Karl Menger
Encyclopedia
Karl Menger was a mathematician
. He was the son of the famous economist Carl Menger
. He is credited with Menger's theorem
. He worked on mathematics of algebras, algebra of geometries, curve and dimension theory, etc. Moreover, he contributed to game theory and social sciences.
and received his PhD from the University of Vienna
in 1924. L. E. J. Brouwer
invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at Harvard University and The Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame
. From 1946 to 1971, he was a professor at Illinois Institute of Technology
in Chicago
. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree.
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His most famous popular contribution was the Menger sponge
(mistakenly known as Sierpinski's sponge), a three-dimensional version of Sierpinski's carpet
. It is also related to the Cantor set
.
With Arthur Cayley
, Menger is considered one of the founders of distance geometry
; especially by having formalized definitions to the notions of angle and of curvature in terms of directly measurable physical quantities, namely ratios of distance values.
The characteristic mathematical expressions appearing in those definitions are Cayley–Menger determinants.
He was an active participant of the Vienna Circle
which had discussions in the 1920s on social science and philosophy. During that time, he proved an important result on the St. Petersburg paradox
with interesting applications to the utility theory in economics
. Later he contributed to the development of game theory
with Oskar Morgenstern
.
, which hosts an annual IIT Karl Menger Lecture and offers the IIT Karl Menger Student Award to a student for exceptional scholarship each year.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He was the son of the famous economist Carl Menger
Carl Menger
Carl Menger was the founder of the Austrian School of economics, famous for contributing to the development of the theory of marginal utility, which contested the cost-of-production theories of value, developed by the classical economists such as Adam Smith and David Ricardo.- Biography :Menger...
. He is credited with Menger's theorem
Menger's theorem
In the mathematical discipline of graph theory and related areas, Menger's theorem is a basic result about connectivity in finite undirected graphs. It was proved for edge-connectivity and vertex-connectivity by Karl Menger in 1927...
. He worked on mathematics of algebras, algebra of geometries, curve and dimension theory, etc. Moreover, he contributed to game theory and social sciences.
Biography
Karl Menger was a student of Hans HahnHans Hahn
Hans Hahn was an Austrian mathematician who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory.-Biography:...
and received his PhD from the University of Vienna
University of Vienna
The University of Vienna is a public university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world...
in 1924. L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at Harvard University and The Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame
University of Notre Dame
The University of Notre Dame du Lac is a Catholic research university located in Notre Dame, an unincorporated community north of the city of South Bend, in St. Joseph County, Indiana, United States...
. From 1946 to 1971, he was a professor at Illinois Institute of Technology
Illinois Institute of Technology
Illinois Institute of Technology, commonly called Illinois Tech or IIT, is a private Ph.D.-granting university located in Chicago, Illinois, with programs in engineering, science, psychology, architecture, business, communications, industrial technology, information technology, design, and law...
in Chicago
Chicago
Chicago is the largest city in the US state of Illinois. With nearly 2.7 million residents, it is the most populous city in the Midwestern United States and the third most populous in the US, after New York City and Los Angeles...
. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree.
Contributions to mathematics
Menger sponge
In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge...
(mistakenly known as Sierpinski's sponge), a three-dimensional version of Sierpinski's carpet
Sierpinski carpet
The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions . Sierpiński demonstrated that this fractal is a universal curve, in that any possible one-dimensional graph, projected onto the two-dimensional...
. It is also related to the Cantor set
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
.
With Arthur Cayley
Arthur Cayley
Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....
, Menger is considered one of the founders of distance geometry
Distance geometry
Distance geometry is the characterization and study of sets of points based only on given values of the distances between member pairs. Therefore distance geometry has immediate relevance where distance values are determined or considered, such as in surveying, cartography and...
; especially by having formalized definitions to the notions of angle and of curvature in terms of directly measurable physical quantities, namely ratios of distance values.
The characteristic mathematical expressions appearing in those definitions are Cayley–Menger determinants.
He was an active participant of the Vienna Circle
Vienna Circle
The Vienna Circle was an association of philosophers gathered around the University of Vienna in 1922, chaired by Moritz Schlick, also known as the Ernst Mach Society in honour of Ernst Mach...
which had discussions in the 1920s on social science and philosophy. During that time, he proved an important result on the St. Petersburg paradox
St. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...
with interesting applications to the utility theory in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
. Later he contributed to the development of game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
with Oskar Morgenstern
Oskar Morgenstern
Oskar Morgenstern was a German-born Austrian-School economist. He, along with John von Neumann, helped found the mathematical field of game theory ....
.
Legacy
Menger's longest and last academic post was at the Illinois Institute of TechnologyIllinois Institute of Technology
Illinois Institute of Technology, commonly called Illinois Tech or IIT, is a private Ph.D.-granting university located in Chicago, Illinois, with programs in engineering, science, psychology, architecture, business, communications, industrial technology, information technology, design, and law...
, which hosts an annual IIT Karl Menger Lecture and offers the IIT Karl Menger Student Award to a student for exceptional scholarship each year.
Further reading
- Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in Grattan-Guinness, I.Ivor Grattan-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
, ed., Landmark Writings in Western Mathematics. Elsevier: 844-55.