'BreathtaklnQ. 8adiou .. announces a new epoch in philosophy.' SLAVOJ ZIZEK o Alain 8adiou + Number and Numbers Number and Numbers Alain Badiou Translated by Robin Mackay polity First published in French as Le Nombre et les lIombres <0 Editions du Seuil, 1990. This English edition © Polity Press, 2008 Reprinted in 2008 Polity Press 65 Bridge Street Cambridge CB2 1 UR, UK. Polity Press 350 Main Street Malden, MA 02148, USA All rights reserved. Except for the quotation of short passages for the purpose of criticism and review, no part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. ISBN-13: 978-07456-3878-2 ISBN-13: 978-07456-3879-9 (pb) A catalogue rewrd for this book is available from the British Library. Typeset in 10.5 on 12 pt Sabon by SNP Best-set Typesetter Ltd., Hong Kong Printed and bound in the United States by Odyssey Press Inc., Gonic, New Hampshire For further information on Polity, visit our website: www.polity.co.uk This book is supported by the French Ministry for Foreign Affairs, as part of the Burgess programme headed for the French Embassy in London by the Institut Fran .. ais du Royaume-Uni. www.frenchbooknews.com r••i institut francais ~ Contents Translator's Preface vii 0 Number Must Be Thought Genealogies: Frege, Dedekind, Peano, Cantor 5 I Greek Number and Modern Number 7 2 Frege 16 3 Additional Note on a Contemporary Usage of Frege 24 4 Dedekind 31 5 Peano 46 6 Cantor: 'Well-Ordered ness' and the Ordinals 52 2 Concepts: Natural Multiplicities 59 7 Transitive Multiplicities 61 8 Von Neumann Ordinals 67 9 Succession and Limit. The Infinite 73 10 Recurrence, or Induction 83 II Natural Whole Numbers 93 vi CONTENTS 3 Ontology of Number: Definition, Order, Cuts, Types 99 12 The Concept of Number: An Evental Nomination 101 ADDITIONAL NOTES ON SETS OF ORDINALS 112 13 Difference and Order of Numbers 115 14 The Concept of Sub-Number 131 15 Cuts: The Fundamental Theorem 139 16 The Numberless Enchantment of the Place of Number 156 4 Operational Dimensions 181 17 Natural Interlude 183 18 Algebra of Numbers 198 Conclusion 209 19 In Conclusion: From Number to Trans-Being 211 Notes 215 Index 231 Translator's Preface Alain Badiou's Number and Numbers, first published two years after his Being and Event, is far from being the specialist work its title might suggest. In fact, it recapitulates and deepens Being and Event's explosion of the pretexts upon which the 'philosophy of mathematics' is reduced to a theoretical ghetto; and their kinship to those reaction ary modes of thought that systematically obscure the most pressing questions for contemporary philosophy. Neither does Number and Numbers balk at suggesting that even the greatest efforts on the part of number-theorists themselves have fallen short of the properly radical import of the question of number. Badiou's astonishing analy ses in the historical section of the book uncover the inextricable bond between philosophical assumptions and mathematical approaches to the problem in these supposedly 'merely technical' works. The aim of Number and Numbers, then, is certainly not to mould the unwill ing reader into a calculating machine, or a 'philosopher of mathemat ics': its exhortation is that we (mathematicians, philosophers, subjects under Capital) systematically think number out of the technical, procedural containment of which its quotidian tyranny, and the abysmal fear it strikes into the heart of the non-mathematician, are but symptoms. Symptoms, needless to say, whose expression within the situation of philosophy is a pronounced distaste for number-as philosopheme - whence its recognisable absence in much 'continental philosophy', except where it is pilloried as the very nemesis of the ontological vocation. So if the 'return of the numerical repressed' proposed here will, by definition, excite a symptomatic resistance, for viii TRANSLATOR'S PREFACE Badiou it alone can clear the way for the proper task of philosophy; as a working-through of the mathematical ontology presented in Being and Eve1Zt, Number and Numbers is a thorough conceptual apprenticeship preparatory to the thinking of the event. For the great thinkers of number-theory at the end of the nine teenth century, the way to an ontological understanding of number was obscured by calculatory and operational aspects. Today, accord ing to Badiou, the political domination of number under capitalism demands that the project be taken up anew: only if contemporary philosophy rigorously thinks through number can it hope to cut through the apparently dense and impenetrable capitalist fabric of numerical relations, to think the event that can 'subtract' the subject from that 'ontic' skein without recourse to an anti-mathematical romanticism. Whilst this doubtless demands 'one more effort' on the part of the non-mathematician, it would be a peevish student of philosophy who, understanding the stakes and contemplating the conceptual vista opened up, saw this as an unreasonable demand - especially when Badiou offers to those lacking in mathematical knowledge the rare privilege of taking a meticulously navigated conceptual shortcut to the heart of the matter. Badiou's remarkable book comprises a number of different works - a radical philosophical treatise, a contribution to number-theory, a document in the history of mathematics, a congenial textbook and a subtle and subversive exercise in political theory - whose intricate interdependencies defy any order of priority. The translator's task is to reproduce, with a foreign tongue, that unique voice that can compel us to 'count as one' these disparate figures. In negotiating this challenge, I have sought to prioritise clarity over adherence to any rigid scheme of translation, except where mathematical terminology demands consistent usage, or where an orthodoxy is clearly already in force within extant translations of Badiou's work. In the latter case, my references have been Oliver Feltham's landmark translation of Being and Event, with which I have sought to harmonise key terms, I Peter Hallward's invaluable A Subject to Truth/ and Ray Brassier and Alberto Toscano's collection of Badiou's Theoretical Writings.3 Apart from these, in translating chapters 2 and 3 I referred closely to Sam Gillespie and Justin Clemens' translation in UMBR(a), Science and Truth (2000). Finally, whilst seeking also to maintain continuity with long-standing English translations of number-theoretical works, some classics in their own right, occasionally the rigour of Badiou's thinking has demanded a re-evaluation of their chosen translations for key terms.4 Translators also find themselves obliged to arbitrate TRANSLATOR'S PREFACE ix between a fidelity to Badiou's in many ways admirable indifference to the pedantic apparatus of scholarly citation, and the temptation to pin down the allusions and quotations distributed throughout his work. Badiou's selection of texts is so discerning, however, that it is hardly a chore to return to them. Having thus had frequent recourse to the texts touched on in Number and Numbers (particularly in the first, historical part), I have seen no reason not to add citations where appropriate. One presumes that those self-conscious styles of philosophical writing that necessitate laboured circumlocutions or terminological preciosity on the part of a translator would for Badiou fall under the sign of 'modern sophistry', taken to task herein, as elsewhere in his work. Nevertheless, the aspiration to universal conceptual transpar ency does not preclude consideration of Badiou as stylist: firstly, as Oliver Feltham has remarked, Badiou's sentences utilise subject/verb order in a characteristic way, and I have retained his tensile syntax whenever doing so does not jeopardise comprehension in translation. Perhaps just as importantly, Badiou does not achieve the deft and good-humoured development of such extremely rich and complex conceptual structures as are found in Number and Numbers without a generous and searching labour on behalf of the reader, not to mention a talent for suspense. Although the later sections of Number and Numbers may seem daunting, I hope to have reproduced Badiou's confident, meticulous, but never stuffy mode of exposition so as to ease the way as much as possible. In fact, in contrast to his own occasionally chilly edicts, I would venture to suggest that here, 'in his element', Badiou allows himself a certain enthusiasm. One certainly does not accompany him on this odyssey without also developing a taste for the 'bitter joy' of Number. This translation slowly came to fruition on the basis of a somewhat impulsive decision; it may not have survived to completion without the enthusiasm and aid of an internationally dispersed group of friends and acquaintances, actual and virtual, with whom I shared the work in progress. I would like to extend my thanks to those who helped by pointing out errors and offering advice on the evolving manuscript: Anindya Bhattacharyya, Ray Brassier, Michael Carr, Howard Caygill, Thomas Duzer, Zachary L. Fraser, Peter Hallward, Armelle Menard Seymour, Reza Negarestani, Robin Newton, Nina Power, Manuela Tecusan, Alberto Toscano, Keith Tilford, David Sneek, and Damian Veal. My thanks also to Alain Badiou for his generous help and encouragement, and to the Institution and Staff of the Bodleian, Taylor Institution, and Radcliffe Science Libraries in Oxford. Part of my work on the translation was undertaken whilst X TRANSLATOR'S PREFACE in receipt of a studentship from the Centre for Research in Modern European Philosophy at Middlesex University, London. My greatest debt of gratitude is to Ruth, without whose love and understanding my battles with incomprehension could not even be staged; and to Donald, a great inspiration, for whom the infinite joys of number still lie ahead. Robin Mackay o Number Must Be Thought 0.1. A paradox: we live in the era of number's despotism; thought yields to the law of denumerable multiplicities; and yet (unless perhaps this very default, this failing, is only the obscure obverse of a concept less submission) we have at our disposal no recent, active idea of what number is. An immense effort has been made on this point, but its labours were essentially over by the beginning of the twentieth century: they are those of Dedekind, Frege, Cantor, and Peano. The factual impact of number only escorts a silence of the concept. How can we grasp today the question posed by Dedekind in his 1888 treatise, Was sind und was sollen die Zahlen?l We know very well what numbers are for: they serve, strictly speaking, for everything, they provide a norm for All. But we still don't know what they are, or else we repeat what the great thinkers of the late nineteenth century - anticipating, no doubt, the extent of their future jurisdiction - said they were. 0.2. That number must rule, that the imperative must be: 'count!' - who doubts this today? And not in the sense of that maxim which, as Dedekind knew, demands the use of the original Greek when retraced: &d 0 av8Qu)3to; &QL8~11'tL~EL2 - because it prescribes, for thought, its singular condition in the matheme. For, under the current empire of number, it is not a question of thought, but of realities. 0.3. Firstly, number governs our conception of the political, with the currency - consensual, though it enfeebles every politics of the