Von Neumann, John

WORKS BY VON NEUMANN

SUPPLEMENTARY BIBLIOGRAPHY

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.”

von Neumann, John

Hungarian Computer Scientist and Mathematician
1903–1957

John Louis von Neumann was one of the great pioneers of computer science and mathematics during the twentieth century. Known for his concept of the stored computer program, he performed work that paved the way for the powerful and ubiquitous electronic computers of the early twenty-first century. His work on the Institute for Advanced Studies (IAS) computers built the foundation for what is now known as the "von Neumann Architecture." This architecture resulted in the development of powerful supercomputers employed by government, universities, and other institutions.

Von Neumann was born December 28, 1903, in Budapest, Hungary, and died February 8, 1957, in Washington D.C. During his youth, he was often referred to as a prodigy, having published his first technical paper at the age of eighteen. He began attending the University of Budapest in 1921, where he studied chemistry, receiving his diploma in chemical engineering in 1925.

In 1930 von Neumann was invited to Princeton University in the United States, and he was one of the original professors when the university established the Institute for Advanced Studies in 1933. He recognized the importance of computers in the field of applied mathematics and other disciplines and was involved in several strategic government research projects during World War II. Indeed, one of the cornerstones of von Neumann's philosophy was to apply computers to fields of study that interested him. His work in the fields of statistics, ballistics, meteorology, hydrodynamics, and game theory was invaluable during World War II.

He contributed his scientific expertise to the Manhattan Project, the first attempt to develop an atomic bomb for military purposes. At the time, it was feared that Nazi Germany would be the first to develop and deploy the atomic bomb and thus win the war. Von Neumann played an important role as an adviser to the U.S. government, and his talent for finding solutions to complex problems proved invaluable on the projects with which he was involved. He also played an important part as a trusted conduit between groups of scientists working on separate projects that were sequestered from one another due to wartime needs of security. Thus, he brought together the talents of scientists working at Los Alamos National Laboratory, the scientists working on the Manhattan Project, and the scientists and engineers working on the first digital computer, Electronic Numerical Integrator and Computer (ENIAC).

After World War II, von Neumann continued to work on government research projects with military applications. His work with supercomputers helped perform the calculations necessary for developing the next generation hydrogen bomb. His ongoing research also led to increasingly capable supercomputers used by the U.S. national laboratories. These proved important for both military and peacetime scientific applications. Hired as a consultant by the IBM Corporation in the 1950s, von Neumann performed duties that involved reviewing proposed and ongoing projects for the company.

Von Neumann is also considered the father of "self replicating systems," systems that could reproduce themselves in a manner not greatly dissimilar from biological life. Von Neumann's concept consisted of two central components: a universal computer and a universal constructor. The universal computer contained the software that directed the universal constructor, and was essentially the central brain of the system. Guided by the universal computer, the constructor was a machine that was fully capable of creating copies of the universal computer and of itself. Once the constructor built another copy of itself, the control software was copied from the original universal computer.

The newly created constructors would then begin to execute the control software, and the process would repeat. The system as a whole is thus self-replicating. The self-replicating concept has been extended to constructors capable of building other objects, depending on the control software employed by the universal computer. The self-replicating machine concept has been explored by scientists at the National Aeronautics and Space Administration (NASA) for building inexpensive and self replicating probes for future space exploration.

One of von Neumann's most famous quotes illustrates the brilliance and depth of his intelligence and personality: "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."

see also Babbage, Charles; Early Computers; Early Pioneers; Government Funding, Research; Hollerith, Herman; Turing, Alan M.

Joseph J. Lazzaro

Bibliography

Goldstine, Herman H. The Computer from Pascal to von Neumann. Princeton, NJ: Princeton University Press, 1972.

von Neumann, Nicholas A. John von Neumann: As Seen by His Brother. Meadowbrook, PA: Private Printing, 1988.

John von Neumann

1903-1957

Hungarian-born American Mathematician

One of the most influential figures of the twentieth century, John von Neumann was also one of the most creative minds of any era. Much of what modern computer users take for granted—for instance, the use of a central processing unit, or CPU—has its roots in von Neumann's foundational computer science work. Of his many other achievements, the most noteworthy include his work on game theory, quantum physics, and the development of the atomic bomb.

Born Janos von Neumann in Budapest, Hungary, on December 28, 1903, the future mathematician adopted the name John when he emigrated to the United States. His father, Max, was a prosperous banker, and he and von Neumann's mother, Margaret, soon recognized that they had a genius in the family; therefore they arranged to have him tutored at home. When he attended the Lutheran Gymnasium for boys, von Neumann's teachers placed him with a tutor from the University of Budapest, mathematician Michael Fekete.

Von Neumann studied in Budapest and Zürich, and also spent a great deal of time in Berlin. In 1926 he earned his Ph.D. in mathematics at the University of Budapest with a dissertation on set theory. He went to work at the University of Berlin as the equivalent of an assistant professor, reportedly the youngest person to hold that position in the university's history. In 1926 von Neumann received a Rockefeller grant to conduct postdoctoral work with mathematician David Hilbert (1862-1943), who long remained a powerful influence, at the University of Göttingen. Already a rising star in the world of mathematics, von Neumann transferred to the University of Hamburg in 1929, the same year he married Mariette Kovesi. The two had a daughter, Marina, in 1935.

In his early work as Hilbert's student, von Neumann assisted his mentor in attempting to show the axiomatic consistency of arithmetic—a project doomed to failure by Gödel's incompleteness theorem. Hilbert was not destined to make significant contributions to quantum physics, but von Neumann's involvement in his teacher's attempts to apply the axiomatic approach to that discipline led him to an abstract unification of Schrödinger's wave theory and Heisenberg's particle theory. He was the first to affect such a union.

During the 1930s, von Neumann went to Princeton, where he became part of the newly formed Institute for Advanced Study. There he developed what came to be known as von Neumann algebras, published The Mathematical Foundations of Quantum Mechanics (1932), still considered essential reading on the subject, and investigated a number of other areas. In 1937 von Neumann, having become a naturalized citizen of the United States, began the first of many projects for the military, acting as a consultant in ballistics research for the army. During the same year, Mariette divorced him, but in 1938 he remarried to Klara Dan, who, like Mariette, was from Budapest.

During the early part of World War II, von Neumann worked on several defense-related projects. In 1943 he became involved in the development of an atomic bomb at Los Alamos, New Mexico. There he convinced J. Robert Oppenheimer (1904-1967) to investigate the use of an implosion technique in detonating the bomb. Simulation of this technique would require extensive calculations, which would be hopelessly time-consuming using old-fashioned methods. It was then that von Neumann began looking into the army's recently developed ENIAC (Electronic Numerical Integrator and Calculator)—the world's first computer.

Von Neumann and others improved on ENIAC with EDVAC (Electronic Discrete Variable Automatic Computer), which incorporated von Neumann's groundbreaking concept of the stored program. This became the foundation for computer design, and later, in developing a computer for scientific use at Princeton, von Neumann established a number of features now essential to all computers: random access-memory (RAM), the use of input and output devices operating in serial or parallel mode, and other elements.

In the midst of his other work, von Neumann became immersed in game theory, a concept that had first intrigued him in Germany. Working with mathematical economist Oskar Morgenstern (1902-1977), who shared von Neumann's conviction that the mathematics of the physical sciences was not adequate for the study of social sciences, von Neumann began applying the idea of an analogy between games and complex decision-making processes. Today, game theory is used in a variety of fields, from business organization to military strategy.

Von Neumann remained involved in defense technology after the war, and contributed to the development of the hydrogen bomb as well as other weapons. He held a number of key positions, including a seat on the Atomic Energy Commission, to which President Dwight D. Eisenhower appointed him in 1954. Eisenhower also awarded von Neumann the Medal of Freedom, one of many prestigious honors he received. Diagnosed with bone cancer in 1955, von Neumann was confined to a wheelchair, but continued to work feverishly. He died on February 8, 1957, in Washington, D.C., at the age of 53.

JUDSON KNIGHT

Von Neumann, John 1903–1957

BIBLIOGRAPHY

John von Neumann (born December 28, 1903 in Hungary, died February 8, 1957 in Washington, D.C.) was a versatile scholar whose path-breaking ideas have enriched various disciplines. In social sciences his contributions to game theory, economic growth, and consumers’ choice are of special importance.

Von Neumann’s talents showed up early, and outstanding mathematicians tutored him individually. In 1923 he entered MSc chemistry studies in Zürich and at the same time studied for a doctoral degree in mathematics in Budapest. In 1926 and 1927, as an assistant to David Hilbert in Göttingen, he laid down the axiomatic foundations of quantum mechanics. His reputation grew rapidly, and he was invited to several universities. He visited Princeton University first in 1929 and became a professor of mathematics at its Institute for Advanced Study in 1933. He became a leading expert on shock and detonation waves, which became significant during World War II (1939-1945), when von Neumann became involved in important projects such as the Manhattan Project. It was mainly the complex nonlinear problems that emerged in these projects that made him realize the importance of computers, and he made key contributions to formulating the basic principles of computer science.

Von Neumann made major scientific contributions to the social sciences as well. As always, he was interested in comprehensive structures, and focused on the core problems in the field. He was the first to prove the existence of equilibrium for two-person zero-sum games in 1928, based on his famous minimax theorem. Using a similar mathematical structure he formulated a multisectoral model of balanced economic growth (first presented in 1932), which was a brilliant mathematical synthesis of some classical ideas concerning the production and price proportions of economic equilibrium. He was the first to employ a fixed-point theorem in the proof of existence of competitive equilibrium, on the one hand, and an explicit duality approach, recognizing the symmetry of the conditions that characterize the choice of optimal activities and the equilibrium price system sustaining it under the conditions of a competitive equilibrium, on the other. His model allows for different theoretical interpretations (classical, Marxist, neoclassical, etc.) indicating its general nature.

Although his model was a prototype of the highly abstract models used in modern economics, he cautioned often against the potential deterioration of such an approach into intellectual games. He saw this danger threatening not only the development of economics but also mathematics itself. He repeatedly criticized economists for not using more appropriate mathematics, and he emphasized the need for more comprehensive tools than those borrowed from classical physics.

Von Neumann set an excellent example for such a novel approach in his work with Oskar Morgenstern on game theory. In their book, Theory of Games and Economic Behavior (1944), they laid down the foundations of modern game theory and initiated a new discipline almost from scratch. It was in this connection that they developed the axiomatic theory of expected utility, which states that under certain conditions the preferences of a rational individual can be represented by a function of the expected utility form. The use of the von Neumann-Morgenstern expected utility function became universal in economics because it is analytically very convenient and its normative character may provide a valuable guide to rational actions. Although he darted only briefly into its domain, von Neumann’s tremendous influence on the development of modern economics has been widely acknowledged.

SEE ALSO Game Theory; Neoclassical Growth Model; Optimal Growth; Utility, Von Neumann-Morgenstern

BIBLIOGRAPHY

Von Neumann, John. 1945. A Model of General Economic Equilibrium. Review of Economic Studies 13: 1–9. (Published first in German, 1937.)

Von Neumann, John, and Oskar Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.

Zalai, Ernő, guest ed. 2004. A Special Issue on John von Neumann. Acta Oeconomica 54 (1): 1–96.

Ernő Zalai

John Von Neumann

The Hungarian-born American mathematician John Von Neumann (1903-1957) was the originator of the theory of games and an important contributor to computer technology.

John Von Neumann was born in Budapest on Dec. 28, 1903. He left Hungary in 1918 and studied at the University of Berlin and the Zurich Institute of Technology. After receiving his doctorate in mathematics from the University of Budapest in 1926, he attended the University of Göttingen for a year. Göttingen enjoyed a tremendous reputation in the mathematical sciences: the great master and inspirer of generations of students, David Hilbert, had not yet retired, the "ex-prodigy" Norbert Wiener was a visiting fellow from the United States, and the university was the meeting ground for many brilliant young scientific intellects. One of Von Neumann's fellow students was the future atomic scientist J. Robert Oppenheimer.

Von Neumann taught mathematics at the University of Berlin (1927-1929) and the University of Hamburg (1929-1930). Then the young Hungarian, like so many others at that time, found refuge in the United States, obtaining a post at Princeton University, where he taught mathematical physics until 1933. He had been working on quantum mechanics for a number of years, and his book on that subject, published in 1932, provided a useful exposition of the mathematical logic of the theory.

However, Von Neumann had already developed a theory which was to be potentially of much greater value but which was not fully developed for nearly 20 years. In 1927 he propounded a mathematical technique for the analysis of conflict, but it was only in 1944 that he and Oskar Morgenstern wrote the celebrated Theory of Games and Economic Behavior, which had a profound influence on the development of strategy in widely differing fields of application. The theory of games is a concept which can be applied to the logic of conflict; it is an attempt to provide a quantitative basis for rational behavior in a situation which has conflict potential. This purely mathematical technique has developed as an important subject of study for its economic, social, political, and military applications.

In 1933 Von Neuman became professor of mathematics at the Institute for Advanced Study in Princeton, a position he held until his death. During World War II he played an important role in the field of applied mathematics devoted to military needs and worked on the motion of compressible fluids caused by explosions. He was a consultant at the Los Alamos Scientific Laboratory (1943-1955), where his extraordinary intellectual grasp coupled with common sense were of considerable influence. Having seen the potential of high-speed machine calculation in these problems, he studied the mathematical logic of computers and their complex technology. The first computer at Princeton was built in 1952 under his guidance. The U.S. Atomic Energy Commission placed him on its Central Advisory Committee in 1952 and made him a commissioner 2 years later. His interest in computer technology continued until his death on Feb. 8, 1957, in Washington, D.C.

Further Reading

Biographical information on Von Neumann appears in the National Academy of Sciences, Biographical Memoirs, vol. 32 (1958), and Shirley Thomas, Men of Space, vol. 1 (1960). □