**John von Neumann**(/vɒn ˈnɔɪmən/; December 28, 1903 – February 8, 1957) was a Jewish born Hungarian and laterAmerican pure and applied mathematician, physicist, inventor, polymath, and polyglot. He made major contributions to a number of fields,

^{[3]}including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry,topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and fluid dynamics), economics (game theory), computing (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.

^{[4]}He was a pioneer of the application of operator theory to quantum mechanics, in the development offunctional 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

^{[3]}

^{[5]}and the concepts of cellular automata,

^{[3]}the universal constructor, and the digital computer.

Von Neumann's mathematical analysis of the structure of self-replication preceded the discovery of the structure ofDNA.

^{[6]}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 theergodic theorem, Princeton, 1931–1932." Along with theoretical physicist Edward Teller and mathematician Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.
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.## Contents

[hide]- 1 Early life and education
- 2 Career and abilities
- 2.1 Beginnings
- 2.2 Set theory
- 2.3 Geometry
- 2.4 Measure theory
- 2.5 Ergodic theory
- 2.6 Operator theory
- 2.7 Lattice theory
- 2.8 Mathematical formulation of quantum mechanics
- 2.9 Quantum logic
- 2.10 Game theory
- 2.11 Mathematical economics
- 2.12 Linear programming
- 2.13 Mathematical statistics
- 2.14 Nuclear weapons
- 2.15 The Atomic Energy Committee
- 2.16 The ICBM Committee
- 2.17 Mutual assured destruction
- 2.18 Computing
- 2.19 Fluid dynamics
- 2.20 Politics and social affairs
- 2.21 On the eve of World War II
- 2.22 Greece versus Rome
- 2.23 Weather systems
- 2.24 Cognitive abilities
- 2.25 Mastery of mathematics

- 3 Personal life
- 4 Later life
- 5 Honors
- 6 Selected works
- 7 See also
- 8 Notes
- 9 References
- 10 Further reading
- 11 External links

## Early life and education[edit]

Von Neumann was born

**Neumann János Lajos**(Hungarian pronunciation: [ˈnojmɒn ˈjaːnoʃ ˈlɒjoʃ]; in Hungarian the family name comes first) Hebrew name Yonah, inBudapest, Austro-Hungarian Empire, to wealthy Jewish parents of the Haskalah.^{[7]}^{[8]}^{[9]}He was the eldest of three brothers. 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 the 1880s. Miksa's father (Mihály b. 1839)^{[10]}and grandfather (Márton)^{[10]}were both born in Ond (now part of the town of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (Margaret Kann) (1881–1936).^{[11]}
Her parents were Jakab Kann II (Pest (now Budapest) 1845–1928) and Katalin Meisels (Munkács, Carpathian Ruthenia c. 1854–1914). In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Francis Joseph. The Neumann family thus acquired the hereditary appellation

*Margittai*, meaning of Marghita. Neumann János became Margittai Neumann János (John Neumann of Marghita), which he later changed to the German Johann von Neumann.
He was an extraordinary child prodigy in the areas of language, memorization, and mathematics. As a 6-year-old, he could divide two 8-digit numbers in his head.

^{[12]}By the age of 8, he was familiar with differential and integral calculus.^{[13]}
Von Neumann was part of a Budapest generation noted for intellectual achievement: he was born in Budapest around the same time as 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).

^{[14]}
John entered the 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.

^{[15]}
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 stationery, are still on display at the von Neumann archive in Budapest.

^{[16]}By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition.^{[17]}
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.

^{[3]}He simultaneously earned a diploma in chemical engineering from the ETH Zürich in Switzerland^{[3]}at his father's request, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavour than mathematics.^{[N 1]}## Career and abilities[edit]

### Beginnings[edit]

Between 1928 and 1932, he taught as a

*Privatdozent*at the University of Berlin.^{[19]}By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month.^{[20]}Von Neumann's reputed powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease.^{[18]}
In 1930, von Neumann was invited to Princeton University, New Jersey. In 1933, he was offered a position on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through; von Neumann remained a mathematics professor there until his death. His mother and his brothers followed John to the United States; his father, Max Neumann, died in 1929. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. In 1937, von Neumann became a United States naturalized citizen and immediately tried to enlist in the US Army Reserve but was rejected because of his age.

^{[21]}In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.### Set theory[edit]

The axiomatization of mathematics, on the model of Euclid's

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

*axiom of foundation*and the notion of*class.*
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.

^{[22]}
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.

^{[22]}However, Gödel had already discovered this consequence, now known as his second incompleteness theorem, and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter.^{[23]}### Geometry[edit]

Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue 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[edit]

See also: Lifting theory

In a series of famous papers, von Neumann made spectacular contributions to measure theory.

^{[24]}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 group-theoretic 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 even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions (anticipating his later work, Mathematical formulation of quantum mechanics, 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.

^{[24]}^{[25]}### Ergodic theory[edit]

Von Neumann made foundational contributions to ergodic theory, in a series of articles published in 1932.

^{[26]}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".^{[24]}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.^{[27]}### Operator theory[edit]

Main article: Von Neumann algebra

Von Neumann introduced the study of rings of operators, through the von Neumann algebras.

^{[28]}A von Neumann algebra is a *-algebra of bounded operators on aHilbert 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. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.

### Lattice theory[edit]

Von Neumann worked on lattice theory between 1937 and 1939. 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."

^{[29]}Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor".^{[29]}
Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."

^{[29]}### Mathematical formulation of quantum mechanics[edit]

See also: von Neumann entropy, Density matrix, Quantum mutual information, von Neumann measurement scheme and Wave function collapse

Quantum mechanics |
---|

Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematical Foundations of Quantum Mechanics.

After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He realized, in 1926, that a state of a quantum system could be represented by a point in a (complex) Hilbert spacethat, in general, could be infinite-dimensional even for a single particle. This is in contrast to a classical system where a state is represented by a point in a (real) phase space with 6N dimensions where N is the number of particles (3 generalized coordinates and 3 conjugate generalized momenta for each particle). In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system. The

*physics*of quantum mechanics was thereby reduced to the*mathematics*of Hilbert spaces and linear operators acting on them.
For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the

*non-commutativity*of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger.
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs. non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 1966, John S. Bell published a paper arguing that the proof contained a conceptual error and was therefore invalid (see the article on John Stewart Bell for more information). However, in 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation, and that von Neumann did not claim that his proof completely ruled out hidden variable theories.

^{[30]}In any case, the proof 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 either requires a*notion of reality*substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity.
In a chapter of

*The Mathematical Foundations of Quantum Mechanics*, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, it never gained acceptance amongst the majority of physicists).^{[31]}
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 logic[edit]

Main article: Quantum logic

In a famous paper of 1936 with Garrett Birkhoff, the first work ever to introduce quantum logics,

^{[32]}von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (*e.g.*, 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. This experimental fact is translatable into logic as the*non-commutativity*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 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 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 a given physical system).

^{[33]}### Game theory[edit]

Von Neumann founded the field of game theory as a mathematical discipline.

^{[34]}Von Neumann proved his minimax theorem in 1928. This theorem establishes that inzero-sum games with perfect information (i.e. in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses, 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 that 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). 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 front-page story. In this book, von Neumann declared that economic theory needed to use functional analytic methods, especially convex sets andtopological 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, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting andseparating hyperplanes and convex set, and fixed-point theory—have been the primary tools of mathematical economics ever since.

^{[35]}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*).^{[36]}
Morgenstern wrote a paper on game theory and thought he would show it to von Neumann because of his interest in the subject. He read it and said to Morgenstern that he should put more in it. This was repeated a couple of times, and then von Neumann became a coauthor and the paper became 100 pages long. Then it became a book.

^{[37]}### Mathematical economics[edit]

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 the Brouwer fixed-point theorem.

^{[34]}Von Neumann's model of an expanding economy considered the matrix pencil*with nonnegative matrices***A**− λ**B****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.^{[38]}^{[39]}^{[40]}
Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices.

^{[41]}The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.^{[42]}^{[43]}^{[44]}^{[45]}^{[46]}This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities,complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself.^{[47]}
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 toKenneth Arrow, in 1983 to Gérard Debreu, and in 1994 to John Nash who used fixed point theorems to establish equilibria for non-cooperative games and forbargaining 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.

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

^{[48]}### Linear programming[edit]

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 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, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.

^{[49]}
Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873), which was later popularized byKarmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squaressubproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming.

^{[49]}### Mathematical statistics[edit]

Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables.

^{[50]}This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic^{[51]}for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.^{[51]}
Subsequently, Denis Sargan and Alok Bhargava

^{[52]}extended the results for testing if the errors on a regression model follow a Gaussian random walk (*i.e.*, possess aunit root) against the alternative that they are a stationary first order autoregression.### Nuclear weapons[edit]

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

^{[3]}
Von Neumann's principal contribution to the atomic bomb 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" (compression).

When it turned out that there would not be enough uranium-235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the plutonium-239 that was available from the Hanford Site. 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 byGeorge Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945.

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.

^{[53]}
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.

^{[54]}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 bySecretary of War Henry L. Stimson.^{[55]}
On July 16, 1945, with numerous other Los Alamos personnel, von Neumann was an eyewitness to the first atomic bomb blast, code named Trinity, conducted as a test of the implosion method device, on the White Sands Proving Ground, 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 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.

^{[53]}
After the war, J. 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."

^{[citation needed]}
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 nuclear fusion.

^{[56]}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.^{[57]}
The Fuchs–von Neumann work was passed on, by Fuchs, to the Soviet Union as part of his nuclear espionage, but it was not used in the Soviets' own, independent development of the Teller–Ulam design. The historian Jeremy Bernstein has pointed out that ironically, "John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made."

^{[57]}### The Atomic Energy Committee[edit]

In 1954 von Neumann was invited to become a member of the Atomic Energy Committee. He accepted this position and used it to further the production of compact H-bombs suitable for Intercontinental ballistic missile delivery. He involved himself in correcting the severe shortage of tritium and lithium 6 needed for these compact weapons, and he argued against settling for the intermediate range missiles that the Army wanted. He was adamant that H-bombs delivered into the heart of enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile wouldn't be a problem with an H-bomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case.

^{[58]}### The ICBM Committee[edit]

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

### Mutual assured destruction[edit]

John von Neumann is credited with the equilibrium strategy of mutual 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). He also "moved heaven and earth" to bring MAD about. His goal was to quickly develop ICBMs and the compact hydrogen bombs that they could deliver to the USSR, and he knew the Soviets were doing similar work because the CIA interviewed German rocket scientists who were allowed to return to Germany, and von Neumann had planted a dozen technical people in the CIA. The Russians believed that bombers would soon be vulnerable, and they shared von Neumann's view that an H-bomb in an ICBM was thene plus ultra of weapons, and they believed that whoever had superiority in these weapons would take over the world, without necessarily using them.

^{[59]}von Neumann was afraid of a "missile gap" and took several more steps to achieve his goal of keeping up with the Soviets:- He modified the ENIAC by making it programmable and then wrote programs for it to do the H-bomb calculations verifying that the Teller-Ulam design was feasible and to develop it further.
- He became a member of the Atomic Energy Committee to speed up the development of a compact H-bomb that would fit in an ICBM.
- He personally interceded to speed up the production of lithium-6 and tritium needed for the compact bombs.
- He caused several separate missile projects to be started, because he felt that competition combined with collaboration got the best results.
^{[60]}

### Computing[edit]

Von Neumann was a founding figure in computing.

^{[62]}Von Neumann's hydrogen bomb work was 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 solutions to complicated problems to be approximated using random numbers. He was also involved in the design of the later IAS machine.
Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, 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 Pennsylvaniaon the EDVAC project, von Neumann wrote an incomplete

*First Draft of a Report on the EDVAC*. The paper, whose premature distribution nullified the patent claims of EDVAC designers J. Presper Eckert and John Mauchly, described a computer architecture in which the data and the program are both stored in the computer's memory in the same address space.^{[63]}
This architecture is to this day the basis of modern computer design, unlike the earliest computers that were "programmed" using a separate memory device such as a paper tape or plugboard.

^{[64]}Although the single-memory, 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 theENIAC computer at the University of Pennsylvania.^{[63]}
John von Neumann also consulted for the ENIAC project. The electronics of the new ENIAC ran at one-sixth the speed, but this in no way degraded the ENIAC's performance, since it was still entirely I/O bound. Complicated programs could be developed and debugged in days rather than the weeks required for plugboarding the old ENIAC. Some of von Neumann's early computer programs have been preserved.

^{[65]}
The next computer that von Neumann designed was the IAS machine at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the RCA Research Laboratory nearby. John von Neumann recommended that the IBM 701, nicknamed

*the defense computer*include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successful IBM 704.^{[66]}^{[67]}
Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953.

^{[68]}However, the theory could not be implemented until advances in computing of the 1960s.^{[69]}^{[70]}
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*.^{[71]}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 theirexponential growth.
Von Neumann's rigorous mathematical analysis of the structure of self-replication (of the semiotic relationship between constructor, description and that which is constructed), preceded the discovery of the structure of DNA.

^{[6]}
Beginning in 1949, von Neumann's design for a self-reproducing computer program is considered the world's first computer virus, and he is considered to be the theoretical father of computer virology.

^{[72]}
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.

^{[73]}
His algorithm for simulating a fair coin with a biased coin

^{[74]}is used in the "software whitening" stage of some hardware random number generators.### Fluid dynamics[edit]

Von Neumann made fundamental contributions in exploration of problems in numerical hydrodynamics. For example, with Robert 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,

^{[75]}and the co-discovery of the ZND detonation model of explosives.^{[76]}### Politics and social affairs[edit]

Von Neumann obtained, at the age of 29, one of the first five professorships at the new Institute for Advanced Study inPrinceton, 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 that was often at variance with the inclinations of his scientific colleagues.

^{[77]}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 US would triumph over totalitarianism from Nazism, Fascism and Soviet Communism.^{[78]}
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 US 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 Senatecommittee hearing he described his political ideology as "violently anti-communist, and much more militaristic than the norm". He was quoted in 1950 remarking, "If you say why not bomb [the Soviets] tomorrow, I say, why not today. If you say today at five o'clock, I say why not one o'clock?"

^{[79]}### On the eve of World War II[edit]

Von Neumann's shrewd prewar analysis is often quoted.

^{[80]}Asked about how France would stand up to Germany he said "Oh, France won't matter." Asked whether the US would enter the war and what their motives would be, he said they would enter the war as a purely defensive measure to protect their interests overseas and would not be motivated by imperialistic ambitions, but that such ambitions could arise after the war. He said the Roman Empire was purely defensive in the early days and only became imperialistic towards the end. He also said that it would not be profitable for the US to sell arms to combatants, because such sales are usually on credit, and such debts are never paid. He said this before 1935 when Roosevelt outlawed such sales.### Greece versus Rome[edit]

He loved the US and got tired of Europeans saying that Europe was cultured like Greece and the US lacked culture, like Rome. He would reply that yes, Greece was cultured and Rome wasn't, but Europe was descended from the Macedonians, who were barbarians.

^{[80]}### Weather systems[edit]

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.^{[81]}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 inducing global warming.^{[82]}^{[83]}### Cognitive abilities[edit]

Von Neumann's ability to instantaneously perform complex operations in his head stunned other mathematicians.

^{[84]}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."^{[85]}Paul Halmos states that "von Neumann's speed was awe-inspiring."^{[13]}Israel Halperin said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car."^{[86]}Edward Teller wrote that von Neumann effortlessly outdid anybody he ever met,^{[87]}and said "I never could keep up with him".^{[88]}Teller also said "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us. Most people avoid thinking if they can, some of us are addicted to thinking, but von Neumann actually enjoyed thinking, maybe even to the exclusion of everything else."^{[89]}
Lothar Wolfgang Nordheim described von Neumann as the "fastest mind I ever met",

^{[84]}and Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius."^{[90]}George Pólya, whose lectures at ETH Zürich von Neumann attended as a student, said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."^{[91]}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:^{[92]}
It's claimed that Von Neumann had a very strong eidetic memory, commonly called "photographic" memory—though such a phenomenon has never been scientifically documented in a human.

^{[18]}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 diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how*A Tale of Two Cities*started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes."^{[93]}
It has been said that von Neumann's intellect was absolutely unmatched. "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man", said Nobel Laureate Hans Bethe of Cornell University.

^{[18]}"It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei in "Selected Letters." Glimm writes "he is regarded as one of the giants of modern mathematics".^{[4]}The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians",^{[94]}while Peter Lax described him as possessing the "most scintillating intellect of this century".^{[95]}### Mastery of mathematics[edit]

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

^{[96]}- A facility with the symbolic manipulation of linear operators;
- An intuitive feeling for the logical structure of any new mathematical theories;
- An intuitive feeling for the combinatorial superstructure of new theories.

## Personal life[edit]

Von Neumann married twice. He married Mariette Kövesi in 1930, just prior to emigrating to the United States. Before his marriage he was baptized a Catholic in 1930 for the sake of his future wife's family.

^{[97]}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.
Von Neumann was initially refused permission to immigrate into the United States, although considered a famous and talented mathematician, but later with the influence of fellow scientists in the US he was able to secure the permit and received US citizenship. Von Neumann predicted the German takeover of Europe, anticipated its consequences for the Jews, and succeeded in ensuring the escape and immigration of his own immediate family along with his second wife Klara's family to the US, in 1938 just before the annexations and battles by Germany and the beginnings of World War II.

^{[98]}
The von Neumanns, Klara and John were very active socially within the Princeton academic community.

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 said that von Neumann had greater expertise in Byzantine history than he did.

^{[18]}
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 three-piece pin-stripe.

^{[78]}Mathematician David Hilbert is reported to have asked at von Neumann's 1926 doctoral exam: "Pray, who is the candidate's tailor?" as he had never seen such beautiful evening clothes.^{[99]}
He was sociable and enjoyed throwing large parties at his home in Princeton,

^{[18]}occasionally twice a week.^{[100]}His white clapboard house at 26 Westcott Road was one of the largest in Princeton.^{[101]}
Despite being a notoriously bad driver, he nonetheless enjoyed driving—frequently while reading a book—occasioning numerous arrests, as well as accidents. WhenCuthbert Hurd hired him as a consultant to IBM, Hurd often quietly paid the fines for his traffic tickets.

^{[102]}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.^{[18]}
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especiallylimericks).

^{[13]}At Princeton he received complaints for regularly playing extremely loud German march music on his gramophone, which distracted those in neighbouring offices, including Albert Einstein, from their work.^{[103]}Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly.^{[18]}
Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo 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.

^{[104]}## Later life[edit]

In 1955, von Neumann was diagnosed with what was either bone or pancreatic cancer.

^{[105]}A von Neumann biographer,Norman Macrae, has speculated that the cancer was caused by von Neumann's presence at the Operation Crossroads nuclear tests held in 1946 at Bikini Atoll.^{[106]}
His mother, Margaret von Neumann, had been diagnosed with cancer in 1936 and died within two weeks. John had eighteen months from diagnosis till death. In this period von Neumann returned to the Roman Catholic faith that had also been significant to his mother after the family's conversion in 1929–1930. John had earlier said to his mother, "There is probably a God. Many things are easier to explain if there is than if there isn't."

^{[107]}
Von Neumann held on to his exemplary knowledge of Latin and quoted to a deathbed visitor the declamation "Judex ergo cum sedebit," and ends "Quid sum miser tunc dicturus? Quem patronum rogaturus, Cum vix iustus sit securus?" (When the judge His seat hath taken ... What shall wretched I then plead? Who for me shall intercede when the righteous scarce is freed?)

^{[107]}^{[108]}
On his death bed, Von Neumann entertained his brother by using his photographic memory to recite from heart, word-for-word the first few lines of each page ofGoethe's

*Faust*.^{[18]}
Von Neumann died a year and a half after the diagnosis of cancer, at the Walter Reed Army Medical Center in Washington, D.C. under military security lest he reveal military secrets while heavily medicated. He was buried at Princeton Cemetery in Princeton, Mercer County, New Jersey.

^{[109]}
While at Walter Reed, he invited a Roman Catholic priest, Father Anselm Strittmatter, O.S.B., to visit him for consultation.

^{[110]}Von Neumann reportedly said in explanation that Pascal had a point, referring to Pascal's Wager.^{[111]}^{[112]}^{[113]}^{[114]}Father Strittmatter administered the last sacraments to him.^{[13]}Some of von Neumann's friends (such as Abraham Pais and Oskar Morgenstern) said they had always believed him to be "completely agnostic."^{[115]}^{[116]}Poundstone: "Of this deathbed conversion, Morgenstern told Heims, "He was of course completely agnostic all his life, and then he suddenly turned Catholic—it doesn't agree with anything whatsoever in his attitude, outlook and thinking when he was healthy." After the religious conversion, Father Strittmatter recalled that von Neumann did not receive much peace or comfort from it, as he still remained terrified of death.^{[117]}## Honors[edit]

- 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.
- The crater von Neumann on the Moon is named after him.
- The John von Neumann Center in Plainsboro Township, New Jersey (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.
^{[118]} - On February 15, 1956, von Neumann was presented with the Presidential Medal of Freedom by President Dwight D. Eisenhower.
- 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 designed by artist Victor Stabin. 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.

### Infopark and Neumann János Street[edit]

Infopark is situated in the 11th district of Budapest, near the Buda side of Rákóczi bridge, in the university neighborhood, across the river from the National Theatre and the Palace of Arts. The streets bordering Infopark are Hevesy György Street, Boulevard of Hungarian Scientists, Street of Hungarian Nobel Prize Winners and Neumann János street.

## Selected works[edit]

- 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 0-691-02893-1. - 1944.
*Theory of Games and Economic Behavior*, with Morgenstern, O., Princeton Univ. Press, online at archive.org. 2007 edition: ISBN 978-0-691-13061-3. - 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 0-08-009566-6 - 1966.
*Theory of Self-Reproducing Automata*, Burks, A. W., ed., University of Illinois Press. ISBN 0-598-37798-0^{[71]} - von Neumann, John (1998) [1960].
*Continuous geometry*. Princeton Landmarks in Mathematics. Princeton University Press. ISBN 978-0-691-05893-1.MR 0120174. - von Neumann, John (1981) [1937]. Halperin, Israel, ed.
*Continuous geometries with a transition probability*.*Memoirs of the American Mathematical Society***34**(252). ISBN 9780821822524. MR 634656.

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