Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1911. Eugene Wigner was a year ahead of von Neumann at the Lutheran School and soon became his friend. This was one of the best schools in Budapest and was part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. The Hungarian school system produced a generation noted for intellectual achievement, which included Theodore von Kármán (born 1881), George de Hevesy (born 1885), Michael Polanyi (born 1891), Leó Szilárd (born 1898), Dennis Gabor (born 1900), Eugene Wigner (born 1902), Edward Teller (born 1908), and Paul Erdős (born 1913). Collectively, they were sometimes known as “The Martians”.

On February 20, 1913, Emperor Franz Joseph elevated John’s father to the Hungarian nobility for his service to the Austro-Hungarian Empire. The Neumann family thus acquired the hereditary appellation Margittai, meaning “of Margitta” (today Marghita, Romania). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosen coat of arms depicting three marguerites. Neumann János became margittai Neumann János (John Neumann de Margitta), which he later changed to the German Johann von Neumann.

According to his friend Theodore von Kármán, von Neumann’s father wanted John to follow him into industry and thereby invest his time in a more financially useful endeavor than mathematics. In fact, his father asked von Kármán to persuade his son not to take mathematics as his major. Von Neumann and his father decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year, non-degree course in chemistry at the University of Berlin, after which he sat for the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor’s set theory. He graduated as a chemical engineer from ETH Zurich in 1926 (although Wigner says that von Neumann was never very attached to the subject of chemistry), and passed his final examinations for his Ph.D. in mathematics simultaneously with his chemical engineering degree, of which Wigner wrote, “Evidently a Ph.D. thesis and examination did not constitute an appreciable effort.” He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.

Building on the work of Felix Hausdorff, in 1924 Stefan Banach and Alfred Tarski proved that given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets that can be reassembled together in a different way to yield two identical copies of the original ball. Banach and Tarski proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But in a 1929 paper, von Neumann proved that paradoxical decompositions could use a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations contains such subgroups, and this opens the possibility of performing paradoxical decompositions using these subgroups. The class of groups von Neumann isolated in his work on Banach–Tarski decompositions was very important in many areas of mathematics, including von Neumann’s own later work in measure theory (see below).

Von Neumann’s habilitation was completed on December 13, 1927, and he started his lectures as a Privatdozent at the University of Berlin in 1928. He was the youngest person ever elected Privatdozent in the university’s history in any subject. By the end of 1927, von Neumann had published 12 major papers in mathematics, and by the end of 1929, 32, a rate of nearly one major paper per month. His powers of recall allowed him to quickly memorize the pages of telephone directories, and recite the names, addresses and numbers therein. In 1929, he briefly became a Privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University.

The formalism of density operators and matrices was introduced by von Neumann in 1927 and independently, but less systematically by Lev Landau and Felix Bloch in 1927 and 1946 respectively. The density matrix is an alternative way to represent the state of a quantum system, which could otherwise be represented using the wavefunction. The density matrix allows the solution of certain time-dependent problems in quantum mechanics.

Von Neumann founded the field of game theory as a mathematical discipline. He proved his minimax theorem in 1928. It establishes 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 to minimize his maximum losses. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.

Von Neumann’s famous 9-page paper started life as a talk at Princeton and then became a paper in German that was eventually translated into English. His interest in economics that led to that paper began while he was lecturing at Berlin in 1928 and 1929. He spent his summers back home in Budapest, as did the economist Nicholas Kaldor, and they hit it off. Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras. Von Neumann found some faults in the book and corrected them–for example, replacing equations by inequalities. He noticed that Walras’s General Equilibrium Theory and Walras’s Law, which led to systems of simultaneous linear equations, could produce the absurd result that profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced the paper.

On New Year’s Day in 1930, von Neumann married Marietta Kövesi, who had studied economics at Budapest University. Von Neumann and Marietta had one child, a daughter, Marina, born in 1935. As of 2017, she is a distinguished professor of business administration and public policy at the University of Michigan. The couple divorced in 1937. In October 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest before the outbreak of World War II.

Before marrying Marietta, von Neumann was baptized a Catholic in 1930. Von Neumann’s father, Max, had died in 1929. None of the family had converted to Christianity while Max was alive, but all did afterward.

With this contribution of von Neumann, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its consistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger axioms that could be used to prove a broader class of theorems. A strongly negative answer to whether it was definitive arrived in September 1930 at the historic Second Conference on the Epistemology of the Exact Sciences of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.

In a series of papers published in 1932, von Neumann made foundational contributions to ergodic theory, a branch of mathematics that involves the states of dynamical systems with an invariant measure. Of the 1932 papers on ergodic theory, Paul Halmos wrote that even “if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality”. By then von Neumann had already written his articles on operator theory, and the application of this work was instrumental in the von Neumann mean ergodic theorem.

Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, in his 1932 work Mathematical Foundations of Quantum Mechanics. After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system.

Von Neumann first proposed a quantum logic in his 1932 treatise Mathematical Foundations of Quantum Mechanics, where he noted that projections on a Hilbert space can be viewed as propositions about physical observables. The field of quantum logic was subsequently inaugurated, in a famous paper of 1936 by von Neumann and Garrett Birkhoff, the first work ever to introduce quantum logics, wherein von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann’s 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., horizontally and vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is added between the other two, the photons will indeed pass through. This experimental fact is translatable into logic as the non-commutativity of conjunction ( A ∧ B ) ≠ ( B ∧ A ) {displaystyle (Aland B)eq (Bland A)} . It was also demonstrated that the laws of distribution of classical logic, P ∨ ( Q ∧ R ) = ( P ∨ Q ) ∧ ( P ∨ R ) {displaystyle Plor (Qland R)=(Plor Q)land (Plor R)} and P ∧ ( Q ∨ R ) = ( P ∧ Q ) ∨ ( P ∧ R ) {displaystyle Pland (Qlor R)=(Pland Q)lor (Pland R)} , are not valid for quantum theory.

In 1933, he was offered a lifetime professorship at the Institute for Advanced Study in New Jersey when that institution’s plan to appoint Hermann Weyl fell through. He remained a mathematics professor there until his death, although he had announced his intention to resign and become a professor at large at the University of California, Los Angeles. His mother, brothers and in-laws followed von Neumann to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname von Neumann. His brothers changed theirs to “Neumann” and “Vonneumann”. Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army’s Officers Reserve Corps. He passed the exams easily but was rejected because of his age. His prewar analysis of how France would stand up to Germany is often quoted: “Oh, France won’t matter.”

Von Neumann’s abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined “hidden variables,” as in classical statistical mechanics. In 1935, Grete Hermann published a paper arguing that the proof contained a conceptual error and was therefore invalid. Hermann’s work was largely ignored until after John S. Bell made essentially the same argument in 1966. In 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann’s proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation and did not claim that his proof completely ruled out hidden variable theories. The validity of Bub’s argument is, in turn, disputed. In any case, Gleason’s theorem of 1957 fills the gaps in von Neumann’s approach.

Von Neumann introduced the study of rings of operators, through the von Neumann algebras. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant. Von Neumann embarked in 1936, with the partial collaboration of F.J. Murray, on the general study of factors classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 “rank among the masterpieces of analysis in the twentieth century”. The direct integral was later introduced in 1949 by John von Neumann.

In a number of von Neumann’s papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. In his 1936 paper on analytic measure theory, he used the Haar theorem in the solution of Hilbert’s fifth problem in the case of compact groups. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

Between 1937 and 1939, von Neumann worked on lattice theory, the theory of partially ordered sets in which every two elements have a greatest lower bound and a least upper bound. Garrett Birkhoff writes: “John von Neumann’s brilliant mind blazed over lattice theory like a meteor”.

Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.

Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). He improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior, written with Oskar Morgenstern. Morgenstern wrote a paper on game theory and thought he would show it to von Neumann because of his interest in the subject. He read it and said to Morgenstern that he should put more in it. This was repeated a couple of times, and then von Neumann became a coauthor and the paper became 100 pages long. Then it became a book. The public interest in this work was such that The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analysis, especially convex sets and the topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.

In a visit to Los Alamos in September 1944, von Neumann showed that the pressure increase from explosion shock wave reflection from solid objects was greater than previously believed if the angle of incidence of the shock wave was between 90° and some limiting angle. As a result, it was determined that the effectiveness of an atomic bomb would be enhanced with detonation some kilometers above the target, rather than at ground level.

When it turned out that there would not be enough uranium-235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann’s idea was implemented. Implosion was the only method that could be used with the plutonium-239 that was available from the Hanford Site. He established the design of the explosive lenses required, but there remained concerns about “edge effects” and imperfections in the explosives. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, this was achieved by George Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945.

On July 16, 1945, von Neumann and numerous other Manhattan Project personnel were eyewitnesses to the first test of an atomic bomb detonation, which was code-named Trinity. The event was conducted as a test of the implosion method device, at the bombing range near Alamogordo Army Airfield, 35 miles (56 km) southeast of Socorro, New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5 kilotons of TNT (21 TJ) but Enrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of torn-up paper as the shock wave passed his location and watching how far they scattered. The actual power of the explosion had been between 20 and 22 kilotons. It was in von Neumann’s 1944 papers that the expression “kilotons” appeared for the first time. After the war, Robert Oppenheimer remarked that the physicists involved in the Manhattan project had “known sin”. Von Neumann’s response was that “sometimes someone confesses a sin in order to take credit for it.”

Von Neumann was a founding figure in computing. Von Neumann was the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged. Von Neumann wrote the 23 pages long sorting program for the EDVAC in ink. On the first page, traces of the phrase “TOP SECRET”, which was written in pencil and later erased, can still be seen. He also worked on the philosophy of artificial intelligence with Alan Turing when the latter visited Princeton in the 1930s.

Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained the hydrogen bomb project. He collaborated with Klaus Fuchs on further development of the bomb, and in 1946 the two filed a secret patent on “Improvement in Methods and Means for Utilizing Nuclear Energy”, which outlined a scheme for using a fission bomb to compress fusion fuel to initiate nuclear fusion. The Fuchs–von Neumann patent used radiation implosion, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design. Their work was, however, incorporated into the “George” shot of Operation Greenhouse, which was instructive in testing out concepts that went into the final design. The Fuchs–von Neumann work was passed on to the Soviet Union by Fuchs as part of his nuclear espionage, but it was not used in the Soviets’ own, independent development of the Teller–Ulam design. The historian Jeremy Bernstein has pointed out that ironically, “John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made.”

For his wartime services, von Neumann was awarded the Navy Distinguished Civilian Service Award in July 1946, and the Medal for Merit in October 1946.

As part of his research into weather forecasting, von Neumann founded the “Meteorological Program” in Princeton in 1946, securing funding for his project from the US Navy. Von Neumann and his appointed assistant on this project, Jule Gregory Charney, wrote the world’s first climate modelling software, and used it to perform the world’s first numerical weather forecasts on the ENIAC computer; von Neumann and his team published the results as Numerical Integration of the Barotropic Vorticity Equation in 1950. Together they played a leading role in efforts to integrate sea-air exchanges of energy and moisture into the study of climate. Von Neumann proposed as the research program for climate modeling: “The approach is to first try short-range forecasts, then long-range forecasts of those properties of the circulation that can perpetuate themselves over arbitrarily long periods of time, and only finally to attempt forecast for medium-long time periods which are too long to treat by simple hydrodynamic theory and too short to treat by the general principle of equilibrium theory.”

The detailed proposal for a physical non-biological self-replicating system was first put forward in lectures von Neumann delivered in 1948 and 1949, when he first only proposed a kinematic self-reproducing automaton. While qualitatively sound, von Neumann was evidently dissatisfied with this model of a self-replicator due to the difficulty of analyzing it with mathematical rigor. He went on to instead develop a more abstract model self-replicator based on his original concept of cellular automata.

Beginning in 1949, von Neumann’s design for a self-reproducing computer program is considered the world’s first computer virus, and he is considered to be the theoretical father of computer virology.

In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group (WSEG), whose function was to advise the Joint Chiefs of Staff and the United States Secretary of Defense on the development and use of new technologies. He also became an adviser to the Armed Forces Special Weapons Project (AFSWP), which was responsible for the military aspects on nuclear weapons. Over the following two years, he became a consultant to the Central Intelligence Agency (CIA), a member of the influential General Advisory Committee of the Atomic Energy Commission, a consultant to the newly established Lawrence Livermore National Laboratory, and a member of the Scientific Advisory Group of the United States Air Force.

Von Neumann entered government service primarily because he felt that, if freedom and civilization were to survive, it would have to be because the United States would triumph over totalitarianism from Nazism, Fascism and Soviet Communism. During a Senate committee hearing he described his political ideology as “violently anti-communist, and much more militaristic than the norm”. He was quoted in 1950 remarking, “If you say why not bomb [the Soviets] tomorrow, I say, why not today? If you say today at five o’clock, I say why not one o’clock?”

The cybernetics movement highlighted the question of what it takes for self-reproduction to occur autonomously, and in 1952, John von Neumann designed an elaborate 2D cellular automaton that would automatically make a copy of its initial configuration of cells. The von Neumann neighborhood, in which each cell in a two-dimensional grid has the four orthogonally adjacent grid cells as neighbors, continues to be used for other cellular automata. Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire moon or asteroid belt would be by using self-replicating spacecraft, taking advantage of their exponential growth.

Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953. However, the theory could not be implemented until advances in computing of the 1960s.

In 1955, von Neumann became a commissioner of the AEC. He accepted this position and used it to further the production of compact hydrogen bombs suitable for Intercontinental ballistic missile (ICBM) delivery. He involved himself in correcting the severe shortage of tritium and lithium 6 needed for these compact weapons, and he argued against settling for the intermediate-range missiles that the Army wanted. He was adamant that H-bombs delivered into the heart of enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile wouldn’t be a problem with an H-bomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case. Despite his disagreement with Oppenheimer over the need for a crash program to develop the hydrogen bomb, he testified on the latter’s behalf at the 1954 Oppenheimer security hearing, at which he asserted that Oppenheimer was loyal, and praised him for his helpfulness once the program went ahead.

In 1955, von Neumann was diagnosed with what was either bone, pancreatic or prostate cancer after he was examined by physicians for a fall, whereupon they inspected a mass growing near his collarbone. The cancer was possibly caused by his radiation exposure during his time in Los Alamos National Laboratory. He was not able to accept the proximity of his own demise, and the shadow of impending death instilled great fear in him. He invited a Catholic priest, Father Anselm Strittmatter, O.S.B., to visit him for consultation. Von Neumann reportedly said, “So long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end,” referring to Pascal’s wager. He had earlier confided to his mother, “There probably has to be a God. Many things are easier to explain if there is than if there isn’t.” Father Strittmatter administered the last rites to him. Some of von Neumann’s friends (such as Abraham Pais and Oskar Morgenstern) said they had always believed him to be “completely agnostic”. Of this deathbed conversion, Morgenstern told Heims, “He was of course completely agnostic all his life, and then he suddenly turned Catholic—it doesn’t agree with anything whatsoever in his attitude, outlook and thinking when he was healthy.” Father Strittmatter recalled that even after his conversion, von Neumann did not receive much peace or comfort from it, as he still remained terrified of death.

On February 15, 1956, von Neumann was presented with the Medal of Freedom by President Dwight D. Eisenhower. His citation read:

Von Neumann was on his deathbed when he entertained his brother by reciting by heart and word-for-word the first few lines of each page of Goethe’s Faust. On his deathbed, his mental capabilities became a fraction of what they were before, causing him much anguish; at times Von Neumann even forgot the lines that his brother recited from Goethe’s Faust. He died at age 53 on February 8, 1957, at the Walter Reed Army Medical Center in Washington, D.C., under military security lest he reveal military secrets while heavily medicated. He was buried at Princeton Cemetery in Princeton, Mercer County, New Jersey.

Shortly before his death from cancer, von Neumann headed the United States government’s top secret ICBM committee, which would sometimes meet in his home. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were sizable, they could be overcome in time. The SM-65 Atlas passed its first fully functional test in 1959, two years after his death. The feasibility of an ICBM owed as much to improved, smaller warheads as it did to developments in rocketry, and his understanding of the former made his advice invaluable.

Von Neumann’s proof inaugurated a line of research that ultimately led, through Bell’s theorem and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics either requires a notion of reality substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity.