**Karl Menger** was an Austrian-American mathematician who worked on algebras, geometries, curve and dimension theory. He also contributed to game theory and social sciences.

- Twenty-five years earlier, Carl Menger had given Franz Joseph a three month course on economics.
- Karl Menger attended the Döblinger Gymnasium in Vienna (1913-1920) where two of his fellow students were Wolfgang Pauli and Richard Kuhn (1900-1967).
- One of the students in Menger's class was Heinrich Schnitzler (1902-1982) who went on to become an actor and film director.
- After Menger graduated from the Döblinger Gymnasium, he entered the University of Vienna in 1920 to study physics.
- At the University of Vienna, Menger attended physics lectures by the theoretical physicist Hans Thirring (1888-1976) who had made significant contributions to the theory of general relativity.
- However Hans Hahn became a lecturer in Vienna in March 1921 and Menger attended a course he gave on What's new concerning the concept of a curve.
- After a week of complete engrossment, he produced a definition of a curve and confided it to fellow student Otto Schreier, who could find no flaw but alerted Menger to recent commentary by Hausdorff and Bieberbach as to the problem's intractability, which Hahn hadn't mentioned.
- Before the seminar's second meeting Menger met with Hahn, who, unaccustomed to giving first-year students a serious hearing, nevertheless listened and after some thought agreed that Menger's was a promising attack on the problem.
- Although Menger was now fascinated in the topic and was encouraged by Hahn to work on it, he was diagnosed with tuberculosis and went to a sanatorium in Aflenz in the mountains of Styria in southern Austria.
- Menger's mathematical investigations, carried out in the sanatorium, led him to a definition of dimension independently of Pavel Urysohn.
- However Urysohn had died in a drowning accident before he could publish his work and Menger was not aware of it.
- The severe lung disease forced Menger to spend more than a year in the sanatorium, but he returned to Vienna with important papers he had written on dimension while in the sanatorium and, advised by Hahn, completed his doctorate in 1924 with his thesis Über die Dimensionalität von Punktmengen Ⓣ(On the dimensionality of point sets).
- This was not the only work Menger undertook at this time.
- By the time he had completed this work, Menger had gained considerable expertise in economics.
- In March 1925 Menger was invited by L E J Brouwer to use his recently won Rockefeller Fellowship to come to the University of Amsterdam where he spent two years working as Brouwer's assistant.
- In 1927 Menger was invited by Hahn to accept the chair of geometry at the University of Vienna when Kurt Reidemeister left for Königsberg.
- Menger was not sorry to leave Amsterdam since he had become involved in a priority dispute with Brouwer and they were not on the best of terms.
- In Vienna, Menger became a member of the Vienna Circle which comprised philosophers, mathematicians and logicians.
- He also started up the Mathematical Colloquium in Vienna in 1928 which was addressed by leading mathematicians and Menger published the Proceedings.
- The 'Kolloquiums' were organised by Karl Menger at the University of Vienna, initially in response to a request from some of the mathematical students there.
- Menger kept a record of the meetings and published the notes through Teubner Verlag or Deuticke.
- There are contributions from Čech, Gödel, Menger, Popper, Tarski, Taussky, von Neumann, Wald and Wiener.
- In 1928 Menger published the book Dimensiontheorie Ⓣ(Dimension theory)..
- The essentials of the dimensionality theory, which has by now attained a considerable perfection through the recent writings of Menger, Hurewicz, P S Aleksandrov and others, have been developed with admirable clarity and completeness in a recently published book by Professor Menger.
- Menger spent the academic year 1930-31 in the United States.
- Before Menger went to the United States, Kurt Gödel had joined his Mathematical Colloquium.
- While in the United States, Menger kept in touch with the Colloquium in Vienna through Georg Nöbeling and also corresponded with Gödel.
- Back in Vienna, Menger published Kurventheorie Ⓣ(Curve Theory) in 1932.
- Menger attended the International Congress of Mathematicians in Zürich in September 1932 when he gave one of the plenary addresses on Neuere Methoden und Probleme der Geometrie Ⓣ(New methods and problems of geometry).
- When Hitler came to power in Germany in 1933, Menger soon realised the problems that lay ahead for Austria.
- Menger left the country a year before the Anschluss.
- The problems had become very real to Menger when, in June 1936, Moritz Schlick (professor of philosophy in Vienna and one of the founders of the Vienna Circle) was shot dead by a student.
- A month later Menger, still stunned by the tragedy, was in Oslo at the International Congress of Mathematicians.
- Menger, however, were politely told that the University of Vienna had no place for them.
- At Notre Dame, Menger was a colleague of Emil Artin who was escaping from the Nazis and spent the year 1937-38 there.
- Menger, with these two colleagues and a couple of others, set up a Ph.D. programme at Notre Dame.
- Menger also organised a Mathematical Colloquium based on the one he had set up at Vienna and began publishing Reports of a Mathematical Colloquium in 1938.
- However after the war began to affect the United States in 1941, academic life was disrupted and Menger's Mathematical Colloquium failed to become as influential as the Vienna Circle had been.
- Rudolf Carnap, one of Menger's colleagues in the Vienna Circle, had also emigrated to the United States and had set up the Chicago Circle.
- Even while Menger was running his own Colloquium at Notre Dame, he still made the effort to attend Carnap's Chicago Circle whenever possible.
- Around this time Menger's interests in mathematics broadened and he began to work on hyperbolic geometry, probabilistic geometry and the algebra of functions.
- Menger's work on geometry failed to have the impact that his work on dimension theory had.
- Also during the war years Menger's contribution to the war effort was teaching calculus to Naval cadets as part of the V-12 Navy College Training Program which ran from 1942 to 1944.
- The chairman of the mathematics department at the Illinois Institute of Technology in Chicago was Lester R Ford whom Menger had known from the time of his 1931 visit to the Rice Institute in Houston, Texas.
- Menger talked to Ford about wanting to move to Chicago and Ford was soon in a position to make him an offer.
- In 1948 Menger went to the Illinois Institute of Technology and he was to remain in Chicago for the rest of his life.
- Menger was solicitous of students.
- We have seen how Menger's interests extended beyond mathematics to philosophy and economics.
- Many important ideas of the Vienna Circle originated with Menger - though they are often attributed to others.
- Generally, Menger could be described as the most logical positivist of them all.

Born 13 January 1902, Vienna, Austria. Died 5 October 1985, Chicago, Illinois, USA.

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Origin Austria

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive