Badiou's 'Being and Event': A Reader's Guide

BADIOU'S BEING AND EVENT A Reader's Guide CHRISTOPHER NORRIS

"

.

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Christopher Norris 2009

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HB: PB: HB: PB:

0-8264-9828-0 0-8264-9829-9 978-0-8264-9828-1 978-0-8264-9829-8

Library of Congress Cataloging-in-Publication Data

Norris, Christopher, 1947Badiou's Being and event: a reader's guide/Christopher Norris. p. cm. Includes index. ISBN-l3: 978-0-8264-9828-1 (HB) ISBN- 1 O: 0-8264-9828-0 (HB) ISBN-l3: 978-0-8264-9829-8 (pbk.) ISBN- 1O: 0-8264-9829-9 (pbk.) 1. Badiou, Alain. Etre et l'evenement. 2. Ontology. 3. Set theory. 4. Events (Philosophy) I. Title. B2430.B273E875 2009 111-dc22 2008045238

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For Wendy

Lewis and Ray Davies

CONTENTS

Acknowledgements Author's Note

viii x

1.

Context

2.

Overview of Themes

14

3.

Reading the Text

37

4.

Further Reading

284

Notes

289

Index

307

ACKNOWLEDGEMENTS

My thanks to colleagues in the Philosophy Section at Cardiff especially Robin Attfield, Pat Clark, Rhian Rattray, Alison Venables and Barry Wilkins - for their friendship and encour­ agement over the past few years. I am more than happy to acknowledge the regular fix of lively talk and intellectual stimu­ lus - as well as good social cheer - that came from my weekly informal seminars with M.A . and Ph.D. students. My interest in Badiou's work was reinforced by meeting and hearing him at the splendid conference 'Wandering with Spinoza' , hosted by the Centre for Ideas at the Victorian College of the Arts, Melbourne, Australia in September

2006 and organized by Elizabeth Presa

and Dimitris Vardoulakis. This was also, as it turned out, some­ thing of a Badiou occasion and confirmed my already strong sense that Being and Event is itself one of those major - 'evental' - occur­ rences in the history of thought that his book so impressively documents. Since then my writing of this Reader's Guide has been helped, sustained and occasionally nudged in some new direc­ tion by discussions with or readings of (among others) Gideon Calder, Sam Gillespie, Paul Gorton, Gerald Gould, Theo Gram­ menos, Peter Hallward, Terry Hawkes, Vesna Main, Billy McMurtrie, Scott Newton, Laurence Peddle, Robert Reay-Jones, Alison Scott-Baumann, Rob Stradling and Rea Wallden. Man­ uel Barbeito Varela invited me to Santiago de Compostela where I was able to present an impromptu version of some of this material and benefited greatly from the students' stimulating company during a week of intensive but none the less enjoyable exchange. Ricky Sebold read through a draft version and offered some exceptionally well-informed and pertinent comments. Reg Coates, Dave Hilldrup, Dave Hume and Dick James - stalwart companions of the control-line handle - did much to keep me on an even keel and to ward off the perils of writerly solitude, as did Tim Andrews, Richard Evans, John Mealing, Gordon May, viii

ACKNOWLEDGEMENTS

Digby Perriam and other friends in the South Bristol Model Air­ craft Club. Sarah Campbell at Continuum Books was quick to come up with the idea of this project when I first mentioned my interest in Badiou, and has provided much welcome help and advice as the work went along. Wendy Lewis, Ray Davies, Di Corker, Margaret Innocent, Lyn Mererid, Beaty Smith and all my comrades in Cor Cochion Caerdydd (the Cardiff Reds Choir) were a great source of hope and a constant inspiration through some bad political times, although I realize now that I've been prefacing books with sentences to much the same syn­ copated upbeat/downbeat effect for the past two decades and more. To Alison, Clare and Jenny - as usual.-,,:- my love and gratitude. Cardiff October 2008

AUTHOR'S NOTE

I should mention that there will have been published, by the time this book appears, an English translation by Alberto Toscano

Logiques des mondes (Paris: Seuil, 2006; Logics of Worlds, London: Continuum, 2009). It is a work fully compara­ ble to Being and Event in scale and philosophical ambition, as

of Badiou's

well as marking a number of significant (at times quite extraor­ dinary) new departures in Badiou's thinking about issues of

vis-a-vis

politics and ontology

developments and formal pro­

cedures in the logico-mathematical domain. In fact there were already clear signs of this new direction in his thinking by the

Briefings on Existence: A Short Treatise on Transitory Ontology (original French publication 1998). This explains the confusion regarding its place in Badiou's oeuvre time of Badiou's

because the new work has been described, since it first appeared and pending its English publication, either as 'Volume Two of ' or as 'a sequel to'

Being and Event. What is clear enough Logiques des mondes is that

-

should say - from a perusal of

I

it

stands as a largely self-sufficient text which takes up many lines of argument from that earlier work but does so from an angle (or variety of angles) so distinctive as truly to constitute a sequel rather than a supplement, continuation, integral part or concluding statement . Given my remit here this is just as well since

Being and Event K2

to say a veritable

is in itself something of an Everest - not

-

for the commentator wishing to convey

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

AUTHOR'S NOTE

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

CHAPTER

1

CONTEXT

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

BADIOU'S BEING AND EVENT

and concepts of understanding, the phenomenal and the nou­ menal or mere inclination (however benign) and the dictates of absolute morallaw.2 In each case, that refusal goes along with his leading claim that ontology - rather than epistemology - must be the starting point for any project that would seek to throw off this inherited burden of strictly (on their own terms) insoluble problems and instead make room for the kind of constructive or creative-exploratory thinking that is philosophy's true vocation. Not that Badiou is for one moment setting the philosopher up as sovereign dispenser of truth, knowledge and wisdom or wishing to re-live past philosophical glories from a time before the other disciplines (the natural, social and human sciences) began to assert their distinctive claims. On the contrary, he never ceases to emphasize that philosophy

is not and in truth has never been

such a self-sufficient enterprise possessed of its own intellectual authority or juridical power whereby it presumes to lay down standards of validity or truth for those other, supposedly less well-regulated subject-domains. In fact it is a central precept of Badiou's work - fully borne out by his own practice in a text like

Being and Event

-

that philosophy can make progress or achieve

worthwhile results only when it acknowledges its crucial depend­ ence on certain extra-philosophical 'conditions' of thought, that is to say, its constant need for involvement with debates and activities outside what counts (on a typecast division of aca­ demic labour) as philosophy pure and proper. For Badiou the most essential conditions are science, politics, art and love, the latter understood largely (though by no means exclusively) in psychoanalytic terms. It is in relation to develop­ ments in these four chief areas of knowledge and experience that philosophy is able to reflect upon the kinds of conceptual or ethical breakthrough that can then be seen to mark a genuine and lasting - rather than merely notional or short-term - stage of advance. Of course the precise nature of that relationship and the different forms that it can take in each case are topics that need a lot more detailed explanation and to which we shall return at various points in this book. For now, what I want to stress is the fact that Badiou seeks neither to aggrandize philosophy's juridical role as was once (not so long ago) the fashion nor yet to play it down - like some present-day postmodernists, neo­ pragmatists and followers of Wittgenstein - to the point of 2

CONTEXT

recommending that philosophers abandon their trade altogether, or else give up their delusions of epistemological grandeur and content themselves with being just one more voice in the 'cul­ tural conversation of mankind'.3 This latter idea is one that Badiou has most emphatically denounced and which he sees as threatening not only the fine-tuned balance of interests between philosophy and its fourfold conditions but also its capacity for critical reflection on issues in the wider intellectual, ethical, cultural and socio-political spheres. On his account there is something highly suspect about the Widespread 'linguistic tum' across various philosophic schools of thought, from the echt­ analytic (Frege-Russell) mode and the reactive 'ordinary lan­ guage' approach with its sources in Wittgenstein or Austin to hermeneutics, post-structuralism, postmodernism and the type of 'strong'-descriptivist or ultra-constructivist thinking espoused by someone like Richard Rorty. Whatever their deep-laid differ-:­ ences these schools have one major premise in common, namely their belief - the hallmark of linguistic philosophy in general that only through recourse to language as the basis of all enquiry could philosophy escape from those false dilemmas induced by the epistemological 'way of ideas' from Descartes to Kant and beyond. For Badiou, the 'linguistic turn' is not - as its proponents would have us believe - a means of liberation from outworn ideas or misconceived pseudo-dilemmas but rather a means of distracting attention from problems that would otherwise occupy the forefront of any philosophical project meriting the name.4 These latter are best thought of as ontological in character since they have to do directly with the question of being - in its vari­ ous kinds or modalities - as distinct from the epistemological question with regard to our knowledge thereof or the linguistic question with regard to what we can say, describe, or justifiably assert concerning it. Those who take the language-first approach should be seen not so much as philosophers in any genuine or properly applicable sense of that term but rather as standing squarely in line of descent from the ancient sophists, or skilled all-purpose rhetoricians, whom Plato charged with promoting mere eloquence above the interests of truth and justice. In treat­ ing language - for which read alternatively 'discourse', 'para­ digm', 'descriptive framework' or 'conceptual scheme' - as their 3

BADIOU'S BEING AND EVENT

ultimate court of appeal these thinkers more or less overtly embrace a pragmatist, conventionalist or communitarian stance whereby 'truth' cashes out for all practical purposes as what's 'good in the way of belief ' , or best suited to convince those (the relevant target-group) whose agreement or good opinion one seeks to win. What is distinctive about Badiou's critique is the fact that he locates this sophistical strain not only in thinkers like Rorty who would happily subscribe to such a characterization of their views but also right across that large tract of present-day philosophic country where the dominant approach is one or another version of the linguistic tum. Above all, as we shall see, it is Wittgenstein's later thinking and its pervasive influence in so many quarters of current aca� demic debate that Badiou identifies as the single most potent source of a loftily dismissive attitude to real, even urgent philo­ sophical problems, one that would have us regard those prob­ lems as involving no more than a regrettable failure to observe the proprieties of this or that 'language-game' or communal 'form of life'. 5 As against such derelictions of its proper role Badiou is at pains to emphasize his point: that although philo­ sophy cannot be pursued to any worthwhile, constructive or humanly beneficial purpose except through exposure to issues and challenges thrown up by one or other of its four 'condi­ tions', nevertheless it - and they - will suffer a lapse into inertly orthodox or passively conformist ways of thought unless philo­ sophy retains a sense of its own relative autonomy and proper (reflective and critical) role. Where this function is exercised to most decisive effect is through the readiness to raise questions or engage issues that directly concern those other disciplines yet which, given various practical constraints, may not rank high (or figure at all) on the list of priorities for anyone doing special­ ist work in the field. Such is most strikingly the case, he argues, with regard to mathematics where there exists a great distance between working mathematicians who mostly take little interest in what philosophers might have to say and philosophers - espe­ cially those in the analytic mainstream - whose approach and whose choice of topics for debate all too often invite just such a negative response. This they do mainly by focusing on problems amenable to their sorts of treatment - prototypically the issue about rule-following - and hence failing to think mathematically 4

CONTEXT

in a way that might put them more closely in touch with the sources of mathematical creativity.6 What sets Badiou's work so decidedly apart is its refusal to engage in suchlike familiar (since intra-philosophical) kinds of debate and its insistence that any 'philosophy of mathematics' worthy of the name will have to manifest not only a strong intellectual command of the subject but also the capacity for conceptual work of a sufficiently high order. In Being and Event this work has to do with developments in post-Cantorian set theory and their potentially transformative effect as Badiou seeks to show - on disciplines, practices and modes of social being as well as of intellectual enquiry far beyond the mathematical as normally (i.e. restrictively) defmed. Most significant here is Cantor's radically innovative way of conceiving infinity not, in quasi-mystical or crypto-theological terms, as that which by very defmition transcends the utmost powers of human reckoning or calculative grasp but rather as belonging to a realm of transfinite numbers, quantities and ratios which can indeed be reckoned with and placed in precisely calcu­ lable relationship one with another.? This discovery - as Badiou most emphatically considers it, rather than a mere paradigm change or shift of working method among mathematicians at a certain point in time - is the heart of his entire philosophical project and will therefore be the focus of detailed commentary later on. For the moment, there are three points that chiefly need explaining. First is the fact that, following Cantor, it became possible to define an infinite set as a set whose members are equinumerous (i.e. stand in a one-to-one equivalence relation) with one of its proper subsets. Thus the infinite set of natural numbers (integers) can be paired off with the infinite set of odd numbers even though the latter is a proper subset of the former or a set that on any 'normal' (finite) method of counting would be missing one integer for every two counted. It follows, sec­ ondly, that there must exist multiple orders or 'sizes' of infmite set whose cardinality or proportion one to another can best be thought of by analogy with that between the integers and even numbers, but which all have the same constitutive feature of numerical equivalence with one of their proper subsets. Thirdly, this means that the development of set-theoretical concepts, procedures and hypotheses can be expected to proceed through �

5

BADIOU'S BEING AND EVENT

a series of further limit-point encounters like that which moti­ vated Cantor's original discovery. In other words, it is a mode of

enquiry that has typically achieved its most notable advances

at the point of coming up against some obstacle, paradox or

hitherto unnoticed anomaly which threatens to undermine its

conceptual foundations.8

.

Such was the problem that Bertrand Russell famously uncov­

ered with regard to self-referring or self-predicative expressions,

that is, the dilemma or logical contradiction that results if one tries to make sense of phrases such as 'the set of all sets' that

are not members of themselves'.9 Russell purported to resolve this problem by adopting a purely stipulative rule, one which

decreed - as a matter of fiat - that such expressions were illicit

since they contained terms belonging to two logically distinct (object-language and meta-linguistic) levels and were therefore

certain to create all manner of confusion. On the other hand,

some set theorists - those who rate higher in Badiou's esteem -

have rejected any fall-back 'solution' along these lines and opted

instead for a highly conjectural, even adventurous but none the

less rigorous and consequent mode of hypothesis formation

which requires the truth of certain as-yet unprovable statements

or theorems. What Badiou brings out very clearly is the way that

mathematicians from Cantor to the present have responded to

such challenges by allowing their thought to be drawn ahead of

itself, so to speak, by that which exceeds its present-best powers

of assured set-theoretical grasp. Advances of this sort may come about either through the impact of some currently recognized

but so far recalcitrant problem or else through the anticipatory

awareness of a problem that has not yet emerged distinctly to view but which they are able to discern as a kind of presumptive

anomaly, or a sense of possibilities at present blocked off by the limits of existing methods and proof-procedures. Hence the cen­

trality to Badiou's project of certain set-theoretical methods

developed by Paul Cohen, in particular those of 'forcing' and

'the generic', which between them provide a means of explaining

how such developments occur.IO

Here again, I must defer a more adequate treatment of these

complicated matters until we reach the relevant stage of my

detailed commentary on the text. The same goes for those other

main topic-areas of

Being and Event - politics, 6

science, art and

CONTEXT

love - set forth by Badiou as the principal 'conditions' that should or must inform any philosophically substantive project of thought. The political dimension is directly linked to his writing on mathematics since the latter provides Badiou not only with the rudiments of a social ontology - that is, a conception of the various structures and modes of interaction that character­ ize human collectivities - but also, more to the point, with a closely analogous means of explaining the modus operandi of various socio-political orders. Thus, in present-day terms, there is a large and precisely specifiable gap' between the claims put forward on behalf of liberal democracy - claims concerning inclusiveness, equality, justice, universal rights, freedoms guaran­ teed by rule of law and so on - and the reality of various blatant exceptions to each and every one of these notional achievements. According to Badiou, set theory provides by far the most ade­ quate conceptual resources when it comes to accounting for this democratic deficit and locating those oppressed or excluded social elements that lack recognition or pass unacknowledged under dominant (official or consensual) ideas of social member­ ship. Thus he sets out to show - utterly improbable as this might seem to readers brought up in orthodox disciplinary ways of thOUght - that there exists a close structural homology between, on the one hand, such pressing issues of socialjustice or political representation and, on the other, the sharply distinguished set-theoretical concepts of part and member, belonging and inclusion, or inconsistent and consistent multiplicity. This latter distinction plays a crucial role in Badiou's case for mathematics as the basis of social and political as well as scien­ tific or physical ontology. For it is precisely through the ever­ present excess of inconsistent over consistent multiplicity - of the parts whose sum total constitutes the truth of a given situ­ ation over the members legally or socially acknowledged as belonging to it - that a politics of radical participant democracy might hope to gain a strengthened critical purchase. In so doing it would fasten on precisely those excluded, disenfranchised or oppressed minorities (such as the sans-papiers or 'paperless', legally unrecognized immigrants in France) who simply don't count as persons for official, electoral or social-welfare purposes. That is to say, they don't figure anywhere in the legal-administra­ tive 'count-as-one', another of those key concepts in Badiou 7

BADIOU'S BEING AND EVENT

that enable him to posit a direct equivalence - not just a loose analogy or suggestive structural kinship - between the two domains of set theory and political philosophy. Thus it is always at those particular locations or 'evental sites' on the margins of the recognized body politic that the dominant order is likeliest to come under strain through a forced encounter with that which exceeds and challenges its sovereign or juridical powers. What then emerges with (at times) revolutionary force is the discrep­ ancy between what Badiou terms the 'state of the situation' and the 'situation' itself. That is to say, there suddenly appears a non­ negotiable rift between the socio-political order as viewed from a legitimist, reformist or social-democratic perspective and that which so far exceeds its grasp as to pose a constant (if hitherto suppressed or dissimulated) threat to its authority. Just as in the case of mathematics, logic and the formal sciences - where signal advances most often occur through the encounter with some deep-laid obstacle or paradox - so likewise in the political con­

text it is always at these points of maximal stress that society

first manifests the signs of imminent structural change. Here also it is a matter of decisive events ('singularities' as Badiou terms them, again in a specific mathematical sense) which could not possibly have been predicted since they found no place in the pre-existent social ontology, yet which afterwards - follow� ing this singular event - necessitate a likewise radical change in our grasp of what had indeed been possible in just that prior situation .

. Here we should recall the title of this book and its pointed

juxtaposition of two terms which between them stake out the philosophic territory that Badiou aims to traverse. Being is the domain of ontology, that is, of whatever can truly be said .con� ceming the existence of entities (whether physical or abstract) which might always transcend the scope of current-best know� ledge - of empirical verification or formal proof - while none the less possessing an objective truth-value that may yet be discov­ ered at some further, more advanced stage of enquiry. Badiou will have much to say about this cardinal distinction between truth and knowledge, one that runs strongly counter to the thinking of assorted empiricists, logical positivists and anti-realists not to mention post-structuralists, neo-pragmatists, social constructiv� ists and others who will have no truck with any such (as they 8

CONTEXT

take it) hopelessly outmoded 'metaphysical'conception. II Hence­ to repeat - his insistence on the power of mathematics, unique in this regard, to take us into speCUlative regions of thought where knowledge may be forced up against its limits at any given time and indeed drawn beyond those limits by a sense of what requires some more or less radical re-thinking of current truth­ claims, methods and procedures. Events, on the other hand, are just those strictly unforeseeable and - as they appear at the time in question - wholly contingent irruptions of the new that may turn out to exert a uniquely powerful and lasting effect but which elude ontological specification precisely insofar as they belong to no existing (i.e. up-to-now thinkable) order of things. To this extent their nature and modes of occurrence cannot be captured by any account that adopts a straightforwardly progressive or developmental conception of truth as that to which knowledge constantly aspires or upon which it steadily converges at the end of enquiry. Rather, what results from a grasp of this evental character - as opposed to the cumulative process of knowledge acquisition - is a sharpened sense of how far it can exceed or transcend the scope of knowledge at any given stage in that process. Such was, for instance, the advent of Cantorian set theory with its totally transformed conception of the infinite and its opening up of a conceptual terrain - a new-found range of onto­ logical resources - whose discovery was not so much a further consequence of previous advances or successes but rather a product of the leap beyond a previous state of logical impasse. Thus it offers a striking example of the way that events in Badiou's strictly dermed sense of that term - have the twofold character of arriving, so to speak, out of the blue or without the kind of partial foreknowledge that attends more regular modes of occurrence and also of requiring that their further develop­ ment be followed through with a rigorous fidelity to their logical or indeed their ethico-political consequences. What marks the genuine event as distinct from its various false surrogates - among which, he thinks, must be counted a great many so-called 'major', 'epochal' or 'world-historical' episodes - is the fact that it exerts this intransigent demand on those who come after and whose elec­ tive or self-imposed task it is to press so far as possible in working out its further (presently obscure or unguessed-at) implications. -

9

BADIOU'S BEING AND EVENT

This is why Badiou insists so resolutely on the return to ontology as first philosophy in place of what he sees as the disastrous wrong turn that thinking has taken, first towards epistemology (thUS privileging issues of the scope and limits of knowledge over questions of truth) and then - worst of all - towards lan­ guage as that which purportedly constitutes the ultimate horizon of human understanding. It is also why he lays such unremitting stress on mathematics as the basis of all ontological enquiry and, accordant with this, on the essentially formal character of those various projects or truth procedures that constitute the otherwise diverse spheres of human scientific, political, ethical and artistic endeavour. Hence Badiou's idea of the 'militant' for truth who seizes - or is seized by - some such intimation of that which lies beyond their present-best powers of attainment or epistemic grasp yet which none the less exerts an intransigent demand upon their will and capacity to seek it out. Perhaps the hardest thing to grasp about his work is how he can maintain that crucial distinc­ tion between being and event, or the domain of ontology and that which intrinsically eludes any pre-established mode of onto­ logical grasp. After all, it is Badiou's chief contention - not only as concerns mathematics and the formal as well as physical sci­ ences but likewise across the whole range of subject-areas where truth is in question - that advances of this order are always pre­ ceded by a formal discovery or the invention of a new, ontologi­ cally ground-breaking procedure which then makes room for their admission to the realm of knowable truths, provable theo­ rems or accredited matters of fact. In which case, it might be asked, how could such advances ever come about were it not that there always already existed a space of possibility opened up for them by some foregoing venture onto new ontologicaf ground? That is to say, how can Badiou think to square these two appar­ ently conflicting claims, first with regard to the prior status of ontology as a matter of absolute philosophic principle, and sec­ ond with regard to the event as, by very definition, that which falls outside and beyond the sphere of any knowable ontology? For there would seem to be a clearly marked logical tension - if not an outright contradiction - between the idea of truth as intrin­ sically a matter for ontological enquiry (since pertaining to the 'being of beings' and whatever can be thought concerning them) 10

CONTEXT

�

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

�

It is o exaggeration to say that this issue touches the quick of

his work not only in philosophy of mathematics but also in each of those other main fields which Badiou takes to comprise phi­ losoph�'s fourfold enabling conditions. On the one hand he is compell�d to insist, albeit with regret, that philosophy cannot

itself claim to be one of those generic truth-procedures since it is neither (like them) a means of access to particular, context­ specific instances of how truth emerges in the course of human

endeavour or enquiry nor again (like mathema!ics) a fIrst-order

discipline engaged in the discovery of hitherto unexplored onto­

logical terrain. Thus the best that philosophy can do is keep a

close eye on the distinction between being and event - or onto­

logy and that which unpredictably emerges to throw those explo­

rations off their preconceived course - while seeking to articulate

and clarify the process through which such events come to pass

and to exert a profoundly transformative effect on one or other

of its own elemental conditions. Still Badiou makes it clea r that he is far from relegating philosophy to the kind of strictly

subordinate or 'under-labourer' role to which it was confined by thinkers - from Locke to the logical positivists - for whom the

natural sciences figured as a model of soundly based, rationally

conducted and knowledge-conducive enquiry. Indeed it is his

contention with regard to the scientists, political theorists, artists and psychoanalysts (the latter here figuring as those who, qua 'experts' in the subject, should have most to tell us concerning

love) that they are all of them crucially reliant on philosophy

when it comes to distinguishing knowledge from truth, or that

which lies

within the compass of their present-best conceptual

grasp from that which, while currently eluding such grasp, none

the less makes its absence felt through the pressure of unresolved problems, dilemmas or failures of demonstrative proof. More­

over and despite his unequivocal belief that mathematics must constitute the basis for any regional ontology remotely adequate -

to the task in hand

-

Badiou also counts the mathematicians

(even the most eminent among them) as standing in need of

philosophic assistance when drawing out the kinds of further implication that their thinking truly warrants. 11

BADIOU'S BEING AND EVENT

This helps to explain his at times quite waspish criticism of those analytic philosophers who focus on their own sorts of intra-disciplinary dispute and thereby avoid any serious engage­ ment with real mathematical challenges. However he also takes issue with 'working mathematicians' - including some whose achievements he holds in the highest regard - who dismiss the philosophers' efforts

tout court

as so much irrelevant or futile

since (as they presume) mathematically under-informed intellec­ tual pretension.12 In his own case (as I have indicated here and will go to argue in more detail) this charge can be shown to miscarry when confronted with Badiou's meticulously reasoned presentation of set-theoretical developments from Cantor to Cohen and also his manifest depth of engagement with issues that can scarcely be considered beneath or outside the mathe­ maticians' concern. Yet by far the most convincing defence of Badiou's position is that which he himself provides in the course of

Being and Event,

that is, his demonstration - in the strictest

(mathematical) sense of that term - that set theory can offer the basis for some highly productive and innovative thinking in subject-areas from which it might seem (most of all to analytic philosophers) as remote as could be. This is why it is one of the very few philosophic works published during the past

50 years

that merit the kind of close-focused exegetical commentary

Republic, Metaphysics, Descartes' Meditations, Spinoza's Ethics, Kant's Critique of Pure Reason or (although Badiou might strongly dissent) Wittgenstein's Philosophical Investigations.

afforded to such acknowledged classics as Plato's Aristotle's

Nor is it in any way coincidental that those same classic texts­ Wittgenstein's apart - are among his chief points of reference in

Being and Event,

whether treated from a broadly sympathetic

and supportive or a mainly critical and diagnostic standpoint. For among the many striking and, I would say, intellectually heartening aspects of

Being and Event

is the way that Badiou

cuts clean across the supposed genre-divide between recent 'continental' thought with its frequent focus on the detailed exegesis of canonical texts and mainstream analytic philosophy with its typical assumption that problems are best, most produc­ tively engaged as if

de novo and with minimal reference to what­

ever has been written about them in the past.

12

CONTEXT

Hence my seemingly extravagant claim for Being and Event as a work that bids fair for admission to the elevated company men­ tioned above. Badiou is a truly original thinker in the sense of that term that these comparisons properly imply, and not just the debased and currently overworked sense which often means either someone who has invented a minor new twist on well-worn philo­ sophic themes or else someone who has come up with a genuinely new line of argument but only by reason of its utter perversity or the sheer unlikelihood that anyone would think

thatY No doubt

there will be some analytic philosophers whose nrst response will be to turn the argument around and deploy it as a tu quoque rejoinder against Badiou or those who put th�_case for his pre­ eminent status in this regard. Thus they will ask what could possi­ l>ly be more perverse than a project of thought that claims to derive substantive truths with respect not only to natural-scientillc but also to political, social, ethical and even aesthetic understand­ ing from an abstract ontology grounded in th,. .::ven more abstract resources of set-theoretical thought. Any ment of set-theoretical enquiry from Frege and Cantor to Co· len), and thence to a renovated concept of truth that would also and inseparably carry along with it a renovated concept of the subject. It is here that Badiou flrst introduces another of Cohen's set-theoretical concepts, namely that of 'forcing', or the process

whereby certain paradoxes, contradictions, aporias or inconsis­ tencies within some existing state of knowledge can later from a more advanced stage of understanding - be seen to have marked precisely those symptomatic stress-points where know­ ledge fell short of a truth beyond its best powers of discernment or conceptual grasp. Again, his chief reason for taking this expository route via such relatively technical areas of set theory and philosophy of mathematics is that it offers not just a sugges­ tive analogy but a close and even (so Badiou would claim) a precise correspondence with the way that major advances are achieved in other flelds of endeavour. So it is that 'being can be supplemented' , or that 'the existence of a truth' can be thought of as 'suspended from the occurrence of an event', one whose implications will be far from clear at the outset since 'the event is only decided as such in the retroaction of an intervention' (p. 1 7) . 33

BADIOU'S BEING AND EVENT

This is how Badiou can think of ontology in realist terms as involving the progressive exploration of territory that is 'there' to be explored - since by no means (as anti-realists would have it) created in or through that very process - and yet as always open to startling new discoveries which themselves require a great deal of thinking-through for a good while after their fIrst, often somewhat tentative or stumbling formulation. Hence his descrip­ tion of Being and Event as a book 'designed to broadcast that an intellectual revolution took place at the beginning of the sixties, whose vector was mathematics, yet whose repercussions extend throughout the entirety of possible thought' (p. 1 6). That revolu­ tion he sees as having resulted from Cohen's development of set theory to the point where it fInally provided philosophy with the formal resources to explain how knowledge could be thOUght to fall short of truth - or truth to run ahead of knowledge in such a way that this discrepancy supplied the very driving force or incentive required to motivate a long-term, intensive and sharply focused project o f enquiry. 'In the category of the generic', Badiou writes, 'I propose a contemporary thinking of these procedures which shows that they are simultaneously indeterminate and complete; because, in occupying the gaps of available encyclopedias, they manifest the common-being, the mUltiple-essence, of the place in which they proceed' (p. 1 7) . Of course this sentence would bear a good deal of analytical unpacking since it encapsulates much of the argu­ ment that Badiou will go on to expound and refIne over the fol­ lowing large tract of closely reasoned, densely allusive, intricately cross-referenced text. For the moment let me note how it phrases his cardinal distinction between truth and knowledge as a mat­ ter of those exploratory (e.g. mathematical, scientifIc, political or artistic) procedures that may be thought of - metaphorically but aptly enough - as tracking what is unknown even to the best, most informed and informative sources through those gaps in the current 'encyclopedia' of knowledge that reveal the existence of missing ('indeterminate') items precisely on account of its not living up to the promise of truly encyclopaedic com­ pleteness. We should also bear in mind his identifIcation of 'common-being' with 'multiple-essence', conveying as it does Badiou's equally cardinal claim that the greatest innovation of -

34

OVERVIEW OF THEMES

set theory - and the yardstick of its various advances from Cantor to Cohen - is its formal demonstration of the fact that no 'consistent multiplicity' (i.e. no dominant ordering or version of the 'count-as-one') can ever fully contain or comprehend the 'inconsistent multiplicity' that always pre-exists and to that extent exceeds or eludes such a count. For it is this central claim that enables and motivates his appeal to mathematics as the basis of ontology and to ontology, in tum, as the basis for what­ ever we can justifiably assert concerning the truth-conditions for our various truth-apt statements, theories, hypotheses or conjectures. As I have said, this puts him very much at odds not only with a swathe of linguistically or hermeneutically oriented move­ ments Nithin recent continental (i.e. mainland-European) thougt c but also with that strain of anti-realist thinking that is a prominent feature of mainstream analytic philosophy of lan­ guage, whether in the Wittgensteinian or the more 'technical', logico-semantic mode.24 The issue of philosophy versus the soph­ ists, rhetoricians and cultural relativists is one that had its first airing .n the dialogues of Plato where it typically came up in conjU' .ction with other main themes of Badiou's work. Among them was the absolute pre-eminence of mathematics as an intel­ lectual discipline, a training ground for philosophers, and (above all) a royal road to the discovery of truths that, once discovered, acquired a priori status or the character of seeming self-evident to reason yet which often had to be arrived at through a lengthy process of constructing hypotheses, devising proof-procedures, pursuing long sequences of hypothetico-deductive argument, or testing theorems and conjectures for any logical anomaly that might arise in the course of those procedures. Despite (as we shall see) his express reservations about being labelled a mathe­ matical Platonist, Badiou nevertheless belongs very firmly on this side of the deep division that is commonly supposed to exist between thinkers of a Platonist and thinkers of an Aristotelian intellectual temperament. 2S So we can now tum to Part I of Being and Event which takes the form of six 'Meditations' on mathematicaVontological themes, and which starts out from a sustained engagement with the dialectics of the one and the many as treated in Plato's later writings.

35

BADIOU'S BEING AND EVENT

Discussion points

On the basis of your reading so far, what do you take to be Badiou's most distinctive or far-reaching proposals with regard to philosophy's role vis-a-vis mathematics and politics? Why does Badiou take such a strong line against the recent 'lin­ guistic tum' across various branches of philosophy and the social or human sciences?

36

CHAPTER 3

READING THE TEXT

PART I. BEING: MULTI PLE AND VOID. PLATO/CANTOR

1 . The one and the many: Plato to Deleuze

Plato's dialogue Parmenides is one of those problematic works ­ 'problematic' at least for scholars and interpreters who wish to extract some fairly straightforward or unambiguous item of doctri'.e - where Platonism (or the collection of ideas generally given thai name) fmds itself exposed to some difficult questions which place it under considerable strain. I For Badiou, conversely, this is a work whose 'revolving doors . . . introduce us to the sin­ gular jC'y of never seeing the moment of conclusion arrive' (p. 23). That j . to say, this is a thoroughly aporetic dial ogue, one that fails t ) reach any decisive or definitive conclusion not through some weakness, dialectical wrong tum or argumentative path not taken but rather through inherent complications in its subject­ matter that would have to wait more than two millennia before mathematicians and logicians came up with the conceptual means to handle them. Those complications have to do with the one and the many, or the issue - central to every aspect of Platonist metaphysics, ontology and epistemology - as to whether the one must be thought to have priority over the many or the many over the one. The former position was adopted most strongly by Plato 's precursor Parmenides who taught that truth and reality must be thought of as timeless, unchanging and altogether beyond the realm of transient sensory experience which could not be grasped intellectually without giving rise to all manner of logical incon­ sistency. Thus the famous paradoxes of time and motion devised by Parmenides' disciple Zeno were designed to make the case for a purely monistic and rationalist conception of being in relation to thought that would escape those paradoxes by denying the rea­ lity of time and change, these latter conceived as mere illusions 37

BADIOU'S BEING AND EVENT

brought about by our habitual over-reliance on the supposed 'evidence' of the senses This ontological distinction was carried across into the Platonist dichotomy of episteme and doxa, or genuine knowledge with its grounding in logos (or the exercise of reason) as opposed to mere opini on or consensus belief, equated with the realm of sensory impressi ons on account of their shared impermanence or chronic liability to change. Many philosophers from Plato down have taken something akin to this position .

,

either as a matter of explicit doctrinal adherence or else as an

inbuilt assumption prerequisite to other main dimensions of their thought Thus for Leibniz, as Badiou notes, what is not a being is not a being', since - onto logically speaking - the condition of existence for any given object is precisely that set of individuat­ ing features or attributes that mark it out as that particular, self­ same object and no other. Such is Leibniz's cardinal precept concerning the 'identity of indiscernibles' and also its comple­ mentary adj unct with regard · to the 'indiscernibility of identi­ cals'. More recently it has found expression in W V. Quine s pithy saying 'no entity without identity', and it remains an item of shared (if often unspoken) belief across some otherwise large divergences of view in present-day philosophic debate. 2 For Badiou, on the contrary, it is far from evident that this issue has been settled - philos ophically resolved - in favour of the one over the many, or of that which can always be subsumed without remainder under some unifying concept as opposed to that which exceeds or eludes any such all-comprehending appli­ cation of the dominant count-as-one. This is why he insists on reopening those questions from Plato that can now be seen, as a result of recent mathematical advances, to belong among the most potentially fruitful topics but also to the pro blems most frequently ignored, suppressed or finessed by thinkers in Plato's wake. Thus Badiou follows Plato's example in making mathe­ matics the basis of ontology, ontology the starting point of all genuinely truth-seeking enquiry, and this ParmenideanIPlatonic issue of the one and the many a privileged means of access to the questions thus posed for philosophical thought. Chief among them is the question whether being can be grasped in unitary terms as that which underlies and brings intelligible form to the passing sensory show or whether, on the contrary, this is j ust a fixed preconception that has typified much philosophical .

'

'

38

READING THE TEXT

thinking from Plato down but which cannot be sustained against

the various objections that rise against it.

Hence Badiou's fascination with the game - the ingenious,

inventive, conceptually resourceful but also consequential and serious game - played out between Plato and his interlocutors.

What Plato glimpsed under pressure froID the kinds of quandary, dilemma or conceptual impasse encountered in these late dia­

logues was the impossibility of thinking such issues through to a

conclusion on his own metaphysically favoured terms. The desired upshot - in line with the doctrine set out in Plato's middle-period

writings - was to vindicate his claim for the existence of supra­

sensory forms or essences that would have their being above and

beyond the transient

flux of perceptual experience and thereby

perforr this imperative unifying function. However, the prob­ •

lems e nerged very clearly as soon as the obvious question arose as to howthe one - the supposedly transcendent and all-embracing

principle of unity - could manifest itself to human understanding except in and through the realm of 'presentation' which was also

(inescapably) the realm where thought encountered multiplicity in all Its forms. Hence the logical come-uppance delivered to

that J armenidean-Platonist hankering for the one through its

encounter with inconsistent multiplicity, that is, with whatever

intrinsically exceeds or disrupts its homogenizing drive. This rendition of Badiou's central theme may strike some readers as excessively wire-drawn, or as just a piece of idle scholastic paradox-spinning. However, it not only goes to the heart (i.e. the

mathematics-based formal core) of his entire philosophical proj­

ect but also provides him with a highly effective point of entry to

those various 'conditioning' fields of thought - from science to politics, art and love - which, as we have seen, compose a major

part of that project. For it is Badiou's claim that the issues first

aired in these ancient Greek disputes are those which continued to exert a powerful hold on the thinking of philosophers over

the next two millennia and have now - since the pioneering work

in the foundations of set theory by thinkers such as Frege, Russell

and (especially) Cantor - been brought to a stage where their true implications are at last coming into view.

Chief among these is the absolute priority of the multiple over

the one, a priority visible in Plato's Parmenides where reasoning

runs up against the aporetic limit or the threatening collapse into 39

BADIOU'S BEING AND EVENT

l ogical inconsistency of any attempt to prove the contrary case. Indeed this dial ogue can be seen to mark the p o int at which Platonist thought turns against its own earlier, distinctly Parmeni­ dean commitments and begins - in however tentative and self resi sting a fashion - to venture onto different, no doubt more vertiginous but also (as Badiou will p roceed to demons trate) philosophically more fertile ground. This is the stage to which thinking attains when it first proposes as a serious candidate for philo so phical acceptance the statement that 'being is not', or (as will later become more clearly expressibl e with the advent of modern set theo ry) that the multip le is that which always and everywhere exceeds the grasp of any unifying function or any mode of kn owledge premised on this or that application of the dominant count-as-one. What mathematical developments since Cantor have at last achieved is a genuine working grasp of those issues that proved so vexatious for Parmenides and Plato, and whose legacy Badiou will trace in Being and Event as it surfaces repeatedly to complicate the thinking of (amo ng others) Aristotle, Descartes, Pascal, Leibniz, Spin oza and Hegel. Thus the one can now be treated as the pro duct of a certain formal operation, that is to say, a pro cedure of counting or grouping that imposes some o rder on an otherwise inchoate since open-ended multiplicity but which is always - and for just that reason - exposed to the potenti ally di sturbing effect of that which finds no place in the exi sting conceptual domain since it exists as a 'supernumerary ' element excluded from the count-as-one. Hence the two main theses which Badiou takes as 'prerequi­ sites for any possible ontology': first that '[t]he multiple from which ontology makes up its situation is compo sed solely of mUltiplicities. There is no one. In other words, every mUltiple is a multiple o f multiples'; and secondly, that '[t]he count-as- one is no more than the system of conditions through which the multiple can be recognized as multiple' (p. 29). An other way of putting this is to draw a distincti on between 'consistent' and 'inconsis­ tent' mUltiplicity. The former is defmed by Badio u as that which results from some preceding count or formal operation, whereas the latter must be thought of as itself pre-exis ting , surpassing and eluding the count-as-one yet also - since of course that o peration must have something to operate on - as providing its necessary starting point or precondition . In which case, Badio u ­

40

READING THE TEXT

argues, we shall have to re-conceive ontology on the model of or as identical with - certain specific situations, that is, certain specifiable stages in the process whereby inconsistent mUltipli­ city is rendered consistent by this or that formal means, before once again turning out to possess elements or subsets that can­ not be reduced to any such principle of order. Thus a 'situation', as Badiou understands it, is defined in essentially liminal or transitive terms as a 'structured presentation' which

partakes

of both consistent and inconsistent multiplicities. 'This duality', he writes, 'is established In the distribution of the count-as-one; inconsistency before, and consistency afterwards'

(p. 25).

And

again, '[s]tructure is what obliges us to consider, -via retroaction, that pre�entation is a multiple (inconsistent) and what authorizes us, via Lllticipation, to compose the terms of the presentation as

units )f a multiple (consistent)

,

(p. 25 ) So it is - through this .

constantly evolving dialectic of containment and uncontainable excess - that thOUght is empowered to transgress and surpass the limits laid

down

by any regnant paradigm or merely

de facto

conse!',sus of belief. In i s absence, thOUght will most likely yield to one or other of

the tI '0 great opposing temptations that Badiou

finds endemic across a whole range of present day disciplines. On the one hand is the lure of that sophistical approach (po st structuralist, post­ modernist, Wittgensteinian, hermeneutic, constructivist or neo­ pragmatist) according to which it makes no sense to suppose the existence of truths beyond those that fall within range of our -

-

present epistemic, cognitive, conceptual or - what this is usually taken to

entail - linguistic-expressive

capacity. On the other,

both provoking and reacting to such sceptical-relativist ideas, there is the older and more characteristic philosophical tempta­ tion that consists in flatly asserting the claim for ontology as a discipline aimed towards matters of absolute, objective truth and just as flatly denying that it ought to be concerned with 'situations' or conjunctures

of

the kind that Badiou constantly

invokes. If the greatest single problem for ontology is that thrown up in the course of Plato's

Parmenides and

repeatedly in various

guises thereafter then the greatest temptation is that which involves 'removing the obstacle by posing that ontology is not actually a situation' (p. 26). Badiou on the contrary describes it as the chief 'wager' of his book that ontological issues cannot be 41

BADIOU'S BEING AND EVENT

treated in all their specificity, complexity and depth except by taking adequate account of how they emerge in given situations and how they respond more or less inventively to various con­ junctural problems or dilemmas. Moreover, this entails that there cannot be a concept of the multiple, at least if by 'concept' is meant a clearly defined - necessary and sufficient - set of condi­ tions for what the term properly denotes. After all, '[o]ne would thus count it [the multiple] as one and being would be lost again', insofar as being is equated by Badiou with whatever eludes our best-present powers of cognitive grasp precisely on account of its having 'no place in the conceptual order laid down by the currently prevailing count-as-one. This is why, for set-theoretical purposes, 'the prescription is . . . totally implicit . . . .it operates such that it is only ever a matter of pure multiples, yet there is no defmed con­ cept of the multiple to be encountered anywhere' (p. 29). All the same - what is perhaps the most difficult aspect of Badiou's thinking for anyone new to his work - it is in conse­ quence of just this ultimate elusiveness or resistance to concep­ tual definition that set theory has displayed such a striking capacity for advancing through and beyond (and by means of) the various problems and paradoxes thrown up in the course of its development to date. Badiou will have a great deal more to say - and with closer reference to specific episodes in that deve­ lopment - regarding this notable progress in its powers of con­ ceptual-ontological grasp and the way that they spring from a repeated encounter with the limits of present understanding linked to an anticipatory sense of what as yet lies beyond reach of any adequate formal or conceptual statement. For the moment there are three main points that we should note by way of pre­ liminary orientation. First is that ontology, as Badiou conceives it, has always to be treated by way of its relation to those various specific 'situations' or conjunctures of thought which determine the scope and limits of knowledge at any given time. Second ­ apparently in conflict with this - is the precept that ontology has to do with questions of truth, rather than knowledge, and hence that it is a gross confusion (albeit one endemic to many present-day movements of philosophic thought) to suppose that ontological issues could ever be settled or even usefully addressed through any kind of context-relative or historically indexed approach. Hence Badiou's third and most crucial claim: that what accounts 42

READING THE TEXT

for the capacity of thought to transcend any current, however limited state of knowledge is the development of certain formal operations through which those limits show up symptomatically by acting as constraints on the quest for a more advanced, con­ ceptually adequate mode of understanding. Just how this occurs - or how it can possibly be thought to occur - is Badiou's theme in later parts of Being and Event where he takes up Cohen's set-theoretical ideas offorcing and the generic as between them offering a means to explain what would other­ wise constitute a strictly unresolvable paradox.3 At this stage he makes the point by explaining how the axiomatic-deductive method in mathematics, logic and the formal sciences can indeed posit truths and anticipate the kinds of procedure required to establir 11 those truths without as yet having actually arrived at any sl .ch procedure or laid it out as a valid (Le. formally ade­ quate) proof. His adherence to the axiomatic method - to a con­ ception of truth as attainable solely by way of certain logically rigorous formal procedures - is a prominent feature of Badiou's thinking with respect not only to mathematics but in each of those otherwise diverse fields where he takes the dialectic of being and event to constitute the driving force of scientific, political and cultural change. It is also what sets him implacably at odds with a great many present-day thinkers for whom such a claim on behalf of truth - let alone on behalf of mathematics as the royal road to truth - would betray nothing more than a quaint attachment to thoroughly antiquated ways of thought. Those thinkers would include the aforementioned company of post­ modernists, post-structuralists, Wittgensteinians, neo-pragmatists and suchlike adepts of the 'linguistic tum' alongside other more highly esteemed antagonists, like Gilles Deleuze, with whom Badiou has taken issue on just these grounds.4 Deleuze stands out - and receives by far the most detailed and respectful of the many critiques that Badiou has devoted to philo­ sophical sparring partners - by reason of his having emphati­ cally espoused an 'open' rather than a 'closed' ontology, one that derives its chief inspiration from the differential calculus and 'intcmsive' (qualitative) rather than 'extensive' (discrete and quantitative) multiplicities, and which thus defines itself squarely against any formal or axiomatic conception of truth. S Moreover, as becomes very clear in Badiou's protracted engagement with 43

BADIOU'S BEING AND EVENT

Deleuze, this difference as regards relatively technical issues in the philosophy of mathematics has wider repercussions for their respective views concerning questions in the natural, social and human sciences and even - as Badiou would be quick to aff111Il ­ matters of an urgently p olitical or ethical character. For it is his contention that in each case the interests of truth are best served by adherence to certain rigo rously specified operative principles. These latter may be axioms, formal procedures, scientific hypoth­ eses, research programmes, p olitical initiatives or the kinds of constraint upon future conduct brought about by the demands of fidelity to some prior commitment undertaken with a view to its unpredictable yet none the less looked-for since partIy antici­ pated outcome. So there is a close connection between , on the one hand, Badi ou 's d is agreement with Deleuze over issues con­ cerning mathematical onto logy and the rival claims of extensive versus intensive (or axiomatic versus differential) modes of thought and, on the other, his disagreement concerning the need for concerted and organized action in the socio -political sphere. This latter, well-publicized difference of views no doubt has its roots in their respective histories of political involvement before and after the watershed events of May 1 968. However, it also has much to do with Deleuze's almost visceral mistrust of large - scale 'molar' (as opposed to small-scale, 'molecular') forms of activist engagement and his broadly anarchistic idea of ' desiring-p rod uction' as the locus of those dis ruptive energie s that alone have the potential to break down existing structures of power and contro1.6 C onversely, it is Badiou's deep-laid con­ viction that in politics as in other, more o bviously formal disci­ plines of thought there is no prospect of genuine advance or po ssibility of truly innovative thinking except by way of rigorous procedures that should always themselves be subject to likewise rigorous assessment at every stage. Closely allied to this is the willingness to venture far-reaching hypotheses that decide the whole course of one's investigative work but for which as yet there exists no adequately formalized proof-procedure, or to stake large claims on some future event which thereby shapes one's every act and decision even though that event is by no means certain - even (as things look at present) unlikely - to transpire. Moreover, it demands that one follows them through with the utmost fidelity and rigour, whether in terms of intellectual commitment to a 44

READING THE TEXT

detailed and demanding programme of research in the formal, physical or human sciences or in terms of a long-term political project entered into and sustained with unyielding commitment against whatever circumstantial odds. So it is not hard to see why Badiou should have singled out Deleuze as one of those exem­ plary figures - along with (among others) Aristotle, St. Paul, Pascal, Leibniz, Spinoza and Heidegger - to whose thinking he is sharply, even passionately opposed in various respects while none the less finding in them just the kinds of adversary with whom he can most productively engage. I have taken this slight detour from the path of strict exegeti­ cal conduct so as to explain (for maybe sceptical readers) how issues in such technical or specialized fields as advanced set the­ ory ar j philosophy of mathematics can bear something more than ! . suggestive, vaguely analogical relation to issues of politi­ cal theory and practice. It comes out in what Badiou has to say about the need for an axiomatic-deductive approach to mathe­ matics and the formal sciences in virtue of its singular capacity to pass beyond the limits of intuitive grasp or presently existing kno\'\ ledge, devise proof-procedures for unproven theorems or conjf ctures, and thereby - when these work out according to that sensi;; of anticipatory grasp - discover new stretches of mathe­ matical or ontological terrain. The latter pair of terms Badiou would consider a pseudo-dichotomy - a distinction without a real difference - since for him mathematical discoveries are also and inseparably stages of advance into so-far unexplored onto­ logical regions. Indeed it is one of philosophy's primary tasks to expound and clarify these ontological advances and thus make up for the marked unwillingness, among most working mathe­ maticians, to do so off their own bat. It is precisely the signal merit of such axiomatic-deductive thought that it allows enquiry to proceed along paths that are clearly marked out or rigorously formalized - where individual terms have a strictly specified 'compositional' role as defined by their place within the overall structure - while avoiding the kinds of premature appeal to some existing state of knowledge or stock of intuitions that would block any prospect of significant further advance. Thus again, '[i]t is clear that only an axiom system can structure a situation in which what is presented is presentation. It alone avoids having to make a one out of the multiple, leaving the latter as what is 45

BADIOU'S BEING AND EVENT

implicit in the regulated consequences through which it mani­ fests itself as multiple' (p. 30). No doubt some explanation is required as to what Badiou means by the rather odd claim that 'what is presented is presen­ tation'. Here again he is referring to that basic set-theoretical precept according to which the one (i.e. fIrst in the series of inte­ gers or natural numbers) is in no sense a primitive term but, on the contrary, results from an certain operation brought to bear on the otherwise open multiplicity - the endless regress of multi­ ples whose terms are themselves multiples whose terms are them­ selves multiples, and so forth - that must be taken to precede and moreover to constitute the very condition of possibility for any such operation. In other words - Badiou's own, from the useful glossary of technical terms appended to Being and Event - pre­ sentation is itself the 'primitive word of metaontology (or of philosophy)' (p. 5 1 9). This is because it specifies only the bare fact of something's having been presented, 'something' in this context taken to denominate the as-yet open or undefined con­ tent of any presentation in general, rather than some determi­ nate content as given by this or that specific application of the count-as-one. This is also to say that '[p]resentation is multiple­ being such as it is effectively deployed', and furthermore that it is 'reciprocal with "inconsistent multiplicity'" since the latter likewise exceeds and eludes any adequate specification. For '[t]he One is not presented, it results, thus making the multiple consist' (p. 5 1 9). In other words, it is only at the stage of the count - in whatever onto-mathematically determined form - that inconsis­ tent multiplicity (presentation as such) is confmed or reduced to the realm of defInite numerical, logical, propositional or other such formally specified content. With all this in mind we are now better placed to see just how high are the stakes when Badiou returns to Plato in Medita­ tion Two and stages a further dialectical encounter which in fact involves four disputants - himself, Plato, Parmenides and Socrates - although their voices are mostly unmarked. This sec­ tion is headed by the Parmenidean epigraph - 'If the one is not, nothing is' - but goes on to tum that dictum around from its manifest sense (that everything must be one because otherwise all would be confused in a multiplicity of fleeting appearances) to what Badiou derives from it by way of set-theoretical reasoning 46

READING THE TEXT

(that since any such ontology founded on the one will break down through the logical contradictions exposed in Plato's dia­ logue therefore it is the nothing that must constitute the starting point of any viable ontology) . For the sceptical reader still seek­ ing evidenre of Badiou's philosophical acumen or power of sus­ tained and concentrated argument there is no better evidence

than that contained in this brief but pregnant chapter. As he

notes, the Platonic dialogue 'is consecrated to an "exercise" of pure thought proposed by the elderly Parmenides to the young Socrates' (p. 3 1). It possesses a quality of intense and hard­ pressed adversarial reasoning that is carried across with no dimi­ nution to Badiou's meticulously reasoned and attentive but also shrewdly transformative reading of the text. Thus his aim is to show - very much against the grain of Parmenides'

(and

also,

more f mbiguously, Plato's) intent - how any attempt to state a doctrine of being premised on the absolute priority of the one will end up despite and against its purpose by implicitly affirming a doctrine of the multiple as prior to the count-as-one. In short, '[w]hat Plato is endeavoring to think here, in a magnificent , dense text, if evidently inconsistent multiplicity, which is to say, pure preser tation, anterior to any one-effect, or to any structure'

(p. 33). We shall soon see enough of its further consequences not least with regard to those political issues of inclusion, exclu­ sion and representation that are rarely absent even from Plato's more arcane metaphysical debates - to recognize just how much is at stake when Badiou presses this ancient dialectic of the one and the many to the point of aporia and ultimate inversion of the Parmenidean doctrine. What emerges from this reading, therefore, is a Plato who enunciates truths that contravene his own express metaphysical commitments and one such truth in particular: that 'in the absence of any being of the one, the multiple in-consists in the presenta­ tion of a mUltiple of multiples with out any foundational stop­ ping point' (p.

33). 'In-consists' is one of those neologisms - in

this case the making of an intransitive verb from the adjective 'inconsistent' - that may strike some readers as overly self-indulgent ' but which in fact serve very well to communiCate a technical (in this case set-theoretical) concept. Thus its point is to capture not only the state of 'inconsistent multiplicity' which Badiou opposes to the order of 'consistent' mUltiples that result from 47

BADIOU'S BEING AND EVENT

some particular mode of presentation or product of the count­ as-one but also the process or

activity whereby any such consis­

tent multiple is subject to the kind of potentially destabilizing force exerted by the various anomalies, discrepancies, exceptions and other problematical instances that find no place in the count. Here again there is a close and unmistakable connection between what Badiou has to say in this fonnal (set-theoretical) register and what he has to say on topics of an ethical, social and political nature. Indeed to put it like this is to understate and misrepresent his claim since, for Badiou, the structures and procedures involved in an adequately theorized social-political ontology are identical to those that emerge from an ontologically adequate account of devel­ opments in set theory. As we shall see later on, it is precisely through the non-coincidence between belonging and inclusion a disparity expressible with greatest precision in set-theoretical tenns but one -

with far-reaching implications as regards those other subject­ domains - that Badiou is able to stake this claim and, moreover, to unfold its implications witha high degree of fonnal rigour as well as extraordinary speculative sweep. It is therefore natural that

Being and Event

should set out by

engaging with Plato's Parmenides as the text wherein - through its prior engagement with Parmenides himself - these questions receive their earliest and (from Badiou's point of view) their ide­ ally complex, dialectical and inconclusive or aporetic treatment. Not that Badiou is out to fault Plato on grounds of mere falla­ cious reasoning or - as so often when analytic philosophers write about thinkers of the past - on grounds of his unfortunately not having had the benefit of those more advanced logical or con­ ceptual resources enj oyed by anyone nowadays approaching the same sorts of problem. Nothing could be further from his prac­ tice of critical commentary than the tone of somewhat pitying fondness that often creeps into such writings in the 'rational­ reconstructive' mode.7 With Plato as with the many other think­ ers discussed in

Being and Event

Badiou inclines more towards

the 'continental' way of according those precursors serious, detailed and respectful attention on their own terms while also very clearly signalling the points where his thought diverges from theirs and the various respects in which later developments most crucially, the advent of set theory - have pointed a way beyond the obstacles they once faced. After all, Plato is scarcely 48

READING THE TEXT

to accept the or of inconsistent over consis­

at fault if his failure (more like: his dogged refusal) priority of the many over the one,

tent multiplicity, is due not so much to some corrigible lapse of reasoning on his part but rather to the inherent difficulty - even the downright impossibility - of clearly and distinctly conceiv­ ing any such. thing. All the same Badiou is very far from rejecting that conception

of philosophy

as wedded to the virtues of conceptual clarity

in the French philosophi­ cal tradition - going back to Descartes :-' and carried on by twen­ tieth-century thinkers such as Gaston Bachelard and Georges Canguilhem.8 Its influence is everywhere apparent in Badiou's

and distinctness, one with deep roots

thought, whether in his detailed reconstructions of the process by whi ;h

conceptual advances come about in mathematics and

the fOf m al sciences or in his singling out just those thinkers - and

just those aspects of their work - that most

strikingly

exemplify

that claim. Hence his strong attachment to the seventeenth­ century rationalists (despite many and various disagreements with them) insofar as they adhered to a conception of scientific, philo! ophical and social progress that placed chief emphasis on the p )wer of reason to think its way through and beyond the limits of common sense, intuitive, uncritical, parochial, passively

of belief. Hence also his productively understood

imbibed or other such customary habits approach to Plato as a thinker more

through a critical-diagnostic reading that treats him as at least partially accountable to later and, in certain respects, more advanced procedures of thought. fied principle

of charity that

If read in this way - on a quali­

maximizes rational content while

also (quite compatibly) making full allowance for explicable error - then

Parmenides

can be seen as a work

that

implicitly

foregrounds its own dilemmas or moments of unresolved con­

looks far beyond its own tem­ intellectual and cultural horizons. That is to say, Plato's statement, 'If the one is not, nothing is' can be left to stand wholly unaltered and yet, in consequence of such a reading, be construed as follows: 'if the one is not, what occurs in the place of " the many" is the pure name of the void, insofar as it alone

ceptual strain, and which thus poral,

subsists as being' (p. 35). Thus the upshot of Plato's exceptionally hard-pressed dialogue

is to show Socrates - as seldom happens, though always (when it 49

BADIOU'S BEING AND EVENT

d oe s) to hi ghly revealing effect - in the role of a thinker caught very much on the hop by the unforeseen dialectical consequences of his own stated po sition . This ironic twist emerges in various ways, among them the change of sen se that certain crucial words undergo as a result of their being caught up in a process of imman ent self-examination or conceptual auto-critique that leads them beyond their established (and no doubt authorially intended) semantic range. They include the pair 1tt..n90c; and 1tot..t.a. . , the former transliterating (or transl ating, since it has fully entere d the English language) as 'plethora', whereas the latter si gnifie s 'the m any ' and has achieved - aptly enough from Badiou's viewpo in t - a rather less 'prop erly ' naturalized status through the phrase 'hoi polloi', 'the common peo ple ' or 'the p leb s' . His poin t (here summarizing rather brutally) is that Plato, for all his contrary intent, cannot help but demonstrate the con­ ceptu al impossibility of thinking 1tt..n 90c; as a fullness or pleni­ tude of being , such as would underwrite the Platonist appeal to that realm of transcendent forms, ideas or essences which consti­ tutes the sole guarantee of truth and knowledge against the ever­ shifting , illusory character of sensory-perceptual experience. What Plato clearly wishes to maintain - and states as such - is the intrinsic and natural pre-eminenceof the one over the many, and hence the necessity (not merely the possibility) of gathering the p letho ric multiplicity into some higher, ultimately truth-preserv­ ing state of union. What the unfolding logic of the dialogue con­ strains him to acknowledge - albeit despite himself - is the opposite consequence that flows from his failure to think thro ugh . with suffi­ the required opposition between 1tt..n90c; and 1tot..t.a. cient clarity or logical rigour to sustain his professed ontological commitment to the absolute priority of the one over the many. 2. Cantor: 'theory of the pure multiple'

Badiou goes on to develop these themes from Plato

in the context

of modern set theory and what he proclaims as its radically trans­ formative or innovative impact on our basic ontological concep­ tions. Set theory has to do with relationships of membership, inclusion and exclusion among numbers or other entities that are taken as forming a unit of assessment for some given purpose. Thus sets are defmed as products of the count-as-one, that is, the classifi­ catory procedure that consists in grouping together a certain range 50

READING THE TEXT

of such entities and treating them as co-members of a single assem­ blage whatever their otherwise diverse natures or properties. The latter point is crucial in mathematical terms but also for Badiou's socio-political thinking since it allows the set theorist - or anyone who has truly absorbed its implications - to ignore any merely con­ tingent or localized differences between such entities and accord them strictly equal status as regards their membership of any given set. Ironically enough, as Badiou notes, it was a point not fully taken by Cantor when he first enounced his 'theory of the pure multiple' and dermed it as follows: 'By set what is understood is the grouping into a totality of quite distinct objects of our intuition or of our thought' (cited, BE, p. 38). 'Without exaggeration', Badiou responds, Car .;or assembles in this definition every single concept whose decomposition is brought about by set theory: the concept of totality, of the object, of distinction, and that of intuition. What makes up a set is not a totalization, nor are its elements objects, nor may distinctions be made in some infmite collec­ tior s of sets (without a special axiom), nor can one possess the slightest intuition of each supposed element of a mod­ estly large set. (p. 38) Of course this series of objections would also disqualify my own attempt at a brief definition as given above and along with it, truth to tell, the majority of such attempts to be found in refer­ ence works and introductory texts. Indeed, if so minded, one can pick out a good few passages elsewhere in Being and Event where Badiou himself can be seen to indulge the same forbidden mode of talk. However, his purpose is to drive home the point that set theory has now progressed to a stage where it is (or should be) no longer necessary to fall back upon such notions, and more­ over that the intervening post-Cantorian sequence of advances which his book sets forth in some detail - were potentially contained within Cantor's inaugural insight. And so it came about, Badiou writes, that '[a] great theory . . . was born, as is customary, in an extreme disparity between the solidity of its reasoning and the precariousness of its central concept' (p. 38). He reinforces this point by then describing the process of increasingly advanced and rigorous formalization whereby set 51

BADIOU'S BEING AND EVENT

theory was progressively uncoupled from all such naive or restric­ tive appeals to a domain of distinct objects and likewise distinct thoughts or intuitions concerning them. Above all, what Badiou seeks to dispel - not only for the benefit of relatively uninformed readers but also in riposte to some philosophers of mathematics who take a contrary view - is the idea of intuition as having any role to play in set-theoretical reasoning. Here he is in agreement with the majority of analytic philosophers who likewise adopt an extensionalist rather than intensionalist approach, that is, one that defines the conditions for membership solely and strictly with reference to the set of those entities (whatever their nature) that fall within the relevant domain, and not in terms of any qualifying attributes or distinctive features that mark them out as fit candidates according to this or that (e.g. intuitive) criterion. 'What was thought of as an "intuition of objects" was recast such that it could only be thought of as the extension of a concept, or of a property, itself expressed in a partially (or indeed com­ pletely, as in the work of Frege and Russell) formalized language' (p. 39). If the latter project ran into problems with Russell's dis­ covery of certain paradoxes at the conceptual heart of set theory, then this was yet another indication that mathematical thinking, once launched on this investigative path, would continue to advance through repeatedly coming up against limits to its pres­ ent (very often intuitive) and pointers to its future (more concep­ tually adequate) state of understanding. In effect, the very confidence initially displayed by Frege and Russell in the power of their logical language (technically speak­ ing: that of the first-order quantified predicate calculus) to offer a complete and perfectly consistent formalization of the set­ theoretical domain was itself a sure sign of the project's being headed for just such an obstacle somewhere along the way. It is precisely this recurrent gesture of containment - this move to control and delimit the scope of enquiry through various tech­ niques of always premature 'totalization' - that Badiou regards as having posed a chief obstacle to progress by evading the radical challenge that set theory presents to every existing ontological or, indeed, socio-political order. Hence his objection not only to the claim that intuition might yield valid insights or conceptual progress (since intuition is most often just the name applied to preconceived habits of belief) but also to that narrowly logicist 52

READING THE TEXT

that the exploratory scope of set theory might be circum­ scribed by a purely formal programme whose terms are sp ecifi­ able in advance and which therefore pre-emptively restricts any future devel opments to what is conceivable at present. So it was, Badiou thinks, that the Frege-Rus s ell project was predestined to run aground on those paradoxes that sprang to view as soon as it encountered its limit-po int condition in that realm of the 'pure multiple ' - or that formally unrestricted set-theoretical domain which required that statements of the relevant class be open to

idea

the test of self-reflexive application.

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

53

BADIOU'S BEING AND EVENT

truth with formal validity and validity, in tum, with and total closure under logical entailment. In that context the acceptable face of self-reference ­ its ubiquitous and therefore un o bjectionable presence - under­ goes a distinct change of expressi on and becomes, in effect, the un-doer of that who le optimistic logici st p roj ect Russell's answer was to make it a stipulative rule that state­ ments in formal l anguages such as those of mathematics or the logical calculus should not be self-referring in a way that gave rise to difficulties of this sort. Rather they could best be averted by a 'Theory of Types' which dis tinguis hed clearly between vari­ ous orders or levels of statement, that is, those belonging to the first-order language of direct or material-mode assertion, those that referred to such first-order statements from a higher logical identifies

the classical ideals of co n sistency

.

level, and so on up through successive stages of increasingly abstract formal (i.e. meta-linguistic) specification. Only thus,

as a crucial component of logic and mathematics - be kept on its path towards an ever more secure, since ever more precisely codified conception of validity or truth. Still his purported 'solu­ tion' to these problems struck many, then and now, as objection­ ably ad hoc and as having more to do with interests of pragmatic

Russell thought, could set theory

-

present-day developments in

or methodological convenience than with principles self-evident to

Indeed, the set-theoretical paradoxes have remained a spur to philosophic thought and a potent source of speCUlative ideas both within mathematics and across a range of other disciplines ever since Russell first discovered them. Their impact was intensi­ fied by various related developments, including - most notably Go"del's undecidability-proof to the effect that any formal system sufficiently complex to generate the axioms of elementary arithme­ tic or first-order logic would necessarily include or entail at least one statement the truth or validity of which could not be proved within the system itself.10 In other words, one could have either truth as matter of rigorous logical procedure or consistency ('complete­ ness') as a matter of intra-systemic coherence but surely not both unless by some manoeuvre, like Russell's, that looked suspiciously like a mere device for saving logico-mathematical appearances. Nevertheless set theory survived these and other challenges through the effort of various thinkers to provide some method of formal restatement in axiomatic terms that would keep the paradoxes reason.

-

54

READING THE TEXT

safely out of view or at least prevent them from doing real harm. During the past century it has become absolutely central to every branch of pure and applied mathematics, as well as to every math­ ematics-based development in the physical and even (in certain contexts) the social and human sciences. Badiou's work is notable for not losing sight of the set-theoretical paradoxes - indeed, for placing them squarely at the centre of its philosophic interests - while regarding them more as an incen­ tive to thought or a spur to renewed intellectual-creative activity than as an obstacle that has to be ignored or set aside if further progress is to be made. Thus although they 'went on to weaken mathematical certainty and provoke a crisis which it would be wrong to imagine over [since] it involves the very essence of mather ,atics', nevertheless - he asserts - the widespread accep­ tance )f Russell's pseudo-solution meant that the problem with this logicist project 'was pragmatically abandoned rather than victoriously resolved' (p. 38). As for Cantor, Badiou sees an effort to 'force a way through' this looming impasse by resorting to quasi­ mystic'd, even theologically inspired notions of absolute infinity as opr osed to the realm of mathematically specifiable transfinite numb ;rs which he himself had discovered, thereby opening up (in his famous phrase) a 'mathematicians' paradise'. Theology makes its re-entry to the otherwise radically de-theologized (since de-transcendentalized) realm of set theory as a result of Cantor's retrograde tendency to equate absolute being ' not with the (consistent) presentation of the multiple', but rather with 'the transcendence through which a divine infinity in-consists, as one, gathering together and numbering any multiple whatso­ ever' (p. 42). On the other hand Badiou is more than willing to credit Cantor with having grasped more vividly than any of his fellow pioneers what also drove him to seek refuge in such 'onto­ theological' notions, namely the upshot of his own discovery when relieved of its inherited metaphysical baggage and pressed to its ultimate, strictly logical end point. Such was the incipient realization, already legible though not fully acknowledged in Cantor's work, that any resultant (set-theoretically derived) concept of 'being' would resist or elude the best efforts of sys­ tematic statement in terms compatible with that whole tradition of thought, whether in its mathematical, philosophic or (what effectively subtends both of these) its crypto-theological aspect. 55

BADIOU'S BEING ANO EVENT

In this respect Cantor stands out as the most striking and, for Badiou's purposes, the most intellectually heroic example of a thinking whose special virtue it is to confront the maximum challenge to its powers of rigorous development - a challenge as much internal or self-generated as brought to bear by external opposition - and thereby gain all the greater strength to overcome its own residual attachments and resistances. It is the same pat­ tern that we have seen emerging in Badiou's dialectical encounter with Plato, and that appears in varied forms whenever he engages with strong precursors such as Aristotle, Descartes, Leibniz Spinoza and Hegel. Thus there are certain aspects of their thin1- . ing that he finds problematic, unacceptable or retrograde while none the less considering their work to possess just this same self­ resistant or - at their most impressive moments - self-transcending quality. It is well worth quoting another lengthy passage from his commentary on Cantor since it captures precisely what BadiOl so values about those few select thinkers who, in his estimatior , achieve that rank. 'Cantor's thought wavers', he writes, between onto-theology - for which the absolute is thought as a supreme infinite being, thus as trans-mathematical, in­ numerable, as a form of the one so radical that no mUltiple can consist therein - and mathematical ontology, in which �onsistency provides a theory of inconsistency, in that what proves an obstacle to it (paradoxical multiplicity) is its point of impossibility, and thus, quite simply,

is not. (p. 42)

What this passage displays most clearly - in his own way of thinking as well as in those aspects of Cantor's thought that he finds exemplary despite their downright contradictory character ­ is the constant interplay of two terms, 'consistent' and 'inconsis­ tent', as the main source of conceptual leverage or (at risk of sounding too Hegelian) the dialectical motor of Badiou's entire project. He views the history of advances in mathematical knowledge as having most often come about through a process whereby various sorts of problem or paradox eventually gave rise to some new concept or agreed-upon way of proceeding which in turn - when its consequences became clear - could be seen to involve a further, deeper, and yet more thought-provoking chal­ lenge to the enterprise in hand. If again this sounds decidedly 56

READING THE TEXT

Hegelian then the impression is not entirely wide of the mark although it does demand qualification in ways that will emerge later on. At any rate Badiou is absolutely firm in his belief that although knowledge must be held distinct from truth - since truth might always transcend the utmost limits of human know­ ability - nevertheless knowledge is attainable, albeit with the strict fallibilist proviso that all and any present claims in that regard might conceivably be subject to future revision or out­ right disconfirmation. Indeed, one of the philosophic traits that lifts his work well clear of post-structuralist, postmodernist and other recent Francophile movements of thought is Badiou's unwavering commitment to the existence of language-independent or culture-transcendent truths and his equally strong rejection of the claim that this is in any sense a sign of dogmatism or entrenched doctrinal adherence. On the contrary, it is only by affirming that commitment and hence by conceding the possibil­ ity of error in even our most deeply held theories, truth-claims or items of belief that we are saved from equating truth tout court with what counts as such for ourselves and fellow members of our own (whether specialized or culture-wide) community. l l Badiou makes this point most concisely in the context of describing those advances in the formal development (or axio­ matization) of set theory that were carried through by post­ Cantorian thinkers such as Zermelo and Fraenkel, the devisers of that particular version - the ZF system - that he adopts mainly on grounds of conceptual economy and ease of exposi­ tory treatment. Nevertheless, as he is keen to impress upon the reader, where set theory is at issue 'axiomatization is not an artifice of exposition, but an intrinsic necessity' (p. 43). That is to say contra a good many Wittgensteinians, neo-pragmatists, intuitionists, conventionalists or anti-realists - what is at stake in that process is not just a matter of finding some more convenient since compactly expressible means of formal presentation for concepts that might otherwise (and perhaps better) have been expressed in something less drastically divergent from the norms of natural or 'ordinary' language. Rather it is the very possibility of thinking beyond that tenacious since intuitively deep-laid and linguistically ingrained mindset which - if not subject to constant rectification and critique through just such axiomatic­ deductive procedures of thought - will persist in presenting us -

57

BADIOU'S BEING AND EVENT

with 'common-sense' ideas and pseudo-solutions to misconceived problems. Here again Badiou stands four-square with the ratio­ nalist tradition of thought from Descartes, Leibniz and Spinoza to its latter-day progeny among both mainstream analytic phi­ losophers who continue the basic P. �ge-Russell programme and also those French thinkers - Bad .elard, Canguilhem, even the deeply Spinoza-influenced Marxis·, theoretician Louis Althusser ­ who stake their projects on the capacity of thought to transcend the deliverances of mere intuition or received (linguistically ensconced) doctrine. 12 By the same token he stands just as squarely opposed to those latter-day 'sophists' - as distinct from the more intellectually reputable since dialectically challenging company of 'anti-philosophers' - who take refuge in just such sources of false assurance or just such appeals to the delusory idea of a wisdom vested in 'ordinary language' a"'.J its associated customary 'forms-of-life'. Nowhere is the fallacy of this way of thinking more clearly shown up, so Badiou maintains, than in the context of develop­ ments in set theory and the problems faced by philosophy of mathematics to the extent that it strives to account for those developments in conceptually adequate terms. His preference for ZF over rival systems has to do precisely with its pressing as far as possible in this direction, avoiding all forms of premature conceptual (or ontological) commitment, and thereby faithfully pursuing what Badiou sees as the path of thought laid out for set-theoretical enquiry. This it does by allowing just one relation between terms - that of belonging, represented by the symbol E - and excluding all reference to other properties that would bring such otiose commitments along with them. The purpose of adopting this austere approach is to avoid the constant tempta­ tion (as witness Cantor's 'theological' tum) of regressing to a more intuitively manageable concept of set theory which contin­ ues, in the classical manner, to distinguish between objects, mul­ tiples, multiples of multiples and so forth. Thus '[w]hen I write "ex belongs to �, ex E �, the signs ex and � are variables from the same list, and can be substituted for by specifically indistinguish­ able terms' (p. 44) . That is to say, on the ZF system it is easier to conceive how thinking can dispense with the intuitively self­ evident distinction between 'individual' objects and groups of objects, or particular (discrete) sets and assemblages composed 58

READING THE TEXT

of multiple sets under some higher level gro uping principle. Moreover it leaves no room for what seems - on a more conser­ vative or intuitive account - the self evident truth that logically there must be a distinction between elements and the sets to which those elements belong or in terms of which they are speci­ fied as elements. Indeed it is at just this p oint that set theory in its more developed forms departs from the ' naive ' or still intuit­ able stage at which Cantor remained through his supposition that to think of sets was necessarily to think of them as entities that differed, ontologically and logically speaking, from the ele­ ments that made them up. In other words, ' [t]he sign E , unbeing of any one, determines, in a uniform manner, the presentation of "something" as indexed to the multiple' (p. 44). What set theory most notably - and to some thinkers most disturbingly - con­ jures up is the prospect of a bad infinity or a multiple that is not composed of so many fIXed or defmable units but must rather be thought of as a 'uniformly pure multiplicity' without any clearly specifiable constituent parts. Here Badiou offers the suggestion that '[i]f one admits, with a grain of salt, Quine's famous formula "to be is to be the value of a variable", one can conclude that the ZF system postulates that there is only one type of presentation of being: the multiple' (p. 44). This remark is best read as conveying a certain sympathy, on Badiou's part, with the 'austere desert landscap es' that Quine famously preferred to the Ivsh vegetation of more ample or profligate ontologies. 13 However, the 'grain of salt' serves to indi­ cate - di stinctly in tension with that - a clear sense of just how restrictive is Quine's echt-analytic desire to prohibit any reckless ontological ventures beyond the safe (,scientifically' validated) ground of a quantified first-order predicate logic coupled with a radically empiricist conception of epistemic warrant. That is to say, Badiou is by no means averse to the formal rigour or the extreme on tological austerity of Quine'S approach, accordant as it is with his own professed aims of giving logic precedence in all .matters of ontological enquiry and moreover restricting such enquiry to what can be said - consistently maintained on the basis of a disciplined investigation into the various set-theoretically thinkable modes of being. However he is sharply at odds with Quine in j ust about every other respect, including his commit­ ment to a rationalist conception of ontology that could scarcely -

-

59

BADIOU'S BEING AND EVENT

be furth ;r from Quine's outlook of radical empiricism. Equally un-Qui'.lean - and likewise reflecting his distinctly 'continental' angle of vision - is Badiou's conception of progress in the fonnal (as well as in certain branches of the physical) sciences as typically powered by conflicts, anomalies or moments of productive fric­ tion between the drive for consistency and that which will always elude or subvert any fully consistent methodology or set of results. It is just this idea or a constant dialectical tension intrinsic to the very nature of '.llought - rather than of problems that crop up periodically and that have to be resolved before thinking can once again proceed along its appointed path - that Badiou fmds most compellingly enacted in the sequence of set-theoretical advances from Cantor down. He makes the point with reference to Zennelo's principle - a main component of the ZF system - that 'a property only determines a multiple under the supposition that there is already a presented multiple' (p. 45). That is to say, any imputed feature or attribute pertaining to some given mem­ ber of some given set and on the basis of which their membership is taken to depend must itself suppose a pre-existent multiplicity sub­ ject to no such selective constraint and therefore - by deftnition more numerous or inclusive., Here again, in this idea of what is suppressed or marginalized by any determinate (e.g. 'democratic') instance of the count-as-one, we may glimpse some of the political or socio-critical implications that Badiou will go on to draw from his set-theoretical elaborations. In fonnal terms, 'Zennelo's axiom system subordinates the induction of a multiple by language to the existence, prior to that induction, of an initial multiple' (p. 45). 'Language' here presumably includes not only those varieties of natural language to which, as we have seen, Badiou accords no authority in such matters but also those fonnal or regimented lan­ guages - like that which he shares with Quine, that is, the language of the ftrst-order quantifted predicate calculus - whose very consis­ tency is such as to ensure that they can serve only in a strictly heu­ ristic, assistive or enabling (though also a strictly indispensable) role. For they will always result from the suppression of - and therefore, at certain crucial junctures, be subject to disruption by - an inconsis­ tent multiplicity that cannot be fully grasped or encompassed but only more-or-Iess drastically reduced to order by any application of the count-as-one. 60

READING THE TEXT

What occurs at such moments is an especially forceful demonstra­ tion of the truth that applies always and everywhere in matters of ontological import but which is mostly concealed repressed or glossed over - by philosophic doctrines or 'common­ sense' ideas premised on the plenitude or the positivity of being. It has to do with the essentially 'subtractive' character of ontological enquiry and the impossibility that thinking should ever fully coin­ cide with the contents of thought as given by intuition, by language, or by any supposedly consistent apparatus of formal concepts that fails (or programmatically declines) to make allowance for that which might always elude its foregone systematic grasp. This is why such truths are visible only in the fissures, contradictions and aporias that mark the great majority of texts in the Western philosophical canon. The exceptions are those very few thinkers � Plato (if unwill­ ingly) and certain set-theoreticians among them - who pressed the dialectic of being and non-being to its logical conclusion and also those philosophers such as Aristotle, Descartes, Leibniz, Spinoza and Hegel who - albeit in radically different ways - bore witness to the limits of a positive ontology through their failure to express or consistently articulate the conditions under which it might be achieved. Hence, according to Badiou, the whole problem of the subtractive suture of set theory to being qua being. It is a problem that language cannot avoid, and to which it leads us by foundering upon its paradoxical dissolution, the result of its own excess. Language - which provides for separations and compositions - cannot, alone, institute the existence of the pure multiple; it cannot ensure that what the theory presents is indeed presentation. (p. 48) This distinction between 'what the theory presents' and 'presen­ tation' in another (ontologically prior) sense of the term takes us to the heart of Badiou's political as well as his 'purely' mathe­ matical and formal-philosophical thinking. 'What the theory presents' is what finds an accredited, duly acknowledged place in those various prevailing systems (mathematical, formal and natural-scientific but also - by more than suggestive analogy political and socio-cultural) that decide what shall count as a member or constituent of some given set, group or class. What the term 'presentation' signifies, on the other hand, is the totality 61

BADIOU'S BEING AND EVENT

of those elements that offer themselves as potential candidates for membership, whether or not that potential is realized by their actually bein p so treated. Hence Badiou's central thesis in the formal (i.e. the ontological and set-theoretical) domain: that even though 'inconsistency is not actually presented as such since all presentation is under the law of the count', nevertheless 'inconsistency as pure multiple is solely the presupposition that prior to the count the onds not' (p.

52). Or rather, it is just because the first of these claims

can be

shown to hold - shown (that is) through Badiou's elaborate work­ ing-through of the set-theoretical paradoxes - that the second claim also goes through. His point, to repeat, is that the one is always the

result

of some such counting operation brought to

bear in the act or through the process of transforming an incon­ sistent into a consistent multiplicity, or deciding which elements shall count as members and which be consigned to the limbo of non-belonging. At the same time this central truth of ontology the truth of its essentially subtractive character - is concealed from most enquirers simply through the fact that

tion

by very defini­

those excluded elements cannot figure within the count-as­

one or be perceived as integral or constituent parts of any existent situation. Thus '[n]othing is presentable in a situation otherwise than under the effect of structure, that is, under the form of the one and its composition in consistent multiplicities' (p.

52). From within the discourse of mathematics and the formal sciences - that is, within the ambit of those disci­

which it follows that only

plines most readily amenable to set-theoretic formalization

-

can

thinking resist the otherwise inevitable tendency to recognize only those elements that make up some known or acknowledged situa­ tion and hence to ignore whatever eludes, escapes or exceeds the prevalent count-as-one. �y situation, seized in its immanence, thus reverses the inaugural axiom of our entire procedure. It states that the one is and that the pure multiple - inconsistency - is not' (p.

52). However - and this claim is absolutely central to what I

shall describe as Badiou's realist ontology, although he might well have certain misgivings about that description - the truth of such a situation is in no way dependent on what we may perceive, rec­ ognize, believe or take ourselves to know concerning it. This conception of truth as always potentially surpassing our best attainable state of knowledge - in the jargon, as 'recognition62

READING THE TEXT

transcendent' or 'epistemically unconstrained' - is one that unites Badiou with many realists in the analytic camp, whatever his differences with them in other regards. 14 Moreover it is one in the absence of which his project would utterly founder since it would lack any means to explain how thought can advance through the process of discovering - rather than inventing those anomalies and conflicts that previously passed unnoticed but which then at a certain point emerged clearly to view and set the conditions in place for re-thinking the issue at hand. What is also required in order for this to Occur is a reversal, however short lived, of the imperative that governs most thinking at most times in most areas of thought, namely that such thinking be conducted very largely in terms of consistent multiplicity or structured situations so as to gain sufficient purchase on its vari­ ous object-domains. Thus '[i]n a non-ontological (thUS non­ mathematical) situation, the multiple is possible only insofar as it is explicitly ordered by the law according to the one of the count'. And again, '[i]nside the situation there is no graspable inconsistency which would be subtracted from the count and thus a-structured' (p. 52). However this restriction may be lifted, to some extent at least, insofar as thinkers in other disciplines acquire the conceptual resources made available by develop­ ments in post-Cantorian set theory and thus come to grasp the basic point: that if the one is what results from some previous operation, then 'of necessity "something" of the multiple does not absolutely coincide with the result' (p. 53). Indeed it is precisely in the need for such an operation - the inability of thought to achieve a proper sense of conceptual pur­ chase except on condition of reducing inconsistent to consistent multiplicity - that the 'something' in question most strongly manifests itself as preceding and exceeding the count-as-one. The 'wavering towards inconsistency' is something that Badiou detects across a wide range of philosophical texts where the overriding drive for system and method - or (in Heidegger's case) for access to a realm of ontologically authentic Being beyond the merely ontic or quotidian - is allowed to subdue any countervailing sense of that which would otherwise resist such appropriation. This applies especially to programmatic thinkers such as Spinoza and Leibniz whose ruling premise is that truth must be expressible in terms of a consistent, logically articulated 63

BAOIOU'S BEING AND EVENT

system of propositions that admits of no internal gaps, discrep­ ancies or other such faults and which thus stands proof against criticism or indeed - by implication - against any further progress beyond its self-achieved stage of advance. Nevertheless, as Badiou will endeavour to show, their projects encounter just the kind of resistance from internal anomalies - most often from unresolved conflicts between the large-scale (purported) logical structure of their argument and its detailed working-out - which is only to be expected given his claim concerning the ultimate prevalence of inconsistent over consistent multiplicity, and hence the ubiqui­ tous (no matter how elusive) remainder or reminder of the 'supernumerary' element that haunts all systematic discourse. 3. Plato and the unpresentable

One striking example that Badiou adduces is Plato's strange and in many ways untypical dialogue Timaeus where he advances a highly speculative piece of metaphysical argument - with distinct echoes of Anaximander and the pre-Socratic thinkers - concern­ ing the emergence of order from chaos, of the many from the one, of phenomenal appearances from that which underlies and makes possible those appearances, and of sexual difference from a prior state of hermaphrodite dual gender. IS Badiou's interest here is not so much in Plato's 'great cosmological construction', although he is strongly drawn to works of this kind (grand efforts in the system-building vein like Spinoza's Ethics or Leibniz's Monadology) while looking out for all the tell-tale points at which their more ambitious or programmatic claims can be seen to break down on recalcitrant details and twists of argumenta­ tion. Rather he is intrigued by what Plato - in the opening para­ graph of Timaeus - refers to as the 'errant cause' and invokes as part of his attempt to explain how the cosmos originally came into being through a combination of chance, necessity and the working-out of a rational scheme or underlying principle of order that would somehow reconcile the warring claims of free­ will and determinism. At this point Plato is led or obliged by the logic of his argument to introduce a term that seems to involve the familiar - if endlessly debated - notion of causation but under a guise (that of 'errancy') that marks it out as no normal or familiar case of the kind. In his Spinoza commentary, later on, Badiou will locate this 'wandering' and strictly 'unpresentable' 64

READING THE TEXT

(since thoroughly anomalous) point of excess - or rather of lack, insofar as it marks what is superfluous and redundant but also, for that reaSon, inadequate to the system's needs - in the Spinozist notion of 'infInite modes ' which, like the Platonic 'errant cause', proves stubbornly resistant to a rational accounting on system­ compatible terms. What tllought thus encounters in its dealing with these texts is the 'void' to which Badiou makes constant reference in his writ­ ing on set-theoretical matters and also - via the series of closely argued connections that I have sketched out briefly above - his thinking in the socio-political sphere. Thus the void is that which cannot be conceived except in subtractive term� . or as a 'non-one' that eludes the count insofar as it both precedes the operation by which 'inconsistent' is converted to 'consistent' multiplicity and remains as a kind of internal exile or - in Badiou's strikin g ana­ logy - a (non-)member of the (non-)collectivity of sans-papiers whose absence from any official tally or electoral roll renders them publ icly invisible. However, this is not to deny that they may yet exert a considerable power of resistance and disruption through their standing as blatant exceptions or counter-e�amples to the much-touted rhetoric of equality, freedom, participant democracy, social inclusion and universal human rights that - at least m the present-day French and similar contexts - forms the constant refrain of governments and political parties from main­ stream left to mainstream right. 16 So there is a political as well as mathematical question involved when Badiou puts the case for a strictly · extensionalist approach that would take absolutely no stock of any attributes, properties or distinguishing (i.e. qualify­ ing) features that might otherwise - on the opposite (intension­ alist) account - play a crucial part in deciding issues of set-theoretical membership. All the more must this apply to the non-members or to those excluded from the count on whatever legal, ethnic, cultural or linguistic grounds since the main re.ason for their being so placed is their imputed lack of those (supposed) identifying features that are taken to mark out members in good. legal, ethnic, or cultural-linguistic standing. Thus, '[i]f the void is thematized, it must be according to the presentation of its errancy, and not in regard to some singularity, necessarily· full, which would distinguish it as one within a differentiating count' (p. 57). For Badiou, there is nothing so politically and ethically 65

BADIOU'S BEING AND EVENT

retrograde - indeed so potentially lethal - as a version of the count that looks to such 'singular' (though type-indicative) fea­ tures as a means. of sorting those that belong from those that belong outside or elsewhere. This is why he can say - at risk of being charged by analytic phi­ losophers with flouting the fact/value distinction or claiming to derive an 'ought' from an 'is' - that 'for the void to become localiz­ able at the level of presentation, and thus for a certain type of intra-situational assumption of being qua being to occur, a

dys­

function of the count is required, which results from an excess-of­ one' (p.

56). The obverse of this is of course the implication that

what is required - ethically or politically - in order to promote the interests of justice in an unjust society is a stronger, theoretically informed grasp of what is wrong with any thinking premised on ideas of identity or difference as intrinsic to the individuals or spe­ cific relationships concerned. In which case those interests could be further advanced through an increased understanding, suchas he provides in Being and Event, of the various set-theoretical con­ cepts (especially the opposed pairs of inclusion/belonging, part! member and inconsistent/consistent multiplicity) along with the relevant formal procedures whereby they might serve to demon­ strate the disparity between any given situation and what that situ­ ation holds in the way of unrecognized, unacknowledged or 'uncounted' parts. This implication Badiou takes to be already contained in Cantor's Theorem - the starting point of all set the­ ory - according to which the power set (Le. the set of all subsets) of any given set has a greater cardinality (i.e. numerical value) than the set itself. Thus the latter will include not only all its members counted one-by-one but also the various combinations (subsets) into which those members may be placed, thus yielding an excess of subsets over set that grows exponentially with the size of the original set right up to the multiple orders of infInity where of course it surpasses the utmost powers of any fInite enumerative reckoning. I ? Paradoxically enough it is the knowledge of this shortfall in the present capacity of thought to encompass such truths or to bring them within the scope of attainable knowledge that Badiou sees as driving the constant quest for more powerful, refIned and elaborate modes of set-theoretical investigation. Nor indeed should this claim seem so very paradoxical if one considers the copious evidence from every branch of the formal, 66

READING THE TEXT

natural and human sciences that new discoveries are most often sparked by some increasingly acute sense of presumptive anom­ aly. That is t6 say, they result from a grasp of possible or likely future development that canno t amount to knowledge - since it cannot so far be specified with adequate precision - yet the pros­ pect of which is logically entailed by certain gaps, defects or dis­ crepancies in the present state of understanding. Hence Badiou's claim, in Being and Event, that the history of advances in set the­ ory exemplifies this process to uniquely impressive and convinc­ ing effect since it shows how such 'advances can come about through the process whereby thinking transcends those obstacles that are constantly placed in its path through its residual attach­ ment to limiting conceptions of its own investigative scope. Among them - and most sharply limiting - is that idea of the relevant criteria for set-theoretical inclusion or exclusion which takes this to be crucially a matter of the possession or non-pos­ session of certain distinctive or qualifying features. On the con­ trary, the proven capacity of set-theoretical reasoning to overcome its various challenges or setbacks to date has resulted from its perfect indifference to any question of what makes up the multiples (and multiples of multiples) that fall within its formal domain. In short, 'the attribute "to-be-a-multiple" tran­ scends the particular mUltiples which are elements of a given multiple' (p.

65). Which is also to say - with fairly obvious rele­

vance to the social and political dimension of Badiou's thought ­ that any issue concerning the status of elements will have nothing to do with their intrinsic natures, real or imputed, but solely with their set-theoretical relation to the multiples that include or exclude them as members and the multiples or elements which they in tum include or exclude. At some risk of simplification one could say that this is the direct equivalent, in formal terms, of Badiou's strongly univer­ salist stance in matters of ethics and politics, and - by the same token - his firm rejection of those various strains of 'radical' difference-thinking (whether in epistemology, ethics, politics, post-colonial discourse or gender studies) that have lately cap­ tured the high ground in critical and cultural theory. 18 Indeed it is the main reason for his choice of mathematics, rather than language, as the basis on which to construct his critical ontology and also his account of those fourfold other subject-domains 67

BADIOU'S BEING AND EVENT

which can be conceptualized only in relation to that ontology even if their major occurrences belong to an order of 'events' that inherently eludes the grasp of any pre-existent ontological scheme. Where language - or the normative appeal to language enjoins us to accept what is presently sayable, describable, or representable as flxing the ultimate scope and limits of truth or intelligibility mathematics on the contrary allows - indeed requires - that we surpass those limits and conceive what may extend that scope beyond anything yet achieved or remotely envisaged. This it does through the presence of those various anomalies, conflicts or unresolved dilemmas that will always exist so long as mathematics remains a live and intrinsically a problem-solving activity of thOUght. But there is also something peculiar to the nature of mathematical (more specifically of set­ theoretical) thinking that reinforces the decisive advantage it enjoys over language as a source of ontological advances and a spur to new insights in those other subject-domains. This is its unique capacity for grasping the 'subtractive' dimension of being, that is, the 'axiom of the void set' according to which 'there exists a set that has no elements', one that is strictly 'unpre­ sentable' in terms of any given ontology or particular instance

of the dominant count-as-one. Moreover it is another basic axiom that the void (or nUll) set is necessarily a member - an integral or constitutive member - of every other set whatsoever. Thus it fig­ ures to Badiou's way of thinking - though also in set-theoretical discourse generally - as that which grounds the entire project (since all those other sets are built up successively by a process of recursive extrapolation from the null set) yet that which by very deflnition eludes the grasp of any defmite, positive or clearly intuitable order

of conceptualization.

Indeed, Badiou claims that this elusiveness can be seen to

complicate a good many of the statements put forth by the pro­ ponents of set theory, including the above-cited (pretty much canonical) formulation to the effect that 'there

exists a set which

has no elements' . For of course any notion of 'existence' here is one that needs qualifying to the point of near-withdrawal since it runs flat up against the seeming contradiction, familiar from Plato's

Parmenides,

involved in all forms of negative existential

statement. However, as we have seen, Badiou is enough of a Hegelian - and

ipso facto

enough of an anti-Kantian - to take 68

READING THE TEXT

this not as a disabling dilemma or a sign that thought is over­ stepping its appointed bounds but rather as a spur to thinking through and beyond what constitutes an impasse only on such a static conception of the scope and limits of reason. According to his alternative dialectical conception it is at just such points of seeming deadlock or aporia that thought typically turns that situation to advantage and achieves the leap to a terra incognita or whole new tract of hitherto unexplored but henceforth richly fertile terrain. Above all such advances have come about through the increasing centrality of that which Badiou designates the 'subtractive' dimension of mathematical thought, namely its encounter with the various crises periodically_induced by what­ ever is excluded from its zone of operations yet continues to exert an unsettling and hence a potentially transformative power. Such was the effect of those early developments in set theory that led

via

an ironic repetition of the paradoxes first aired in

Plato's Parmenides to a recognition of the null set as both the founding concept and, so to speak, the immanent driving force of all further advances, refinements and discoveries. It is at just this point, according to Badiou, that 'the subtractive character of being causes the intuitive distinction between elements and sets to break down' (p. 67). What we are thereby obliged to think ­ conceptually 'forced', as Badiou phrases it, deploying this term in the technically precise sense proposed by Cohen - is the abso­ lute and principled impossibility of drawing that intuitively self­ evident and yet, in this context, logically insupportable distinction. Which is also to say, in more positive terms, that what we are thereby enabled to conceive is the possibility that thinking can advance through a grasp of non-intuitive (even strongly counter­ intuitive) truths that are manifest precisely in those conflicts, inconsistencies, conceptual tensions, or failures of logical coher­ ence which signal the advent of a crisis in knowledge but also and solely on that condition - the prospect of new discovery. Badiou is well aware that such ideas can very easily be miscon­ strued or tendentiously deployed in the service of quasi-mystical or negative-theological claims. The former have their chief source in the ancient Greek precedent of Pythagorean numerological doctrine - itself a strong influence on Platonist metaphysics and epistemology - whereas the latter, as we have seen, fmd expres­ sion through Cantor as well as in the much-quoted statements 69

BADIOU'S BEING AND EVENT

of a good few other mathematicians and scientists. Badiou addresses this (to him) highly suspect tendency in one of the more cryptic paragraphs of Being and Event where he seeks to clarify the truly 'remarkable conclusion' arrived at by this stage: that 'it is because the one is not that the void is unique' (p. 69). Its uniqueness makes it a 'proper name' in the sense that, unlike all other set-theoretical entities, it can be picked out or specified ­ 'marked', in Badiou's terminology - only by an act of pure des­ ignation or ostension, rather than by some identifying list of its attributes, properties or status with regard to other sets (that is, all sets whatsoever) in which it is included as a member or in which it is included as a member. This gave rise, understandably, to a certain mystique of ineffability that brought it within the rarefied ambit of negative-theological thought. If those early mathematicians 'had to shelter their own audacity behind the character of a forgotten language' then this was in one sense an irrational regression to theological or mystical ideas that had no legitimate place in mathematical thought. Nevertheless their sense of the numinous and feelings of awe in this regard were justi­ fied at least insofar as they recognized, however 'dully', the power of thought when it learns to proceed through the via negativa of a quest for truth that renounces any positive grounding in the sup­ posed warrant of sensory-perceptual, commonsense-intuitive, or even (what passes for) conceptual or a priori self-evidence. Badiou is a resolutely secular thinker though one who takes pretty much for granted the present-day obsolescence of the monotheistic religions - at least for the purposes of serious (truth-based or truth-oriented) philosophical, moral and cultural debate - rather than stoking the fires by adopting a high-profile atheist stance. All the same, he does fmd something exemplary for his own purposes in the figure of a religious 'militant' such as St. Paul, one whose crusading zeal (whatever its far from benign historical and cultural consequences) marked not only an irrup­ tive event of the first magnitude but also - accordant with Badiou's staunchly universalist precepts - a belief in the charac­ ter of truth as transcending all merely national (e.g. Hebrew or Greek) ideas of cultural-linguistic belonging. 19 In Pascal likewise, as we shall see, he discovers the paradigm case of an 'anti-philosopher' whose commitment to the precept that reason should always give way before the dictates of religious faith 70

READING THE TEXT

would appear to be drastically at odds with his own thinking (pp. 21 2-22). Yet, according to Badiou, Pascal's work - in math­ ematics and logic but also in its ethical and even its existential­ theological aspect - is much better viewed in light of its power to challenge, provoke and ultimately vindicate philosophy's universalist claims. For it is here, in the kinship between mathe­ matics and a supra-cultural conception of political justice, that we should seek to make sense of Badiou's above-cited statement to the effect that any rigorous reflection on the void as it figures in the progress of mathematical thought will reveal such thought as 'liminal to language' in ways that are analogous (though cru­ cially not identical) to the pseudo-logic of negative theology. That the latter is indeed a pseudo-logic and shown up as such by contrast with the history of demonstrable gains or advances in the field of mathematical enquiry is explained, for Badiou, by the fact that negative theology. Always involves a covert appeal to that which supposedly (if indescribably) sustains or underlies the various predicates that fail to capture its essence. Thus it represents a falling-back into that 'presentifying' mode of thought - that deep-laid set of ontological assumptions premised on the positive rather than the negative or subtractive nature of being - which had for so long acted as an obstacle to thought in the realm of mathematics and the formal sciences. What is required in order to remove that obstacle is a strictly axiomatic truth-procedure which 'cannot propose anything in particular', and can therefore 'neither be a matter of the one, which is not, nor of the composed mUltiple, which is never anything but a result of the count, an effect of structure' (p. 66).

4. Plenum and void: why Aristotle hated a vacuum It is at this stage in the unfolding of his argument that Badiou turns to Aristotle as one of those thinkers with whom he is pro­ foundly at odds in relation to some basic metaphysical and onto­ logical commitments but whose thinking he typically treats as a challenge to be met so far as possible on shared argumentative ground rather than dismissed out of hand. The main point at issue is Aristotle's denial that there could possibly exist any such thing as a void or vacuum in nature.20 This was a doctrine - on essentially metaphysical grounds - that continued to enjoy the 71

BADIOU'S BEING AND EVENT

authority of something like holy writ throughout the medieval and renaissance periods in European thought, and which began to lose ground only with the rise of early modem science and its rejection of scholastic dogma in favour of experimental methods that involved (in this case) the construction of increasingly effec­ tive vacuum pumps. In recent years this received view has itself been subject to challenge from 'strong' sociologists of knowledge who reject the idea of scientific progress as a piece of self-serving mythology and who view the issue of experimental physics versus Aristotelian doctrine - or (in the classic encounter of this kind) of the seventeenth-century physicist Robert Boyle versus the philosopher Thomas Hobbes - as a case of two conflicting ideo­ logies, rather than a conflict between scientific method and a legacy of outworn metaphysical baggage.21 Badiou's aim i s not so much to adjudicate in this quarrel (which has surely been set­ tled on physical-scientific terms) but to shift the whole ground of debate and define what precisely Aristotle meant when he wrote of the non-existence of the void as a matter of metaphysi­ cally ordained or a priori demonstrable truth. 'For the Greek', Badiou writes, 'the void is not an experimental difference but rather an ontological category, a supposition relative to what naturally proliferates as figures of being' (p. 71). That is to say, no merely 'artificial' creation of a void, such as Boyle was much later to achieve albeit, on his own admission, to a very imper­ fect or partial degree owing to the drastic limitations of available technology - could possibly count as refuting the Aristotelian doctrine, based as that doctrine was on a quite different order of purely ex hypothesi speculative reasoning that brooked no such empirical counter-evidence. Badiou's point here is not, of course, to uphold Aristotle's claim against the massive, strictly unignorable weight of accu­ mulated physical-scientific evidence nor indeed - as should be obvious by now to defend it as a matter of prior metaphysical or ontological commitment. After all, nothing could be further from Badiou's absolute and principled insistence on the essen­ tially subtractive nature of ontology - the primacy of that which eludes specification in positive ('presentifying') terms - than the Aristotelian doctrine that 'nature abhors a vacuum', or that talk of the void mu st involve either scientific absurdity or logical self-contradiction. However this is to commit something like a -

-

72

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basic category-mistake since, in Aristotle's thinking as Badiou understands it, 'the

artificial

[i.e. experimental] production of

a void is not an adequate response to the question of whether nature allows, according to its own opening forth, "a place where nothing is" to occur, because such is the Aristotelian definition of the void' (p.

7 1). He is drawing quite explicitly on Heidegger's

reading of Aristotle here, and this is indeed - so far as I recall - the most markedly Heideggerian passage of commentary anywhere in Badiou's work. All the same it is not so purely echt-Heideggerian as to conjure

a depth-hermeneutic · realm of metaphysically

unencumbered Being that would constitute both the primordial, long-forgotten source and the concealed onto19gical ground of all those merely ontical beings that science has taken for its object-domain. On the contrary, what most engages · Badiou's interest is Aristotle's axiomatic mode of reasoning from first prin­ ciples which, although sharply opposed to his own, can none the less be seen to have certain implications that are strikingly germane to much of what Badiou has to say on his three main topics of the void, infmity and the ev�nt. Thus if Aristotle's theses have not been refuted on their own metaphysical ground by subsequent advances in physical-scientific understanding then this is not because, as Heidegger would have it, the physical sciences (like philosophy) have long been mort­ gaged to a technocratic will-to-power and

an epochal oblivion him in

of Being but rather because Aristotle's reasoning places

the company of those - Badiou included - 'for whom the void is in truth the mUne of being, and so can neither be cast into doubt nor established via the effects of an experiment' (p. 72). That is, any refutation of Aristotle's doctrine at the level of ontological enquiry here in question - as distinct from the physical-scientific level where its claims can no longer stand up - will have to go by way of a critical engagement with the underlying logic or onto­ logie of Aristotelian physics. It will therefore not be required to take sides on the kind of issue nowadays engaged between real­ ists who suppose Boyle to have got it right - or at least to have been very much on the right scientific track - in asserting the pos­ sibility of a vacuum and those on the opposite (strong-sociological or cultural-constructivist) wing who consider Boyle's claim and Hobbes's denial of it to be strictly on a par as regards their truth­ content since each was the product of a certain ideological or 73

BADIOU'S BEING AND EVENT

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

ipso facto

to invite all the massive conceptual, metaphysical and ontolo­ gical problems that come in its train. Chief among them are the problems of spatio-temporal

indifference (since the void admits un-measure (since it likewise

of no distinctions in this regard),

prevents any meaningful comparison between different dimen­ sions, velocities, or other such quantitative attributes), and above all

infinity

(by reason of its forcing thought to confront that

intrinsically disturbing since - for Aristotle and even for Cantor ­ inherently excessive or paradox-inducing idea). This is why, as Badiou cautions, we should not be 'led astray' by physics 'in the modern sense', that is, by the otherwise spectacular self-evidence of past and presently continuing progress in the physical sciences. What Aristotle is asking us to think is something quite different: namely, that 'every reference to the void produces an excess over the count-as-one, an irruption of inconsistency, which propa­ gates - metaphysically - within the situation at infinite speed' (p. 75) . In which case - he concludes - 'the void is incompatible with the slow order in which every situation re-ensures, in their place, the multiples that it presents'. It can only be conceived in terms of a potentially subversive or destabilizing threat to the entire onto-metaphysico-epistemological structure of thought that constitutes Aristotelian philosophy in its various specific regions of enquiry. When Aristotle says that 'the void bears no ratio to the full, such that neither does movement [in the void]', or again that 'there is no ratio in which the void is exceeded by bodies, just as there is no ratio between the nothing and num­ ber', Badiou would scarcely take issue with the content of these 74

READING THE TEXT

statements - bearing out as they do his own central theses with regard to the subtractive nature of ontology - but only with Aristotle's belief that they constitute a case by reductio ad absur­ dum against the possibility of a void. Thus he fully endorses the reasoning behind this Aristotelian doctrine - 'to my mind, the ensemble of these remarks is entirely coherent' (p. 74) - but takes it to demonstrate just the opposite of Aristotle's intended point, namely that the impossibility in question is that of adequately conceiving the void in terms of a 'presentifying' metaphysics and ontology, rather than a concep­ tual impossibility tout court. That we can grasp this distinction now where Aristotle couldn't, despite being carried so far towards it by the rigour and consistency of his own logic, is mainly the result of our coming at the issue from a standpoint informed by set-theoretical concepts and techniques, above all with regard to infInity and transfInite mathematics. Moreover it enables uS to read the works of earlier thinkers - among them (most impor­ tantly for Badiou) Plato, Aristotle, Leibniz, Spinoza, Rousseau, Hegel and Heidegger - in such a way as to locate the various symptomatic tensions, conflicts and blind-spots of unexamined presupposition that mark their discourse where it touches on themes that involve some dealing with those problematic topoi and is thus forced up against its own conceptual limits. I should make it clear - lest anything I have said give rise to the contrary impression - that Badiou is as far as possible from implying-1hat his approach to these thinkers could fairly be described as a 'psychoanalysis of philosophy'. Indeed it is one of his leading claims that philosophy should know its proper place not only in the sense of accepting a strictly ancillary role vis-a.-vis the ground-breaking ontological work of mathematics but also in the more positive sense of maintaining its crucial measure of autonomy vis-a.-vis those various enabling 'conditions' which might otherwise compromise its critical independence or intel­ lectual integrity. All the same, as will emerge more explicitly in Part VIII of Being and Event when he turns to Freudian-Lacanian themes, Badiou's practice of symptomatic reading is one that has much in common with the kinds of critical-philosophic dis­ course that resulted from the structuralist-inspired rapproche­ ment between Marxism and psychoanalysis from the mid-1960s on. That rapprochement came about very largely through the 75

BADIOU'S BEING AND EVENT

intensive theoretical labours of Louis Althusser, and it is thus worth noting that Badiou - despite his express opposition to the linguistic tum in its structuralist as well as its Wittgensteinian and other forms - is among the very few major present-day think­ ers who have retained at.1east a qualified allegiance to the Althus­ serian project.22 Indeed this is one striking example of what Badiou means by the fidelity to certain truth-procedures (whether in mathematics, science, politics or art) where the upshot - the prospect of their being carried through to a successful conclu­ sion - may be highly uncertain, even subject at times to severe or calamitous setbacks, yet where the stakes are sufficiently high to warrant that kind of long-term commitment. This is why he is so drawn to thinkers, like St. Paul and Pascal, with whom he would appear to have little in common po,litically, ethically or philosophically. What they share is the distinctive coupling of a well-nigh existentialist conception of truth as a matter of authen­ tic individual dedication to the project in hand with a strongly universalist claim to the effect that, should the project at last be vindicated, then its truth or validity conditions will apply across every kind of social, cultural, political, ethnic or other such restrictive boundary. This conjunction of seemingly opposite doctrines will appear less strained or downright contradictory if one reflects on the way that mathematical discoveries - such as Andrew Wiles's celebrated recent proof of Fermat's Last Theorem after more than three centuries of intensive effort by numerous dedicated individuals - may well involve personal commitment on a quite heroic scale of mental and physical endurance and yet, once established, hold good without regard to any such (now) extra­ neous facts about their psycho-biographical genesis.23 Not that Badiou would subscribe unreservedly to the distinction - com­ monly advanced or assumed among analytic philosophers of science - between 'context of discovery' and 'context of justifi­ cation', or the various circumstances under which some discov­ ery may have come about and the various standards of empirical, logical, inductive, predictive or causal-explanatory justification that led to its widespread acceptance in the scientific community.24 His approach differs from theirs mainly in the much greater weight he attaches to questions of historical development that is, the genealogy of set-theoretical concepts, techniques and 76

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proof-procedures - and also, consistent with that, his greater emphasis on how such discoveries can be shown to have occurred through certain highly specific procedures of thought on the part of certain mathematicians confronted with certain, likewise specific problems or obstacles to progress. On the one hand this is why he can cross disciplinary boundaries so as to stress the closely analogous relationship between cutting-edge work in mathematics, requiring as it does the highest degree of dedication to a given truth-procedure, and those other chief fields of human endeavour where the demands of rigorous and consequent thought go along with the demand for unswerving fidelity or single-minded commitment. On the other hand, it is why he comes out in such firm opposition to the idea that truth in any of those subject-areas might properly be thought of as 'relative to' or 'constructed by' the various languages, discourses, cultural communities, more-or-Iess specialized (e.g. mathematical) 'forms of life', and so forth, which supposedly constitute the ultimate horizon of intelligibility for those who inhabit them. We are now better placed to take stock of his comments con­ cerning Aristotelian physics (or metaphysics) and the mistake of supposing that Aristotle's denial of the void has been - or could in principle be - refuted by modern science. This is not to place Badiou in the company of those above-mentioned 'strong' socio­ logists who would insist that the principle of 'parity of esteem' be carried so far that we refrain from judgement as regards the (notional) truth of the issue between Boyle and Hobbes and instead seek an explanation for both of their conflicting views in the particular socio-cultural-political context wherein those views took rise. Rather - and contrary to any such flat-out rela­ tivist approach - what Badiou wants us to see is that when Aristotle asserted the non-existence of the void with such (as it turned out) misplaced confidence and vigour he did so for rea­ sons that had nothing to do with empirical or experimental proof and everything to do with his clear understanding that its exis­ tence, or mere possibility, would utterly wreck his entire concep­ tion of the cosmic and natural order of things. That Aristotle got it wrong in physical-scientific terms Badiou would not for one moment deny. That his denial of the void on metaphysical grounds was also an error - and one with negative repercussions for the history of thought right down to Hobbes and beyond - is 77

BADIOU'S BEING AND EVENT

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

Being and Event

has taken us from Plato to Aristotle, that is, from Badiou's medi­ tations on a thinker with whom he feels a strong (if qualified) intellectual kinship to his diagnostic reading of a thinker with whose central claims he is profoundly at odds yet whose reason­ ing he finds both consequent and all the more revealing for its blind-spots of questionable presupposition. Having 'placed' his thought in relation to theirs and having used the resultant three­ sided comparison as a means to draw out some of Badiou's most important and distinctive themes we can now proceed to engage 78

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

Badiou's strong universalist stance in ethics and politics goes sharply against the present-day emphasis, among cultural and critical theorists, on the need to show maximum possible respect for differences of creed, tradition, cultural background, value priority or gender orientation. Are you . persuaded by his arguments in support of this position What exactly is the relationship between mathematics, ontology and political thought as Badiou conceives theIii'f

PART II. BEING: EXCESS, STATE OF THE SITUATION, ONE/MULTI PLE, WHOLE/PARTS OR e /cJ'

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

In the four Meditations that make up Part II Badiou sets out some basic set-theoretical concepts and procedures before explaining their wider (extra-mathematical) pertinence and then, typically, engaging with a past thinker - in this case Spinoza - whose con­ trasting claims can be seen to throw his own into sharper relief. What none the less gives this part its strong sense of a developing and tightly structured argument is Badiou's constant circling back to the relationship between being and event, or the domain of ontology (with its basis in mathematics) and the domain of events (taken as denoting whatever exceeds the bounds of any pre­ existent ontology and establishes new terms for the conduct of future investigation). I shall therefore focus on the salient themes ­ in particular the set-theoretical concepts along with their emer­ gent political implications - and also offer some background commentary on Badiou's project in the wider context of present­ day philosophical (including Anglo-American analytic) thought. Meditation Seven is entitled 'The Point of Excess' and takes us directly to the heart of Badiou's mathematically based concep­ tion of ontology as applied to issues · in the formal, physical, social and human sciences. In other words it carries forward the discussion that began with his intensely dialectical staging of the difference between Platonist and Aristotelian ontologies 79

BADIOU'S BEING AND EVENT

and that was all along rehearsed - though now more explicitly in relation to developments in post-Cantorian set theory. l Most fundamental here are the relations of belonging (symbolized E ) and of inclusion (symbolized c), the former indicating that 'a multiple is counted as an element in the presentation of another multiple', whereas the latter signifies 'that a multiple is a sub-multiple of another multiple' (p. 8 1). That is to say, as con­ cerns the relation of belonging it is here a matter of some multi­ ple (X that forms an element of some other multiple � such that (X is 'presented' by the count-as-one or the existing 'situation' as prescribed or dictated by �. Thus E is the 'unique foundational sign of set theory' since it establishes the possibility of all those relations (among them inconsistent, anomalous, contradictory or suchlike problematic relations) that constitute both an obsta­ cle to thought and the means by which thinking typically achieves its most decisive stages of advance. In the case of inclusion, con­ versely, multiple (X is taken to include all the subsets (i.e. constit­ uent multiples) of � and � is thus defined as itself a subset of (X and yet - as Cantor showed with respect to the different 'sizes' of infinity - a subset that must be thought of as equinumerous with (X or as capable of having its mem bers paired off one-for-one with the members of (X. 2 For clarity's sake Badiou uses the term 'element' to signify belonging and 'subset' to signify inclusion, although these should not be taken to mark any further, that is, ontological distinction. His principal concern -here and through­ out Being and Event - is to show how such seemingly abstract considerations in the realm of pure mathematics can have a direct (not merely suggestive, obl ique or analogical) bearing on matters outside that realm. So it is with the dualism of belonging and inclusion which, Badiou says, 'directs, step by step, the entire thought of quantity and . . . the great orientations of thought, prescribed by being itself' (p. 82). And again, more specifically, '[i]n one case (the case E ), the multiple falls under the count-as-one which is the other multiple. In the other case (the case c), every element pre­ sented by the first multiple is also presented by the second' (p. 82). If the former (in Badiou's clearly specified terminology) 'pres­ ents' certain elements as 'belonging' to a given 'situation' while others are thereby excluded from it, the latter should be thought of as 'representing' all the subsets included in a given 'state of 80

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the situation'., that is to say, as involving no such selective mem­ bership conditions. It is precisely through the constant possibil­ ity of a rift, a mismatch or breakdown of structural reciprocity between these two basic conditions of belonging and inclusion that there also emerges the potential for significant change - for revolutions of thought or theoretically informed practice - in the various spheres to which Badiou applies his dialectic of being and event. Above all, as we shall see shortly, it has to do with that breakdown in the order of commonsense-intuitive mathe­ matical grasp signalled by the discovery of the power-set axiom, namely that 'if a set a exists (is presented) then there also exists the set of all its subsets', a set whose numerical value must clearly exceed that of set a by an order of magnitude that increases exponentially with the size of set a itself and which generates multiple orders of infinity as soon as mathematics goes trans­ finite in the wake of Cantor's revolution. This means that the restrictive conditions on belonging which defmed the membership of a must now be lifted or redefined so as to acknowledge the existence of � - the power-set of a - whose numerical value far exceeds anything admissible under those same restrictive conditions. What the power-set axiom requires us to think is the effect of that strictly ubiquitous 'point of excess' that will always signal the existence of a gap between belonging and inclusion, the situation and the state of the situation, or consis­ tent multiplicity (as presented by the dominant count-as-one) and inconsistent multiplicity (as re-presented by all those supposedly non-belonging but none the less included subsets). This can also be phrased in terms of the rift between structure and meta­ structure, or again (in Badiou's chosen terminology) between the elements of a set and its numerically ' l arger' multiple of subsets. That those quote-marks are required around the term 'larger' is one consequence of Cantor's showing that seemingly different 'sizes' of infmite set - like 'all the integers' and 'all the even integers' - could be counted off one-for-one against each other without limit and could not, therefore, be thought of as larger or smaller in any such straightforward, self-evident or intuitive terms. However, it follows from the power-set axiom that in a different, mathematically definable sense the subsets of any given multiple will be larger (numerically greater) than the multiple itself and that when this axiom is extended to the 81

BADIOU'S BEING AND EVENT

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

-

82

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with some pre-existent state of knowledge or conceptual scheme) still there is a clear sense in which Badiou's whole project rests on ontological foundations and indeed requires them just in order to make that same distinction. More precisely, what he sees as philosophy's proper task is not that of making ontolo­ gical discoveries or exploring new ontological regions on its own account - since this is a role best left to the mathematicians but rather that of pursuing a 'meta-ontological' enquiry that expounds, clarifies and draws out the consequences (some of them decidedly extra-mathematical) of any results thus obtained. One can therefore see why he lays such stress on the claim, contra Wittgenstein and Heidegger, that 'mathematics thinks' insofar as it involves a creative, inventive and truth-disclosing activity of thought that cannot be reduced either (following Wittgenstein) to a mere assemblage of vacuous since purely self-confirming logical tautologies nor again (following Heidegger) to a mere expression of the techno-scientific-metaphysical will-to-power over nature and humankind alike.4 Indeed one fascinating aspect of Badiou's work is the way he pursues a selective exegetical path among the many thinkers - from Parmenides, Plato and Aristotle, via Descartes and Pascal, to Frege, Russell, Heidegger, Wittgen­ stein and others - against whose projects he measures his own with varying degrees of perceived affinity or (very often in the same thinker) perceived differences of view. In each case his basic argument is that we can and should read these thinkers differently in light of the epochal advance brought about by Cantor's inaugural discoveries in set theory and their subsequent development by mathematicians and some (very few) academic or professional philosophers. Most crucial here is the power-set axiom since it establishes the principle - the point of departure not only for Badiou's mathe­ matically based critical ontology but also for his thinking on mat­ ters of political, scientific and ethical import - that no instance of the count-as-one, whatever its claim to universal inclusivity, could ever contain (or purport to represent) those endlessly proliferat­ ing subsets of multiples revealed by a grasp of that axiom. Badiou follows his 'technical' rendition of the case as concerns set theory and its formal structure with a sentence that effectively re-states that case in a register whose normative modality seems to waver between laying ciown the necessary (non-negotiable) terms and 83

BADIOU'S BEING AND EVENT

conditions for any adequate address to these matters and pre­ senting something like an ethical injunction to observe, respect, or act upon those terms and conditions. 'Once this is admitted', he writes, 'one is required to think the gap between simple presen­ tation and this species of re-presentation which is the count-as­ one of subsets' (p. 85). That requirement clearly derives its imperative force from the various formal demonstrations, from Cantor down, of how set theory achieves its most signal advances by a procedure of thinking through-and-beyond the various obstacles - mostly of a commonsense-intuitive kind, like that initially provoked by the power-set axiom - which have risen against it. To this extent it is normative in the sense that it pre­ scribes what properly counts as a correct or valid application of the formal procedure concerned. That is to say, it belongs to the domain of mathematical truth where the issue of fidelity has nothing to do with ethics or the moral virtues and everything to do with domain-specific requirements such as consistency, rigour, demonstrative force, logical explicitness and so forth. However it also belongs to that other dimension where the term 'fidelity' does have a strong ethical toning and where issues of truth cannot be held entirely apart from matters of truthfulness or intellectual virtue. Not that I would wish to place Badiou in the company of those who have argued for a virtue-based epistemology with its roots in the tradition of ethical thought descending from Aristotle and taken up lately by philosophers in quest of some alternative to the current range of often sharply conflicting (e.g. deontologi­ cal versus consequentialist) views.s In brief, this is an approach that envisages no possible solution to the 'problem of knowledge' in its various forms except by instancing the sorts of jointly moral and cognitive-investigative qualities, dispositions or intel­ lectual traits that best, most reliably conduce to the advance­ ment of human understanding. These would typically range from epistemic virtues like perceptual acuity or sensory-intuitive 'feel' for the physical items or properties under investigation to epistemologically salient aspects of intellectual character such as dedication, perseverance, open-mindedness, respect for the evidence, willingness to test even one's most cherished or firmly held beliefs against that evidence, and a well-developed capacity for self-criticism. That Badiou has the utmost regard for those 84

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virtues is clear enough from his writing about certain exemplary figures - such as the mathematicians Jean Cavailles and Albert Lautman, both of whom sacrificed their lives as members of the French Resistance - whose actions and work can be seen to have manifested them in the highest degree.6 All the same he is very careful not to confuse truth with truthfulness, or issues of the kind: 'is statement x true ' or justified according to the strictest standards of evidence or of formal (e.g., set-theoretical) proce­ dure?' with issues of the kind: 'has statement x been arrived at by someone (or some community of like-Diinded enquirers) possessed of all the relevant, truth-conducive or knowledge-promoting virtues?'. That these two conditions may often be satisfied by the same statement - since the latter is defined as one that should predictably give rise to the former - is everywhere implicit in Badiou's account of those particular discoveries or stages of advance, in set theory and other domains, that have typically occurred through a combination of rigorous thinking with the courage to defend them against the weight of established doc­ trine or commonsense-intuitive belief. Still they cannot be run together or this distinction effectively collapsed - as advised by some proponents of a virtue-based epistemology - without consequently opening the way to all manner of sceptical, social­ constructivist or cultural-relativist ideas. For it is no great distance from this invocation of the intellectual virtues (however carefully or strongly specified) to the idea that any such appeal is reliant on the existence of a certain socially accepted or culture-specific conception of just what constitutes a virtuous epistemic practice. The next pair of Meditations, Eight and Nine, bear the somewhat forbidding titles 'The State, or Metastructure, and the , Typology of Being (normality, singularity, excrescence) and 'The State of the Historical-Social Situation' . All the same read­ ers should not be tempted to skip since there is, as those titles suggest, substantive ethical and socio-political as well as 'purely' philosophic content in this formal demonstration of his central claim as concerns set theory and its implications for the various domains of applied ontological enquiry. Indeed they should if possible be read at a sitting since they run to just and between them offer the clearest account in

1 8 pages in all Being and Event

of how the three main dimensions of Badiou's work (crudely put: the mathematics, the ontology and the politics) should rather 85

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

state of the situation to signify

'that by means

of which the structure of a situation - of any structured presen­ tation whatsoever - is counted as one, which is to say the one of the one-effect itself' (p. 95). This affinity will be explained, he promises, in Meditation Nine when the focus switches more explicitly to politics and when these so far 'metaphoric' connec­ tions or suggestive cross-domain analogies acquire a more detailed working-out in conceptual-analytic terms. The link is accomplished chiefly through that same technical coinage, 'state of the situation', which Badiou variously defmes as 'count-of-the-count', 'metastructure', or 'that by means of which the structure of a situation is, in turn, counted as one'. His point is that this involves a 'second count', a further operation - of the kind that has its place

in all formal, logical or set-theoretical sys­

tems - whereby the frrst count is subject to a duplicate reckoning 86

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so as to confIrm its consistency and comprehensive grasp, or so as to ensure that nothing has gone uncounted by the prior opera­ tion. To this extent, he claims, 'concrete analysis converges with the philosophical theme' since in both cases the thesis can be verifled, whether through a process of formal (mathemati­ cal or logical) analysis, a project of applied natural-scientifIc research, or a study of social and political structures along with their inherently problematic modes of self-legitimation. These latter can always be shown to involve the duplication of a fIrst­ order count that establishes the terins of belonging or member­ ship by a second-order count - a re-presentation - which purports to secure the former's status as a valid (since all-encompassing and non-exclusionary) c,ompte rendu but which in fact, upon closer inspection, turns out to pose a strictly inescapable challenge to that very claim. Thus 'the structure of the count is reduplicated in order to verify itself, to vouch that its effects, for the entire duration of its exercise, are complete, and to unceasingly bring the one into being within the un-encounterable danger of the void' (p. 94). Such - one may fairly deduce - is the perceived or the generally dominant effect of this second-order count during periods of relative stability in various domains such as Kuhnian 'normal' (as opposed to 'revolutionary') science, or mathematics when it settles to working through the implications of some pre­ vious major advance, or social existence when subject to no urgent or strongly disruptive forces of political unrest. 8 It is also, in set-theoretical terms, what ensures the seeming predominance of 'consistent' over 'inconsistent' mUltiplicity, or of that which submits to the count-as-one over that which eludes or exceeds its calculative grasp. By this means it seeks to guarantee 'that there is no possibility of that disaster of presentation ever occurring which would be the presentational occurrence, in torsion, of the structure's own void' (p. 94). However, as should be evident from what has gone before, such stability - or the appearance thereof - is maintained only through the constant suppression, disavowal, or holding at bay of those anomalous instances that would otherwise mark the irruption of inconsistent multiplicity. For this shows (despite Russell's purported 'solution' to the set-theoretical paradoxes) how precarious is any structure that resorts to some higher level, meta-structural concept

in

order to legitimize its claims or any 87

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

2. State, subject, representation So when we reach the end of Meditation Eight Badiou has placed before us a developed terminology and a set of clearly marked conceptual distinctions that promise a means of accomplishing the passage - the impossible passage, as some would have it from issues within mathematics and the formal sciences to issues in the socio-political domain. He has done so, to repeat, by pur­ suing the logic of those set-theoretical terms and operations that bring out the constant disruptive or destabilizing effect of incon­ sistent upon consistent multiplicity, of that which is included in 88

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any given multiple upon that which is taken as bel ongi ng to it, all those uncounted parts of a certain situation upon the

and of

count-as-one that purports to represent its every last element. Thus, despite his above-mentioned caveat regarding its 'meta­ phoric' character, Badiou's usage of the term 'state' at this point of his exposition - in particular his claim (p.

101) that 'in ontology,

the state's "anti-void" functions are not guaranteed ' - is sure to evoke its social-po litical as well as its up-to-now pre dominantly

technical, i.e., set-theoretical sense. So likewise with Badiou's sum­ the close of Meditation Eight, that ­

mary remark, again towards

according to the terms and conditions of membership laid down by an existing state -

'inclusion must not arise onthe basis of any other principle of counting than that of belonging' (p. 101). While this still has primary reference to certain regulative precepts in the discourse of mathematics (in Kuhnian terms, those which charac­ terize periods of 'normal', problem-solving or non-revolutionary work) it can scarcely be read without evo king a whole range of analogous political situations. Among them - most emphatically is that of the displaced, immigrant, socially excluded and disenfran­ chised minorities whose cause Badiou has taken up through his involvement with various extra-parliamentary activist groUps.9 These themes

are developed more fully in Meditation Nine,

where in effect (althOUgh without declaring as much) he withdraws

the cautionary note and allows that these points of cross-reference between mathematics and politic s are indeed something more than merely metaphorical. Its purpose is to specify more exactly ­ by way of adequate conceptUalization rather than suggestive analogy - the order of relationship involved. As so often it is Aristotle who provides Badiou with his opening example of a thinker.whose basic orientation - whose entire apparatus of meta­

physical, ontological, ethical, political and scientific concepts stands squarely opp osed to his own (since in various ways committed to denying the existence or possibility of the void) yet for just that reason throws the relevant issues into sharp relief. Badiou makes that case with respect to what he sees, along with other recent commentato rs, as the close afImity between Aristotle and Marx in certain crucial respects. IO Chief among them is

Aristotle's clear recognition that any dealing the state might have with those under its jurisdiction is not a matter of relating to individuals, to persons, or even to 'subjects' in a sense of that 89

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term connoting the possession of particular traits beyond their generic (structural) placement as 'subject to' the power and authority of the state. 1 1 Rather - at its most fundamental - it has to be conceived in terms of precisely such an undifferentiating structural relationship maintained across and despite all distinc­ tions between one and another individual. This emphasis on the generic as opposed to the specific (i.e. experiential or existential) aspects of the relation - or quasi-rela­ tion - between 'subject' and state is one that Badiou has taken from, or developed through critical engagement with, a wide range of philosophical precursors from Plato and Aristotle to Pascal, Spinoza, Rousseau, Hegel, Marx, Husser! and Heidegger. More directly, it shows his continued allegiance to certain aspects of Althusserian 'structuralist' Marxism, in particular Althusser's conception of the subject as 'interpellated' by the ruling ideo­ logy or as always, inescapably caught up in those structures of 'imaginary' misrecognition that grant it the illusory sense of individual subjecthood. 1 2 Also, as we shall see later on, it reflects his likewise continued commitment to the central claims of Lacanian psychoanalysis. That is, Badiou subscribes to Lacan's cardinal and again broadly structuralist idea that the ego as sup­ posed locus of identity or conscious, reflective selfhood is in fact a mere epiphenomenon of discourse, a 'plaything' of the uncon­ scious whose ubiquitous workings (like those of ideology in Althusser's account) are there to be glimpsed - but never truly or fully comprehended - in the verbal exchange between analyst and analysand.13 However, what is most decisive here is the example of mathematics and its demands upon the conduct of disciplined enquiry both within and beyond the formal sciences. Above all it is the set-theoretical requirement that thought should be concerned with universal rather than particular, struc­ tural or combinatorial rather than discrete, and - in the techni­ cal sense of these terms current among logicians and philosophers of language - an extensional rather than intensional approach to issues of sense and reference. On this account the operative sense of a term is fixed entirely by the range of those objects, whether physical or abstract, to which it applies or extends and not by any­ thing specific about those objects (their distinctive properties, qual­ ities or attributes) that marks them out as rightfully falling under the term in question. That is to say, an extensionalist conception of 90

READING THE TEXT

sets and their membership conditions is one that rigorously excludes any thought of whatever might otherwise be taken to dis­ tinguish potential or candidate members, and thus to determine which items qualify for inclusion. There has been much debate among mathematicians as regards these rival approaches. Thus the advocates of extensionalism mostly espouse a realist principle according to which set-theoretic statements or theorems have their truth-value fIxed by the way things stand in mathematical reality, while the advocates of intensionalism more often incline towards an anti-realist position whereby truth becomes a matter of epistemic warrant and there is no making sense of the claim that there exist such objective truth-makers beyond our furthest powers of proof or ascertain­ ment. 14 Badiou adopts an extensionalist, that is, a strictly non­ differentiating ontology of sets, subsets, elements, parts, members and so forth, which leaves no room for any kind of qualitative distinction and which therefore conceives their various orders of relationship in purely numerical terms. It is on this basis that he is able to mount what most analytic philosophers would think an exorbitantly far-fetched claim, that is, to derive from certain axioms and theorems of set theory not only a generalized social ontology but also a conception of political justice grounded in the clear-cut distinction between two set-theoretical concepts, namely those of belonging and inclusion. Whence, as I have said, his mathematically derived and ethico-politically strengthened conviction that these interests can best be served - or the claims of justice best advanced - only by a radically egalitarian and universalist outlook opposed to any form of identity politics, or any notion of difference (whether ethnic, national, gender-based, or cultural-linguistic) as having a signifIcant role to play in such matters. Here it is worth noting a certain calculated ambiguity about Badiou's usage of those two key terms 'inclusion' and 'belong­ ing'. On the one hand it is a thesis basic to his political-activist project in the simplest terms, that of 'counting those who aren't counted' - that the multiplicity of elements or parts included within a given multiple may far exceed the multiplicity of mem­ bers taken as 'properly' belonging to it or accorded the rights (along with the concomitant responsibilities) of full participant status. In which case, it would seem, this excess of inclusion over -

91

BADIOU'S BEING AND EVENT

belonging is the axiom of set theory most strongly conducive to the interests of a radical-emancipatory politics that would push so far as po ssible towards ensuring that the status of belonging was extended to all those included - though as yet not acknowl­ edged to enjoy such status - within some given so cio-political situation. On the other hand it is clear from Badiou's formula­ tions as cited above that inclusion is also the p rinciple of reck­ oning which the State applies in its false claim to treat everyo ne on equal terms, that is, in its claim to respect each and every person who figures in the dominant, supposedly comprehensive count-as - o ne - whatever their rank, class or socio-cultural p osi­ tion - as equally 'belonging ' to the all-inclusive, democratically constituted body politic . Thus '[t]he "voter", for example, is not the subj ect John Doe, it is rather the part that the separated structure of the State re-presents, according to its own one; that is, it is the set whose sole element is John Doe and not the mul­ tiple whose immediate-one is "John Doe'" (p. 1 07). There is a certain tension in his thinking here - even a sense of conceptual strain - between 'inclusion' as that which offers a merely formal or abstract (hence delusive) notion of equality and as that which affords a powerful means to conceptualize and thereby think a way beyond the drastic shortfall in every hitherto existent version of liberal or social democracy. Nor is this surprising, it might be said, given the notorious adaptability of mathematical (or pseudo­ mathematical) techniques to all manner of p o litically or ideol ogi­ cally loaded persuasive ends. However it is precisely against such abuses - so Badiou maintains - that one can muster the unique probative force of those set-theoretical concepts, axioms and procedures that he lays out for detailed inspection in Being and Event. That is, they can most ap tly serve to demonstrate the kinds and degrees of democratic deficit - or p olitical disenfranchise­ ment - that typify the workings of ' liberal democracy ' , whether US-style or in various forms throughout Western Europe and the ex-Soviet bloc. Thus his point about the State's obsessive concern with inclu­ sion is that this goes along not only with a systematic disregard for issues of belonging (or of how individuals relate to their con­ ditions of socio-political existence) but also with a narrowed and bureaucratic sense - brought about by rigid application of the count-as-one - of what inclusion properly signifies. 92

READING THE

TEXT

The individual is always - patiently or impatiently - subject to this elementary coercion, to this atom of constraint which constitutes the possibility of every other type of constraint, including inflicted death . . . . Any consistent subset is imme­ diately counted and considered by the State, for better or worse, because it is matter for representation. On the other hand, despite the protestations and declarations to the con­ trary, it is always evident that in the end, when it is a matter of people's lives which is to say, of the multiple whose one they . have received - the State is not concerned. Such is the ulti­ mate and ineluctable depth of its separation. (pp. 1 07-8) -

It is characteristic of Badiou's thought that the passage should convey such a clear sense of moral and political passion - of mixed anger and sadness at the cost in terms of human wellbeing of such avowedly inclusive but in fact highly selective or exclu­ sionary practices - while none the less making its point through a mode of argument deriving from certain, as it might seem, decidedly abstract set-theoretical procedures. However, this widely held idea (at least among arts-and-humanities types) of mathe­ matical discourse as a realm of soulless abstractions devoid of human or creative content is one that Badiou has rejected with great energy and eloquence, most often by instancing the rich profusion of previously unthought-of entities that mathemati­ cians have discovered by dint of their jointly conceptual and imaginative powers.IS At the same time he is keen to insist against any kindred notion of mathematics as inherently remote from political concerns or the interests of social justice - that one very effective way to bring home the wrongs inflicted by unjust or oppressive (even if self-styled 'democratic') regimes is to deploy the conceptual resources of set theory as a means of clearly specified and logically precise articulation. That is, they offer the strongest purchase for an account of how presently existing forms of pseudo-democratic governance operate to ensure the belonging of all and only those members or subjects in good standing who qualify according to the count-as-one. What set theory makes possible is a formalized reckoning with just that excess of inclusion over acknowledged or recognized belonging - or of uncounted subsets over any count according to presently accepted rules and conventions - which constitutes 93

BADIOU'S BEING AND EVENT

the basis, in theory and principle, of a properly motivated chal­ lenge to the socio-political status quo. Nor should it be thought that this is an overly, even inhumanly 'abstract' approach to issues that can have so decisive a bearing on the welfare, the life-hopes and the very survival of so many people across such a range of real-world oppressive situations. After all, as Badiou pointedly remarks, it is not for nothing that governments, when an emblem of their void wanders about - generally, an inconsistent or riot­ ing crowd - prohibit 'gatherings of more than three people', which is to say they explicitly declare their non-tolerance of the one of such 'parts', thus proclaiming that the function of the State is to number inclusions such that consistent belongings be preserved. (p. 1 09) What is striking here is Badiou's ability to move straight across with such consummate ease - though without the least sense of tricksy semantics or conceptual legerdemain - between a regis­ ter of set-theoretical 'abstractions' and a language of engage­ ment with directly political or social-activist concerns. That he is able to do so against all manner of likely objections - including various analytic updates on the fact/value dichotomy or the sup­ posed logical impossibility of deriving a normative-evaluative 'ought' from a purely constative 'is' - should be put down to Badiou's highly focused and concentrated sequence of arguments up to this point, rather than to any evasive strategy or vaguely analogical habit of thinking on his part. More specifically, it has to do with his singular gift for locating just those erstwhile stress­ points yet also (and for just that reason) points of subsequent radical advance within the history of set-theoretical thought which lend themselves to re-statement in political terms. If poli­ tics is always, by very definition, 'an assault against the State, whatever the mode of that assault might be, peaceful or violent', then such a claim finds support in the proven capacity of formal reasoning to 'mobilize the singular multiples against the normal multiples by arguing that excrescence is intolerable' (p. 1 10). An 'excrescence' is defined, in this context, as a term that is 'represented by' the state of some given situation while not 'pre­ sented in' the situation itself, or again -what amounts to the same 94

READING THE TEXT

thing - as 'included in' that same situation while not recognizedly 'belonging to' it for social-political-administrative purposes. It is therefore a locus of maximal challenge to the structures of State-accredited authority and power that in turn derive their appearance of legitimate, that is, 'democratic' warrant from the count-as-one and its fallacious claim to represent a consistent multiplicity of free and equal subjects, rather than - as emerges through Badiou's analysis - an inconsistent multiplicity that in truth subtends and belies that false appearance. What is uniquely capable of breaking through this ideological fa�ade is the sudden irruption of those same 'excrescences' or stubbornly resistant, hence 'intolerable' instances of failure (or refusal) to be counted as belonging to some given self-image of the fully democratic since all-inclusive body politic. Thus 'excrescence'· · takes its place as another key term - alongside 'void', 'event' and 'evental site' - in the range of set-theoretically derived concepts whereby Badiou seeks to understand and explain how it is possible for something radically new to occur (whether a great advance in mathematics, discovery in science, or episode of major political change) despite and against all the odds of received belief or entrenched socio-political power. For this is the point at which his project hinges between its ontological orientation, that is, its concern to explicate the structures and modalities of being con­ ceived not in static but dynamic or constantly self-transformative terms, and on the other hand its orientation towards an idea of the event as involving a more drastic (and wholly unpredictable) break with whatever may seem to have led up to it from a subse­ quent or post-evental viewpoint. Hence Badiou's sceptical - some would say cynical - attitude concerning the prospects for genuine as opposed to merely cos­ metic change through any 'democratically' sanctioned parliamen­ tary means or any process that exists under present conditions in the United States or countries belonging to the (so-called) Euro­ pean 'community'. Hence also, conversely, his passionate convic­ tion that alternative means can be found whereby to circumvent, outflank or break through what he cites as the phenomenon despairingly noted by Lenin and other revolutionary thinkers, namely the 'obscene persistence' of the State despite the best efforts of those ranged against it. All the same such despair of the State's ever being brought to let go of its established powers 95

BADIOU'S BEING AND EVENT

and prerogatives should by no means be thought to entail a simi­ lar despair with regard to the existence of other, extra-parlia­

mentary or non-State-involving means by which to bring about that desirable upshot. 'Rather than a warrior beneath the walls of the State, a political activist is a patient watchman of the void instructed by the event, for it is only when grappling with the event (p.

that

the

State

blinds

itself

to

its

own

mastery'

1 1 1). The power of the State is exercised - and perpetuated ­

only on condition of ignoring, discounting or (at the limit) forcibly suppressing those whose very marginality or social invis­ ibility gives them the countervailing power which comes of their excluded, anomalous and hence potentially subversive or desta­ bilizing presence.

If granted recognition even in this outcast role

as 'excrescences' or not-to-be-tolerated others they would be strongly placed to expose both the arbitrary, unjust nature of the State's claim to power and the weakness of that claim when subject to challenge on ethico-juridical and indeed, according to Badiou's central thesis, on formally specifiable grounds.

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

vis-a-vis

the order of 'being', or for that which intervenes - in 96

READING THE TEXT

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

97

BADIOU'S BEING AND EVENT

Gilles Deleuze. 19 On Deleuze's account all that creaky Spinozist scaffolding should best be ignored and the Ethics be read not for its (pseudo-)demonstrative logical structure but rather for the moments of passional intensity and highly charged personal reflection that erupt at various points of the text, especially in the scholia. Not that Badiou is in the least inclined to ignore this 'other' Spinoza, or to overlook the signs of a restless, unruly, desirous physical being that Deleuze places very much at stage­ centre. On the contrary, his reading makes much of those stress­ points, anomalous passages and other such crucial (though often disregarded) junctures in the Ethics where the supposedly seamless progression from stage to stage in its structure of argu­ ment is interrupted by moments of a strikingly different, highly charged emotive or passionate character. Thus Badiou, no less than Deleuze, rejects any reading that would focus solely on its logical (or quasi-geometrico-deductive) structure at the cost of downplaying - or ignoring - that other, intensely affective dimension of Spinoza's life and work. Indeed, it is crucial to his own thinking that Spinoza's resolutely monist ontology - his conception of mind and body or thought and matter as two 'attributes' of the self-same substance - should be prey to just such uncontrollable intrusions not only from the realm of pas­ sional experience but also from the world of contingent histori­ cal

and

socio-political events.

After all,

Badiou's entire

philosophic project involves precisely this cardinal distinction between, on the one hand, the order of being as revealed or dis­ covered through enquiry into the set-theoretical foundations of ontology and, on the other, the order of events as that which inherently eludes any such account and which sets new stan­ dards - new fidelity conditions - for the exercise of thought in its other, for example, political, artistic and ethical spheres. Spinoza famously broke off his work on the Ethics in order to write the

Tractatus Theologico-Politicus

and thereby intervene,

to the best of his powers, in the crisis of conflicting religious as well as political allegiances which at that time threatened to overthrow the Dutch Free Republic.20 Badiou's reading gains credence from this salient fact, along with the extent to which Spinoza's passions, both positive and negative, were so often evoked by his intense participation in this struggle to preserve the hard-won freedoms of thOUght and speech. On the other 98

READING THE TEXT

hand - against Deleuze - he holds that we shall underrate the ethical and political as well as the philosophic force of Spinoza's thought if we treat its geometrico-deductive mode of presenta­ tion as just a handy formal device or a means of achieving maxi­ mum rhetorical and argumentative effect. What Badiou finds so intriguing about Spinoza is precisely this unique combination of a mind flXed upon truths that are taken to hold sub specie

aeternitatis,

or as always potentially transcending the compass

of time-bound human cognition, with an intelligence keenly and deeply aware of its temporal (e.g. cultural-historical and socio­ political) involvements. Spinoza thus stands as a test-case and notable precursor for the two major theses that between them motivate Badiou's philo­ sophical project. His thinking prefigures what Badiou has to say - with the advantage of set-theoretical, Marxist, psychoana­ lytic and other kinds of informative hindsight - concerning mathematics as the basis of all ontology and the event as that which redefines our intellectual and ethico-political responsibil­ ities

vis-a-vis some there�ter strictly binding (since truth-pursu­

ant) obligation. In this respect he manages to straddle the two major camps of recent French Spinoza interpretation. On the one side were those - like Althusser and the early Balibar - who recruited him to the cause of a 'structuralist' or critical-rational­ ist Marxism conceived very much in the Spinozist manner of a quest for truth and knowledge ideally unclouded by the effects of false, deceptive, imaginary or ideological belief.21 On the other were those, including Deleuze, who reacted strongly against this idea (as much with regard to issues in present-day politics as to issues in Spinoza scholarship) and who swung right across to the opposite" extreme of a reading that emphasized the philosophically exorbitant character of Spinoza's thought and its affinity with such notions as desiring-production, libidinal economy or 'deterritorialised' energy-flows. 22 As I have shown

elsewhere, each of these drastically opposed readings is able to claim a good measure of exegetical warrant through the direct appeal to certain strongly supportive passages in Spinoza's text.23 However, what is conspicuously missing from both inter­ pretations - and what Badiou sets out to provide - is an ade­ quate account of how the method of reasoning more geometrico relates to Spinoza's treatment of the passions (positive and 99

BADIOU'S BEING AND EVENT

negative) along with his response to the pressures and prospects of historical-political life. Thus Spinoza is a central figure in Badiou's genealogy of modem thought since he, like B adiou, was above all concerned to understand the relations between truth and knowledge, the­ ory an d practice, reason as that which aspires to a timeless (pro­ totypically mathematical) order of truth and reason as subject to practical constraints when re quired to adapt itself to chang­ ing historical and socio-political conditions. However - and this is where Badiou parts company with Deleuze - we shall be in no position to appreciate the strength or intensity of Spinoza's political passions and convictions unless we are willing to mea­ sure them against the demonstrative force of his reasoning more geometrico and not treat the latter as a mere excrescence or a misconceived attempt to achieve scientific credibility for some otherwise highly dubious premises and conclusions. And again, we shall fail to grasp an important aspect of that reasoning namely its role as both a critical check upon those passions and a motivating source for them - if we adopt a ultra-rationalist position, like Althusser's, that very largely ignores both the affec­ tive dimension and the circumstantial details of Spinoza's con­ joint life-and-thought. To be sure, there is a strong case to be made for viewing Spinoza as a thinker far ahead of his time and one who moreover managed to elaborate a proto-Marxist theory of truth, subjectivity and ideological misrecognition. This he achieved - so Althusser maintains - through his distinction between the 'first' and 'second' kinds of knowledge, that is, his account of how 'confused' or 'imaginary' ideas should properly give way to their 'adequate', clear and distinct (since rigorously theorized) replacements.24 Yet as Althusser's critics have been quick to point out it is hard to extract any convincing account of political agency or motivation from his high-structuralist account of how subjects are passively interpellated by - or recruited to this or that prevailing ideological formation.2s What is lacking in his general approach to these matters, as likewise in his reading of Spinoza, is what Badiou most importantly aims to provide: an account of how philosophy might reconcile the claims of conceptual rigour, clarity and precision with an openness to the contingency and unpredictability of real-world events, whether

100

READING THE TEXT

the ethical, political, philosophical, artistic or personal (espe­ spheres Hence the prominent position of Spin oza as an elective pre­ cursor to Badiou's philosophical project, despite the extreme contrast between Spinoza's radical monism and Badiou's com­ mitment to an equally radical conception of inconsistent multi­ plicity as that which precedes, subtends and surpasses any unity imposed upon it by various operations of the 'count-as-one'. Hence also the lesson for those who might b e temp ted to dismiss Badiou's writing on set-theoretical themes as at best a somewhat fanciful diversion and at worst a display of gratuitous expertise in a discipline utterly remote from his home�ground interests, that is, politics, ethics, aesthetics and psychoanalysis. They would be wrong about this for a number of reasons, among them - as I have said - the high sophistication and conceptual range of Badiou's mathematical thought and the extent to which his ontol ogical (i.e. set-theoretical) concerns intersect with his treat­ ment of those other themes For it is just Badiou's point that they constitute the chief enabling 'conditions' for a project that would keep its sights firmly fixed on the standard of truth while none the less taking adequate account of those various kinds of event that can always intervene in such a way as radically to redefme what qualifies as thinking, acting or living in accor­ dance with that same standard This is why he is so critical, even contemptuous, of much that passes for philosophy of mathemat­ ics in the recent analytic tradition, focused as it often is either on in

cially amorous)

.

.

.

narrowly technical or on hyper-inflated issues - such as the seem­

ingly endless debate around rule-following - that (in

his view)

merely trivialize the subject and deflect thinking from other,

philosophically as well as mathematically more challenging no matter how qual­ ified, of the Frege-Russell iogicist programme that would seek to derive all the basic truths of mathematics from a handful of set theoretical axioms and strictly deductive procedures of proof and demonstration.27 It is here that Badiou's thinking comes closest to Spinoza even though he rejects the Spinozist idea of a single, undifferentiated order of being (interchangeably 'God' or 'nature') that contains or subsumes all its various 'modes' of o bjects, beings and events.28 So likewise with the two 'attributes of mind and body which

paths.26 It is also why he rejects any version,

-

'

1 01

BADIOU'S BEING AND EVENT

Spinoza conceives - in company with some present-day physical­ ist or central-state-materialist philosophers of mind - as merely epiphenomenal, that is to say, as products of our humanly limited powers of apprehension.29 So it is not surprising that Badiou whose ontology starts out from the notion of infinitely multiple infinities constrained by the stipulative count-as-one in its vari­ ous forms - should make a point of staking his distance from Spinoza as the philosopher most committed to a radically moni­ stic or anti-dualist, that is, anti-Cartesian metaphysics of mind and world. After all, it was Badiou who caused considerable upset among the followers of Deleuze by claiming that the latter especially in his thinking about issues in mathematics - betrayed all the symptoms of a covert attachment to the ultimate univoc­ ity of being and truth, despite his overt celebration of difference, heterogeneity or multiple and endlessly proliferating 'lines of flight'.30 I shall not here attempt an adjudication of the issue between Badiou and Deleuze except to say that it reflects their very different views with respect to crucial topics in the history of philosophy from Plato down and - most crucially of all - the relationship between mathematics, philosophy and the various 'conditions' that constitute philosophy's means of access to truth. If there is one point at least on which they agree it is the strict impossibility of thinking the multiple without reference to the count-as-one as that which seeks, albeit vainly, to comprehend the multiple and thereby get a purchase on what would other­ wise exceed its utmost capacities of rational grasp. Thus despite his rejection of Spinoza's monist ontology Badiou can subscribe unreservedly to the Spinozist dictum 'For we have a true idea', and moreover to the Spinozist claim - on the face of it one with radically monist implications that 'the order of things' and the 'order of ideas' are in fact one and the same order under different descriptions or aspects.31 This follows from his acceptance of mathematical Platonism, construed in a non­ standard way according to which contra the sceptics, anti­ realists and conventionalists - there is simply no distinguishing the object-domain of mathematical entities and truths from the various procedures or acts of thOUght whereby they are brought within range of discovery or formal-demonstrative proof. 32 What saves this conception from the much-touted Platonist 'dilemma ' of objective truth versus humanly attainable knowledge is -

-

1 02

READING THE TEXT

Badiou's refusal (with good Platonist warrant) to allow any such gap to open up in the first place, along with his equally firm insistence on the way that mathematics - as our paradigm case of truth-oriented thought - typically achieves its most signal advances through a constant dialectic of problem-creating and problem-resolving initiatives. It is in the restless movement between these poles that mathematics exhibits both its own capacity for creative self-renewal and the extent to which its vari­ ous formal procedures bear upon other fields of human experi­ ence, knowledge and enquiry. This is why, as we have seen, Badiou takes his cue in matters ontological from those passages in Plato's Sophist and Parmenides where Socrates· most directly confronts the aporias of the one and the many, and where think­ ing sets out on the long and tortuous path that will eventually lead to the paradoxes of classical set theory and the various attempts (by Russell and others) to resolve or at any rate defuse those paradoxes. Here as elsewhere - for instance in his com­ mentaries on Spinoza and Hegel - Badiou's is essentially a diag­ nostic reading which aims to draw out those symptomatic moments of recalcitrant, resistant or non-assimilable sense that signal the presence of a counter-logic at odds with the thinker's overt professions of intent. Above all it is Spinoza who provokes this ambivalent response since Badiou is strongly drawn to certain aspects of Spinozist thOUght - chief among them his axiomatic-deductive style of rea­ soning more geometrico - while none the less rejecting his radically monist conception of mind and nature as two 'attributes' of the self-same substance, itself manifested in the various 'modes' whose seeming multiplicity belies their true nature as so many aspects or phenomenal appearances thereof.33 For there is an obvious conflict between the central claim of Spinozist ethics, that is, that true free­ dom lies in the acceptance of an all-encompassing (even if in large part humanly unknowable) order of necessity and Badiou's great care to distinguish the realms of being and event. Thus 'Spinoza represents the most radical attempt ever in ontology to identify structure and metastructure, to assign the one-effect directly to the state, and to in-distinguish between belonging and inclusion' (p.

1 1 3). Such is the result of Spinoza's radically monist conception

when consistently applied to issues in the social, political and ethi­ cal spheres as well as in realms - such as the formal and physical 1 03

BADIOU'S BEING AND EVENT

sciences - where it fails to account for the creative-transformative power of human intellect and also for the way that such advances are achieved against, despite, or (sometimes) as an unlooked-for result of contingent factors that fmd no place in the kindsof neatly tailored account that typify the 'official' histories of those disci­

In short, 'it is clear that [Spinoza's] is the philosophy par excellence whichforecloses the void', and therefore that Badiou will

plines.

need to demonstrate some flaw - some logical inconsistency or fail­ ure to meet its own conceptual requirements - in the Spinozist sys­ tem of thought. This he does by focusing on the notion of 'infmite mode' which Spinoza introduces as a means of bridging those var­ ious dualisms (god and nature, infmite and fInite, mind and body, essence and accident, freedom and necessity) that would otherwise lack any possible means of reconciliation. His argument here is extremely compressed and resistant to any kind of summary account. Sufficient to say that he shows this idea of 'infInite modes' to bear the full weight of Spinoza's problematical (indeed strictly unthinkable since logically self­ contradictory) doctrine and reveal where that doctrine runs up against the need to acknowledge what it cannot accept in keeping with its own governing precepts. This involves an intensive critical engagement with the whole apparatus of Spinozist thought and, in particular, its strenuous though (as Badiou thinks) its inherently self-refuting attempt to encompass the transcendence of all those vexing antinomies through a radically monist conception - 'god or nature',

deus sive natura

-

whereby freedom would achieve its

true meaning as the knowledge or acceptance of necessity. Thus Spinoza's strategy is doomed to fail and 'the void, whose meta­ structural or divine closure should ensure that it remains in-exis­ tent and unthinkable, is well and truly named and placed by Spinoza under the concept of infznite mode' (p. the defmition of

mode

1 1 3). That is to say,

as that which pertains to a vast though

fInite variety of objects presented under either their physical (bodily) or mental (intellectual) attributes cannot possibly be rec­ onciled with talk of the infInite, this latter having its legitimate place only in the Spinozist discourse on that which in essence - or by very defInition - transcends or exceeds any such limiting con­ ception.

It is here that Spinoza's monist doctrine (deus sive natura)

can be seen to come up against its limit: that is, 'the necessity of having to invoke a void term, whose name without a testifIable 1 04

READING THE TEXT

referent ('infinite mode') inscribes errancy in the deductive chain' (p. 120). Or again, the 'great lesson' of Spinoza's thought can be expressed in the following terms: 'even if, via the position of a supreme count-as-one which fuses the state of a situation and the situation (i.e. metastructure and structure, or inclusion and belonging), you attempt to annul excess and reduce it to a unity of the presentative axis, you will not be able to avoid the errancy of the void' (p. 1 20). Thus it is hard to conceive how Spinoza's purported deriva­ tion of ethics from ontology - entailmg his idea of 'freedom' as in truth nothing other than the wise acceptance of an all-embrac­ ing necessity - could possibly be reconciled with Badiou's demand for a clear and principled distinction between the realms of being and event. The former (i.e. ontological) domain is that which Spinoza went some way towards charting through his insistence on the discipline of argument more geometrico and hence his historically precocious understanding - one that would not be fully borne out until the advent of modem set theory that mathematics provides the sole adequate basis for a critical ontology able to account for our knowledge of the growth of knowledge across the formal, physical and social-scientific domains. However what Spinoza's system crucially lacked was any allowance - on its own terms, any means of making allowance - for the evental dimension wherein human purposes, values and commitments come up against the sheerly contingent character of real-world circumstance. Here it is worth noting that Badiou reads Spinoza very much on Spinozist terms, that is, not in the fideist mode of commentary that would simply report and endorse his ideas but rather through the kind of 'symptomatic' approach that Spinoza himself (much to the horror of his piously orthodox contemporaries) applied in his reading of scriptural texts.34 From this standpoint it is possible to see 'that the infinite mode is where Spinoza designates, despite himself - and thus with the highest unconscious awareness of his task - the point (excluded everywhere by him) at which one can no longer avoid the supposition of a subject' (p. 1 1 3). A 'subject', that is, in the precise sense specified by Badiou: one whose existence is predi­ cated on or synonymous with the project that commands their utmost fidelity and which thus defines their very condition of subjecthood. 105

BADIOU'S BEING AND EVENT

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

What do you find most striking or revealing about Badiou's con­ ception of the human subject and its role in relation to issues of truth and knowledge? How do you understand Badiou's closely connected claims with regard to the state, the situation, the void, inclusion and belonging? PART I I I . BEING: NATU RE AN D INFINITY. HEIDEGGERlGALI LEO 1 . The Greek inauguration: poetry or mathematics?

Part m of Being and Event contains Meditations Eleven to Fifteen and covers a large, not to say giddying amount of philosophic, cultural-his­ torical and - increasingly from this point on - mathematical ground. It begins with a section entitled 'Nature: Poem or Matheme?' in which Badiou reflects at greater length on a topic flISt raised in his Introduc­ tion, namely the issue between poetry and mathematics as claimants to the title of 'flISt philosophy'. That is, he sets out to contest Heidegger's claim for the role of poetry (rather than mathematics or the physical sci­ ences) as having marked the inaugural moment - indeed the very condi­ tion of emergence - for ancient Greek thought and all that followed in its cultural-historical wake. This claim was the basis of Heidegger's late 'turn' towards an outlook that placed less emphasis on the idea of reso­ lutely facing-up to the call of Being and counselled in stead a mood of Gelassenheit, of receptive openness to truths vouchsafed by poets whose language (especially the German language through its supposed 1 06

READING THE TEXT

deep afflnity with the ancient Greek) was still capable of bearing wit­ ness to that otherwise long-forgotten source. I Hence the depth-henneneutic approach whereby Heidegger claims to think his way back through and beyond the various accretions of 'Western metaphysics' from Plato and Aristotle down to Husser!. 2 Chief among them is the drastically reduced, denatured and inadequate conception of truth that starts out with Plato's idealist doctrine of transcendent fonns or ideas, that fmds its next major statement in Aristotle's theory of truth as homoiosis or correspondence, and thereafter undergoes a suc­ cession of modifications and refmements which on Heidegger's account do nothing to redeem those defects. 3. What they share is a failure to allow for that further, deeper, more authentically revealing since primordial dimension of truth that the Greeks especially certain of the pre-Socratics - invoked in the name of aletheia, a tenn that translates literally as 'unforgetting', or retriev­ ing from oblivion truths that had once been known to the soul but then overlaid (as the story goes in its more mystical Pythago­ rean, neo-Platonic or proto-Wordsworthian guise) by false or distracting appearances. For Heidegger, this failure is redeemable only through a deconstruction (Aubbau or 'unbuilding') of the various inherited concepts that have held philosophy in thrall, along with an ear and mind sufficiently attuned to the ways in which poetic language may communicate beyond any level of meaning that might be accessed by conceptual analysis or plain­ prose paraphrase. Moreover - and it is here that Badiou most emphatically takes leave of Heidegger - such thinking would go furthest towards restoring the kind of receptive or responsive awareness that might yet save us from the depredations of a technocratic reason with its joint source in the genealogy of Western post-Socratic philosophic rationalism and post-Galilean, that is, mathematics-based physical science.4 This diagnosis can only strike Badiou as a gross distortion of intellectual history brought about partly by an under-valuation of the disciplined yet none the less creative activity of thought manifest in mathematics and partly by a basic misconception of poetry's past and continuing role in that history. For Badiou, one may conjecture, it is just another version of the old logical­ positivist doctrine that accepted as genuinely meaningful only that class of empirically vacuous statements (or tautologies) 1 07

BADIOU'S BEING AND EVENT

whose truth was self-evident purely in virtue of their logical form and that other class of statements whose truth or untruth could be established by observation or empirical warrant.s Thus Heidegger's disdain for mathematics and the physical sciences no matter how disguised by his talk of their predestined emer­ gence from the epoch of Western metaphysics - can be seen as just the mirror-image of that positivist disdain for any kind of di scourse (including poetry) which failed to meet either require­ ment. Badiou is just as far from sharing Heidegger's hostile atti­ tude towards the sciences as he is from sharing this positivist idea of the sciences, physical or formal, as philosophy's sole source of guidance and therefore of poetry (along with other such 'emotive' or merely ' metaphysical ' modes of expression) as lack­ ing any properly assignable truth-value.6 Above all he is deter­ mined to show - as against the former mistaken supposition - that mathematics has creative resources in plenty as well as the conceptual or demonstrative rigour whereby to bear out Plato's claim for its status as first philosophy. In this respect Badiou stands firmly opposed to those philos­ ophers, among them Heidegger and Wittgenstein , who go so far as to deny that mathematics involves ' thinking' in any proper sense of the term, that is, any sense that would not confuse think­ ing with mere calculation , mechanical reasoning or rule-gov­ erned formal procedure. ? Thus he rejects the echt- Heideggerian idea that those poetic resonances, once strongly conveyed and still distantly audible in the Greek word -55. 'Who is Nietzsche?', (trans.) Alberto Toscano, Pli: Warwick Journal of Philosophy, No. 1 1 (200 1 ), pp. 1-10. 'Logic of the Site', (trans.) Steve Corcoran and Bruno Bosteels, Dia­ critics, Vol. 33, No. 3 (2003), pp. 141-50. 'Some Replies to a Demanding Friend', (trans.) Peter Hallward, in Think Again: Alain Badiou and the Future of Philosophy (London: Continuum, 2004), pp. 232-7. 'The Adventure of French Philosophy', New Left Review, No. 35 (2005), pp. 67-77. 'Democratic Materialism and the Materialist Dialectic', Radical Philos­ ophy, No. 1 30 (2005), pp. 20-4. 'Lacan and the pre-Socratics', (ed.) Slavoj Zizek, in Lacan: The Silent Partners (London: Verso, 2006), pp. 7-1 6. 3. ENG LISH-LANGUAGE BOOKS AND ARTICLES ABOUT BADIOU

Ashton, Patil, A. 1. Bartlett and Justin Clemens (eds), The Praxis of Alain Badiou (Victoria: Re.Press, 2006). Balibar, Etienne, 'The History of Truth: Alain Badiou in French Philosophy', in Hallward (ed.) (2004), pp. 21-38. Barker, Jason, Alain Badiou: A Critical Introduction (London: Pluto Press, 2002). Bensaid, Daniel, 'Alain Badiou and the Miracle of the Event', in Hallward (ed.) (2004), pp. 94-1 05. Bosteels, Bruno, 'Alain Badiou's Theory of the SUbject: The Recommence­ ment of Dialectical Materialism? (Parts I and 2)', Pli: Warwick Joumal of Philosophy, No. 12 (2001), pp. 200-29 and No. 1 3 (2002), pp. 173-208. Bosteels, Bruno, 'On the Subject of the Dialectic', in Hallward (ed.), (2004), pp. 1 50-64. Bosteels, Bruno, 'Can Change Be Thought? A Dialogue with Alain Badiou', in Riera (ed.) (2005), pp. 237-6 1 . Brassier, Ray, 'Nihil Unbound: Remarks on Subtractive Ontology and Thinking Capitalism', in Hallward (ed.) (2004), pp. 50-8. Brassier, Ray, 'Badiou's Materialist Epistemology of Mathematics', Angelaki: Journal of the Theoretical Humanities, Vol. 10, No. 2 (2005), pp. 1 35-50. Clemens, Justin, 'Platonic Meditations', Pli; Warwick Journal of Phi­ losophy, No. 1 1 (2001 ), pp. 200-29. Clemens, Justin, The Romanticism of Contemporary Theory: Institution, Aesthetics, Nihilism (Aldershot: Ashgate, 2003). Copjec, Joan, 'Gai Savoir Sera: The Science of Love and the Insolence of Chance', in Riera (ed.) (2005), pp. 1 19-35.

286

FURTHER READING

Critchley, Simon, 'Demanding Approval: On the Ethics of Alain Badiou', Radical Philosophy, Vol. 1 00 (2000), pp. 1 �27. Critchley, Simon, 'On the Ethics of Alain Badiou', in Riera (ed.) (2005), pp. 2 1 5-35. de Beistegui, Miguel, 'The Ontological Dispute: Badiou, Heidegger, and Deleuze', in Riera (ed.) (2005), pp. 45-58. Desanti, Jean Toussaint, 'Some Remarks on the Intrinsic Ontology of Alain Badiou', in Hallward (ed.) (2004), pp. 59-66. Dews, Peter, 'Uncategorical Imperatives: Adorno, Badiou and the Ethical Turn', Radical Philosophy, No. I I I (2002), pp. 33-7. Dews, Peter, 'States of Grace: The Excess of the Demand in Badiou's Ethics of Truths', in Hallward (ed.) (2004), pp. 1 O�19. Diittmann, Alexander Garcia, 'What Remains of Fidelity after Serious Thought', in Hallward (ed.) (2004), pp. 202-7. Eagleton, Terry, 'Subjects and Truths', New LefrR'eview, No. 9 (200 1), pp. 1 55-60. Feltham, Oliver, Alain Badiou: Live Theory (London: Continuum, 2008). Fraser, Zachary Luke, 'The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus', Cosmos and History, Vol. 1 , Nos. 1-2 (2006), pp. 94-1 33. Gillespie, Sam, 'Placing the Void: Badiou on Spinoza', Angelaki: Journal of the Theoretical Humanities, Vol. 6, No. 3 (December 200 1), pp. 63-77. Gillespie, Sam, 'Beyond Being: Badiou's Doctrine of Truth', Communi­ cation and Cognition, Vol. 36, No. 1-2 (2003), pp. 5-30. Gillespie, Sam, The Mathematics of Novelty: Badiou's Minimalist Meta­ physics (Victoria: re.press, 2008). Hallward, Peter, Badiou: A Subject to Truth (Minneapolis: University of Minnesota Press, 2003). Hallward. Peter (ed.), Think Again: Alain Badiou and the Future of Phi­ losophy (London: Continuum, 2004). Hallward, Peter, 'Depending on Inconsistency: Badiou's Answer to the "Guiding Question of All Contemporary Philosophy"', Polygraph, Vol. 17 (2005), pp. 1 1-25. Hewlett, Nick, 'Engagement and Transcendence: The Militant Philo­ sophy of Alain Badiou', Modern and Contemporary France, Vol. 1 2, No. 3 (2004), pp. 335-52. Ingram, James D. , 'Can Universalism Still Be Radical? Alain Badiou's Politics of Truth', Constellations, Vol. 1 2, No. 4 (2005), pp. 56 1-73. Kaufman, Eleanor. 'Why the Family Is Beautiful (Lacan against Badiou)', Diacritics, Vol. 32, Nos. 3/4 (2002), pp. 1 35-5 1 . Laclau, Emesto, � Ethics o f Militant Engagement', in Hallward (ed.) (2004), pp. 120-37. Legerc1e, Jean-Jacques, 'Cantor, Lacan, Mao, Beckett, Meme Combat: The Philosophy of Alain Badiou', Radical Philosophy, No. 93 ( 1 999), pp. � 1 3 .

287

BADIOU'S BEING AND EVENT

Jean-Jacques, 'Badiou's Poetics', in Hallward (ed.) (2004), pp. 208-17. MacCannell, Juliet Flower, �lain Badiou: Philosophical Outlaw', in Riera (ed.) (2005), pp. 1 37-84. Macherey, Pierre, 'The Mallarme of Alain Badiou', in Riera (ed.) (2005), pp. 1 09-1 5. Madarasz, Norman, 'On Alain Badiou's Treatment of Category Theory in View of a Transitory Ontology', in Riera (ed.) (2005), pp. 23-43. May, Todd, 'Badiou and Deleuze on the One and the Many', in Hallward (ed.) (2004), pp. 67-76. Mount, B. Madison, 'The Cantorian Revolution: Alain Badiou on the philosophy of set theory', Polygraph, No. 17 (2005), pp. 41-9 1 . Nancy, Jean-Luc, 'Philosophy without Conditions', in Hallward (ed.) (2004), pp. 39-49. Norris, Christopher, 'Some Versions of Platonism: Mathematics and Ontology According to Badiou', Philosophical Frontiers, Vol. 3, No. 1 (January to June 2008), pp. 1-26. Norris, Christopher, �ain Badiou: Truth, Mathematics and the Claim of Reason', PU: The Warwick Journal of Philosophy, Vol. 19 (2008), pp. 1 89-217. Noys, Benjamin, 'The Provocations of Alain Badiou', Theory, Culture and Society, Vol. 20, No. 1 (2003), pp. 1 23-32. Ranciere, Jacques, �esthetics, Inaesthetics, Anti-Aesthetics', in Hallward (ed.) (2004), pp. 2 1 8-3 1 . Riera, Gabriel (ed.), Alain Badiou.· Philosophy and its Conditions . (New York: State University of New York Press, 2007). Smith, Daniel W, 'Mathematics and the Theory of Multiplicities: Badiou and Deleuze Revisited', Southern Journal of Philosophy, Vol. 4 1 , No. 3 (2003), pp. 41 1-49. Smith, Daniel W, 'Badiou and Deleuze on the Ontol ogy of Mathemat­ ics', in Hallward (ed.) (2004), pp. 77-93. Smith, Brian A., 'The Limits of The Subj ect in Badiou's Being and Event', Cosmos and History, Vol. 1 , Nos. 1-2 (2006), pp. 1 34-58. Strathausen, Carsten, 'The Badiou-Event', Polygraph, No. 1 7 (2005), pp. 275-93. Toscano, Alberto, 'From the State to the World? Badiou and Anti­ Capitalism', Communication and Cognition, Vol. 37, Nos. 3-4 (2004), pp. 1 99-223. Toscano, Alberto, 'Communism as Separation', in Hallward (ed.) (2004), pp. 1 38-49. Wilkens, Matthew, 'Introduction: The Philo sophy of Alain Badiou' , Polygraph, No 1 7 (2005), pp. 1-9. ZiZek, Slavoj, 'Psychoanalysis in Post-Marxism: The Case of Alain B adiou ', South A tlantic Quarterly, Vol. 97, No. 2 (1 998), pp. 235-6 1 . Zizek, Slavoj, 'Is There a Politics o f Subtraction? Badiou versus Lacan', Communication and Cognition , Vol. 36, Nos. 1-2 (2003), pp. 1 03-1 9. Zizek, Slavoj, 'From Purification to Subtraction: Badiou and the Real', in Hallward (ed.) (2004), pp. 1 65-8 1 . Ler;:ercle,

.

288

NOTES

1 . CONTEXT

1 Alain Badiou, Being and Event (trans.) Oliver Feltham (London: Continuum, 2005). All further references throughout this book will be given parenthetically by BE and page number in the text. 2 For further discussion, see Graham"Bird (ed.), A Companion to Kant (Oxford: Blackwell, 2006); Ruth F. Chadwick and Clive Cazeaux (eds), Immanuel Kant: Critical Assessments 4 vols (London: Rout­ ledge, 1 992); Paul Guyer (ed.), The Cambriage Companion to Kant (Cambridge: Cambridge University Press, 1 992). 3 See especially Richard Rorty, Contingency, Irony and Solidarity (Cambridge: Cambridge University Press, 1989); also Christopher Norris, What's Wrong with Postmodernism? (Brighton: Harvester, 1990) and The Truth About Postmodernism (Oxford: Blackwell, 1 993). 4 See for instance Michael Devitt, Realism and Truth, 2nd edn (Oxford: Blackwell, 1 986) and Norris, Philosophy of lAnguage and the Chal­ lenge to Scientific Rea/ism (London: Routledge, 2004). 5 See especially Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1951). 6 See for instance Paul Benacerraf and Hilary Putnam (eds), The Phi­ losophy of Mathematics: Selected Essays, 2nd edn (Cambridge: Cambridge University Press, 1 983); also W. D. Hart (ed.), The Phi­ losophy of Mathematics (Oxford: Oxford University Press, 1 996). On the vastly overworked topic of rule-following see Ludwig Wittgen­ stein, Philosophical Investigations (trans.) G. E. M. Anscombe (op. cit.), Sections 20 1-92 passim; Saul Kripke, Wittgenstein on Rules and Private lAnguage: An Elementary Exposition (Oxford: Blackwell, 1 982); Alexander Miller and Crispin Wright (eds), Rule-following and Meaning (Chesham: Acumen, 2002). For Badiou's decidedly adverse view of much of this work, see 'Ontology is Mathematics', in Theoretical Writings. All subsequent references to works of Badiou other than Being and Event will be given by title only since full publi­ cation details may be found in Chapter 4, 'Further Reading'. 7 For some highly illuminating discussion, see Badiou, Metapolitics, Polemics, and Century. 8 For a clear and highly relevant survey, see Michael Potter, Set Theory and its Philosophy: A Critical Introduction (Oxford: Oxford University Press, 2004). 9 Bertrand Russell, Introduction to Mathematical Philosophy (London: Allen & Unwin, 1930). 10 Paul 1. Cohen, Set Theory and the Continuum Hypothesis (New York: W. A. Benjamin, 1 966). 289

NOTES

1 1 For an account of these developments, see Norris, Against Relativism: Deconstruction Philosophy of Science and Critical Theory (Oxford: Blackwell, 1 997) and Philosophy of Language and the Challenge to Scientific Realism (London: Routledge, 2004). 1 2 See especially the various remarks to this effect in Badiou, 'Onto­ logy is Mathematics' (Note 6). 1 3 For further argument in support of this claim, see Norris, 'Change, Conservation and Crisis-Management in the Discourse of Analytic Philosophy', in Language, Logic and Epistemology: A Modal-realist Approach (Basingstoke: Palgrave-Macmillan, 2004), pp. 227-66. ,

2. OVERVIEW OF THEMES

1 See especially Immanuel Kant, Political Writings (ed.) Hans Reiss (Cambridge: Cambridge University Press, 1970) and Kant on His­ tory (ed.) L.w. Beck (Indianapolis: Bobbs-Merrill, 1 963). 2 For further discussion of these various developments, see Christo­ pher Norris, Against Relativism: Philosophy of Science, Deconstruc­ tion and Critical Theory (Oxford: Blackwell, 1 997) and Philosophy of Language and the Challenge to Scientific Realism (London: Routledge, 2004). 3 For a useful sampling, see Richard Rorty (ed.), The Linguistic Turn (Chicago, IL: University of Chicago Press, 1 967). 4 See especially Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1953) and On Certainty (ed. and trans.) Anscombe and G. H. von Wright (Blackwell, 1 969). 5 See Rorty (ed.), The Linguistic Turn (op. cit.). 6 See especially Rudolf Camap, 'The Elimination of Metaphysics through Logical Analysis of Language', in A. 1. Ayer (ed.), Logical Positivism (New York: Free Press, 1 959), pp. 60-8 1 ; also - for some fascinating background history - Michael Friedman, A Parting of the Ways: Carnap, Cassirer and Heidegger (Chicago: Open Court, 2000). 7 Martin Heidegger, Being and Time (trans.) John Mcquarrie and Edward Robinson (Oxford: Blackwell, 1 980). 8 Nancy Cartwright, Jordi Kat, Lola Fleck and Thomas Uebel, Otto Neurath: Scien ce Between Philosophy and Politics (Cambridge: Cambridge University Press, 1 996). 9 See Note 6. 10 Heidegger, Poetry, Language and Thought (trans.) A. Hofstadter (New York: Harper & Row, 1971). 11 See especially Michael Dummett, Elements of Intuitionism (Oxford: Oxford University Press, 1 977) and Truth and Other Enigmas (London: Duckworth, 1978); also - for a critical review of these devel­ opments - Norris, Truth Matters: Realism, Anti-realism, and Response dependence (Edinburgh: Edinburgh University Press, 2002). -

290

NOTES

12 For a detailed study, see J. Alberto Coffa, The Semantic Tradition from Kant to Carnap: To the Finland Station (Cambridge: Cambridge University Press, 1991). 1 3 These issues receive a good critical airing in Ian Hacking, The Social Construction of What? (Cambridge, MA: Harvard Univer­ sity Press, 2000). See also, from a range of perspectives, David Bloor, Knowledge and Social Imagery, 2nd edn (Chicago : Univer­ sity of Chicago Press, 1991); Harry Collins and Robert Evans, Re-thinking Expertise (Chicago, UP., 2007); Kenneth J. Gergen, Realities and Relationships: Soundings in Social Construction (Har­ vard, UP., 1 994); Norris, Against Relativism (op. cit.) and Philosophy · of Language and the Challenge to Scientific Realism (London: Routledge, 2004) 14 See for instance entries for Chapter 1 , Con��xt , Note 6. 1 5 Paul J. Cohen, Set Theory and the Continuum Hypothesis (New York: W. A. Benjamin, 1 966). 16 For some informative accounts, see Catherine Belsey, Critical Prac­ tice (London: Methuen, 1981); Richard Harland, Superstructuralism: The Philosophy of Structuralism and Post-structuralism (London: Routledge, 1 99 1 ); Mark Poster, The Mode of Information: Post­ structuralism and Social Context (Chicago, IL: University of Chicago Press, 1990); Robert Young (ed.), Untying the Text: A Post-structuralist Reader (London: Routledge & Kegan Paul, 1 98 1). 1 7 See especially Jacques Lacan, EcrUs: A Selection (trans.) A. Sheri­ dan-Smith' (London: Tavistock, 1 977); also (for an excellent short introduction) Malcolm Bowie, Lacan (London: Fontana, 1991). 1 8 See for instance Louis Althusser, For Marx (trans.) Ben Brewster (London: New Left Books, 1 969); also Gregory Elliott, Althusser: The Detour of Theory (London: Verso, 1 987). 19 Jean Paul Sartre, A Critique of Dialectical Reason, Vol. 1 , Theory of Practical Ensembles (trans.) A. Sheridan-Smith (London: New Left Books, 1 976) and Vol. 2 (trans.) Quintin Hoare (London: Verso, 2006). 20 See Notes 7 and 10; also Heidegger, The Question Concerning Tech­ nology, and Other Essays (trans.) W. Lovitt (New York: Harper & Row, 1 977). 21 For further discussion see Richard Wolin (ed.), The Heidegger Controversy (New York: Columbia University Press, 1 99 1 ). 22 See for instance Iris Marion Young, Justice and the Politics of Differ­ ence (princeton, NJ: Princeton University Press, 1 990); also - for a critical review of such thinking across various disciplines - Norris, Truth and the Ethics of Criticism (Manchester: Manchester University Press, 1 994). 23 Immanuel Levinas, Totality and Infmity (trans.) A. Lingis (Pittsburgh, PA: Duquesne University Press, 1 969). 24 See Note 1 1 ; also Rorty (ed.), The LinguistiC Turn (op cit.). 25 For a strikingly heterodox view, see Lloyd P. Gerson, Aristotle and Other Platonists (Ithaca, NY: Cornell University Press, 2005). .

'

'

-

.

291

NOTES

3. READING THE TEXT

Part I. Being: multiple and void. Plato/Cantor I

2 3

4 S

6

7

Plato, Parmenides (trans.) Mary L. Gill and Paul Ryan (Indianap o­ lis: Hackett, 1 996). See also F. M. Cornfo rd . Plato and Parmenides: Parmenides' Way of Truth and Plato's Parmenides (trans. and intro­ duction) Cornford (London: Routledge & Kegan Paul, 1 939) and John A. Palmer, Plato 's Reception of Parmenides (Oxfo rd : Clarendon Press, 2002). W. V. Quine, Ontological Relativity and Other Essays (New York: Columbia University Press, 1 969), p. 23. Paul J. Cohen, Set Theory and the Continuum Hypothesis (New York: w.A. Benjamin, 1 966). See Abraham A. Fraenkel , Elements of Set Theory, rev. edn (Amsterdam: North-Holland, 1 973) for a main source of Badiou's thinking about issues of set-theoretical ontology; als o Michael Potter, Set Theory and its Philosophy: A Critical Intro­ duction (Oxford: Oxford University Press, 2004). B adiou , Deleuze: The Clamor of Being. See Gilles Deleuze, The Logic of Sense (trans.) Mark Lester (London : Athlone Press, 1 990) and Difference and Repetition (trans.) Paul Patton (Athlone, 1 994). Gilles Deleuze and Felix Guattari, Anti-Oedipus: Capitalism and Schizophrenia (trans.) Robert Hurley, Mark Seem and Helen Lane (London : Continuum, 2004). See especially Bertrand Russell, A History of Western Philosophy (London: Al len & Unwin, 1961); also - for some highly illuminating commentary on these issues of interpretatio n - Jonathan Bennett,

Learningfrom Six Philosophers: Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume, Vols I and 2 (Oxford: Clarendon Press, 200 1). 8 See Gaston Bachelard, Laformation de l'esprit scientifzque (paris: Vrin. 1938); The Philosophy of No (trans.) G. C. Waterston (New York: Orion Press, 1969); The New Scientific Spirit. (trans.) Arthur Goldhammer (Boston: Beacon Press, 1984) . See also Mary Tiles, Bachelard' Science and Objectivity (Cambridge: Cambridge University Press, 1984). 9 See Note 3; also Bertrand Russell, Introduction to Mathematical Philosophy (London : Allen & Unwin . 1 930). 1 0 Kurt Godel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems (trans.) B. Meltzer (New York: Basic B ooks, 1 9 62) . 1 1 For further arguments to this effect, see Stathis Psillos, Scientific Realism: How Science Tracks Truth (London: Routledge, 1 999); J. Aron s on , R. Harre and E. Way, Realism Rescued: How SCientific Progress is Possible (London: Duckworth, 1 994) ; Michael Devitt, Realism and Truth, 2nd edn (Oxford : Blackwell, 1 986); Jarrett Leplin (ed.), Scientific Realism (Berkeley & Los Angeles: University of California Press, 1984); Christopher Norris, Language, Logic and Epistemology: A Modal-Realist Approach (Manchester: Manchester University Press, 2004).

292

NOTES

12 See Note 8; also Norris, Spinoza and the Origins of Modern Critical Theory (Oxford : Blackwell, 1 991). 1 3 See Quine, 'On What There Is', in From a Logical Point of View, 2nd edn (Cambridge, MA: Harvard University Press, 1 961), pp. 1-19. 14 See Note I I . 1 5 Plato, Timaeus and Critias (cd. and trans.) Desmond E . Lee (Har­ mondsworth: Pengui n , 1 974); also Reason and Necessity: Essays on Plato's Timaeus (ed.) M. R. Wright (Lond on : Duckworth, 2000). 1 6 See especially Badiou, Metapo!itics, Polemics, and Century. 1 7 For further discussion, see A. W. Moore The Infinite (London: Routledge, 2000) and Potter, Set Theory and its Philosophy (op. cit.). 18 See for instance Iris Marion Young, Justice and the Politics of Dif­ ference (princeton, NJ: Princeton Univex:sity Press, 1 990); also Recognition and Difference: Politics, Identity, Multiculture (cds) Scott Lash and Mike Featherstone (London: Sage, 2002); Chris Weedon, Feminism, Theory and the Politics of Difference (Oxford: Blackwell, 1 999). 19 Badiou, Saint Paul: The Foundation of Universalism. 20 Aristotle Physics (trans ) Robin Waterfield (Oxford: Oxford Uni­ versity Press, 1 996); also Articles on Aristotle, Vol 1 , Science (eds) Jonathan Barnes, Malcolm Schofield and Richard Sorabj i (London: Duckworth, 1975). 2 1 Steven Shapin and Simon Schaffer, Leviathan and the Air-Pump: Hobbes, Boyle and the Experimental Life (Princeton, NJ: Princeton University Press, 1 985); also Norris, 'Why Strong Sociologists Abhor a Vacuum: Shapin and Schaffer on the Boyle/H obbes controversy', in Against Relativism: Philosophy of Science, Deconstruction and Critical Theory (Oxford: Blackwell, 1 997), pp. 265-94. 22 See Althusser, For Marx (op. cit.) and Essays in Self-criticism (trans.) Ben Brewster (London: New Left Books: 1 976); also Althusser and Etienne Balibar, Reading Capital (trans.) Brewster (New Left Books, 1 970). 23 Simon Singh, Fermat's Last Theorem: The Story of a Riddle that Confounded the World's Greatest Minds for 358 Years (London: Fourth Estate, 1 997). 24 For the classic exposition of this 'two contexts approach, see Hans Reichenbach, Science and Prediction (Chicago: University of Chicago Press, 1938). ,

,

.

.

'

Part II. Being: excess, state of the situation, 1

one/multiple, whole/parts or e tc? See A. A. Fraenkel and R. Bar-Hillel, Foundations of Set Theory (Amsterdam: North Holland 1 958); Azriel Levy, Basic Set Theory (New York: Dover, 2002); Patrick Suppes, Axiomatic Set Theory (New York: Courier Dover, 1 972). -

,

293

NOTES

2 See also Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, NJ: Princeton University Press, 1 990). 3 See various texts and passages to this effect in Badiou, Infznite Thought, Handbook of Inaesthetics, Metapolitics, and Century. 4 See especially Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1953) and On Cer­ tainty, (ed. and trans.) Anscombe and G. H. von Wright (Oxford: Blackwell, 1 969); also Martin Heidegger, 'The Question Concerning Technology' and Other Essays (trans.) William Lovitt (New York: Harper & Row, 1 977). For Badiou's most explicit and powerful critique of such ideas, see the essays collected in 'Ontology is Math­ ematics', Section One of Theoretical Writings, pp. 3-93. 5 For relevant discussion, see Lorraine Code, Epistemic Responsibility (Hanover, NH: University Press of New England, 1 987); M. DePaul and L. Zagzebski (eds), Intellectual Virtue: Perspectives from Ethics and Epistemology (Oxford: Oxford University Press, 2002); M. Steup (ed.) Knowledge, Truth, and Duty; Essays On Epistemic Justification, Responsibility, and Virtue (Oxford University Press, 2002). 6 See Badiou, 'Ontology is Mathematics' (op. cit.). 7 See especially Badiou, Metapolitics and Century; also Polemics. 8 Thomas S. Kuhn, The Structure of Scientific Revolutions, 2nd edn (Chicago: University of Chicago Press, 1970). 9 See entries under Note 7. 1 0 See for instance C. B. MacPherson, The Political Theory of Posses­ sive Individualism (Oxford: Oxford University Press, 1 962). 1 1 For a closely though not precisely analogous argument, see Louis Althusser, 'Ideology and Ideological State Apparatuses: Notes Toward an Investigation', in 'Lenin and Philosophy' and Other Essays (trans.) Ben Brewster (London: New Left Books, 1 97 1 ), pp. 1 27-86. 12 Ibid; see also Althusser, For Marx (trans.) Ben Brewster (London: Allen Lane, 1 969). 13 Jacques Lacan, Ecrits: a selection (trans.) Alan Sheridan-Smith (London: Tavistock, 1 977). 1 4 For further discussion, see Dale Jacquette (ed.), Philosophy of Logic: An Anthology (Oxford: Blackwell, 2002). 1 5 See especially Badiou, 'Ontology is Mathematics' (op. cit.). 16 See for instance Kurt Mueller-Vollmer (ed.), The Hermeneutics Reader (Oxford: Blackwell, 1 988); Richard Rorty (ed.), The Linguistic Turn (Chicago: University of Chicago Press, 1 967); Robert Stainton (ed.), Perspectives in the Philosophy of Language: A Concise Anthol­ ogy (New York: Broadview Press, 2000). 17 For some useful discussion of this focal shift from the 'way of ideas' to the 'way of words' (or from epistemology to philosophy of lan­ guage), see Ian Hacking, Why Does Language Matter to Philosophy? (Cambridge: Cambridge University Press, 1 975). 18 Spinoza, Ethics, in The Collected Writings of Spinoza (trans.) Edwin Curley (Princeton, NJ: Princeton University Press, 1 985). 294

NOTES

19 See Badiou, Deleuze: The Clamor of Being; also Gilles Deleuze,

Spinoza: Practical Philosophy (trans.) Robert Hurley (San Francisco: City Lights Books, 1 988) and Expressionism in Philosophy: Spinoza (trans.) Martin Joughin (New York: Zone Books, 1 992), along with the various references to Spinoza in Deleuze and Felix Guattari, A Thousand Plateaus (Capitalism and Schizophrenia, Vol. 2) (trans.) Brian Massumi (Minneapolis: University of Minnesota Press, 1 987). 20 See Etienne Balibar, Spinoza and Politics (London: Verso, 1 998); Jonathan Israel, Radical Enlightenment: Philosophy and The Making of Modernity, 1650-1 750 (Oxford: Oxford University Press, 2002); Yirmiyahu Yovel, Spinoza and Other Heretics, Vol. 1 , The Marrano of Reason, and Vol. 2, The Adventures of Immanence (Princeton NY: Princeton University Press, 1 989). 21 See for instance Louis Althusser, For Marx (trans.) Ben Brewster (op. cit.) and 'Elements of Self-criticism', in Essays in Self-critiCism (London: New Left Books, 1976), pp. 1 0 1 -6 1 ; Althusser and Etienne Balibar, Reading Capital (trans.) Brewster (New Left Books, 1 970); Balibar, Spinoza and Politics (London: Verso, 1 988); Pierre Mach­ erey, In a Materialist Way: Selected Essays (ed.) Warren Montag, (trans.) Ted Stolze (Verso, 1 998). 22 See Note 19. 23 Christopher Norris, Spinoza and the Origins of Modern Critical Theory (Oxford: Blackwell, 1 99 1 ). ,

24 Spinoza, Ethics (op. cit.) . 25 See for instance Ted Benton, The Rise and Fall of Structural Marx­ ism (London: New Left Books, 1 984) and Gregory Elliott, Althusser: The Detour of Theory (London: Verso, 1 987). 26 See Note 1 5. 27 See especially Badiou, 'Ontology is Mathematics' (op. cit.). 28 Spinoza, Ethics (op. cit.). 29 For a classic statement of the hard-line case, see D. M. Armstrong, A Materialist Theory of the Mind (London: Routledge & Kegan Paul, 1 968); also Paul M. Churchland, Scientific Realism and the Plasticity of Mind (Cambridge : Cambridge University Press, 1 979). 30 See Note 19. 3 1 Spinoza, Ethics (op. cit.). 32 Badiou, 'Ontology is Mathematics' (op. cit.). 33 Spinoza, Ethics (op. cit.); Badiou, 'Spin9za's Closed Ontology', in Theoretical Writings, pp. 8 1-93; also Norris, Spinoza and the Origins of Modern Critical Theory (op. cit. ) 34 See Note 20. .

Part I I I . Being: nature and infinity. Heidegger/Galileo

1 See especially Martin Heidegger, Poetry, Language and Thought (trans.) Albert Hofstadter (New York: Harper & Row, 1 971); also Early Greek Thinking (trans.) David Krell and Frank Capuzzi (Harper & Row, 1975) and Heidegger: Basic Writings (ed.) Krell (Harper & Row, 1 977). 295

NOTES

2 Martin Heidegger, Being and Time (trans.) John Mcquarrie and Edward Robinson (Oxford: Blackwell, 1 980). 3 For a useful historical-comparative account of these various con­ 4

5 6

ceptions, see Barry Allen, Truth in Philosophy (Cambridge, MA: Harvard University Press, 1 993). See for instance Heidegger, ' The Question Concerning Technology' and Other Essays (trans.) William Lovitt (New York: Harper & Row, 1 977) and 'The End of Philosophy and the Task of Thinking', in Time and Being (trans.) Joan Stambaugh (Harper & Row, 1 972). For further discussion, see A. 1. Ayer (ed.), Logical Positivism (New York: Free Press, 1 958). Some literary critics, among them I. A. Richards, tried to make the best of this situation by embracing the idea of poetry as an 'emo­ tive', that is, strictly non-cognitive or non-truth-apt mode of discourse See Richards, Principles of Literary Criticism (London: Kegan, Paul, Trench & Trubner, 1924). See especially Ludwig Wittgenstein, Remarks on the Foundations of Mathematics (ed.) G. H. von Wright, R. Rees and G. E. M. Ans­ combe (Oxford: Blackwell, 1 956); Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1 954); Crispin Wright, Wittgenstein on the Foundations of Mathematics (London: Duckworth, 1 980); Heidegger, 'The End of Philosophy and the Task of Thinking' and ' The Question Concerning Technology' (see Note 4). See Note l . See especially Badiou, Theoretical Writings, Polemics, and The Century; also De quoi Sarkozy est-il le nom? Paul Cohen, Set Theory and the Continuum Hypothesis (New York: Addison-Wesley, 1 966). Michael Dununett, Truth and Other Enigmas (London: Duckworth, 1978); Neil Tennant, Anti-realism and Logic (Oxford: Clarendon Press, 1 987); also - for a strongly opposed line of argument - Christopher Norris, Truth Matters: Realism, Anti-realism and Response-depen­ dence (Edinburgh: Edinburgh University Press, 2002). Fo r further discussion of these issues, see Michael Devitt, Realism and Truth, 2nd edn (Oxford: Blackwell, 1 986); also Norris, Episte­ mology: Key Concepts in Philosophy (London: Continuum, 2005). See Notes 1 and 2; also Christopher Norris, 'Settling Accounts: Heidegger, de Man and the Ends of Philosophy', in What's Wrong With Postmodernism: Critical Theory and the Ends of Philosophy (Hemel Hempstead: Harvester-Wheatsheaf, 1 990), pp. 222-83. See also Jean-Luc Nancy and Philippe Lacoue-Labarthe, The Literary Absolute: The Theory of Literature in German Romanticism (trans.) Philip Barnard and Cheryl Lester (Albany, NY: State University of New York Press, 1 988). Heidegger, Poetry, Language and Thought (op. cit.). Friedrich Nietzsche, The Birth of Tragedy out of the Spirit of Music (trans.) Shaun Whiteside, (ed.) Michael Tanner (Hannondsworth : Penguin, 1 993). .

7

8 9 10 11

12 13

14

15

16

296

NOTES

1 7 See for instance Jean-Jacques Leyercle, 'Badiou's Poetics', in Peter Hallward (ed.), Think Again: Alan Badiou and the Future of Philosophy (London: Continuum, 2004), pp. 25-47. 1 8 See Notes 1 3 and 14. 19 Tom Rockmore and Joseph Margolis (eds), The Heidegger Case: On Philosophy and Politics (Philadelphia, PA: Temple University Press, 2008); also Rockmore, On Heidegger's Nazism and Philosophy (Berkeley & Los Angeles: University of California Press, 1 992). 20 For an argument to somewhat similar effect, see Paul de Man, 'Heidegger's Exegeses of Holderlin', in Blindness and Insight: Essays in the Rhetoric of Contemporary Criticism, 2nd edn (London: Methuen, 1 983), pp. 24(H)6. 21 By far the most convenient source is the online, freely downloadable English translation of Mallarme's 'Un CQUP de des' by A. S. Kline which includes an introduction, the poet's original preface, and a complete French text. TItis may be accessed at [http://www. tonykline. co.ukIPITBRlFrenchlMallarmeUnCoupdeDes.htm]. See also Ste­ phane Mallarme, Selected Poetry and Prose (ed.) M. A. Caws (New York: New Directions, 1 982). For further detailed commen­ tary, see Malcolm Bowie, Mallarme and the Art of Being Difficult, 2nd edn (Cambridge: Cambridge University Press, 2008). 22 For a wide-ranging and highly perceptive survey of the field, see Graham Dunstan Martin, Language, Truth and �oetry: Some Notes Toward a Philosophy of Literature (Edinburgh: Edinburgh Univer­ sity Press, 1 975). 23 See for instance Gaston Bachelard, Le rationalisme applique (paris: Presses Universitaires de France, 1 949); The New Scientific Spirit (Boston: Beacon Press, 1 984); Georges Canguilhem, Ideology and Rationality in the History of the Life Sciences (trans.) A. Goldham­ mer (Cambridge, MA: MIT Press, 1 988); also Mary Tiles, Bachelard: Science and Objectivity (Cambridge: Cambridge University Press, 1 984). 24 For a useful survey of these developments in a broad intellectual­ historical context, see Peter Sedgwick, Descartes to Derrida: An Introduction to European Philosophy (Oxford: Blackwell, 200 1). 25 See especially Louis Althusser, For Marx (trans.) Ben Brewster (London: Allen Lane, 1 969) and Jacques Lacan, Ecrits: a selection (trans.) Alan Sheridan-Smith (London: Tavistock, 1977). 26 For a comparable critique of Heidegger's depth-hermeneutic and phenomenologically grounded approach to these questions, see Dale Jacquette, Ontology (Montreal: McGill-Queen's University Press, 2003). 27 Badiou, Saint Paul: The Foundation of Universalism. 28 See Note 1 7. 29 For the most extreme statement of this high-formalist view, see Veronica Forrest-Thompson, Poetic Artifice: A Theory of Twenti­ eth-century Poetry (Manchester: Manchester University Press, 1 978). 297

NOTES

30 For an exceptionally acute and wide-ranging commentary on the history of thinking about the infinite, see A. W. Moore, The Infinite (London: Routledge, 200 1). 3 1 For a first-rate online survey of these various schools of thought together with a useful selective bibliography, see Leon Horsten, 'Phi­ losophy of Mathematics', The Stanford Encyclopaedia of Philosophy [http://plato.stanford.edulentries/PhilOSOPhy-mathematicsl]· 3 2 ImmanueI Kant, CritiqUe of Pure Reason (trans.) N. Kemp Smith (London: Macmillan, 1964). 33 See especially Badiou, Manifesto for Philosophy and Theoretical Essays. 34 See Moore, The Infinite (op. cit.). 35 See Alexandre Koyre, From the Closed World to the Infmite Universe (New York: Harper & Row, 1 958). 36 On the rule following issue, see Ludwig Wittgenstein, Philosophical Investigations (op. cit.), Sections 20 1-92 passim; Saul Kripke, Witt­ genstein on Rules and Private Language: An Elementary Exposition (Oxford: Blackwell, 1 982); Alexander Miller and Crispin Wright (eds), Rule-following and Meaning (Chesham: Acumen, 2002). 37 Badiou makes this point with great argumentative as well as polemi­ cal force in the sequence of essays entitled 'Ontology is Mathematics', Section One of Theoretical Writings, pp. 3-93. 38 For a detailed �rvey of this history of thought, see 1. Alberto Coffa, The Semantic Tradition from Kant to Carnap: To the Vienna Station (Cambridge : Cambridge University Press, 199 1 ). 39 For a detailed discussion, see Gregory H. Moore, Zermelo's Axiom of Choice: Its Origins, Development and Influence (Berlin & New York: Springer Verlag, 1 982). 40 See Note 36. 41 G. W. F. Hegel, Science of Logic (trans.) A. V. Miller (London: Allen & Unwin, 1 959). 42 Hegel, Phenomenology of Mind (trans.) A. V. Miller (Oxford: Clarendon Press, 1 987). 43 Ibid. 44 For more detailed discussion see Moore, Infinity (op. cit.). 45 Stuart Barnett (ed.), Hegel After Derrida (London: Routledge, 1 998); Christopher M. Gemerchak, The Sunday of the Negative: Reading Bataille Reading Hegel (Albany, NY: State University of New York Press, 2003); Alexandre Kojeve, Introduction to the Read­ ing of Hegel (New York: Basic Books, 1969). 46 For the best-known and most widely influential instance, see Jean­ Franyois Lyotard, The Postmodern Condition: A Report on Knowledge (trans.) Geoff Bennington and Brian Massumi (Manchester: Manchester University Press, 1984). 47 See Note 25; also Pierre Macherey, In a Materialist Way: Selected Essays (ed.) Waren Montag, (trans.) Ted Stolze (London: Verso, 1 998). 48 See Note 25. 298

NOTES

Part IV. The event: history and ultra-one I G. W. F. Hegel, Phenomenology of Mind(trans.) A. V. Miller (O xfo rd :

Clarendon Press, 1 987). 2 Jean-Paul Sartre, A Critique of Dialectical Reason, Vol. I , Theory of Practical Ensembles (trans.) A. Sheridan-Smith (London: New Left Books, 1976) and Vol. 2 (trans.) Quintin Hoare (London: Verso, 2006). 3 See especially Badiou, Metapolitics and Polemics; also 'Philosophy and Politics' and 'Ontology and Politics: An Interview with Alain Badiou', in Infmite Thought, pp. 69-78 and 1 69-94. 4 Thomas S. Kuhn, The Structure 'of Scientific Revolutions (Chicago: University of Chicago Press, 1 970). 5 Roland Barthes, Mythologies (trans.) Annette Lavers (London: Paladin, 1 972). 6 Jean-Franc;ois Lyotard, The Postmodern Condition: A Report on Knowledge (trans.) Geoff Bennington and B rian Massumi (Man­ chester: Manchester University Press, 1 984). 7 See for instance Badiou, Metapolitics and The Century; also De quoi Sarkozy est-il le nom? 8 Badiou Manifesto for Philosophy. 9 For an early example of the kind, see Daniel Bell, The End of Ideol­ ogy: On the Exhaustion of Political Ideas in the Fifties (Glencoe, IL: Free Press, 1960); also more recently - Francis Fukuyama, The End of History and the Last Man (London: Hamish Hamilton, 1992). 10 See Kurt Mueller-Vollmer (ed.), The Hermeneutics Reader (Oxford: Blackwell, 1988); Richard Rorty (ed.), The Linguistic Turn (Chicago: University of Chicago Press, 1967); Robert Stainton (ed ), Perspec­ tives in the Philosophy of Language: A Concise Anthology (New York: Broadview Press, 2000). 1 1 See for instance - for a range of views - Keith Jenkins (ed.), The Postmodern History Reader (London: Routledge, 1 997). 1 2 See for instance Franc;ois Furet, Interpreting the French Revolution (trans.) Elborg Forster (Cambridge: Cambridge University Press, 1968); Furet and Mona Ozouf (eds), A Critical Dictionary of the French Revolution (trans.) Arthur Goldhammer (Cambridge, MA : Harvard University Press, 1 989). 1 3 Badiou, Infmite Th ough t, pp. 46-7. 14 Timothy Williamson, Knowledge and its Limits (Oxford: Oxford Uni­ versity Press, 2000). I should say that Williamson arrives at these questions - in particular the issue between realists and anti-realists con­ cerningthe(non-)existenceor(in)conceivabilityofunknownlunknowable truths - from a standpoint notably akin to Badiou's, despite their other­ wise large differences of philosophic idiom and orientation. Thus Williamson similarly sets out to show, contra anti-realists like Dwnmett, how we can indeed make sense - logically and epistemologi­ cally speaking - of the claim that such truths may be known to exist, and moreover that certain conditions may be specified for the kinds of ,

-

.

299

NOTES

15

16

17 18 19

advance in our powers of epistemic or cognitive grasp that would (even if at an impossible stretch for our present best state of understanding) bring them within reach. 'Once we acknowledge that the domain [that of unknowability] is non-empty, we can explore more effectively its extent. In order to be able to set a limit to knowledge, we do not have to fwd both sides of the limit knowable. Although, trivially, we cannot know that which we cannot know, we can know that we cannot know something' (pp. 300-01). And conversely, though by the same token, we can know or excogitate a great many things that we are not consciously aware of in the Cartesian sense of having them transparently, directly, reflectively or occurrently present to mind. Williamson is fully in agree­ ment with Badiou on these two major points, as likewise in adopting a 'classical' (i.e. bivalent or non-intuitionist) approach to mathematics, logic and the formal sciences. Note 10; also Ian Hacking, Why Does Language Matter to Philo­ sophy? (Cambridge: Cambridge University Press, 1 975). Jacques Lacan, Ecrits: A Selection (trans.) Alan Sheridan-Smith (London: Tavistock, 1 977). Hilary Kornblith (ed.), Epistemology: Internalism and Externalism (Oxford: Blackwell, 2001). See in particular Alvin Goldman, Pathways to Knowledge: Private and Public (Oxford: Oxford University Press, 2002). See Note 1 1 ; also Frank Ankersmit and Hans Kellner (eds), A New Philosophy of History (London: Reaktion, 1 995).

Part V. The event: intervention and .fidelity. Pascal/choice; Holderlin/deduction

2 3 4 5 6 7 8 9

10

For a representative selection, see Blaise Pascal, Pensees and Other Writings (ed.) A. and H. Levi (Oxford: World Classics, 1 999). Badiou, Manifesto for Philosophy. Pascal, Pensees (op. cit.). See also Badiou, Manifesto for Philosophy. See Badiou, 'Philosophy and Mathematics: Infinity and the End of Romanticism', in Theoretical Writings, pp. 2 1-38. For a sampling of views, see A. J. Ayer (ed.), Logical Positivism (New York: Free Press, 1 958). See for instance John Hick, Faith and Knowledge (London: Macmillan, 1 967). Badiou, Saint Paul: The Foundation of Universalism See Alan Hajek's excellent discussion of Pascal's Wager, download­ able from the Stanford Encyclopedia of Philosophy at [http://plato. stanford.eduJentries/pascal-wagerl]. See for instance 1. S. Mill, An Examination of Sir William Hamilton's Philosophy, and of the Principal Philosophical QuestiOns Discussed in his Writing (London: Longmans, Green & Dyer, 1 878) and Bertrand Russell, Why I am Not a Christian, and Other Essays on Religion and 300

NOTES

Related Subjects (ed.) Paul Edwards (London: Allen & Unwin, 1 957). 1 1 See Nicholas Hammond (ed.), The Cambridge Companion to Pascal (Cambridge: Cambridge University Press, 2003) and Robert J. Nelson, Pascal: Adversary and Advocate (Cambridge, MA: Harvard University Press, 1 982). 12 For some illuminating commentary, see A. W. Moore, The Infinite (London: Routledge, 200 1). 13 Immanuel Kant, Critique of Pure Reason (trans.) N. Kemp Smith (London: Macmillan, 1 964). 14 This distinction was frrst formulated by Hans Reichenbach in his book Experience and Prediction (Chicago, IL: University of Chicago Press, 1 938). 15 For further argument to this effect, see Christopher Norris, Against Relativism: Philosophy of Science, Deconstruction and Critical The­ ory (Oxford: Blackwell, 1 997) and Philosophy of Language and the Challenge to Scientific Realism (London: Routledge, 2004). 16 On this topic see especially Bernard Williams, Truth and Truthful­ ness (Princeton, NJ: Princeton University Press, 2004). 17 G. H. Moore, Zerlemo's Axiom of Choice: Its Origins, Development and Influence (New York: Springer Verlag, 1 982). 18 For further discussion, see E. Mendelson, Introduction to Mathe­ matical Logie, 4th edn (London: Chapman & Hall, 1 997). 19 Badiou, Deleuze: The Clamor of Being. 20 For a lucid survey, see G. Hughes and M. Cresswell, A New Intro­ duction to Modal Logic (London: Routledge, 1 996). 21 See Saul Kripke, Naming and Necessity (Oxford: Blackwell, 1 98 1); Hilary Putnam, Mind. Language and Reality (Cambridge: Cambridge University Press, 1975); Stephen P. Schwartz (ed.), Naming. Necessity and Natural Kinds (Ithaca, NY: Cornell University Press, 1 975). 22 For a thorough treatment of these topics, see especially Irving M. Copi and Carl Cohen, Introduction to Logic (New York: Prentice­ Hall, 2005); also - for a more concise exposition W. H. Newton-Smith, Logic: An Introductory Course (London: Routledge, 1 985). 23 W. V. Quine, Philosophy of Logic, 2nd edn (Cambridge, MA: Har­ vard University Press, 1986). 24 Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1 953). 25 For a useful discussion, see Anthony Weston, A Handbookfor Argu­ ments (Indianapolis, IN: Hackett, 200 1), p. 48ff. 26 Michael Dummett, Truth and Other Enigmas (London: Duckworth, 1 978) and Elements of Intuitionism, 2nd edn (Oxford: Oxford University Press, 2000). 27 Deleuze: The Clamor of Being 28 See Note 26; also - for Ii. dissenting view - Norris, Truth Matters: Realism. Anti-Realism. and Response-Dependence (Edinburgh: Edin­ burgh University Press, 2002). -

301

NOTES

29 For further discussion, see Crispin Wright, Realism, Meaning and Truth (Oxford: Blackwell, 1 987) and Truth and Objectivity (Cam­ bridge, MA: Harvard University Press, 1 992). 30 G. W. F. Hegel, Hegel's Logic (trans.) William Wallace (Oxford: Clarendon Press, 1975). Part VI. Quantity and knowledge. The discernible (or constructible): LeibnizlGodel 1 See especially Paul Benacerraf, 'What Numbers Could Not Be', in Benacerraf and Hilary Putnam (eds), The Philosophy of Mathematics: Selected Essays, 2nd edn (Cambridge: Cambridge University Press, 1 983), pp. 272-94; also W. D. Hart (ed ), The Ph ilosophy of Mathe­ matics (Oxford: Oxford University Press, 1996) and Hilary Putnam, Mathematics, Matter and Method (Cambridge University Press, 1 975). 2 For more on 'bald naturalism', see John McDowell, Mind and World (Cambridge, MA: Harvard University Press, 1 994). 3 Richard Harland, Superstructuralism: The Philosophy of Structural­ ism and Post-structuralism (London: Methuen, 1 987). 4 See Note 1; also Michael Dummett, Elements of IntUitionism, 2nd edn (Oxford: Clarendon Press, 2000) and Truth and Other Enigmas (London: Duckworth, 1978). 5 See especially the essays collected in Badiou, 'Ontology is Mathe­ matics', Section One of Theoretical Writings, pp. 3-93. 6 See Notes 1 and 4. 7 See Gilles Deleuze, Difference and Repetition (trans. ) Paul Patton (London: Continuum, 2004); Deleuze and Felix Guattari, The Anti­ Oedipus: Capitalism and Schizophrenia, Vol. 1 (trans.) Brian Massumi (Minneapolis: University of Minnesota Press, 1 987); Badiou, Deleuze: The Clamor of Being; also Todd May, 'Badiou and Deleuze on the One and the Many', in Peter Hallward (ed.), Think Again: Alain Badiou and the Future of Philosophy (London: Continuum, 2004), pp. 67-76. 8 Deleuze and Guattari, Anti-Oedipus (op. cit.). 9 Dale Jacquette, Ontology (Montreal: McGill-Queen's University Press, 2002). 10 Martin Heidegger, Being and Time (trans.) J. Mcquarrie and E. Robinson (Oxford: Blackwell, 1 962). 1 1 Jacquette, Ontology (op. cit.), p. 29. 12 Ibid, p. 79. 13 Ibid, p. 42. 14 Ibid, p. 45. 15 See G. W. Leibniz, New Essays on Human Understanding (trans. and ed.) Peter. Remnant and Jonathan Bennett (Cambridge: Cambridge University Press, 1 996); The Monadology and Other Philosophical Writings (trans.) Robert Latta (Oxford: Oxford University Press,

302

NOTES

16 17 18 19

1925); Philosophical Essays (trans.) Roger Ariew and Daniel Garber (Indianapolis, IN: Hackett, 1 989). See Note 1 5. See Gottlob Frege, Conceptual Notation and Related Articles (trans. and ed.) T. W. Bynum (Oxford: Clarendon Press, 1972). See Leibniz, Monadology (op. cit.). See Note 7. P art V I I . The generic: indiscernible and truth - the event: P. J. Cohen

1 See especially Jerrold 1. Katz,' Realistic Rationalism (Cambridge, MA: M.I. T. Press, 1998). 2 See Michael Dummett, Truth and Other Enigmas (London: Duck­ worth, 1 978); also - from a range of more-or-1ess qualified anti-realist standpoints - Crispin Wright, Truth and Objectivity (Cambridge, MA: Harvard University Press, 1992); Paul Benacerraf, 'What Num­

bers Could Not Be', in Benacerraf and Hilary Putnam (eds), The Philosophy of Mathematics: Selected Essays, 2nd edn (Cambridge: Cambridge University Press, 1 983), pp. 272-94; Neil Tennant, The Taming of the True (Oxford: Clarendon Press, 2002). 3 See for instance Michael Devitt, Realism and Truth, 2nd edn (Oxford: Blackwell, 1 986); Norris, Truth Matters: Realism, Anti-realism and Response-dependence (Edinburgh: Edinburgh University Press, 2002); Stathis Psillos, Scientific Rea/ism: How Science Tracks Truth (London: Routledge, 1 999). 4 Paul J. Cohen, Set Theory and the Continuum Hypothesis (New York: w.A. Benjamin, 1 966). 5 Badiou, 'The Subtraction of Truth', in Theoretical Writings,

pp. 97-1 60. 6 See for instance Richard Rorty (ed.), The Linguistic Turn: Essays

7 8 9 10 11

12 13

in Contemporary Philosophical Method (Chicago: University of Chicago Press, 1967). See Note 2; also Dummett, The Logical Basis of Metaphysics (London: Duckworth, 1991) and The Seas of Language (Oxford: Clarendon Press, 1 993). See Notes 2 and 7. Roland Barthes, Mythologies (trans.) Annette Lavers (London: Jonathan Cape, 1 972), 1 12. Jacques Lacan, Bcrits: A Selection (trans.) Alan Sheridan-Smith (London: Tavistock, 1 977). Jean-Jacques Rousseau, The Social Contract (trans.) 1. H. Tozer (London: Wordsworth Classics, 1998). Martin Heidegger, Being and Time (trans.) John Mcquarrie and Edward Robinson (Oxford: Blackwell, 1 962). For further discussion see David Boucher and Paul Kelly (eds), The Social Contract from Hobbes to Rawls (London: Routledge, 1 994).

303

NOTES

14 For a reading of Rousseau that is comparable in certain respects, see Jacques Derrida, Of Grammatology (trans.) G. C. Spivak (Baltimore, MD: Johns Hopkins University Press, 1 976). 1 5 See Note 1 3 ; also Gerald F. Gaus, Justificatory Liberalism: An Essay on Epistemology and Political Theory (New York: Oxford University Press, 1996). 1 6 William James, The Will to Believe and Other Essays in Popular Philosophy (New York: Dover, 1 956). 17 John Stuart Mill, On Liberty (ed.) Gertrude Himmelfarb (Har­ mondsworth: Penguin, 1 985). 18 Kurt Godel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems (trans.) B. Meltzer (New York: Basic Books, 1 962). 1 9 See especially Kurt Godel, 'What Is Cantor's Continuum Problem?', in Benacerraf and Putnam (eds), Philosophy of Mathematics (op. cit.), pp. 470-85; also Jerrold J. Katz, Realistic Rationalism (op. cit.). 20 See also Roger Penrose, The Emperor's New Mind (Oxford: Oxford University Press, 1 990). 2 1 On this hugely overworked topic, see Ludwig Wittgenstein, Philosophical Investigations (trans.) G. E. M. Anscombe (Oxford: Blackwell, 1 953), Sections 20 1 -92 passim; Saul Kripke, Witt­ genstein on Rules and Private Language: An Elementary Exposition (Oxford: Blackwell, 1 9 82); Alexander Miller and Crispin Wright (eds), Rule-following and Meaning (Chesham: Acumen, 2002) . 22 See especially Edmund Hussed, Logical Investigations, 2 vols (trans.) J. N. Finlay (New York: Humanities Press, 1 970). 23 Gottlob Frege, 'Review of Dr. E. Husserl's Philosophy of Arith­ metic', (trans.) E. W Kluge, Mind, Vol. 8 1 (July 1 972), pp. 321-37. 24 See for instance Donald Davidson, 'The Method of Truth in Meta­ physics', in Inquiries into Truth and Interpretation (Oxford: Clarendon Press, 1 984), pp. 1 99-214.

Part VI I I . Forcing: truth and the subject. Beyond Lacan

See for instance Anthony J. Cascardi, The Subject of Modernity (Cambridge: Cambridge University Press, 1 992); E. Cadava, P. Connor and J.-L. Nancy (eds), What Comes After the Subject? (London: Routledge, 1 99 1); Paul Smith, Discerning the Subject (Minneapolis: University of Minnesota Press, 1 988). 2 See Note 1; also - for a vigorous literary-theoretical take on this topic - Sean Burke, The Death and Return of the Author (Edinburgh: Edinburgh University Press, 1 992). 3 Baruch Spinoza, Ethics (trans.) Edwin Curley (London: Penguin, 1 996); also Christopher Norris, Spinoza and the Origins of Modern Critical Theory (Oxford: Blackwell, 1991). 1

304

NOTES

4 For a commanding overview, see Franyois Dosse,

History of Struc­ turalism, 2 vo1s (trans.) Deborah Glassman (Minneapolis: University

5

6 7 8

9 10

11 12

l3

14 15 16 17

18 19

20

of Minnesota Press, 1 997). See especially Jacques Lacan, Ecrits (trans.) Alan Sheridan (London: Tavistock, 1 977) and Louis Althusser, For Marx (trans.) Ben Brewster (Harmondsworth: Penguin, 1 969) . . See especially E. P. Thompson, The Poverty of Theory, 2nd edn (London: Merlin, 2004). See Michel Henry, The Genealogy of Psychoanalysis (trans.) Douglas Brick (Stanford, CA: Stanford University Press, 1 993) and David Macey, Lacan in Contexts (London: Verso, 1 988). For a good synoptic treatment,' see Peter Sedgwick, Descartes to Derrida: An Introduction to European Philosophy (Oxford: Blackwell, 2001); also Gary Gutting, French Philq�ophy in the Twentieth Century (Cambridge: Cambridge University Press, 2001 ) . Lacan, 'The Insistence o f the Letter in th e Unconscious', (trans.) Jacques Ehrmann, Yale French Studies, Nos. 3617 (1966), pp. 1 12-47. Donald Davidson, 'The Method of Truth in Metaphysics', in Inqui­ ries into Truth and Interpretation (Oxford: Clarendon Press, 1 984), pp. 1 99-214. Paul Cohen, Set Theory and the Continuum Hypothesis (New York: w.A. Benjamin, 1 966). Kurt Gode1, The Consistency of the Continuum-Hypothesis (prince­ ton, NJ: Princeton University Press, 1 940). For further discussion see 1. Ferreiros, 'The Notion of Cardinality and the Continuum Hypothesis', in Labyrinth of Thought: The History of Set Theory and its Role in Modern Mathematics (Basel: Birkhauser, 1 999), pp. 1 7 1-214. See 1. R. Lucas, The Conceptual Roots of Mathematics (London: Routledge, 2000) and Penelope Maddy, 'Believing the Axioms, I', Journal of Symbolic Logic, Vol. 53, No. 2 (1 988), pp. 48 1-5 1 1 . For further discussion see Thomas 1. Jech, The Axiom of Choice (New York: Dover, 1 973). See especially Richard Rorty, Objectivity, Relativism, and Truth (Cambridge: Cambridge University Press, 1 991). See Notes 5 and 9; also Lacan, The Language of the Self: The Function of Language in Psychoanalysis (trans.) Anthony Wilden (Baltimore, MD: Johns Hopkins Press, Baltimore, 1 968); The Seminar XI, The Four Fundamental Concepts of Psychoanalysis (ed.) Jacques-Alain Miller, (trans.) Alan Sheridan (New York: w.w. Norton, 1 977); The Seminar XX; Encore: On Feminine Sexuality, the Limits of Love and Knowledge (ed.) Miller, (trans.) Bruce Fink (New York: Norton, 1 998). Lacan, 'The Insistence of the Letter' (op. cit.). See for instance Raymond Tallis, Not Saussure: A Critique of Post­ Saussurean Literary Theory (Basingstoke: Macmillan, 1 988). On post-structuralism and its re1ativistlconstructivistlanti-realist excesses, see Christopher Norris, Uncritical Theory: Postmodernism, 305

NOTES

21 22

23 24 25 26

Intellectuals and The Gulf War (London: Lawrence and Wishart, 1 992) and Reclaiming Truth: Contribution to a Critique of Cultural Relativism (Lawrence and Wishart, 1 996). See Rene Descartes, 'Rules for the Direction of the Mind' in Descartes: Philosophical Writings (trans.) Peter Geach and G. E. M. Anscombe (London: Thomas Nelson, 1954), pp. 1 53-80. Immanuel Kant, Critique of Pure Reason (trans.) N. Kemp Smith (London: Macmillan, 1 964). See Chapter I, Book II, Division II of Part II, 'Transcendental Doctrine of Elements'. See especially Jean-Paul Sartre, The Transcendence of the Ego (trans.) Forrest Williams and Robert Kirkpatrick (New York: Noonday Press, 1 948). See Rorty, Philosophy and the Mirror of Nature (Oxford: Blackwell, 1 980). Kant, Critique of Pure Reason (op. cit.). For a perceptive commentary on this system of 'rotating chairman­ ship' between the various Kantian faculties, see Gilles Deleuze,

Kant's Critical Philosophy: The Doctrine of the Faculties (trans.) Hugh Tomlinson and Barbara HabbeIjam (London: Athlone Press, 1 984). 27 For more on this semantic history, see Raymond Williams, Keywords: A Vocabulary of Culture and Society, rev. edn (Oxford: Oxford Uni­ versity Press, 1 985). 28 Rorty, Philosophy and the Mirror of Nature (op. cit.); also Ian Hacking Why Does Language Matter to Philosophy? (Cambridge: Cambridge University Press, 1975). 29 See especially Badiou, Ethics. .

,

INDEX

absolute infinity 5 5 absolute knowledge (Hegel) 145, 206 activist engagement 44 agency 28-30 Althusser, Louis 58, 90, 99, 100, 1 49, 263-4 Althusserian structural Marxism 1 21-2 analytic philosophy 1 2, 1 5 , 21-2, 1 6 1 , 1 82 Anaximander 64 ancient Greece 23 culture of 1 1 5--16 drama 201-2 philosophy I OCr 1 5 anomalies 1 90, 1 98-9 anti-Cartesian metaphysics 1 02 anti-humanism 121-2 anti-philosophers 58, 1 6Cr7 anti-realism 22-3, 34, 35, 9 1 , 1 1 3, 1 83-90, 22 3, 225, 23 1 Aristotle 6 1 , 71-9, 83-4, 89-90, 1 06 , 1 07 , 1 29 , 1 64 , 243 arts 2, 1 9, 26, 27, 1 09, 1 9 1 , 202, 203 astronomy 24 atheism 1 69 Austin, 1. L. 3, 1 6 authenticity 1 25 autonomy 29 axiomatic-deductive reasoning 43, 45, 1 1 1 , 200, 254 axiomatic truth-procedure 7 1 axiom o f choice 1 38-43, 1 74-83, 268

Bachelard, Gaston 49, 58 bad infmity 146, 149 Barthes, Roland 1 55-6 Bataille, Georges 148 being 8-9, 1 1 , 37-9, 6 1 , 63, 64, 73, 26Cr7 vs. beings 2 1 concept o f 55 impasse of 269 subtractive character 68 being/event dichotomy 2Cr7, 32-3, 79-80, 1 85, 210 belonging 48, 58, 78, 79-88, 92, 103, 1 32, 1 55, 1 80, 1 8 1 bivalence 1 86 Boyle, Robert 72, 73, 77 breakthrough discoveries 1 9-20 Canguilhem, Georges 49, 58 Cantor, Georg 5, 39, 1 12, 1 28-9, 1 3Cr7, 1 67, 1 86, 1 92, 1 93, 1 95-7, 219, 225, 237-8, 267-8 Cantor's the orem 50-64, 66 cardinality 1 96, 267 cardinals 1 95, 1 96, 205 Camap, Rudolf 20-1 causal explanations 1 79 causation 64 Cavailles, Jean 84 change 37-9 chaos, order from 64 choice, axiom of 138-43, 1 74-83, 268 Christianity 1 34, 1 67-72 citizenship 244 cogito 27 1 , 273-4, 276, 280

307

INDEX

Cohen, Paul 6, 26, 33, 1 l 3, 22 1 , 225, 230-6, 25 1-2, 265-8 Cold War propaganda 1 4- 1 5 collectives and collectivity 27-8 combinations 2 1 0 commonalities 3 1 common-being 34-5 consciousness 1 26, 145, 1 6 1 , 275 conscious subject 1 2 1 conscious thought 1 62 consistency 1 55 consistent multiplicity 40-1 , 47-8, 62-5, 8 1 , 88, 1 75, 190, 227 constructivism 4 1 , 222, 223, 225 containment 52 context of discovery 76, 1 72, 1 77, 200, 2 1 6, 261 context of justification 76, 1 72, 1 77, 200, 2 1 6 continental philosophy 12, 1 5, 2 1 , 22, 35, 48, 1 82 contingency 2 1 0-1 1 continuum hypothesis 268 Copernican -Galilean astronomy 24 Copernican revolution l 3 1 , 277 cosmos, creation of 64 count-as-one 40-2, 46, 47, 50, 60-2, 68, 78, 79-89, 92-3, 94, 101-2, 104, 1 1 2, 147, 1 75, 224-6 counterintuitive truths 69 counter-state 1 8 1 count-of-the-count 86 critical theory 32, 67

Critique of Dialectical Reason

(Sartre) 1 53, 1 54, 1 56 crypto-theology l 35 cuI tural-aesthetic nationalism 1 1 6 cultural construction 1 55-6 cultural left 1 7 cultural relativism 1 5, 1 6-1 7, 1 8

cultural theory 3 1 , 67 culture-transcendent truths 57 Davidson, Donald 267 deductive reasoning 1 87 Deleuze, Gilles 43-5, 97-8, 99-1 00, 1 02, 1 78, 200-1 , 203 depth-hermeneutic approach 20-1 , 1 09 Descartes, Rene 30, 49, 58, 6 1 , 1 2 1 , 1 35, 1 66, 270-83 determinism 28, 96, 1 59, 2 1 1 diagonalization (Cantor) 1 96-7, 207, 2 1 9 difference 1 8, 3 1 , 5 1 , 66, 9 1 differential calculus 43, 20 1 , 2 1 5 discernibles 222 discoveries 1 9-20, 67, 69, 1 1 2-14, 21 1 discovery, context of 76-7, 1 73-4, 1 77, 1 99-200, 2 1 6, 26 1 disenfranchisement and social injustice 7-8, 89 disputed class (Dummett) 1 97 double-negation-elimination 1 84, 1 87 doxa 38 drama 1 1 6, 20 1 , 202 dualisms 86, 104 false 1-2 natural/non-natural 1 52 phenomenal/noumenal 1 3 1 Dummett, Michael 1 83, 1 86, 197 Dutch Free Republic 98 Easton, W B. 1 99 Easton's Theorem 1 99 ego 1 62 ego-psychology 229, 28 1 Einsteinian Relativity 24 empiricism 60 end-of-history thesis 1 57, 1 64 end-of-ideology thesis 1 57, 1 64 308

INDEX

forms of life (Wittgenstein) 3 1 , 58, 1 8 1 , 203 Fraenkel, Abraham 57 freedom 29, 104-- 5 free will 1 59 'free-world' liberalism 14-- 1 5 Frege, Gottlob 1 6, 39, 52, 53, 1 0 1 French philosophy 32, 49, 1 36, 149 French Resistance 84 French Revolution 1 52, 1 56, 1 60, 1 65 Freud, Sigmund 20, 27 1 , 272 Freudianism 229.

episteme 38 epistemically constrained truth-values 1 83-4 epistemology 10, 30, 84--5 , 1 61-4, 258 epochal events 1 56, 1 60, 1 9 1 , 2 1 1 errancy 64 eschatological verificationism 1 67 ethical judgements 1 62 ethics 67, 1 03, 1 77, 203 Ethics (Spinoza) 97-8 evental site 1 1 8, 1 79, 1 90, 195, 226 event(s) 9, 1 1 , 96, 1 55, 1 58-9 being and 32-3, 79-80, 266 epochal 1 56, 1 60, 1 9 1 , 21 1 historical 1 64 past 1 58-9 poem as 1 22-7 situations and 1 59, 1 68 truth and 1 60--5 unpredictability of 125-6 excess, theorem of the point of 8 1 excluded middle 1 84, 1 85, 1 87 exclusion 50, 67 excrescences 94-6, 1 53, 1 90 existence 38, 68, 1 93 existentialism 33 extensionalist approach (set theory) 52, 65, 90-- 1 fact/value distinction 66, 94 faith 1 66-8, 1 69-70 falsehood 1 84--5 Fermat's Last Theorem 76 fidelity 84, 1 09, 1 25, 140, 1 59, 1 64, 1 77-82, 1 99, 200, 225 finitism 1 34 first-order statements 54 forcing 33-4, 43, 1 1 3, 2 19-24, 232-5, 236-8, 265-6 formal languages 60--1

Galileo 1 32, 1 33, 1 35, 1 36, 1 92 general ontology 208-9 general will 241 , 242 generic, the (Cohen) 43, 1 1 3, 1 90, 206, 221-7, 233, 265 generic extension 256-7, 269 generic procedure 27, 29 German culture 1 1 6 God 1 34, 1 35, 1 4 1 , 1 67, 1 69-7 1 , 206, 2 1 3-14 GMel, Kurt 54--5 , 254, 259, 267-8 grand narratives 1 56 Greek drama 1 1 6, 201 , 202 Greek philosophy 1 06-1 5 group-in-fusion (Sartre) 1 53 Harland, Richard 1 94 Hegel, George Wilhelm Friedrich 6 1 , 1 29, 143-52, 1 56, 1 8 1 , 1 87 Heidegger, Martin 1 9-2 1 , 23, 30--1 , 73, 1 06-8, 1 1 4, 1 1 6, 1 1 8-19, 1 21-2, 208, 242-3 heliocentric hypothesis 1 3 5 hermeneutics 3, 4 1 , 270 historical judgements 1 63-4 historical revisionism 1 60 historical situations 1 56

309

INDEX

quantitative 1 44 religious belief and 1 67, 1 69-70 'sizes' of 1 93 supra-rational idea of 1 70 infinite modes 1 04-5 infinite sets 8 1 , 1 29-30, 267 multiple orders of 5 subsets of 5 internalism 1 62-3 intrinsic size 195 intuition 5 1 , 52, 1 2 1 intuitionists 22-3 intuition of objects 52

history 1 5 1-60 of the present 1 64 representing 1 60-5 Hobbes, Thomas 72, 77-8, 241 Holderlin, Friedrich 2 1 , 1 1 5-2 1 homogeneity o f nature 1 3 1 human agency 28-30 human rights 1 5 Hume, David 1 87 Husserl, Edmund (and phenomenology) 1 0Cr7, 1 2 1 -2, 255 hypotheses, far-reaching 44 identicals, indiscernibility of 2 1 2, 214 identity o f indiscemibles 3 8 identity politics 9 1 imaginary misrecognition 105 impasse of being 269 inclusion 48, 50, 66, 78, 79-89, 9 1 , 92, 1 03, 1 55, 1 80-1 , 1 92 incompleteness theorem (GOdel) 254 inconsistency 47-8, 62, 87-8 inconsistent multiplicity 39, 40- 1 , 4Cr9, 60, 62-4, 8 1 , 87, 1 75, 1 90, 227 indiscemibility 21 9-27, 25 1-9 indiscernibles 38, 1 90, 2 1 2, 214, 233-4, 265 indiscernment 269 individual 92 infinite/infinity 5, 55, 66, 74, 1 92, 204, 205-6 absolute 55 axiom of choice and 1 38-43 'bad' 146, 149 concept of 1 28-37, 142 Hegel on 143-5 1 of nature 1 35-7 paradoxes of 1 36 positive 145

Jacquette, Dale 208-9, 2 1 0 justification, context o f 76 Kant, Immanuel 1 , 1 5, 23, 30, 1 3 1 , 1 72, 277-8 knowledge 1 6 1 -2 absolute (Hegel) 145, 206 acquisition of 1 99-200, 25 1-9 advances in 56-7, 1 1 2-14, 202 consciousness and 1 6 1 generic procedure for 27 humanly attainable 278 limits of 23 mathematical 56-7, 1 1 3, 1 33, 1 83, 1 97, 220, 232-3, 245, 255 present-best state of 1 87, 1 92, 1 98, 220-8 problem of 84 transcendence of current 43 truth and 8-9, 23, 24-5, 34, 57, 62, 99, 1 02, 1 83-9, 1 92, 2 1 9-28, 230-2, 251-8, 261-2 Koyre, Alexander 1 36 Kuhn, Thomas 1 54

31 0

INDEX

Lacan, Jacques 20, 149, 1 62, 229, 263-5, 270-83 Lacanian psychoanalysis 28, 90, 122, 229-30, 272, 28 1 language 10, 23, 35, 67, 68, 149-50, 203, 262-3 depth-hermeneutical idea of 1 09 formal 60-1 logical 210-12 of mathematics 1 32-3 natural 57-8, 60, 1 32-3 ordinary 3, 58, 1 1 0, 1 50 philosophy of 1 63, 258 poetic 2 1 , 23, 109 of the situation 220 truth and 97 language-first approach 3-4, 223, 273 language-games (Wittgenstein) 1 5-1 6, 3 1 , 1 8 1 , 203 language-independent truths 57 Lautman, Albert 84 leap of faith 1 70, 200 Leibniz, Gottfried 24, 38, 58, 6 1 , 63, 64, 1 63, 1 8 1 , 210-1 8 Lenin, Vladimir 20, 95 les evenements 1 5 Levinas, Emmanuel 3 1 liberal democracy 7 , 92 liberal-humanism 29 liberalism, 'free-world' 14-1 5 liberal pluralism 1 6-1 7 life-forms 1 5-16, 3 1 , 58, 1 8 1 , 203 limit ordinal 142-3 linguistic constructivism 28 linguistic philosophy 3-4 linguistic tum 3-4, 1 5-1 6, 43, 1 57, 1 62, 2 1 2, 223, 261 literary criticism 1 1 8 Locke, John 241 logic 212

anti-realism and 1 83-9 modal 1 79-80 logical language 52, 210-12 logical rules 1 8 1 logico-semantic distinctions 2 1 logos 38 love 2, 26, 27, 202, 270 Lyotard, Jean-Fran�ois 3 1 , 1 56 Macherey, Pierre 149 Mallarme, Stephane 109, 1 1 5-1 6, 1 1 9-27, 142 many, pri0rity of over the one 49, 50 Marx, Karl 1 5, 20, 89 Marxism 75, 85, 90, 99, 1 00, 1 22, 199, 228-9 mathematical knowledge 56-7, 1 1 3, 1 33, 1 83, 1 97, 220, 232-3, 245, 255 mathematical thought, subtractive dimension of 69 mathematical truth 22-3, 219-20 mathematics 4-5, 9, 67, 68, 203 advances in 19, 56-7, 1 82, 200, 230 axiomatic-deductive approach to 45 as basis of ontological inquiry 1 0, 30, 32, 38-49, 67-8, 9S-9, 1 87, 1 92, 21 1 discourse of 23, 1 90-1 Hegel on 144-5 1 history of 144 language of 1 32-3 philosophy and 24-6, 30-1 , 10 1 poetry and 1 06-1 5, 1 20 politics and 7-8, 89, 94, 1 10, 1 12 pre-eminence of 35 membership (set theory') 50-1 , 52, 60, 78, 1 55 meta-narrative 148 meta-ontological truths 25, 83

31 1

INDEX

meta-ontology 25, 30, 3 1 , 46, 83 metaphysical determinism 2 1 1 metaphysics 20, 96, 1 06, 107, 1 20 metastructure 8 1 , 86 militants of truth 1 0, 1 8, 1 25, 1 40, 1 54, 1 68, 1 99, 2 1 7, 260 Mill, John Stuart 1 69, 247 mobilization 1 53, 1 54 modal logic 1 79-80 monist onto logy 96, 98, 1 03, 1 04, 2 1 2 monotheism 1 34 multiple(s) 67-8 comparison of 1 92-3 concept of the 42 indiscemibility of 25 1-2 multiplicity of 1 28-37, 1 97, 2 1 3 natural 1 3 1 -2, 1 93, 194 priority of over the one 39-40, 49 theory of the pure 50-64 typology of the 1 1 2 mUltiplicity 35, 39, 40, 46 consistent vs. inconsistent 39, 40-1 , 46-9, 60, 62-4, 8 1 , 87, 1 75, 1 90, 227 infinite, of multiples 1 28-37 of multiples 1 97, 2 1 3

necessitarianism 1 59 necessity-operator (modal logic) 1 80 negative theology 7 1 , 206 neopragmatism 4 1 Neurath, Otto 20 Nietzsche, Friedrich 1 1 6 non-being 6 1 , 1 1 1 , 208 non-bel onging 22 1 non-contradiction 2 1 2 non-Euclidean geometry 24 non-intuitive truths 69 non-natural 1 5 1-2 normality 1 14, 1 1 5 Nouveaux Philosophes 1 57 null set 68 objective truth 21-2, 1 02, 257 objectivist realism 253-4 objectJsubject dialectic 1 99, 200, 269 ontological advances 45 ontological inquiry 23-4 mathematics as basis of 1 0, 30, 32, 38-49, 67-8, 98-9, 1 87, 1 92, 2 1 1 subtractive character of 6 1 , 62 ontology 2, 8-1 1 , 1 7, 30, 33-4, 41-3, 79-80, 1 63, 1 82, 1 92-3, 206-7 general 208-9 limit of 200 meta-ontology 25, 30, 3 1 , 46, 83 monist 96, 98, 1 03, 1 04, 2 1 2 open 43 philosophy and 24-5 prerequisites for 40 realist 62 social-political 48 subtractive 72, 222-8, 25 1 oppressed minorities 7-8 order, from chaos 64

nationalism 1 1 6-18 National Socialism 1 1 8, 243 naturalism 1 89-20 1 natural language 57-8, 60, 1 32-3 natural multiples 1 3 1 -2, 1 93, 1 94 natural numbers 1 36 natural sciences 1 9, 1 9 1 , 203 natural situations 1 56 nature 1 08, 1 14, 1 1 5, 1 32, 1 55 homogeneity of 1 3 1 infinity o f 1 34-5, 1 36, 141 vs. non-natural 1 5 1-2 Nazis 243 Nazism 1 1 8, 243

31 2

INDEX

ordinals 1 30-1 , 1 3 1 , 141-2, 1 95 ordinary language 3, 58, 1 1 0, 1 50 orientations 20 1-10 paradox into concept 255 Parmenides (plato) 37-4 1 , 46-9,

68, 69 Pascal, Blaise 70-1 , 76, 1 65-74, 217 past events 1 58, 1 59 phenomenal/noumenal dualism 1 3 1 phenomenology 33

Phenomenology of Mind (Hegel) 144, 145, 1 48 philosophy Badiou's view of 2 5, 1 1 1 2, 24-5, 33 continental 2, 1 2, 1 5, 2 1 , 35, 48, 1 82 first 1 06- 1 5 o f language 1 63, 258 linguistic 3-4 mathematics and 24-6, 30-1 , -

-

101 meta-ontological role o f 30-1 ontology and 24-5 psychoanalysis of 75 relationship to other disciplines 1 9-20 task of 33 physics 74, 77, 1 34 Plato 23, 24, 30, 35, 56, 6 1 , 1 02, 1 06-7, 1 20, 277

Parmenides 37-4 1 , 46-9, 68-9 Timaeus 64

Platonism 37, 1 02, 254 pluralism 1 8 1 poetic intuition 121 poetry 20, 2 1 , 23 of H61derlin 1 1 5-22 of Mallarme 1 1 5, 1 22-7 mathematics and 1 06-1 5, 1 20

point of impossibility 78 political disenfranchisement 92 political emancipation 86 political judgements 1 62, 1 63 political justice 7 1 politically oppressed, mobilization of 1 54 political philosophy Kant's 1 5 set theory and 7-8 political realism 1 57-8 political representation 7 political theory 45 politics 2, 7, 19, 26, 27, 44, 67, 202, 203 mathematics and 7-8, 89, 94, 1 1 0, 1 1 2 rethinking of 1 5 1-60 Rousseau on 241-5 1 positive infmity 1 45 post-Cantorian set theory 19-20, 128 post-eventa1 truth 228-40 postmodernism 3, 41 post-structuralism 3, 28, 4 1 , 1 2 1 , 1 56, 1 57 power set 66, 8 1 , 83, 84, 1 29 practical syllogisms 1 64 presentation 45-6, 47, 6 1 , 1 38, 1 55, 201 primitive terms 46 pseudo-dichotomy 45 psychoanalysis 20, 28, 1 2 1 , 122, 1 62, 229-30, 264-5, 27 1 , 272, 28 1 psychoanalysis of philosophy 75 pure multiple 53 Pythagorean numerology 69 quantitative infinity 144 quantity 1 93 Quine, W. V. 38, 59-60

31 3

INDEX

radical empiricism 60 radicalism 1 57 rational autonomy 1 59 rationalist metaphysics 96 rationalists 49, 58 rational reconstruction (philosophylhistory of ideas) 48 Rawls, John 242 realist ontology 62 reason 49, 1 66-7, 211 , 214, 2 1 5 reasoning axiomatic-deductive 43, 45, 1 1 1 , 200, 254 deductive 1 87 more geometrico 97, 99, 1 00, 103-5 recognition-transcendent truth 62-3 reductio ad absurdum 1 83, 1 84, 1 88 reference 90 regime of interpretation 20 relativism 1 7, 24 religion 70, 1 66-70 religious belief 1 66-70 representation 88, 93, 1 55, 20 1 revolutionary praxis 1 53 revolutions 1 53-4, 1 56-9, 1 76 rights 29 Rorty, Richard 3-4, 3 1 , 1 66, 269, 270, 275 Rousseau, Jean-Jacques 241-5 1 rule following (Wittgenstein) 1 3 8-43 rules 1 37, 1 8 1 Russell, Bertrand 6 , 1 6, 39, 52, 53, 54, 55, 87, 1 0 1 , 1 28, 1 69 Russian Revoluti on 1 52-3 -

sans-papiers

7, 65 Sartre, Jean-Paul 28, 1 2 1 , 1 22, 1 53, 1 54, 1 56, 2 1 4 Saussure, Ferdinand de 263

Scanlon, Thomas 242 science 2, 26, 27, 1 78, 202 advances in 72, 1 83-9 history of 1 36 self-conscious subject 1 2 1 , 145, 274, 275 self-consciousness 1 6 1 selfhood 90 self-referring expressio ns 6, 54 semio-linguistics 263 sense 90 sense-certainty 144, 145 senses, evidence of the 38 September 1 1 , 200 1 1 60 sets vs. combinations 2 1 0 elements of 8 1 infinite 5, 8 1 , 1 29, 267 membership in 50, 5 1 , 52, 60, 65 null 68 power set 8 1 , 83, 1 29 subsets of 1 28, 1 29 transitive sets 1 3 1-2 void 68 without elements 68 set theory 5-7, 23, 26, 42, 1 86, 20 1 , 206, 207, 259 advances in 57, 67 advent of 9, 48 challenge of 52-3 impact of 83 modem 50-2, 105 paradoxes in 52-5 philosophy and 1 2 political philosophy and 7-8 post-Cantorian 1 9, 1 28 ZF system 57-60, 267 singularities 8, 1 1 7, 1 53, 1 54, 1 95, 214 situations 8, 4 1 , 42, 80, 8 1 , 228-40 events and 1 59, 1 68

31 4

INDEX

Social Contract (Rousseau) 241-5 1 social democracy 1 1 2 social justice 7, 66, 93, 242 social ontology 7, 8, 48 societal change 8 socio-political sphere 65 Socrates 49-50, 103 sophistry 24 sophists 58, 1 66, 203 Soviet Union 1 56, 1 60 Spinoza, Baruch 58, 6 1 , 63, 64,

79, 96-1 06, 1 63, 1 94, 2 1 1 , 241-2, 261 stability 87-8 Stalin, Joseph 1 56, 1 78 state 88, 89-90, 92, 95--6, 1 55 state of the situation 8, 80, 8 1 ,

86, 197 201 , 234, 25 1 -

st. Paul 70, 76, 125, 1 66, 1 68 strong programme (sociology of knowledge) 15, 24, 72, 77 structuralism 28, 90, 99, 121-2,

149-50, 1 56, 264 structure 8 1 Structure of Scientific Revolutions (Kuhn) 1 54 concept of 260, 278 conscious 121 link between truth and 30-1 relationship between event and 33 role of 244 theory of the 259-70 truth and 234-5, 261-2 subjectivity 22, 1 8 1-2 subject/object dialectic 1 99,

sufficient reason, principle of

(Leibniz) 2 1 2, 214

supernumerary elements 40, 1 75,

1 90-1 'superstructuralism' 1 94 supra-sensory forms (plato) 39 'talking cure' (psychoanalysis) 229 theology 55, 1 34-5 . negative 7 1 , 206 theorem of the point of excess 8 1-2 theory of the pure multiple 50-64 theory of the subject 259-70 Theory of Types' (Russell) 54, 1 28 Thermidorean political culture 1 57 Timaeus (Plato) 64 time 37-8 totalization 5 1

Tractatus Theologico-Politicus (Spinoza) 98 transcendence 194-5, 205 transfinite numbers 5, 55 transformative thought 201-10 transitive sets 1 3 1-2 transmissible thinking I I I Trotsky, Leon 1 56, 1 78 truth absolute 193 attainment of 43 axiomatic conception of 43 concept of 107 culture-transcendent 57 epistemically constrained 1 1 3,

1 83-4

200, 269

event and 1 60-5 falsehood and 184-5 knowledge and 8-9, 23, 24-5,

subject of truth 234-5 subjects 89 subsets 66, 1 28, 1 29 subtractive ontology 72, 222-8,

34, 57, 62, 99, 102, 1 83-9, 192, 219-28, 230-2, 25 1-8, 261-2

25 1

315

INDEX

truth (Cont'd) language and 97 language-first approach to 3-4 language-independent 57 link between subject and 30- I mathematical 22-3, 2 1 9-20 meta-ontological 25 militants of 10, 1 8, 1 25, 140, 1 54, 1 68, 1 99, 2 1 7, 260 non-intuitive 69 objective 21-2, 1 02, 257 ontological enquiry and 10-1 1 ontology and 42-3 post-evental 228-40 subject and 234-5, 261-2 subjectivity and 1 80-1 vs. truthfulness 1 73 universal 1 8, 3 1 , 70 verification-transcendent 1 83 193 truth-event 32, 269 truthfulness 84-5, 1 73, 261 truth-procedures 1 1 , 76, 1 88-9, 232, 259, 26 1 , 265, 268, 27 1 , 278-8 1 'two cultures' controversy 1 1 0 tyranny of the majority 247

United States 95 unity 39-40 universalism 67, 70, 9 1 universal truths 1 8, 3 1 universe, infinite 1 36 unknown, the 1 1 3 un-measure 74 vacuum, impossibility of (Aristotle) 71-9 validity 54, 266 veracity 277 veridical, concept of 222, 224 verification-transcendent truth-values 1 83 193 Vienna Circle 19, 20 virtue-based epistemology 84, 85 void 65-6, 7 1-9, 88, 1 10, 1 90, 214-1 5 void set 68 Western metaphysics 20, 1 06, 1 07, 1 20 Wiles, Andrew 76 Williamson, Timothy 1 6 1 Wittgenstein, Ludwig 3, 4, 1 5- 1 6, 3 1 , 4 1 , 1 08, 1 50-1 , 1 66, 1 8 1 , 1 82, 223 working mathematicians 24, 25 world-transfonnative events 1 6 1

ultra-one 1 60, 1 63, 1 67, 1 79, 1 90, 214 unconscious 1 62, 263, 265, 27 1 , 272 'Un coup de des' (Mallanne) 1 1 9, 1 22-1 26 undecidability-proof (Godel) 54

Zeno 38 Zennelo, Ernst 57 Zennelo's principle 60 ZF system 57-60, 267

31 6