From Dedekind to Gödel: Essays on the Development of the Foundations of Mathematics PDF
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FROM DEDEKIND TO GODEL SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METIIODOLOGY, AND PHILOSOPHY OF SCIENCE Managing Editor: JAAKKO HINTIKKA, Boston University Editors: DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University a/California, Berkeley TIIEO A.F. KUIPERS, University ofGroningen, The Netherlands PATRICK SUPPES, Stanford University, California JAN WOLEN-SKI, Jagiellonian University, Krakow, Poland VOLUME 251 FROM DEDEKIND .. TO GODEL Essays on the Development of the Foundations of Mathematics Edited by JAAKKO HINTIKKA Boston University Library of Congress Cataloging-in-Publication Data Fram Dedekind ta Godel essays an the develapment of the faundatians of mathematics I ed1ted by Jaakka H1nt1kka. p. CIII. -- (Synthese l1brary ; v. 251) Inc 1u des index. ISBN 978-90-481-4554-6 ISBN 978-94-015-8478-4 (eBook) DOI 10.1007/978-94-015-8478-4 1. Mathematics--Faundatians. I. Hintikka, Jaakka, 1929- II. Series. OA8.6.F76 1995 511--dc20 95-16293 ISBN 978-90-481-4554-6 Printed on acid-free paper AII Rights Reserved © 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint of the hardcover 18t edition 1995 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 Of by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS Preface vii JAAKKO HINTIKKA Tracking Contradictions in Geometry: The Idea of a Model from Kant to Hilbert 1 JUDSON WEBB Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics 21 JAAKKO HINTIKKA Kronecker on the Foundations of Mathematics 45 HAROLD M. EDWARDS The Mysteries of Richard Dedekind 53 DAVID CHARLES MCCARTY Frege's Letters 97 CLAIRE ORTIZ HILL Frege's Principle 119 RICHARD G. HECK, JR. Husserl and Hilbert on Completeness 143 CLAIRE ORTIZ HILL Hahn's Uber die nichtarchimedischen Grossensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them 165 PHILIP EHRLICH The Origins of Russell's Paradox: Russell, Couturat, and the Antinomy of Infinite Number 215 GREGORY H. MOORE v vi T ABLE OF CONTENTS The Emergence of Descriptive Set Theory 241 AKIHIRO KANAMORI Chance Against Constructibility 263 JAN VON PLATO Thoralf Skolem, Hermann Weyl and 'Das Gefiihl der Welt als begrenztes Ganzes' 283 WILLIAM Boos On Tarski's Background 331 JAN WOLENSKI Wittgenstein and Ramsey on Identity 343 MATHIEU MARION On Saying What You Really Want to Say: Wittgenstein, Godel, and the Trisection of the Angle 373 JULIET FLOYD Godel and Husser! 427 DAGFINN F0LLESDAL Index of Names 447 Index of Subjects and Titles 455 JAAKKO HINTIKKA PREFACE Burton Dreben once said that the worst known period in the history of philosophy was one hundred years ago. As far as the history of the foundations of mathematics is concerned, this dictum might at first sight seem to be belied by the prolific output of a veritable industry of books and papers on Frege and to a considerably lesser extent on Cantor and Russell. This volume was nevertheless originally inspired by a con viction that Dreben's mot is applicable also to the foundations of mathematics. Admittedly, Frege and Russell forged the logical tools that have become indispensable for all serious work in the foundations of mathe matics. They also put forward a large number of ideas about mathematics. But from this it does not follow that their direct contributions to foun dational studies were of the same order of magnitude as their contribution to the development of logic. For one thing, both Frege and Russell worked under severely restrictive assumptions which, among other things, made it hard for them to use any model-theoretical methods or insights in their work. In general, the impact of logicians like Frege and Russell on how working mathematicians themselves have looked upon their subject matter and the problems mathematicians were seriously concerned with has recently been vastly overrated by philosophers. It is, for instance, turning out that although Frege himself was a professing mathemati cian, his grasp of many of the foundational issues that mathematicians of his day were actually debating was not very firm. And even if this charge against Frege should tum out to be exaggerated, it can only be shown to be so through a study of those mathematicians who were at the cutting edge of the work of deepening the foundations of mathematics in the late nineteenth and early twentieth centuries. They include, among others, Weierstrass, Dedekind, Kronecker, early Hilbert, Borel and his compatriots, Hermann Weyl, etc. They also constitute an important part of the background of such later figures as Godel and Tarski. Furthermore, many of the ideas that were discussed among the mathematical foun dationalists have received but scant attention recently. They include the concepts of arbitrary function and arbitrary sequence, the anticipation vii laakko Hintikka (ed.), Essays on the Development of the Foundations of Mathematics, vii-ix. © 1995 Kluwer Academic Publishers. Vlll J AAKKO HINTIKKA of model-theoretical ideas by Hilbert's axiomatical approach to mathe matical theories, maximality assumptions (as, e.g., in Hilbert's "Axiom of Completeness"), and the precise role of set theory in the foundations of mathematics. The conference whose proceedings this volume includes had as one of its aims to raise the consciousness of philosophers of the situation. It was an ecumenical rather than a sectarian meeting, however. Even though we perhaps did not give equal time to the fans of Frege and Russell, we wanted to have their contributions, too, presented and dis cussed at the meeting. In any case, we hope that such papers as those of Ehrlich, Webb and Hintikka show that there are viewpoints on the history of the foundations of mathematics in (roughly) 1850-1930 that have not received their philosophical due, and which also affect the historical study of this period of mathematics. It is also forgotten sometimes that philosophers other than analytic ones were seriously interested in the foundations of mathematics. The papers by Claire Hill and Dagfinn F0llesdal relate some ideas of Husserl' s to the development of the foundations of mathematics. Other papers published here as a part of the proceedings of the April 1992 meeting explore important but relatively neglected episodes or lines of development in the history of foundational studies. I will not insult their authors by trying to summarize their clear and forceful exposi tions, which speak eloquently enough for themselves. Two of the papers published here were not read at the 1992 meeting. Of them, Juliet Floyd's paper was presented at the Boston Colloquium for the Philosophy of Science only a couple of weeks later. It is included here because it broadens the spectrum of thinkers considered here and also because it relates Wittgenstein's ideas in an unusually interesting way to actual work in logic and in the foundations of mathematics. The other genetically unrelated paper is by Mathieu Marion. It is included here because it is a natural sequence to Hintikka's paper. In other words, Marion takes the contrast between standard and nonstandard interpretations of higher-order logics, which Hintikka considers as one of the main themes in the history of the foundations of mathematics, and shows that it was the precise bone of contention in an intensive but largely private and consequently little known controversy between Ludwig Wittgenstein and Frank Ramsey. Risto Vilkko's work in preparing the indexes is gratefully acknowl edged. For the record, the conference from which this volume originated PREFACE ix convened on April 5-7, 1992 at Boston University as a part of the 1991-92 program of the Boston Colloquium in the Philosophy of Science. It was supported by the Dibner Institute, by the Division of Logic, Methodology and Philosophy of Science of IUHPS, and by Boston University. This support is here acknowledged most gratefully and warmly. Without it, this volume would not exist. JUDSON WEBB TRACKING CONTRADICTIONS IN GEOMETRY: THE IDEA OF A MODEL FROM KANT TO HILBERT This paper explores such questions as who actually discovered non euclidean geometry, who actually believed in its consistency and why, and who can be said to have proved it to be free of contradiction. To this end I will analyze some views and results if ten or so philosophers and mathematicians from Kant to Hilbert. One main theme is that without some rudimentary idea of a model, the discovery and establishment of non-euclidean geometry would not have been possible. Another is that only the notion of a model enabled thinkers to conceive of properties of logical inference such as soundness and completeness of axioms and/or rules. These themes are surprisingly difficult to articulate clearly without compromising historical accuracy, but I believe that in most cases the attempt to do so leads to a better understanding of the writers involved. It was a commonplace of older Kantian scholarship that the dis covery of non-euclidean geometry undermined his theory of the synthetic a priori status of geometry. It is a commonplace of newer Kant schol arship that he already knew about non-euclidean geometry from his friend Lambert, one of the early pioneers of this geometry, and that in fact its very possibility only reinforces Kant's doctrine that euclidean geometry is synthetic a priori because only its concepts are constructible in intuition. The principal passage cited as evidence of such knowledge is one in which Kant illustrates his Postulate of Empirical Thought stating that "that which agrees with the formal conditions of experience, that is, with the conditions of intuition and of concepts is possible". But to establish possibility requires synthetic construction in addition to non contradiction, as Kant explains in connection with the concept of a biangle: It is indeed a necessary logical condition that a concept of the possible must not contain any contradiction; but this is by no means sufficient to determine the objective reality of the concept, that is, the possibility of such an object as is thought through the concept. Thus there is no contradiction in the concept of a figure which is enclosed within two straight lines since the concept of two straight lines and of their coming together contain no negation of a figure. The impossibility arises not from the concept itself, but in con nection with its construction in space, that is, from the condition of space and its determinations. CA220)I laakko Hintikka (ed.), Essays on the Development of the Foundations of Mathematics, 1-20. © 1995 Kluwer Academic Publishers.
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