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{{Redirect|Von Neumann}}
{{eastern name order|Neumann János Lajos}}
{{Infobox scientist
|name              = John von Neumann
|image            = JohnvonNeumann-LosAlamos.gif
|image_size        = 220px
|caption          = Von Neumann in the 1940s
|birth_name        = Neumann János Lajos
|birth_date        = {{birth date|mf=yes|1903|12|28}}
|birth_place      = [[Budapest]], [[Austria-Hungary]]
|death_date        = {{death date and age|mf=yes|1957|2|8|1903|12|28}}
|death_place      = [[Walter Reed General Hospital]]<br>[[Washington, D.C.]]
|residence        = United States
|citizenship      =
|nationality      = Hungarian and American
|fields            = [[Computer science]], [[economics]], [[mathematics]], [[physics]]
|workplaces        = [[University of Berlin]]<br />[[Princeton University]]<br />[[Institute for Advanced Study]]<br />[[Los Alamos National Laboratory|Site Y, Los Alamos]]
|alma_mater        = [[Eötvös Loránd University|University of Pázmány Péter]]<br />[[ETH Zürich]]
|doctoral_advisor  = [[Lipót Fejér]]
|academic_advisors = [[László Rátz]]
|doctoral_students = [[Donald B. Gillies]]<br />[[Israel Halperin]]
|notable_students  = [[Paul Halmos]]<br>[[Clifford Hugh Dowker]]
|known_for        = {{collapsible list|title={{nbsp}}|[[Abelian von Neumann algebra]]<br>[[Affiliated operator]]<br>[[Amenable group]]<br>[[Arithmetic logic unit]]<br>[[viscosity|Artificial viscosity]]<br>[[Axiom of regularity]]<br>[[Axiom of limitation of size]]<br>[[Backward induction]]<br>[[Blast wave|Blast wave (fluid dynamics)]]<br>[[Bounded set (topological vector space)]]<br>[[Carry-save adder]]<br>[[Class (set theory)]]<br>[[Quantum decoherence|Decoherence theory]]<br>[[Computer virus]]<br>[[Commutation theorem]]<br>[[Continuous geometry]]<br>[[Direct integral]]<br>[[Doubly stochastic matrix]]<br>[[Dual problem|Duality Theorem]]<br>[[Density matrix]]<br>[[Durbin–Watson statistic]]<br>[[Game theory]]<br>[[Hilbert's fifth problem]]<br>[[Hyperfinite type II factor]]<br>[[Ergodic theory]]<br>[[EDVAC]]<br>[[nuclear weapon design|explosive lenses]]<br>[[Lattice (order)|Lattice theory]]<br>[[Lifting theory]]<br>[[Inner model]]<br>[[Inner model theory]]<br>[[Interior point method]]<br>[[Mutual assured destruction]]<br>[[Merge sort]]<br>[[Middle-square method]]<br>[[Minimax|Minimax theorem]]<br> [[Monte Carlo method]]<br>[[Normal-form game]]<br>[[Pointless topology]]<br>[[Polarization identity]]<br>[[Pseudorandomness]]<br>[[Pseudorandom number generator|PRNG]]<br>[[Quantum mutual information]]<br>[[Radiation implosion]]<br>[[Rank ring]]<br>[[Operator theory]]<br>[[Operation Greenhouse]]<br>[[Self-replication]]<br>[[Hardware random number generator#Software whitening|Software whitening]]<br>[[Standard probability space]]<br>[[Stochastic computing]]<br>[[Subfactor]]<br>[[von Neumann algebra]]<br>[[von Neumann architecture]]<br>[[Von Neumann bicommutant theorem]]<br>[[Von Neumann cardinal assignment]]<br>[[Von Neumann cellular automaton]]<br>von Neumann constant (two of them)<br>[[Interpretation of quantum mechanics#von Neumann/Wigner interpretation: consciousness causes the collapse|Von Neumann interpretation]]<br>[[Measurement in quantum mechanics#von Neumann measurement scheme|von Neumann measurement scheme]]<br>[[Von Neumann ordinal|Von Neumann Ordinals]]<br>[[Von Neumann universal constructor]]<br>[[Von Neumann entropy]]<br>[[von Neumann Equation]]<br>[[Von Neumann neighborhood]]<br>[[Von Neumann paradox]]<br>[[Von Neumann regular ring]]<br>{{nowrap|[[Von Neumann–Bernays–Gödel set theory]]}}<br>[[Spectral theory|Von Neumann spectral theory]]<br>[[Von Neumann universe]]<br>[[Von Neumann conjecture]]<br>[[Von Neumann's inequality]]<br>[[Stone–von Neumann theorem]]<br>[[Von Neumann's trace inequality]]<br>[[Von Neumann stability analysis]]<br>[[Quantum statistical mechanics]]<br>[[Von Neumann extractor]]<br>[[Mean ergodic theorem|Von Neumann ergodic theorem]]<br>[[Ultrastrong topology]]<br>[[Von Neumann–Morgenstern utility theorem]]<br>[[ZND detonation model]]}}
|author_abbrev_bot =
|author_abbrev_zoo =
|influences        =
|influenced        =
|awards            = [[Bôcher Memorial Prize]] (1938), [[Enrico Fermi Award]] (1956)
|signature        = johnny von neumann sig.gif
|footnotes        =
}}
'''John von Neumann''' ({{IPAc-en|v|ɒ|n|_|ˈ|n|ɔɪ|m|ən}}; December 28, 1903 – February 8, 1957) was a Hungarian-born American pure and applied [[mathematician]] and [[polymath]]. He made major contributions to a number of fields,<ref name="NYT">{{cite web|author=[[Ed Regis (author)|Ed Regis]]|title=Johnny Jiggles the Planet|url=http://query.nytimes.com/gst/fullpage.html?res=9E0CE7D91239F93BA35752C1A964958260|work=[[The New York Times]]|date=1992-11-08|accessdate=2008-02-04}}</ref> including mathematics ([[foundations of mathematics]], [[functional analysis]], [[ergodic theory]], [[geometry]], [[topology]], and [[numerical analysis]]), physics ([[quantum mechanics]], [[hydrodynamics]], and [[fluid dynamics]]), [[economics]] ([[game theory]]), [[computer science]] ([[Von Neumann architecture]], [[linear programming]], [[Von Neumann universal constructor|self-replicating machines]], [[stochastic computing]]), and [[statistics]].<ref name="Glimm, p. vii">Glimm, p. vii</ref> He was a pioneer of the application of [[operator theory]] to [[quantum mechanics]], in the development of [[functional analysis]], a principal member of the [[Manhattan Project]] and the [[Institute for Advanced Study]] in [[Princeton, New Jersey|Princeton]] (as one of the few originally appointed), and a key figure in the development of [[game theory]]<ref name="NYT"/><ref>{{cite book|last=Nelson|first=David|title=The Penguin Dictionary of Mathematics|pages=178–179|isbn=0-14-101077-0|year=2003|publisher=Penguin|location=London}}</ref> and the concepts of [[cellular automata]],<ref name="NYT"/> the [[von Neumann universal constructor|universal constructor]], and the [[digital computer]].


Von Neumann's mathematical analysis of the structure of [[self-replication]] preceded the discovery of the [[structure of DNA]].<ref name=Rocha_lec_notes>{{cite web |last=Rocha|first=L.M.|title=Lecture Notes of I-585-Biologically Inspired Computing Course, Indiana University|contribution=Von Neumann and Natural Selection.|url=http://informatics.indiana.edu/rocha/i-bic/pdfs/ibic_lecnotes_c6.pdf}}</ref> In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of [[operator theory]], Berlin 1930 and Princeton 1935–1939; on the [[ergodic theorem]], Princeton, 1931–1932." Along with Hungarian-born American theoretical physicist [[Edward Teller]] and Polish 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.
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==Early life and education==
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Von Neumann was born '''Neumann János Lajos''' ({{IPA-hu|ˈnojmɒn ˈjaːnoʃ ˈlɒjoʃ}}; in Hungarian the family name comes first) in [[Budapest]], [[Austro-Hungarian Empire]], to wealthy [[Jewish]] parents.<ref>Doran, p. 1</ref><ref>Nathan Myhrvold, [http://www.time.com/time/magazine/article/0,9171,21839,00.html "John von Neumann".] [[Time (magazine)|Time]], March 21, 1999. Accessed September 5, 2010</ref><ref>Blair, p. 104</ref> He was the eldest of three brothers. His father, Neumann Miksa (Max Neumann) was a [[banker]], who held a [[doctor of law|doctorate in law]]. He had moved to Budapest from [[Pécs]] at the end of the 1880s. Miksa's father (Mihály b. 1839)<ref name="Mihaly">{{cite web|title=Mihály Neumann|url=http://www.geni.com/people/Mih%C3%A1ly-Neumann/6000000010710485708|publisher=[[Geni.com]]}}</ref> and grandfather (Márton)<ref name="Mihaly" /> were both born in Ond (now part of the town of [[Szerencs]]), [[Zemplén county]], northern Hungary. John's mother was Kann Margit (Margaret Kann).<ref>MacRae, pp. 37–38</ref>
 
Her parents were Jakab Kann II (Pest (now Budapest) 1845–1928) and Katalin Meisels ([[Munkács]], [[Kárpátalja]] c. 1854–1914). In 1913, his father was elevated to the nobility for his service to the [[Austria-Hungary|Austro-Hungarian empire]] by Emperor [[Franz Josef]]. The Neumann family thus acquiring the hereditary title ''margittai'', Neumann János became margittai Neumann János (John Neumann of Margitta), which he later changed to the German Johann von Neumann. János, nicknamed "Jancsi" (Johnny), was 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.<ref>Poundstone, William, ''Prisoner's Dilemma'', New York: Doubleday 1992</ref> By the age of 8, he was familiar with differential and integral [[calculus]].<ref name="Halmos, P.R 1973 pp. 382">{{cite journal|author=Halmos, P.R.|title=The Legend of von Neumann|journal=The American Mathematical Monthly-volume= 80|issue= 4–year=1973|pages=382–394|jstor=2319080|doi=10.2307/2319080}}</ref>
 
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).<ref>Doran, p. 2</ref>
 
John entered the German-speaking Lutheran high school [[Fasori Gimnázium|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.<ref>Glimm, p. 5</ref>
 
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.<ref>MacRae, p. 70</ref> By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of [[ordinal number]]s, which superseded [[Georg Cantor]]'s definition.<ref>Nasar, Sylvia, ''A Beautiful Mind'', London 2001, p. 81 ISBN 0743224574.</ref>
 
He received his [[Doctor of Philosophy|Ph.D.]] in [[mathematics]] (with minors in [[experimental physics]] and [[chemistry]]) from [[Eötvös Loránd University|Pázmány Péter University]] in Budapest at the age of 22.<ref name="NYT"/> He simultaneously earned a diploma in [[chemical engineering]] from the [[ETH Zurich]] in Switzerland<ref name="NYT"/> 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.{{#tag:ref|''Life Magazine'' stated that he received both his undergraduate degree and his PhD at the age of 21.<ref name="Life Magazine 1957, pages 89-104"/>|group="N"}}
 
==Career and abilities==
 
===Beginnings===
Between 1926 and 1930, he taught as a ''[[Privatdozent]]'' at the [[Humboldt University of Berlin|University of Berlin]], the youngest in its history.{{#tag:ref|While teaching as a professor he was frequently mistaken to be a student.|group="N"}} 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.<ref>MacRae, p. 145</ref> Von Neumann's powers of [[Eidetic memory|speedy, massive memorization and recall]] allowed him to recite volumes of information, and even entire directories, with ease.<ref name="Life Magazine 1957, pages 89-104"/>
 
In 1930, von Neumann was invited to [[Princeton University]], [[Princeton, New Jersey|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, having died in 1929. He [[anglicize]]d his first name to John, keeping the German-aristocratic surname of von Neumann. In 1937, von Neumann became a [[Naturalized citizen#United States|naturalized citizen]] of the U.S. In 1938, he was awarded the [[Bôcher Memorial Prize]] for his work in analysis.
 
===Set theory===
{{Refimprove section|date=April 2007}}
{{See also|Von Neumann–Bernays–Gödel set theory#History}}
 
The axiomatization of mathematics, on the model of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', had reached new levels of rigor and breadth at the end of the 19th century, particularly in arithmetic (thanks to the [[Peano axioms|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|naive]] [[set theory]] suffered a setback due to [[Russell's paradox]] (on the set of all sets that do not belong to themselves).
 
The problem of an adequate axiomatization of [[set theory]] was resolved implicitly about twenty years later (by [[Ernst Zermelo]] and [[Abraham Fraenkel]]). Zermelo and Fraenkel provided [[Zermelo–Fraenkel set theory]], 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 (set theory)|class]].''
 
[[File:NeumannVonMargitta.jpg|thumb|350px|Excerpt from the university calendars for 1928 and 1928/29 of the [[Humboldt University of Berlin|Friedrich-Wilhelms-Universität Berlin]] announcing Neumann's lectures on axiomatic set theory and logics, problems in quantum mechanics and special mathematical functions.]]
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 model]]s'') 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 [[Gödel's incompleteness theorems|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.<ref name="John2005">{{cite book|author=John von Neumann|editor=Miklós Rédei|title=John von Neumann: Selected letters|publisher=[[American Mathematical Society]]|series=History of Mathematics|volume=27|year=2005|page=123|isbn=0-8218-3776-1}}</ref>
 
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.<ref name="John2005" /> 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.<ref>{{cite book|author=John von Neumann|editor=Miklós Rédei|title=John von Neumann: Selected letters|publisher=[[American Mathematical Society]]|series=History of Mathematics|volume=27|year=2005|page=124|isbn=0-8218-3776-1}} "Many thanks for your letter and your reprint. As you have established the unprovability of consistency as a natural continuation and deepening of your earlier results, I clearly won't publish on this subject."</ref>
 
===Geometry===
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 algebra]]s with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the [[hyperfinite type II factor]].
 
===Measure theory===
{{See also|Lifting theory}}
 
In a series of famous papers, von Neumann made spectacular contributions to [[measure theory]].<ref name="measure">{{cite journal|title=Von Neumann on measure and ergodic theory|author=[[Paul R. Halmos]]|journal=[[Bull. Amer. Math. Soc.]]|volume=64|issue=3, Part 2|year=1958|pages=86–94|url=http://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10203-7/S0002-9904-1958-10203-7.pdf|doi=10.1090/S0002-9904-1958-10203-7}}</ref> 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 function]]s).
 
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.<ref name="measure"/><ref>{{cite journal|first=J.|last=von Neumann|title=Die Einfuhrung Analytischer Parameter in Topologischen Gruppen|journal=[[Annals of Mathematics]]|volume=34|issue=1|series=2|year=1933|pages=170–179|doi=10.2307/1968347}}</ref>
 
=== Ergodic theory ===
Von Neumann made foundational contributions to [[ergodic theory]], in a series of articles published in 1932.<ref>Two famous papers are below: {{cite journal|first=John|last=von Neumann|title=Proof of the Quasi-ergodic Hypothesis|year=1932|journal=Proc Natl Acad Sci USA|volume=18|pages=70–82|doi=10.1073/pnas.18.1.70|pmid=16577432|issue=1|pmc=1076162|bibcode=1932PNAS...18...70N }}.
{{cite journal|first=John|last=von Neumann|title=Physical Applications of the Ergodic Hypothesis|year=1932|journal=Proc Natl Acad Sci USA|volume=18|pages=263–266|doi=10.1073/pnas.18.3.263|pmid=16587674|issue=3|pmc=1076204|jstor=86260|bibcode=1932PNAS...18..263N}}.
{{cite journal|authorlink=Eberhard_Hopf|first=Eberhard|last=Hopf|title=Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung|year=1939|journal=Leipzig Ber. Verhandl. Sächs. Akad. Wiss.|volume=91|pages=261–304}}.</ref> 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".<ref name="measure"/> By then von Neumann had already written his famous articles on [[operator theory]], and the application of this work was instrumental in the [[Ergodic theory#Mean ergodic theorem|von Neumann mean ergodic theorem]].<ref>[[Michael C. Reed]], [[Barry Simon]], ''Methods of Modern Mathematical Physics, Volume 1: Functional Analysis'', [[Academic Press]]; Revised edition (1980)</ref>
 
===Operator theory===
{{main|Von Neumann algebra}}
 
Von Neumann introduced the study of rings of operators, through the [[von Neumann algebra]]s.<ref>D.Petz and M.R. Redi, [http://books.google.com/books?id=MY2_V2BfP5cC&pg=PA163 ''John von Neumann And The Theory Of Operator Algebras''], in ''The Neumann compendium'', [[World Scientific]], 1995, pp. 163–181 ISBN 9810222017.</ref> A von Neumann algebra is a [[*-algebra]] of [[bounded operators]] on a Hilbert space that is closed in the weak operator topology and contains the [[identity operator]].
 
The [[von Neumann bicommutant theorem]] shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
 
The [[direct integral]] was introduced in 1949 by John von Neumann. 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===
[[Garrett Birkhoff]] writes: "John von Neumann's brilliant mind blazed over [[lattice theory]] like a meteor".<ref name="Garrett Birkhoff 1958, page 50-5">{{cite journal|url=http://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10192-5/S0002-9904-1958-10192-5.pdf%7C|author=Garrett Birkhoff|title=Von Neumann and lattice theory|journal= Bull. Amer. Math. Soc.|volume=64|issue=3|isbn=0821810251|year=1958|pages= 50–56|doi=10.1090/S0002-9904-1958-10192-5}}</ref> 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."<ref name="Garrett Birkhoff 1958, page 50-5"/>
 
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 [[Ideal (ring theory)|right-ideals]] of a suitable [[Von Neumann regular ring|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."<ref name="Garrett Birkhoff 1958, page 50-5"/>
 
===Mathematical formulation of quantum mechanics===
{{See also|von Neumann entropy|Density matrix|Quantum mutual information|Measurement in quantum mechanics#von Neumann measurement scheme|label 4 = von Neumann measurement scheme|Wave function collapse}}
{{quantum}}
 
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 ''Mathematische Grundlagen der Quantenmechanik''.
 
After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a 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 [[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 theory|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 make the mistake that Bell claimed he had made.<ref>{{cite journal|title=Von Neumann's 'No Hidden Variables' Proof: A Re-Appraisal|year=2010|last1=Bub|first1=Jeffrey|journal=[[Foundations of Physics]]|volume=40|issue=9–10|pages=1333–1340|bibcode=2010FoPh...40.1333B|doi=10.1007/s10701-010-9480-9|arxiv=1006.0499}}</ref> 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 collapse|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).<ref>von Neumann, John. (1932/1955). ''Mathematical Foundations of Quantum Mechanics''. Princeton: [[Princeton University Press]]. Translated by Robert T. Beyer.</ref>
 
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 [[Interpretations of quantum mechanics|interpretation of the theory]], and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.
 
===Quantum logic===
In a famous paper of 1936, the first work ever to introduce quantum logics,<ref>[[Dov M. Gabbay]], John Woods, [http://books.google.com/books?id=3TNj1ZkP3qEC&pg=PA205 ''The Many Valued and Nonmonotonic Turn in Logic''], [[Elsevier]], 2007, pp. 205–217 ISBN 0444516239.</ref> von Neumann first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work. But in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters which are polarized perpendicularly (e.g., one horizontally and the other vertically), and therefore, [[a fortiori]], it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession. But if the third filter is added ''in between'' the other two, the photons will indeed pass through. And this experimental fact is translatable into logic as the ''non-commutativity'' of conjunction <math>(A\land B)\ne (B\land A)</math>. It was also demonstrated that the laws of distribution of classical logic, <math>P\lor(Q\land R)=(P\lor Q)\land(P\lor R)</math> and
<math>P\land (Q\lor R)=(P\land Q)\lor(P\land R)</math>, 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 ''<math>A \land (B\lor C)= A\land 1 = A</math>'', while <math>(A\land B)\lor (A\land C)=0\lor 0=0</math>.
 
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).<ref>''Philosophical Papers: Volume 3, Realism and Reason'', [[Hilary Putnam]], [[Cambridge University Press]], 27 December 1985, p. 263</ref>
 
===Game theory===
Von Neumann founded the field of [[game theory]] as a mathematical discipline. Von Neumann proved his [[minimax theorem]] in 1928. This theorem establishes that in [[zero-sum|zero-sum games]] with [[perfect information]] (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy which will result in the minimization of his maximum loss.
 
Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). 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 analysis|functional analytic]] methods, especially [[convex set]]s and [[topology|topological]] [[fixed point theorem (disambiguation)|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 lattice]]s. Von Neumann's functional-analytic techniques—the use of [[dual vector space|duality pairings]] of [[real vector space]]s to represent prices and quantities, the use of [[supporting hyperplane|supporting]] and [[separating hyperplane theorem|separating hyperplanes]] and convex set, and fixed-point theory—have been the primary tools of mathematical economics ever since.<ref>{{cite book|last=Blume|first=Lawrence&nbsp;E.|authorlink=Lawrence E. Blume|chapter=Convexity|year=2008c|title=[[The New Palgrave Dictionary of Economics]]|editor=[[Steven N. Durlauf|Durlauf, Steven N.]] and Blume, Lawrence E|publisher=[[Palgrave Macmillan]]|edition=Second|url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508|doi=10.1057/9780230226203.0315}}
* {{cite book|author=Blume, Lawrence E|chapter=Convex programming|year=2008cp|authorlink=Lawrence E. Blume|title=The New&nbsp;Palgrave Dictionary of Economics|publisher=Palgrave Macmillan|edition=2nd|url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000348|doi=10.1057/9780230226203.0314}}
* {{cite book|last=Blume|first=Lawrence&nbsp;E.|authorlink=Lawrence E. Blume|editor2-link=Lawrence E. Blume|chapter=Duality|year=2008d|title=The New&nbsp;Palgrave Dictionary of Economics|editor-first=Steven&nbsp;N.|editor-last=Durlauf|editor2-first=Lawrence&nbsp;E|editor2-last=Blume|publisher=Palgrave Macmillan|edition=Second|url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626|doi=10.1057/9780230226203.0411}}
* {{cite book|first1=Jerry|last1=Green|first2=Walter&nbsp;P.|last2=Heller|chapter=1 Mathematical&nbsp;analysis and&nbsp;convexity with applications to economics|pages=15–52|url=http://www.sciencedirect.com/science/article/B7P5Y-4FDF0FN-5/2/613440787037f7f62d65a05172503737|doi=10.1016/S1573-4382(81)01005-9|title=Handbook of mathematical&nbsp;economics, Volume&nbsp;I|editor1-link=Kenneth Arrow |editor1-first=Kenneth&nbsp;Joseph|editor1-last=Arrow|editor2-first=Michael&nbsp;D|editor2-last=Intriligator|series=Handbooks in economics|volume=1|publisher=North-Holland Publishing&nbsp;Co.|location=Amsterdam|year=1981|isbn=0-444-86126-2|mr=634800}}
* {{cite book|last=Mas-Colell|first=A.|authorlink=Andreu Mas-Colell|chapter=Non-convexity|title=The New Palgrave: A Dictionary of Economics|editor1-first=John|editor1-last=Eatwell |editor1-link=John Eatwell|editor2-first=Murray|editor2-last=Milgate |editor2-link=Murray Milgate|editor3-first=Peter|editor3-last=Newman|editor3-link=Peter Kenneth Newman|publisher=Palgrave Macmillan|year=1987|edition=first|doi=10.1057/9780230226203.3173|pages=653–661|url=http://www.econ.upf.edu/~mcolell/research/art_083b.pdf}}
* {{cite book|last=Newman|first=Peter|authorlink=Peter Kenneth Newman|chapter=Convexity|title=The New Palgrave: A Dictionary of Economics|editor1-first=John|editor1-last=Eatwell|editor2-first=Murray|editor2-last=Milgate|editor3-first=Peter|editor3-last=Newman|editor3-link=Peter Kenneth Newman|publisher=Palgrave Macmillan|year=1987c|edition=first|doi=10.1057/9780230226203.2282<!-- SNAFU at NP? 30 January 2011-->|url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000453}}
* {{cite book|last=Newman|first=Peter|authorlink=Peter Kenneth Newman|chapter=Duality|title=The New Palgrave: A Dictionary of Economics|editor1-first=John|editor1-last=Eatwell|editor2-first=Murray|editor2-last=Milgate|editor3-first=Peter|editor3-last=Newman|editor3-link=Peter Kenneth Newman|publisher=Palgrave Macmillan|year=1987d|edition=first|doi=10.1057/9780230226203.2412<!-- SNAFU at NP? 30 January 2011-->|url=http://www.dictionaryofeconomics.com/article?id=pde1987_X000626}}</ref> 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'').<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Projects/MacQuarrie/Chapters/Ch4.html|title=Mathematics and Chess|author=John MacQuarrie|publisher=School of Mathematics and Statistics, [[University of St Andrews]], Scotland|accessdate=2007-10-18|quote=Others claim he used a method of proof, known as 'backwards induction' that was not employed until 1953, by von Neumann and Morgenstern. [[Ken Binmore]] (1992) writes, Zermelo used this method way back in 1912 to analyze Chess. It requires starting from the end of the game and then working backwards to its beginning. (p.32)}}</ref>
 
===Mathematical economics===
Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of [[Brouwer's fixed point theorem]]. Von Neumann's model of an expanding economy considered the [[generalized eigenvalue problem|matrix pencil]]&nbsp;'' '''A'''&nbsp;−&nbsp;λ'''B''' '' with nonnegative matrices&nbsp;'''A''' and '''B'''; von Neumann sought [[probability vector|probability]] [[generalized eigenvector|vectors]]&nbsp;''p'' and&nbsp;''q'' and a positive number&nbsp;''λ'' that would solve the [[complementarity theory|complementarity]] equation
:''p''<sup>T</sup>&nbsp;('''A''' − ''λ''&nbsp;'''B''')&nbsp;''q'' = 0,
along with two inequality systems expressing economic efficiency. In this model, the ([[transpose]]d) 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 [[generalized eigenvalue problem|solution]] ''λ'' represents the growth factor which is 1 plus the [[economic growth|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.<ref>For this problem to have a unique solution, it suffices that the nonnegative matrices&nbsp;'''A''' and&nbsp;'''B''' satisfy an [[Perron–Frobenius theorem|irreducibility condition]], generalizing that of the [[Perron–Frobenius theorem]] of nonnegative matrices, which considers the (simplified) [[eigenvalue|eigenvalue problem]]
: '''A''' − λ '''I''' ''q'' = 0,
where the nonnegative matrix''&nbsp;'''A''''' must be square and where the [[diagonal matrix]]''&nbsp;'''I''' ''is the [[identity matrix]]. Von Neumann's irreducibility condition was called the "whales and [[Wrangler (University of Cambridge)|wranglers]]" hypothesis by David Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good. Weaker "irreducibility" conditions were given by [[David Gale]] and by [[John George Kemeny|John Kemeny]], [[Oskar Morgenstern]], and [[Gerald L. Thompson]] in the 1950s and then by [[Stephen M. Robinson]] in the&nbsp;1970s.</ref><ref>David Gale. ''The theory of linear economic models''. [[McGraw–Hill]], New York, 1960.</ref><ref>{{cite book|last1=Morgenstern|first1=Oskar|authorlink1=Oskar Morgenstern|last2=Thompson|first2=Gerald L.|authorlink2=Gerald L. Thompson|title=Mathematical theory of expanding and contracting economies|series=Lexington Books| publisher=[[D. C. Heath and Company]]|year=1976|location=Lexington, Massachusetts|pages=xviii+277|isbn=0-669-00089-2 }}</ref>
 
Von Neumann's results have been viewed as a special case of [[linear programming]], where von Neumann's model uses only nonnegative matrices.<ref>[[Alexander Schrijver]], ''Theory of Linear and Integer Programming''. [[John Wiley & Sons]], 1998, ISBN 0-471-98232-6.</ref> The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.<ref>
* {{cite book|last=Rockafellar|first=R. Tyrrell|title=Monotone processes of convex and concave type|series=Memoirs of the American Mathematical Society|publisher=American Mathematical Society|location=Providence, R.I.|pages=i+74|issue=77|isbn=0-8218-1277-7|year=1967}}
* {{cite book|last=Rockafellar|first=R. T.|authorlink=R. Tyrrell Rockafellar|chapter=Convex algebra and duality in dynamic models of production|title=Mathematical models in economics (Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972)|editor=Josef Loz and Maria Loz|pages=351–378|publisher=[[North-Holland Publishing]] and [[Polish Academy of Sciences]] (PAN)|location=Amsterdam|year=1974}}
* {{cite book|authorlink=R. Tyrrell Rockafellar|first=R. T.|last=Rockafellar|title=Convex analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1970 (Reprint 1997 as a Princeton classic in mathematics)|isbn=0-691-08069-0}}
</ref><ref>{{cite book|title=John Von Neumann and modern economics|editor=Mohammed Dore, [[Sukhamoy Chakravarty]], [[Richard Murphy Goodwin|Richard Goodwin]]| publisher=Oxford: Clarendon|year=1989|page=261|author=[[Kenneth Arrow]], [[Paul Samuelson]], [[John Harsanyi]], [[Sidney Afriat]], [[Gerald L. Thompson]], and [[Nicholas Kaldor]].}}</ref><ref>[[Yinyu Ye]]. Chapter 9.1 [http://books.google.com/books?id=RQZd7ru8cmMC&pg=PA277 "The von Neumann growth model"], pp. 277–299 in ''Interior point algorithms: Theory and analysis''. Wiley. 1997 ISBN 0471174203.</ref> This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of [[fixed-point theorems]], [[linear inequalities]], [[Linear programming#Complementary slackness|complementary slackness]], and [[dual problem|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.<ref>''Contributions to von Neumann's Growth Model'', Proceedings of a Conference Organized by the Institute for Advanced Studies Vienna, Austria, 6 and 7 July 1970, Prof. Dr. Gerhart Bruckmann and Prof. Dr. Wilhelm Weber (eds), ISBN 978-3-662-22738-1 (Print) 978-3-662-24667-2 (Online), Springer Verlag, September 21, 1971, {{doi | 10.1007/978-3-662-24667-2}}.</ref>
 
The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of Nobel prizes in 1972 to [[Kenneth Arrow]], in 1983 to [[Gérard Debreu]], and in 1994 to [[John Forbes Nash, Jr.|John Nash]] who used fixed point theorems to establish equilibria for [[noncooperative game]]s and for [[bargaining problem]]s in his Ph.D. thesis. Arrow and Debreu also used linear programming, as did Nobel laureates [[Tjalling Koopmans]], [[Leonid Kantorovich]], [[Wassily Leontief]], [[Paul Samuelson]], [[Robert Dorfman]], [[Robert Solow]], and [[Leonid Hurwicz]].
 
===Linear programming===
Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in [[linear programming]], after [[George B. Dantzig]] described his work in a few minutes, when an impatient von Neumann asked him to get to the point. Then, Dantzig listened dumbfounded while von Neumann provided an hour lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.<ref name="George B 2003">George B. Dantzig and Mukund N. Thapa. 2003. ''Linear Programming 2: Theory and Extensions''. [[Springer-Verlag]] ISBN 1441931406.</ref>
 
Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873) which was later popularized by [[Karmarkar's algorithm]]. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative [[least squares]] subproblem with a convexity constraint ([[orthogonal projection|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.<ref name="George B 2003"/>
 
===Mathematical statistics===
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.<ref>{{cite journal|author=von Neumann, John|year=1941|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177731677|title=Distribution of the ratio of the mean square successive difference to the variance|journal=[[Annals of Mathematical Statistics]]|volume=12|pages=367–395|jstor=2235951|doi=10.1214/aoms/1177731677|issue=4}}</ref> This ratio was applied to the residuals from regression models and is commonly known as the [[Durbin–Watson]] statistic<ref name="jstor.org">{{cite journal|author=Durbin, J., and Watson, G. S.|year=1950|title=Testing for Serial Correlation in Least Squares Regression, I|journal=[[Biometrika]]|volume=37|pages=409–428|pmid=14801065|issue=3–4|doi=10.2307/2332391}}</ref> for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order [[autoregression]].<ref name="jstor.org"/>
 
Subsequently, [[John Denis Sargan]] and [[Alok Bhargava]]<ref>{{cite journal|author=Sargan, J.D. and Bhargava, Alok |year=1983|jstor=1912252 |title=Testing residuals from least squares regression for being generated by the Gaussian random walk|journal=[[Econometrica]]|volume=51|pages=153–174|doi=10.2307/1912252}}</ref> extended the results for testing if the errors on a regression model follow a Gaussian [[random walk]] (i.e. possess a [[unit root]]) against the alternative that they are a stationary first order [[autoregression]].
 
===Nuclear weapons===
[[File:John von Neumann ID badge.png|thumb|250px|von Neumann's wartime [[Los Alamos National Laboratory|Los Alamos]] ID badge photo.]]
 
Beginning in the late 1930s, von Neumann developed an expertise in [[explosion]]s—phenomena which are difficult to model mathematically. During this period von Neumann was the leading authority of the mathematics of [[shaped charges]]. This led him to a large number of military consultancies, primarily for the Navy, which in turn led to his involvement in the [[Manhattan Project]]. The involvement included frequent trips by train to the project's secret research facilities in [[Los Alamos National Laboratory|Los Alamos, New Mexico]].<ref name="NYT"/>
 
Von Neumann's principal contribution to the [[atomic bomb]] itself was in the concept and design of the [[nuclear weapon design|explosive lenses]] needed to compress the [[plutonium]] core of the [[Trinity test]] device and the "[[Fat Man]]" weapon that was later dropped on [[Nagasaki, Nagasaki|Nagasaki]]. While von Neumann did not originate the "[[implosion-type nuclear weapon|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 [[U235]] to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the [[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 by [[George 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.<ref name="Critical Assembly">{{cite book|title=Critical Assembly: A Technical History of Los Alamos during the Oppenheimer Years, 1943–1945|coauthors=Paul W. Henriksen, Roger A. Meade, Catherine Westfall|year=1993|publisher=Cambridge University Press|location=Cambridge, UK|isbn=0-521-44132-3|author=Lillian Hoddeson, with contributions from [[Gordon Baym]] }}</ref>
 
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, Nagasaki|Nagasaki]] as the [[Atomic bombings of Hiroshima and Nagasaki|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.<ref name="Rhodes">{{cite book|last=Rhodes|first=Richard|title=The Making of the Atomic Bomb|year=1986|location=New York|publisher=Touchstone [[Simon & Schuster]]|isbn=0-684-81378-5 }}</ref> The cultural capital [[Kyoto]], which had been spared the [[firebombing]] inflicted upon militarily significant target cities like [[Bombing of Tokyo|Tokyo in World War II]], was von Neumann's first choice, a selection seconded by Manhattan Project leader General [[Leslie Groves]]. However, this target was dismissed by [[United States Secretary of War|Secretary of War]] [[Henry Stimson]].<ref name="Groves">{{cite book|last=Groves|first=Leslie|title=Now It Can Be Told: The Story of the Manhattan Project|year=1962|location=New York|publisher=[[Da Capo Press]]|isbn=0-306-80189-2}}</ref>
 
On July 16, 1945, with numerous other Los Alamos personnel, von Neumann was an eyewitness to the [[Trinity test|first atomic bomb blast]], conducted as a test of the implosion method device, 35 miles (56&nbsp;km) southeast of [[Socorro, New Mexico|Socorro]], [[New Mexico]]. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5 [[ton|kilotons]] of [[TNT equivalent|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.<ref name="Critical Assembly" />
 
After the war, [[Robert Oppenheimer]] remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."
 
Von Neumann continued unperturbed in his work and became, along with [[Edward Teller]], one of those who sustained the [[Thermonuclear weapon|hydrogen bomb project]]. He then collaborated with [[Klaus Fuchs]] on further development of the bomb, and in 1946 the two filed a secret patent on "Improvement in Methods and Means for Utilizing Nuclear Energy", which outlined a scheme for using a fission bomb to compress fusion fuel to initiate a [[thermonuclear reaction]].<ref>Herken, pp.&nbsp;171, 374</ref> 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.<ref name="Bernstein2010">{{cite journal|last1=Bernstein|first1=Jeremy|title=John von Neumann and Klaus Fuchs: an Unlikely Collaboration|journal=Physics in Perspective|volume=12|page=36|year=2010|doi=10.1007/s00016-009-0001-1|bibcode = 2010PhP....12...36B}}</ref>
 
The Fuchs–von Neumann work was passed on, by Fuchs, to the USSR as part of his [[nuclear espionage]], but it was not used in the 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."<ref name="Bernstein2010" />
 
===The ICBM Committee===
In 1955, von Neumann became a commissioner of the [[United States Atomic Energy Commission|United States Atomic Energy Program]]. Shortly before his death, when he was already quite ill, von Neumann headed the US government's top secret von Neumann [[Intercontinental ballistic missile]] (ICBM) committee. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were sizeable, 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.
 
===Mutually assured destruction===
John von Neumann is credited with the equilibrium strategy of [[mutually assured destruction]], providing the deliberately humorous acronym, MAD. (Other humorous acronyms coined by von Neumann include his computer, the [[MANIAC I|Mathematical Analyzer, Numerical Integrator, and Computer]] – or MANIAC).
 
===Computer science===
[[File:Nobili Pesavento 2reps.png|right|thumb|400px|The first implementation of von Neumann's self-reproducing universal constructor.<ref name=Pesavento1995>{{cite journal|journal=[[Artificial Life (journal)|Artificial Life]]| title=An implementation of von Neumann's self-reproducing machine|year=1995|first=Umberto|last=Pesavento|volume=2|issue=4|pages=337–354|publisher=[[MIT Press]]|url=https://web.archive.org/web/20070418081628/http://dragonfly.tam.cornell.edu/~pesavent/pesavento_self_reproducing_machine.pdf|format=[[PDF]]|doi=10.1162/artl.1995.2.337|pmid=8942052}}</ref> Three generations of machine are shown, the second has nearly finished constructing the third. The lines running to the right are the tapes of genetic instructions, which are copied along with the body of the machines. The machine shown runs in a 32-state version of von Neumann's cellular automata environment.]]
 
Von Neumann was a founding figure in computer science.<ref>Goldstine, pp. 167–178.</ref> 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 [[Algorithmically random sequence|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 number]]s, 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 Pennsylvania]] on the [[EDVAC]] project, von Neumann wrote an incomplete ''[[First Draft of a Report on the EDVAC]]''. The paper, whose public distribution nullified the patent claims of EDVAC designers [[J. Presper Eckert]] and [[John William 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.<ref name="ENIAC museum">The name for the architecture is discussed in [http://www.library.upenn.edu/exhibits/rbm/mauchly/jwm9.html John W. Mauchly and the Development of the ENIAC Computer], part of the online [http://www.seas.upenn.edu/~museum/ ENIAC museum], in Robert Slater's computer history book, ''Portraits in Silicon'' (MIT Press, 1989), and in Nancy Stern's book ''From ENIAC to UNIVAC'' (Digital Press,1981).</ref>
 
John von Neumann also consulted for the [[ENIAC]] project, when ENIAC was being modified to contain stored programs. Since the modified ENIAC was fully functional by 1948 and the EDVAC wasn't delivered to [[Ballistics Research Laboratory]] until 1949, one could argue that ENIAC was the first computer to use a stored program in production runs. John von Neumann also designed the instruction set or op codes for the modified ENIAC, and he should be given credit for this. The electronics of the new ENIAC ran 6 times slower, but this in no way degraded the ENIAC's performance, since it was still entirely [[I/O bound]]. On the other hand, complicated programs could be developed and debugged in days rather than weeks, which is one of the advantages of storing the entire program in the computers electronics. The crucial point is whether or not a new paper tape has to be produced, using a slow and error prone tape punch every time the program is altered in any way, and then read in mechanically. A program is typically modified many times before it reaches its final form.
 
This architecture is to this day the basis of modern computer design, unlike the earliest computers that were 'programmed' by altering the electronic circuitry. Although the 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 the ENIAC at the University of Pennsylvania.<ref name="ENIAC museum"/>
 
[[Stochastic computing]] was first introduced in a pioneering paper by von Neumann in 1953.<ref>{{cite conference|last=von Neumann|first=J.|title=Probabilistic logics and the synthesis of reliable organisms from unreliable components|booktitle=The Collected Works of John von Neumann|publisher=Macmillan|year=1963|isbn=978-0-393-05169-8}}</ref> However, the
theory could not be implemented until advances in computing of the 1960s.<ref>{{cite conference|last1=Petrovic|first1= R.|last2=Siljak|first2=D.|title=Multiplication by means of coincidence|year=1962|booktitle=ACTES Proc. of 3rd Int. Analog Comp. Meeting}}</ref><ref>{{cite journal|last=Afuso|first=C.|title=Quart. Tech. Prog. Rept|location=[[Department of Computer Science, University of Illinois, Urbana-Champaign]], Illinois|year=1964}}</ref>
 
Von Neumann also created the field of [[cellular automaton|cellular automata]] without the aid of computers, constructing the first [[self-replication|self-replicating]] automata with pencil and graph paper. The concept of a [[Von Neumann universal constructor|universal constructor]] was fleshed out in his posthumous work ''Theory of Self Reproducing Automata''.<ref name=TSRA>{{cite book|year=1966|author=John von Neumann|editor=[[Arthur W. Burks]]|title=Theory of Self-Reproducing Automata|publisher=[[University of Illinois Press]]|place=Urbana and London|isbn=0-598-37798-0}} [http://www.history-computer.com/Library/VonNeumann1.pdf PDF reprint]</ref> Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire [[natural satellite|moon]] or [[asteroid belt]] would be by using self-replicating machines, taking advantage of their [[exponential growth]].
 
Von Neumann's rigorous mathematical analysis of the structure of [[self-replication]], preceded the discovery of the structure of DNA.<ref name="Rocha_lec_notes" />
 
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.<ref>Éric Filiol, [http://books.google.com/books?id=CZGLFf6IhCIC&pg=PA19 ''Computer viruses: from theory to applications, Volume 1''], [[Birkhäuser]], 2005, pp. 19–38 ISBN 2287239391.</ref>
 
[[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.<ref>{{cite book|last=Knuth|first=Donald|authorlink=Donald Knuth|year=1998|title=The Art of Computer Programming: Volume 3 Sorting and Searching|page=159|isbn=0-201-89685-0|publisher=[[Addison–Wesley]]|location=Boston}}</ref>
 
His algorithm for simulating a [[fair coin]] with a biased coin<ref>{{cite journal|last=von Neumann|first=John|year=1951|title=Various techniques used in connection with random digits|journal=[[National Bureau of Standards]] Applied Math Series|volume=12|page=36}}</ref> is used in the "software whitening" stage of some [[hardware random number generator]]s.
 
===Fluid dynamics===
Von Neumann made fundamental contributions in exploration of problems in numerical [[hydrodynamics]]. For example, with [[R. D. Richtmyer]] he developed an algorithm defining ''artificial [[viscosity]]'' that improved the understanding of [[shock wave]]s. It is possible that we would not understand much of astrophysics, and might not have highly developed [[Jet engine|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 wave]]s). 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 wave]]s,<ref>Neumann, John von, "The Point Source Solution," John von Neumann. Collected Works, A. J. Taub (ed.), Vol. 6 [Elmsford, N.Y.: Pergamon Press, 1963], pp. 219–237</ref> and the co-discovery of the [[ZND detonation model]] of explosives.<ref>{{cite book|last=von Neumann|first=J.|contribution=Theory of Detonation Waves. Progress Report to the National Defense Research Committee Div. B, OSRD-549|date=April 1, 1942. PB 31090|editor=Taub, A. H.|title=John von Neumann: Collected Works, 1903–1957|volume=6|publisher=[[Pergamon Press]]|location=New York|year=1963|isbn=978-0-08-009566-0}}</ref>
 
===Politics and social affairs===
[[File:Julian Bigelow.jpg|thumb|John von Neumann at The Princeton Institute for Advanced Study (Left to right: Julian Bigelow, Herman Goldstine, J. Robert Oppenheimer, and John von Neumann).]]Von Neumann obtained at the age of 29 one of the first five professorships at the new [[Institute for Advanced Study]] in [[Princeton, New Jersey]] (another had gone to [[Albert Einstein]]). He was a frequent consultant for the [[Central Intelligence Agency]], the [[United States Army]], the [[RAND Corporation]], [[Standard Oil]], [[General Electric]], [[IBM]], and others.
 
Throughout his life von Neumann had a respect and admiration for business and government leaders; something which was often at variance with the inclinations of his scientific colleagues.<ref>Mathematical Association of American documentary, especially comments by Morgenstern regarding this aspect of von Neumann's personality</ref> Von Neumann entered government service (Manhattan Project) primarily because he felt that, if freedom and civilization were to survive, it would have to be because the U.S. would triumph over totalitarianism from the right ([[Nazism]] and [[Fascism]]) and totalitarianism from the left ([[Soviet Communism]]).<ref name="whitman">{{cite web|url=http://256.com/gray/docs/misc/conversation_with_marina_whitman.shtml|title=Conversation with Marina Whitman|publisher=Gray Watson (256.com)|accessdate=2011-01-30|ref=whiteman}}</ref>
 
As president of the von Neumann Committee for Missiles, and later as a member of the [[United States Atomic Energy Commission]], from 1953 until his death in 1957, he was influential in setting U.S. scientific and military policy. Through his committee, he developed various scenarios of nuclear proliferation, the development of intercontinental and submarine missiles with atomic warheads, and the controversial strategic equilibrium called [[mutual assured destruction]]. During a [[United States Senate|Senate]] committee hearing he described his political ideology as "violently [[Anti-communism|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?".<ref>Blair, p. 96.</ref>
 
===Weather systems===
Von Neumann's team performed the world's first numerical [[weather forecast]]s on the ENIAC computer; von Neumann published the paper ''Numerical Integration of the Barotropic Vorticity Equation'' in 1950.<ref>{{cite journal|author=Charney JG, Fjörtoft R and von Neumann J|year=1950|title=Numerical Integration of the Barotropic Vorticity Equation|journal=[[Tellus A]]|volume= 2|pages= 237–254|doi=10.1111/j.2153-3490.1950.tb00336.x|issue=4}}</ref> 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]].<ref>MacRae, p. 332</ref><ref>Heims, pp. 236–247.</ref>
 
===Cognitive abilities===
Von Neumann's ability to instantaneously perform complex operations in his head stunned other mathematicians.<ref name="Neumann, Princeton University Press 1980, page 171">Goldstine, p. 171.</ref> [[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."<ref>Eugene Wigner, [http://books.google.com/books?id=Dn4zVoy6QeAC&pg=PA129 ''Historical and Biographical Reflections and Syntheses''], Springer 2002, p. 129 ISBN 3540572945.</ref> [[Paul Halmos]] states that "von Neumann's speed was awe-inspiring."<ref name="Halmos, P.R 1973 pp. 382"/> [[Israel Halperin]] said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car."<ref>Michael Kaplan, Ellen Kaplan, ''Chances are–: adventures in probability'', Viking 2006</ref> [[Edward Teller]] wrote that von Neumann effortlessly outdid anybody he ever met,<ref>''Darwin Among the Machines: the Evolution of Global Intelligence'', Perseus Books, 1998, George Dyson, 77</ref> and said "I never could keep up with him".<ref>''John von Neumann'', by Edward Teller, [[The Bulletin of the Atomic Scientists]], April 1957, p. 150.</ref>
 
[[Lothar Wolfgang Nordheim]] described von Neumann as the "fastest mind I ever met",<ref name="Neumann, Princeton University Press 1980, page 171"/> and [[Jacob Bronowski]] wrote "He was the cleverest man I ever knew, without exception. He was a genius."<ref>Jacob Brownowski, ''The Ascent of Man'', BBC 1976, p. 433 ISBN 1849901155.</ref> [[George Pólya]], whose lectures at [[ETH Zurich]] von Neumann attended as a student, said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."<ref>Miodrag Petković, [http://books.google.com/books?id=pmSftwkAocAC&pg=PA157 ''Famous puzzles of great mathematicians''], American Mathematical Soc., 2009, p. 157 ISBN 0821848143.</ref> 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:<ref>[http://mathworld.wolfram.com/TwoTrainsPuzzle.html Fly Puzzle (Two Trains Puzzle)]</ref>
 
{{quote|Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 mph. At the same time a fly that travels at a steady 15 mph starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover? The slow way to find the answer is to calculate what distance the fly covers on the first, northbound, leg of the trip, then on the second, southbound, leg, then on the third, etc., etc., and, finally, to sum the [[infinite series]] so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann, "All I did was sum the infinite series."<ref name="Halmos, P.R 1973 pp. 382"/>}}
 
Von Neumann had a very strong [[eidetic memory]], commonly called 'photographic' memory.<ref name="Life Magazine 1957, pages 89-104"/> [[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 ''[[The Tale of Two Cities]]'' started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes."<ref>Goldstine, p. 167.</ref>
 
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 A. Bethe]] of [[Cornell University]].<ref name="Life Magazine 1957, pages 89-104">Blair, pp. 89–104.</ref> "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." [[James Glimm|Glimm]] writes "he is regarded as one of the giants of modern mathematics".<ref name="Glimm, p. vii"/> The mathematician [[Jean Dieudonné]] called von Neumann "the last of the great mathematicians",<ref>''[[Dictionary of Scientific Biography]]'', ed. [[C. C. Gillispie]], [[Scribners]], 1981</ref> while [[Peter Lax]] described him as possessing the "most scintillating intellect of this century".<ref>Glimm, p. 7</ref>
 
==Personal life==
Von Neumann married twice. He married Mariette Kövesi in 1930, just prior to emigrating to the United States. They had one daughter (von Neumann's only child), [[Marina von Neumann Whitman|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 von Neumann|Klara Dan]], whom he had met during his last trips back to Budapest prior to the outbreak of [[World War II]]. The von Neumanns were very active socially within the Princeton academic community.
 
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.<ref name="Life Magazine 1957, pages 89-104" />
 
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.<ref name="whitman" /> 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.<ref>{{cite web|url=http://www.nytimes.com/2012/05/06/books/review/turings-cathedral-by-george-dyson.html?nl=books&emc=booksupdateema4_20120504|title=Unleashing the Power|date=May 4, 2012|work=[[The New York Times]]|author=[[William Poundstone]]}}</ref>
 
He was sociable and enjoyed throwing large parties at his home in Princeton,<ref name="Life Magazine 1957, pages 89-104"/> occasionally twice a week.<ref>MacRae, pp. 170–171</ref> His white [[Clapboard (architecture)|clapboard]] house at 26 Westcott Road was one of the largest in [[Princeton, New Jersey|Princeton]].<ref>Ed Regis. ''Who Got Einstein's Office?: Eccentricity and Genius at the Institute for Advanced Study''. [[Perseus Books]] 1988 p. 103 ISBN 0671699237.</ref>
 
Despite being a notoriously bad driver, he nonetheless enjoyed driving (frequently while reading a book)—occasioning numerous arrests as well as accidents. When [[Cuthbert Hurd]] hired him as a consultant to IBM, Hurd often quietly paid the fines for his traffic tickets.<ref name="cch">{{cite web|title=An Interview with Cuthbert C. Hurd|author=Nancy Stern|publisher= [[Charles Babbage Institute]], University of Minnesota|date= January 20, 1981|url=http://www.cbi.umn.edu/oh/pdf.phtml?id=159|accessdate=June 3, 2010 }}</ref> 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.<ref name="Life Magazine 1957, pages 89-104"/>
 
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed [[Jewish humor|Yiddish]] and [[Off-color humor|"off-color" humor]] (especially [[limericks]]).<ref name="Halmos, P.R 1973 pp. 382"/> At Princeton he received complaints for regularly playing extremely loud German [[marching music]] on his gramophone, which distracted those in neighbouring offices, including [[Einstein]], from their work.<ref>MacRae, p. 48</ref> 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 TV playing loudly.<ref name="Life Magazine 1957, pages 89-104"/>
 
Von Neumann's closest friend in the United States was the Polish 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.<ref>''From Cardinals To Chaos: Reflections On The Life And Legacy Of Stanislaw Ulam'', Necia Grant Cooper, Roger Eckhardt, Nancy Shera, [[CUP Archive]], 1989, Chapter: "The Lost Cafe" by [[Gian-Carlo Rota]], [http://books.google.com/books?id=yv43AAAAIAAJ&pg=PA26 pp. 26–27] ISBN 0521367344.</ref>
 
==Death==
[[File:John von neumann tomb 2004.jpg|thumb|right|Von Neumann's gravestone]]
 
In 1955, von Neumann was diagnosed with what was either [[bone cancer|bone]] or [[pancreatic cancer]].<ref>While there is a general agreement that the initially discovered bone tumor was a secondary growth, sources differ as to the location of the primary cancer. While Macrae gives it as pancreatic, the ''Life'' magazine article says it was prostate.</ref> A von Neumann biographer, [[Norman Macrae]], has speculated that the cancer was caused by von Neumann's presence at the [[Operation Crossroads|Operation Crossroads nuclear tests]] held in 1946 at [[Bikini Atoll]].<ref>MacRae, p. 231.</ref>
 
Von Neumann died a year and a half later, at the [[Walter Reed Army Medical Center]] in [[Washington, D.C.]] under military security lest he reveal military secrets while heavily medicated. On his death bed, he entertained his brother with word-for-word recitations of the first few lines of each page of [[Goethe's Faust|Goethe's ''Faust'']].<ref name="Life Magazine 1957, pages 89-104"/> He was buried at [[Princeton Cemetery]] in [[Princeton, New Jersey|Princeton]], [[Mercer County, New Jersey|Mercer County]], [[New Jersey]].<ref>John von Neumann at [[Find a Grave]] [http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=7333144]</ref>
 
While at Walter Reed, he invited a [[Roman Catholic]] priest, Father Anselm Strittmatter, [[Benedictine|O.S.B.]], to visit him for consultation.<ref>The question of whether or not von Neumann had formally converted to Catholicism upon his marriage to Mariette Kövesi (who was Catholic) is addressed in Halmos, P.R. "The Legend of von Neumann", ''[[The American Mathematical Monthly]]'', Vol. 80, No. 4. (April 1973), pp. 382–394. He was baptised Roman Catholic, but certainly was not a practicing member of that religion after his divorce.</ref> Von Neumann reportedly said in explanation that [[Blaise Pascal|Pascal]] had a point, referring to [[Pascal's wager]].<ref>{{cite book|title=John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More|year=1992|publisher=American Mathematical Soc.|isbn=9780821826768|author=Norman MacRae|edition=2|accessdate=7 October 2012|page=379|quote=But Johnny had earlier said to his mother, "There probably is a God. Many things are easier to explain if there is than if there isn't." He also admitted jovially to Pascal's point: so long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end.}}</ref><ref>{{cite book |title=Key Ideas in Economics |year=2003 |publisher=[[Nelson Thornes]] |isbn=9780748770816 |first1=Robert |last1=Dransfield |first2=Don |last2=Dransfield |accessdate=26 June 2012 |page=124 |quote=He was brought up in a Hungary in which anti-Semitism was commonplace, but the family were not overly religious, and for most of his adult years von Neumann held agnostic beliefs.}}</ref><ref>{{cite book|title=Musings Of The Masters: An Anthology Of Mathematical Reflections|year=2005|publisher=MAA|isbn=9780883855492|author=Raymond George Ayoub|editor=Raymond George Ayoub|accessdate=26 June 2012|page=170|quote=On the other hand, von Neumann, giving in to Pascal's wager on his death bed, received extreme unction.}}</ref><ref>Marion Ledwig. [http://sammelpunkt.philo.at:8080/1647/1/ledwig.pdf "The Rationality of Faith"], citing MacRae, p. 379.</ref> Father Strittmatter administered the [[Last Rites|last sacraments]] to him.<ref name="Halmos, P.R 1973 pp. 382"/> Some of Von Neumann's friends, having always known him as "completely agnostic", believed that his religious conversion was not genuine since it did not reflect his attitudes and thoughts when he was healthy.<ref>{{cite book|title=J. Robert Oppenheimer: A Life|year=2006|publisher=[[Oxford University Press]]|isbn=9780195166736|author=Abraham Pais|accessdate=23 September 2012|page=109|quote=He had been completely agnostic for as long as I had known him. As far as I could see this act did not agree with the attitudes and thoughts he had harbored for nearly all his life. On February 8, 1957, Johnny died in the Hospital, at age 53.}}</ref> Even after his conversion, Father Strittmatter recalled that von Neumann did not receive much peace or comfort from it as he still remained terrified of death.<ref>{{cite book|title=Prisoner's Dilemma|year=1993|publisher=[[Random House Digital]], Inc.|isbn=9780385415804|author=William Poundstone|accessdate=26 June 2012|quote=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." The conversion did not give von Neumann much peace. Until the end he remained terrified of death, Strittmatter recalled.}}</ref>
 
==Honors==
* 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 [[Institute of Electrical and Electronics Engineers|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 (crater)|von Neumann]] on the [[Moon]] is named after him.
* The John von Neumann Computing Center in Princeton, New Jersey ({{Coord|40.348695|N|74.592251|W|type:landmark|name=John von Neumann Computing Center}}) was named in his honour.
* The professional society of Hungarian computer scientists, [[John von Neumann Computer Society]], is named after John von Neumann.<ref>{{cite web|title=Introducing the John von Neumann Computer Society|publisher=John von Neumann Computer Society|url=http://www.njszt.hu/neumann/neumann.head.page?nodeid=210|accessdate=2008-05-20}}</ref>
* On February 15, 1956, Neumann was presented with the [[Presidential Medal of Freedom]] by President [[Dwight Eisenhower]].
* On May 4, 2005 the [[United States Postal Service]] issued the ''[[Victor Stabin#Stamps|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.
 
===Info Park and Neumann János Street===
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==
* 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, [http://archive.org/download/theoryofgamesand030098mbp/theoryofgamesand030098mbp.pdf online at archive.org]. 2007 edition: ISBN 978-0-691-13061-3.
* 1945. ''First Draft of a Report on the EDVAC'' [http://systemcomputing.org/turing%20award/Maurice_1967/TheFirstDraft.pdf 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'', [[Arthur W. Burks|Burks, A. W.]], ed., University of Illinois Press. ISBN 0-598-37798-0<ref name=TSRA/>
* {{cite book|last1=von Neumann|first1=John|title=Continuous geometry|origyear=1960|url=http://books.google.com/books?id=onE5HncE-HgC|publisher=[[Princeton University Press]]|series=Princeton Landmarks in Mathematics|isbn=978-0-691-05893-1|mr=0120174|year=1998}}
* {{cite book|last1=von Neumann|first1=John|editor1-last=Halperin|editor1-first=Israel|title=Continuous geometries with a transition probability|origyear=1937|url=http://books.google.com/books?id=ZPkVGr8NXugC|mr=634656|year=1981|journal=Memoirs of the American Mathematical Society|issn=0065-9266|volume=34|issue=252|isbn=9780821822524}}
 
==See also==
{{Portal|Set theory}}
{{colbegin|2}}
* [[List of things named after John von Neumann]]
* [[Self-replicating spacecraft]]
* [[Stone–von Neumann theorem]]
* [[Von Neumann–Bernays–Gödel set theory]]
* [[Von Neumann algebra]]
* [[Von Neumann architecture]]
* [[Von Neumann bicommutant theorem]]
* [[Von Neumann conjecture]]
* [[Von Neumann entropy]]
* [[Von Neumann programming languages]]
* [[Von Neumann regular ring]]
* [[Von Neumann universal constructor]]
* [[Von Neumann universe]]
* [[Von Neumann's trace inequality]]
{{colend}}
 
;PhD Students
* [[Donald B. Gillies]], Ph.D. student<ref name=genealogy>{{MathGenealogy |id=53213}}. Accessed 2011-03-05.</ref>
* [[Israel Halperin]], Ph.D. student<ref name=genealogy/><ref>While Israel Halperin's thesis advisor is often listed as [[Salomon Bochner]], this may be because "Professors at the university direct doctoral theses but those at the Institute do not. Unaware of this, in 1934 I asked von Neumann if he would direct my doctoral thesis. He replied Yes." (Israel Halperin, "The Extraordinary Inspiration of John von Neumann", ''[[Proceedings of Symposia in Pure Mathematics]]'', vol. 50 (1990), pp. 15–17).</ref>
 
==Notes==
 
===Footnotes===
{{Reflist|group=N}}
 
===Citations===
{{Reflist|colwidth=30em}}
 
==References==
{{refbegin|2}}
* Blair, Clay, Jr. [http://books.google.com/books?id=rEEEAAAAMBAJ&pg=PA89 ''Passing of a Great Mind''], ''Life Magazine'', 25 February 1957
* {{cite book|last=Doran|first=Robert S.|coauthors =John Von Neumann, [[Marshall Harvey Stone]], [[Richard V. Kadison]], American Mathematical Society|title=Operator Algebras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John Von Neumann and Marshall H. Stone|url=http://books.google.com/?id=m5bSoD9XsfoC&pg=PA1|publisher=[[American Mathematical Society]] Bookstore|year=2004|isbn=978-0-8218-3402-2}}
* [[Herman Goldstine|Goldstine, Herman]] [http://books.google.com/books?id=jCSpiVBH5W0C&pg=PA167 ''The Computer from Pascal to von Neumann''], Princeton University Press, 1980, ISBN 0691023670.
* {{cite book|last=Heims|first=Steve J.|title=John von Neumann and Norbert Wiener, from Mathematics to the Technologies of Life and Death|year=1980|publisher=MIT Press|location=Cambridge, Massachusetts|isbn=0-262-08105-9}}
* {{cite book|last=Herken|first=Gregg|title=Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller|year=2002|isbn=978-0-8050-6588-6}}
* [[James Glimm|Glimm, James]]; Impagliazzo, John; [[Isadore Singer|Singer, Isadore Manuel]] [http://books.google.com/books?id=XBK-r0gS0YMC&pg=PA15 ''The Legacy of John von Neumann''], American Mathematical Society 1990 ISBN 0-8218-4219-6
* {{cite book|last=Israel|first=Giorgio|coauthors=Ana Millan Gasca|title=The World as a Mathematical Game: John von Neumann, Twentieth Century Scientist|year=1995}}
* {{cite book|last=MacRae|first=Norman|authorlink=Norman Macrae|title=John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More|year=1992|publisher=Pantheon Press|isbn=0-679-41308-1|url=http://books.google.com/books?id=OO-_gSRhe-EC&pg=PA37}}
* {{cite book|last=Slater|first=Robert|title=Portraits in Silicon|pages=23–33|isbn=0-262-69131-0|year=1989|publisher=MIT Press|location=Cambridge, Mass.}}
{{refend}}
{{FOLDOC}}
 
==Further reading==
* Aspray, William, 1990. ''John von Neumann and the Origins of Modern Computing''.
* Chiara, Dalla, Maria Luisa and Giuntini, Roberto 1997, ''La Logica Quantistica'' in Boniolo, Giovani, ed., ''Filosofia della Fisica'' (Philosophy of Physics). Bruno Mondadori.
* Halmos, Paul R., 1985. ''I Want To Be A Mathematician'' Springer-Verlag
* Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903–1957). "Teil 1: Lehrjahre eines jüdischen Mathematikers während der Zeit der Weimarer Republik". In: ''Informatik-Spektrum'' 29 (2), S. 133–141.
* Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903–1957). "Teil 2: Ein Privatdozent auf dem Weg von Berlin nach Princeton". In: ''Informatik-Spektrum'' 29 (3), S. 227–236.
* Heims, Steve J., 1980. ''John von Neumann and [[Norbert Wiener]]: From Mathematics to the Technologies of Life and Death'' [[MIT Press]]
* [[William Poundstone|Poundstone, William]]. ''Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb''. 1992.
* Redei, Miklos (ed.), 2005 ''John von Neumann: Selected Letters'' American Mathematical Society
* [[Stanislaw Ulam|Ulam, Stanislaw]], 1983. ''Adventures of a Mathematician'' Scribner's
* Vonneuman, Nicholas A. ''John von Neumann as Seen by His Brother'' ISBN 0-9619681-0-9
* 1958, ''Bulletin of the American Mathematical Society 64''.
* 1990. ''Proceedings of the American Mathematical Society Symposia in Pure Mathematics 50''.
* [http://books.nap.edu/html/biomems/jvonneumann.pdf John von Neumann 1903–1957], biographical memoir by S. Bochner, [[United States National Academy of Sciences|National Academy of Sciences]], 1958
 
;Popular periodicals
* ''[[Good Housekeeping|Good Housekeeping Magazine]]'', September 1956 ''Married to a Man Who Believes the Mind Can Move the World''
 
;Video
* ''John von Neumann, A Documentary'' (60 min.), [[Mathematical Association of America]]
 
==External links==
{{Commons|János Lajos Neumann|John von Neumann}}
{{Wikiquote|John von Neumann}}
* {{MacTutor Biography|id=Von_Neumann}}
* [http://www.findarticles.com/p/articles/mi_m0IMR/is_3-4_79/ai_113139424 von Neumann's contribution to economics]—''International Social Science Review''
* [http://scholar.google.com.au/citations?user=6kEXBa0AAAAJ&hl=en von Neumann's Scholar Google profile]
* [http://purl.umn.edu/107206 Oral history interview with Alice R. Burks and Arthur W. Burks], [[Charles Babbage Institute]], [[University of Minnesota]], Minneapolis. [[Alice Burks]] and [[Arthur Burks]] describe [[ENIAC]], [[EDVAC]], and [[IAS computer]]s, and John von Neumann's contribution to the development of computers.
* [http://purl.umn.edu/107714 Oral history interview with Eugene P. Wigner], [[Charles Babbage Institute]], University of Minnesota, Minneapolis.
* [http://purl.umn.edu/107493 Oral history interview with Nicholas C. Metropolis], [[Charles Babbage Institute]], University of Minnesota.
* [http://plato.stanford.edu/entries/qt-nvd/ Von Neumann vs. Dirac] — from ''[[Stanford Encyclopedia of Philosophy]]''
* [http://www.sandia.gov/careers/fellowships.html#jon John von Neumann Postdoctoral Fellowship] at [[Sandia National Laboratories]]
* [http://www.itconversations.com/shows/detail454.html Von Neumann's Universe], audio talk by [[George Dyson (science historian)|George Dyson]]
* [http://www.stephenwolfram.com/publications/recent/neumann/ John von Neumann's 100th Birthday], article by [[Stephen Wolfram]] on von Neumann's 100th birthday.
* [http://alsos.wlu.edu/qsearch.aspx?browse=people/Neumann,+John+von Annotated bibliography for John von Neumann] from the [[Alsos Digital Library for Nuclear Issues]]
* [http://nik.bmf.hu/ Budapest Tech Polytechnical Institution – John von Neumann Faculty of Informatics]
* [http://elm.eeng.dcu.ie/~alife/von-neumann-1954-NORD/ John von Neumann speaking at the dedication of the NORD], December 2, 1954 (audio recording)
* [http://www.presidency.ucsb.edu/ws/index.php?pid=10735 The American Presidency Project]
* [http://jewish.hu/view.php?clabel=neumann_janos Jewish.hu: Famous Hungarian Jews]
* {{cite book|title=John von Neumann (1903–1957)|url=http://www.econlib.org/library/Enc/bios/Neumann.html|work=[[The Concise Encyclopedia of Economics]]|edition=2nd|series=[[Library of Economics and Liberty]]|publisher=[[Liberty Fund]]|year=2008 }}
* {{Findagrave|7333144}}
{{Set theory}}
{{Authority control|VIAF=99899730|LCCN=n/50/021202}}
 
<!-- Metadata: see [[Wikipedia:Persondata]] -->
{{Persondata
| NAME              = von Neumann, John
| ALTERNATIVE NAMES =Neumann János Lajos Margittai (Hungarian)
| SHORT DESCRIPTION =[[Mathematician]] and [[polymath]]
| DATE OF BIRTH    = December 28, 1903
| PLACE OF BIRTH    = [[Budapest]], [[Austria-Hungary]]
| DATE OF DEATH    = February 8, 1957
| PLACE OF DEATH    = [[Washington DC]], United States
}}
{{DEFAULTSORT:Vonneumann, John}}
[[Category:1903 births]]
[[Category:1957 deaths]]
[[Category:20th-century mathematicians]]
[[Category:American agnostics]]
[[Category:American mathematicians]]
[[Category:American nuclear physicists]]
[[Category:American people of Hungarian-Jewish descent]]
[[Category:American Roman Catholics]]
[[Category:Austrian nobility]]
[[Category:Burials at Princeton Cemetery]]
[[Category:Cancer deaths in Washington, D.C.]]
[[Category:Carl-Gustaf Rossby Research Medal recipients]]
[[Category:Cellular automatists]]
[[Category:Computer designers]]
[[Category:Enrico Fermi Award recipients]]
[[Category:ETH Zurich alumni]]
[[Category:Functional analysts]]
[[Category:Game theorists]]
[[Category:Hungarian agnostics]]
[[Category:Hungarian computer scientists]]
[[Category:Hungarian emigrants to the United States]]
[[Category:Hungarian inventors]]
[[Category:Hungarian Jews]]
[[Category:Hungarian mathematicians]]
[[Category:Hungarian nobility]]
[[Category:Hungarian nuclear physicists]]
[[Category:Institute for Advanced Study faculty]]
[[Category:Jewish agnostics]]
[[Category:Jewish American scientists]]
[[Category:Lattice theorists]]
[[Category:Manhattan Project people]]
[[Category:Mathematical economists]]
[[Category:Mathematical physicists]]
[[Category:Measure theorists]]
[[Category:Medal for Merit recipients]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:Mental calculators]]
[[Category:Monte Carlo methodologists]]
[[Category:American operations researchers]]
[[Category:People from Pest, Hungary]]
[[Category:People who emigrated to escape Nazism]]
[[Category:Presidential Medal of Freedom recipients]]
[[Category:Presidents of the American Mathematical Society]]
[[Category:Princeton University faculty]]
[[Category:Researchers in stochastics]]
[[Category:Set theorists]]

Latest revision as of 01:09, 2 January 2015


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