comm-menger.qxp 4/22/98 9:44 AM Page 558

Karl Menger Seymour Kass

Karl Menger died on October 5, 1985, in Chicago. dents were Nobel Laureates Richard Kuhn and Except in his native Austria [5], no obituary no- Wolfgang Pauli. He entered the University of Vi- tice seems to have appeared. This note marks ten enna in 1920 to study physics, attending the lec- years since his passing. tures of physicist Hans Thirring. Hans Hahn Menger’s career spanned sixty years, during joined the mathematics faculty in March 1921, which he published 234 papers, 65 of them be- and Menger attended his seminar “News about fore the age of thirty. A partial bibliography ap- the Concept of Curves”. In the first lecture Hahn pears in [15]. His early work in geometry and formulated the problem of making precise the topology secured him an enduring place in math- idea of a curve, which no one had been able to ematics, but his reach extended to logic, set the- articulate, mentioning the unsuccessful attempts ory, differential geometry, calculus of variations, of Cantor, Jordan, and Peano. The topology used graph theory, complex functions, algebra of in the lecture was new to Menger, but he “was functions, economics, philosophy, and peda- completely enthralled and left the lecture room gogy. Characteristic of Menger’s work in geom- in a daze” [10, p. 41]. After a week of complete etry and topology is the reworking of funda- engrossment, he produced a definition of a curve mental concepts from intrinsic points of view and confided it to fellow student Otto Schreier, (curve, dimension, curvature, statistical metric who could find no flaw but alerted Menger to re- spaces, hazy sets). A few of these and some of cent commentary by Hausdorff and Bieberbach his other accomplishments will be mentioned as to the problem’s intractability, which Hahn here. hadn’t mentioned. Before the seminar’s second Menger was born in Vienna on January 13, meeting Menger met with Hahn, who, unaccus- 1902, into a distinguished family. His mother, tomed to giving first-year students a serious Hermione, was an author and musician; his fa- hearing, nevertheless listened and after some ther, Carl Menger, well known as a founder of thought agreed that Menger’s was a promising the Austrian School of Economics, was tutor in attack on the problem. economics to Crown Prince Rudolph (the ill- Inspired, Menger went to work energetically, starred Hapsburg heir-apparent, played by but having been diagnosed with tuberculosis, he Charles Boyer in “Mayerling”). left Vienna to recuperate in a sanatorium in the From 1913 to 1920 Menger attended the mountains of Styria, an Austrian counterpart of Doblinger Gymnasium in Vienna, where he was “The Magic Mountain”. He wrote later that at the recognized as a prodigy. Two of his fellow stu- same time Kafka was dying in another sanato- rium. In Styria he elaborated his theory of curves Seymour Kass is professor of mathematics at the Uni- and dimension, submitting a paper to Monat- versity of Massachusetts, Boston. His e-mail address is shefte fur Mathematik und Physik in 1922 which [email protected]. contained a recursive definition of dimension in

558 NOTICES OF THE AMS VOLUME 43, NUMBER 5 comm-menger.qxp 4/22/98 9:44 AM Page 559

a separable metric space. P. S. Urysohn simul- ner; Chelsea reprint, 1967). It contains Menger’s taneously and independently of Menger devel- n-Arc Theorem: Let G be a graph with A and B oped an equivalent definition. The two disjoint n-tuples of vertices. Then either G Menger/Urysohn definition has become the cor- contains n pairwise disjoint AB-paths (each con- nerstone of the theory [6]. necting a point of A and a point of B), or there After receiving a Ph.D. in 1924, Menger, in- exists a set of fewer than n vertices that sepa- terested in L. E. J. Brouwer’s fundamental work rates A and B. Menger’s account of the theorem’s in topology and wanting to clarify his thought origins appears in [11], an issue of the Journal about intuitionism, which he saw as the coun- of Graph Theory dedicated to his work. In an in- terpart in mathematics to Ernst Mach’s posi- troductory note Frank Harary calls it “the fun- tivism in science, accepted Brouwer’s invitation damental theorem on connectivity of graphs” and to Amsterdam, where he remained two years as “one of the most important results in graph docent and Brouwer’s assistant. Though he found theory”. Variations of Menger’s theorem and Brouwer testy, he retained warm feelings about some of its applications are given in Harary’s his stay in Amsterdam and cited the good Graph Theory (Addison-Wesley, 1969). Brouwer did for some young mathematicians In that same year Menger joined philosopher and “the beautiful experience of watching him Moritz Schlick’s fortnightly discussion group, listen to reports of new discoveries” [15]. which became known as the Vienna Circle [10], In 1927, at age 25, Menger accepted Hahn’s [18]. In 1931 in tandem with the Circle, Menger invitation to fill Kurt Reidemeister’s chair in ran a mathematical colloquium in which Rudolph geometry at the University of Vienna. During Carnap, Kurt Gödel, Alfred Tarski, Olga Taussky, the year 1927–28, in preparing a course on the and others took part. Gödel (also a Hahn student) foundations of projective geometry, Menger set first announced his epoch-making incomplete- out to construct an axiomatic foundation in ness results at the colloquium. Menger edited the terms of the principal features of the theory: the series Ergebnisse eines Mathematischen Kollo- operations of joining and intersecting, which quiums in the years 1931–37. Additionally, dur- Garrett Birkhoff later called lattice operations. ing this period a series of public lectures in sci- Menger was critical of Hilbert’s and Veblen’s ence was held. Einstein and Schroedinger were formulations: Hilbert’s requiring a different prim- among the speakers. Admission was charged, itive for each dimension; Veblen’s giving points and the money generated was used to support a distinguished role not played in the theory it- students. self, since hyperplanes play the same role as In the posthumously published Reminiscences points. Von Neumann in [17] took a line paral- of the Vienna Circle and the Mathematical Col- lel to Menger’s, seeking “to complete the elimi- loquium [10] Menger gives perceptive and lucid nation of the notion of point (and line and plane) accounts of the philosophical and cultural at- from geometry”. He refers to Menger as “the mosphere in Vienna, the personalities that first to replace distinct classes of ‘undefined en- streamed through the Circle and those who par- tities’ by a unique class which consists of all lin- ticipated in the colloquium, and the topics dis- ear subspaces of the given space, an essential cussed and issues that engrossed the partici- part of his system being the axiomatic require- pants. There are chapters on Wittgenstein and ment of a linear dimensionality function.” Menger a moving account of Menger’s long association subsequently gave a self-dual set of axioms for with Gödel. There is also an account of Menger’s the system. time at Harvard and The Rice Institute (1930–31), In 1928 he published Dimensiontheorie (Teub- where he met most of the leading U.S. mathe- ner). Fifty years later, J. Keesling wrote: “This maticians of the day. Though he recognized book has historical value. It reveals at one and E. H. Moore as the father of American mathe- the same time the naiveté of the early investi- matics, Percy Bridgman and Emil Post made the gators by modern standards and yet their re- strongest impression on him. Menger regarded markable perception of what the important re- Bridgman as the successor to Mach. And he had sults were and the future direction of the theory” great admiration for Post, who (with Haskell [7]. The authors of [10] illustrate the remark Curry) is a source for Menger’s later work in the with Menger’s theorem that every n-dimensional algebra of functions. separable metric space is homeomorphic to part In the early 1930s Menger developed a notion of a certain “universal” n-dimensional space, of general curvature of an arc A in a compact which can in turn be realized as a compact set convex metric space. Consider a triple of points in (2n +1)-dimensional Euclidean space. The of A, where A is an ordered continuum, not nec- universal 1-dimensional space, (the “Menger uni- essarily described by equations or functions. versal curve” or “Menger sponge”), appears in The triangle inequality implies the existence of Mandelbrot’s The Fractal Geometry [9] and in [2]. three points in the Euclidean plane isometric to Menger’s Kurventheorie appeared in 1932 (Teub- the given triple, and their Menger curvature is

MAY 1996 NOTICES OF THE AMS 559 comm-menger.qxp 4/22/98 9:44 AM Page 560

the reciprocal of the radius of the circumscrib- it stimulated research. During the 1960s he lec- ing circle. This curvature is zero if and only if tured to high school students on the subject one of the points is between the other two. “What is x?”. Menger defined the curvature at a point of A to His fourth child was born in 1942, and the oth- be the number (if it exists) from which the cur- ers were then reaching school age. Menger was vature of any three sufficiently close points in drawn to Chicago, where Carnap and others the Euclidean plane differs arbitrarily little. Nu- were developing a sort of “Chicago Circle” and merous results and modifications of Menger’s where his children’s education would benefit concept were obtained by his student Franz Alt from a cosmopolitan environment. L. R. Ford, and by Gödel [4]. The extension to higher-di- whom he had met at Rice in 1931, had become mensional manifolds was achieved by Menger’s IIT mathematics department chairman. Thus in student Abraham Wald (later a distinguished 1948 Menger came to IIT and remained in statistician), who obtained a fundamentally new Chicago the rest of his life. way of introducing Gaussian curvature. Menger’s In the note “Statistical Metrics” [14] Menger comment: “This result should make geometers made a contribution to resolving the interpre- realize that (contrary to the traditional view) the tative issue of quantum mechanics. He proposed fundamental notion of curvature does not de- transferring the probabilistic notions of quan- pend on coordinates, equations, parametriza- tum mechanics from the physics to the under- tions, or differentiability assumptions. The lying geometry. He showed how one could re- essence of curvature lies in the general notion place a numerical distance between points p of convex metric space and a quadruple of points and q by a distribution function Fpq whose in such a space” [12]. value Fpq(x) at the real number x is interpreted Menger had a lifelong interest in economics as the probability that the distance between p [15]. Oscar Morgenstern reports that Menger’s and q is less than x. Studies of such spaces by paper “Das Unsicherherheitsmoment in der numerous authors followed, as did the book Wertlehre” (1934) played a primary role in per- Probabilistic metric spaces (North Holland, 1983), suading von Neumann to undertake a formal by Berthold Schweizer and Abe Sklar. treatment of utility [8]. Two essays of Menger’s In 1951 Menger introduced the concept of a appear in Economic activity analysis, the 1954 “hazy set” [16], in which the element-set relation Princeton Economics Research Project’s collec- is replaced by the probability of an element be- tion of essays edited by Morgenstern. longing to a set. Hazy sets were rediscovered and With Hitler’s coming to power in 1933, Aus- renamed “fuzzy sets” in [1] and have become the trian agitation for unification with Germany in- subject of a broad research area. tensified. With the resulting turmoil and street In the war years Menger taught calculus to fu- violence and Chancellor Dolfuss taking dictato- ture navy officers in the V-12 program, which rial powers in 1934, the vigorous intellectual prompted him to rethink the foundations of the life in Vienna atrophied. Extreme pro-German subject. It led to the monograph Algebra of “nationalists” ruled the faculty and the student analysis [13] and some papers which aimed to body of the university, and the Circle was dis- systematize and clarify the foundations of analy- paraged and maligned. Hahn, a progressive force, sis, and it rekindled his interest in the algebra had died in 1934, and in June 1936 Schlick, of functions. It also acquainted him with the founder of the Circle, was shot dead by a de- calculus textbooks of the time, which he found ranged student. Still stunned by the tragedy, made scant distinction between f and f (x), and Menger attended the 1936 International Con- in which specific functions, used everywhere in gress of Mathematicians in Oslo, becoming one calculus—like the identity function and selector of its vice-presidents. He described the deterio- functions for ordered n-tuples—were not iden- rating situation to friends and associates. Shortly tified. But Menger was operating in the frame- thereafter he received a cable offering him a work of the Vienna Circle and the emphasis professorship at the University of Notre Dame. there on clear thought and expression. One He and his family arrived in South Bend in 1937. might guess that he found an assault on the In March 1938, the month of the Anschluss, traditional calculus irresistible. Thus Menger’s Menger resigned his professorship in Vienna. attempt to reform calculus: Calculus, a modern He did not return until the 1960s. approach (Ginn, 1955). Written in characteristi- European intellectuals who fled Europe for cally vigorous style, it was a radical revision of refuge in America were sometimes uncomfort- textbooks of the period, scrapping some tradi- able with American ways and unaccustomed to tional notation. It received a lengthy, thought- teaching elementary courses. Though central ful, and cautiously favorable review in the European in dress, manner, and style, Menger felt Monthly by H. E. Bray [3] but was never accorded at home in America and enjoyed teaching un- serious attention. He sent a copy to Einstein, who dergraduates, believing that when properly done, replied that he liked it and recognized the need

560 NOTICES OF THE AMS VOLUME 43, NUMBER 5 comm-menger.qxp 4/22/98 9:44 AM Page 561

for some clarity in notation, but advised against [8] H. W. Kuhn and A. W. Tucker, John von attempting too much “housecleaning”. Neumann’s work on the theory of games, That his book was ignored saddened Menger’s Bull. Amer. Math. Soc. 64 (1958). later years. When Menger addressed the foun- [9] B. B. Mandelbrot, The fractal geometry of dations of dimension theory, topology, projec- nature, W. H. Freeman, 1982. tive n-space, or differential geometry, attention [10] K. Menger, Reminiscences of the Vienna Cir- was paid by the best mathematical minds of his cle and the Mathematical Colloquium, generation: Hahn, Brouwer, von Neumann, Gödel. (L. Golland, B. McGuinness, and A. Sklar, The failure of the calculus endeavor strained eds.), Kluwer Academic Publishers, Dort- his relations with the mathematical community. mund, 1994. Menger has been described as a fiery per- [11] ———, On the origin of the n-arc theorem, J. sonality. As a junior faculty member at IIT in the Graph Theory 5 (1981), 341–350. 1960s, I found him gracious, charming, and vi- [12] ———, The formative years of Abraham Wald vacious. Menger was solicitous of students. From and his work in geometry, Ann. of Math. Sta- his early days in Vienna onward he invited stu- tistics 23 (1952), 14–20. dents and faculty to his home. In Chicago it in- [13] ———, Algebra of analysis, Notre Dame Math. cluded a tour of his decorative tile collection, Lectures, vol. 3, Univ. Notre Dame, IN, 1944. which lined the walls of his living room. And he [14] ———, Statistical metrics, Proc. Nat. Acad. Sci. sometimes invited doctoral students for early U.S.A. 28 (1942). morning mathematical walks along Lake Michi- [15] ———, Selected papers in logic and founda- gan. tions, didactics, economics, Reidel, 1979. His office was a showplace of chaos, the desk- [16] ———, Ensembles flous et fonctions aleatoires, top covered with a turbulent sea of papers. He CRP 232 (1951), 2001–2003. knew the exact position of each scrap. On the [17] J. von Neumann, Continuous geometry, Proc. telephone he could instruct a secretary exactly Nat. Acad. Sci. U.S.A. 22 (1936), 92–100. how to locate what he needed. Once, in his ab- [18] K. Sigmund, A philosopher’s mathematician: sence, a new secretary undertook to “make Hans Hahn and the Vienna Circle, The Math- order”, making little stacks on his desk. Upon his ematical Intelligencer 17 (1995), 6–29. return, discovering the disaster, he nearly wept, because “Now I don’t know where anything is.” Menger liked America. He even liked the Marx Brothers. I once met him emerging from the Clark Theater in Chicago, where “A Night at the Opera” was playing. Still suffused with laughter, all he could say was, “Funny! Funny!” Though born into a family with ties to the Aus- trian crown, Menger did not like establishments. His work shows that he could shed traditional ways as called for. He was a peerless mathe- matician and an independent and original spirit.

References [1] R. Bellman, R. Kalabe, and L. Zadeh, Ab- straction and pattern design, J. Math. Anal. Appl. 13 (1966), 1–7. [2] L. Blumental and K. Menger, Studies in geometry, W. H. Freeman, 1970. [3] H. E. Bray, Math. Monthly of America, vol. 61 (1954), 483–493. [4] K. Gödel, Collected works, vol. 1, Oxford Univ. Press, 1986, pp. 272–281. [5] E. Hlawka and K. Menger, Sonderabdruck aus dem Almanach der Osterreichischen Akademie der Wissenschaften, 136 Jahrang, 1986. [6] W. Hurewicz, Dimension theory, Princeton Univ. Press, 1941. [7] J. Keesling, Bull. Amer. Math. Soc. 83 (1958), 203.

MAY 1996 NOTICES OF THE AMS 561