*Compiled by Greg Fasshauer*

- Menger's Life
- Menger's 100th Birthday (Jan.13 2002)
- The Menger Sponge
- Bibliography of Menger's Work
- Quote by Karl Menger
- Karl Menger's Ph.D. Students
- Principal Dates
- Pictures of Karl Menger
- Karl Menger's Fields of Research
- References
- Karl Menger's Voice

### Karl Menger's Fields of Research

- Theory of Curves and Dimension Theory
- A General Theory of Length and the Calculus of Variations
- Probabilistic Metric
- New Foundations for the Bolyai-Lobachevsky Geometry
- A New Approach to Calculus
- Multiderivatives
- Foundations of Mathematics and Logical Tolerance
- Ethics. Formal Studies of Human Relations
- Geometry of General Metric Spaces (Convexity. Geodesics. Characterization of Euclidean Sets.)
- Coordinate-Free Treatment of Curvature
- Algebra of Geometry. Normed Lattices. Comprehensive Treatment of Diverse Mathematical Structures. Duality
- Algebra of Functions. Interconnection of Algebraic Systems
- Analysis of the Idea of Variables. A Theory of the Application of Analysis to Science
- Analytic Functions
- Economics: Introduction of Probability into the Theory of Value. The Logic of the Laws of Return

## Quote

*Nicht etwa, daß bei größerer Verbreitung des Einblickes in die Methode der Mathematik notwendigerweise viel mehr Kluges gesagt würde als heute, aber es würde sicher viel weniger Unkluges gesagt.
- Karl Menger*

[Not that, if one were to spread the insight into the methods of mathematics more widely, this would necessarily result in many more intelligent things being said than today, but certainly many fewer unintelligent things would be said.]

## Principal Dates

1902 | born in Vienna |

1920-1924 | studied at the University of Vienna; Ph.D. in Mathematics |

1925-1927 | docent at the University of Amsterdam |

1927-1936 | professor of geometry at the University of Vienna |

1930-1931 | visiting lecturer at Harvard University and The Rice Institute |

1931-1937 | founder of the Ergebnisse eines Mathematischen Kolloquiums |

1937-1946 | founder of Reports of a Mathematical Colloquium, 2nd series, and Notre Dame Mathematical Lectures [1944-1][1944-6] |

1946-1971 | professor of mathematics, Illinois Institute of Technology, Chicago |

1951 | visiting lecturer at the Sorbonne, Paris |

1961 | guest lecturer in several European universities |

1964 | visiting professor, University of Arizona and Ford Institute, Vienna |

1968 | visiting professor, Middle East Technical University, Ankara |

1971 | professor emeritus in Chicago |

1975 | Austrian Cross of Honor for Science and Art First Class |

1983 | Doctorate of Humane Letters and Sciences, Illinois Institute of Technology |

1985 | died in Highland Park near Chicago |

## Biography

Karl Menger was one of the leading mathematicians of the 20th century. Mario Bunge, in the review of a collection of Menger's papers [Sel], says:

*The author's publications span a half-century period and they cover an amazing variety of fields, from logic to set theory to geometry to analysis to the didactics of mathematics to economics to Gulliver's interest in mathematics... This selection should be of interest to foundational workers, mathematics teachers, philosophers, and all of those who suffer from nostalgia for the times when mathematicians, like philosophers, were interested in all conceptual problems.*

Karl Menger was born on January 13, 1902 in Vienna. His father was the famous Austrian economist Carl Menger (1840-1921) who was one of the founders of marginal utility theory [Gru]. His mother Hermione Andermann, who was thirty years younger than her husband, was a noted novelist who was also interested in music.

As a young man Karl explored his talents for literature. In fact, in 1920 Menger was in his last year at the Döblinger Gymnasium in Vienna where he was a classmate of Heinrich Schnitzler's whose father Arthur was a well-known dramatist (two other classmates were the future Nobel Laureates Wolfgang Pauli and Richard Kuhn). Between the years 1920 and 1931 there are several entries concerning Menger in Arthur Schnitzler's diary. None of the remarks concerning Menger's writing are very favorable. In fact, from 1920 until 1923 Menger was working on a drama about the apocryphal Pope Joan. Schnitzler refers to Menger's writing as *unsuccessful, not favorable*, and remarks that Menger has *no literary ambitions* other than wanting to complete this one drama. Schnitzler, besides noting Menger's genius and talents (for mathematics and physics), refers to Menger as a person who is *not quite normal*, a *strange fellow*, a *megalomaniac*, and *suicidal.*

Menger entered the University of Vienna in the fall of 1920 to study physics. However, when he heard Hans Hahn lecture on *Neueres über den Kurvenbegriff* (What's new concerning the concept of a curve) in March of 1921 his interests were redirected towards mathematics. In the aforementioned lecture Hahn pointed out that there was at that time no satisfactory definition of a curve. In particular he presented the failed attempts of Georg Cantor, Camille Jordan and Giuseppe Peano. Moreover, such authorities as Felix Hausdorff and Ludwig Bieberbach had declared this problem to be virtually unsolvable. Within a few days Menger, an undergraduate student with no other mathematical background information about the problem other than what Hahn had told him, came up with a solution, and presented it to Hahn. Menger's interest in this topic led to his work in curve and dimension theory.

In a lecture given at the University of Chicago in 1976 Menger recalls these events (click here to listen to a part of this lecture).

While working late at night and in damp rooms Menger fell severely ill with tuberculosis. He was sent to a sanatorium in Aflenz (Styria), in the Austrian Alps (from fall 1921-April 1923) and ended up doing much of his fundamental work while recovering there. During this time both of his parents passed away.

At the same time Menger was formulating his ideas, the young Russian mathematician Pavel Urysohn (born in 1898 and died in a drowning accident in 1924) also constructed a theory of dimension in the years 1921 and 1922. According to George Temple [100]:

*Menger's definition is undoubtedly simpler and more general than Urysohn's, and the question of priority is of minor importance.*

After Menger returned from the sanatorium he continued his studies under Hahn, and received his doctorate in 1924.

In March 1925 Menger went to Amsterdam to work with the eminent mathematician L.E.J. Brouwer. In Amsterdam Menger continued to work on curve theory and dimension theory, and also gained insight into logic and the foundations of mathematics. In particular, he was exposed to Brouwer's intuitionistic interpretation of science and mathematics. Witold Hurewicz, who had worked on dimension theory under Menger in Vienna, followed him to Amsterdam in the spring of 1926. After earning his habilitation in the fall of 1926 Menger returned to the University of Vienna (after two somewhat quarrelsome years with Brouwer) and was appointed *Extraordinarius für Geometrie* in February 1927 (replacing Kurt Reidemeister, who had left to take a chair in Königsberg).

In the fall of 1927 Karl Menger became a member of the famous Vienna Circle. This group of about three dozen philosophers, logicians, mathematicians, as well as natural and social scientists was started by Moritz Schlick, Otto Neurath and Hans Hahn and became known to the public in 1929 with the publication of a manifesto entitled *The Scientific World View. The Vienna Circle*. The meetings of the Circle took place in a rather dingy room on the ground floor of the building that housed the mathematical and physical institutes, in the Boltzmanngasse. The original members included

- Rudolf Carnap (philosopher, left Vienna in 1931 to go to Prague, and then emigrated to the United States in 1936, where he started the
*Chicago Circle*at the University of Chicago), - Herbert Feigl (philosopher, student of Schlick's, emigrated to the United States in 1931, founded the Minnesota Center for Philosophy of Science),
- Philipp Frank (mathematician, went to Harvard in 1938),
- Kurt Gödel (mathematician, a student of Hahn's, went to Princeton in 1938),
- Hans Hahn (professor of mathematics at the University of Vienna, and Menger's Ph.D. thesis advisor; died of complications related to cancer surgery July 24 1934 at the age of 35 in Vienna),
- Victor Kraft (philosopher, made contributions toward establishing ethics as a science),
- Karl Menger,
- Otto Neurath (philosopher with interests in logic, optics, the economic theory of value),
- Theodor Radakovic (assistant professor of mathematics at the Polytechnical Institute in Vienna, former student of Hahn's),
- Kurt Reidemeister (professor of geometry at the University of Vienna; departed in 1927)
- Moritz Schlick (professor of philosophy at the University of Vienna, one of the main forces behind the Vienna Circle; was shot dead by a student in 1936),
- Friedrich Waismann (studied mathematics and philosophy under Schlick).

Some of the more or less regular guests included

- Alfred Jules Ayer (philosopher and logician) [also see Wikipedia],
- Gustav Bergmann (mathematician),
- Carl G. Hempel (philosopher) [also see Wikipedia],
- Felix Kaufmann (philosopher of law, worked for an oil company),
- Richard von Mises (mathematician),
- Ernest Nagel (philosopher and logician),
- John von Neumann (mathematician and one of the first computer scientists, click here or here),
- Olga Hahn-Neurath (logician, sister of Hans Hahn and wife of Otto Neurath),
- Rose Rand (philosopher and logician),
- Hans Reichenbach (philosopher),
- Olga Taussky-Todd (mathematician),
- Alfred Tarski (logician),
- Willard Van Orman Quine (philosopher),
- Edgar Zilsel (philosopher).

The Circle also benefitted from frequent contacts with

Menger's more elaborate comments about some of the above listed members can be found in [Rem, pp. 54-73].

The work done by the members of the Vienna Circle may be considered as some of the most important and most influential of thought in the twentieth century. However, the intellectual success was severely hampered by the political happenings at the time. In fact, the Vienna Circle found a tragic end with the *Anschluss* in March of 1938.

Here is some more information on the Vienna Circle (Institute Vienna Circle or click here, here or here).

In parallel to the Vienna Circle Menger started a Mathematical Colloquium at the University of Vienna in 1928. Some of the participants and frequent lecturers included

- Franz Alt (one of Menger's Ph.D. students; emigrated to the United States in 1938; was a founding member of the Association for Computing Machinery (ACM) in 1947; later deputy director of American Institute of Physics in the 1970s),
- Nachman Aronszajn,
- Leonhard M. Blumenthal,
- Karol Borsuk,
- Rudolf Carnap,
- Eduard Cech,
- Kurt Gödel,
- Hans Hornich (specialized in complex analysis and potential theory, became a professor at the Technical University of Vienna),
- Marston Morse,
- John von Neumann (click here or here),
- Georg Nöbeling (student of Menger's; went to Erlangen, Germany, in 1933, where he became a professor in 1942; president of the German Mathematical Society (Deutsche Mathematiker-Vereinigung) in the 1950s; passed away in 2008 shortly after his 100th birthday)
- R. G. Putnam,
- Karl Schlesinger (economist and banker),
- Alfred Tarski,
- Olga Taussky-Todd (went first to Göttingen, where she helped publish David Hilbert's Collected Works; then moved to England, where she married the mathematician John Todd; finally ended up with a most distinguished mathematical career in the United States),
- Abraham Wald (student of Menger's who obtained his Ph.D. in 1930 - after taking only three courses; started in geometry, but then turned to mathematical economics working with Oscar Morgenstern; emigrated to the United States in 1938 with the help of Menger and studied statistics; 1944 professor at Columbia University; one of the founders of sequential analysis and statistical decision theory; died in a plane crash in India in 1950),
- G. T. Whyburn
- Norbert Wiener

The proceedings of the Colloquium (from 1928/29 through 1935/36) were edited and published by Menger (with the help of Kurt Gödel, Georg Nöbeling, Abraham Wald and Franz Alt) in the *Ergebnisse eines Mathematischen Kolloquiums*. This collection of papers contains path-breaking papers by Menger, Gödel, Tarski, Wald, Wiener, John von Neumann, and many others. In particular, the field of mathematical economics was profoundly influenced by the discussion of equilibrium equations by Schlesinger and the response by Wald in March of 1934 [Erg, pp.280-288], which as Menger said [translated from Erg, p.290] marked

*an end of the period in which economists simply formulate equations, without worrying about existence or uniqueness of their solutions.*

Another paper of fundamental importance was contributed by John von Neumann [Erg, p.453], who presented a generalization of Brouwer's *Fixed Point Theorem*.

Karl Menger spent the academic year 1930/31 in the United States visiting Harvard University and the Rice Institute in Houston, Texas. During this visit he met many prominent American mathematicians and philosophers of the time. In [Rem, Ch. XIII] he recalls his meetings with George David Birkhoff, Marston Morse, Henry Maurice Sheffer, Paul Weiss, Norbert Wiener, Edward Huntington, Alfred Whitehead, Willard Van Orman Quine, Percy Bridgman, Josef Schumpeter (an economist), Charles W. Morris, Lester R. Ford, and, in Chicago, Eliakim Hasting Moore. Menger went for long hikes with Henry Maurice Sheffer, as well as with Norbert Wiener, during which they had long discussions. Among the philosophers he was most impressed by Percy Bridgman, whom he referred to as a *modern reincarnation of [Ernst] Mach* [Rem, p.166]. Karl Menger also had great respect for E. H. Moore of whom he says [Rem, p.170]

*If there has been a father of American mathematics, then it certainly was Moore, the teacher of Birkhoff, Oswald Veblen, Robert Lee Moore, and most American mathematicians of their age who later became prominent.*

Menger kept in touch with his students and the Kolloquium in Vienna through Georg Nöbeling, who wrote him in early 1931 of the groundbreaking work of Kurt Gödel [Göd]. Menger promptly interrupted the lecture series he was giving at the Rice Institute to report about Gödel's discovery. Menger says [Rem, p.203]

*Thus the mathematicians at Rice Institute were probably the first group in America to marvel at this turning-point of logic and mathematics.*

In 1936 Menger attended the International Congress of Mathematics in Oslo, and was elected one of its vice presidents. He described the deteriorating situation in Vienna to friends and colleagues. Soon thereafter he was offered a position at the University of Notre Dame, Indiana.

In 1937 Menger went to the United States and accepted the position at Notre Dame. Initially Menger had only asked for an extended leave of absence from the University in Vienna, but after the war he was not invited back. Karl Sigmund writes about the situation:

*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.*

Gödel visited Menger at Notre Dame, but Menger was not able to convince him to stay. At Notre Dame Menger started the Ph.D. program in the mathematics department (together with Arthur Milgram, Paul Pepper, John Kelley, and Emil Artin, see [Hop]), and he organized a series of *Notre Dame Mathematical Lectures* (the 2nd volume of which, Emil Artin's *Galois Theory*, was rather well-known). Menger also started a *Mathematical Colloquium* (shaped after the one in Vienna), and published the related *Reports of a Mathematical Colloquium, Second Series* which appeared between 1938 and 1946. However, WWII affected academic life in the United States, and the success of the Mathematical Colloquium was limited.

In 1935 Karl Menger had married Hilda Axamit, an actuarial student. They had 4 children. Karl Jr., born in 1936, Rosemary and Fred, twins born in 1937, and Eve, born in 1942.

In 1946 Karl Menger was invited to join the newly founded Illinois Institute of Technology by the chairman of the mathematics department Lester R. Ford, who had been at the Rice Institute at the time of Menger's visit there in 1931. Just a few years earlier Eduard Helly had also been called to IIT - but he died shortly after he accepted his position in 1943.

Rudolf Carnap and others had started a *Chicago Circle* at the University of Chicago, and Menger tried to participate as often as possible, even while still at Notre Dame in South Bend, Indiana. Karl Menger spent the rest of his life in Chicago.

During the war years Menger had been heavily involved in the teaching of Calculus to Naval cadets. This was one of the reasons that much of his work in the 50s and 60s was concerned with mathematics education. Among the things he wrote was a Calculus textbook in which he proposed a number of major changes in mathematical formalism and notation aimed at facilitating the teaching of basic mathematics.

His booklet "You Will Like Geometry" was used as a guide to the IIT geometry exhibit at the Museum of Science and Industry in the early 1950s and included Menger's famous "Taxicab Geometry" explanation. In the introduction to "You Will Like Geometry", Menger wrote,

*'Impossible,' you say, 'Geometry is a bore. It has been dead and petrified for centuries.' But you are wrong. Geometry is amazing and ingenious and beautiful and profound; and most important, it is alive and growing. Just follow the growth of the geometric world of plane figures through the ages.*

Also in the 1950s, Menger appeared on local TV and radio programs to talk about mathematics; appeared several times in the Chicago Tribune as an expert on such topics as why students find math difficult ("Johnny Is Puzzled By the X: That's Why He Hates Math, Expert Says"), and lectured to local high school science teachers on geometry.

In 1951/52 Menger spent a sabbatical at the Sorbonne in Paris, and in 1963 he returned to Austria for the first time since he had left in 1937. In 1968 he was a visiting professor at the Middle East Technical University in Ankara, Turkey. In 1971 he was elected a corresponding member of the Austrian Academy of Sciences.

In 1971 Karl Menger became professor emeritus at IIT.

On June 2, 1975, in a ceremony at IIT, the Austrian Consul in Chicago presented the Austrian Cross of Honor for Science and Art First Class to Karl Menger (by then Professor Emeritus). This pleased Menger immensely, since his father had received the same honor many years before.

IIT honored Karl Menger with a Doctorate of Humane Letters and Sciences in December of 1983.

Karl Menger loved music and modern architecture. He collected decorative tiles from all over the world. He disliked wine, but enjoyed sweet liqueurs. Though not a vegetarian, 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. Menger liked to take long walks, and sometimes he invited doctoral students for early morning walks along Lake Michigan. Menger liked America. He even liked the Marx Brothers.

Karl Menger died in his sleep on October 5, 1985 at the home of his daughter Rosemary and son-in-law Richard Gilmore in Highland Park, Illinois.

A short biography of Karl Menger can also be found at the History of Mathematics server at the University of St. Andrews, Scotland.

A very interesting article about Karl Menger was written by Seymour Kass and published by the American Mathematical Society [Kas].

An excellent article about Karl Menger and his Mathematisches Kolloquium in Vienna was written by Louise Golland (an IIT alumn) and Karl Sigmund.

Wikipedia: Karl Menger

## Karl Menger's 100th Birthday Celebration

To commemorate the 100th anniversary of Karl Menger's birth, a conference was held on April 11-12, 2002 in Vienna: Mengerfest (organized by Karl Sigmund and the Austrian Mathematical Society), with presentations by

- Georg Winckler (Rektor Universität Wien),
- Peter Schuster (Vizepräsident Österreichische Akademie der Wissenschaften),
- F.R. McMorris (Chair, Department of Applied Mathematics, IIT),
- Hans Kaiser (Vizerektor, Technische Universität Wien, Vorsitzender ÖMG-Landessektion Wien),
- Harald Rindler (Vorstand, Institut für Mathematik, Universität Wien),
- Walter Benz (Hamburg),
- Abe Sklar (IIT, Chicago),
- Tony Crilly (University of Middlesex),
- Alan Moran (University of Middlesex),
- Ludwig Reich (Universität Graz),
- Karl Sigmund (Universität Wien),
- Lester Senechal (Mount Holyoke),
- Ioan James (University of Oxford),
- Dirk van Dalen (University of Utrecht),
- Bert Schweizer (University of Massachusetts).

## The Menger Sponge

Probably the most popular creation of Karl Menger is the so-called Menger sponge (sometimes wrongly referred to as Sierpinski's sponge). It can be considered as the three-dimensional analog of the *Cantor set* (1D) and the *Sierpinski square* (2D). The Menger sponge appears in many modern books on fractals (e.g. [Man]).

### Construction of the Menger sponge

Take a cube, divide it into 27 = 3 x 3 x 3 smaller cubes of equal size and remove the cube in the center along with the six cubes that share faces with it. You are left with the eight small corner cubes and twelve small edge cubes holding them together. Now, imagine repeating this process on each of the remaining 20 cubes. Repeat again. And again...

In a more abstract setting the Menger sponge is also referred to as the *Menger universal curve*. This comes from Menger's work in dimension theory.

A Menger Sponge, made from business cards.

The animation below shows the Menger sponge of depth 3 created in Maple.

## Karl Menger's Fields of Research

### Curve and Dimension Theory

In his book *Dimensionstheorie* (published in 1928) Menger gave the following recursive definition of the dimension of an abstract set [Rem, p.50]:

A set *S* in a Cartesian space (of any dimension) is *n*-dimensional if,

- each point of
*S*is contained in neighborhoods as small as may be desired with whose frontiers*S*has at most (*n*-1)-dimensional intersections; and - for at least one point of
*S*the frontier of each sufficiently small neighborhood has at least an (*n*-1)-dimensional intersection, [and, to start the process] the empty set [is assigned] the dimension -1.

One of Menger's theorems in this area states that

Every *n*-dimensional separable metric space is topologically equivalent to part of a certain universal *n*-dimensional space, which can in turn be realized as a compact set in (2*n*+1)-dimensional Euclidean space.

The universal one-dimensional curve (or - as a compact set in three-dimensional space - the Menger sponge) is shown above.

This theorem was generalized by Georg Nöbeling and is known as the "Menger-Nöbeling Embedding Theorem".

In 1932 Menger published *Kurventheorie* which contains the famous *n*-Arc Theorem:

Let *G* be a graph with *A* and *B* two disjoint *n*-tuples of vertices. Then either *G* contains *n* pairwise disjoint *AB*-paths (each connecting a point of *A* and a point of *B*), or there exists a set of fewer than *n* vertices that separates *A* and *B*.

This theorem was referred to as one of the fundamental theorems in graph theory by Frank Harary; modern graph theorists call it Menger's Theorem. The history of this theorem was presented by Menger in [n-Arc] in 1981.

### Geometry of General Metric Spaces

Menger was the first to introduce Fréchet's definiton of metric spaces in geometry. The result was called *metric geometry* by Menger, and it included theories of betweenness, geodesic lines and curvature of curves and of surfaces in abstract metric spaces.

One of Menger's theorems on isometric imbeddings is

A metric space *R* can be imbedded in Hilbert space *H* if and only if *R* is separable and every set of *n*+1 (*n*=2,3,4,...) distinct points of *R* can be imbedded in *R _{n}*.

Schoenberg showed how this theorem is at the basis of the notion of positive definite functions in abstract metric spaces [Schoe].

### A General Theory of Length and the Calculus of Variations

### Coordinate-Free Treatment of Curvature

In the 1930s Menger formulated a definition of curvature of an arc *A* without referring to an underlying coordinate system.

Let *A* be an arc in a compact convex metric space. Consider a triple (*q*,*r*,*s*) of points of *A*. The *Menger curvature* of *A* is given by the reciprocal of the radius of the circumscribing circle of the three points. The curvature is zero if and only if one of the points is between the other two. The *curvature of A at a point p* is now given by the limit of the Menger curvature as the three points (*q*,*r*,*s*) become arbitrarily close to *p* (see [Stu, p.320]).

These ideas were extended later to higher-dimensional manifolds primarily by Menger's student Abraham Wald. Of this work Menger says (see [Wald]):

*This result should make geometers realize that (contrary to the traditional view) the fundamental notion of curvature does not depend on coordinates, equations, parametrizations, or differentiability assumptions. The essence of curvature lies in the general notion of convex metric space and a quadruple of points in such a space.*

### Probabilistic Metric

The idea of introducing probabilistic notions into geometry was also one that occupied Menger's thoughts. His motivation came from the idea that positions, distances, areas, volumes, etc., all are subject to variations in measurement in practice. And, as, e.g., quantum mechanics implies, even in theory some measurements are necessarily inexact. In 1942 Menger published a note entitled *Statistical Metrics* [Sta]. In this note he explained how to replace the numerical distance between two points *p* and *q* by a function *F _{pq}* whose value

*F*at the real number

_{pq}(x)*x*is interpreted as the probability that the distance between

*p*and

*q*is less than

*x*. Originally Menger had planned to collaborate with his former student Abraham Wald on this subject, but Wald was killed in a plane crash in India in 1950. However, others, such as Berthold Schweizer (a former student of Menger's) and Abe Sklar (a colleague of Menger's, and now Professor Emeritus of mathematics at IIT), took up the work and developed what is now called the theory of

*probabilistic metric spaces*[Pro].

### Algebra of Geometry

Menger's so-called *algebra of geometry* (see e.g. [AG]) played an important role in John von Neumann's mathematical foundations of quantum mechanics. Menger was one of the first to investigate lattice structures.

### Hyperbolic Geometry

In hyperbolic geometry Menger formulated an axiomatic foundation which was independent of, and simpler than, any possible one for Euclidean geometry. However, this work did not attract as much attention as his work in curve and dimension theory did.

### Algebra of Functions

Menger's book *Algebra of Analysis* was a direct result of his experience with the existing literature for teaching calculus. Menger found that the foundations of analysis needed to be systematized and clarified. In particular, he was bothered by the fact that many textbooks did not emphasize the role of functions appropriately. There was no clear conceptual distinction between a function *f* and its value *f(x)*. And other functions, such as the identity function, played no explicit role at all. In particular, Menger introduced a property of multivariate functions called *superassociativity*. Today algebraic structures in which this property holds are referred to as *Menger algebras*. Menger algebras have found applications, among other areas, in logic, and, to Menger's delight, in geometry.

### A New Approach to Calculus

Karl Menger enjoyed teaching undergraduates. He believed that, when properly done, it stimulated research. In the late 1950s he lectured to high school students on the subject *What is x?* [1956-5].

During WWII Menger taught calculus to future navy officers. This experience led him to rethink the foundations of calculus. He published the book *Algebra of Analysis*, and also wrote his own calculus book: *Calculus. A Modern Approach*. One of the reasons for writing the calculus book was the fact that Menger felt that the traditional way of presenting the subject was deeply flawed. Menger sent a copy of the book to Einstein, who liked it and applauded the attempt to clarify notation. However, he advised against too much "housecleaning". Even though viewed favorably by some, the book was seen as too radical a reform of the subject by most people, and the project failed. This was a great disappointment to Menger.

One of the most striking (and often overlooked) pedagogical innovations in the book is the development of a complete "miniature calculus" which discusses all of the basic features, including the *Fundamental Theorem*, without introducing limits.

Menger's calculus book was republished in 2007 as a Dover edition: [Dover] [Amazon]

### Analysis of the Idea of Variables

### Multiderivatives

### Analytic Functions

### Foundations of Mathematics and Logical Tolerance

In 1951 Menger was the first to introduce the idea of fuzzy sets (which he called *hazy sets*). The fundamental idea behind a hazy set is to replace the element-set relation by the probability of an element belonging to a set. This concept was later "rediscovered" [BKZ], and then attracted a lot of attention.

### Economics

The prominent economist Oscar Morgenstern claims (see [Kas]) that Menger's 1934 paper *Das Unsicherheitsmoment in der Wertlehre* played a primary role in persuading John von Neumann to undertake a formal treatment of utility [KT]. The collection [Sel] contains an entire chapter on economics. In particular, various aspects of the *Petersburg Game* are studied by Menger. This game goes as follows (see [Sel, p.260]):

There are two players, *A* and *B*. The game begins with *B* placing a certain bet with *A*. Then a coin is flipped. If it shows heads, then *B* receives $1 from *A*, and the game is over. If it shows tails, it is flipped again. If the second toss turns up heads, then *B* receives $2 from *A*, and the game ends. The coin is flipped until head occurs for the first time. If this happens at the *n*th toss (i.e., if the first *n*-1 tosses all come up tails, but the *n*th toss is heads), then *B* receives $2^{n-1} from *A* and the game ends. Depending on whether *n*=1,2,3,4,..., *B* wins $2^{0} = $1, $2^{1} = $2, $2^{2} = $4, $2^{3} = $8, ... What bet should *B* be willing to make?

### Ethics

During the politically tension filled 1930s Menger found it difficult to concentrate on mathematics. He attempted to develop a formal concept of ethics which had the same relation to traditional ethics as formal logic had to traditional logic. For more on Menger's excursions into ethics see [Leo].

In 1934 Menger wrote a little book *Moral, Wille und Weltgestaltung* (Morality, Decision, and Social Organization) on the application of simple mathematical notions to ethical problems which he translated into English in 1974 [Mor]. Menger's last paper [Soc] was an extension of this work.

Karl Popper said about this book:

*This is one of a few books in which the author attempts to depart from the stupid talk in ethics.*

Oscar Morgenstern later used Menger's book as a starting point for his work on game theory together with John von Neumann (see the section on economics).

## Students

- Franz Alt (Ph.D. 1932, University of Vienna,
*Metrische Definition der Krümmung einer Kurve*), - Kurt Gödel (took some courses from Menger),
- Joseph Bernard Harkin (Ph.D. 1968, IIT,
*Uniform Stieltjes Integrals*, co-advised by Abe Sklar), - Hans Hornich (Ph.D. 1929, University of Vienna,
*Über einen zweigradigen Zusammenhang*, co-advised by Wilhelm Wirtinger), - Witold Hurewicz (took some courses from Menger, Ph.D. 1926, University of Vienna,
*Über eine Verallgemeinerung des Borelschen Theorems*), - Joseph Landin (Ph.D. 1946, Notre Dame,
*Axiomatic Theory of a Singular Non-Euclidean Plane*), - Michel A. McKiernan (Ph.D. 1956, IIT,
*The Functional Differential Equation dF(^(:-1:-F))*), - Georg Nöbeling (Ph.D. University of Vienna, thesis published 1931),
- Berthold Schweizer (Ph.D. 1956, IIT,
*Jacobi Series and the Numerical Solution of Eigenvalue Problems*), - Richard Sielaff (Ph.D. 1959, IIT,
*On Compositive Functions of Matrices*), - Helen Skala (Ph.D. 1969, IIT,
*Projective Structures*), - Mary van Straten (Ph.D. 1947, Notre Dame,
*The Topology of the Configurations of Desargues and Pappus*). - Abraham Wald (Ph.D. 1930, University of Vienna).
- Herbert Ian Whitlock (Ph.D. 1967, IIT,
*The Algebra of Multiplace Functions under Composition*),

Also see Karl Menger's page at The Mathematics Genealogy Project.

## Reference

Listed below are the references cited above. Some of this material was used in the compilation of the information presented here.

A complete bibliography of Menger's work is here.

[100] George Temple, *100 Years of Mathematics*, Springer, 1981.

[BKZ] R. Bellman, R. Kalabe and L. Zadeh, *Abstraction and pattern design*, J. Math. Anal. Appl. (13) 1966, 1-7.

[Erg] *Karl Menger: Ergebnisse eines Mathematischen Kolloquiums*, Egbert Dierker and Karl Sigmund, *Exact Thought in a Demented Time: Karl Menger and his Viennese Mathematical Colloquium*, The Mathematical Intelligencer Vol.22 (1), 2000.

[Göd] Kurt Gödel, *Über Vollständigkeit und Widerspruchsfreiheit*, [Erg] 3 (1931/32).

[Gru] Carl Menger, *Grundsätze der Volkswirtschaftslehre*, 1871.

[Hop] Arthur J. Hope, *The Story of Notre Dame: Notre Dame - One Hundred Years*, Notre Dame University, 1999.

[Kas] Seymour Kass, *Karl Menger*, Notices of the Amer. Math. Soc. (43) 1996, 558-561.

[KT] H.W. Kuhn and A.W. Tucker, *John von Neumann's work on the theory of games*, Bull. Amer. Math. Soc. (64) 1958.

[Leo] R.J. Leonard, *Ethics and the Excluded Middle: Karl Menger and Social Science in Interwar Vienna*, Isis (89), 1998, 1-26.

[Man] Benoit B. Mandelbrot, The Fractal Geometry of Nature, W.H.Freeman, 1982.

[Pro] Berthold Schweizer and Abe Sklar, *Probabilistic Metric Spaces*, North Holland, 1983.

[Rem] *Karl Menger: Reminiscences of the Vienna Circle and the Mathematical Colloquium*, Louise Golland, Brian McGuinness and Abe Sklar (eds.), Vienna Circle Collection Vol.20, Kluwer, 1994.

[Schoe] Iso J. Schoenberg, *On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space*, Annals of Math. (38), 1937, 787-793.

[Sig] Karl Sigmund, *Musil, Perutz, Broch: Wiener Literaten und ihre Neigung zur Mathematik*, Neue Zürcher Zeitung (international edition), 8/9 March 1997, p. 49.

[Wien2001] Karl Sigmund, *"Kühler Abschied von Europa" - Wien 1938 und der Exodus der Mathematik*, Ausstellungskatalog, Sept.17-Oct.20, 2001, University of Vienna.