Karl Menger's father, Carl Menger (1840-1921) was a famous economist, a professor at the University of Vienna, and founder the Austrian School of Economics. Karl's mother was Hermine Andermann (1869-1924), a journalist, author and musician. Carl was Roman Catholic and Hermine was Jewish so, although they lived together, the two could not marry (all marriages in Austria at this time were religious ceremonies). When their only child Karl (the subject of this biography) was born in 1902 this was seen as unacceptable by the Viennese social conventions and Carl was forced to withdraw from public life. He stopped teaching at the University of Vienna as soon as his son was born and he resigned his chair at the University a year later. Carl applied to Emperor Franz Joseph of Austria to have his son Karl made legitimate and his request was eventually granted. 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). It is worth noting that Kuhn was awarded the Nobel Prize for Chemistry in 1938 and Pauli was awarded the Nobel Prize for Physics in 1945. One of the students in Menger's class was Heinrich Schnitzler (1902-1982) who went on to become an actor and film director. Heinrich Schnitzler was the son of the famous author Arthur Schnitzler (1862-1931) and, at this stage, Menger had ideas of writing dramas. This was not an easy time to be growing up in Vienna for, after the outbreak of World War I, in September 1914 the Döblinger Gymnasium building was converted into a war hospital, and teaching was transferred to a secondary school building in the Krottenbachstrasse. In February 1916 the war hospital closed and pupils returned to the original building. The years following the end of World War I in 1918 were particularly hard at the school. Political events such as the end of the monarchy forced major educational changes and the Gymnasium became a state school. After Menger graduated from the Döblinger Gymnasium, he entered the University of Vienna in 1920 to study physics. He had not given up his idea of writing dramas, for at this time he began to write a drama about the apocryphal Pope Joan.
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. Seymour Kass writes :-
In the first lecture Hahn formulated the problem of making precise the idea of a curve, which no one had been able to articulate, mentioning the unsuccessful attempts of Cantor, Jordan, and Peano. The topology used in the lecture was new to Menger, but he "was completely enthralled and left the lecture room in a daze" . 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. This was not the only work Menger undertook at this time. His father, Carl Menger, had died while Menger was in the sanatorium and had left the second edition of his book Grundsätze der Volkswirthschaftslehre (1st edition 1871) unfinished. Menger completed his father's work and supervised the publication of the second edition which appeared in 1922. 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. He wrote from Amsterdam to his girlfriend Hilda Axamit, an actuarial mathematics student, in Vienna :-
I am lecturing on the calculus of variations. Personally, I am occupied by geometry of all kinds, furthermore by epistemology. I hope I'll get the energy to put together my views about the problem of truth. In the last weeks, I have had so many ideas that I don't have any time at all to write them down, and run away every evening to distract myself ... in order not to overwork. Apart from that, I curse the fact that I am not in Vienna but rather here. I can't get used to living here, and I will try my best to leave here forever in the month of June.
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. However, he certainly respected Brouwer :-
Though he found Brouwer testy, he retained warm feelings about his stay in Amsterdam and cited the good Brouwer did for some young mathematicians and "the beautiful experience of watching him listen to reports of new discoveries".
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. Steve Abbott, reviewing a 1998 reprinting of the Proceedings, writes :-
The 'Ergebnisse eines Mathematischen Kolloquiums' were published in Vienna between 1929 and 1937. The eight issues included many contributions by highly respected mathematicians but suffered from a limited distribution, and few complete sets remain. This book reprints all the articles (in German) along with chapters (in English) surveying the important developments in economics, logic, topology and geometry that were reported in the 'Ergebnisse'. 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. The meetings included a mixture of lectures, discussions of unsolved problems and reviews of recent work. Menger kept a record of the meetings and published the notes through Teubner Verlag or Deuticke. ... It is the stellar collection of 'Kolloquium' speakers that justifies the reprinting of the papers. There are contributions from Čech, Gödel, Menger, Popper, Tarski, Taussky, von Neumann, Wald and Wiener.
In 1928 Menger published the book Dimensiontheorie.. Paul Althaus Smith (1900-1980) writes in a review :-
There is an important phase in the development of modern point set theoretical geometry which has been closely associated with the concept of dimensionality, - we refer to the attempt to create precise mathematical meaning for the simple geometric spaces of our intuition in terms of primitive non-arithmetical concepts. That the idea of dimensionality should have come into play and itself have been studied and made precise is indeed natural, since the curves, surfaces, and solids of our experience furnish the very basis for our intuitive ideas of dimensionality. ... [A definition of dimension] was introduced by Professor Karl Menger of Vienna and developed in a sequence of shorter papers characterized by their elegance and generality. 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.
Of course, more sophisticated ideas on dimension continue to be developed. However, James Kesling, in a 1977 review of another work, wrote of Menger's Dimensiontheorie:-
It reveals at one and the same time the naiveté 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.
Menger spent the academic year 1930-31 in the United States. He visited Harvard University and the Rice Institute in Houston, Texas. 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. For example, in early 1931 he wrote:-
I am replying to your charming letter in a moving train and therefore by typewriter. Nöbeling had already written to me last autumn about your great discovery. I read your article with the utmost interest and immediately delivered a report on it here. I rank your achievement among the greatest of modern logic and send you my heartiest congratulations.
Back in Vienna, Menger published Kurventheorie in 1932. Wallace Alvin Wilson writes :-
The book under discussion is to be the second volume of a work entitled 'Mengentheoretische Geometrie'. In addition to being the first treatise on the theory of curves and their topological properties, the book performs the further service of gathering together in an organic whole much of the research work on this subject done since the war - work which, from the nature of its piecemeal appearance, must have seemed to many to be of a pointless character.
This book contains Menger's 'n-Arc Theorem', described by Frank Harary as:-
... one of the most important results in graph theory ... the fundamental theorem on connectivity in graphs.
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. He became engaged to his girlfriend Hilda Axamit and in 1934 they spent a holiday together in Ramsau and Strobl. They married on 5 December 1934 and had four children; Karl Jr (born 9 July 1936), twins Rosemary and Fred (born 13 December 1937), and Eve (born 1942). When Hitler came to power in Germany in 1933, Menger soon realised the problems that lay ahead for Austria. Steve Abbot, writing about Menger's Colloquium, explains :-
The Ergebnisse was criticised at the time for having 'too many Jewish contributors'. With the rise of the Nazis in Germany, it was only a matter of time before the political situation in Austria forced the end of the Kolloquium meetings. 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. He explained to everyone how the situation in Vienna was deteriorating fast. Soon after this he was offered a chair at the University of Notre Dame, Indiana and he went to the United States in 1937 to take up the post. At this stage he kept open his chair in Vienna but, in March 1938, as a result of the political situation in Austria, he resigned his chair in Vienna. Certainly at this stage he was still expecting to return to Vienna after the war but, as Karl Signund writes:-
After the war, the reconstruction of the bombed-out State Opera was accorded highest priority by democratic new Austria. Men like ... 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. Also on the staff at Notre Dame was Paul Milton Pepper (1909-2010) who had just completed his doctorate at the University of Cincinnati. 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. He arranged a visit by Gödel to Notre Dame but failed to persuade him to accept a post there. 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. The Reports stopped publication in 1946. 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. This is almost certainly because geometry, at this time, was a rather neglected area of mathematics, particularly in the United States. 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. This led to his interest in mathematical education and, during the 1950s and 1960s, he wrote articles on mathematical education and published books with new ideas on teaching calculus, geometry and other branches of mathematics.
An article by Menger on teaching is available at THIS LINK
reviews of Karl Menger's books are at THIS LINK.
The visits that Menger made to Carnap's Chicago Circle led to him feeling that Chicago would be both a better place for him to work and a better place for his children to be educated. 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. Seymour Kass, who was a colleague of Menger's at Chicago during the 1960s, writes :-
Menger has been described as a fiery personality. As a junior faculty member at IIT in the 1960s, I found him gracious, charming, and vivacious. Menger was solicitous of students. From his early days in Vienna onward he invited students and faculty to his home. In Chicago it included a tour of his decorative tile collection, which lined the walls of his living room. And he sometimes invited doctoral students for early morning mathematical walks along Lake Michigan. His office was a showplace of chaos, the desktop covered with a turbulent sea of papers. He knew the exact position of each scrap. On the telephone he could instruct a secretary exactly how to locate what he needed. Once, in his absence, a new secretary undertook to "make order", making little stacks on his desk. Upon his return, discovering the disaster, he nearly wept, because "Now I don't know where anything is."
Menger taught in Chicago until he retired in 1971.
We have seen how Menger's interests extended beyond mathematics to philosophy and economics. Donald Gillies, in his fascinating paper on 'Karl Menger as a Philosopher', writes :-
Menger is not always given his due recognition as a philosopher. In fact ..., many important ideas of the Vienna Circle originated with Menger - though they are often attributed to others. Moreover, Menger's own philosophy of mathematics (for which I have coined the name: 'laissez-faire formalism') is the implicit philosophy of many working mathematicians. Generally, Menger could be described as the most logical positivist of them all.
In Austrian Marginalism and Mathematical Economics (1973), Menger writes:-
Since I am the son of the author of the 'Grundsätze' as well as a mathematician, 'two souls reside within my breast '. I hope that this fact will help me in discussing various points of the controversy objectively.
Karl Sigmund, reviewing Menger's paper, writes:-
By bringing together economists and mathematicians, Menger had played a catalytic role for which he was uniquely predisposed by his upbringing and education.
In  his interests are described as follows:-
He had a great love of music. ... he built up a notable collection of decorative tiles from all over the world. ... he ate meat sparingly, particularly in his last years. But he was always glad to sample cuisines, from Cuban to Ethiopian, that were new to him. He liked baked apples.
Article by: J J O'Connor and E F Robertson
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