Karl Menger: Ergebnisse eines Mathematischen Kolloquiums

470 Pages·1998·23.067 MB·German-English

by  Gérard Debreu (auth.), Univ.-Prof. Dr. Egbert Dierker, Univ.-Prof. Dr. Karl Sigmund (eds.)

Karl Menger
Ergebnisse eines Mathematischen Kolloquiums
Herausgegeben von/Edited by
E. Dierker, K. Sigmund
Mit Beiträgen vonlWith contributions by
J. W. Dawson jr., R. Engelking, W. Hildenbrand
Geleitwort von/Foreword by
G. Debreu
Nachwort von! Afterword by
F. Alt
Springer-Verlag Wien GmbH
Univ.-Prof. Dr. Egben Dierker
Institut für Wirtschaftswissenschaften. Universität Wien
Hohenstaufengasse 9. A-IOIO Wien
Univ.-Prof. Dr. Karl Sigmund
Institut für Mathematik, Universität Wien
Strudlhofgasse 4. A-I09O Wien
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© 1998 Springer-Verlag Wien
Ursprünglich crsch1cncn be1 Sprlllgcr-VcrlagfWicn 1998
Softcover reprint of Ihe hardcover 1S I edition 1998
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Gedruckt auf säurefreiem, chlorfrei gebleichtem Papier - TCF
SPIN: 10637930
ISBN 978-3-7091-7330-5 ISBN 978-3-7091-6470-9 (eBook)
DOI 10.1007/978-3-7091-6470-9

Preface
The Ergebnisse eines Mathematischen Kolloquiums were published by Karl
Menger (with the collaboration of Kurt Godel, Georg Nobeling, Abraham
Wald and Franz Alt) in eight issues during the years 1929 to 1937. They con
stitute a major source text for the history of scientific ideas in the 'Thirties,
containing path-breaking papers by Menger, Godel, Tarski, Wald, Wiener,
John von Neumann and many others. Due to political circumstances, the
distribution of the Ergebnisse was very limited, and only few complete sets
have survived to this day.
This volume contains the full text. We have reprinted the German
original, adding English commentaries written by outstanding experts.
John Dawson jr. comments the contributions on mathematical logic,
Ryszard Engelking the works on topology and Werner Hildenbrand the
papers on mathematical economics. Furthermore this volume contains a
biography of Karl Menger, a survey written by Menger on the major
topological and geometrical ideas of the Kolloquium, and an introduction
by Nobel Prize winner Gerard Debreu. We have also listed those papers
of the Ergebnisse which have been translated into English, and added
short biographical notes on Menger and his collaborators. Prof Franz
Alt, one of the co-editors of the Ergebnisse, was kind enough to provide
us with an Afterword, and Mrs. Louis Golland with a photograph of
Menger.
We wish to thank Doz. Friedrich Stadler for suggesting to reprint the
Ergebnisse, Professors L. Schmetterer and E. Streissler for helpful advice,
Dr. E. Kohler from the Vienna Circle and Professor R. Leonard for provid
ing us with important informations. We also thank Dr. M. Stoltzner for his
translation of Menger's survey article and the Fonds zur Forderung der
wissenschaftlichen Forschung for support. We are greatly indebted to Pro
fessors Bert Schweizer and Abe Sklar, who were students, then collabora
tors of Menger and kindly volunteered to read the biographical introduc
tion to this volume. Very special thanks, finally, go to Professor Franz Alt,
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Preface
one of the co-editors of the Ergebnisse, who kindly provided us with help
ful suggestions and clarifications.
E. Dierker
K. Sigmund
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Table of Contents
Foreword: Economics in a mathematics colloquium
by Gerard Debreu ...................................................... 1
Menger's Ergebnisse - a biographical introduction
by Karl Sigmund ....................................................... 5
Logical contributions to the Menger colloquium
by John W. Dawsonjr. ................................................. 33
Topology in the Ergebnisse
by Ryszard Engelking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
An exposition of Wald's existence proof
by Werner Hildenbrand ................................................ 51
On the direction of ideas and the principal tendencies of the Vienna
Mathematical Colloquium
by Karl Menger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A table of contents of the Ergebnisse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Reprint of "Die Ergebnisse eines Mathematischen Kolloquiums,
Vol. 1-8" ................................................................ 89
Biographical notes on the editors of the Ergebnisse ................... 465
Editorial notices ........................................................... 467
Afterword
by Franz Alt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 469
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Foreword: Economics in a mathematics colloquium
Gerard Debreu
In stark contrast to the short eight years of its existence, the colloquium
that met in Vienna from 1928 to 1936 had a long lasting influence on eco
nomic theory. At first, the reaction of the economics profession to the new
ideas presented in Karl Menger's seminar was slow, and fifteen years elap
sed before the strength of their impact was felt. Many explanations can be
found for the delay that took place from Karl Schlesinger's paper given
orally in 1934, to the Linear Programming Conference held in Chicago in
1949, a prelude to the Activity Analysis of Production and Allocation
Monograph edited by Tjalling C. Koopmans, and published in 1951. Only
a few of those explanations can be listed.
The world had already taken the path to a global conflict, which could
be foreseen, with almost blinding clarity, in the abrupt end of the collo
quium in 1936. In that world, the Proceedings was not common reading
among economists, and it was easily available neither from specialized
libraries, nor from colleagues. The title did not promise any economic con
tents. The language alone was a powerful deterrent. The place from which
it came seemed distant. The main participants were, or were to become,
mathematicians, and statisticians of the most brilliant magnitudes, but their
names were little known to economists. A complete understanding of their
mathematics required a high level of preparation. Their mode of reasoning
was unforgivingly rigorous, at a time when logical rigor was not believed
to be a condition for theorizing about economics. Partial as this list will
remain, it must include as an especially forbidding obstacle to the rapid
and full recognition of their contributions, the novelty of the questions they
asked, and began to answer. Yet the ideas that Karl Schlesinger, Abraham
Wald and John von Neumann published in volumes 6, 7, and 8 of the Pro
ceedings in 1935, 1936, 1937, deserved immediate notice.
The starting signal for the development that would bring a profound
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E. Dierker et al. (eds.), Karl Menger
© Springer-Verlag/Wien 1998
Gerard Debreu
transformation to mathematical economics was given by Schlesinger. In
the first paper on economics of the Colloquium, presented on March 19,
1934 (and published in 1935), he raised a question aimed at the center of
Walrasian theory. In his Elements d' Economie Politique Pure (1874-77),
Leon Walras did not inquire whether the equations with which he descri
bed the general equilibrium of an economy actually had a solution, let
alone a solution in which all the variables had the proper signs. The
mathematics available at that time did not permit a satisfactory answer, and
he had to content himself with the giant step he had taken. But 60 years
went by without any alteration of the argument for the existence of a Gene
ral Equilibrium that the master of the School of Lausanne proposed, and
that his disciples repeated for six decades. The reason given for existence
was simply a count of equations and unknowns. Showing that their num
bers were equal established no more, however, than the plausibility of a
positive answer to Schlesinger's question. And yet, long before his query
was formulated, one of the main mathematical tools with which it was
eventually settled was provided in 1910-11 by L. E. 1. Brouwer (Mathe
matische Annalen), in the form of his powerful fixed point theorem.
In his paper (on March 19, 1934), Schlesinger suggested a modifica
tion of Leon Walras' (and Gustav Cassel's) equations that soon turned out
to be essentiaL Walras (and Cassel) assumed that the list of free commodi
ties (i.e., for which demand is smaller than supply) was given a priori.
Schlesinger wrote that this condition was inadmissible, that the list of free
commodities should be determined by the system of equations itself, each
one of the free commodities having a zero price.
Immediately after that correct formulation was given, Wald proved the
existence of a GE (also on March 19, 1934), and, at the next session of the
Colloquium, on November 6, 1934, established existence under much
weaker conditions. His two papers, providing the first proofs of existence
for a GE marked an important moment in the history of mathematical eco
nomICS.
With a fifteen years lag, an echo was finally heard in Chicago in 1949,
and from that time on, events began to unfold more rapidly. First, however,
one must go back to Menger's Colloquium where still another major con
tribution was published in 1937 by von Neumann as the last article in the
last volume of the Proceedings. His paper was presented orally first in
English at the Mathematics Colloquium of Princeton University in the
winter of 1932, next printed in German in the Ergebnisse, finally that ver
sion was translated into English and appeared in the 1945 Review of Eco-
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Economics in a mathematics colloquium
nomic Studies as "A Model of General Equilibrium". Von Neumann's con
tribution reached many goals; one of the most important, in his own
appraisal, was his "lemma" (in the English translation), or "Satz" (in the
German original). That "lemma" concerns a generalization of a Brouwer
Fixed Point to what is now known as a Kakutani Fixed Point. John von
Neumann's "lemma", and Shizuo Kakutani's theorem can almost immedi
ately be derived from each other. That von Neumann thought his "lemma"
to be important can be read in his article's German title "Uber ein
okonomisches Gleichungssystem und eine Verallgemeinerung des Brou
werschen Fixpunktsatzes". His generalization turned out to be essential for
the existence proofs presented in December 1952 (and published in Eco
nometrica in 1954) by Kenneth Arrow and Gerard Debreu, and by Lionel
McKenzie. Yet his "lemma" was unnecessarily powerful for his purpose,
in spite of John von Neumann's assertion that "the mathematical proof
is possible only by means of a generalization of Brouwer's Fix-Point
Theorem". Elementary proofs of von Neumann's central result that do
not call on fixed point theorems were given later by several authors, in
particular by David Gale (in "Linear Inequalities and Related Systems",
edited in 1956 by H.W. Kuhn and A. W. Tucker). Thus the main mathe
matical tool for the proof of existence of a GE owes it origin to an accident.
Kakutani then stated von Neumann's "lemma" in a form that made it
easier to use, and proved it in a simpler manner in 1941 (Duke Mathemat
ical Journal). Nine years later, in a single page note, one of the shortest
influential papers in science, John Nash defined, and proved the existence
of, a "Nash equilibrium" for finite N-person games using Kakutani's gen
eralization (Proceedings of the National Academy of Sciences of the USA,
1950). Nash's article was short, but it was also fundamental, as the Nobel
committee in economics recognized in 1994. By itself it could destroy the
myth that forceful economic ideas can be fully expressed only with great
prolixity.
Nash's introduction of Kakutani's theorem in economics in 1950
played a major role in the solutions of the existence problem proposed in
1954 by Arrow and Debreu, and by McKenzie.
In the meantime, Morton Slater presented, in Chicago, a little known
Cowles Commission Discussion Paper (Math 403, November 7, 1950)
"Lagrange Multipliers Revisited". There he applied the von Neumann
Kakutani result to his study of a class of constrained maximum problems
that had been considered by Kuhn and Tucker. In a different context from
Nash's article with which it was simultaneous, Slater presented another
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