Table Of ContentREMINISCENCES OF THE VIENNA CIRCLE
AND THE MATHEMATICAL COLLOQUIUM
VIENNA CIRCLE COLLECTION
HENK L. MULDER, University ofA msterdam, 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 ofA msterdam, 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 1s t 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 19 21
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. This was proved for n =·1 in
the 1928 book, where a sketch of a proof for all other n was also
given. (Complete proofs for all n were given by Georg N6beling and
Description: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
