 John von Neumann

"Von Neumann" redirects here. For other uses, see Von Neumann (disambiguation).The native form of this personal name is Neumann János. This article uses the Western name order.
John von Neumann (English pronunciation: /vɒn ˈnɔɪmən/) (December 28, 1903 – February 8, 1957) was a HungarianAmerican mathematician and polymath who made major contributions to a vast number of fields,^{[1]} including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis, hydrodynamics, and statistics, as well as many other mathematical fields. He is generally regarded as one of the greatest mathematicians in modern history.^{[2]} The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians",^{[3]} while Peter Lax described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century,^{[4]} and Hans Bethe stated "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man".^{[5]} Even in Budapest, in the time that produced geniuses like Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913), his brilliance stood out.^{[6]}
Von Neumann was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory^{[1]}^{[7]} and the concepts of cellular automata,^{[1]} the universal constructor, and the digital computer. Von Neumann's mathematical analysis of the structure of selfreplication preceded the discovery of the structure of DNA.^{[8]} In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "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." Along with Teller and Stanisław Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.
Biography
The eldest of three brothers, von Neumann was born Neumann János Lajos (Hungarian pronunciation: [ˈnojmɒn ˈjaːnoʃ ˈlɒjoʃ]; in Hungarian the family name comes first) on December 28, 1903 in Budapest, AustroHungarian Empire, to wealthy Jewish parents.^{[9]}^{[10]}^{[11]} His father, Neumann Miksa (Max Neumann) was a banker, who held a doctorate in law. He had moved to Budapest from Pécs at the end of 1880s. His mother was Kann Margit (Margaret Kann).^{[12]} In 1913, his father was elevated to the nobility for his service to the AustroHungarian empire by Emperor Franz Josef. The Neumann family thus acquiring the hereditary title margittai, Neumann János became margittai Neumann János (John Neumann of Margitta), which he later changed to the German Johann von Neumann.
János, nicknamed "Jancsi" (Johnny), was a child prodigy in the areas of language, memorization, and mathematics. By the age of six, he could exchange jokes in Classical Greek, memorize telephone directories on sight, and display prodigious mental calculation abilities.^{[13]} As a 6 year old, he would astonish onlookers by instantly dividing two 8digit numbers in his head, producing the answers to a decimal point.^{[14]} By the age of 8, he had attained mastery in calculus.^{[15]} He entered the Germanspeaking Lutheran high school Fasori Evangelikus Gimnázium in Budapest in 1911. Although his father insisted he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him 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. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears.^{[16]} Szegő subsequently visited the von Neumann house twice a week to tutor the child prodigy. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out with his father's stationary, are still on display at the von Neumann archive in Budapest.^{[17]} 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 Cantor's.^{[18]} He received his Ph.D. in mathematics (with minors in experimental physics and chemistry) from Pázmány Péter University in Budapest at the age of 22.^{[1]} He simultaneously earned a diploma in chemical engineering from the ETH Zurich in Switzerland^{[1]} at the behest of his father, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavour than mathematics.
Between 1926 and 1930, he taught as a Privatdozent at the University of Berlin, the youngest in its history. By the end of year 1927 Neumann had published twelve major papers in mathematics, and by the end of year 1929, thirtytwo papers, at a rate of nearly one major paper per month.^{[12]}
In 1930, Von Neumann was invited to Princeton University, New Jersey, and, subsequently, was one of the first four people selected for the faculty of the Institute for Advanced Study (two of the others being Albert Einstein and Kurt Gödel), where he remained a mathematics professor from its formation in 1933 until his death. His father, Max von Neumann had died in 1929. But his mother, and his brothers followed John to the United States. He anglicized his first name to John, keeping the Austrianaristocratic surname of von Neumann.
In 1937, von Neumann became a naturalized citizen of the U.S. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.
Von Neumann married twice. He married Mariette Kövesi in 1930, just prior to emigrating to the United States. They had one daughter (von Neumann's only child), Marina, who is now a distinguished professor of international trade and public policy at the University of Michigan. The couple divorced in 1937. In 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II. The von Neumanns were very active socially within the Princeton academic community.
In 1955, von Neumann was diagnosed with what was either bone or pancreatic cancer.^{[19]} A von Neumann biographer Norman Macrae has speculated: "It is plausible that in 1955 the thenfiftyoneyearold Johnny's cancer sprang from his attendance at the 1946 Bikini nuclear tests."^{[20]} Von Neumann died a year and a half later. While at Walter Reed Hospital in Washington, D.C., he invited a Roman Catholic priest, Father Anselm Strittmatter, O.S.B., to visit him for consultation. This move shocked some of von Neumann's friends in view of his reputation as an agnostic.^{[21]} Von Neumann, however, is reported to have said in explanation that Pascal had a point, referring to Pascal's wager.^{[22]} Father Strittmatter administered the last sacraments to him.^{[23]} He died under military security lest he reveal military secrets while heavily medicated. von Neumann was buried at Princeton Cemetery in Princeton, Mercer County, New Jersey.^{[24]} On his death bed, he entertained his brother with a word for word memory of Goethe's Faust.^{[25]}
Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital and later published in book form as The Computer and the Brain, gives an indication of the direction of his interests at the time of his death.
Set theory
The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigor 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 geometry (thanks to David Hilbert). At the beginning of the twentieth 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).
The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later (by Ernst Zermelo and Abraham Fraenkel). Zermelo and Fraenkel provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics: But they did not explicitly exclude the possibility of the existence of a set that belong 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.
The axiom of foundation established 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, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession (hence excluding 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 later became an essential instrument in set theory.
The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo/Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.
With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress 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 which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time. But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency.^{[26]} It is precisely this consequence which has attracted the most attention, even if Gödel originally considered it only a curiosity, and had derived it independently anyway (it is for this reason that the result is called Gödel's second theorem, without mention of von Neumann.)
Geometry
Von Neumann founded the field of continuous geometry. It followed his pathbreaking 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]. 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.
Measure theory
See also: Lifting theoryIn a series of famous papers, Von Neumann made spectacular contributions to measure theory.^{[27]} The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially grouptheoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space." In a number of von Neumann's papers, the methods of argument he employed are considered 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 (anticipating his later work on almost periodic functions).
In the 1936 paper on analytic measure theory, von Neumann used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups.^{[28]}^{[29]}
Ergodic theory
Von Neumann made foundational contributions to ergodic theory, in a series of articles published in 1932, which have attained legendary status in mathematics.^{[30]} Of the 1932 papers on ergodic theory, Paul Halmos writes that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality".^{[31]} By then von Neumann had already written his famous articles on operator theory, and the application of this work was instrumental in the Von Neumann mean ergodic theorem.^{[32]}
Operator Theory
Main article: Von Neumann algebraVon Neumann introduced the study of rings of operators, through the von Neumann algebras.^{[33]} A von Neumann algebra is a *algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator.
The Von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
The direct integral was introduced in 1949 by John von Neumann in one of the final papers in the series On Rings of Operators. One of von Neumann's arguments was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.
Probability Theory
Von Neumann's work on measure theory and operators led him to introduce a number of concepts in probability theory: for example, the standard probability space.
Lattice theory
Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor".^{[34]} Von Neumann worked on lattice theory between 193739. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "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." 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 rightideals 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 writingtable in a bathrobe."^{[35]}
Mathematical formulation of quantum mechanics
See also: von Neumann entropy, Density matrix, Quantum mutual information, von Neumann measurement scheme, and Wave function collapseQuantum mechanics
Introduction
Mathematical formulationsFundamental conceptsExperimentsDoubleslit · Davisson–Germer
Stern–Gerlach · Bell's inequality
Popper · Schrödinger's cat
Elitzur–Vaidman bomb tester
Quantum eraser
Delayed choice quantum eraser
Wheeler's delayed choiceFormulationsEquationsInterpretationsde Broglie–Bohm
Consciousnesscaused
Consistent histories · Copenhagen
Ensemble · Hidden variables
Manyworlds · Objective collapse
Pondicherry · Quantum logic
Relational · Stochastic
TransactionalScientistsBell · Bohm · Bohr · Born · Bose
de Broglie · Dirac · Ehrenfest
Everett · Feynman · Heisenberg
Jordan · Kramers · von Neumann
Pauli · Planck · Schrödinger
Sommerfeld · Wien · WignerVon Neumann was the first to rigorously establish a mathematical framework for quantum mechanics with his work Mathematische Grundlagen der Quantenmechanik.
After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a socalled Hilbert space, analogous to the 6N dimension (N is the number of particles, 3 general coordinate and 3 canonical momentum for each) phase space of classical mechanics but with infinitely many dimensions (corresponding to the infinitely many possible states of the system) instead: the traditional physical quantities (e.g., position and momentum) could therefore be represented as particular linear operators operating in these spaces. The physics of quantum mechanics was thereby reduced to the mathematics of the linear Hermitian operators on Hilbert spaces.
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 noncommutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger, and culminated in his 1932 book Mathematische Grundlagen der Quantenmechanik.
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs. nondeterminism and in the book he presented a proof according to which quantum mechanics could not possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics. However, in 1966 it was discovered that this proof contained a conceptual error (see the article on John Stewart Bell for more information). The proof nonetheless inaugurated a line of research that ultimately led, through the work of Bell in 1964 on Bell's Theorem, and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics requires a notion of reality substantially different from that of classical physics.
In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the socalled 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 (although this view was accepted by Eugene Wigner, it never gained acceptance amongst the majority of physicists).^{[36]}
Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formalism of problems in quantum mechanics which underlies the majority of approaches and 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
Quantum logics
In a famous paper of 1936, the first work ever to introduce quantum logics,^{[37]} von Neumann first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work. But in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters which are polarized perpendicularly (e.g. one horizontally and the other vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession. But if the third filter is added in between the other two, the photons will indeed pass through. And this experimental fact is translatable into logic as the noncommutativity of conjunction . It was also demonstrated that the laws of distribution of classical logic, and , are not valid for quantum theory. The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically determinate, while each of its members are necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semiintegral spin (angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g. x and y) results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the x direction is positive." By the principle of indeterminacy, the value of the spin in the direction y will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of y is positive" nor the proposition "the spin in the direction of y is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of y is positive or the spin in the direction of y is negative" must be true for ɸ. In the case of distribution, it is therefore possible to have a situation in which , while .
Von Neumann proposes to replace classical logics, with a logic constructed in orthomodular lattices, (isomorphic to the lattice of subspaces of the Hilbert space of the physical system).^{[38]}
Game theory
Von Neumann founded the field of game theory as a mathematical discipline. Von Neumann's proved his minimax theorem in 1928. This theorem establishes that in zerosum 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 (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy which will result in the minimization of his maximum loss. 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). Another result he proved during his German period was the nonexistence of a static equilibrium. An equilibrium can only exist in an expanding economy. Paul Samuelson edited an anniversary volume dedicated to this short German paper in 1972 and stated in the introduction that von Neumann was the only mathematician ever to make a significant contribution to economic theory.
Von Neumann 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). The public interest in this work was such that The New York Times ran a frontpage story. In this book, von Neumann declared that economic theory needed to use functional analytic methods, especially convex sets and topological fixed point theorem, rather than the traditional differential calculus, because the maximum–operator did not preserve differentiable functions. Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, nondifferentiability, and vector lattices. Von Neumann's functionalanalytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex set, and fixedpoint theory—have been the primary tools of mathematical economics ever since.^{[39]} Von Neumann was also the inventor of the method of proof, used in game theory, known as backward induction (which he first published in 1944 in the book coauthored with Morgenstern, Theory of Games and Economic Behaviour).^{[40]}
Mathematical economics
Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point theorem. 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
 p^{T} (A − λ B) q = 0,
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. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.^{[41]}^{[42]}^{[43]} Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices.^{[44]} The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.^{[45]}^{[46]}^{[47]} This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixedpoint theorems, linear inequalities, complementary slackness, and saddlepoint duality.
The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of Nobel prizes in 1972 to Kenneth Arrow, in 1983 to Gérard Debreu, and in 1994 to John Nash who used fixed point theorems to establish equilibria for noncooperative games and for bargaining problems in his Ph.D thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans, Leonid Kantorovich, Wassily Leontief, Paul Samuelson, Robert Dorfman, Robert Solow, and Leonid Hurwicz.
Linear programming
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, after George B. Dantzig described his work in a few minutes, when an impatient von Neumann asked him to get to the point. Then, Dantzig listened dumbfounded while von Neumann provided an hour lecture on convex sets, fixedpoint theory, and duality, conjecturing the equivalence between matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of 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 zerovector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interiorpoint method of linear programming.^{[48]}
Mathematical statistics
Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of mean square successive difference to the variance for normally distributed variables.^{[49]} This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic^{[50]} for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression. Subsequently, John Denis Sargan and Alok Bhargava^{[51]} 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. Von Neumann's contributions to statistics have had a major impact on econometric methodology.
Nuclear weapons
Beginning in the late 1930s, von Neumann began to take more of an interest in applied (as opposed to pure) mathematics. In particular, he developed an expertise in explosions—phenomena which are difficult to model mathematically. 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 in Los Alamos, New Mexico.^{[1]}
Von Neumann's principal contribution to the atomic bomb itself was in the concept and design of the explosive lenses needed to compress the plutonium core of the Trinity test device and 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" meaning compression. When it turned out that there would not be enough U235 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 plutonium239 that was available from the Hanford site. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, 5% was achieved by George Kistiakowsky, and the construction of the Trinity bomb was completed in July 1944.
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.^{[52]}
Beginning in the spring of 1945, along with four other scientists and various military personnel, von Neumann was included in the target selection committee 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.^{[53]} The cultural capital Kyoto, which had been spared the firebombing inflicted upon militarily significant target cities like Tokyo in World War II, was von Neumann's first choice, a selection seconded by Manhattan Project leader General Leslie Groves. However, this target was dismissed by Secretary of War Henry Stimson.^{[54]}
On July 16, 1945, with numerous other Los Alamos personnel, von Neumann was an eyewitness to the first atomic bomb blast, conducted as a test of the implosion method device, 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, but Enrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of tornup 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.^{[52]}
After the war, Robert Oppenheimer remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."
Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained the hydrogen bomb project. He then 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 a thermonuclear reaction.^{[55]} The Fuchs–von Neumann patent used radiation implosion, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design. Their work was, however, incorporated into the "George" shot of Operation Greenhouse, which was instructive in testing out concepts that went into the final design. The Fuchs–von Neumann work was passed on, by Fuchs, to the USSR as part of his nuclear espionage, but it was not used in the Soviet's 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."^{[56]}
The ICBM Committee
In 1955, von Neumann became a commissioner of the United States Atomic Energy Program. Shortly before his death, when he was already quite ill, von Neumann headed the top secret von Neumann ICBM committee. 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 indeed formidable, they could be overcome in time. The SM65 Atlas passed its first fully functional test in 1959, two years after his death.
MAD
John von Neumann is credited with the strategy of Mutually assured destruction, providing the deliberately humorous acronym, MAD. (Other humorous acronyms coined by von Neumann include his computer, the Mathematical Analyzer, Numerical Integrator, and Computer  or MANIAC).
Computer science
Von Neumann was a founding figure in computer science.^{[58]} 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 complicated problems to be approximated using random numbers. Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middlesquare 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, and also noted that when it went awry it did so obviously, unlike methods which could be subtly incorrect.
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, which was widely distributed, 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 to this day the basis of modern computer design, unlike the earliest computers that were 'programmed' by altering the electronic circuitry. Although the singlememory, stored program architecture is commonly called von Neumann architecture as a result of von Neumann's paper, the architecture's description was based on the work of J. Presper Eckert and John William Mauchly, inventors of the ENIAC at the University of Pennsylvania.^{[59]}
Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953.^{[60]} However, the theory could not be implemented until advances in computing of the 1960s.^{[61]} ^{[62]}
Von Neumann also created the field of cellular automata without the aid of computers, constructing the first selfreplicating automata with pencil and graph paper. The concept of a universal constructor was fleshed out in his posthumous work Theory of Self Reproducing Automata.^{[63]} Von Neumann proved that the most effective way of performing largescale mining operations such as mining an entire moon or asteroid belt would be by using selfreplicating machines, taking advantage of their exponential growth.
Donald Knuth cites von Neumann as 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 together.^{[64]}
His algorithm for simulating a fair coin with a biased coin^{[65]} is used in the "software whitening" stage of some hardware random number generators.
Fluid dynamics
Von Neumann made fundamental contributions in exploration of problems in numerical hydrodynamics. For example, with R. D. Richtmyer he developed an algorithm defining artificial viscosity that improved the understanding of shock waves. It is possible that we would not understand much of astrophysics, and might not have highly developed jet and rocket engines without the work of von Neumann. A problem was that 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.
Other well known contributions to fluid dynamics included the classic flow solution to blast waves^{[66]}, and the codiscovery of the ZND detonation model of explosives.^{[67]}
Politics and social affairs
Von Neumann obtained at the age of 29 one of the first five professorships at the new Institute for Advanced Study in Princeton, New Jersey (another had gone to Albert Einstein). He was a frequent consultant for the Central Intelligence Agency, the United States Army, the RAND Corporation, Standard Oil, General Electric, IBM, and others.
Throughout his life von Neumann had a respect and admiration for business and government leaders; something which was often at variance with the inclinations of his scientific colleagues.^{[68]} Von Neumann entered government service (Manhattan Project) primarily because he felt that, if freedom and civilization were to survive, it would have to be because the U.S. would triumph over totalitarianism from the right (Nazism and Fascism) and totalitarianism from the left (Soviet Communism).^{[69]}
As President of the Von Neumann Committee for Missiles, and later as a member of the United States Atomic Energy Commission, from 1953 until his death in 1957, he was influential in setting U.S. scientific and military policy. Through his committee, he developed various scenarios of nuclear proliferation, the development of intercontinental and submarine missiles with atomic warheads, and the controversial strategic equilibrium called mutual assured destruction. During a Senate committee hearing he described his political ideology as "violently anticommunist, and much more militaristic than the norm". He was quoted in 1950 remarking, "If you say why not bomb [Russia] tomorrow, I say, why not today. If you say today at five o’clock, I say why not one o’clock?".^{[70]} As a result, he partly inspired the character of 'Doctor Strangelove' in Doctor Strangelove.
Weather control
Von Neumann's team performed the world's first numerical weather forecasts on the ENIAC computer; von Neumann published the paper Numerical Integration of the Barotropic Vorticity Equation in 1950.^{[71]} Von Neumann's interest in 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), thereby raising global temperatures.^{[72]}
Personality
Von Neumann had a wide range of cultural interests. Since the age of six, von Neumann had been fluent in Latin and ancient Greek, and he held a lifelong passion for ancient history, being renowned for his prodigious historical knowledge. A professor of Byzantine history once reported that von Neumann had greater expertise on Byzantine history than he did.^{[73]} Von Neumann took great care over his clothing, and would always wear formal suits, once riding down the Grand Canyon astride a mule in a threepiece pinstripe.^{[69]} He was extremely sociable and, during his first marriage, he enjoyed throwing large parties at his home in Princeton,^{[74]} occasionally twice a week.^{[75]} His white clapboard house at 26 Westcott Road was one of the largest in Princeton.^{[76]} Despite being a notoriously bad driver, he nonetheless 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.^{[77]} He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.^{[78]}
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "offcolor" humor (especially limericks).^{[23]} At Princeton he received complaints for regularly playing extremely loud German marching music on his gramophone, which distracted those in neighbouring offices, including Einstein, from their work.^{[79]} Von Neumann's closest friend in America was the Polish mathematician Stanislaw Ulam. A later friend of Ulam's, GianCarlo Rota writes: "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 hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend.^{[80]}
Cognitive and mnemonic abilities
Von Neumann's ability to instantaneously perform complex operations in his head stunned other mathematicians.^{[81]} Eugene Wigner wrote that, seeing von Neumann's mind at work, "one had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch."^{[82]} Paul Halmos states that "von Neumann's speed was aweinspiring."^{[83]} Israel Halperin said: "Keeping up with him was... impossible. The feeling was you were on a tricycle chasing a racing car."^{[84]} Edward Teller wrote that von Neumann effortlessly outdid anybody he ever met,^{[85]} and said "I never could keep up with him".^{[86]} Lothar Wolfgang Nordheim described von Neumann as the "fastest mind I ever met",^{[87]} and Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius."^{[88]} George Pólya, whose lectures at ETH Zurich 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. He'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."^{[89]} 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:
Two bicyclists start twenty 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, northbound, leg of the trip, then on the second, southbound, 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 infinite series."^{[90]}Von Neumann had a photographic memory.^{[91]} Herman Goldstine writes: "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 dimunition in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how The 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."^{[92]}
Honors
 The John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences (INFORMS, previously TIMSORSA) is awarded annually to an individual (or group) who have made fundamental and sustained contributions to theory in operations research and the management sciences.
 The IEEE John von Neumann Medal is awarded annually by the IEEE "for outstanding achievements in computerrelated science and technology."
 The John von Neumann Lecture is given annually at the Society for Industrial and Applied Mathematics (SIAM) by a researcher who has contributed to applied mathematics, and the chosen lecturer is also awarded a monetary prize.
 The crater Von Neumann on the Moon is named after him.
 The John von Neumann Computing Center in Princeton, New Jersey (40°20′55″N 74°35′32″W / 40.348695°N 74.592251°W) was named in his honour.
 The professional society of Hungarian computer scientists, John von Neumann Computer Society, is named after John von Neumann.^{[93]}
 On February 15, 1956, Neumann was presented with the Presidential Medal of Freedom by President Dwight Eisenhower.
 On May 4, 2005 the United States Postal Service issued the American Scientists commemorative postage stamp series, a set of four 37cent selfadhesive stamps in several configurations. The scientists depicted were John von Neumann, Barbara McClintock, Josiah Willard Gibbs, and Richard Feynman.
 The John von Neumann Award of the Rajk László College for Advanced Studies was named in his honour, and has been given every year since 1995 to professors who have made an outstanding contribution to the exact social sciences and through their work have strongly influenced the professional development and thinking of the members of the college.
Selected works
 1923. On the introduction of transfinite numbers, 346–54.
 1925. An axiomatization of set theory, 393–413.
 1932. Mathematical Foundations of Quantum Mechanics, Beyer, R. T., trans., Princeton Univ. Press. 1996 edition: ISBN 0691028931.
 1944. Theory of Games and Economic Behavior, with Morgenstern, O., Princeton Univ. Press. 2007 edition: ISBN 9780691130613.
 1945. First Draft of a Report on the EDVAC TheFirstDraft.pdf
 1963. Collected Works of John von Neumann, Taub, A. H., ed., Pergamon Press. ISBN 0080095666
 1966. Theory of SelfReproducing Automata, Burks, A. W., ed., Univ. of Illinois Press.
 von Neumann, John (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 9780691058931, MR0120174, http://books.google.com/books?id=onE5HncEHgC
 von Neumann, John (1981) [1937], Halperin, Israel, ed., "Continuous geometries with a transition probability", Memoirs of the American Mathematical Society 34 (252), ISSN 00659266, MR634656, http://books.google.com/books?id=ZPkVGr8NXugC
See also
 Stone–von Neumann theorem
 Von Neumann–Bernays–Gödel set theory
 Von Neumann algebra
 Von Neumann architecture
 Von Neumann bicommutant theorem
 Von Neumann conjecture
 Von Neumann entropy
 Von Neumann programming languages
 Von Neumann regular ring
 Von Neumann universal constructor
 Von Neumann universe
 Von Neumann's trace inequality
 Selfreplicating spacecraft
 PhD Students
 Donald B. Gillies, Ph.D. student^{[94]}
 Israel Halperin, Ph.D. student^{[94]}^{[95]}
Notes
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} Ed Regis (19921108). "Johnny Jiggles the Planet". The New York Times. http://query.nytimes.com/gst/fullpage.html?res=9E0CE7D91239F93BA35752C1A964958260. Retrieved 20080204.
 ^ Glimm, p. vii
 ^ Dictionary of Scientific Bibliography, ed. C. C. Gillispie, Scibners, 1981
 ^ Glimm, p. 7
 ^ Life Magazine, 25th February 1957, Passing of a Great Mind, by Clay Bair JR. pages 89104
 ^ Doran, p. 2
 ^ Nelson, David (2003). The Penguin Dictionary of Mathematics. London: Penguin. pp. 178–179. ISBN 0141010770.
 ^ Rocha, L.M., "Von Neumann and Natural Selection.", Lecture Notes of I585Biologically Inspired Computing Course, Indiana University, http://informatics.indiana.edu/rocha/ibic/pdfs/ibic_lecnotes_c6.pdf
 ^ Doran, p. 1
 ^ Nathan Myhrvold, "John von Neumann". Time, March 21, 1999. Accessed September 5, 2010
 ^ Clay Blair, Jr. "Passing of a Great Mind". Life, February 25, 1957; p. 104
 ^ ^{a} ^{b} Norman Macrae (June 2000). John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. AMS Bookstore. pp. 37–38. ISBN 9780821826768. http://books.google.com/books?id=OO_gSRheEC&pg=PA37. Retrieved 24 March 2011.
 ^ William Poundstone, Prisoner's dilemma (Oxford, 1993), introduction
 ^ Poundstone, William, Prisoner's Dilemma, New York: Doubleday 1992
 ^ Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394
 ^ Glimm, p. 5
 ^ John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, by Norman Macrae, American Mathematical Soc., 2000, page 70
 ^ Nasar, Sylvia, A Beautiful Mind, London 2001, page 81
 ^ While there is a general agreement that the initially discovered bone tumor was a secondary growth, sources differ as to the location of the primary cancer. While Macrae gives it as pancreatic, the Life magazine article says it was prostate.
 ^ Macrae, p. 231.
 ^ The question of whether or not von Neumann had formally converted to Catholicism upon his marriage to Mariette Kövesi (who was Catholic) is addressed in Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394. He was baptised Roman Catholic, but certainly was not a practicing member of that religion after his divorce.
 ^ Marion Ledwig. "The Rationality of Faith", citing Macrae, p. 379.
 ^ ^{a} ^{b} Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394
 ^ John von Neumann at Find a Grave[1]
 ^ Life Magazine, 25th February 1957, Passing of a Great Mind, by Clay Bair JR. pages 89104
 ^ John von Neumann (2005). Miklós Rédei. ed. John von Neumann: Selected letters. History of Mathematics. 27. American Mathematical Society. p. 123. ISBN 0821837761.
 ^ VON NEUMANN ON MEASURE AND ERGODIC THEORY. PAUL R. HALMOS, Bull. Amer. Math. Soc. Volume 64, Number 3, Part 2 (1958), 8694.
 ^ von Neumann, J. (1933), "Die Einfuhrung Analytischer Parameter in Topologischen Gruppen", Annals of Mathematics, 2 34 (1): 170–179
 ^ VON NEUMANN ON MEASURE AND ERGODIC THEORY. PAUL R. HALMOS, Bull. Amer. Math. Soc. Volume 64, Number 3, Part 2 (1958), 8694.
 ^ Two famous papers are below: von Neumann, John (1932), "Proof of the Quasiergodic Hypothesis", Proc Natl Acad Sci USA 18 (1): 70–82, Bibcode 1932PNAS...18...70N, doi:10.1073/pnas.18.1.70, PMC 1076162, PMID 16577432, http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1076162. von Neumann, John (1932), "Physical Applications of the Ergodic Hypothesis", Proc Natl Acad Sci USA 18 (3): 263–266, Bibcode 1932PNAS...18..263N, doi:10.1073/pnas.18.3.263, JSTOR 86260, PMC 1076204, PMID 16587674, http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1076204. Hopf, Eberhard (1939), "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung", Leipzig Ber. Verhandl. Sächs. Akad. Wiss. 91: 261–304.
 ^ VON NEUMANN ON MEASURE AND ERGODIC THEORY PAUL R. HALMOS, Bull. Amer. Math. Soc. Volume 64, Number 3, Part 2 (1958), 8694.
 ^ I: Functional Analysis : Volume 1 by Michael Reed, Barry Simon,Academic Press; REV edition (1980)
 ^ John von Neumann And The Theory Of Operator Algebras, D.Petz and M.R. Redi, in The Neumann compendium, World Scientific, 1995, page 163181
 ^ Garrett Birkhoff, VON NEUMANN AND LATTICE THEORY, John Von Neumann 19031957, J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 505
 ^ Garrett Birkhoff, VON NEUMANN AND LATTICE THEORY, John Von Neumann 19031957, J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 505
 ^ von Neumann, John. (1932/1955). Mathematical Foundations of Quantum Mechanics. Princeton: Princeton University Press. Translated by Robert T. Beyer.
 ^ The Many Valued and Nonmonotonic Turn in Logic, Dov M. Gabbay, John Woods, Elsevier, 2007, pages 205217
 ^ Philosophical Papers: Volume 3, Realism and Reason, Hilary Putnam, Cambridge University Press, 27 Dec 1985, page 263
 ^ Blume, Lawrence E. (2008c). "Convexity". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.0315. http://www.dictionaryofeconomics.com/article?id=pde2008_C000508.
 Blume, Lawrence E. (2008cp). "Convex programming". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.0314. http://www.dictionaryofeconomics.com/article?id=pde2008_C000348.
 Blume, Lawrence E. (2008d). "Duality". In Durlauf, Steven N.; Blume, Lawrence E. The New Palgrave Dictionary of Economics (Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.0411. http://www.dictionaryofeconomics.com/article?id=pde1987_X000626.
 Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In Arrow, Kenneth Joseph; Intriligator, Michael D. Handbook of mathematical economics, Volume I. Handbooks in economics. 1. Amsterdam: NorthHolland Publishing Co.. pp. 15–52. doi:10.1016/S15734382(81)010059. ISBN 0444861262. MR634800. http://www.sciencedirect.com/science/article/B7P5Y4FDF0FN5/2/613440787037f7f62d65a05172503737.
 MasColell, A. (1987). "Nonconvexity". In Eatwell, John; Milgate, Murray; Newman, Peter. The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. pp. 653–661. doi:10.1057/9780230226203.3173. http://www.econ.upf.edu/~mcolell/research/art_083b.pdf.
 Newman, Peter (1987c). "Convexity". In Eatwell, John; Milgate, Murray; Newman, Peter. The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. doi:10.1057/9780230226203.2282. http://www.dictionaryofeconomics.com/article?id=pde1987_X000453.
 Newman, Peter (1987d). "Duality". In Eatwell, John; Milgate, Murray; Newman, Peter. The New Palgrave: A Dictionary of Economics (first ed.). Palgrave Macmillan. doi:10.1057/9780230226203.2412. http://www.dictionaryofeconomics.com/article?id=pde1987_X000626.
 ^ John MacQuarrie. "Mathematics and Chess". School of Mathematics and Statistics, University of St Andrews, Scotland. http://wwwgroups.dcs.stand.ac.uk/~history/Projects/MacQuarrie/Chapters/Ch4.html. Retrieved 20071018. "Others claim he used a method of proof, known as 'backwards induction' that was not employed until 1953, by von Neumann and Morgenstern. Ken Binmore (1992) writes, Zermelo used this method way back in 1912 to analyze Chess. It requires starting from the end of the game and then working backwards to its beginning. (p.32)"
 ^ For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem
 A − λ I q = 0,
 ^ David Gale. The theory of linear economic models. McGraw–Hill, New York, 1960.
 ^ Morgenstern, Oskar; Thompson, Gerald L. (1976). Mathematical theory of expanding and contracting economies. Lexington Books. Lexington, Massachusetts: D. C. Heath and Company. pp. xviii+277. ISBN 0669000892.
 ^ Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0471982326.
 ^
 Rockafellar, R. Tyrrell. Monotone processes of convex and concave type. Memoirs of the American Mathematical Society. Providence, R.I.: American Mathematical Society. pp. i+74. ISBN 0821812777.
 Rockafellar, R. T. (1974). "Convex algebra and duality in dynamic models of production". In Josef Loz and Maria Loz. Mathematical models in economics (Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972). Amsterdam: NorthHolland and Polish Adademy of Sciences (PAN). pp. 351–378.
 Rockafellar, R. T. (1970 (Reprint 1997 as a Princeton classic in mathematics)). Convex analysis. Princeton, NJ: Princeton University Press. ISBN 0691080690.
 ^ Kenneth Arrow, Paul Samuelson, John Harsanyi, Sidney Afriat, Gerald L. Thompson, and Nicholas Kaldor. (1989). Mohammed Dore, Sukhamoy Chakravarty, Richard Goodwin. ed. John Von Neumann and modern economics. Oxford:Clarendon. p. 261.
 ^ Chapter 9.1 "The von Neumann growth model" (pages 277–299): Yinyu Ye. Interior point algorithms: Theory and analysis. Wiley. 1997.
 ^ George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. SpringerVerlag.
 ^ von Neumann, John. (1941). "Distribution of the ratio of the mean square successive difference to the variance". Annals of Mathematical Statistics, 12, 367–395. (JSTOR)
 ^ Durbin, J., and Watson, G. S. (1950) "Testing for Serial Correlation in Least Squares Regression, I." Biometrika 37, 409–428.
 ^ Sargan, J.D. and Alok Bhargava (1983). "Testing residuals from least squares regression for being generated by the Gaussian random walk". Econometrica, 51, p. 153–174.
 ^ ^{a} ^{b} Lillian Hoddeson ... . With contributions from Gordon Baym ...; "Lillian Hoddeson, Paul W. Henriksen, Roger A. Meade, Catherine Westfall (1993). Critical Assembly: A Technical History of Los Alamos during the Oppenheimer Years, 1943–1945. Cambridge, UK: Cambridge University Press. ISBN 0521441323.
 ^ Rhodes, Richard (1986). The Making of the Atomic Bomb. New York: Touchstone Simon & Schuster. ISBN 0684813785.
 ^ Groves, Leslie (1962). Now It Can Be Told: The Story of the Manhattan Project. New York: Da Capo. ISBN 0306801892.
 ^ Herken, pp. 171, 374
 ^ Bernstein, Jeremy (2010). "John von Neumann and Klaus Fuchs: an Unlikely Collaboration". Physics in Perspective 12: 36. Bibcode 2010PhP....12...36B. doi:10.1007/s0001600900011.
 ^ Pesavento, Umberto (1995), "An implementation of von Neumann's selfreproducing machine" (PDF), Artificial Life (MIT Press) 2 (4): 337–354, doi:10.1162/artl.1995.2.337, PMID 8942052, http://web.archive.org/web/20070418081628/http://dragonfly.tam.cornell.edu/~pesavent/pesavento_self_reproducing_machine.pdf
 ^ The Computer from Pascal to von Neumann, Princeton University Press, 1980, Herman Heine Goldstine, page 167178
 ^ The name for the architecture is discussed in John W. Mauchly and the Development of the ENIAC Computer, part of the online ENIAC museum, in Robert Slater's computer history book, Portraits in Silicon, and in Nancy Stern's book From ENIAC to UNIVAC.
 ^ von Neumann, J. (1963). "Probabilistic logics and the synthesis of reliable organisms from unreliable components". The Collected Works of John von Neumann. Macmillan. ISBN 9780393051698.
 ^ Petrovic, R.; Siljak, D. (1962). "Multiplication by means of coincidence". ACTES Proc. of 3rd Int. Analog Comp. Meeting.
 ^ Afuso, C. (1964), Quart. Tech. Prog. Rept., Department of Computer Science, University of Illinois, Urbana, Illinois
 ^ John von Neumann (1966). Arthur W. Burks. ed. Theory of SelfReproducing Automata. Urbana and London: Univ. of Illinois Press. ISBN 0598377980. PDF reprint
 ^ Knuth, Donald (1998). The Art of Computer Programming: Volume 3 Sorting and Searching. Boston: Addison–Wesley. p. 159. ISBN 0201896850.
 ^ von Neumann, John (1951). "Various techniques used in connection with random digits". National Bureau of Standards Applied Math Series 12: 36.
 ^ Neumann, John von, "The point source solution," John von Neumann. Collected Works, edited by A. J. Taub, Vol. 6 [Elmsford, N.Y.: Permagon Press, 1963], pages 219  237
 ^ von Neumann, J. (1963), "Theory of detonation waves. Progress Report to the National Defense Research Committee Div. B, OSRD549", in Taub, A. H., John von Neumann: Collected Works, 19031957, 6, New York: Pergamon Press, ISBN 9780080095660
 ^ Mathematical Association of American documentary, especially comments by Morgenstern regarding this aspect of von Neumann's personality
 ^ ^{a} ^{b} "Conversation with Marina Whitman". Gray Watson (256.com). http://256.com/gray/docs/misc/conversation_with_marina_whitman.shtml. Retrieved 20110130.
 ^ Life Magazine, 25th February 1957, Passing of a Great Mind, by Clay Bair JR. p. 96
 ^ Charney, Fjörtoft and von Neumann, 1950, Numerical Integration of the Barotropic Vorticity Equation Tellus, 2, 237254
 ^ Macrae, p. 332; Heims, pp. 236–247.
 ^ Life Magazine, 25th February 1957, Passing of a Great Mind, by Clay Bair JR. pages 89104
 ^ Life Magazine, 25th February 1957, Passing of a Great Mind, by Clay Bair JR. pages 89104
 ^ Macrae, pp. 170–171
 ^ Ed Regis. Who Got Einstein's Office?: Eccentricity and Genius at the Institute for Advanced Study. Perseus Books 1988 p 103
 ^ Nancy Stern (January 20, 1981). "An Interview with Cuthbert C. Hurd". Charles Babbage Institute, University of Minnesota. http://www.cbi.umn.edu/oh/pdf.phtml?id=159. Retrieved June 3, 2010.
 ^ Life Magazine, 25th February 1957, Passing of a Great Mind, by Clay Bair JR. pages 89104
 ^ John von neumann: the scientific genius who pioneered the Modern Computer, Game Theory, Nuclear Deterrence, Norman Macrae, American Mathematical Soc., 2000, page 48
 ^ From cardinals to chaos: reflections on the life and legacy of Stanislaw Ulam, Necia Grant Cooper, Roger Eckhardt, Nancy Shera, CUP Archive, 1989, Chapter: The Lost Cafe by GianCarlo Rota, pages 2627
 ^ The Computer from Pascal to von Neumann, Princeton University Press, 1980, Herman Heine Goldstine, page 171
 ^ Historical and Biographical Reflections and Syntheses, Eugene Wigner, Springer 2002, page 129
 ^ Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394
 ^ Chances are: adventures in probability, Michael Kaplan, Ellen Kaplan, Viking 2006
 ^ Darwin Among the Machines: the Evolution of Global Intelligence, Perseus Books, 1998, George Dyson, 77
 ^ John von Neumann, by Edward Teller, The Bulletin of the Atomic Scientists, April 1957, page 150.
 ^ The Computer from Pascal to von Neumann, Princeton University Press, 1980, Herman Heine Goldstine, page 171
 ^ The Ascent of Man, Jacob Brownowski, BBC 1976, page 433
 ^ Famous puzzles of great mathematicians, American Mathematical Soc., 2009, Miodrag Petković, page 157
 ^ Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394
 ^ Life Magazine, 25th February 1957, Passing of a Great Mind, by Clay Bair JR. pages 89104
 ^ The Computer from Pascal to von Neumann, Princeton University Press, 1980, Herman Heine Goldstine, page 167
 ^ "Introducing the John von Neumann Computer Society". John von Neumann Computer Society. http://www.njszt.hu/neumann/neumann.head.page?nodeid=210. Retrieved 20080520.
 ^ ^{a} ^{b} John von Neumann at the Mathematics Genealogy Project.. Accessed 20110305.
 ^ While Israel Halperin's thesis advisor is often listed as Salomon Bochner, this may be because "Professors at the university direct doctoral theses but those at the Institute do not. Unaware of this, in 1934 I asked von Neumann if he would direct my doctoral thesis. He replied Yes." (Israel Halperin, "The Extraordinary Inspiration of John von Neumann", Proceedings of Symposia in Pure Mathematics, vol. 50 (1990), pp. 15–17).
References
This article was originally based on material from the Free Online Dictionary of Computing, which is licensed under the GFDL.
 Doran, Robert S.; John Von Neumann, Marshall Harvey Stone, Richard V. Kadison, American Mathematical Society (2004). Operator Algebras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John Von Neumann and Marshall H. Stone. American Mathematical Society Bookstore. ISBN 9780821834022. http://books.google.com/?id=m5bSoD9XsfoC&pg=PA1.
 Heims, Steve J. (1980). John von Neumann and Norbert Wiener, from Mathematics to the Technologies of Life and Death. Cambridge, Massachusetts: MIT Press. ISBN 0262081059.
 Herken, Gregg (2002). Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller. ISBN 9780805065886.
 Glimm, James; Impagliazzo, John; Singer, Isadore Manuel The Legacy of John von Neumann, American Mathematical Society 1990 ISBN 0821842196
 Israel, Giorgio; Ana Millan Gasca (1995). The World as a Mathematical Game: John von Neumann, Twentieth Century Scientist.
 Macrae, Norman (1992). John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Pantheon Press. ISBN 0679413081.
 Slater, Robert (1989). Portraits in Silicon. Cambridge, Mass.: MIT Press. pp. 23–33. ISBN 0262691310.
Further reading
 Aspray, William, 1990. John von Neumann and the Origins of Modern Computing.
 Chiara, Dalla, Maria Luisa and Giuntini, Roberto 1997, La Logica Quantistica in Boniolo, Giovani, ed., Filosofia della Fisica (Philosophy of Physics). Bruno Mondadori.
 Goldstine, Herman, 1980. The Computer from Pascal to von Neumann.
 Halmos, Paul R., 1985. I Want To Be A Mathematician SpringerVerlag
 Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903–1957). Teil 1: Lehrjahre eines jüdischen Mathematikers während der Zeit der Weimarer Republik. In: InformatikSpektrum 29 (2), S. 133–141.
 Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903–1957). Teil 2: Ein Privatdozent auf dem Weg von Berlin nach Princeton. In: InformatikSpektrum 29 (3), S. 227–236.
 Heims, Steve J., 1980. John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death MIT Press
 Macrae, Norman, 1999. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Reprinted by the American Mathematical Society.
 Poundstone, William. Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb. 1992.
 Redei, Miklos (ed.), 2005 John von Neumann: Selected Letters American Mathematical Society
 Ulam, Stanisław, 1983. Adventures of a Mathematician Scribner's
 Vonneuman, Nicholas A. John von Neumann as Seen by His Brother ISBN 0961968109
 1958, Bulletin of the American Mathematical Society 64.
 1990. Proceedings of the American Mathematical Society Symposia in Pure Mathematics 50.
 John von Neumann 1903–1957, biographical memoir by S. Bochner, National Academy of Sciences, 1958
 Popular periodicals
 Good Housekeeping Magazine, September 1956 Married to a Man Who Believes the Mind Can Move the World
 Life Magazine, February 25, 1957 Passing of a Great Mind
 Video
 John von Neumann, A Documentary (60 min.), Mathematical Association of America
External links
 O'Connor, John J.; Robertson, Edmund F., "John von Neumann", MacTutor History of Mathematics archive, University of St Andrews, http://wwwhistory.mcs.standrews.ac.uk/Biographies/Von_Neumann.html.
 von Neumann's contribution to economics — International Social Science Review
 Oral history interview with Alice R. Burks and Arthur W. Burks, Charles Babbage Institute, University of Minnesota, Minneapolis. Alice Burks and Arthur Burks describe ENIAC, EDVAC, and IAS computers, and John von Neumann's contribution to the development of computers.
 Oral history interview with Eugene P. Wigner, Charles Babbage Institute, University of Minnesota, Minneapolis. Wigner talks about his association with John von Neumann during their school years in Hungary, their graduate studies in Berlin, and their appointments to Princeton in 1930. Wigner discusses von Neumann's contributions to the theory of quantum mechanics, and von Neumann's interest in the application of theory to the atomic bomb project.
 Oral history interview with Nicholas C. Metropolis, Charles Babbage Institute, University of Minnesota. Metropolis, the first director of computing services at Los Alamos National Laboratory, discusses John von Neumann's work in computing. Most of the interview concerns activity at Los Alamos: how von Neumann came to consult at the laboratory; his scientific contacts there, including Metropolis; von Neumann's first handson experience with punched card equipment; his contributions to shockfitting and the implosion problem; interactions between, and comparisons of von Neumann and Enrico Fermi; and the development of Monte Carlo methods. Other topics include: the relationship between Alan Turing and von Neumann; work on numerical methods for nonlinear problems; and the ENIAC calculations done for Los Alamos.
 Von Neumann vs. Dirac — from Stanford Encyclopedia of Philosophy.
 John von Neumann Postdoctoral Fellowship – Sandia National Laboratories
 Von Neumann's Universe, audio talk by George Dyson
 John von Neumann's 100th Birthday, article by Stephen Wolfram on Neumann's 100th birthday.
 Annotated bibliography for John von Neumann from the Alsos Digital Library for Nuclear Issues
 Budapest Tech Polytechnical Institution – John von Neumann Faculty of Informatics
 John von Neumann speaking at the dedication of the NORD, December 2, 1954 (audio recording)
 The American Presidency Project
 John Von Neumann Memorial at Find A Grave
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