John von Neumann: The Pioneer of ‘Game Theory’

He was an excellent representation of twentieth century mathematics and his influence was unbelievably wide within and outside mathematics.

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John von Neumann (Source: Wikimedia)

John von Neumann was one of the most influential figures of 20th century who made far-reaching contributions to the areas of mathematics, mathematical physics and economy. He was a Hungarian-American mathematician and a genius who pioneered the famous ‘Game Theory’, modern computer designing, equilibrium model of economy, and much more. He was also a key figure in the development of atomic bomb during the World War II.

EARLY LIFE

John von Neumann was born on December 28, 1903 in a wealthy family of Budapest, Hungary. He was the eldest of three brothers. His father, Neumann Miksa, was a lawyer for a bank. In his childhood, Neumann learnt German, English, French and Italian because his father believed that grasp on other languages was very important. At the age of six he was capable of speaking Ancient Greek. He had a great interest in History and Mathematics since his early years.

EDUCATION

From 1911 to 1921 he attended the Lutheran Fasori Evangélikus Gimnáziuma famous elite high school in Budapest. This school had produced a number of intellectual minds including Eugene Wigner, John Harsanyi and Edward Teller. Eugene Wigner, a Nobel Prize winner in Physics, was a senior of Neumann and when he was asked why Hungary of his generation had produced so many geniuses, Wigner replied that Neumann was the only genius.

By the age of 19, Neumann had published two research papers in mathematics

Neumann’s exceptional talent in Mathematics was discovered by his mathematics teacher, László Rátz. On Ratz’s request, a professional mathematician was hired for private tutoring of John Neumann. He was then tutored by a renowned analyst Gábor Szegő from 1915-1916. He was also taught by József Kürschák and Lipót Fejér. At Gimnázium, Neumann received education on the highest professional level by the renowned intellectuals of that time. By the age of 19, Neumann had published two research papers in mathematics. His first research paper was a joint paper published with well-known mathematician Michael Fakete while his second paper gave the modern definition of ordinal numbers which supplant George Cantor’s definition.

After graduating from high school, Neumann’s father decided to enroll him in chemical engineering program at Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland. At that time chemical engineering was a popular field and financially a more tempting endeavor than mathematics. That consideration weighed heavily in the eyes of his father. For this purpose, he first took a two year non-degree course in chemistry at University of Berlin and after that he sat for the entrance exam at the acclaimed ETH Zürich which he passed in 1923. Along with his education at ETH, Neumann also studied mathematics in Budapest and in 1926, soon after his graduation from ETH, he received his PhD degree in mathematics from University of Budapest with minors in experimental physics and chemistry. His doctoral thesis was on axiomatization of set theory.

CAREER

In 1926, he went to University of Gottingen on Rockefeller fellowship and there he worked as David Hilbert’s assistant. Hilbert was a pioneer in functional analysis. Neumann’s initial work was on set theory and logic and by the end of 1927, he had published twelve major papers in mathematics. But at Gottingen, Neumann became more interested in quantum mechanics and functional analysis. Hilbert himself was strongly attracted to this field. In 1927, one of von Neumann’s groundbreaking papers appeared on surface discussing the mathematical foundation of quantum mechanics as a joint publication with Hilbert and L. Nordheim.

His two research papers on the widely known ‘Game Theory’ appeared in 1928. These papers contained the formulation and proof of the minimax theorem. The theorem states that in zero-sum games with perfect information i.e. in which players know at each time all moves that have taken place so far, there exists a pair of strategies for both players that allows each player to minimize their losses. In 1944 Neumann published the results in which he extended the minimax theorem to include games with more than two players and games with imperfect information.

His two research papers on the widely known ‘Game Theory’ appeared in 1928

In 1929, Neumann worked as non-tenured faculty (Privatdozent) at University of Hamburg for a short duration of time. In the same year he was appointed as visiting lecturer at Princeton University before being offered a permanent professorship at the University in 1930, which von Neumann rejected. Finally, in 1933, he accepted an offer to become one of the six permanent professors at Institute of Advanced Studies (IAS) established in Princeton; the other five professors included Albert Einstein, James Alexander, Marston Morse, Hermann Weyl and Veblen. He served as a professor at Princeton until his last days.

His initial work on algebras of operators was published in his paper in 1929. He also introduced study of rings of operator through von Neumann algebra. From 1936 to 1940, he published six papers on the study of factors classification of von Neumann algebra with the collaboration of F.J. Murray. These papers are ranked among the masterpieces of analysis in twentieth century and at least four field medals have been awarded for new results on or in connection with von Neumann algebras: to A. Connes, M. Kontsevich, V. F. R. Jones and E. Witten. Furthermore, one of the major scientific contributions of Neumann was his work on Ergodic Theory. It is a branch of Mathematics that studies dynamical systems with an invariant measure. During the year 1932, Neumann published various papers on ergodic theory and made fundamental contributions to this field.

During early thirties due to oppressive political atmosphere in Europe, he decided to settle in U.S

In 1937, Neumann acquired nationality of United States. In 1938, he was awarded the prestigious Bôcher Prize for his work on almost periodic function on groups. In those years his interest shifted to the study of supersonic and turbulent flow of fluids. Because of his interest in atomic side, he was hired as a consultant at Ballistic Research Laboratory of the Army Ordinance Department in 1937.

Von Neumann contributed in a large number of fields and touched upon theory of functions of real variable, topology, measure theory, lattice theory, continuous groups, almost periodic functions, representation of groups, quantum logic etc. In 1937, his work on a model of general economic equilibrium was published. Based on his work on economic model and game theory, in 1944 Neumann published a book, Theory of Games and Economic Behavior with Oskar Morgenstern as the co-author.

NEUMANN’S CONTRIBUTIONS DURING WORLD-WAR II

During World War II, von Neumann became increasingly involved in US military-related research and governmental consulting activities. He was one of the mathematicians and scientists recruited by US Government to work on defence and military projects. He also made a major contribution in the Manhattan Project.  To acknowledge his services during the war to the United States, he was awarded with Medal of Merit and Distinguished Service Award in 1946.

VON NEUMANN COMPUTER

After the war, Neumann received offers from different universities including MIT and University of California Los Angeles (UCLA). During that time, computer design was the main focus of Neumann’s academic research.  MIT’s offer included a fund for electronic computer at MIT but Neumann wanted to make IAS the home of electronic computer development project. After securing IAS’s support for the computer project Neumann declined MIT’s offer.

It is known that in August 1944, Neumann got a chance to meet Herman Goldstein who was working in Philadelphia on the development of Electronic Numerical Integrator and Computer (ENIAC).  Soon Neumann got into the work and presented new ideas. His major contributions were elaboration of the principles of programmable computer (von Neumann Computers) and the implementation of these principles in the construction of IAS machine to be built at Princeton University. He also initiated the research in cellular automata and probabilistic automata.

POST-WAR LIFE

Neumann’s post-war consultancy activities expanded tremendously both in public and private sector. In 1954, he was selected as a commissioner at US Atomic Energy Commission. After his job there he didn’t intend to return to IAS and in 1956 he was again offered a position at MIT and UCLA. In March 1956, he finally decided to accept a position as a professor at UCLA but unfortunately, he was never able to take the position because he died of cancer on February 8, 1957 in Washington D.C. He is buried in Princeton cemetery.

He was an excellent representation of twentieth century mathematics and his influence was unbelievably wide within and outside mathematics. In his 1947 article he stated:

I think it is a relatively good approximation to truth that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of his own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science.

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