Logicism, Intuitionism, and Formalism What has Become of Them?
The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called
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SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Editor-in-Chief:
VINCENT F. HENDRICKS, Roskilde University, Roskilde, Denmark JOHN SYMONS, University of Texas at El Paso, U.S.A.
Honorary Editor:
JAAKKO HINTIKKA, Boston University, U.S.A.
Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands THEO A.F. KUIPERS, University of Groningen, The Netherlands TEDDY SEIDENFELD, Carnegie Mellon University, U.S.A. PATRICK SUPPES, Stanford University, California, U.S.A. ´ JAN WOLENSKI, Jagiellonian University, Krak´ow, Poland
VOLUME 341
LOGICISM, INTUITIONISM, AND FORMALISM WHAT HAS BECOME OF THEM? Edited by
Sten Lindstr¨om Ume˚a University, Sweden
Erik Palmgren Uppsala University, Sweden
Krister Segerberg Uppsala University, Sweden
and
Viggo Stoltenberg-Hansen Uppsala University, Sweden
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Editors Prof. Sten Lindstr¨om Ume˚a University Dept. Historical, Philosophical and Religious Studies SE-901 87 Ume˚a Sweden [email protected]
Prof. Erik Palmgren Uppsala University Department of Mathematics Box 480 751 06 Uppsala Sweden [email protected]
Prof. Krister Segerberg Uppsala University Department of Philosophy Box 627 751 26 Uppsala Sweden [email protected]
Prof. Viggo Stoltenberg-Hansen Uppsala University Department of Mathematics Box 480 751 06 Uppsala Sweden [email protected]
ISBN: 978-1-4020-8925-1
e-ISBN: 978-1-4020-8926-8
DOI 10.1007/978-1-4020-8926-8 Library of Congress Control Number: 2008935522 Springer Science+Business Media B.V. 2009 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.
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Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Preface
The present anthology has its origin in two international conferences that were arranged at Uppsala University in August 2004: “Logicism, Intuitionism and Formalism: What has become of them?” followed by “Symposium on Constructive Mathematics”. The first conference concerned the three major programmes in the foundations of mathematics during the classical period from Frege’s Begriffsschrift in 1879 to the publication of G¨odel’s two incompleteness theorems in 1931: The logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert’s formalist and proof-theoretic programme. The main purpose of the conference was to assess the relevance of these foundational programmes to contemporary philosophy of mathematics. The second conference was announced as a satellite event to the first, and was specifically concerned with constructive mathematics—an active branch of mathematics where mathematical statements—existence statements in particular—are interpreted in terms of what can be
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