism, th e sicklyallies of flambo yant nihilism . For the truth is that mathematics does notunderstand th e meaning of the claim '1 canno t know' . The mathemalica lrealm does not ackno wledge the existence of spiritualist categories such asthose of the unthinka ble an d the un th ough t, supposedly exceedi ng themeagre resour ces of human reason; or of those scep tical catego ries whichclaim we cannot fever provide a definitive solu tion to a problem or a definitiveanswer to a serious q uestion.

The other sciences an: not so reliable in th is regard . Quentin M dllassouxhas conv incingly argued tha t physics provides no bulwark aga inst sp iri tualist(which is to say obscurantist) speculation, and biology - th at wild empiricismdi sgu ised as science - even less so. Only in math ematics can one uneq uivo-cally maintain th at if though t can form ulat e a problem , it can and will serveit, regard less of how long it takes. Fo r it is also in mathematics th at themaxim ' Keep going!', the o nly maxim requi red in eth ics, has the greatestweight. How else are we to expl ain the fact that the so lu tion to a pro blemformu lated by Fermat more than three cen tu ries ago can be d iscoveredtoday? Or that todny's mathematician s are still actively eng aged in proving ordisp roving conjec tures first p roposed by the G reeks more th an two thousandyears ago? T here can be no doubt that ma the matics conceived in the grandstyle is wa rlike, po lemical, fearsome. And it is by donning the con tempo rarymatheme like a coot of armour that 1 have undcnaken, alone at firs t, to undothe disastrous co nseq uences of philosophy's ' linguistic tur n '; to demarcat e

Humpty Dumpty sat on the wall ,Hu mpty Dumpty had a great faJl.All the king's horses and all the king 's menCo uldn' t put H um pt y together again!

17t-tathemancs and Philosophy

ph ilosophy from phenomenological religiosity; 10 re-found the metaphysicalIriad of be ing , event and subject; to take a stand against poetic prophesying;ro identify gen eric multiplicities as th e on to logical form of the true; to assigna place to La can ian forma lism; and , more recen tly, to articulate the logic ofappearing .

Le t'S say that, as far as we're concern ed , mathematics is always mo re orless equivalent to the bulldozer with wh ich we remo ve the ru bble matprevents us from constructing new edifices in the open air.

T he principal di fficulty probably resides in th e assumption th at mathema-tical compe tence requires years of ini tiation. Whence th e temptation, for thephilosophical de magogue, either to ignore mathematics altoge the r or act as ifthe most pr imitive rud imen ts arc eno ugh in orde r to un derstand what isgoing on there. In th is regard , Ka nt set a very bad example by encouraginggenerations of philosoph ers to believe th at th ey could grasp the essence ofmathematical judgement th rough a single example like 7 + 5 ,. 12. Th is is abit like someo ne saying that one can grasp the relation between philosoph yand poetry by recit ing:

Afte r all, th is is JUSt a bunc h of verses, JUSt as 7 + 5 .. 12 is JUSt a bun ch ofnu mbers .

It is striking that , whether one considers a phi losophical tex t written in thelinle sty le or one wri tten in th e grand style, no justi fication wha tsoever seems10 be req uired for quoting poetry, but no-one wou ld ever d ream of quoting apiece of ma thematical reasoning . No-one seems to consider it acceptable todispense with Hold cnin o r Rim bau d or Pessoa in favour of HumptyDumpty, or to d itch Wagner for Ju lio Iglesias. But as soon as it is II questionuf mathe matics, the reader eit her simply loses interest o r immediately assoc i-ates it with the little style, which is to say, with epistemology, th e histo ry ofscience, specialization.

T his was not Phuo's point of view, nor that of any of the great philoso-phers. Plato very often qu otes poe try, but he also quotes theorems, onesWhich are probably deemed relatively easy by today's standards, bUI werecertain ly dema nding when Plato was writing: th us, in the M e,/O for instance,the construct ion of th e sq uare whose surface is double that of a given sq uare.

I claim the righ t 10 q uote instances of mathema tica l reasoning, providedthe)' arc ap propriate to the philosophical theses in the context of which they

Theoretical Wri tings16

ar e being inscribed, and th e knowledge req uired for understanding them hasalread y been made available 10 the reader . Gi ve us an example, I hear yousay. But I'm nOI going to give you an exam ple: of an example, because I 'vealready provided hundreds of real examples, int egra ted into th e movement ofthough t. So I will men tion two of these movements instead : th e presentationof Dedekind 's doctrine of nu mber in Chapter 4 of Nwnber and N llmbers,l 9and the conside ration of the point of excess in M editation 7 of Being andEvew.20 Consult them, read them, using the reminders, cross- references andth e glossary I have provided in each book . And anyone who still cla ims notto understa nd should write to me telling me exactly wha t it is they don ' tunderstand - otherwise, I fear, we' re simp ly dea ling with excuses for th ereader 's laziness. Ph ilosophers are able 10 understand a fragm ent by Anaxi-man der, an elegy by Rilke, a seminar on th e real by Lacan , but not th e2.500year-o ld proof that th ere are an infin ity of prime numbers . This is anuna ccep table, anti-philosophical state of affairs ; one wh ich only serves theinterests of th e partisans of the little style.

I have spoken of bulldozers and rubble. Wh ich cont emporary ru ins do Ihave in mind? 1 think H egel saw it befo re anyone else: ultimately, mathe-maries proposes a new concept of th e infinite. And on th e basis of th isconcept. it allows for an immanen tization of the infin ite, separating it fromth e One of theology. H egel also saw that the algebraists of his time, likeEuler and Lagran ge, had not quite grasped this: it is only with Baron Cauchyth at the th orn y issue of the limit of a series is finally settled , and not untilCantor th at ligh t is finally thrown on th e august question of the actualinfin ite. H egel th ought th is confus ion was due to the fact tha t th e 'true'concept of the infinite belonged to speculation, 50 that mathematics wasmerely Irs unconscious bea rer, its unwitti ng midwife . The truth is that themathema tical revolution - th e rendering explicit of what had always beenim plicit within mathematics since the tim e of th e Greeks, wh ich is to say, thethorough-going rationalizat ion of the infini te - was yet to come, and in asen se will always be yet to co me, since we still do not know how to effect areasonable ' forcing' of th e kind of infini ty proper to the continuum . Never-theless, we do know why mathematics rad ically subverts bo th empiricistmod erat ion and elegan t scepticism: math emati cs teaches us tha t there is noreason wha tsoeve r to confine th inking within the ambi t of finitude . Withma thematics we know that , as Hegel woul d have said, th e infin ite is nearby.

Yet someo ne mig ht ob ject: 'Well th en, since we already know the resul t,why not just be satisfied with it and leave it at that ? Wh y continue with thearid labou r of familiariz ing ourselves with new axioms, un precede ntedproofs, difficu lt concepts and inconceivably abstract th eories?' Because theinfinite, such as mathematics renders it amenable to th e philosophical will , is

not a fixed and irreversible acqu rstucn. T he historicity of mathematics isnothing but the labour of the infinite, its ongoing and unpred ictablerc-cxposition . A revolut ion, whether French or Bolshevik. cannot exhaust th efonnal concept of emancipation, even though it presents its real; simi larly,inc mathematical ava tars of th e thought of th e infin ite do not exhaust th e

~pccu l3ti ve concept of infinite thought. The confrontation with mathematicsmust constan tly be reconst ituted because the idea of th e infinite only mani-fests itself th rough the mo ving surface of its mathematical reconfigu rations.T his is all the more essent ial given that our ideas of the fini te, and hence orthe philosophical virtualities latent in finitude, become retroac tively d isplacedand reinvigorated through th ose crises. revo lu tions and changes of heart thataffect th e ma th ematical schema of the infin ite. The latt er is a mo ving front. astruggle as silent as it is relentless, where noth ing - no more th ere than else-where - ann ou nces the advent of perpetual peace .

What do th e following notions have in common as regards th eir subtlestconsequences for thinking: the infinity of prime numbers as conceived by theGreeks. the fact that a fun ction tends toward infin ity, th e infinitely sma ll innon-stan dard ana lysis. regular o r singular infinite cardinals, the existence of anumbe r-object in a topos, th e way in which an ope rator grasps and projectsan untctalizable collection of algebraic structures omo a family of sets - no t10 ment ion hu nd reds of other theoret ical formulations. concepts, model s anddetermin ations? Probably someth ing th at has to do with the fact that theinfinite is th e intimate law of th ough t. its naturally anti-natu ral med ium . Butin anoth er regard, th ey have noth ing at all in common. N othing that wou ldallow one merely to reit erate and maintain a simplified , allusive relation withmathematics. This is because. in th e word s of my late friend G illes Chatelcr,lIle mathema tical elaboration of though t is not of th e orde r of a mere linearunfold ing or straigh tforward logical consequence. It comp rises decisive butpreviously un known gest ures." One mu st begin again, because mathematicsis always beginning again and transforming its abstract panoply of concepts.Onc has to begin study ing. wri ting and understanding again that which is infact the hard est th ing in th e world to understand and whose abstraction isthc most inso lent, because thc philosop hical struggle against th e alliance offinitude and obscurantism will onl y be rekindled through th is recommence-mcnt .

T his is why Mallu rm e was wrong on at least one poin t. Lik e evcry greatI'K>Ct, M:al1ar mc was engaged in a tacit rivalry with ma thematics. He wasIrying to show that a densely imagistic poe tic line, when art iculated with inthe bare cadences of thinking, comprises as much if not more truth than thecxtre-linguisrlc inscr iption of the mathcme. This is why he could wri te, in :Isketch for lgirur:

18 Tbeorencal Wri t ings r-tathemaucs and Philosophy 19

20 Theoretical Writings

Infinity is born of cha nce, which you have denied . You , expi red mathem a-tician s - 1, absolute projection . Shou ld end in Infinity ,22

T he idea is clear: Mallurmc accuses mathematicians of denying chance andthereby of fixing the infini te in the heredi tary rigidi ty of calculation . InIgitur, th at rig idity is symbolized by th e fami ly. Whence the poetic, ann-math emarlcnl operation which, M allar me believes, binds infinit y to cha nceand is symbolized by the d ice-th row. On ce the dice have been cast , andregard less of the resul ts , ' in fin ity escapes the family,.n This is why th e math-ematicians expire, and the ab stract conception of the infinite along withthem, in favour of tha t impersonal absolute now represent ed by the hero.

But what M allarme has failed to see is how the o perations through whichmathemati cs has reconfigu red the conceptio n of the infin ite are constantlyaffirming chance th rough th e cont ingency of th eir reco mmcncemem . II is upto philosophy to gath er togeth er or con join th e poet ic affir mat ion of infinitydrawn metaphor ically from chance, and the mathematical construction of theinfin ite, d raw n formally from an axioma tic int u ition . As a resu lt, the in junc-tion to mathematical beauty intersec ts with the injunction to poetic truth.And vice versa.

T here is a very brief poem by Alvar o De Campos, one of the heteron ymsused by Fe rnand o Pcssoa. De Campos is a scientist and engineer and hispoem succi nctly summarizes everyth ing I ha ve been saying. You should beable to memorize it right away. Here it is:

Newton 's binomial is as beautiful as the Vcnus de M ilo .T he truth is few people notice iL24

Style - grand sty le - simply consis ts in noti cing it .

CHAPTER 2

Philosophy and MathematicsInfinity and the End of Romanticism

\'>;that docs th e tide 'ph ilosophy and mathematics' imply about the relationbetween these two disciplines? Does it indicate a difference? An influence? Aboundary? Or perhaps an indifference? For me it im plies none of these. Iunders tand it as impl ying an identification of the mod alities according towhich mat hematics, ever since it s Greek inception, has been a condit ion forphilosop hy; an identification of the figur es th at have historically ent angledmathematics in the determination of the space proper to philosophy.

From a purely descriptive perspective , three of these modali ties or figur escan be distingui shed :

Operating from the perspective of philo sophy, th e first modality sees inmathematic s an approximation, or preliminary peda gogy, for questionsthat are otherwise the province of philosop hy. One acknowledges inmat hematics a certa in aptitude for thin king 'firs t pri nciples' , or forknowledge of being and truth ; an aptitude that becomes fully realized inph ilosophy. We will call thi s th e ontological modali ty of the relationbet ween phi losophy and mathematics.

- T he second modality is th e one that treats math ematics as a regionaldiscipline, an area of cognition in gene ral. Philosophy then sets O UI toexamine what grounds this region al ch aracter o f mathematics. It willboth classify mathematics with in a table of forms of knowled ge, andreflect on the guaran tees (of truth or correc tness) for the discipline thathas been so classified. We will call th is the epistemolog ical modality.

- Finally, the th ird mod ality posits that mathematics is ent irely discon-ncctcd from the questions, o r qu estioning, p rop er to philosophy.Accordi ng to th is vision of thi ngs , math ematics is a register of languagegames, a formal type, or a singular grammar. In an y case, mathematicsdocs not t hillk an ything . In its most rad ical fo rm , th is o rien tationsubsumes mathematics within a generalized techn ics that carries out anun thinking manipulation of being , a levelling of being as pur e standing-

"Theoreti cal Wr itIngs Philosophyand r-tatbemanc s 23

reserve. We will call this modality the , ri/;cal modality , because itaccomplishes a cr itical d isjunct ion between the realm proper to mathe-maries on the one hand, and that of th inking as wh at is at slake in ph ilo-sophy o n the other .

The qu estion I would like: 10 ask is the following; how do things standtoday as far as the aniculation o f th ese: three mod ali ties is conce rned? Howarc we: to situate philosophy's mathematical cond ition from th e pe rspective ofphilosophy? And the thesis I wish to uphold takes the fonn of a ges ture:whereby mathematics is to be re-em angled in to philosophy's innermost st ruc-ture; a st ructure from whic h it has , in actuality, been excluded. I What isrequi red today is a new conditioning of ph ilosoph y by mathematics, a condi-tioning which we are do ubly late in putti ng in to place: both late with respectto what mathem atics itself indicates, and late with respect to th e minimalrequiremen ts necessary for th e con tinuation of ph ilosophy. What is u lti-ma tely at stake here can be form ulated in terms of the followi ng question ,which weighs upon us and threatens to exhaust us: can we be de livered,ji llollj' de livered, fro m our su bjection to Romanticism?

J. THE DISJ UN C TION OF MATHEMATICS A SPHILOSOPHICA LLY CONSTI TU T IVE OF

ROMANTICISM

Up 10 and including Kant, ma thematics and philoso phy were reciprocallyen tan gled, to the exten t that Kant himself (followi ng Descartes. Leibniz,Spinoza, and many others) Still sees in the m ythic name of Thales a commonorigin for math em atics and know ledge in general. For all these philosophers ,it is absolutely clea r that math ematics alone allowed the inaugural breakwi th superstition and ignorance. Mathem atics is fo r th em that singu larform of thinking which has interrupted the S()f)ereigtlty of myth. We oweLO it the first fo rm of sel f-su fficien t think ing. independent of any sacredpostu re of enunci at ion ; in oilier word s, th e firs t form of enti rely secul arizedth ink ing.

But the phi losophy of Rom an ticism - and Hegel is decisive in this regard -carried out an almost complete disentanglem ent of philosophy and rnath c-maries. It shaped the conviction that philosoph y can and mu st deploy athinking th at docs not at an y moment internalize mathem atics as cond itionfor that deployment. I maintain that thi s disentanglemen t can be iden tified asth e Romantic speculative gestu re par excelle nce; to the po int that it re tro-actively d etermined the Classical age of ph ilosophy as one in wh ich the

philosoph ical text continued to be intrinsica lly cond itioned by math em aticsIII vario us ways.

T he positivist and cmpiricist approaches , which have been highly influe n-tial during the last two centuri es , merel y invert the Romantic specu lativegesture. The clai m that science constitu tes the one and on ly paradigm forthe pos iti..-ity of knowledge can be made on ly from with in the com pleteddisentanglemen t of philosophy and the sciences. The anti-philosophical\'crdi et returned by the various forms of positivism oven ums the anti-scientific verd ict returned by the various forms of Romantic ph iloso ph y, butfails 10 in terrogate its initial premise. It is stri king tha t Heidegger andCam ap disagree about everythi ng, except th e idea that it is incumbent uponus to inhabi t and activ ate the end of metaphysics. This is because for bothHcidegger and Camap, the name 'metaphysics' designates the Classical eraof philosophy, the era in which mathemati cs and phi losophy were still reci-procally entang led in a general representat ion of the opera tions of thought.Carnap wants to purify the scien tific ope rat ion , while H eidegger wishes tooppose to science - in wh ich he perceives th e nihili st manifestation ofmeta phys ics - a pa th of th ink ing modelled on poe try . In th is sense, bo thremain heirs to th e Romanti c ges tu re of d isen tanglem en t, albeit in differ entregis ters .

This perspective sheds ligh t on th e way in which vario us form s of positi -vism and empiricism - as we ll as that refined fo rm of soph istry representedby Wingenstei n - remai n inca pable of identifying mathem atics aJ a type oftJri"ki,w, even at a time when any attempt to characterize ir as somethingelse (as a gam e. a grammar, etc.) constitutes an affront to the availableevidence as well as to the sen sibility of every mathematician. Essentially,both logical positivism and Anglo- American lingui stic sophistry claim - butwithout th e Roman tic force that would acoompany a lucid awareness ofthei r claim - that science is a techn iq ue for which mathematics p rovidesthe grammar , or that ma thematics is a gam e and th e only important thingis to iden tify its rule. W hatcver th e case ma y be , mathematics docs notthink. T he on ly ma jor d ifference between th e Romantic founders of what IWould call th e second modern era (the first bei ng the Classical one) and th eposit ivists o r mod ern soph ists , is tha t the former preserve the ideal ofthinking (in art , or phi losoph y), whi le the latt er on ly ad mi t forms of know-ledge.

A sign ifican t aspec t of the issue is th at, for a great sophist like Win gen-st ein, it is po int less [Q etlter into mathematics. Wittgenstein , more casual inth is respect than Hegel , p roposes merely to 'b rush u p agains t' mathematics,to cast an eye upo n it from afa r, the way an artis t migh t gaze upon somechess players:

But the troubl e is that mathem atics, which is an exem plary discipline oftho ugh t, does not lend itself to any kin d of description and is not represen-table in terms of the cartograph ic metaphor o f a coun try to which on e couldpay a qu ick visit. And in any case , it is impossible to be lazy in mathematics.It is possibly the on ly kind of thinking in whic h the sligh tes t lap se in concen-tration entails the disappearance, pure and simple, of what is be ing though tabout. Whence th e fact that W ittgen stein is cont inuo usly speaking of some-thing other than mathematics. He speaks o f the impression he has of it fro mafar and, more profound ly, of its symptomatic role in his ow n itinerary. Butthi s descriptive and symptomatolo gical treatmen t takes it for granted thatphilosop hy can keep mathematics at a distan ce. T his is exact ly thc standardeffect th at the Romantic ges tu re o f disentang lem ent seeks to achieve.

Wha t is the crucial presupposition for the ges ture whereby Hegel and h issuccessors managed to effect th is lon g-lasting disjunction between mathe-matics on the one hand and philosophical d iscou rse on the other? In myopin ion , this presupposition is tha t of historicism, which is to say, the tem per-alizuticn o f the concep t. II was the newfound certain ty tha t infinite or truebeing could only be apprehend ed through it s own tem porality that led th eRom antics to depose mathematics from its localization as a cond ition forphilosophy. T hus the idea l and arem poral characte r of math em atical thinkingfigu red as the central argument in this deposit ion . Romantic specu lationopposes time and life as temporal ecstesis to th e abstract and empty etern ityof math ematics . If time is the ' existence of the conce p t', then mathematics isunworthy of that concep t.

It could also be said that German Rom ant ic philosophy, whic h produced -the ph ilosophical means and the techn iques o f thought required for histori-cism, es tablished the idea th at the genu ine in fini te on ly man ifest s itself as ahorizonal structu re f or the historicity of the finitude of existence, But bot h th erepresen ta tion o f the limi t as a horizon and the theme of fini tude arc ent irelyforeign to mat he ma tics, whose own conce p t of the limit is tha t of a present-poi nt and whose think ing requ ires the p resuppo sit ion of the infin ity of itssite. For historicism , of wh ich Roman ticism is the philosoph eme, math e-mat ics, which links the infinite to the bounded power of the Jetter and whosever y ac ts repeal an y invocation of time, could no longer be accorded a para-

2. ROMAN TICISM CON TINUES TO BE THE S ITE FOROUR THINKIN G TO DA Y, A ND THIS CON TIN UA TION

REN DERS THE THEME OF THE DEA TH OF GODIN EFFECTUAL

Jigmatic Sta tus, whet hcr it be with regard to certainty or with regard totruth .

We will here call ' Roman tic' an y d isposition of th inking which determinesthe infini te within the Open, or as horizonal correlate for a histo ricit y offinitude. T oday in parti cu lar, what essentially subsists of Roman ticism is thetheme of fin itude. T o re-imricatc mathem atics and philosophy is also, andperhaps above all , to hav e done with finitude, wh ich is the p rin cipal contem -pora ry resid ue of the Rom antic specu lat ive gestu re.

2SPhilosophy and r-tathema ucs

The q uest ion of ma them atics, and of its localization by ph ilosop hy, has thesingular me rit of providi ng us with a profound insight into the nature of ou rown time. Beyond th e claim s - no t so much heroic as empty - about an ' irre-ducible modernity', a ' novelty still needi ng to be thought' , the persist en ce ofthe disju nction between mathematics and phi losophy seems to indicate th atRomantici sm's his toricist core contin ues to fun ction as the fundamentalhorizon for ou r think ing . The Rom an tic gesture stil l holds sway over usinsofar as the infinite continues to fun ction as a horizon al correlativ e andopening for th e histori city of finitude. Our mode rni ty is Romantic to theexten t that it remains caught up in the temporal iden tification of the concept.As a result, ma the matics is here rep resented as a condi tion for philosophyonly from the stand poin t of a rad ical d isjunctive gesture, which persists inOppos ing the historical life of thought and the concep t to the empty andformal eternity of mathem at ics.

Basically, if one cons iders the status ascri bed to poetry and mathematics byPlato, one see s how , eve r since Romanticism , they have swap ped p laces ascond itions. Plat o wanted to banis h poe ts and on ly allow geometers access tophilosophy. Today, it is the poem that lies at the heart o f thc philosophicald ispos ition an d the matheme that is excluded from it . In our time, it ismathe matics which , altho ugh acknow ledg ed in its scient ific (i.e. tec hn ical)aspect, is left to lang uish in a condition of exile and neglect by philosophcrs.Mathematics has been redu ced to a gram matical she ll wherein sophists canpu rsue their linguistic exerci ses, or 10 a morose area of specialization forcobwebbed epistemo log ists. Meanwh ile, the aura of the poem - seeminglysince Nietzsche, but act ually since Hegel - glows eve r brig hter. Nothingilluminates contemporary philosop hy 's fundamental ami-Platon ism more

Theore tical Wrltmgs

The ph ilosopher mu st twist and turn about so as [Q pa ss by the math cm a-tical problems, and not run up agains t one - wh ich would have to besolved be fore he could go further.His labour in philosophy is as it were an idleness in mathematics.I t is not that a new build ing has to be erec ted, or that a new bridge has tobe bu ilt , but th at the geography as it now is, has to be d cscribed .f

24

vivid ly than its paten t reversal of th e Platonic system of conditions for philo-asophy.

BUI if this is the case, then the question that concerns us here has nothingto do with postmodc rnism. For the modem epoch comprises two periods, theClassical and the Rom antic, and our question regards post-romanticism . Howcan we get out of Romanticism without lapsing into a neoclassical reaction ?This is the real pr ob lem , one whose genuine pertinence becomes apparen tonce we start to see how, behind th e th eme of 'the end of th e avant -gardes',the postmodc rn merely dissimulates a classical-romant ic cclccricisrn. If wewish for a more precise formula tion of th is parti cular problem, an cxarntna-[ion of the link be tween philosoph y and mathematics is the only valid path Iknow of. It is the only standpoint from whi ch one has a ch ance of cuttingstraigh t to th e heart of the matter, which is no thi ng othe r than the critique offinitude.

T ha t th is critique is urgently required is confirmed by th e spectacle - alsovery Romant ic - of the increasing collusion between phi losophy (o r wha tpasses for ph ilosoph y) and religions of all kinds, since the collapse of Marxistpo litics. Can we reall y be surprised at so-and-so's rabbinical Judaism, or so-and-so's conversion to Is lam, or another 's thin ly veiled Ch ristian devotion.given that everything we hear boil s down to this: that we are 'consigned tofinitude' and arc 'eutmially marta!' ? Wh en it comes to crushing the infamyof superstition, it has always been necessary to invoke th e solid secularetern ity of the sciences. But how can this be done wilhin ph ilosophy if thedisentanglement of mathema tics and philosophy leaves beh ind Presence andthe Sacred as the only things th at make our be ing-mortal beara ble?

T he tru th is that th is di sen tanglement defuses th e Nic tzschea n proc lama-tion of th e death of God. We do not possess the wh erewithal to be atheists solong as the theme of finitude governs ou r thi nking.

In the deployment of the Romantic figu re, th e infinite, whi ch becomes theOpen as site for the temporalization of finitud e, remains beho lden to th e Onebecause it remains beholden to history. As long as finitude remains theult imat e determi nati on of existence, God abides. He abides as that whosed isappearance continues to hold sway over us, in the form of the abandon-ment, the derel iction, or the leaving-behind of Being.

T here is a very tenacious and profound link betwee n the d isentanglementof ma thematics and philosophy an d th e preservat ion, in the inverted ordiver ted form of fin itude, of a non-appropriable or unnameable horizon ofimmo rtal d ivini ty. 'Only a God can save us' , H eidegger cour ageouslyproclaim s, but once mathematics has been deposed, even tho se witho ut hiscourage conti nue to ma intain II taci t God th rough the lack of being engen-dered by our co-extensiveness with time.

Descartes was more of an ath eist than we are, because ete rni ty was notsomcthing he lacked. Little by little, a genera lized historicism is smotheringus bencalh a d isgusting venee r of sancti fication.

When it comes 10 the effectiveness, if not th e pr oclamation of the death ofGod, the contemporary quandary in which we find ourselves is a function ofthe fact tha t philosophy's neglect of math ematical th inking delivers theinfin ite, thro ugh the med ium of history, ove r to a new avatar of the One.

Onl y by rela ting the infinite back 10 a neu tral banality, by insc ribi ngetern ity in the mathcmc alone, by sim ultaneo usly abandon ing historicism an dfinitude , docs it become possible to think with in a rad ically dcconsecratedrealm. Henceforth, the finite, wh ich continues 10 be in th rall to an eth icalaura and to be grasped in the pa thos of mortal- being, mu st only be conceivedof as a tr ut h' s di fferential incis ion with in the banal fabric of infinity.

The contemporary prerequisite for a desecration of thought - which, it isall tOO apparent, remains to be accompli shed - resides in a complete d isman-tling of the historicist schema. T he infini te must be submitted 10 th emath cme's simple and transpa ren t deductive chains, subtracted from alljurisdiction by th e One, stripped of its hcrizonal function as the corr elate offinitude and released from th e metaph or of the Open.

And it is at this point, in which thought is subjected to extreme tension, th atmath ematics summons us. Our imperative consists in forging a new modalityfor the ent anglemen t of mathematics and phi losoph y, a modality thr oughwhich the Romantic gesture th at continues to govern us will be termi nated .

Mathematics has shown that it has th e resources to deploy a perfect lyprecise conception of th e infinite as ind ifferent mult iplicity . This 'ind ifferen-riarion' of the infin ite, its pos t- Can tc rian treatment as mere number , thepluraliza tion of its concept (th ere are an infinity of different infinit ies) - allthis has rende red the infinite bana l; it has term inated th e pregnant latency offinitude and allowed us to realize that every situation (ourselves included) isinfinite. And it is this evcn tal capacity proper to math ematical thought thatfinally enjoins us to link it to the philosoph ical proposition .

It is in this sense that I have invoked a ' Platonism of the multiple' as aprogra mme for philosop hy tod ay.

The usc of the term ' Platonism' is a provocat ion, o r banner, th rough whichIt) proclaim the closure of the Romantic gestu re and th e necessity of declaringonce more: ' M ay no-one who is not a geometry enter here' - once it has beenacknowledged tha t the non-geom eter remains in thrall to the tenets ofRomanuc disjunction and the pa tho s of finitud e.

T he usc of the term 'multiple' indicates tha t th e infinite must be under-Stood as indifferent mu ltip lici ty, as the pure material of being.

The conjunction of th ese tWO terms proclaims ther the death of God can be

26 Theoretical Wr itings Philosophyand Mathematics 27

3. PLATO CA RR IES OUT A PHIL OS OPHICA LDEPLOYMENT OF MATHEMATICS AT THE FRONTIER

BET WEEN THOUGHT A N D THE FREEDOM OFTHOUGHT

r..nd ered ope ra tive without privation , th at th e infin ite can be un tcthered fro mthe One, that histo ricism is terminated , and tha t etern ity can be regainedwithin time without the need fo r consecration .

In order 10 inaugura te such a programme, we will have to look back towardth e history of the qu estion. J sha ll punctuate th is his tory at th e t wo extremi-ties of its arch: :1It one extreme stan ds Plato, who exiles the poe m andpromotes th e malhcmc; while at th e OthCT stands Hegel, who invents th eRomantic ges tu re in philosoph y and is the th inker of th e abasemen t ofmath ematics.

The theorizing concern ing be ing and the in tell igible wh ich is sustained bythe science (i pin i mil of the d ialec tic is clea rer tha n tha t susta ined by whatarc known as the scie nces lu chl/il . if is certa in ly the case tha t those whotheorize acco rdin g to the se sciences, wh ich have hypotheses as their princi-pies, arc obl iged to p roceed discur sively ra the r than empirically. Bu tbec ause th eir intui ting rem ains dependen t on these hypotheses and has nomeans of accessing the princip le, they do no t see m to you to po ssess thein te llection of what they theorize, which nevertheless, in so far as it is iIlu-nunared by the principle, concerns the imeJligibi lity o f the en tity. It seem s10 me you characte rize the procedure of geometers and their ilk as discur-

29PhIlosophyand r-ta hen aucs

si\'c (dial/oial , which is not how you characterize intellection . This d iscu r-s i \"t.~ness lies midway between (metax uJ o pinion LdoxaJ and int ellect (nousl . }

I. For Plato, mathem at ics is a condi tio n for th inking or theorizing in generalbecause it consti tu tes a break with doxa or opi nion. This much is fam iliar.But wh at needs to be emphas ized is that mathem ati cs is the only point ofrupture w ith doxa thac is give n as exist ing, or constitute d, The existence ofmathematics is ult imately what constitutes its absolute singu larity . Every-thing else that exists remains prisoner to opinion , but not ma th ematics.So the effective. historical, independent existence of mathem aticsprovi des a paradi gm for the pom'bili ly of b reaking with opin ion.

Of cou rse, th ere is d ialecti cal convers ion, which for Plato is a supe riorform of b reaking with doxo. But no one can say whether dialect icalconvers ion, which is the essence of th e ph ilosophical d isposition , exists. Itis held up as a proposal o r project. ra ther tha n as some thi ng act ua llyexisti ng. D ialectics is a programme, o r initiat ion. while mathematics is anexisting, avai lab le procedure. D ialecti cal conversion is the (eventual)poin t at which th e Platoni c tex t touches the rea l. But th e on ly po in t o fextern al suppo rt for the b reak with doxa - in th e form of something thatalready exist s - is constitu ted by mathema tics and mathematics alone.

Having sa id thi s, th e singu larity o f mathematics constantly and un fai-ling ly p rovokes opin ion, wh ich is th e reign of th e doxa. Whence thecons tant broadsides against th e 'abstract' or ' inh uman ' na tu re of mathe-mati cs. Wh enever one seeks a rea l, existing basis for a thinking thatbreaks with every form of opi nio n , one can always resort to mathematics.Ultimately , th is singu larity proper to math ematics is consens ual, becauseeveryone recogn izes the re isn't - and cannot be - such a th ing as ma tbe-m:nical opinion (which is no t to rule out the exi stence of opinion s, gene r-ally un favou rab le, about mathem atics - q uite the cont rary). M athem at icsexh ib its - lind therei n lies its 'a ristocra tic ' aspec t - an irrem edi abl edisco n tinu ity with regard to every sort of immedia cy proper 10 uoso.

Conversely, it may legitimately be assumed tha t every negative opinionabo ut ma thema tics constitutes , whether exp licitly or implicitly, a defenceof the rights o f opinion, a p lea fo r the immediate sove reignty of doxa.Romanticism , I believe, is gu ilty of this sin . As historicism, it has no

In exam in ing what is of significance fo r us in this text - i.e. the relation ofconjuncr ionfd is jun etl0n between mathem atics and ph ilosophy - I willproceed by delineating the four fun da mental characteristics that structure thematrix for every conceivab le rela tion between these tWO d ispo sit ions ofthought.

Theorencal Writ ings28

Plato is ob viously th e one who deployed a fundamental entanglement ofmathematics and ph ilosop hy in all its ramifications. He prod uced a ma trixfor cond itioning in which th e th re