Von Neumann studied chemistry at the University of Berlin and in 1926 earned a degree in chemical engineering at Technische Hochschule in Zürich. That same year he received a Ph.D. in mathematics from the University of Budapest, with a dissertation on set theory. His axiomatization left a permanent mark on the subject, and his definition of ordinal numbers, published when he was 20 years old, was universally adopted. Von Neumann was a privatdozent (lecturer) at Berlin in 1926–29 and at the University of Hamburg in 1929–30. During this time he worked mainly on quantum physics and operator theory. Largely because of his work, quantum physics and operator theory can be viewed as two aspects of the same subject.

In 1930 von Neumann was a visiting lecturer at Princeton University; he was appointed professor there in 1931. In 1932 he gave a precise formulation and proof of the “ergodic hypothesis” of statistical mathematics. His book on quantum mechanics, The Mathematical Foundations of Quantum Mechanics, published in 1932, remains a standard treatment of the subject. In 1933 he became a professor at the newly founded Institute for Advanced Study in Princeton, keeping that position until his death. Meanwhile, he turned his attention to the challenge made in 1900 by a German mathematician, David Hilbert, who proposed 23 basic theoretical problems for 20th-century mathematical research. Von Neumann solved a special case of Hilbert's fifth problem, the case of compact groups.

In the second half of the 1930s, the main part of von Neumann's publications, written partly in collaboration with F.J. Murray, was on “rings of operators” (now called Neumann algebras). These concepts became one of the most powerful tools in the study of quantum physics. An important outgrowth of rings of operators is “continuous geometry.” Von Neumann saw that what really determines the character of the dimensional structure of a space is the group of rotations that the structure allows. The groups of rotations associated with rings of operators make possible the description of space with continuously varying dimensions.

About 20 of von Neumann's 150 papers are in physics; the rest are distributed more or less evenly among pure mathematics (mainly set theory, logic, topological group, measure theory, ergodic theory, operator theory, and continuous geometry) and applied mathematics (statistics, numerical analysis, shock waves, flow problems, hydrodynamics, aerodynamics, ballistics, problems of detonation, meteorology, and two nonclassical aspects of applied mathematics, games and computers). His publications show a break from pure to applied research around 1940.

During World War II, von Neumann was much in demand as a consultant to the armed forces and to civilian agencies. His two main contributions were his espousal of the implosion method for bringing nuclear fuel to explosion and his participation in the development of the hydrogen bomb.

The mathematical cornerstone of von Neumann's theory of games is the “minimax theorem,” which he stated in 1928; its elaboration and applications are in the book he wrote jointly with Oskar Morgenstern in 1944, Theory of Games and Economic Behavior. The minimax theorem says that, for a large class of two-person games, there is no point in playing. Either player may consider, for each possible strategy of play, the maximum loss that he can expect to sustain with that strategy and then choose as his “optimal” strategy the one that minimizes the maximum loss. If a player follows this reasoning, then he can be statistically sure of not losing more than that value called the minimax value. Since—this is the assertion of the theorem—that minimax value is the negative of the one, similarly defined, that his opponent can guarantee for himself, the long-run outcome is completely determined by the rules.

In computer theory, von Neumann did much of the pioneering work in logical design, in the problem of obtaining reliable answers from a machine with unreliable components, the function of “memory,” machine imitation of “randomness,” and the problem of constructing automata that can reproduce their own kind. One of the most striking ideas, to the study of which he proposed to apply computer techniques, was to dye the polar ice caps so as to decrease the amount of energy they would reflect—the result could warm the Earth enough to make the climate of Iceland approximate that of Hawaii.

The “axiomatic method” is sometimes mentioned as the secret of von Neumann's success. In his hands it was not pedantry but perception; he got to the root of the matter by concentrating on the basic properties (axioms) from which all else follows. His insights were illuminating and his statements precise.