Von Neumann, John
Von Neumann, John
John von Neumann, mathematician, was born in Budapest in 1903 and died in Washington, D.C., in 1957. He was the first of the great creative mathematicians to devote major effort to the social sciences. After studying in Budapest and Zurich, von Neumann became a Privatdozent in Berlin; in 1931 he received an appointment at Princeton University, and in 1933 he joined the Institute for Advanced Study in Princeton, where he remained for the rest of his life. In 1955, on leave from the institute, he was made a member of the U.S. Atomic Energy Commission. For his scientific work and public services he received several honorary doctorates, academy memberships, prizes, medals, and other distinctions.
Von Neumann’s genius ranged over many areas of pure mathematics as well as applied fields. He made important contributions to the axiomatics of set theory, mathematical logic, Hilbert space theory, operator theory, group theory, and measure theory. He proved the ergodic theorem, established a continuous geometry without points, introduced almost-periodic functions on groups, and at the end of his life was much concerned with nonlinear differential equations. In addition, he had a consuming interest in numerical applications, ranging from the development of new computing techniques to the study of the mathematical validity of large-scale numerical operations as they are carried out by modern electronic computers.
Von Neumann’s work in physics was manifold. In his Mathematical Foundations of Quantum Mechanics (1932), a study of enduring significance, he laid a firm basis for this new field by the first comprehensive use and development of Hilbert space. In his study “The Logic of Quantum Mechanics” (see von Neumann & Birkhoff 1936) he revealed the inner logical structure of quantum mechanics and suggested that each science has its own specific logic. Von Neumann’s influence was felt in hydrodynamics, mechanics of continua, astrophysics, and meteorology. In statistics he made contributions to trend analysis, and he developed the Monte Carlo method. He established the logical basis for electronic computer design and built the first of the truly modern flexible machines. He was also concerned with the development of a logical theory of automata and proved the possibility of a self-reproducing machine. This work (1966) is closely related to his “Probabilistic Logics” (1956).
Von Neumann’s work had great importance for the social sciences. For example, he opened up entirely new avenues in mathematical economics. In 1928 he published a fundamental paper on the theory of games of strategy in which the now famous minimax theorem was proved for the first time. This theorem establishes that, in a two-person zero-sum game with finite numbers of strategies, there always exist optimal strategies for each player. Each player is assumed to choose a strategy independently, and in ignorance, of his opponent’s choice. Selection of an optimal strategy is shown to involve the selection of proper probabilities of adopting each of the pure strategies available. [See Game theory.]
This work was developed further in Theory of Games and Economic Behavior (von Neumann & Morgenstern 1944). The theory was extended to n-persons (n?3) and to cases where the sum of winnings by all players is a constant different from zero or is variable. The Theory of Games also developed a theory of individual choice in situations of risk, which has given rise to an extensive literature on utility. Game theory, besides analyzing games proper, is taken as a model for economic and social phenomena; it applies to all situations where the participants do not control or know the probability distributions of all variables on which the outcome of their acts depends, situations that therefore cannot be described as ordinary maximum or minimum problems (even allowing for side conditions). Since the publication of the Theory of Games, hundreds of books and papers by many authors in many countries have furthered and applied the theory.
In 1937 von Neumann wrote on the general equilibrium of a uniformly expanding closed economy under conditions of constant returns to scale in production and unlimited supply of natural resources. Employing the minimax theory, he proved that the economy’s expansion factor must equal the interest factor. The linear production relations in the model include linear inequalities and take full account of alternative processes and of indirect production among industries. In these respects, the model is the forerunner of linear programming and activity analysis, both of which are related to game theory by virtue of the minimax theorem. This work, together with that of Abraham Wald, marked the beginning of a new period in mathematical economics. [See Economic equilibrium.] Von Neumann showed that the representation of an economic system requires a set of inequalities since, for example, for any good, both the amount produced and the price must necessarily be nonneg-ative. A solution of the system must satisfy the inequality constraints, and the existence of a solution is not ensured merely by the equality of the number of unknowns and the number of equations.
A fundamental element in von Neumann’s mathematical work is the close relation of his thought to the physical and social sciences. He was firmly convinced that the greatest stimulus for mathematics has always come from the mathematician’s involvement with empirically given problems; the simultaneous development of calculus and mechanics is the most striking example. He also believed that the mathematical treatment of the social sciences must be quite different from that of the physical sciences. His profound involvement with the social sciences and his very good knowledge of the natural sciences give special weight to his judgment that these two types of science have different mathematical structures. He expected the mathematical study of social phenomena to bring about the development of new mathematical techniques. He took the largely combinatorial approach of game theory as an indication that the time when this would happen might still be remote.
While von Neumann was primarily interested in the mathematical problems of the physical sciences, he nevertheless had a profound concern for the social sciences, which he considered to be in a state comparable to that of physics prior to Newton. This concern expressed itself also in his interest in history and politics, two fields in which he read widely. He had great influence on his contemporaries not only through the large amount of his published work but also through his many contacts with scientists all over the world.
Oskar Morgenstern
[See alsoGame theory; Programming.]
WORKS BY VON NEUMANN
(1928) 1959 On the Theory of Games of Strategy. Volume 4, pages 13-42 in A. W. Tucker and R. Duncan Luce (editors), Contributions to the Theory of Games. Princeton Univ. Press. ? First published in German.
(1932) 1955 Mathematical Foundations of Quantum Mechanics. Investigations in Physics, No. 2. Princeton Univ. Press. ? First published in German.
(1936) 1962 Von Neumann, John; and Birkhoff, Garrett The Logic of Quantum Mechanics. Volume 4, pages 105-125 in John von Neumann, Collected Works. Edited by A. H. Taub. New York: Pergamon.
1937 Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes. Ergebnisse eines mathematischen Kolloquiums 8:73-83.
(1944) 1964 Von Neumann, John; and Morgenstern, OskarTheory of Games and Economic Behavior. 3d ed. New York: Wiley.
1950 Functional Operators. 2 vols. Annals of Mathematical Studies, Nos. 21-22. Princeton Univ. Press.
(1956) 1963 Probabilistic Logics and the Synthesis of Reliable Organisms From Unreliable Components. Volume 5, pages 329-378 in John von Neumann, Collected Works. Edited by A. H. Taub. New York: Pergamon.
(1958) 1959 The Computer and the Brain. New Haven: Yale Univ. Press. ? Published posthumously.
1960 Continuous Geometry. Princeton Mathematical Series, No. 25. Princeton Univ. Press. ? Published posthumously.
1966 Theory of Self-reproducing Automata. Edited and completed by A. W. Burks. Urbana: Univ. of Illinois Press. ? Published posthumously.
Collected Works. Edited by A. H. Taub. 6 vols. New York: Pergamon, 1961-1963.
SUPPLEMENTARY BIBLIOGRAPHY
Behnke, Heinrich; and Hermes, Hans 1957 Johann von Neumann: Ein grosses Mathematikerleben unserer Zeit. Mathematisch-physikalische Semesterberichte 5: 186-190.
Bochner, S. 1958 John von Neumann, December 28, 1903-February 8, 1957. Volume 32, pages 438-457 in National Academy of Sciences, Washington, D.C., Biographical Memoirs. Washington: The Academy. ? Includes a ten-page bibliography.
John von Neumann, 1903-1957. 1958 American Mathematical Society, Bulletin 64, no. 3, part 2.
Kuhn, H. W.; and Tucker, A. W. 1958 John von Neumann’s Work in the Theory of Games and Mathematical Economics. American Mathematical Society, Bulletin 64, no. 3, part 2:100-122.
Morgenstern, Oskar 1958 Obituary: John von Neumann, 1903-1957. Economic Journal 68:170-174.
Ulam, S. 1958 John von Neumann, 1903-1957. American Mathematical Society, Bulletin 64, no. 3, part 2: 1-49. ? See especially the bibliography on pages 42-48. See also pages 48-49, “Abstracts of Papers Presented to the American Mathematical Society.”