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THE WESTERN ONTARIO SERIES IN PHILOSOPHY OF SCIENCE A SERIES OF BOOKS IN PHILOSOPHY OF MATHEMATICS AND NATURAL SCIENCE, HISTORY OF SCIENCE, HISTORY OF PHILOSOPHY OF SCIENCE, EPISTEMOLOGY, PHILOSOPHY OF COGNITIVE SCIENCE, GAME AND DECISION THEORY

Managing Editor WILLIAM DEMOPOULOS

Department of Philosophy, University of Western Ontario, Canada Department of Logic and Philosophy of Science, University of California/Irvine Assistant Editors DAVID DEVIDI

Philosophy of Mathematics, University of Waterloo ROBERT DISALLE

Philosophy of Physics and History and Philosophy of Science, University of Western Ontario WAYNE MYRVOLD

Foundations of Physics, University of Western Ontario Editorial Board University of Western Ontario Hebrew University of Jerusalem JEFFREY BUB, University of Maryland PETER CLARK, St. Andrews University JACK COPELAND, University of Canterbury, New Zealand JANET FOLINA, Macalester College MICHAEL FRIEDMAN, Stanford University CHRISTOPHER A. FUCHS, Perimeter Institute for Theoretical Physics, Waterloo, Ontario MICHAEL HALLETT, McGill University WILLIAM HARPER, University of Western Ontario CLIFFORD A. HOOKER, University of Newcastle, Australia AUSONIO MARRAS, University of Western Ontario JÜRGEN MITTELSTRASS, Universität Konstanz THOMAS UEBEL, University of Manchester JOHN L. BELL,

YEMINA BEN-MENAHEM,

VOLUME 76

Giovanni Sommaruga Editor

Foundational Theories of Classical and Constructive Mathematics

123

Editor Giovanni Sommaruga Dept. of Humanities Social and Political Sciences Chair for philosophy ETH, Zurich Switzerland [email protected]

ISSN 1566-659X ISBN 978-94-007-0430-5 e-ISBN 978-94-007-0431-2 DOI 10.1007/978-94-007-0431-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011922884 c Springer Science+Business Media B.V. 2011  No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The present book project grew out of the Swiss Society for Logic and Philosophy of Science (SSLPS) annual meeting on “Foundational theories of mathematics” which was held in Freiburg (Switzerland) on October 11/12 2006. John Bell and Gerhard Jäger, both participating in different functions in this meeting, responded to this book project with great enthusiasm and fueled its evolution with recurrent positive feedbacks. I’m happy to have had the opportunity over the years to discuss with them Foundations of Mathematics (FOM) and many other hot topics as well as some less hot ones. I’m grateful to Oxford University Press for permission to reprint Section I.2 of Penelope Maddy’s book Naturalism in Mathematics (Oxford: Clare