# Neumann, John von (1903–1957)

# NEUMANN, JOHN VON

*(1903–1957)*

American mathematician, physicist, and economist John von Neumann was born in Budapest, Hungary. He showed an early precocity in mathematics and was privately tutored in the subject; his first paper was written before he was eighteen. He studied at the universities of Berlin, Zürich, and Budapest and received his doctorate in mathematics from Budapest in 1926, almost simultaneously with an undergraduate degree in chemistry from Zürich. After serving as *Privatdozent* at Berlin, he accepted a visiting professorship at Princeton in 1930. Following three years there, he became a professor of mathematics at the Institute for Advanced Study, a position that he held for the rest of his life. In 1955 he was appointed one of the commissioners of the U.S. Atomic Energy Commission, on which he served brilliantly until his death.

Von Neumann made fundamental contributions to mathematics, physics, and economics. Furthermore, these contributions were not disjointed and separate but arise from a common point of view regarding these fields.

Mathematics was always closest to his heart, and it is the field to which he contributed the most. His earliest significant work was in mathematical logic and set theory, topics that occupied him from 1925 to 1929. His accomplishments were of two sorts; they concerned the axiomatics of set theory and David Hilbert's proof theory. In both of these subjects he obtained results of extra-ordinary importance. He became the first to set up an axiomatic system of set theory that satisfied the two conditions of allowing the development of the theory of the whole series of cardinal numbers and employing axioms that are finite in number and are expressible in the lower calculus of functions. This work contained a full classification of the significance of the axioms with regard to the elimination of the paradoxes. With regard to Hilbert's proof theory, von Neumann clarified the concept of a formal system considerably.

His work on the theory of Hilbert space and operators on that space was probably stimulated by what he had done on rigorous foundations for quantum theory. Essentially, von Neumann demonstrated that the ideas originally introduced by Hilbert are capable of constituting an adequate basis for the physical consideration of quantum theory and that there is no need for the introduction of new mathematical schemes for these physical theories. Von Neumann's papers on these subjects constitute about one-third of his printed work and have stimulated extensive research by other mathematicians.

Von Neumann was one of the founders of the theory of games; since the publication of von Neumann's first paper in 1928 it has become an important combinational theory, applied and developed with continuing vigor. Von Neumann's first paper contains rigorous definitions of the concepts of pure strategy (a complete plan, formulated prior to the contest, that makes all necessary decisions in advance) and of mixed strategy (the use of a chance device to pick the strategy for each contest). The central theorem in this theory, the minimax theorem, was not only enunciated and proved by von Neumann but in his hands became a powerful tool for obtaining new methods for combinatorial problems.

A decade after this fundamental paper was written, von Neumann began a collaboration with Oskar Morgenstern that led to *The Theory of Games and Economic Behavior,* a book that has decisively affected the entire subject of operations research.

Von Neumann's principal interest in his later years was in the possibilities and theory of the computing machine. He not only conceived the concept of the so-called stored program computer in 1944 but he made three other signal contributions. First, he recognized the importance of computing machines for mathematics, physics, economics, and industrial and military problems; second, he translated this insight into active sponsorship of a machine (it was called Johniac by his collaborators) that served as a model for several important computers; third, he was one of the authors of a series of papers that provided a theoretical basis for the logical organization and functioning of computers. These papers set out the complete notion of the flow diagram and contained the genesis of many programming techniques.

** See also ** Computing Machines; Decision Theory; Game Theory; Hilbert, David; Mathematics, Foundations of; Proof Theory; Quantum Mechanics; Set Theory.

## Bibliography

Much of von Neumann's work, with the exception of certain still-classified reports or papers that are essentially duplicates of other works, is published in *John von Neumann, Collected Works,* 6 vols. (New York: Macmillan, 1961–1963). Other important books by von Neumann are *Continuous Geometry,* 2 vols. (Princeton, NJ: Institute for Advance Study, 1936–1937); *Mathematical Foundations of Quantum Mechanics* (Princeton, NJ: Princeton University Press, 1955); *The Computer and the Brain* (New Haven, CT: Yale University Press, 1958). For a survey of von Neumann's life and work, see the memorial volume *John von Neumann, 1903–1957, Bulletin of the American Mathematical Society* 64 (3:2) (May 1958).

*Herman H. Goldstine (1967)*

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