Remembering Karl Menger
INTRODUCTION
On April 9, 2007, the Department of Applied Mathematics hosted a half-day event to honor Professor Karl Menger, an outstanding mathematician of the 20th century and former faculty member at IIT. Professor Karl Sigmund of the University of Vienna delivered the Inaugural IIT Karl Menger Lecture on "Menger, Games, and Morals" to a crowded Wishnick Hall Auditorium. The lecture was preceded by a reception and followed by a ceremony awarding the IIT Karl Menger Student Award for exceptional scholarship to Mike McCourt (Applied Mathematics and Physics, 5th year undergraduate).
Earlier, guests attended a lunch, campus tour and department update - featuring reminiscences by guests of Applied Mathematics Professor Jerry Frank, who is retiring this year after 30 years of service to IIT. The day ended with a dinner at the University Club at MTCC. Throughout the day, about 60 alumni from the 1940s to the 2000s, faculty, faculty emeriti, students and other friends joined in for the various activities. They included Professor Franz Alt, age 96, who was a student of Menger in Vienna in the 1930s. The event was made possible with the generous support of the Menger family, IIT Department of Applied Mathematics, IIT Department of Institutional Advancement, and IIT alumni. Major news from the day's events included a challenge from Professor Frank to match donations to the Menger Fund up to $20,000 for the next year to help ensure the continuation of the IIT Karl Menger Lecture and Awards.
WELCOME
At the lecture, IIT President Lew Collens welcomed the audience by stating that this event is "particularly important in the history of IIT, because recognizing members of the IIT community who have contributed to the intellectual life of the world is something that should be done more often." Collens went on to describe Menger as "a giant in the history of IIT, and of mathematics" and mentioned his 25 year career at IIT as well as his deep commitment to teaching. Menger, a professor at IIT from 1946 to 1971, taught all levels of mathematics and was an adviser to many master's and doctoral-degree students. Since his retirement, many people study or work in the current Applied Mathematics Department in part because of Menger's legacy.
Menger worked on algebras, curve and dimension theory, geometries, game theory and social sciences. He helped to create distance geometry. He is probably best known for being the first to give a rigorous definition of dimension, and second best known for creating the Menger Sponge. He was active in the Vienna Circle in the 1920s and left Europe for the U.S. with the rise of fascism in the 1930s. He taught at the University of Notre Dame for about 10 years before coming to IIT in 1946.
Abe Sklar, a colleague of Menger's, and now Professor Emeritus of mathematics at IIT, introduced Professor Karl Sigmund - an internationally known mathematician and pioneer of evolutionary game dynamics. Recently, Sigmund has been interested in the history of mathematics, especially the Vienna Circle, and has co-edited the works of Karl Menger.
FEATURED SPEAKER KARL SIGMUND
In his lecture, Sigmund talked about Menger as a young man entertaining the notion of being a playwright, and finding his way into Arthur Schnitzler's diary (a well-known dramatist at the time) in which Menger is referred to as "highly gifted, may be a genius, eccentric, and (with laughter from the audience) a megalomaniac." Menger eventually entered the University of Vienna in the fall of 1920 where his interests focused on the field of mathematics.
It is in this field that Menger revealed his genius specifically by investigating approaches to social ethics through mathematics. This led to the publication of the booklet "Moral, Wille und Weltgestaltung" (Morality, Decision, and Social Organization) on the application of simple mathematical notions to ethical problems. Oscar Morgenstern later used Menger's book as a starting point for his work on game theory together with John von Neumann. It is in game theory and its applications in biology (such as the evolution of cooperation) and economics (experimental economics and even the e-commerce interactions on ebay.com) where Menger's theories find practical and modern uses.
GAME THEORY
Sigmund demonstrated some examples of game theory through the ultimatum game, an experimental situation designed to investigate emergent social behavior. Emergent behaviors can be shown numerically through the tendency of humans to reciprocate in certain social situations depending on available information such as reputation. Sigmund described direct reciprocation by using the common phrase, "I'll scratch your back if you scratch mine." Sigmund then described indirect reciprocation by modifying the phrase, "I'll scratch your back if you scratch somebody." These abstract descriptions demonstrate real-world events such as web-based social news aggregators like digg.com which rely on the participation of volunteer users to participate in the creation and promotion of content on a popular website.
It is often questioned why people would want to participate in the effort of creating content on websites like digg.com, but game theory can go a long way to show why these users may want to participate in the community. As the community verifies the reputation of certain users, that reputation lends validity to those users creating cooperation around these users. If users never help the community with their participation, no cooperation will arise around these users, effectively removing them from the community. It is in this interaction that reciprocation leads to the creation of content relevant to the cooperating community, benefiting the community as well as the users who contribute to the community.
CONCLUSION
Sigmund concluded his lecture by reminding the audience that anonymous online indirect reciprocal interactions, such as the commenting systems on e-commerce sites like ebay.com can be investigated using game theory. Sigmund then reminded the audience that game theory is a direct descendant of Karl Menger's booklet, "Moral, Wille und Weltgestaltung" which was published in 1934.
IIT KARL MENGER STUDENT AWARD
Prof Sigmund answered questions for a few minutes and then presented the IIT Karl Menger Student Award for exceptional scholarship to Mike McCourt (BCPS, 5th year undergraduate). McCourt was described as an exceptional scholar who makes classrooms he attends come alive. As an undergraduate, McCourt has research experience as well as a publication to his name. McCourt also has started an IPRO and has received an NSF graduate fellowship. Two other students in the Applied Mathematics Department have been awarded summer 2007 research scholarships, Zhao Zhang (graduate student in Applied Mathematics) who will be studying financial risk with Prof. Tom Bielecki, and Jeffrey Stanford (undergraduate student in CS and Applied Mathematics) who will be working with Prof. Rob Ellis. An extensive collection Menger's life's work can be found at the link below as well as community submitted memories of Karl Menger on the Applied Mathematics Department website.
