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Naturalising Badiou Naturalising Badiou Mathematical Ontology and Structural Realism Fabio Gironi IRC Postdoctoral Research Fellow in Philosophy, University College Dublin, Ireland © Fabio Gironi 2015 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2015 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries ISBN: 978–1–137–46346–3 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. To Jasper, my companion at the bonfire Contents Acknowledgements i x Note about Citations xii Introduction 1 1 Badiou’s Mathematical Ontology 9 1.1 Death(s) of God: for axiomatic immanence 9 1.2 Subtractive ontology 12 1.3 Set theory 21 1.4 The Void 27 1.5 Secularising infinity 29 2 The Ontological and the Empirical: Naturalist Objections 3 4 2.1 Pythagoreanism and materialism 36 2.2 Philosophy of mathematics to the rescue 4 1 2.3 The applicability of mathematics 4 5 2.4 Naturalist explanations of mathematics 4 9 2.5 Two naturalising strategies 57 2.6 The matherialist option 60 3 Taking a Stance on Realism and Naturalism 63 3.1 Scientific realism 6 4 3.2 The realist stance 6 7 3.3 Transgressive naturalism 72 3.4 Historicist excursus: Collingwood and Bachelard 7 5 3.5 The realist’s task 80 4 Structural Realisms 86 4.1 Scientific realism, its adbuctive defence ... and its discontents 86 4.2 Epistemic structural realism 90 4.3 Ontic structural realism 93 4.4 Mathematical structuralism 102 4.5 Structures of what? 108 vii viii Contents 5 Truth and Randomness 112 5.1 Against fundamental ontology 1 12 5.2 Idealism and rationality 119 5.3 Truth as abductive process 124 5.4 Metamathematics and randomness 1 35 Conclusion 1 43 Notes 149 Bibliography 2 10 Index 2 31 Acknowledgements I would like to enumerate a long and impressive list of fellow students and prominent scholars to thank for the priceless insights I have drawn from long conversations with them, over late-night glasses of single malt and endless walks in the misty countryside. However, this work is mostly the product of solitary struggle, mainly involving sparkling water and a comfy armchair, and such a list is thus very short indeed. Not being the most conversational type, my most productive work is done in isolation, with the sole silent, yet not mute, company of books. Of the few individuals who have been instrumental in the successful completion of this book my gratitude goes, first of all, to Chris Norris. As one of the handful of scholars worldwide capable of following and assessing my conceptual journeys across the lands of analytic and conti- nental philosophy, his guidance has always been attentive but unob- trusive. He has time and again frustrated my hopes to try and refer to a book he hadn’t already read, he has voiced his disagreements with some of my ideas as a peer, without ever pressuring me to recant them, and he often seemed to have more faith in my project than I ever did. The breadth of his interests and knowledge, coupled with his humble and unassuming attitude towards everyone, has been an outstanding example of scholarly virtue to me. I have been immensely privileged to be his student. For good or bad, my intellectual trajectory was radically modified by the discovery, sometime in early 2009, of a highly active internet community of philosophers that was flourishing around the then-novel movement of ‘speculative realism’. Whatever the intrinsic merits of this dubiously unitary movement, the virtual encounter with many bright young thinkers loosely associated with it has been tremendously impor- tant for the development of my thought. It forced me continuously to try to catch-up with brighter minds, a chase (still in process) which time and again shattered my fragile self-esteem, and punctuated by the occasional, dreaded gaze into the gaping abyss where The Books You Haven’t Read Yet lurk. In one way or another several of these dozens of virtual acquaintances have influenced me. Inevitably, however, some of these had a greater impact on my philosophical commitments, a set that luckily comes to largely intersect with those I now have the privilege to call friends. Pete Wolfendale has for me variously taken on the garb of ix