Karl Menger

{{about|the mathematician|his father, the economist|Carl Menger}}

{{Infobox scientist

| name = Karl Menger

| image = Karl Menger 1970 Shimer College Wiki.jpg

| caption =

| birth_date = {{birth date|1902|1|13|mf=y}}

| birth_place = Vienna, Austria-Hungary

| death_date = {{death date and age|mf=yes|1985|10|5|1902|1|13}}

| death_place = Highland Park, Illinois, U.S.

| ethnicity =

| fields = Mathematics

| workplaces = Illinois Institute of Technology
University of Notre Dame
University of Vienna

| alma_mater = University of Vienna (PhD, 1924)

| doctoral_advisor = Hans Hahn

| academic_advisors =

| thesis_title = Über die Dimensionalität von Punktmengen

| thesis_url =

| thesis_year = 1924

| doctoral_students = Abraham Wald
Witold Hurewicz
Georg Nöbeling

| notable_students =

| known_for = Menger characterization theorem
Menger curvature
Menger space
Menger sponge
Menger's theorem
Menger–Nöbeling theorem
Cayley–Menger determinant

| influences =

| influenced =

| awards =

| signature =

| footnotes =

}}

Karl Menger ({{IPA|de|ˈmɛŋɐ|lang}}; January 13, 1902 – October 5, 1985) was an Austrian-born American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low-regularity ("rough") curves and regions; in graph theory, he is credited with Menger's theorem. Outside of mathematics, Menger has substantial contributions to game theory and social sciences.

Biography

Karl Menger was a student of Hans Hahn 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 (IIT) in Chicago. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree.{{cite web|title=About Karl Menger |url=https://www.iit.edu/applied-math/about/remembering-karl-menger/about-karl-menger|access-date=2025-03-22|publisher=Illinois Institute of Technology}}

Contributions to mathematics

Image:Menger sponge (IFS).jpg

His most famous popular contribution was the Menger sponge (mistakenly known as Sierpinski's sponge), a three-dimensional version of the Sierpiński 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 of 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 published an influential result{{Cite journal|last=Menger|first=Karl|date=1934-08-01|title=Das Unsicherheitsmoment in der Wertlehre|journal=Zeitschrift für Nationalökonomie|language=de|volume=5|issue=4|pages=459–485|doi=10.1007/BF01311578|s2cid=151290589|issn=1617-7134}} on the St. Petersburg paradox with applications to the utility theory in economics; this result has since been criticised as fundamentally misleading.Peters, O. and Gell-Mann, M., 2016. Evaluating gambles using dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(2), p.023103 Later he contributed to the development of game theory with Oskar Morgenstern.

Menger was a founding member of the Econometric Society.

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.{{cite web|title=Remembering Karl Menger|url=http://www.iit.edu/csl/events/archive/remembering_menger.shtml|access-date=2009-03-26|publisher=Illinois Institute of Technology|archive-url=https://web.archive.org/web/20090402052624/http://www.iit.edu/csl/events/archive/remembering_menger.shtml|archive-date=2009-04-02|url-status=dead}}

Menger's memoirs inspired his granddaughter Kirsten Menger-Anderson to write the 2025 novel The Expert of Subtle Revisions, which featured a fictionalized Vienna Circle.{{cite book |last1=Menger-Anderson |first1=Kirsten |title=The Expert of Subtle Revisions |date=2025 |publisher=Crown |location=New York |isbn=978-0593798300 |page=241}}