Mathematics of the Transcendental: Onto-logy and being-there

291 Pages·2014·3.1 MB·English

by  Alain Badiou

MATHEMATICS
OF THE
TRANSCENDENTAL
ALSO AVAILABLE FROM BLOOMSBURY
Being and Event, Alain Badiou
Conditions, Alain Badiou
Infinite Thought, Alain Badiou
Logics of Worlds, Alain Badiou
Theoretical Writings, Alain Badiou
Theory of the Subject, Alain Badiou
MATHEMATICS
OF THE
TRANSCENDENTAL
Alain Badiou
Edited, translated and with an
introduction by A. J. Bartlett
and Alex Ling
Bloomsbury Academic
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First published 2014
© Alain Badiou, 2014
This English translation © Bloomsbury Publishing PLC, 2014
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical, including
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without prior permission in writing from the publishers.
Alain Badiou has asserted his right under the Copyright, Designs and Patents
Act, 1988, to be identified as Author of this work.
No responsibility for loss caused to any individual or organization acting on
or refraining from action as a result of the material in this publication can be
accepted by Bloomsbury Academic or the author.
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN: ePDF: 978-1-4411-5043-1
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress.
Typeset by Fakenham Prepress Solutions, Fakenham, Norfolk NR21 8NN
TABLE OF CONTENTS
Translators’ Introduction: The Categorial Imperative 1
PART ONE TOPOS, OR LOGICS OF
ONTO-LOGY: AN INTRODUCTION FOR
PHILOSOPHERS
11
1 General Aim 13
2 Preliminary Definitions 17
3 The Size of a Category 21
4 Limit and Universality 27
5 Some Fundamental Concepts 29
6 Duality 37
7 Isomorphism 41
8 Exponentiation 45
9 Universe, 1: Closed Cartesian Categories 51
10 Structures of Immanence, 1: Philosophical
Considerations 55
11 Structures of Immanence, 2: Sub-Object 59
12 Structures of Immanence, 3: Elements of an Object 63
13 ‘Elementary’ Clarification of Exponentiation 67
14 Central Object (or Sub-Object Classifier) 71
15 The True, the False, Negation and More 77
16 The Central Object as Linguistic Power 85
17 Universe, 2: The Concept of Topos 89
18 Ontology of the Void and Difference 95
19 Mono., Epi., Equ., and Other Arrows 99
20 Topoi as Logical Places 113
21 Internal Algebra of 1 123
22 Ontology of the Void and Excluded Middle 141
23 A Minimal Classical Model 147
24 A Minimal Non-Classical Model 151
PART TWO BEING THERE: MATHEMATICS
OF THE TRANSCENDENTAL
163
Introduction 165
A. Transcendental Structures 171
B. Transcendental Connections 183
B.1. Connections between the transcendental and set-theoretic
ontology: Boolean algebras 183
B.2. Connections between the transcendental and logic in its
ordinary sense (propositional logic and first order predicate
logic) 195
vi TABLE OF CONTENTS
B.3. Connection between the transcendental and the general
theory of localizations: Topology 202
C. Theory of Appearing and Objectivity 217
D. Transcendental Projections: Theory of Localization 235
E. Theory of Relations: Situation as Universe 249
Appendix: On Three Different Concepts of Identity Between Two
Multiples or Two Beings 265
Translator’s Endnotes 269
Index 277
TABLE OF CONTENTS vii

TRANSLATORS’
INTRODUCTION:
THE CATEGORIAL
IMPERATIVE
There is no difference between what I have done and what
such philosophers as Plato, Descartes, Leibniz, or Hegel have
done, a hundred times over since the very origins of our
discipline: reorganizing a thorough, if not creative, knowledge of
mathematics, by means of all the imagining powers of language.
BEING AND EVENT, XIV
I hope that I say nothing imprecise in mathematics, but also
nothing that is mathematically proffered.
THEORY OF THE SUBJECT, 209–10
The two works made available here for the first time in either English or
French – Topos, or Logics of Onto-Logy: An Introduction for Philosophers
and Being-There: Mathematics of the Transcendental – represent Badiou’s
early forays into category theory (and in particular the sub-branch of
topos theory) from two vastly different philosophical angles. While the
former comes hot on the heels of the publication of his first ‘great book’,
Being and Event, the latter captures in painstaking detail the conceptual
development of what would ultimately become the belated ‘sequel’
to this foundational work, Logics of Worlds. As such, they catalogue
Badiou’s philosophico-mathematical trajectory over a period of nearly two
decades – a period of astonishing philosophical and artistic productivity