Philosophy of Mathematics Today
Mathematics is often considered as a body of knowledge that is essen tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of cl
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Episteme A SERIES IN THE FOUNDATIONAL, METHODOLOGICAL, PHILOSOPHICAL, PSYCHOLOGICAL, SOCIOLOGICAL, AND POLITICAL ASPECTS OF THE SCIENCES, PURE AND APPLIED V O L U M E 22
Editor: Mario Bunge Foundations and Philosophy of Science Unit, McGill University Advisory Editorial Board: Raymond Boudon, Sociology, Maison des Sciences de VHomme, Paris George Bugliarello, Engineering, Polytechnic Institute of New York Bruno Fritsch, Professor emeritus of the Swiss Federal Insitute of Technology, Zürich Ivan T. Frolov, Philosophy and Social Sciences, USSR Academy of Science, Moscow Erwin Hiebert, History of Science, Harvard University Philip Kitcher, Philosophy, University of California, Davis Nicholas Rescher, Philosophy, University of Pittsburgh Michael Ruse, Philosophy and Zoology, University ofGuelph Raimo Tuomela, Philosophy, University of Helsinki Paul Weingartner, Philosophy, Salzburg University
PHILOSOPHY OF MATHEMATICS TODAY Edited by EVANDRO AGAZZI University ofFribourg, Switzerland University ofGenova, Italy, and President of the Swiss Society of Logic and Philosophy of Science
and GYÖRGY D A R V A S Symmetrion - The Institute for Advanced Symmetry Studies, Budapest, and The Hungarian Academy of Sciences, Budapest, Hungary
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C L P . Catalogue record for this book is available from the Library of Congress
ISBN 978-94-010-6400-2 ISBN 978-94-011-5690-5 (eBook) DOI 10.1007/978-94-011-5690-5
Printed on acid-free paper
A l l Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 Softcover reprint of the hardcover 1st edition 1997 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
CONTENTS
E. Agazzi, G. Darvas, Introduction
vii
GENERAL PHILOSOPHICAL PERSPECTIVES
1
F. Mir6 Quesada, Logic, Mathematics, Ontology M. Borga, From Certainty to Fallibility in Mathematics? M. Bunge, Moderate Mathematical Fictionism P. Weingartner, Language and Coding-Dependency ofResults in Logic and Mathematics G. G. Granger, What is a Profound Result in Mathematics? R. Thorn, The Hylemorphic Schema in Mathematics
3 39 51
FOUNDATIONAL APPROACHES S. Mac Lane, Categorical Foundations of the Protean Character ofMathematics J. P. Marquis, Category Theory and Structuralism in Mathematics: Syntactical Considerations G. H. Milller, Reflection in Set Theory. The Bemays-Levy Axiom System C. Parsons, Structuralism and the Concept of Set W. Sieg, Aspects ofMathematical Experience A Ishirnoto, Logicism Revisited in the Propositional Fragment of LeSniewski's Ontology
73 89 101
115
117 123 137 171 195
219
vi
CONTENTS
THE APPLICABILITY OF MATHEMATICS
233
E. Agazzi, The Relation of Mathematics to the Other Sciences G. M. Prosperi, Mathematics and Physics E. Scheibe, The Mathematical Overdetermination of
