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John von Neumann


John von Neumann (Hungarian Margittai Neumann János Lajos) (December 28, 1903 - February 8, 1957) mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics (of explosions), statistics and many other mathematical fields as one of history's outstanding mathematicians.1 Most notably, von Neumann was a pioneer of 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 the co-creator of game theory and the concepts of cellular automata and the universal constructor. 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.

Quantum physicsQuantum mechanics

Introduction to…
Mathematical formulation of…

Fundamental concepts

Decoherence · Interference
Uncertainty · Exclusion
Transformation theory
Ehrenfest theorem · Measurement


Double-slit experiment
Davisson-Germer experiment
Stern-Gerlach experiment
EPR paradox · Popper's experiment Schrödinger's cat


Schrödinger equation
Pauli equation
Klein-Gordon equation
Dirac equation

Advanced theories

Quantum field theory
Wightman axioms
Quantum electrodynamics
Quantum chromodynamics
Quantum gravity
Feynman diagram


Copenhagen · Ensemble
Hidden variables · Transactional
Many-worlds · Consistent histories
Quantum logic
Consciousness causes collapse


Planck · Schrödinger
Heisenberg · Bohr · Pauli
Dirac · Bohm · Born
de Broglie · von Neumann
Einstein · Feynman
Everett · Others


Early years

The oldest of three brothers, von Neumann was born Neumann János Lajos (in Hungarian the family name comes first) in Budapest, Hungary, to a Jewish family. His father was Neumann Miksa (Max Neumann), a lawyer who worked in a bank. His mother was Kann Margit (Margaret Kann).

János, nicknamed "Jancsi" (Johnny), was an extraordinary prodigy. At the age of only six, he was able to divide two 8-digit numbers in his head.

He entered the German speaking Lutheran Gymnasium in Budapest in the year 1911. In 1913, his father was rewarded with ennoblement for his service to the Austro-Hungarian empire, the Neumann family acquiring the Hungarian mark of 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 to the United States; this in the early 1930s, after Hitler's rise to power in Germany. He anglicized Johann to John, he kept the Austrian-aristocratic surname of von Neumann, whereas his brothers adopted surnames Vonneumann and Neumann (using the de Neumann form briefly when first in the US).

Although von Neumann unfailingly dressed formally, he enjoyed throwing extravagant parties and driving hazardously (frequently while reading a book, and sometimes crashing into a tree or getting arrested).2 He once reported one of his many car accidents in this way: "I was proceeding down the road. The trees on the right were passing me in orderly fashion at 60 miles per hour. Suddenly one of them stepped in my path."3 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), 4 and persistently gaze at the legs of young women (so much so that female secretaries at Los Alamos often covered up the exposed undersides of their desks with cardboard).5

Higher education, years in Germany

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 his diploma 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 endeavor than mathematics. Between 1926 and 1930 he was a private lecturer in Berlin, Germany.

By age 25 he had published 10 major papers, and by age 30, nearly 36.6

Years at Princeton University

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 the others 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.

Marriage and family

Von Neumann married twice. He married Mariette Kövesi 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." 7 Von Neumann agreed to convert to Catholicism in order to marry and remained a Catholic until his death. The couple divorced in 1937. He then married Klara Dan in 1938. Von Neumann had one child, by his first marriage, a daughter named Marina. She is a distinguished professor of international trade and public policy at the University of Michigan.

Cancer and death

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 or in later work on nuclear weapons at Los Alamos, New Mexico. (Fellow nuclear pioneer Enrico Fermi had died of stomach cancer in 1954.) Von Neumann died within a few months of the initial diagnosis, in excruciating pain. The cancer had spread to his brain, inhibiting mental ability. When at Walter Reed Hospital in Washington, D.C., he invited Roman Catholic priest (Father Anselm Strittmatter), who administered him the last Sacraments.8 He died under military security lest he reveal military secrets while heavily medicated. John von Neumann was buried at Princeton Cemetery in Princeton, Mercer County, New Jersey.

He wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. He was developing a theory of the structure of the human brain before he died.

Controversial notions

Von Neumann entertained notions which would now trouble many. His love for meteorological prediction led him to dream of manipulating the environment by spreading colorants on the polar ice caps in order to enhance 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.910


The axiomatization of set theory was resolved (by Ernst Zermelo and Abraham Frankel) by way of a series of principles that allowed for the construction of all sets used in the actual practice of mathematics, but it did not explicitly exclude the possibility of the existence of sets that 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.11

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. 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

Quantum mechanics

After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics.'12 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 uncertainty 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 elegant and satisfactory by mathematicians). This approach was formulated in 1930 by Paul Dirac.

In any case, von Neumann's abstract treatment permitted him also to confront the 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, demonstrated that quantum physics requires 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.


Up until the 1930s, economics involved a great deal of mathematics and numbers, but almost all of this was either superficial or irrelevant. It was used, for the most part, to provide uselessly precise formulations and solutions to problems which were intrinsically vague. Economics found itself in a state similar to that of physics of the seventeenth century: still waiting for the development of an appropriate language in which to express and resolve its problems. While physics 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 zero sum games involving perfect information (in which players know a priori 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). When examining every possible strategy, a player must consider all the possible responses of the player's adversary and the maximum loss. 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 and not equations, 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 (which he first published in 1944 in the book co-authored with Morgenstern, Theory of Games and Economic Behaviour).13


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

After obtaining U.S. citizenship, von Neumann took an interest in 1937 in applied mathematics, and then developed an expertise in explosives. 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.

Von Neumann took part in the 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. The lens shape design work was completed by 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.14

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.15 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, but this target was dismissed by Secretary of War Henry Stimson, who had been impressed with the city during a visit while Governor General of the Philippines.16

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 littering scraps of torn-up paper as the shock wave passed his location and watching how far they scattered. The actual power of the explosion had been between 20 and 22 kilotons.14

After the war, Robert Oppenheimer remarked that the physicists involved in the Manhattan Project had "known sin." Von Neumann's rather arch 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 the sustainers of the hydrogen bomb project. He then collaborated with spy 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. 17. Though this was not the key to the hydrogen bomb - the Teller-Ulam design - it was judged to be a move in the right direction.

Computer science

Von Neumann's hydrogen bomb work was also played out in the realm of computing, where he and Stanislaw 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 for the ENIAC, von Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, 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 on the EDVAC project, von Neumann wrote an incomplete set of notes titled the First Draft of a Report on the EDVAC. The paper, which was widely distributed, described a computer architecture in which data and program memory are mapped into the same address space. This architecture became the de facto standard and can be contrasted with a so-called Harvard architecture, which has separate program and data memories on a separate bus. Although the single-memory architecture became commonly known by the name von Neumann architecture as a result of von Neumann's paper, the architecture's conception involved the contributions of others, including J. Presper Eckert and John William Mauchly, inventors of the ENIAC at the University of Pennsylvania.18 With very few exceptions, all present-day home computers, microcomputers, minicomputers and mainframe computers use this single-memory computer architecture.

Von Neumann also created the field of cellular automata without the aid of computers, constructing the first 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. Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire moon or asteroid belt would be by using self-replicating machines, taking advantage of their exponential growth.

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 merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged together.19 His algorithm for simulating a fair coin with a biased coin20 is used in the "software whitening" stage of some hardware random number generators.

He also engaged in exploration of problems in numerical hydrodynamics. 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 that work. The problem 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 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, IBM, and others.

During a Senate committee hearing he described his political ideology as "violently anti-communist, and much more militaristic than the norm." As President of the Von Neumann Committee for Missiles at first, and later as a member of the United States Atomic Energy Commission, starting from 1953 up 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 (aka the M.A.D. doctrine).


The John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences (INFORMS, previously TIMS-ORSA) 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 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, and the chosen lecturer is also awarded a monetary prize.

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

The John von Neumann Computing Center in Princeton, New Jersey was named in his honor. 6

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.

The John von Neumann Award of the Rajk László College for Advanced Studies was named in his honor, and is given every year from 1995 to professors, who had on outstanding contribution at the field of exact social sciences, and through their work they had a heavy influence to the professional development and thinking of the members of the college.

See also

  • Edward Teller
  • Manhattan Project
  • Nuclear weapon
  • Quantum mechanics
  • Robert Oppenheimer


  1. ↑ John von Neumann. MSN Encarta. Retrieved November 17, 2007.
  2. ↑ 1quotes of wisdom. Retrieved November 17, 2007.
  3. ↑ John von Neumann. Bellevue Community College. Retrieved November 17, 2007.
  4. ↑ 2007/02/john_von_neuman.htmlrightcoast.typepad.com. Retrieved November 17, 2007.
  5. ↑ rightcoast.typepad.com/rightcoast/ 2007/02/john_von_neuman.html rightcoast.typepad.com. Retrieved November 17, 2007.
  6. ↑ 2reference.com Retrieved November 17, 2007.
  7. ↑ 3 Retrieved November 17, 2007.
  8. ↑ P.R. Halmos, 1973. "The Legend of Von Neumann." The American Mathematical Monthly 80(4): 382-394.
  9. ↑ Norman Macrae. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. (New York, NY: Pantheon Press 1992).
  10. ↑ Steve J. Heims. John von Neumann and Norbert Wiener, from Mathematics to the Technologies of Life and Death. (Cambridge, MA: MIT Press. 1980).
  11. ↑ 4 Retrieved November 17, 2007.
  12. ↑ 5 Retrieved November 17, 2007.
  13. ↑ John MacQuarrie, Mathematics and Chess. University of St Andrews, Scotland. "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". Retrieved November 17, 2007.
  14. 14.0 14.1 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).
  15. ↑ Richard Rhodes. 1986. The Making of the Atomic Bomb. (New York, NY: Touchstone (Simon & Schuster). ISBN 0684813785).
  16. ↑ Leslie Groves. 1962. Now It Can Be Told: The Story of the Manhattan Project. (New York, NY: Da Capo. ISBN 0306801892).
  17. ↑ Gregg Herken. Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller. (New York, NY: Henry Holt and Co., 2002), 171, 374
  18. ↑ John W. Mauchly and the Development of the ENIAC Computer. Penn Libraries. Retrieved November 17, 2007.
  19. ↑ Donald Knuth, 1998. The Art of Computer Programming: Volume 3 Sorting and Searching. (Reading, MA: Addison-Wesley. ISBN 0201896850).
  20. ↑ John von Neumann, 1951. Various techniques used in connection with random digits. National Bureau of Standards Applied Math Series 12:36.


  • Groves, Leslie. 1962. Now It Can Be Told: The Story of the Manhattan Project. New York, NY: Da Capo. ISBN 0306801892
  • Heims, Steve J. 1980. John von Neumann and Norbert Wiener, from Mathematics to the Technologies of Life and Death. Cambridge, MA: MIT Press. ISBN 0262081059.
  • Herken, Gregg. 2002. Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller. New York, NY: Henry Holt and Co. ISBN 0805065881.
  • Knuth, Donald. 1998. The Art of Computer Programming: Volume 3 Sorting and Searching. Reading, MA: Addison-Wesley. ISBN 0201896850
  • Hoddeson, Lillian, Paul W. Henriksen, Roger A. Meade, and Catherine Westfall. 1993. Critical Assembly: A Technical History of Los Alamos during the Oppenheimer Years, 1943-1945. Cambridge, UK, and New York: Cambridge University Press. ISBN 0521441323.
  • Macrae, Norman. 1992. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. New York, NY: Pantheon Press. ISBN 0679413081.
  • Rhodes, Richard. 1986. The Making of the Atomic Bomb. New York, NY: Touchstone (Simon & Schuster. ISBN 0684813785
  • Slater, Robert. 1987. Portraits in Silicon. Cambridge, MA: MIT Press. ISBN 0262691310.

Further reading

  • van Heijenoort, Jean. 1967. A Source Book in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard Univ. Press.
  • von Neumann, John, R.T. Beyer, trans. 1996. Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press. ISBN 0691028931.
  • von Neumann, John, and Oskar Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press.
  • von Neumann, John, edited and completed by Arthur W. Burks. 1966. Theory of Self-Reproducing Automata. Univ. of Illinois Press. ASIN: B00178N9MU


  • Aspray, William. 1990. John von Neumann and the Origins of Modern Computing. Cambridge, MA: MIT Press. ISBN 0262011212.
  • Goldstine, Herman. 1993. The Computer from Pascal to von Neumann. Princeton, NJ: Princeton University Press. ISBN 0691023670.
  • Hashagen, Ulf. 2006. Johann Ludwig Neumann von Margitta (1903-1957). Teil 1: Lehrjahre eines jüdischen Mathematikers während der Zeit der Weimarer Republik. Informatik-Spektrum 29(2):133-141.
  • Hashagen, Ulf. 2006: Johann Ludwig Neumann von Margitta (1903-1957). Teil 2: Ein Privatdozent auf dem Weg von Berlin nach Princeton. Informatik-Spektrum 29(3):227-236.
  • Macrae, Norman. 1999. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Providence, RI: American Mathematical Society. ISBN 0821820648.
  • Poundstone, William. 1992. Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb. New York, NY: Doubleday. ISBN 0385415672.

External links

All links retrieved May 24, 2018.

  • John J. O'Connor and Edmund F. Robertson. John von Neumann at the MacTutor archive.
  • Von Neumann vs. Dirac. Stanford Encyclopedia of Philosophy.
  • Von Neumann's Universe. Audio talk by George Dyson. IT Conversations.
  • John von Neumann at the Mathematics Genealogy Project.