John von Neumann

For other people named John Neumann, see John Neumann (disambiguation).

John (Janos) von Neumann
John von Neumann in the 1940s.
Born	December 28, 1903 Budapest, Austria-Hungary
Died	February 8, 1957 Washington DC, USA
Nationality	US -Hungarian
Alma mater	University of Pázmány Péter ETH Zurich
Known for	von Neumann algebra von Neumann architecture Game theory
Awards	Enrico Fermi Award 1956
Scientific career
Fields	Mathematics and physics
Institutions	Los Alamos University of Berlin Princeton University
Doctoral advisor	Leopold Fejer
Doctoral students	Israel Halperin

John von Neumann (Neumann János Lajos Margittai) (December 28, 1903 in Budapest, Austria-Hungary – February 8, 1957 in Washington DC, USA) was a Hungarian-born mathematician and polymath who made contributions to quantum physics, functional analysis, set theory, economics, computer science, topology, numerical analysis, hydrodynamics (of explosions), statistics and many other mathematical fields as one of world history's outstanding mathematicians. Most notably, von Neumann was a pioneer of the modern digital computer and the application of operator theory to quantum mechanics (see von Neumann algebra), a member of the Manhattan Project and the Institute for Advanced Study at Princeton (as one of the few originally appointed — a group collectively referred to as the "demi-gods"), and creator of game theory and the concept of cellular automata. Along with Edward Teller and Stanislaw 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 names have the family name first) in Budapest, Hungary to Neumann Miksa (Max Neumann), a lawyer who worked in a bank, and Kann Margit (Margaret Kann). Growing up in a non-practising Jewish family, János, nicknamed "Jancsi", was an extraordinary prodigy. At the age of six, he could divide two 8-digit numbers in his head and converse with his father in ancient Greek. János was already very interested in math, the nature of numbers and the logic of the world around him. By eight, he had mastered calculus; by twelve he was at the graduate level in mathematics reading such books as Emile Borel's "Theorie des Fonctions". His interests as a child were not confined to the realm of mathematics, though, and accounts tell of how, by eight years of age, he had completed his reading of all forty-four volumes of the universal history, which resided in the family’s library. Indeed, he could memorize pages on sight - a gift that later would continue to surprise even Nobel Prize laureates. He loved to invent mechanical toys, and was an expert on the Civil War, the trial of Joan of Arc, and Byzantine history.

He entered the Lutheran Gymnasium in Budapest in 1911. In 1913, his father purchased a title, and the Neumann family acquired the Hungarian mark of nobility Margittai, or the Austrian equivalent von. Neumann János therefore became János von Neumann — a name that he later changed to the German Johann von Neumann. After teaching as history's youngest Privatdozent of the University of Berlin from 1926 to 1930, he, his mother, and his brothers emigrated after Hitler's rise to power from Germany to the United States in the 1930s. Curiously, while he anglicized Johann to John , he kept the Austrian-aristocratic surname of von Neumann, whereas his brothers adopted the different surnames of Vonneumann and Newman.

Although von Neumann unfailingly dressed formally, with suit and tie, he enjoyed throwing the most extravagant parties and driving hazardously (frequently while reading a book, and sometimes crashing into a tree or getting himself arrested as a consequence). He was a profoundly committed hedonist who liked to eat and drink heavily (it was said that he knew how to count everything, except calories), tell dirty stories and very insensitive jokes (e.g. "bodily violence is a displeasure done with the intention of giving pleasure"), and insistently gaze at the legs of young women (so much so that the female secretaries at Los Alamos were often compelled to cover up the exposed undersides of their desks with sheets of paper or cardboard.)

He received his Ph.D. in mathematics (with minors in experimental physics and chemistry) from the University of Budapest at the age of 23. He simultaneously earned a degree in chemical engineering from the ETH Zurich in Switzerland at the behest of his father, who wanted his son to invest his time in a more financially viable endeavour than mathematics. Between 1926 and 1930 he was a private lecturer in Berlin, Germany.

By age 25, he had published ten major papers; and by age 30, nearly three dozen.

Von Neumann was invited to Princeton, New Jersey in 1930, and was one of four people selected for the first faculty of the Institute for Advanced Study (two of which were Albert Einstein and Kurt Gödel), where he was a mathematics professor from its formation in 1933 until his death.

From 1936 to 1938, Alan Turing was a visitor at the Institute, where he completed a Ph.D. dissertation under the supervision of Alonzo Church at Princeton. This visit occurred shortly after Turing's publication of his 1936 paper "On Computable Numbers with an Application to the Entscheidungsproblem" which involved the concepts of logical design and the universal machine. Von Neumann must have known of Turing's ideas but it is not clear whether he applied them to the design of the IAS machine ten years later.

In 1937, he became a naturalized citizen of the United States. In 1938 von Neumann was awarded the Bôcher Memorial Prize for his work in analysis.

Von Neumann was married twice. His first wife was Mariette Kövesi, whom he married in 1930. When he proposed to her, he was incapable of expressing anything beyond "You and I might be able to have some fun together, seeing as how we both like to drink." Von Neumann agreed to convert to Catholicism to placate her family. The couple divorced in 1937, and then he married his second wife, Klara Dan, in 1938. Von Neumann had one child, by his first marriage, his daughter Marina. Marina later married and now is a distinguished professor of both international trade and public policy at the University of Michigan.

Von Neumann was diagnosed with bone cancer or pancreatic cancer in 1957, possibly caused by exposure to radioactivity while observing A-bomb tests in the Pacific, and possibly in later work on nuclear weapons at Los Alamos, New Mexico. (Fellow nuclear pioneer Enrico Fermi had died of bone cancer in 1954.) Von Neumann died within a few months of the initial diagnosis, in excruciating pain. The cancer had also spread to his brain, drastically cutting his ability to think, previously his sharpest and most cherished tool. As he lay dying in Walter Reed Hospital in Washington, D.C., he shocked his friends and acquaintances by asking to speak with a Roman Catholic priest. He died under military security lest he inadvertently reveal military secrets while heavily medicated.

He had written 150 published papers in his life: 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. When he died, he was developing a theory of the structure of the human brain.

Von Neumann entertained notions which would now trouble many. His love for meteorological prediction led him to dreaming of manipulating the environment by, for example, spreading artificial colorants on the polar ice caps in order to enhance the absorption of solar radiation (by reducing the albedo) and thereby raise global temperatures. He also favored a preemptive nuclear attack on the USSR, believing that doing so could prevent it from obtaining the atomic bomb.

Logic

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 Richard Dedekind and Giuseppe Peano) and geometry (thanks to David Hilbert). At the beginning of the twentieth century, however, set theory, the new branch of mathematics invented by Georg Cantor, and thrown into crisis by Bertrand Russell with the discovery of his famous paradox (on the set of all sets which do not belong to themselves), had not yet been formalized. Russell's paradox consisted in the observation that if the set x (of all sets which are not members of themselves) was a member of itself, then it must belong to the set of all sets which do not belong to themselves, and therefore cannot belong to itself; on the other hand, if the set x does not belong to itself, then it must belong to the set of all sets which do not belong to themselves, and therefore it must belong to itself.

The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later (thanks to Ernst Zermelo and Abraham Frankel) by way of a series of principles which allowed for the construction of all sets used in the actual practice of mathematics, but which did not explicitly exclude the possibility of the existence of sets which belong to themselves. In his doctoral thesis of 1925, von Neumann demonstrated how it was possible to exclude this possibility in two complementary ways: 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 Frankel, 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.) In order 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. While, on the Zermelo/Frankel approach, the axioms impede the construction of a set of all sets which do not belong to themselves, on 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 of 1930 at the historical 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: the usual axiomatic systems are unable to demonstrate their own consistency. 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.)

Quantum mechanics

At the International Congress of Mathematicians of 1900, David Hilbert presented his famous list of twenty-three problems considered central for the development of the mathematics of the new century: the sixth of these was the axiomatization of physical theories. Among the new physical theories of the century the only one which had yet to receive such a treatment by the end of the 1930's was quantum mechanics. In fact, QM found itself, at this time, in a condition of foundational crisis similar to that of set theory at the beginning of the century, facing problems of both philosophical and technical natures: on the one hand, its apparent non-determinism had not been reduced, as Albert Einstein believed it must be in order to be satisfactory and complete, to an explanation of a deterministic form; on the other, there still existed two independent but equivalent heuristic formulations, the so-called matrix mechanical formulation due to Werner Heisenberg and the wave mechanical formulation due to Erwin Schrödinger, but there was not yet a single, unified satisfactory theoretical formulation.

After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of QM. He immediately realized, in 1926, that a quantum system could be considered as a point in a so-called 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 famous indeterminacy principle of Heisenberg, 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, and culminated in the 1932 classic The Mathematical Foundations of Quantum Mechanics. However, physicists generally ended up preferring another approach to that of von Neumann (which was considered extremely elegant and satisfactory by mathematicians). This approach was formulated in 1930 by Paul Dirac and was based upon a strange type of function (the so-called Dirac delta function) which was harshly criticized by von Neumann.

In any case, von Neumann's abstract treatment permitted him also to confront the extremely hot-button foundational issue of determinism vs. non-determinism and in the book he demonstrated a theorem according to which quantum mechanics could not possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics. This demonstration contained a conceptual error, but it helped to inaugurate a line of research which, through the work of John Stuart Bell in 1964 on Bell's Theorem and the experiments of Alain Aspect in 1982, eventually demonstrated that quantum physics does indeed require a notion of reality substantially different from that of classical physics.

In a complementary work of 1936, von Neumann proved (along with Garrett Birkhoff) that quantum mechanics also requires a logic substantially different from the classical one. For example, light (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 non-commutativity of conjunction $(A\land B)\neq (B\land A)$. It was also demonstrated that the laws of distribution of classical logic, $P\lor (Q\land R)=(P\lor Q)\land (P\lor R)$ and $P\land (Q\lor R)=(P\land Q)\lor (P\land R)$, 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 semi-integral 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 x 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 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 $A\land (B\lor C)=A\land 1=A$, while $(A\land B)\lor (A\land C)=0\lor 0=0$.

Economics

Up until the 1930s, the field of economics seemed to involve the utilization of a great deal of mathematics and numbers; but almost all of this was either superficial or irrelevant. It was used, for the most part, in order to provide uselessly precise formulations and solutions to problems which were, in fact, intrinsically vague. Economics found itself in a state similar to that of the physics of the 17th century: still waiting for the development of an appropriate language in which to express and resolve its problems. While physics, of course, had found its language in the infinitesimal calculus, von Neumann proposed the language of game theory and a general equilibrium theory for economics.

His first significant contribution was the minimax theorem of 1928. This theorem establishes that in certain so-called zero sum games involving perfect information (in which, that is, players knows a priori both the strategies of their opponents as well as their consequences), there exists one strategy which allows both players to minimize their maximum losses (hence the name minimax). In particular, when examining every possible strategy, a player must consider all the possible responses of the player's adversary and the maximum loss that could be derived. The player then plays out the strategy which will result in the minimization of this maximum loss. Such a strategy, which minimizes the maximum loss, is called optimal for both players just in case their minimaxes are equal (in absolute value) and contrary (in sign). If the common value is zero, the game becomes pointless.

Von Neumann eventually improved and extended the minimax theorem to include games involving imperfect information and games with more than two players. This work culminated in the 1944 classic Theory of Games and Economic Behavior (written with Oskar Morgenstern). This resulted in such public attention that The New York Times did a front page story, the likes of which only Einstein had previously earned.

Von Neumann's second important contribution in this area was the solution, in 1937, of a problem first described by Leon Walras in 1874: the existence of situations of equilibrium in mathematical models of market development based on supply and demand. He first recognized that such a model should be expressed through disequations (as is done today) and not equations (as had been the previous practice), and then he found a solution to Walras problem by applying a fixed-point theorem derived from the work of Luitzen Brouwer. 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 and, in 1983, to Gerard Debreu.

Von Neumann was also the inventor of the method of proof, used in game theory, known as backward induction (first published in 1944 in the book co-authored with Morgenstern, Theory of Games and Economic Behaviour).

Armaments

In 1937 von Neumann, having recently obtained his US citizenship, began to take an interest in problems in applied mathematics. He rapidly became one of the top experts in the field of explosives, and he committed himself to a very large number of military consultancies, primarily for the Navy (it seems possible that he preferred socializing with admirals rather than generals because the former tended to enjoy drinking liquor while the latter preferred coffee.)

One noted result in the field of explosions was the discovery that bombs of large dimension are more devastating if they detonate before touching the soil because of the additional force caused by waves of detonation (the media maintained more simply that von Neumann had discovered that it is better to miss a target than to hit it). The most famous (or infamous) application of this discovery occurred on the 6th and 9th of August 1945, when two nuclear weapons were detonated above the soils of Hiroshima and Nagasaki, at the precise altitude calculated by von Neumann himself in order that they would produce the most extensive damage possible.

John von Neumann's wartime Los Alamos ID badge photo.

Von Neumann had been brought into the Manhattan Project in order to help design the explosive lenses needed to compress the plutonium core of the Trinity test device and the "Fat Man" weapon dropped on Nagasaki.

From a political point of view, von Neumann was a member of the committee whose job it was to select potential "targets". Von Neumann's first choice, the city of Kyoto, was dismissed out of hand by the Secretary of War Henry Stimson.

After the war, Robert Oppenheimer had famously remarked that the physicists had "known sin" as a result of their development of the first atomic bombs. Von Neumann's rather cynical reply was that "sometimes someone confesses a sin in order to take credit for it." In any case, he continued unperturbed in this work, and eventually became, along with Edward Teller, one of the most convinced sustainers of the successive project of the construction of the hydrogen bomb. Von Neumann had collaborated with the spy Klaus Fuchs on hydrogen bomb development, and the two filed a secret patent on "Improvement in Methods and Means for Utilizing Nuclear Energy" in 1946, which outlined a scheme for using an exploding fission bomb to compress fusion fuel before attempting to initiate a thermonuclear reaction. (Herken, pp. 171, 374). Though this was not the key to the successful hydrogen bomb design — the Teller-Ulam design — it was later judged to have been a step in the right direction, even though it was not immediately pursued.

Von Neumann's hydrogen bomb work was also in the realm of computing, where he and Stanislaw Ulam developed computer simulations on von Neumann's new electronic digital computers for the necessary hydrodynamic computations. It was during this time that he contributed to the development of the Monte Carlo method, which allowed very complicated problems to be approximated through the use of random numbers. Because using lists of "truly" random numbers was extremely slow for the ENIAC, von Neumann developed a crude form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized in hindsight as overly crude, von Neumann was aware of this at the time: he justified it as being faster (in terms of computing time) than any other method at his disposal at the time, and also noted that when it went awry it did so very obviously, unlike methods which could be subtlely incorrect.

In 1952, the first hydrogen bomb shot, Ivy Mike, was detonated on the Eniwetok Atoll.

Computer science

Von Neumann gave his name to the von Neumann architecture used in almost all computers, because of his publication of the concept; though many feel that this naming ignores the contributions of J. Presper Eckert, John William Mauchly and others who worked on the concept during their work on EDVAC. Virtually every home computer, microcomputer, minicomputer and mainframe computer is a von Neumann machine. He also created the field of cellular automata without computers, constructing the first examples of self-replicating automata with pencil and graph paper. The concept of a universal constructor was fleshed out in his posthumous work Theory of Self Reproducing Automata. The term "von Neumann machine" alternatively refers to self-replicating machines. Von Neumann proved that the most effective way large-scale mining operations such as mining an entire moon or asteroid belt could be accomplished is through the use of self-replicating machines, to take advantage of the exponential growth of such mechanisms.

In addition to his work on computer architecture, he is credited with at least one contribution to the study of algorithms. Donald Knuth cites von Neumann as the inventor, in 1945, of the well-known merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged together.

He also engaged in exploration of problems in the field of numerical hydrodynamics. With R. D. Richtmyer he developed an algorithm defining artificial viscosity, that proved essential to understanding many kinds of shock waves. It can fairly be said that we would not understand much of astrophysics, and we might not even have highly developed jet and rocket engines, without that work. The problem to be solved was that when computers solve hydrodynamic or aerodynamic problems, they try to put too many computational grid-points at regions of sharp discontinuity (shock waves). The artificial viscosity was a mathematical trick to slightly smooth the shock transition without sacrificing basic physics.

Politics and social affairs

Von Neumann had experienced a lightning-like academic career similar to the velocity of his own intellect, obtaining at the age of twenty-nine one of the first five professorships at the newly born Institute for Advanced Study at Princeton (another had gone to Albert Einstein). He seemed compelled, therefore, to seek out other fields of interest in order to satisfy his ambitious personality, and he found this outlet in his collaboration with the American military-industrial complex. He was a frequent consultant for the CIA, the US Military, the RAND Corporation, Standard Oil, IBM, and others.

During a Senate committee hearing, he once described his political ideology as, in his own words "violently anti-communist, and much more militaristic than the norm." As President of the so-called Von Neumann Committee for Missiles at first, and as a member of the restricted Commission for Atomic Energy later, starting from 1953 up until his death in 1957, he was the scientist with the most political power in the US. Through his committee, he developed various scenarios of nuclear proliferation, the development of intercontinental and submarine missiles with atomic warheads, and the highly controversial strategic equilibrium called Mutually Assured Destruction. In a word, he was the mind behind the scientific aspects of the Cold War which conditioned the Western world for forty years.

Honors

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U.S. postage stamp commemorating von Neumann

The John von Neumann Theory Prize of the Institute for Operations Research and Management Science (INFORMS, previously TIMS-ORSA) is awarded annually to an individual (or sometimes 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 computer-related 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; the chosen lecturer is also awarded a monetary prize.

Von Neumann, a crater on Earth's Moon, is named after John von Neumann.

The professional society of Hungarian computer scientists, Neumann János Számítógéptudományi Társaság, is named after John von Neumann.

On May 4, 2005 the United States Postal Service issued the American Scientists commemorative postage stamp series, a set of four 37-cent self-adhesive stamps in several configurations. The scientists depicted were John von Neumann, Barbara McClintock, Josiah Willard Gibbs, and Richard Feynman.

References

This article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.

• Steve J. Heims, 1980. John von Neumann and Norbert Wiener, from Mathematics to the technologies of life and death. MIT Press.
• Gregg Herken, 2002. Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller.
• Israel, Giorgio, and Gasca, Ana Millan, 1995. The World as a Mathematical Game: John von Neumann, Twentieth Century Scientist.
• Norman Macrae, 1992. John von Neumann.

Further reading

• Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press.
  • 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 U. Press.
• 1944. (with 'Oskar Morgenstern) Theory of Games and Economic Behavior. Princeton Univ. Press.
• 1966. (with Arthur W. Burks) Theory of Self-Reproducing Automata. Univ. of Illinois Press.

Secondary:

• Norman Macrae, 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.
• Aspray, William, 1990. John von Neumann and the Origins of Modern Computing.
• Dalla Chiara, 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.
• Poundstone, William. Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb. 1992.
• 1958, Bulletin of the American Mathemetical Society 64.
• 1990. Proceedings of the American Mathematical Society Symposia in Pure Mathematics 50.

Students

• Donald B. Gillies, PhD student of John von Neumann.
• John P. Mayberry, PhD student of John von Neumann.
• John Forbes Nash Jr., student of John von Neumann.

See also

• Von Neumann algebra
• Von Neumann conjecture
• Stone–von Neumann theorem
• Von Neumann–Bernays–Gödel set theory
• Von Neumann universe
• Von Neumann bicommutant theorem
• Von Neumann regular ring
• Von Neumann architecture
• Universal constructor

External links

• O'Connor, John J.; Robertson, Edmund F., "John von Neumann", MacTutor History of Mathematics Archive, University of St Andrews
• John von Neumann's contribution to economic science — By Maria Joao Cardoso De Pina Cabral, International Social Science Review, Fall-Winter 2003
• Von Neumann vs. Dirac — from Stanford Encyclopedia of Philosophy.
• Edward Teller talking about von Neumann on Peoples Archive.
• A Discussion of Artificial Viscosity
• Von Neumann's Universe, audio talk by George Dyson from ITConversations.com
• [1], article by Stephen Wolfram on Neumann's 100th birthday.
• John von Neumann at the Mathematics Genealogy Project
• His biography at Hungary.hu
• Bibliography of von Neumann's works
• Annotated bibliography for John von Neumann from the Alsos Digital Library for Nuclear Issues