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Karl Menger's Vienna Colloquium (1928-36)
In the history of mathematics and economics, Karl Menger (1902-85) is a fairly anonymous figure. This, perhaps, for a few reasons. Although he was a prodigy, Karl was also the son of another great mind, Carl Menger (1840-1921). Carl (the father) is considered the founder of Austrian Economics. Their names are, as you can imagine, often a point of confusion. Although Karl (the son) made contributions to the algebra of geometries, curve and dimension theory, game theory and graph theory, his most lasting contribution to science might just be the Colloquium he arranged in Vienna in the period 1928-36. This is where Gödel first announced his incompleteness theorem, where von Neumann first presented his purely topological proof of the existence of general competitive equilibria and where game theory began to grow into a field of its own. Indeed, some go so far as to claim that the flourishing of mathematical economics in Vienna would not have taken place without Menger’s Colloquium.
This week’s newsletter is about Karl Menger and his Mathematisches Kolloquium.
Early Life (1902-20)
Karl was born in Vienna on the 13th of January 1902, the only son of his parents Hermione Andermann and Carl. His mother was an author and musician who must have been quite a bit younger than her husband, Carl. Indeed, Karl’s father was 62 years old at the time of his birth. Roughly 32 years prior he had been the author of the book Principles of Economics (1871) which argued the classical cost-based theories of value were inferior to the theory of marginality — which argues that prices are determined at the margin. The book would spark a revolution in economics and the birth of what we now know as the Austrian School of economic thought, which (Carl) Menger is justifiably considered the founder of.
In the years from 1913-20 Karl (the son) attended the Doblinger Gymnasium in Vienna, where he was early recognized as a prodigy (Kass, 1996). He entered the University of Vienna in 1920 aiming to study physics. As the story goes, Menger switched his focus to mathematics after attending a lecture by mathematician Hans Hahn (1879-1934):
“In March of 1921, just after Hahn’s arrival in Vienna, I read a notice that he would conduct a two-hour advanced seminar every Wednesday during the spring semester, entitled Neueres über den Kurvenbegriff. At that time I was at the end of my first semester at the university and primarily a student of physics. I was doubtful that i would be able to follow the advanced mathematical discussions; and I had never seen Hahn before. But after some deliberation, I decided to try.”
The topic of the lecture was ‘News about the concept of curves’. In his presentation, Hahn pointed out that there was, as of yet, no mathematically satisfactory definition of a curve. “When speaking of curves one thinks of thread-like objects, wire models, and the like. But anyone who try to make that idea precise, Hahn said, would encounter great difficulties.” (Menger, 1994).
“I was completely enthralled; and when after that short introduction, Hahn set out to develop the principal tools used in those earlier attempts — the basic concepts of Cantor’s point set theory, all totally new to me — I followed with the utmost attention”
Menger, 19 years old at the time, left the seminar room “in a daze”. After a weekend of “complete engrossment” with the problem, Menger felt he had arrived at what appared to be a simple and complete solution. After conferring with a friend who was a mathematician, he decided to bring his definition to Hahn. “An hour before the beginning of the second seminar meeting, I had to muster all my courage before knocking on the door of Hahn’s office to tell him that I thought I could solve the problem he had formulated the week before”.
It is worth noting that it would have been highly unusual for an undregraduate student in Central Europe to call upon a professor to discuss his or her ideas at the time. Indeed, there was hardly any contact between non-graduate/Ph.D. students and professors at the University of Vienna at this time. “Hahn, who hardly looked up from the book he was reading when I entered, became more and more attentive as I went on. At the end, after some thought, he said that this would indeed be a workable definition, and asked me where I had learned so much about point sets and topology.“
Menger replied that he was a physics student at the end of his first semester, and that he had not heard about topology (invented by Poincaré thirty years before). He said that his definition used only concepts defined by Hahn the week before. Menger later wrote that “Hahn had not realized that two hours of his excellent presentation of basic concepts were sufficient to make them operative even in the mind of someone totally unfamiliar with the field”.
“He [Hahn] nodded rather encouragingly and I left”
Menger continued working on his definition through a bout of tuberculosis that eventually confined him to a sanatorium in the mountains of Austria. Before being committed he deposited, according to Temple (1981) “a sealed envelope containing his definition of a curve and a suggestion of his concept of dimension […] with the Vienna Academy of Sciences.” For some unknown reason, the envelope was not opened until 1926, when its contents were finally published.
Menger had spent a year in the sanitorium, but emerged cured and in 1924 received his Ph.D. in mathematics from the University of Vienna (supervised by Hahn) before leaving for Amsterdam in 1925 to work as a dozent under L. E. J. Brouwer (1881-1966). He did so for two years before returning in 1927 to assume the role of Professor of Geometry at the University of Vienna. One of his first students there would later be rather famous. In Menger’s own words:
“In the fall of 1927, after a stay of two and a half years in Amsterdam, I accepted the chair of geometry at the University of Vienna and returned to my home town. […] Soon after assuming my position […] I offered a quite well attended one-semester course which was on dimension theory. The name of one of the students who had enrolled was Kurt Gödel. He was a slim, unusually quiet young man.”
The life of Kurt Gödel is the topic of the former Privatdozent newsletter ‘Kurt Gödel’s Brilliant Madness’, released on June 21st 2021:
The Vienna Circle
Menger was in 1927 asked by Hahn to join the now famous discussion group The Vienna Circle chaired by Moritz Schlick (1882-1946). As Menger later wrote
“A remarkable feature of Vienna in those years was the existence of numerous Kreise (circles, in the sense of discussion groups), some with, some without, direct ties with the academic world. […] I knew about these circles since most of them included one or another of my friends; but personally I did not belong to any circle, except one that was non-scientific. Until one day in the fall of 1927, Hahn said to me: ‘I hope you can join our philosophical circle around Schlick. We meet informally about every other Thursday evening in Schlick’s Institute.”
Schlick was a philosopher and is considered the father of logical positivism, the view that “only statements verifiable through direct observation or logical proof are meaningful in terms of conveying truth value, information or factual content”. Schlick had been part of Menger’s doctoral examination, but the two hadn’t met since (Menger, 1994).
“I doubt that he remembers me”.
“I have already spoken to him”, Hahn said. “He remembers you well and is pleased with the idea that you might join us”.
According to Menger, Hahn, Schlick and Rudolf Carnap (1891-1970) wanted him to attend the meetings of the Cicle because they were interested in the methodology Menger had used in his work on curves and dimension. “On several occasions, these matters came up in discussions of the Circle and they have [since] been referred to in the writings of Carnap and Popper” (Menger, 1994).
The Vienna Colloquium (1928-36)
It was students of Menger at the University of Vienna who in 1928-29 asked their professor to arrange a mathematisches kolloquium:
“In these meetings, topics and results in their [students’] and my fields of interest were reported and discussed. We followed the unconstrained style of the Schlick Circle; but from the fall of 1929 on I kept a Protokollbuch — minutes of a sort — something which Schlick, as far as I know, unfortunately never did.”
Among the first non-students at Menger’s Colloquium was Gödel, who at that point was a Ph.D. student working on under Hahn. He first attended at the end of 1929. According to Menger:
“From then on he was a regular participant, not missing a single meeting when he was in Vienna and in good health. From the beginning, he appeared to enjoy these gatherings and poke even outside of them with members of the group. […] He was a spirited participant in discussions on a large variety of topics. Orally, as well as in writing, he always expressed himself with the greatest precision and at the same tim with the utmost brevity. In nonmathematical conversations he was very withdrawn.”
Other attendees at the time included Georg Nöbeling (1907-2008), Franz Alt (1910-2011) and Olga Taussky-Todd (1906-95). In the summer of 1929, Menger visited Warsaw where he became acquainted with mathematician Alfred Tarski (1901-83) and his collaborator Alfred Lindenbaum (1904-1941). Menger invited Tarski to give three lectures at the Colloquium, two of which were on logic and to which Menger also invited the members of the Vienna Circle. After one of the lectures, Gödel approached Menger about setting up a meeting with Tarski so that he could share with him his proof establishing the completeness of first-order logic. Tarski reportedly showed great interest in the result.
Contributions to Mathematical Economics
The most consequential results presented at Menger’s Colloquium were likely those related to economics. In particular, the mathematization of economics. Although a mathematician by training, Menger had also gained a considerable expertise in economics during his undergraduate years. Indeed, at 20 years old, following the passing of his father Carl he completed one of his father’s unfinished manuscript for the second edition of the book Principles of Economics, which was published in 1922.
The first attendee to present work relevant to economics at Menger’s Colloquium was a Viennese banker by the name of Karl Schlesinger (1889-1938). His paper was titled Über die Produktionsgleichungen der ökonomischen Wertlehre (“On the Production Equations of Economic Value Theory“) and was presented orally on the 19th of March 1934. In it, Schlesinger suggested a modification of Léon Walras (1834-1910)’ set of equations describing a general equilibrium, which he (Walras) had never proved to have solutions except arguing that they should have based on the number of equations and unknowns (Debreu, 1998). For this and other reasons, the ‘Walrasian model’ had fallen by the wayside until it was resurrected by Swedish economist Gustav Cassel (1866-1945) when he used it as inspiration for his 1918' treatise Theoretische Sozialökonomie (“The Theory of Social Economy”), translated to English in 1923.
Schlesinger’s insight was to point out that the supply of free commodities (of which demand is smaller than supply) should not be assumed have a number a priori, but rather that their supply (like all other supply) should be determined by the system of equations of general equilibrium, each free commodity having a price of zero (Debreu, 1998). This modification would turn out to be essential.
Abraham Wald’s Existence Proofs (1934)
Like Oskar Morgenstern (1902-77), Schlesinger had received tutoring lessons in mathematics from Menger’s Ph.D. student Abraham Wald (1902-50). Schlesinger (seemingly) trivial insight inspired Wald (who was attending the Colloquium where he presented his claims) to, that same day, prove the existence of the general equilibrium which had evaded Walras. His first attempt, given somewhat strict conditions was presented on March 19th. He soon after also presented a proof given much weaker conditions, on November 6th. His papers—one using a model based on a system of exchange and the other based on a model of production and exchange—were published in the proceedings of the Colloquium in 1934:
Wald, A. (1934a). Über die eindeutige positive Lösbarkeit der neuen Produktionsgleichungen, ("On the Unique Non-Negative Solvability of the New Production Functions"). Ergebnisse eines mathematischen Kolloquiums, 1933-34.
Wald, A. (1934b). Über die Produktionsgleischungen der ökonomischen Wertlehre (2. Mitteilung), (“On the Production Equations of Economic Value Theory (Part 2)“). Ergebnisse eines mathematischen Kolloquiums, 1934-35.
Wald’s two papers signify an important moment in the history of mathematical economics. Although the existence of the Walrasian equilibrium had been proposed as early as in the 1870s, the mathematical tools available to Walras never permitted him to show that such solutions must exist, given reasonable assumptions. Although the assumptions of Wald’s model are not very reasonable according to modern standards (including the assumption that there was only one consumer in the market), he was the first to solve the equilibrium problem that had plagued economists up until that point (Düppe & Weintraub, 2014).
"Accordingly, to a follower of Menger, the determination of economic equilibrium would not merely involve the determination of the prices of those goods which have prices (as in Walras); it should also involve the determination of which goods are to have prices and which are to be free. The weakness of the Walras-Cassel lies in the implied assumption that the whole amount of each available factor is utilized; once that assumption is dropped, the awkwardness of the construction....can be shown to disappear." — John Hicks, 1960
A mathematician by training, Wald had little awareness of the significance of his findings for economics. To him, it was a mathematical problem with a mathematical solution and limited mathematical implications. Indeed, his boss and pupil Morgenstern reportedly had to ‘badger’ Wald to write a summary of his two papers, which were later published for an audience of economists in the prestigious Zeitschrift für Nationalökonomie (where Morgenstern served as editor from 1930-38) in 1936. The paper was also later translated to English by Otto Eckstein and published in the prestigious journal Econometrica:
Wald, A. 1936. Über einige Gleichungssysteme der mathematischen Ökonomie (“On some systems of Equations in Mathematical Economics”), Zeitschrift für Nationalökonomie 7(5). p. 637-670.
The assumptions underlying Wald’s proofs would later be greatly improved by economists Kenneth Arrow (1921-2017), Gérard Debreu (1921-2004) and Lionel W. McKenzie (1919-2010), one of Morgenstern’s later Ph.D. students at Princeton University. Their extensions now constitute the foundation of modern general equilibrium theory, the so-called Arrow-Debreu model.
John von Neumann’s Existence Proof (1936)
Wald, Schlesinger and Cassel were not the only attendees at Menger’s Colloquium who had their mind set on advancing economics via mathematization. John von Neumann (1903-57), who seems to have made only a single appearance at the Colloquium, in 1936 presented a paper titled Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes (“On a System of Economic Equations and a Generalization of Brouwer’s Fixed Point Theorem“). The paper was published in the 1937 proceedings to the Colloquium. Although it dealt with the same kind of model as that of Wald’s (inspired by the Walras-Cassel System and modified by Schlesinger), there is no reason to believe that von Neumann had drawn his inspiration from these events two years prior (Kurz, 1999). Indeed, von Neumann had left Europe for America in 1930, joining the Institute for Advanced Study as one of its six founding professors alongside Albert Einstein (1879-1955), Marston Morse (1892-1977), Oswald Veblen (1880-1960) and Hermann Weyl (1885-1955). The paper he presented at Menger’s Colloquium was first presented in the Mathematical Seminar at Princeton University in 1932 (Kurz and Salvadori, 1993), where the IAS was housed until 1939.
von Neumann, ever the polymath, had preoccupied himself with economics as early as in 1926 while still an apprentice of David Hilbert (1862-1943) in Göttingen. His first contribution to economics, indeed the paper that spurred the founding of modern game theory, was his 1928 paper:
in which the idea of mixed-strategy equilibria in two-person zero-sum games was first introduced. The paper and its content was the topic of the Privatdozent newsletter ‘John von Neumann’s Minimax Theorem’, released on March 26th 2021:
Surviving records from the Colloquium and from Menger, Wald and Morgenstern appear to imply that Wald too presented a fixed-point proof of a general equilibrium model at the Colloquium in 1935 (like that presented by von Neumann two years later), but that it could not be printed due to limited space in the proceedings of 1937 (Düppe & Weintraub, 2015).
Overall, the papers published in the proceedings of the Colloquium, the “Ergebnisse”, had an important and lasting effect in many fields related to mathematics and statistics, including mathematical economics. This despite the fact that the circulation of the proceedings was very limited. Indeed, only a few complete sets have survived to this day. The results of Ergebnisise were however later gathered and published in a volume entitled Egrebnisse eines Mathematischen Kolloquiums (“The Results from the Mathematical Colloquium“) which includes chapters in English by Gödel’s posthumous biographer J.W. Dawson, 1983 Nobel Laureate Debreu, professor of mathematics Karl Sigmund (1945-) and mathematician Franz Alt (1910-2011). The book is available as:
Menger, K. 1998. Ergebnisse eines Mathematischen Kolloquiums*. Springer-Verlag Wien GmbH.
Menger’s Later Life
Inspired by the deteriorating political situation in Austria, Menger wrote a book on ethics published by Springer in 1934. Following the death (at the age of 54) of his doctoral advisor Hahn in 1934 and the murder of Schlick by a former student in 1936, the Vienna Circle came to its end in 1935. Menger’s Colloquium continued on for another two years before itself being disbanded in 1936 as more and more of its members left Austria for Britain and the United States. The same year, Menger attended the International Congress of Mathematics in Oslo, being elected its president. “Here friends and associates urged Menger to leave Austria. Menger obtained a professorship at Notre Dame, and the family settled in South Bend, Indiana in 1937”
Menger liked the United States. Unlike many of his fellow émigrés he enjoyed teaching elementary subjects such as trigonometry, regarding teaching (when properly done) not only as pleasurable in itself but also as a stimulus (rather than a hindrance) to creative research (Golland et al, 1994). Safely in America, he helped his former student Wald escape post-Anschluss Austria and arranged for Gödel to visit Notre Dame in 1939.
Menger married an actuarial student by the name of Hilda Axamit in 1936. They had four children: Karl Jr. in 1936, twins Rosemary and Fred in 1937 and Eve in 1942. He remained at Notre Dame until 1946 before being hired at the Illinois Institute of Technology in 1946. He died in his sleep on October 5th 1985 in the home of his daughter Rosemary in Highland Park, Illinois at 83 years old.
Those interested in reading more about Karl Menger and the Vienna Colloquium are encouraged to acquire the book:
Menger, K., Golland, L. McGuinness, B. & Sklar, A. 1994. Reminiscences of the Vienna Circle and the Mathematical Colloquium*. Kluwer Academic Publishers.
Thank you for reading Privatdozent. Have a great weekend!
John von Neumann’s Minimax Theorem, March 26th 2021
Oskar Morgenstern’s Transformation (1925-38), August 27th 2021
The Unparalleled Genius of John von Neumann, May 19th 2021
Gödel’s Constitutional Quarrel, June 14th 2021
Kurt Gödel’s Brilliant Madness, June 21st 2021
Debrau, G. 1998. Economics in a mathematical colloquium. In: Menger, K. (ed.) Ergebnisse eines Mathematischen Kolloquiums*. p. 1-4. Springer-Verlag Wien GmbH.
Düppe, T. & Weintraub, E.R. 2015. Losing Equilibrium: On the Existence of Abraham Wald’s Fixed-Point Proof of 1935.
Kass, S. 1996. Karl Menger. Noices of the American Mathematical Society 43(5), p. 558-561.
Kurz, H.D. 1999. Book Reviews. The European Journal of the History of Economic Thought 6(4), p. 628-630.
Kurz, H.D. & Salvadori, N. 1993. Von Neumann’s growth model and the ‘classical’ tradition. The European Journal of the History of Economic Thought 1(1), p. 129-160.
Menger, K. 1994. Karl Menger. Reminiscences of the Vienna Circle and the Mathematical Colloquium. Kluwer Academic Publishers.
Menger, K. 1998. Ergebnisse eines Mathematischen Kolloquiums*. Springer-Verlag Wien GmbH.
Mutoh, I. 2003. Mathematical economics in Vienna between the Wars. Advances in Mathematical Economics 5, p. 167-195.
Sigmund, K. 1998. Menger’s Ergebnisse — a biographical introduction. In: Menger, K. (ed.) Ergebnisse eines Mathematischen Kolloquiums*. p. 1-4. Springer-Verlag Wien GmbH.
Temple, G. (1981). 100 Years of Mathematics. Springer.
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