Platonism, Naturalism, and Mathematical Knowledge PDF
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ROUTLEDGE STUDIES IN THE PHILOSOPHY OF SCIENCE Platonism, Naturalism, and Mathematical Knowledge James Robert Brown Platonism, Naturalism, and Mathematical Knowledge Routledge Studies in the Philosophy of Science 1 Evolution, Rationality 9 Models, Simulations, and Cognition and Representations A Cognitive Science for the Paul Humphreys Twenty-First Century and Cyrille Imbert Edited by António Zilhão 10 Platonism, Naturalism, and 2 Conceptual Systems Mathematical Knowledge Harold I. Brown James Robert Brown 3 Nancy Cartwright’s Philosophy of Science Edited by Stephan Hartmann, Carl Hoefer, and Luc Bovens 4 Fictions in Science Philosophical Essays on Modeling and Idealization Edited by Mauricio Suárez 5 Karl Popper’s Philosophy of Science Rationality without Foundations Stefano Gattei 6 Emergence in Science and Philosophy Edited by Antonella Corradini and Timothy O’Connor 7 Popper’s Critical Rationalism A Philosophical Investigation Darrell Rowbottom 8 Conservative Reductionism Michael Esfeld and Christian Sachse Platonism, Naturalism, and Mathematical Knowledge James Robert Brown NEW YORK LONDON First published 2012 by Routledge 711 Third Avenue, New York, NY 10017 Simultaneously published in the UK by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2012 Taylor & Francis The right of James Robert Brown to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Typeset in Sabon by IBT Global. Printed and bound in the United States of America on acid-free paper by IBT Global. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Brown, James Robert. Platonism, naturalism, and mathematical knowledge / James Robert Brown. p. cm. — (Routledge studies in the philosophy of science ; 10) Includes bibliographical references and index. 1. Platonists. 2. Naturalism. 3. Mathematics—Philosophy. 4. Science—Philosophy. I. Title. B517.B76 2011 146—dc23 2011021978 ISBN13: 978-0-415-87266-9 (hbk) ISBN13: 978-0-203-15535-6 (ebk) To the memory of my father Andrew Herbert Brown 1917–1993 Contents Preface ix Acknowledgments xi 1 Mathematical Explanation 1 2 What is Naturalism? 30 3 Perception, Practice, and Ideal Agents: Kitcher’s Naturalism 48 4 Just Metaphor? Lakoff’s Language 71 5 Seeing with the Mind’s Eye: The Platonist Alternative 91 6 Semi-Naturalists and Reluctant Realists 111 7 A Life of Its Own? Maddy and the Autonomy of Mathematics 133 Afterword 159 Notes 163 Bibliography 169 Index 177 Preface To be natural is such a diffi cult pose to keep up. —Oscar Wilde Naturalism is on the march. In almost every philosophical outlet and every philosophical fi eld, one encounters naturalist accounts of the subject matter at hand. Naturalism is a thriving research programme—at least it’s thriv- ing in the sense that it has abundant advocates who celebrate its ostensible achievements. Whether it’s also thriving in the more important sense—mak- ing genuine progress—is quite another matter. It is as cheerfully accepted in some circles as postmodernism is cheerfully accepted in others. However, this book takes a dim view of its prospects and argues that naturalism is a mistake. I don’t know if naturalism is wrong-headed about everything, but it is most certainly misguided about mathematics. Of course, naturalism has much going for it—not least its name. Shake- speare was wrong about a few things, including nomenclature. If a rose were known as the fart fl ower, would any of us give it to a lover? Dare anyone object to the natural diet or to acting naturally? It’s like objecting to nature itself. And who, after all, wants to be unnatural, especially since so many “unnatural acts” have delivered their perpetrators to the gallows or the fi ery stake. Remember Sodom and Gomorrah—even supernatural God doesn’t like unnatural acts. But I suppose realists and constructivists have similarly benefi ted from a good choice of names—“Hey, I’m realistic and my advice is always con- structive.” All of which shows that evidence, not etymology, must decide these issues. Of course, naturalists think they’re in great shape when it comes to evidence, since they have all the testimony of natural science itself in support of their naturalism. Needless to say, I don’t believe that for a minute. But I will allow that the burden is on me and other anti-naturalists to show otherwise. And that’s what the bulk of this book is about: the case against mathematical naturalism. In particular, it makes the case against specifi c naturalists: Philip Kitcher, Willard Quine, Penelope Maddy (twice, since she has proposed two distinct naturalist views), George Lakoff, and others. Is it exhaustive? No. But I hope a suffi ciently diverse sample has been covered to lead to the plausible conclusion that any version of natural- ism is likely hopeless. This volume is mostly negative, but not completely so. From time to time a positive case will be made for naturalism’s great rival, Platonism. There
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