# John von Neumann

**John von Neumann** (; Hungarian: *Neumann János Lajos*, pronounced [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was regarded as perhaps the mathematician with the widest coverage of the subject in his time^{[2]} and was said to have been "the last representative of the great mathematicians who were equally at home in pure and applied mathematics".^{[3]}^{[4]} He integrated pure and applied sciences.

Von Neumann made major contributions to many fields, including mathematics (foundations of mathematics, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, geometry, and numerical analysis), physics (quantum mechanics, hydrodynamics, nuclear physics and quantum statistical mechanics), economics (game theory and general equilibrium theory), computing (Von Neumann architecture, linear programming, numerical meteorology, scientific computing, self-replicating machines, stochastic computing), and statistics. He was a pioneer of the application of operator theory to quantum mechanics in the development of functional analysis, and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer.

Von Neumann published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on special mathematical subjects or non-mathematical ones.^{[5]} His last work, an unfinished manuscript written while he was in the hospital, was later published in book form as *The Computer and the Brain*.

His analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a shortlist of facts about his life he submitted to the National Academy of Sciences, he wrote, "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."^{[6]}

During World War II, von Neumann worked on the Manhattan Project with theoretical physicist Edward Teller, mathematician Stanislaw Ulam and others, problem-solving key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. He developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon and coined the term "kiloton" (of TNT) as a measure of the explosive force generated. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, eventually becoming commissioner, and consulted for many organizations including the United States Air Force, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project, and the Lawrence Livermore National Laboratory. As a Hungarian émigré, concerned that the Soviets would achieve nuclear superiority, he designed and promoted the policy of mutually assured destruction to limit the arms race.

In honor of his achievements and contributions to the modern world, he was named in 1999 the Financial Times Person of the Century, as a representative of the century's characteristic ideal that the power of the mind could shape the physical world, and of the "intellectual brilliance and human savagery" that defined the 20th century.^{[7]}^{[8]}^{[9]}

Von Neumann was born on December 28, 1903, to a wealthy, acculturated and non-observant Jewish family. His Hungarian birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names are equivalent to John Louis in English.^{[citation needed]}

Von Neumann was born in Budapest, Kingdom of Hungary, which was then part of the Austro-Hungarian Empire.^{[10]}^{[11]}^{[12]} He was the eldest of three brothers; his two younger siblings were Mihály (English: Michael von Neumann; 1907–1989) and Miklós (Nicholas von Neumann, 1911–2011).^{[13]} His father, Neumann Miksa (Max von Neumann, 1873–1928) was a banker, who held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s.^{[14]} Miksa's father and grandfather were both born in Ond (now part of the town of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (English: Margaret Kann);^{[15]} her parents were Jakab Kann and Katalin Meisels of the Meisels family.^{[16]} Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest; von Neumann's family occupied an 18-room apartment on the top floor.^{[17]}

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.^{[18]}

Von Neumann was a child prodigy. When he was six years old, he could divide two eight-digit numbers in his head^{[19]}^{[20]} and could converse in Ancient Greek. When the six-year-old von Neumann caught his mother staring aimlessly, he asked her, "What are you calculating?"^{[21]}

When they were young, governesses taught von Neumann, his brothers and his cousins. Von Neumann's father believed that knowledge of languages other than their native Hungarian was essential, so the children were tutored in English, French, German and Italian.^{[22]} By the age of eight, von Neumann was familiar with differential and integral calculus,^{[23]} but he was particularly interested in history. He read his way through Wilhelm Oncken's 46-volume world history series *Allgemeine Geschichte in Einzeldarstellungen* (*General History in Monographs*).^{[24]} A copy was contained in a private library Max purchased. One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor.^{[25]}

Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1914.^{[26]} Eugene Wigner was a year ahead of von Neumann at the Lutheran School and soon became his friend.^{[27]} 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).^{[28]} Collectively, they were sometimes known as "The Martians".^{[29]}

Although Von Neumann's father insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give von Neumann advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő.^{[27]} On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears.^{[30]} Some of von Neumann's instant solutions to the problems that Szegő posed in calculus are sketched out on his father's stationery and are still on display at the von Neumann archive in Budapest.^{[27]} By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition.^{[31]} At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.^{[32]}

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.^{[33]} 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,^{[34]} which he passed in September 1923.^{[35]} At the same time, von Neumann also entered Pázmány Péter University in Budapest,^{[36]} as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory.^{[37]}^{[38]} 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),^{[39]} 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."^{[39]} He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.^{[40]}

Von Neumann's habilitation was completed on December 13, 1927, and he began to give lectures as a *Privatdozent* at the University of Berlin in 1928.^{[41]} He was the youngest person ever elected *Privatdozent* in the university's history in any subject.^{[42]} 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.^{[43]} In 1929, he briefly became a *Privatdozent* at the University of Hamburg, where the prospects of becoming a tenured professor were better,^{[44]} but in October of that year a better offer presented itself when he was invited to Princeton University.^{[45]}

On New Year's Day 1930, von Neumann married Marietta Kövesi, who had studied economics at Budapest University.^{[45]} Von Neumann and Marietta had one child, a daughter, Marina, born in 1935. As of 2021 Marina is a distinguished professor emerita of business administration and public policy at the University of Michigan.^{[46]} The couple divorced on November 2, 1937.^{[47]} On November 17, 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest before the outbreak of World War II.^{[48]}^{[49]}

In 1930, before marrying Marietta, von Neumann was baptized into the Catholic Church.^{[50]} 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.^{[51]}

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 failed.^{[52]} 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.^{[53]} His mother, brothers and in-laws followed von Neumann to the United States in 1939.^{[54]} Von Neumann anglicized his first name to John, keeping the German-aristocratic surname von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann".^{[18]} 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.^{[55]} His prewar analysis of how France would stand up to Germany is often quoted: "Oh, France won't matter."^{[56]}

Klara and John von Neumann were socially active within the local academic community.^{[57]} His white clapboard house at 26 Westcott Road was one of Princeton's largest private residences.^{[58]} He always wore formal suits. He once wore a three-piece pinstripe while riding down the Grand Canyon astride a mule.^{[59]} Hilbert is reported to have asked, "Pray, who is the candidate's tailor?" at von Neumann's 1926 doctoral exam, as he had never seen such beautiful evening clothes.^{[60]}

Von Neumann held a lifelong passion for ancient history and was renowned for his historical knowledge. A professor of Byzantine history at Princeton once said that von Neumann had greater expertise in Byzantine history than he did.^{[61]} He knew by heart much of the material in Gibbon's Decline and Fall and after dinner liked to engage in various historial discussions. Ulam noted that one time while driving south to a meeting of the American Mathematical Society, von Neumann would describe even the minutest details of the battles of the Civil War that occurred in the places they drove by.^{[62]}

Von Neumann liked to eat and drink. His wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks).^{[23]} He was a non-smoker.^{[63]} In Princeton, he received complaints for regularly playing extremely loud German march music on his phonograph, which distracted those in neighboring offices, including Albert Einstein, from their work.^{[64]} Von Neumann did some of his best work in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.^{[65]} Despite being a notoriously bad driver, he enjoyed driving—frequently while reading a book—occasioning numerous arrests as well as accidents. When Cuthbert Hurd hired him as a consultant to IBM, Hurd often quietly paid the fines for his traffic tickets.^{[66]}

Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota, wrote, "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in the hospital, every time Ulam visited, he came prepared with a new collection of jokes to cheer him up.^{[67]} Von Neumann believed that much of his mathematical thought occurred intuitively; he would often go to sleep with a problem unsolved and know the answer upon waking up.^{[65]} Ulam noted that von Neumann's way of thinking might not be visual, but more aural.^{[68]}

In 1955, von Neumann was diagnosed with what was either bone, pancreatic or prostate cancer^{[69]}^{[70]} after he was examined by physicians for a fall, whereupon they inspected a mass growing near his collarbone.^{[71]} The cancer was possibly caused by his radiation exposure during his time in Los Alamos National Laboratory.^{[71]} He was not able to accept the proximity of his own demise, and the shadow of impending death instilled great fear in him.^{[72]} He invited a Catholic priest, Father Anselm Strittmatter, O.S.B., to visit him for consultation.^{[73]}^{[71]} 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."^{[74]}^{[75]}^{[76]} Father Strittmatter administered the last rites to him.^{[23]} Some of von Neumann's friends, such as Abraham Pais and Oskar Morgenstern, said they had always believed him to be "completely agnostic".^{[75]}^{[77]} 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."^{[78]} 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.^{[78]}

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*.^{[12]} 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*.^{[71]} 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 of Nassau Presbyterian Church in Princeton, Mercer County, New Jersey.^{[79]}^{[80]}

The axiomatization of mathematics, on the model of Euclid's *Elements*, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and in geometry, thanks to Hilbert's axioms.^{[81]} But at the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves).^{[82]} The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the *axiom of foundation* and the notion of *class.*^{[81]}

The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration called the *method of inner models*, which became an essential instrument in set theory.^{[81]}

The second approach to the problem of sets belonging to themselves took as its base the notion of class, and defines a set as a class that belongs to other classes, while a *proper class* is defined as a class that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is a *proper class*, not a set.^{[81]}

Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the ordinal and cardinal numbers as well as the first strict formulation of principles of definitions by the transfinite induction".^{[83]}

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,^{[84]} 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).

With the aforementioned contributions of von Neumann to sets, 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.

Building on the work of Ackermann, von Neumann began attempting to prove (using the finistic methods of Hilbert's school) the consistency of first-order arithmetic. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on induction).^{[85]} He continued looking for a more general proof of the consistency of classical mathematics using methods from proof theory.^{[86]}

A strongly negative answer to whether it was definitive arrived in September 1930 at the historic 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.^{[87]}

Less than a month later, von Neumann, who had participated in the Conference, communicated to Gödel an interesting consequence of his theorem: that the usual axiomatic systems are unable to demonstrate their own consistency.^{[87]} Gödel had already discovered this consequence, now known as his second incompleteness theorem, and sent von Neumann a preprint of his article containing both theorems.^{[88]} Von Neumann acknowledged Gödel's priority in his next letter.^{[89]} He never thought much of "the American system of claiming personal priority for everything."^{[90]} However von Neumann's method of proof differed from Gödel's, as his used polynomials to explain consistency.^{[91]}^{[92]} With this discovery, von Neumann ceased work in mathematical logic and foundations of mathematics and instead spent time on problems connected with applications.^{[93]}

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.^{[94]} 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".^{[95]} 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.^{[95]}

In measure theory, the "problem of measure" for an n-dimensional Euclidean space **R**^{n} may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of **R**^{n}?"^{[95]} The work of Felix Hausdorff and Stefan Banach had implied that the problem of measure has a positive solution if *n* = 1 or *n* = 2 and a negative solution (because of the Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character":^{[95]} the existence of a measure could be determined by looking at the properties of the transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."^{[95]}

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.^{[95]} A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions".^{[96]} He proved this in the positive, and in later papers with Stone discussed various generalizations and algebraic aspects of this problem.^{[97]} He also proved by new methods the existence of disintegrations for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for compact groups.^{[96]} He had to create entirely new techniques to apply this to locally compact groups.^{[95]} He also gave a new proof for the Radon–Nikodym theorem.^{[98]} His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.^{[95]}^{[99]}

Using his previous work on measure theory von Neumann made several contributions to the theory of topological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic functions to arbitrary groups.^{[100]} He continued this work with another paper in conjunction with Bochner that improved the theory of almost periodicity to include functions that took on elements of linear spaces as values rather than numbers.^{[101]} In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis in relation to these papers.^{[102]}^{[103]}

In a 1933 paper, he used the newly discovered Haar measure in the solution of Hilbert's fifth problem for the case of compact groups.^{[95]}^{[104]} The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of linear transformations and found that closed subgroups of a general linear group are Lie groups.^{[105]} This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem.^{[106]}^{[96]}

Von Neumann was the first one to come up with an “abstract” Hilbert space in a formal and axiomatic fashion. It was defined as a complex vector space with a Hermitian scalar product, with the corresponding norm being both separable and complete. He continued with the development of the spectral theory of operators in Hilbert space in 3 seminal papers between 1929 and 1932.^{[107]} For twenty years von Neumann was considered the 'undisputed master' of this area.^{[96]} These developments were primarily prompted by needs in quantum mechanics where von Neumann realized the need to extend the spectral theory of Hermitian operators from the bounded to the unbounded case.^{[108]} Other major achievements in these papers include a complete elucidation of spectral theory for normal operators, a generalisation of Riesz’s presentation of Hilbert’s spectral theorems at the time, and the discovery of Hermitian operators in a Hilbert space, as distinct from self-adjoint operators, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. In addition he wrote a paper detailing how the usage of infinite matrices, common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory lead to his most profound invention in pure mathematics, the study of von Neumann algebras and in general of operator algebras.^{[109]}

In other work in functional analysis von Neumann was also the first mathematician to apply new topological ideas from Hausdorff to Hilbert spaces. He also gave the first general definition of locally convex spaces.^{[110]} His later work on rings of operators lead to him revisiting his earlier work on spectral theory and providing a new way of working through the geometric content of the spectral theory by the use of direct integrals of Hilbert spaces.^{[111]}

Von Neumann founded 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.^{[112]} The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant.^{[113]} After elucidating the study of the commutative algebra case, von Neumann embarked in 1936, with the partial collaboration of F.J. Murray, on the noncommutative case, 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".^{[3]} The direct integral was later introduced in 1949 by John von Neumann for his work on operator theory.^{[114]} His work here lead on to the next two major topics.

Von Neumann founded the field of continuous geometry.^{[115]} It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., *n*, it can be an element of the unit interval [0,1]. Earlier, Menger and Birkhoff had axiomatized complex projective geometry in terms of the properties of its lattice of linear subspaces. Von Neumann, following his work on rings of operators, weakened those axioms to describe a broader class of lattices, the continuous geometries.
While the dimensions of the subspaces of projective geometries are a discrete set (the non-negative integers), the dimensions of the elements of a continuous geometry can range continuously across the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.^{[116]}^{[117]}

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".^{[118]}

Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices (properties that arise in the lattices of subspaces of inner product spaces): "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity."^{[118]}

Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of *n* ≥ 4 pairwise perspective elements, is isomorphic with the lattice ℛ(*R*) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."^{[118]}

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.^{[119]} This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic^{[120]} for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.^{[120]}

Subsequently, Denis Sargan and Alok Bhargava extended the results for testing if the errors on a regression model follow a Gaussian random walk (*i.e.*, possess a unit root) against the alternative that they are a stationary first order autoregression.^{[121]}

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*.^{[148]} 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.^{[149]}

The *physics* of quantum mechanics was thereby reduced to the *mathematics* of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the *non-commutativity* of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger.^{[149]} When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a self-adjoint operator and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?"^{[150]}

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.^{[151]} Hermann's work was largely ignored until after John S. Bell made essentially the same argument in 1966.^{[152]} 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.^{[153]} The validity of Bub's argument is, in turn, disputed.^{[154]} In any case, Gleason's theorem of 1957 fills the gaps in von Neumann's approach.

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.^{[155]}

In a chapter of *The Mathematical Foundations of Quantum Mechanics*, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter. He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer. Although this view was accepted by Eugene Wigner,^{[156]} the Von Neumann–Wigner interpretation never gained acceptance among the majority of physicists.^{[157]} The Von Neumann–Wigner interpretation has been summarized as follows:^{[158]}

The rules of quantum mechanics are correct but there is only one system which may be treated with quantum mechanics, namely the entire material world. There exist external observers which cannot be treated within quantum mechanics, namely human (and perhaps animal) *minds*, which perform measurements on the brain causing wave function collapse.^{[158]}

Though theories of quantum mechanics continue to evolve, there is a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches that can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.^{[148]}

Viewing von Neumann's work on quantum mechanics as an part of the fulfilment of Hilbert's sixth problem, noted mathematical physicist A. S. Wightman said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. In the publication of his 1932 book, quantum mechanics became a mature theory in the sense it had a precise mathematical form, which allowed for clear answers to conceptual problems.^{[159]}

Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy. The von Neumann entropy is the cornerstone in the development of quantum information theory, while the Shannon entropy applies to classical information theory. This is considered a historical anomaly, as Shannon entropy might have been expected to be discovered before Von Neumann entropy, given the latter's more widespread application to quantum information theory. But Von Neumann discovered von Neumann entropy first, and applied it to questions of statistical physics. Decades later, Shannon developed an information-theoretic formula for use in classical information theory, and asked von Neumann what to call it. Von Neumann said to call it Shannon entropy, as it was a special case of von Neumann entropy.^{[161]}

The formalism of density operators and matrices was introduced by von Neumann^{[162]} in 1927 and independently, but less systematically by Lev Landau^{[163]} and Felix Bloch^{[164]} 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.

The von Neumann measurement scheme, the ancestor of quantum decoherence theory, represents measurements projectively by taking into account the measuring apparatus which is also treated as a quantum object. The 'projective measurement' scheme introduced by von Neumann led to the development of quantum decoherence theories.^{[165]}^{[166]}

As Hilary Putnam writes, von Neumann replaced classical logic with a logic constructed in orthomodular lattices (isomorphic to the lattice of subspaces of the Hilbert space of a given physical system).^{[169]}

Von Neumann made fundamental contributions in the field of fluid dynamics.

Von Neumann's contributions to fluid dynamics included his discovery of the classic flow solution to blast waves,^{[170]} and the co-discovery (independently of Yakov Borisovich Zel'dovich and Werner Döring) of the ZND detonation model of explosives.^{[171]} During the 1930s, von Neumann became an authority on the mathematics of shaped charges.^{[172]}

Later with Robert D. Richtmyer, von Neumann developed an algorithm defining *artificial viscosity* that improved the understanding of shock waves. When computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of *artificial viscosity* smoothed the shock transition without sacrificing basic physics.^{[173]}

Von Neumann soon applied computer modelling to the field, developing software for his ballistics research. During WW2, he arrived one day at the office of R.H. Kent, the Director of the US Army's Ballistic Research Laboratory, with a computer program he had created for calculating a one-dimensional model of 100 molecules to simulate a shock wave. Von Neumann then gave a seminar on his computer program to an audience which included his friend Theodore von Kármán. After von Neumann had finished, von Kármán said "Well, Johnny, that's very interesting. Of course you realize Lagrange also used digital models to simulate continuum mechanics." It was evident from von Neumann's face, that he had been unaware of Lagrange's Mécanique analytique.^{[174]}

While not as prolific in physics as he was in mathematics, he nevertheless made several other notable contributions to it. His papers with Subrahmanyan Chandrasekhar on the statistics of a fluctuating gravitational field generated by randomly distributed stars were considered a *tour de force*.^{[175]} With A. H. Taub and Veblen he wrote a paper extending the Dirac equation to projective relativity, maintaining invariance with regards to coordinate, spin, and gauge transformations, as a part of early research into potential theories of quantum gravity in the 1930s.^{[176]}

Von Neumann founded the field of game theory as a mathematical discipline.^{[177]} 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.^{[178]}

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.^{[179]} 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.^{[177]}

Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been the primary tools of mathematical economics ever since.^{[180]}

Von Neumann raised the intellectual and mathematical level of economics in several influential publications. For his model of an expanding economy, he proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem.^{[177]} Von Neumann's model of an expanding economy considered the matrix pencil * A − λB* with nonnegative matrices

**A**and

**B**; von Neumann sought probability vectors

*p*and

*q*and a positive number

*λ*that would solve the complementarity equation

along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector *p* represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution *λ* represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate.^{[181]}^{[182]}

Von Neumann's results have been viewed as a special case of linear programming, where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists with interests in computational economics.^{[183]}^{[184]}^{[185]} This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself.^{[186]}

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.^{[187]}

Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.^{[188]}

Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming.^{[189]}

Von Neumann was a founding figure in computing.^{[190]} 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.^{[191]}^{[192]} 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.^{[192]} He also worked on the philosophy of artificial intelligence with Alan Turing when the latter visited Princeton in the 1930s.^{[193]}

Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and Stanisław Ulam developed simulations on von Neumann's digital computers for the hydrodynamic computations. During this time he contributed to the development of the Monte Carlo method, which allowed solutions to complicated problems to be approximated using random numbers.^{[194]}

Von Neumann's algorithm for simulating a fair coin with a biased coin is used in the "software whitening" stage of some hardware random number generators.^{[195]} Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."^{[196]} Von Neumann also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect.^{[196]}

While consulting for the Moore School of Electrical Engineering at the University of Pennsylvania on the EDVAC project, von Neumann wrote an incomplete *First Draft of a Report on the EDVAC*. The paper, whose premature distribution nullified the patent claims of EDVAC designers J. Presper Eckert and John Mauchly, described a computer architecture in which the data and the program are both stored in the computer's memory in the same address space. This architecture is the basis of most modern computer designs, unlike the earliest computers that were "programmed" using a separate memory device such as a paper tape or plugboard. Although the single-memory, stored program architecture is commonly called von Neumann architecture as a result of von Neumann's paper, the architecture was based on the work of Eckert and Mauchly, inventors of the ENIAC computer at the University of Pennsylvania.^{[197]}

Von Neumann consulted for the Army's Ballistic Research Laboratory, most notably on the ENIAC project,^{[198]} as a member of its Scientific Advisory Committee.^{[199]}
The electronics of the new ENIAC ran at one-sixth the speed, but this in no way degraded the ENIAC's performance, since it was still entirely I/O bound. Complicated programs could be developed and debugged in days rather than the weeks required for plugboarding the old ENIAC. Some of von Neumann's early computer programs have been preserved.^{[200]}

The next computer that von Neumann designed was the IAS machine at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the RCA Research Laboratory nearby. Von Neumann recommended that the IBM 701, nicknamed *the defense computer*, include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successful IBM 704.^{[201]}^{[202]}

Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953.^{[203]}
However, the theory could not be implemented until advances in computing of the 1960s.^{[204]}^{[205]}

Von Neumann's rigorous mathematical analysis of the structure of self-replication (of the semiotic relationship between constructor, description and that which is constructed), preceded the discovery of the structure of DNA.^{[207]}

Von Neumann created the field of cellular automata without the aid of computers, constructing the first self-replicating automata with pencil and graph paper.

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.^{[208]}^{[209]} 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.^{[210]}

Subsequently, the concept of the Von Neumann universal constructor based on the von Neumann cellular automaton was fleshed out in his posthumously published lectures *Theory of Self Reproducing Automata*.^{[211]} Ulam and von Neumann created a method for calculating liquid motion in the 1950s. The driving concept of the method was to consider a liquid as a group of discrete units and calculate the motion of each based on its neighbors' behaviors.^{[212]} Like Ulam's lattice network, von Neumann's cellular automata are two-dimensional, with his self-replicator implemented algorithmically. The result was a universal copier and constructor working within a cellular automaton with a small neighborhood (only those cells that touch are neighbors; for von Neumann's cellular automata, only orthogonal cells), and with 29 states per cell. Von Neumann gave an existence proof that a particular pattern would make infinite copies of itself within the given cellular universe by designing a 200,000 cell configuration that could do so.

[T]here exists a critical size below which the process of synthesis is degenerative, but above which the phenomenon of synthesis, if properly arranged, can become explosive, in other words, where syntheses of automata can proceed in such a manner that each automaton will produce other automata which are more complex and of higher potentialities than itself.

Von Neumann addressed the evolutionary growth of complexity amongst his self-replicating machines.^{[213]} His "proof-of-principle" designs showed how it is logically possible, by using a general purpose programmable ("universal") constructor, to exhibit an indefinitely large class of self-replicators, spanning a wide range of complexity, interconnected by a network of potential mutational pathways, including pathways from the most simple to the most complex. This is an important result, as prior to that it might have been conjectured that there is a fundamental logical barrier to the existence of such pathways; in which case, biological organisms, which do support such pathways, could not be "machines", as conventionally understood. Von Neumann considers the potential for conflict between his self-reproducing machines, stating that "our models lead to such conflict situations",^{[214]} indicating it as a field of further study.^{[211]}^{: 147 }

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.^{[215]}

Von Neumann investigated the question of whether modelling evolution on a digital computer could solve the complexity problem in programming.^{[214]}

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.^{[216]}

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.^{[217]} 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;^{[217]} von Neumann and his team published the results as *Numerical Integration of the Barotropic Vorticity Equation* in 1950.^{[218]} Together they played a leading role in efforts to integrate sea-air exchanges of energy and moisture into the study of climate.^{[219]} 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."^{[220]}

Von Neumann's research into weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice caps to enhance absorption of solar radiation (by reducing the albedo),^{[221]}^{[222]} thereby inducing global warming.^{[221]}^{[222]} Von Neumann proposed a theory of global warming as a result of the activity of humans, noting that the Earth was only 6 °F (3.3 °C) colder during the last glacial period, he wrote in 1955: "Carbon dioxide released into the atmosphere by industry's burning of coal and oil - more than half of it during the last generation - may have changed the atmosphere's composition sufficiently to account for a general warming of the world by about one degree Fahrenheit."^{[223]}^{[224]} However, von Neumann urged a degree of caution in any program of intentional human weather manufacturing: "What *could* be done, of course, is no index to what *should* be done... In fact, to evaluate the ultimate consequences of either a general cooling or a general heating would be a complex matter. Changes would affect the level of the seas, and hence the habitability of the continental coastal shelves; the evaporation of the seas, and hence general precipitation and glaciation levels; and so on... But there is little doubt that one *could* carry out the necessary analyses needed to predict the results, intervene on any desired scale, and ultimately achieve rather fantastic results."^{[224]}

"The technology that is now developing and that will dominate the next decades is in conflict with traditional, and, in the main, momentarily still valid, geographical and political units and concepts. This is a maturing crisis of technology... The most hopeful answer is that the human species has been subjected to similar tests before and it seems to have a congenital ability to come through, after varying amounts of trouble."

The first use of the concept of a singularity in the technological context is attributed to von Neumann,^{[225]} who according to Ulam discussed the "ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue."^{[226]} This concept was fleshed out later in the book *Future Shock* by Alvin Toffler.

Beginning in the late 1930s, von Neumann developed an expertise in explosions—phenomena that are difficult to model mathematically. During this period, von Neumann was the leading authority of the mathematics of shaped charges. This led him to a large number of military consultancies, primarily for the Navy, which in turn led to his involvement in the Manhattan Project. The involvement included frequent trips by train to the project's secret research facilities at the Los Alamos Laboratory in a remote part of New Mexico.^{[36]}

Von Neumann made his principal contribution to the atomic bomb in the concept and design of the explosive lenses that were needed to compress the plutonium core of the Fat Man weapon that was later dropped on Nagasaki. While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly".^{[227]}

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.^{[228]} He established the design of the explosive lenses required, but there remained concerns about "edge effects" and imperfections in the explosives.^{[229]} His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry.^{[230]} 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.^{[231]}

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.^{[232]}^{[233]}

Von Neumann, four other scientists, and various military personnel were included in the target selection committee that was responsible for choosing the Japanese cities of Hiroshima and Nagasaki as the first targets of the atomic bomb. Von Neumann oversaw computations related to the expected size of the bomb blasts, estimated death tolls, and the distance above the ground at which the bombs should be detonated for optimum shock wave propagation and thus maximum effect. The cultural capital Kyoto, which had been spared the , was von Neumann's first choice,^{[234]} a selection seconded by Manhattan Project leader General Leslie Groves. However, this target was dismissed by Secretary of War Henry L. Stimson.^{[235]}

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.^{[236]} It was in von Neumann's 1944 papers that the expression "kilotons" appeared for the first time.^{[237]} 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."^{[238]}

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.^{[239]} 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.^{[240]} 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."^{[240]}

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.^{[241]}

In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group (WSEG),^{[242]} 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.^{[243]} 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.^{[242]}

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.^{[244]}^{[245]} 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.^{[246]}

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.^{[247]}

Von Neumann is credited with developing the equilibrium strategy of mutual assured destruction (MAD). He also "moved heaven and earth" to bring MAD about. His goal was to quickly develop ICBMs and the compact hydrogen bombs that they could deliver to the USSR, and he knew the Soviets were doing similar work because the CIA interviewed German rocket scientists who were allowed to return to Germany, and von Neumann had planted a dozen technical people in the CIA. The Soviets considered that bombers would soon be vulnerable, and they shared von Neumann's view that an H-bomb in an ICBM was the ne plus ultra of weapons; they believed that whoever had superiority in these weapons would take over the world, without necessarily using them.^{[248]} He was afraid of a "missile gap" and took several more steps to achieve his goal of keeping up with the Soviets:

Von Neumann's assessment that the Soviets had a lead in missile technology, considered pessimistic at the time, was soon proven correct in the Sputnik crisis.^{[250]}

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.^{[59]} 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?"^{[251]}

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

Dr. von Neumann, in a series of scientific study projects of major national significance, has materially increased the scientific progress of this country in the armaments field.
Through his work on various highly classified missions performed outside the continental limits of the United States in conjunction with critically important international programs, Dr. von Neumann has resolved some of the most difficult technical problems of national defense.^{[252]}

Even when dying of cancer, von Neumann continued his work while he still could. Lewis Strauss, who at the time was chairman of the AEC and a close friend, described some of his final memories of von Neumann in his memoir.

Until the last, he continued to be a member of the Commission and chairman of an important advisory committee to the Defense Department. On one dramatic occasion, I was present at a meeting at Walter Reed Hospital, where, gathered around Johnny’s bedside were the Secretary of Defense and his deputies, the Secretaries of the three Armed Services, and all the military Chiefs of Staff. The central figure was a young man who but a few years before had come to the United States States as an immigrant.^{[253]}

A full list of consultancies given by von Neumann is as follows.^{[254]}^{[255]}^{[256]}

Gian-Carlo Rota wrote in his famously controversial book, *Indiscrete Thoughts*, that von Neumann was a lonely man that had trouble relating to others except on a strictly formal level.^{[257]} Françoise Ulam described how she never saw von Neumann in anything but a formal suit and tie,^{[258]} His daughter wrote in her memoirs that she believed her father was motivated by two key convictions, one, that every person had the responsibility to make full use of their intellectual capacity, and two, that there is a critical importance of an environment of political freedom in order to pursue the first convinction. She added that he "enjoyed the good life, liked to live well, and counted a number of celebrities among his friends and colleagues". He was also very concerned with his legacy, in two aspects, the first being the durability of his intellectual contributions to the world,^{[259]} and the secondly the life of his daughter. His brother, Nicholas noted that John tended to take a statistical view of the world, and that characterized many of his views on the world.^{[260]} His encyclopedic knowledge on history did not help him in this point of view, nor did his work in game theory. On the other hand, Stan Ulam described his warmth this way, "Quite independently of his liking for abstract wit, he had a strong appreciation (one might say almost a hunger) for the more earthy type of comedy and humor". He delighted in gossip and dirty jokes. Conversations with friends on scientific topics could go on for hours without respite, never being a shortage of things to discuss, even when leaving von Neumann's speciality in mathematics.^{[261]} Von Neumann was not a quiet person either, he enjoyed going to and hosting parties in his home, Churchill Eisenhart recalls in an interview that von Neumann could attend parties until the early hours of the morning, then the next day right at 8:30 could be there on time and deliver clear, lucid lectures. Graduate students would try to copy von Neumann in his ways, however they did not have any success.^{[262]}

He was also known for always being happy to provide others with scientific and mathematical advice, even when the recipient did not later credit him.^{[263]} However, he did not particularly like it when he felt others were challenging him and his brilliance, being a very competitive person.^{[264]}^{[265]} A story went at the Aberdeen Proving Ground how a young scientist had pre-prepared a complicated expression with solutions for several cases. When von Neumann came to visit, he asked him evaluate them, and for each case would give his already calculated answer just before Johnny did. By the time they came to the third case it was too much for him and he was upset until the joker confessed.^{[266]}

In addition to his speed in mathematics, he was also a quick speaker, with Banesh Hoffmann noting that it made it very difficult to take notes, even in shorthand.^{[267]} Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing strongly on technical ones. In addition, he maintained his knowledge of languages he learnt in his youth, becoming some what of a linguist. He knew Hungarian, French, German and English fluently, and maintained at least conversational level Italian, Yiddish, Ancient Latin and Greek. His Spanish was less perfect, but once on a trip to Mexico he tried to create his own "neo-Castilian" mix of English and Spanish.^{[268]}

Rota, in describing von Neumann's relationship with his friend Stanislaw Ulam, wrote that von Neumann had "deep-seated and recurring self-doubts". However, he had an "incomparably stronger technique", despite describing Ulam as the more creative mathematician.^{[257]} Ulam, in his obituary of von Neumann, likewise wrote that von Neumann "lacked a certain self-confidence", particularly where he felt others possessed qualities that he himself did not have, noting in his view those qualities were "relatively simple-minded powers of intuition of new truths, or the gift for a seemingly irrational perception of proofs or formulation of new theorems." He continued on in a more positive note, describing how he was adept in dimensional estimates and did algebraical or numerical computations in his head without the need for pencil and paper, often impressing physicists who needed the help of physical utensils. His impression of the way von Neumann thought was that he did not visualise things physically, instead treated properties of objects as some logical consequence of an underlying fundamental physical assumption.^{[269]} Albert Tucker described von Neumann's overall interest in things as problem oriented, not even that, but as he "would deal with the point that came up as a thing by itself."^{[270]}

Herman Goldstine compared his lectures to being on glass, smooth and lucid. You would sit down and listen to them and not even feel the need to write down notes because everything was so clear and obvious, however once one would come home and try understand the subject, you would suddenly realize it was not so easy. By comparison, Goldstine thought his scientific articles were written in a much harsher manner, and with much less insight.^{[263]} Another person who attended his lectures, Albert Tucker, described his lecturing as "terribly quick" and said that people often had to ask von Neumann questions in order to slow him down so they could think through the ideas he was going through, even if his presentation was clear they would still be thinking of the previous idea when von Neumann moved on to the next one. Von Neumann knew about this and was grateful for the assistance of his audience in telling him when he was going too quickly.^{[270]} Halmos described his lectures as "dazzling", with his speech clear, rapid, precise and all encompasing. He would cover all approaches to the subject he was speaking on and relate them to each other. Like Goldstine, he also described how everything seemed "so easy and natural" in lectures and a puzzled feeling once one tried to think over it at home.^{[271]}

His work habits were rather methodical, after waking up and having breakfast at the Nassau Club, he would visit the Institute for Advanced Study and begin work for the day. He would continue working for the entirety of the day, including after going home at five. Even if he was entertaining guests or hosting a party he would still spend some time in his work room working away, still following the conversation in the other room where guests were. Although he went to bed at a reasonable time he would awaken late in the night, two or three in the morning by which time his brain had thought through problems he had in the previous day and begin working again and writing things down. He placed great importance on writing down ideas he had in detail.^{[272]}

Goldstine also writes of many quirks of intuition von Neumann had. One such quirk was that one time von Neumann had asked to review an old paper he had not published because he believed there was an error there yet he could not find it. After Goldstine found it, he exclaimed, "Damn it, of course. There is some instinct that kept me from publishing that paper and it must have been a realization that I had a mistake somewhere in it, but I just never knew where it was." Another one was his ability to lecture off old material many years after he had originally given it, Goldstine's example was based on material von Neumann had written in German but was now lecturing on in English, with Goldstine noting that the lecture was almost word for word, symbol for symbol the same. A final example Goldstine writes about was that one time von Neumann had difficulty proving something related to the bounds of eigenvalues, and some time later Goldstine saw in a paper in the Math Reviews where someone had proved a related theorem and described the theorem to von Neumann, who was then able to come to the blackboard and write down a proof. Goldstine says that just knowing that a proof was possible allowed von Neumann to see how to write it down even when previously he had difficulty.^{[273]} Likewise when he had difficulties he would not labor on and struggle with them as soon as he found them, instead he would go home and sleep on it and come back later with a solution.^{[274]}

Von Neumann was asked to write an essay for the layman describing what mathematics is. He explained that mathematics straddles the world between the empirical and logical, arguing that geometry was originally empirical, but Euclid constructed a logical, deductive theory. However, he argued, that there is always the danger of straying too far from the real world and becoming irrelevant sophistry.^{[275]}^{[276]}^{[277]}

At times he could be ignorant of the standard mathematical literature, it would at times be easier to rederive basic information he needed rather than chase references. He did not 'write down' to a specific audience, but rather he wrote it exactly as he saw it. Although he did spend time preparing for lectures, often it was just before he was to present them, and he rarely used notes, instead jotting down points of what he would discuss and how long he would spend on it.^{[271]}

Nobel Laureate Hans Bethe said "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man",^{[24]} and later Bethe wrote that "[von Neumann's] brain indicated a new species, an evolution beyond man".^{[281]} Seeing von Neumann's mind at work, Eugene Wigner wrote, "one had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch."^{[282]} Paul Halmos states that "von Neumann's speed was awe-inspiring."^{[266]} Israel Halperin said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car."^{[283]} Edward Teller admitted that he "never could keep up with him".^{[284]} Teller also said "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."^{[285]} Peter Lax wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics".^{[286]}

When George Dantzig brought von Neumann an unsolved problem in linear programming "as I would to an ordinary mortal", on which there had been no published literature, he was astonished when von Neumann said "Oh, that!", before offhandedly giving a lecture of over an hour, explaining how to solve the problem using the hitherto unconceived theory of duality.^{[287]}

Lothar Wolfgang Nordheim described von Neumann as the "fastest mind I ever met",^{[288]} and Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius."^{[289]} George Pólya, whose lectures at ETH Zürich von Neumann attended as a student, said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."^{[290]} Eugene Wigner writes: "'Jancsi,' I might say, 'Is angular momentum always an integer of *h?* ' He would return a day later with a decisive answer: 'Yes, if all particles are at rest.'... We were all in awe of Jancsi von Neumann".^{[291]} Enrico Fermi told physicist Herbert L. Anderson: "You know, Herb, Johnny can do calculations in his head ten times as fast as I can! And I can do them ten times as fast as you can, Herb, so you can see how impressive Johnny is!"^{[292]}

Halmos recounts a story told by Nicholas Metropolis, concerning the speed of von Neumann's calculations, when somebody asked von Neumann to solve the famous fly puzzle:^{[293]}

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 mph. At the same time a fly that travels at a steady 15 mph starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover? The slow way to find the answer is to calculate what distance the fly covers on the first, southbound, leg of the trip, then on the second, northbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained.

The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles.

When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann, "All I did was sum the geometric series."^{[294]}

Eugene Wigner told a similar story, only with a swallow instead of a fly, and says it was Max Born who posed the question to von Neumann in the 1920s.^{[295]}

Similarly, when the first computers he was helping develop were completed, simple tests like "what is the lowest power of 2 that has the number 7 in the fourth position from the end?" were conducted to ensure their accuracy. For modern computers this would take only a fraction of a second but for the first computers Johnny would race against them in calculation, and win.^{[266]}

Accolades and anecdotes were not limited to those from the physical or mathematical sciences either, neurophysiologist Leon Harmon, described him in a similar manner, "Von Neumann was a true genius, the only one I’ve ever known. I’ve met Einstein and Oppenheimer and Teller and—who’s the mad genius from MIT? I don’t mean McCulloch, but a mathematician. Anyway, a whole bunch of those other guys. Von Neumann was the only genius I ever met. The others were supersmart .... And great prima donnas. But von Neumann’s mind was all-encompassing. He could solve problems in any domain. ... And his mind was always working, always restless."^{[296]}

Even for writer Arthur Koestler, who was not an academic, von Neumann was "one of the few people for whom Koestler entertained not only respect but reverence, and he shared Koestler’s Central European addiction to abstruse philosophical discussions, political debate, and dirty jokes. The two of them derived considerable pleasure from discussing the state of American civilization (was it in crisis or simply at the stage of adolescence?), the likely future of Europe (would there be war?), free will versus determinism, and the definition of pregnancy (“the uterus taking seriously what was pointed at it in fun”)."^{[297]}

Von Neumann was also noted for his eidetic memory, particularly for symbols. Herman Goldstine wrote:

One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how *A Tale of Two Cities* started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.^{[298]}

Von Neumann was reportedly able to memorize the pages of telephone directories. He entertained friends by asking them to randomly call out page numbers; he then recited the names, addresses and numbers therein.^{[24]}^{[299]}

"It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei in *John von Neumann: Selected Letters*.^{[300]} James Glimm wrote: "he is regarded as one of the giants of modern mathematics".^{[301]} The mathematician Jean Dieudonné said that von Neumann "may have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions",^{[3]} while Peter Lax described him as possessing the "most scintillating intellect of this century".^{[302]} In the foreword of Miklós Rédei's *Selected Letters*, Peter Lax wrote, "To gain a measure of von Neumann's achievements, consider that had he lived a normal span of years, he would certainly have been a recipient of a Nobel Prize in economics. And if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too. So the writer of these letters should be thought of as a triple Nobel laureate or, possibly, a 3+1⁄2-fold winner, for his work in physics, in particular, quantum mechanics".^{[303]} Rota writes that "he was the first to have a vision of the boundless possibilities of computing, and he had the resolve to gather the considerable intellectual and engineering resources that led to the construction of the first large computer" and consequently that "No other mathematician in this century has had as deep and lasting an influence on the course of civilization."^{[304]}

Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: "Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods." He went on to explain that the three methods were:

Eugene Wigner wrote that "Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique."^{[306]} Likewise Halmos noted that while von Neumann knew lots of mathematics, the most notable gaps were in algebraic topology and number theory, describing a story of how von Neumann once was walking by and saw something on the blackboard he didn't understand. Upon asking Halmos told him it was just the usual identification for a torus. While elementary even for modern graduate students this kind of work never crossed his path and thus he didn't know it.^{[307]}

One time he admitted to Herman Goldstine that he had no facility at all in topology and he was never comfortable with it, with Goldstine later bringing this up when comparing him to Hermann Weyl, whom he thought was deeper and broader than von Neumann.^{[274]} Similarly Albert Tucker said he never saw von Neumann work on anything he would call "topological" and described how once von Neumann was giving a proof of a topological theorem, which he thought, while ingenious, was the kind of proof an analyst would give rather than someone who worked on combinatorial topology.^{[270]}

Towards the end of his life he deplored to Ulam the fact that it no longer felt possible for anyone to have more than passing knowledge of one-third of the field of pure mathematics.^{[308]} In fact in the early 1940s Ulam himself concocted for him at his suggestion a doctoral style examination in various fields in order to find weaknesses in his knowledge. He did find them, with von Neumann being unable to answer satisfactorily a question each in differential geometry, number theory, and algebra. "This by the way may also tend to show that doctoral exams have little permanent meaning" was his conclusion. However while Weyl turned down an offer to write a history of mathematics of the 20th century, arguing that no one person could do it, Ulam thought Johnny could have aspired to do so.^{[309]}

In his biography of von Neumann, Salomon Bochner describes how much of von Neumann's works in pure mathematics involved finite and infinite dimensional vector spaces in one way or another, which at the time, covered much of the total area of mathematics. However he pointed out this still did not cover an important part of the mathematical landscape, in particular, anything that involved geometry "in the global sense", topics such as topology, differential geometry and harmonic integrals, algebraic geometry and other such fields. In these fields he said von Neumann worked on rarely, and had very little affinity for it in his thinking.^{[106]}

Likewise Jean Dieudonné noted in his biographical article that while he had an encyclopedic background, his range in pure mathematics was not as wide as Poincaré, Hilbert or even Weyl. His specific genius was in analysis and combinatorics, with combinatorics being understood in a very wide sense that described his ability to organize and axiomize complex works a priori that previously seemed to have little connection with mathematics. His style in analysis was not of the traditional English or French schools but rather of the German one, where analysis is based extensively on foundations in linear algebra and general topology. As with Bochner, he noted von Neumann never did significant work in number theory, algebraic topology, algebraic geometry or differential geometry. However, for his limits in pure mathematics he made up for in applied mathematics, where his work certainly equalled that of legendary mathematicians such as Gauss, Cauchy or Poincaré. Dieudonné notes that during the 1930s when von Neumann's work in pure mathematicians was at its peak, there was hardly an important area he didn't have at least passing acquaintance with.^{[96]}

A list of the following awards and honors was drawn from various biographic statements given by von Neumann.^{[322]}^{[323]}^{[256]}

Collections of von Neumann's published works can be found on and . A full list of his works as of 1995 can be found in .