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Reminiscences of the Vienna Circle and the Mathematical Colloquium Vienna Circle Collection REMINISCENCES OF THE VIENNA CIRCLE AND THE MATHEMATICAL COLLOQUIUM VIENNA CIRCLE COLLECTION HENK L. MULDER, University ofAmsterdam, Amsterdam, The Netherlands ROBERT S. COHEN, Boston University, Boston, Mass., U.S.A. BRIAN MCGUINNESS, University of Siena, Siena, Italy RUDOLF HALLER, Charles Francis University, Graz, Austria Editorial Advisory Board ALBERT E. BLUMBERG, Rutgers University, New Brunswick, N.J., U.S.A. ERWIN N. HIEBERT, Harvard University, Cambridge, Mass., U.S.A JAAKKO HINTIKKA, Boston University, Boston, Mass., U.S.A. A. J. Kox, University ofAmsterdam, Amsterdam, The Netherlands GABRIa NUCHELMANS, University of Leyden, Leyden, The Netherlands ANTHONY M. QUINTON, All Souls College, Oxford, England J. F. STAAL, University of California, Berkeley, Calif., U.S.A. FRIEDRICH STADLER, Institute for Science and Art, Vienna, Austria VOLUME 20 VOLUME EDITOR: BRIAN McGUINNESS KARLMENGER REMINISCENCES OF THE VIENNA CIRC LE AND THE MATHEMATICAL COLLOQUIUM Edited by LOUISE GOLLAND, BRIAN McGUINNESS and ABE SKLAR ..... SPRINGER-SCIENCE+BUSINESS" MEDIA, B.V. Library of Congress Cataloging-in-Publication Data Menger, Karl, 1902- Reminiscences of the Vienna Circle and the Mathematical Colloquium I by Karl Menger edited by Louise Gol land, Brian McGuinness, Abe Sk Iar. p. cm. -- (Vienna Circle collection ; v. 20) Includes index. ISBN 978-0-7923-2873-5 ISBN 978-94-011-1102-7 (eBook) DOI 10.1007/978-94-011-1102-7 1. Mathematics--Austria--Vienna--History--20th century. 2. Vienna circle. 3. Menger, Karl, 1902- I. Golland, Louise, 1942- II. McGuinness, Brian. III. Sklar, A. IV. Title. V. Series. OA27.A9M46 1994 510' .9436' 1309042--dc20 94-5014 ISBN 978-0-7923-2873-5 Printed an acid-free paper AII Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1st edition 1994 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. TABLE OF CONTENTS Introduction vii Karl Menger's Contribution to the Social Sciences - A Note by Lionello Punzo xxi Editor's Note on the Text xxvii The Historical Background II The Cultural Background 8 III The Philosophical Atmosphere in Vienna 18 IV Why the Circle invited me. The Theory of Curves and Dimension Theory 38 V Vignettes of the Members of the Circle in 1927 54 VI Reminiscences of the Wittgenstein Family 74 VII Ludwig Wittgenstein's Austrian Dictionary 83 VIII Wittgenstein's Tractatus and the Early Circle 89 IX On the Communication of Metaphysical Ideas. Wittgenstein's Ontology 104 X Wittgenstein. Brower, and the Circle 129 XI Discussions in the Circle 1927-30 140 XII Poland and the Vienna Circle 143 XIII The United States 1930-31 158 XIV Discussions in the Circle 1931-34 174 XV The Circle on Ethics 178 XVI Moritz Schlick's Final Years 194 Memories of Kurt GOdel 200 Index of Names 237 Karl Menger (1902-1985) Bibliopolis (Napels), 1987. Reprinted with pennission. INTRODUCTION Karl Menger was born in Vienna on January 13, 1902, the only child of two gifted parents. His mother Hermione, nee Andermann (1870-1922), in addition to her musical abilities, wrote and published short stories and novelettes, while his father Carl (1840-1921) was the noted Austrian economist, one of the founders of marginal utility theory. A highly cultured man, and a liberal rationalist in the nine­ teenth century sense, the elder Menger had witnessed the defeat and humiliation of the old Austrian empire by Bismarck's Prussia, and the subsequent establishment under Prussian leadership of a militaristic, mystically nationalistic, state-capitalist German empire - in effect, the first modern "military-industrial complex." These events helped frame in him a set of attitudes that he later transmitted to his son, and which included an appreciation of cultural attainments and tolerance and respect for cultural differences, com­ bined with a deep suspicion of rabid nationalism, particularly the German variety. Also a fascination with structure, whether artistic, scientific, philosophical, or theological, but a rejection of any aura of mysticism or mumbo-jumbo accompanying such structure. Thus the son remarked at least once that the archangels' chant that begins the Prolog im Himmel in Goethe's Faust was perhaps the most viii INTRODUCTION beautiful thing in the German language "but of course it doesn't mean anything." By the time the young Karl Menger entered the University of Vienna in the fall of 1920, he had absorbed these attitudes and discovered in himself an aptitude for science. His original intention was to study physics, and he did in fact attend lectures by Hans Thirring. It was not long, however, before he, at first tentatively then definitely, switched to mathematics. Much later, he jokingly remarked that the occasion for the definite abandonment of a possible career in physics was his unsuccessful attempt to read Hermann Weyl's classic book on relativity Raum-Zeit~Materie. More to the point was his discovery that he could make, and in fact had made, an impor­ tant contribution to mathematics. The story is quite dramatic, and Menger himself has told it in the chapter, "Why the Circle Invited Me. The Theory of Curves and Dimension Theory". Here is a capsule version: In the spring of 1921 he went to a lecture by the mathematician Hans Hahn, newly arrived in Vienna, on "What's new concerning the concept of a curve". In the lecture, Hahn pOinted out that there was as yet no mathematically satisfactory definition of a curve. Menger pondered the question for several days, came up with what seemed like a reasonable definition and took it to Hahn. It must be emphasized that at that time, and indeed until quite recently, it was unusual for an undergraduate student in Central Europe to call upon a professor. Hahn listened, was impressed, and encouraged his visitor to keep working on the definition and on the inextricably linked notion of the dimension of a geometric object in general. Menger did so, even after he developed tuberculosis in May 1921. By that fall, it was necessary for him to enter a sanatorium in the mountains of Styria. Before doing so, he deposited, according to INTRODUCTION ix a footnote on p. 132 of George Temple's book 100 Years of Math­ ematics (Springer, 1981) "a sealed envelope containing his definition of a curve and a suggestion of his concept of dimension ... with the Vienna Academy of Sciences ... The envelope was opened in 1926 and its contents published .. Meanwhile [he 1 had also published ... papers [in 1 1923, 1924, 1926 .. " These dates would be irrelevant if they did not bear on a question of priority. It happened that, concurrently with Menger and quite independently, the brilliant young Russian mathematician Pavel S. Uryson (1898-1924; the name generally appears as Paul Urysohn in Western publications) had developed an entirely equivalent though technically more complicated, definition of dimension. To quote again from Temple, p. 132: "According to Alexandroff['s 1925 obituary], Urysohn constructed his theory of dimension during the years 1921-2 and during this period communicated his results to the Mathematical Society of Moscow. A preliminary notice was published in 1922 and a full account in 1925 . Menger's definition is undoubtedly simpler and more general than Urysohn's, and the question of priority is of minor importance." Urysohn tragically drowned while swimming off the coast of France; he will be remem­ bered in mathematics not only for dimension theory, but for other fundamental contributions, including some such as "Urysohn's Lemma" and the "Urysohn Metrization Theorem" to which his name is inseparably attached. After almost a year at the sanatorium, Menger emerged com­ pletely cured and came back to Vienna. In 1924 he received his Ph.D. from the University of Vienna, and in March 1925, he left for Amsterdam, where he was to remain as a docent at the university for two years. He worked with the eminent and idiosyncratic mathemati­ cian who had invited him there, L. E. J. Brouwer. x INTRODUCTION Despite his increasingly strained relations with Brouwer, the years in Amsterdam were fruitful ones for him. He continued his work in dimension and curve theory and gained insight into logic and alternative ways of looking at mathematics as a whole. In 1927 he returned to the University of Vienna as Professor of Geometry. He was asked to join the discussion group which had been organized during his absence around the philosopher Moritz Schlick later known as the Vienna Circle. Significant for mathematics was the Mathematical Colloquium which he organized. There faculty, students, and visitors could report on their own work, discuss out­ standing problems, and listen to reviews of recent publications. The eight volumes of the Ergebnisse eines mathematischen Kolloquiums, covering the academic years from 1928-29 through 1935-36, were edited by Menger, and various of Menger's students, including Georg N6beling, Kurt G6del and Abraham Wald. Menger capped his work on dimension with the book Dimensionstheorie (1928, Teubner). Fifty years later, J. Keesling wrote in the Bulletin of American Mathematical Society (v. 83, p. 203): 'This book has historical value. It reveals at one and the same time the naivete of the early investigators by modern standards and yet their remarkable perception of what the important results were and the future direction of the theory. Copies are difficult to obtain now, but it is worth the effort." One of these "important results" is worth citing. It states that every n-dimensional separable metric space is homeomorphic (topologically equivalent) to part of a certain 'universal' n-dimensional space, which can in turn be realized as a compact set in (2n + 1)-dimensional Euclidean space.
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