Number and Numbers - Alain Badiou - E-Book

Number and Numbers E-Book

Alain Badiou

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Beschreibung

The political regime of global capitalism reduces the world to an endless network of numbers within numbers, but how many of us really understand what numbers are? Without such an understanding, how can we challenge the regime of number? In Number and Numbers Alain Badiou offers an philosophically penetrating account with a powerful political subtext of the attempts that have been made over the last century to define the special status of number. Badiou argues that number cannot be defined by the multiform calculative uses to which numbers are put, nor is it exhausted by the various species described by number theory. Drawing on the mathematical theory of surreal numbers, he develops a unified theory of Number as a particular form of being, an infinite expanse to which our access remains limited. This understanding of Number as being harbours important philosophical truths about the structure of the world in which we live. In Badiou's view, only by rigorously thinking through Number can philosophy offer us some hope of breaking through the dense and apparently impenetrable capitalist fabric of numerical relations. For this will finally allow us to point to that which cannot be numbered: the possibility of an event that would deliver us from our unthinking subordination of number.

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Table of Contents

Title page

Copyright page

Translator's Preface

Notes

0: Number Must Be Thought

Notes

1: Genealogies: Frege, Dedekind, Peano, Cantor

1: Greek Number and Modern Number

Notes

2: Frege

Notes

3: Additional Note on a Contemporary Usage of Frege

Notes

4: Dedekind

Notes

5: Peano

Notes

6: Cantor: ‘Well-Orderedness’ and the Ordinals

Notes

2: Concepts: Natural Multiplicities

7: Transitive Multiplicities

Notes

8: Von Neumann Ordinals

Notes

9: Succession and Limit. The Infinite

Notes

10: Recurrence, or Induction

Notes

11: Natural Whole Numbers

3: Ontology of Number: Definition, Order, Cuts, Types

12: The Concept of Number: An Evental Nomination

Additional Notes on Sets of Ordinals

Notes

13: Difference and Order of Numbers

Notes

14: The Concept of Sub-Number

Notes

15: Cuts: The Fundamental Theorem

Notes

16: The Numberless Enchantment of the Place of Number

Notes

4: Operational Dimensions

17: Natural Interlude

Notes

18: Algebra of Numbers

Notes

19: In Conclusion: From Number to Trans-Being

Notes

Index

End User License Agreement

Guide

Cover

Table of Contents

Start Reading

Preface

CHAPTER 1

Index

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Copyright page

First published in French as Le Nombre et les nombres © Editions du Seuil, 1990.

This English edition © Polity Press, 2008

Polity Press

65 Bridge Street

Cambridge CB2 1UR, 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 record 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 Great Britain by MPG Books Limited, Bodmin, Cornwall

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 Français du Royaume-Uni.

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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 reactionary 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 analyses 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 unwilling reader into a calculating machine, or a ‘philosopher of mathematics’: 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 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 Event, 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 nineteenth century, the way to an ontological understanding of number was obscured by calculatory and operational aspects. Today, according 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,1 with which I have sought to harmonise key terms, Peter Hallward's invaluable A Subject to Truth,2 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 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 transparency 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 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

Notes

1

    London: Continuum, 2005.

2

    Minneapolis: University of Minnesota Press, 2003.

3

    London: Continuum, 2004; in particular, ch. 5, ‘The Being of Number’, represents an extremely condensed gloss of the present work.

4

    See for example ch. 2n 4.

0Number 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 conceptless 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?1 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: ἀεὶ ὁ ἄνθϱωπος ἀϱιθμητίζει2 – 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 thinkable – of suffrage, of opinion polls, of the majority. Every ‘political’ convocation, whether general or local, in polling-booth or parliament, municipal or international, is settled with a count. And every opinion is gauged by the incessant enumeration of the faithful (even if such an enumeration makes of every fidelity an infidelity). What counts – in the sense of what is valued – is that which is counted. Conversely, everything that can be numbered must be valued. ‘Political Science’ refines numbers into sub-numbers, compares sequences of numbers, its only object being shifts in voting patterns – that is, changes, usually minute, in the tabulation of numbers. Political ‘thought’ is a numerical exegesis.

0.4.    Number governs the quasi-totality of the ‘human sciences’ (although this euphemism can barely disguise the fact that what is called ‘science’ here is a technical apparatus whose pragmatic basis is governmental). Statistics invades the entire domain of these disciplines. The bureaucratisation of knowledges is above all an infinite excrescence of numbering.

At the beginning of the twentieth century, sociology unveiled its proper dignity – its audacity, even – in the will to submit the figure of communitarian bonds to number. It sought to extend to the social body and to representation the Galilean processes of literalisation and mathematisation. But ultimately it succumbed to an anarchic development of this enterprise. It is now replete with pitiful enumerations that serve only to validate the obvious or to establish parliamentary opportunities.

History has drawn massively upon statistical technique and is – even, in fact above all, under the auspices of academic Marxism – becoming a diachronic sociology. It has lost that which alone had characterised it, since the Greek and Latin historians, as a discipline of thought: its conscious subordination to the real of politics. Even when it passes through the different phases of reaction to number – economism, sociologism – it does so only to fall into their simple inverse: biography, historicising psychologism.

And medicine itself, apart from its pure and simple reduction to its scientific Other (molecular biology), is a disorderly accumulation of empirical facts, a huge web of blindly tested numerical correlations.

These are ‘sciences’ of men made into numbers, to the saturation point of all possible correspondences between these numbers and other numbers, whatever they might be.

0.5.    Number governs cultural representations. Of course, there is television, viewing figures, advertising. But that's not the most important thing. It is in its very essence that the cultural fabric is woven by number alone. A ‘cultural fact’ is a numerical fact. And, conversely, whatever produces number can be culturally located; that which has no number will have no name either. Art, which deals with number only in so far as there is a thinking of number, is a culturally unpronounceable word.

0.6.    Obviously, number governs the economy; and there, without a doubt, we find what Louis Althusser would have called the ‘determination in the last instance’ of its supremacy. The ideology of modern parliamentary societies, if they have one, is not humanism, law, or the subject. It is number, the countable, countability. Every citizen is expected to be cognisant of foreign trade figures, of the flexibility of the exchange rate, of fluctuations in stock prices. These figures are presented as the real to which other figures refer: governmental figures, votes and opinion polls. Our so-called ‘situation’ is the intersection of economic numericality and the numericality of opinion. France (or any other nation) can only be represented on the balance-sheet of an import–export business. The only image of a country is this inextricable heap of numbers in which, we are told, its power is vested, and which, we hope, is deemed worthy by those who record its mood.

0.7.    Number informs our souls. What is it to exist, if not to give a favourable account of oneself? In America, one starts by saying how much one earns, an identification that is at least honest. Our old country is more cunning. But still, you don't have to look far to discover numerical topics that everyone can identify with. No one can present themselves as an individual without stating in what way they count, for whom or for what they are really counted. Our soul has the cold transparency of the figures in which it is resolved.

0.8.    Marx: ‘the icy water of egotistical calculation’.3 And how! To the point where the Ego of egoism is but a numerical web, so that the ‘egotistical calculation’ becomes the cipher of a cipher.

0.9.    But we don't know what a number is, so we don't know what we are.

0.10.    Must we stop with Frege, Dedekind, Cantor or Peano? Hasn't anything happened in the thinking of number? Is there only the exorbitant extent of its social and subjective reign? And what sort of innocent culpability can be attributed to these thinkers? To what extent does their idea of number prefigure this anarchic reign? Did they think number, or the future of generalised numericality? Isn't another idea of number necessary, in order for us to turn thought back against the despotism of number, in order that the Subject might be subtracted from it? And has mathematics simply stood by silently during the comprehensive social integration of number, over which it formerly had monopoly? This is what I wish to examine.

Notes

1

  [Dedekind, R.,

Was sind und was sollen die Zahlen

(Braunschweig: F. Vieveg, 1888); English translation

The Nature and Meaning of Numbers

, in Beman, W. W. (ed., trans.),

Essays on the Theory of Numbers

(La Salle, IL: Open Court, 1901; reprinted NY: Dover 1963). Badiou's reference is to the translation by J. Milner, with H. Sinaceur's introduction,

Les Nombres, que sont-ils et à quoi servent-ils?

(Paris: Navarin, 1979). All references given below are to the numbered sections of Dedekind's treatise. – trans.]

2

  [

aei o anthropos arithmetizei

– ‘man is always counting’ – in Dedekind, ‘Numbers’, Preface to the first edn. – trans.]

3

  [Marx and Engels,

Communist Manifesto

, translated by S. Moore, with introduction and notes by G. S. Jones (London: Penguin, 2002), p. 222. – trans.]

1Genealogies: Frege, Dedekind, Peano, Cantor