Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

, the son of the economist

Carl Menger Carl Menger von Wolfensgrün (; ; 28 February 1840 – 26 February 1921) was an Austrian economist and the founder of the Austrian School of economics. Menger contributed to the development of the theories of marginalism and marginal utility ...

. In mathematics, Menger studied the theory of

algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...

and the

dimension theory In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...

of low- regularity ("rough") curves and regions; in

graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, he is credited with

Menger's theorem In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of Vertex (graph theory), vertices. Pro ...

. Outside of mathematics, Menger has substantial contributions to

game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...

and social sciences.

Biography

Karl Menger was a student of Hans Hahn and received his PhD from the

University of Vienna The University of Vienna (german: Universität Wien) is a public research university located in Vienna, Austria. It was founded by Duke Rudolph IV in 1365 and is the oldest university in the German-speaking world. With its long and rich histor ...

in 1924.

L. E. J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...

invited Menger in 1925 to teach at the

University of Amsterdam The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university located in Amsterdam, Netherlands. The UvA is one of two large, publicly funded research universities in the city, the other being ...

. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

and the

Rice Institute The International Rice Research Institute (IRRI) is an international agricultural research and training organization with its headquarters in Los Baños, Laguna, in the Philippines, and offices in seventeen countries. IRRI is known for its work ...

. From 1937 to 1946 he was a professor at the

University of Notre Dame The University of Notre Dame du Lac, known simply as Notre Dame ( ) or ND, is a private Catholic research university in Notre Dame, Indiana, outside the city of South Bend. French priest Edward Sorin founded the school in 1842. The main campu ...

. From 1946 to 1971, he was a professor at

Illinois Institute of Technology Illinois Institute of Technology (IIT) is a private research university in Chicago, Illinois. Tracing its history to 1890, the present name was adopted upon the merger of the Armour Institute and Lewis Institute in 1940. The university has prog ...

(IIT) in

Chicago (''City in a Garden''); I Will , image_map = , map_caption = Interactive Map of Chicago , coordinates = , coordinates_footnotes = , subdivision_type = Country , subdivision_name ...

. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree.

Contributions to mathematics

His most famous popular contribution was the

Menger sponge In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Si ...

(mistakenly known as Sierpinski's sponge), a three-dimensional version of the

Sierpiński carpet The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is a generalization of the Cantor set to two dimensions; another is Cantor dust. The technique of subdividing a shape into smaller copies of ...

. It is also related to the

Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...

. With

Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific United Kingdom of Great Britain and Ireland, British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, C ...

, Menger is considered one of the founders of

distance geometry Distance geometry is the branch of mathematics concerned with characterizing and studying sets of points based ''only'' on given values of the distances between pairs of points. More abstractly, it is the study of semimetric spaces and the isom ...

; especially by having formalized definitions of the notions of ''angle'' and of ''curvature'' in terms of directly measurable

physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...

, namely ratios of ''distance'' values. The characteristic mathematical expressions appearing in those definitions are

Cayley–Menger determinant In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its ...

s. He was an active participant of the

Vienna Circle The Vienna Circle (german: Wiener Kreis) of Logical Empiricism was a group of elite philosophers and scientists drawn from the natural and social sciences, logic and mathematics who met regularly from 1924 to 1936 at the University of Vienna, cha ...

, which had discussions in the 1920s on social science and philosophy. During that time, he published an influential result on the

St. Petersburg paradox The St. Petersburg paradox or St. Petersburg lottery is a paradox involving the game of flipping a coin where the expected payoff of the theoretical lottery game approaches infinity but nevertheless seems to be worth only a very small amount to t ...

with applications to the

utility theory As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...

economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...

; this result has since been criticised as fundamentally misleading. Later he contributed to the development of

with

Oskar Morgenstern Oskar Morgenstern (January 24, 1902 – July 26, 1977) was an Austrian-American economist. In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory as applied to the social sciences and strategic decis ...

Legacy

Menger's longest and last academic post was at the Illinois Institute of Technology, which hosts an annual IIT Karl Menger Lecture and offers the IIT Karl Menger Student Award to an exceptional student for scholarship each year.

Notes

External links

* * {{DEFAULTSORT:Menger, Karl 1902 births 1985 deaths 20th-century American mathematicians Austrian mathematicians Vienna Circle Harvard University staff Rice University staff Duke University faculty University of Notre Dame faculty Illinois Institute of Technology faculty Austrian emigrants to the United States University of Vienna alumni Scientists from Vienna

Biography

Contributions to mathematics

Legacy

See also

Notes

Further reading

External links