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{{Short description|Hungarian and American mathematician and physicist (1903–1957)}}
{{Hungarian name|Neumann János Lajos}}
{{Good article}}
{{Use mdy dates|date=March 2015|cs1-dates=ly}}
{{Infobox officeholder
| name                = John von Neumann
| image               = JohnvonNeumann-LosAlamos.gif
| image_size          = 
| caption             = von Neumann in the 1940s
| office              = Member of the<br/>[[United States Atomic Energy Commission]]
| president           = [[Dwight D. Eisenhower]]
| term_start          = March 15, 1955
| term_end            = February 8, 1957
| predecessor         = [[Eugene M. Zuckert]]
| successor           = [[John Stephens Graham|John S. Graham]]
| birth_name          = Neumann János Lajos
| birth_date          = {{birth date|mf=yes|1903|12|28}}
| birth_place         = [[Budapest]], Kingdom of Hungary
| death_date          = {{death date and age|mf=yes|1957|2|8|1903|12|28}}
| death_place         = [[Washington, D.C.]], U.S.
| citizenship         = {{ubl|Hungary|United States}}
| module              = {{Infobox scientist
|embed=yes
| fields            = [[Logic]], [[mathematics]], [[mathematical physics]], [[theoretical physics]], [[statistics]], [[economics]], [[computer science]], [[Mathematical and theoretical biology|theoretical biology]], [[chemistry]], [[computing]]
| workplaces        = {{ubl|[[University of Göttingen]]|[[Humboldt University of Berlin|University of Berlin]]|[[University of Hamburg]]|[[Princeton University]]|[[Institute for Advanced Study]]|[[Los Alamos Laboratory]]|[[National Defense Research Committee]]|[[United States Department of Defense]]|[[United States Atomic Energy Commission]]}}
| alma_mater        = {{ubl|[[Eötvös Loránd University|Pázmány Péter University]]|[[Humboldt University of Berlin|University of Berlin]]|[[ETH Zurich|ETH Zürich]]}}
| thesis_title      = Az általános halmazelmélet axiomatikus felépítése (The axiomatic construction of general set theory)
| thesis_year       = 1925
| doctoral_advisor  = {{ubl |[[Lipót Fejér]]}}
| academic_advisors = {{ubl |[[László Rátz]] |[[Gábor Szegő]] |[[Michael Fekete]] |[[József Kürschák]]{{sfn|Dyson|2012|p=48}} |[[David Hilbert]] |[[Erhard Schmidt]]<ref>{{cite book |author-last1=Israel |author-first1=Giorgio |author-link1=:it:Giorgio Israel |author-last2=Gasca |author-first2=Ana Millan |title=The World as a Mathematical Game: John von Neumann and Twentieth Century Science |series=Science Networks. Historical Studies |publisher=Birkhäuser |location=Basel |year=2009 |volume=38 |doi=10.1007/978-3-7643-9896-5 |isbn=978-3-7643-9896-5 |oclc=318641638 |url=https://link.springer.com/book/10.1007/978-3-7643-9896-5 |page=14}}</ref> |[[Hermann Weyl]] |[[George Pólya]]{{sfn|Goldstine|1980|p=169}}}}
| doctoral_students = {{ubl|[[Donald B. Gillies]]|[[Israel Halperin]]{{#tag: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."<ref name=halperin1990>{{harvc |last1=Halperin |first1=Israel |author-link=Israel Halperin |year=1990 |chapter=The Extraordinary Inspiration of John von Neumann |in1=Glimm |in2=Impagliazzo |in3=Singer |page=16}}</ref>}}|[[Friederich Ignaz Mautner|Friederich Mautner]]<ref name=genealogy>{{MathGenealogy |id=53213}}. Retrieved 2015-03-17.</ref>
}}
| notable_students  = {{ubl|[[Eugene Wigner]]{{sfn|Szanton|1992|p=130}}|[[Paul Halmos]]|[[Peter Lax]]|[[Benoit Mandelbrot]]<ref>{{cite journal|url=http://www.cambridge-systems.com/content/QFBenoitMandelbrotTribute.pdf|archive-url=https://web.archive.org/web/20140714163938/http://www.cambridge-systems.com/content/QFBenoitMandelbrotTribute.pdf|url-status=usurped|archive-date=July 14, 2014|title=Benoit B. Mandelbrot (1924–2010): a father of Quantitative Finance|last=Dempster|first= M. A. H.|journal=Quantitative Finance|volume=11|issue=2|date=February 2011|pages= 155–156|doi=10.1080/14697688.2011.552332|s2cid=154802171}}</ref>}}
| known_for         = [[Mathematical formulation of quantum mechanics]], [[Game theory]], [[Spectral theory]], [[Ergodic theory]], [[von Neumann algebras]], [[List of things named after John von Neumann]]
| author_abbrev_bot = 
| author_abbrev_zoo = 
| influences        = 
| influenced        =
| awards            = {{ubl|[[Bôcher Memorial Prize]] (1938)|[[Navy Distinguished Civilian Service Award]] (1946)|[[Medal for Merit]] (1946)|[[Medal of Freedom (1945)|Medal of Freedom]] (1956)|[[Enrico Fermi Award]] (1956)|[[Carl-Gustaf Rossby Research Medal]] (1957)}}
| signature         = johnny von neumann sig.gif
| footnotes         = 
| spouse            = {{ubl|{{marriage|Marietta Kövesi|1930|1937|reason=div}}|{{marriage|[[Klára Dán von Neumann|Klára Dán]]|1938}}}}
| children          = [[Marina von Neumann Whitman]]
| siglum            = 
}}
| resting_place       = [[Princeton Cemetery]]
}}

'''John von Neumann''' ({{IPAc-en|v|ɒ|n|_|ˈ|n|ɔɪ|m|ən}} {{respell|von|_|NOY|mən}}; {{langx|hu|Neumann János Lajos}} {{IPA|hu|ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ|}}; December 28, 1903&nbsp;– February&nbsp;8, 1957) was a Hungarian and American [[mathematician]], [[physicist]], [[computer scientist]] and [[engineer]]. Von Neumann had perhaps the widest coverage of any mathematician of his time,{{sfn|Rédei|1999|p=7}} integrating [[Basic research|pure]] and [[Applied science#Applied research|applied sciences]] and making major contributions to many fields, including [[mathematics]], [[physics]], [[economics]], [[computing]], and [[statistics]]. He was a pioneer in building the mathematical framework of [[quantum physics]], in the development of [[functional analysis]], and in [[game theory]], introducing or codifying concepts including [[Cellular automaton|cellular automata]], the [[Von Neumann universal constructor|universal constructor]] and the [[Computer|digital computer]]. His analysis of the structure of [[self-replication]] preceded the discovery of the structure of [[DNA]].

During [[World War II]], von Neumann worked on the [[Manhattan Project]]. He developed the mathematical models behind the [[explosive lens]]es used in the [[implosion-type nuclear weapon]].{{sfn|Macrae|1992}} Before and after the war, he consulted for many organizations including the [[Office of Scientific Research and Development]], the [[United States Army|Army's]] [[Ballistic Research Laboratory]], the [[Armed Forces Special Weapons Project]] and the [[Oak Ridge National Laboratory]].{{sfn|Aspray|1990|p=246}} At the peak of his influence in the 1950s, he chaired a number of [[United States Department of Defense|Defense Department]] committees including the [[Teapot Committee|Strategic Missile Evaluation Committee]] and the [[ICBM]] Scientific Advisory Committee. He was also a member of the influential [[United States Atomic Energy Commission|Atomic Energy Commission]] in charge of all atomic energy development in the country. He played a key role alongside [[Bernard Schriever]] and [[Trevor Gardner]] in the design and development of the United States' first [[Intercontinental ballistic missile|ICBM]] programs.{{sfn|Sheehan|2010}} At that time he was considered the nation's foremost expert on [[nuclear weapon]]ry and the leading defense scientist at the [[United States Department of Defense|U.S. Department of Defense]].

Von Neumann's contributions and intellectual ability drew praise from colleagues in physics, mathematics, and beyond. Accolades he received range from the [[Medal of Freedom (1945)|Medal of Freedom]] to a [[Von Neumann (crater)|crater on the Moon]] named in his honor.

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== Life and education ==
=== Family background ===
Von Neumann was born in [[Budapest]], Kingdom of Hungary (then part of the [[Austria-Hungary|Austro-Hungarian Empire]]),<ref>{{cite book |editor-last1=Doran |editor-first1=Robert S. |editor-link1=Robert S. Doran |editor-link2=Richard Kadison |editor-first2=Richard V. |editor-last2=Kadison |title=Operator Algebras, Quantization, and Noncommutative Geometry: A Centennial Celebration Honoring John von Neumann and Marshall H. Stone |publisher=American Mathematical Society |location=Washington, D.C. |year=2004 |url=https://bookstore.ams.org/conm-365 |isbn=978-0-8218-3402-2 |page=1}}</ref><ref name= Time>{{cite news |last=Myhrvold |first=Nathan |author-link=Nathan Myhrvold |date=March 21, 1999 |url=http://www.time.com/time/magazine/article/0,9171,21839,00.html |archive-url=https://web.archive.org/web/20010211124237/http://www.time.com/time/magazine/article/0,9171,21839,00.html |url-status=dead |archive-date=February 11, 2001 |title=John von Neumann |newspaper=[[Time (magazine)|Time]]}}</ref>{{sfn|Blair|1957|p=104}} on December 28, 1903, to a wealthy, non-observant [[Jewish]] family. His birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names are equivalent to John Louis in English.{{sfn|Bhattacharya|2022|p=4}}

He was the eldest of three brothers; his two younger siblings were Mihály (Michael) and Miklós (Nicholas).{{sfn|Dyson|1998|p=xxi}} His father Neumann Miksa (Max von Neumann) was a banker and held a [[doctor of law|doctorate in law]]. He had moved to Budapest from [[Pécs]] at the end of the 1880s.{{sfn|Macrae|1992|pp=38–42}} Miksa's father and grandfather were born in Ond (now part of [[Szerencs]]), [[Zemplén County]], northern Hungary. John's mother was Kann Margit (Margaret Kann);{{sfn|Macrae|1992|pp=37–38}} her parents were Kann Jákab and Meisels Katalin of the [[Meisel family|Meisels family]].{{sfn|Macrae|1992|p=39}} Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest; von Neumann's family occupied an 18-room apartment on the top floor.{{sfn|Macrae|1992|pp=44–45}}

On February 20, 1913, [[Emperor Franz Joseph]] elevated John's father to the Hungarian nobility for his service to the Austro-Hungarian Empire.<ref name="archives.hungaricana.hu">{{cite web
|title       = Neumann de Margitta Miksa a Magyar Jelzálog-Hitelbank igazgatója n:Kann Margit gy:János-Lajos, Mihály-József, Miklós-Ágost {{!}} Libri Regii {{!}} Hungaricana
|url         = https://archives.hungaricana.hu/en/libriregii/hu_mnl_ol_a057_72_1096/?list=eyJxdWVyeSI6ICJuZXVtYW5uIn0
|access-date = 2022-08-08
|website     = archives.hungaricana.hu
|language    = Hungarian
}}</ref> The Neumann family thus acquired the hereditary appellation ''Margittai'', meaning "of Margitta" (today [[Marghita]], Romania). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosen [[coat of arms]] depicting three [[Argyranthemum|marguerites]]. Neumann János became margittai Neumann János (John Neumann de Margitta), which he later changed to the German Johann von Neumann.{{sfn|Macrae|1992|pp=57–58}}

=== Child prodigy ===
Von Neumann was a [[child prodigy]] who at six years old could divide two eight-digit numbers in his head<ref>{{cite book |last=Henderson |first=Harry |title=Mathematics: Powerful Patterns Into Nature and Society |publisher=Chelsea House |location=New York |year=2007 |isbn=978-0-8160-5750-4|oclc=840438801 |page=30 }}</ref>{{sfn|Schneider|Gersting|Brinkman|2015|p=28}} and converse in [[Ancient Greek]].<ref>{{cite book |last=Mitchell |first=Melanie |author-link=Melanie Mitchell |title=Complexity: A Guided Tour |publisher=Oxford University Press |year=2009 |isbn=978-0-19-512441-5 |oclc=216938473 |page=124}}</ref> He, his brothers and his cousins were instructed by governesses. Von Neumann's father believed that knowledge of languages other than their native [[Hungarian language|Hungarian]] was essential, so the children were tutored in [[English (language)|English]], [[French (language)|French]], [[German (language)|German]] and [[Italian (language)|Italian]].{{sfn|Macrae|1992|pp=46–47}} By age eight, von Neumann was familiar with [[differential calculus|differential]] and [[integral calculus]], and by twelve he had read [[Émile Borel|Borel's]] ''La Théorie des Fonctions''.{{sfn|Halmos|1973|p=383}} He was also interested in history, reading [[Wilhelm Oncken]]'s 46-volume world history series {{lang|de|Allgemeine Geschichte in Einzeldarstellungen}} (''General History in Monographs'').{{sfn|Blair|1957|p=90}} One of the rooms in the apartment was converted into a library and reading room.{{sfn|Macrae|1992|p=52}}

Von Neumann entered the Lutheran [[Fasori Gimnázium|Fasori Evangélikus Gimnázium]] in 1914.{{sfn|Aspray|1990}} [[Eugene Wigner]] was a year ahead of von Neumann at the school and soon became his friend.{{sfn |Macrae |1992 |pp=70–71}}

Although von Neumann's father insisted that he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give von Neumann advanced instruction. At 15, he began to study advanced calculus under the analyst [[Gábor Szegő]].{{sfn|Macrae|1992|pp=70–71}} On their first meeting, Szegő was so astounded by von Neumann's mathematical talent and speed that, as recalled by his wife, he came back home with tears in his eyes.<ref>Impagliazzo, John; [[James Glimm|Glimm, James]]; [[Isadore Singer|Singer, Isadore Manuel]] [https://books.google.com/books?id=XBK-r0gS0YMC&pg=PA15 ''The Legacy of John von Neumann''], American Mathematical Society, 1990, p. 5, {{ISBN|0-8218-4219-6}}.</ref> By 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of [[ordinal number#Von Neumann definition of ordinals|ordinal numbers]], which superseded [[Georg Cantor]]'s definition.<ref name=Nasar-p81>{{cite book |last=Nasar |first=Sylvia |author-link=Sylvia Nasar |year=2001 |title=A Beautiful Mind : a Biography of John Forbes Nash, Jr., Winner of the Nobel Prize in Economics, 1994 |location=London |publisher=Simon & Schuster |isbn=978-0-7432-2457-4  |url=https://archive.org/details/beautifulmindli00nasa |page=81}}</ref> At the conclusion of his education at the gymnasium, he applied for and won the Eötvös Prize, a national award for mathematics.{{sfn|Macrae|1992|p=84}}

=== University studies ===
According to his friend [[Theodore von Kármán]], von Neumann's father wanted John to follow him into industry, and asked von Kármán to persuade his son not to take mathematics.<ref>von Kármán, T., & Edson, L. (1967). The wind and beyond. Little, Brown & Company.</ref> Von Neumann and his father decided that the best career path was [[chemical engineer]]ing. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year, non-degree course in chemistry at the [[University of Berlin]], after which he sat for the entrance exam to [[ETH Zurich]],{{sfn|Macrae|1992|pp=85–87}} which he passed in September 1923.{{sfn|Macrae|1992|p=97}} Simultaneously von Neumann entered [[Eötvös Loránd University|Pázmány Péter University]], then known as the University of Budapest, as a [[Doctor of Philosophy|Ph.D.]] candidate in [[mathematics]].<ref name="NYT">{{cite news |last=Regis |first=Ed |author-link=Ed Regis (author) |title=Johnny Jiggles the Planet |url=https://query.nytimes.com/gst/fullpage.html?res=9E0CE7D91239F93BA35752C1A964958260 |work=The New York Times |date=November 8, 1992 |access-date=2008-02-04}}</ref> For his thesis, he produced an [[axiomatization]] of [[Georg Cantor#Set theory|Cantor's set theory]].<ref name="Neumann1928">{{cite journal|last1=von Neumann|first1=J. |title=Die Axiomatisierung der Mengenlehre|journal=Mathematische Zeitschrift|volume=27|issue=1|year=1928|pages=669–752|issn=0025-5874|doi=10.1007/BF01171122|s2cid=123492324 |language=de}}</ref>{{sfn|Macrae|1992|pp=86–87}} In 1926, he graduated as a [[chemical engineer]] from ETH Zurich and simultaneously passed his final examinations ''[[summa cum laude]]'' for his Ph.D. in mathematics (with minors in [[experimental physics]] and chemistry) at the University of Budapest.<ref>{{cite book |last1=Wigner |first1=Eugene |author-link=Eugene Wigner |year=2001 |chapter=John von Neumann (1903–1957) |editor-last1=Mehra |editor-first1=Jagdish |editor-link=Jagdish Mehra |title=The Collected Works of Eugene Paul Wigner: Historical, Philosophical, and Socio-Political Papers. Historical and Biographical Reflections and Syntheses |publisher=Springer |location=Berlin |doi=10.1007/978-3-662-07791-7 |isbn=978-3-662-07791-7 |page=128}}</ref>{{sfn|Pais|2000|p=187}}

He then went to the [[University of Göttingen]] on a grant from the [[Rockefeller Foundation]] to study mathematics under [[David Hilbert]].{{sfn|Macrae|1992|pp=98–99}} [[Hermann Weyl]] remembers how in the winter of 1926–1927 von Neumann, [[Emmy Noether]], and he would walk through "the cold, wet, rain-wet streets of [[Göttingen]]" after class discussing [[hypercomplex number]] systems and their [[Representation theory|representations]].<ref>{{cite book |last1=Weyl |first1=Hermann |editor1-last=Pesic |editor1-first=Peter |title=Levels of Infinity: Selected Writings on Mathematics and Philosophy |date=2012 |publisher=Dover Publications |isbn=978-0-486-48903-2 |page=55 |edition=1}}</ref>

== Career and private life ==
[[File:NeumannVonMargitta.jpg|thumb|upright=1.6|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 the theory of functions II, axiomatic set theory and mathematical logic, the mathematical colloquium, review of recent work in quantum mechanics, special functions of mathematical physics and Hilbert's proof theory. He also lectured on the theory of relativity, set theory, integral equations and analysis of infinitely many variables.]]

Von Neumann's [[habilitation]] was completed on December 13, 1927, and he began to give lectures as a ''[[Privatdozent]]'' at the University of Berlin in 1928.<ref>{{cite journal |last=Hashagen |first=Ulf |author-link=:de:Ulf Hashagen |title=Die Habilitation von John von Neumann an der Friedrich-Wilhelms-Universität in Berlin: Urteile über einen ungarisch-jüdischen Mathematiker in Deutschland im Jahr 1927 |journal=[[Historia Mathematica]] | volume=37 |issue=2 |pages=242–280 |date=2010 |doi=10.1016/j.hm.2009.04.002|doi-access=free}}</ref> He was the youngest person elected ''Privatdozent'' in the university's history.<ref>{{cite book |last1=Dimand |first1=Mary Ann |last2=Dimand |first2=Robert |title=A History of Game Theory: From the Beginnings to 1945 |date=2002 |publisher=Routledge |location=London |isbn=9781138006607 |url=https://www.routledge.com/The-History-Of-Game-Theory-Volume-1-From-the-Beginnings-to-1945/Dimand-Dimand/p/book/9781138006607 |page=129}}</ref> He began writing nearly one major mathematics paper per month.{{sfn |Macrae |1992 |p=145}} In 1929, he briefly became a ''Privatdozent'' at the [[University of Hamburg]], where the prospects of becoming a tenured professor were better,{{sfn|Macrae|1992|pp=143–144}} then in October of that year moved to [[Princeton University]] as a visiting lecturer in [[mathematical physics]].{{sfn|Macrae|1992|pp=155–157}}

Von Neumann was baptized a Catholic in 1930.{{sfn|Bochner|1958|p=446}} Shortly afterward, he married Marietta Kövesi, who had studied economics at Budapest University.{{sfn|Macrae|1992|pp=155–157}} Von Neumann and Marietta had a daughter, [[Marina von Neumann Whitman|Marina]], born in 1935; she would become a professor.<ref>{{cite web |url=http://fordschool.umich.edu/faculty/marina-whitman |title=Marina Whitman |publisher=The Gerald R. Ford School of Public Policy at the University of Michigan |access-date=January 5, 2015 |date=2014-07-18 }}</ref> The couple divorced on November 2, 1937.<ref>"Princeton Professor Divorced by Wife Here". ''Nevada State Journal''. November 3, 1937.</ref> On November 17, 1938, von Neumann married [[Klara Dan von Neumann|Klára Dán]].{{sfn|Heims|1980|p=178}}{{sfn|Macrae|1992|pp=170–174}}

In 1933 Von Neumann accepted a tenured professorship at the [[Institute for Advanced Study]] in New Jersey, when that institution's plan to appoint [[Hermann Weyl]] appeared to have failed.{{sfn|Macrae|1992|pp=167–168}} His mother, brothers and in-laws followed von Neumann to the United States in 1939.{{sfn|Macrae|1992|pp=195–196}} Von Neumann [[Anglicisation|anglicized]] his name to John, keeping the German-aristocratic surname [[von]] Neumann.{{sfn|Macrae|1992|pp=57–58}} Von Neumann became a [[Naturalization#United States|naturalized U.S. citizen]] in 1937, and immediately tried to become a [[lieutenant]] in the U.S. Army's [[Officers Reserve Corps]]. He passed the exams but was rejected because of his age.{{sfn|Macrae|1992|pp=190–195}}

Klára and John von Neumann were socially active within the local academic community.{{sfn|Macrae|1992|pp=170–171}} His white [[clapboard]] house on Westcott Road was one of Princeton's largest private residences.<ref>{{cite book |last=Regis |first=Ed |author-link=Ed Regis (author) |year=1987 |title=Who Got Einstein's Office?: Eccentricity and Genius at the Institute for Advanced Study |publisher=Addison-Wesley |isbn=978-0-201-12065-3 |oclc=15548856 |location=Reading, Massachusetts  |url=https://archive.org/details/whogoteinsteinso00regi |page=103}}</ref> He always wore formal suits.<ref name="whitman"/> He enjoyed [[Jewish humour|Yiddish]] and [[Off-color humor|"off-color"]] humor.{{sfn|Halmos|1973|p=383}} In Princeton, he received complaints for playing extremely loud German [[March (music)|march music]];{{sfn|Macrae|1992|p=48}} Von Neumann did some of his best work in noisy, chaotic environments.{{sfn|Blair|1957|p=94}} According to [[Churchill Eisenhart]], von Neumann could attend parties until the early hours of the morning and then deliver a lecture at 8:30.<ref>{{cite interview |last=Eisenhart |first=Churchill |subject-link=Churchill Eisenhart|interviewer=William Apsray |title=Interview Transcript #9 - Oral History Project |date=1984 |publisher=Princeton Mathematics Department |location=New Jersey |url=https://web.math.princeton.edu/oral-history/c9.pdf |access-date=3 April 2022 |page=7}}</ref>

He was known for always being happy to provide others of all ability levels with scientific and mathematical advice.{{r|halperin1990}}{{sfn|Goldstine|1985|p=7}}<ref>{{cite book |last1=DeGroot |first1=Morris H. |author-link1=Morris H. DeGroot |year=1989 |chapter=A Conversation with David Blackwell |editor-last1=Duren |editor-first1=Peter |editor-link1=Peter Duren |title=A Century of Mathematics in America: Part III |publisher=American Mathematical Society |isbn=0-8218-0136-8 |page=592}}</ref> Wigner wrote that he perhaps supervised more work (in a casual sense) than any other modern mathematician.{{sfn|Szanton|1992|p=227}} His daughter wrote that he was very concerned with his legacy in two aspects: his life and the durability of his intellectual contributions to the world.<ref>{{harvc |last1=von Neumann Whitman |first1=Marina |author-link=Marina von Neumann Whitman |year=1990 |chapter=John von Neumann: A Personal View |in1=Glimm |in2=Impagliazzo |in3=Singer |page=2}}</ref>

Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing on technical ones. [[Herbert York]] described the many "Von Neumann Committees" that he participated in as "remarkable in style as well as output". The way the committees von Neumann chaired worked directly and intimately with the necessary military or corporate entities became a blueprint for all [[United States Air Force|Air Force]] long-range missile programs.{{sfn|York|1971|p=18}} Many people who had known von Neumann were puzzled by his relationship to the military and to power structures in general.{{sfn|Pais|2006|p=108}} [[Stanisław Ulam]] suspected that he had a hidden admiration for people or organizations that could influence the thoughts and decision making of others.{{sfn|Ulam|1976|pp=231-232}}

He also maintained his knowledge of languages learnt in his youth. He knew Hungarian, French, German and English fluently, and maintained a conversational level of Italian, Yiddish, Latin and Ancient Greek. His Spanish was less perfect.{{sfn|Ulam|1958|pp=5-6}} He had a passion for and encyclopedic knowledge of ancient history,{{sfn|Szanton|1992|p=277}}{{sfn|Blair|1957|p=93}} and he enjoyed reading [[Ancient Greece|Ancient Greek]] historians in the original Greek. Ulam suspected they may have shaped his views on how future events could play out and how human nature and society worked in general.{{sfn|Ulam|1976|pp=97,102,244-245}}

Von Neumann's closest friend in the United States was the mathematician [[Stanisław Ulam]].<ref>{{cite book |last=Rota |first=Gian-Carlo |author-link=Gian-Carlo Rota |editor-last=Cooper |editor-first=Necia Grant |editor2-last=Eckhardt |editor2-first=Roger |editor3-last=Shera |editor3-first=Nancy |year=1989 |contribution=The Lost Cafe |pages=23–32 |title=From Cardinals To Chaos: Reflections On The Life And Legacy Of Stanisław Ulam |publisher=Cambridge University Press |isbn=978-0-521-36734-9 |oclc=18290810 }}</ref> Von Neumann believed that much of his mathematical thought occurred intuitively; he would often go to sleep with a problem unsolved and know the answer upon waking up.{{sfn|Blair|1957|p=94}} Ulam noted that von Neumann's way of thinking might not be visual, but more aural.{{sfn|Macrae|1992|p=75}} Ulam recalled, "Quite independently of his liking for abstract wit, he had a strong appreciation (one might say almost a hunger) for the more earthy type of comedy and humor".{{sfn|Ulam|1958|pp=4-6}}

=== Illness and death ===
[[File:John von neumann tomb 2004.jpg|thumb|right|Von Neumann's gravestone]]

In 1955, a mass was found near von Neumann's collarbone, which turned out to be cancer originating in the [[Bone tumor|skeleton]], [[pancreatic cancer|pancreas]] or [[prostate cancer|prostate]]. (While there is general agreement that the tumor had [[metastasis]]ed, sources differ on the location of the primary cancer.)<ref>While Macrae gives the origin as pancreatic, the ''Life'' magazine article says it was the prostate. Sheehan's book gives it as testicular.</ref><ref>{{cite web |title=The Unparalleled Genius of John von Neumann |first=Jørgen |last=Veisdal |url=https://medium.com/cantors-paradise/the-unparalleled-genius-of-john-von-neumann-791bb9f42a2d |date=November 11, 2019 |access-date=November 19, 2019 |publisher=Medium }}</ref> The malignancy may have been caused by [[Absorbed dose|exposure]] to [[Ionizing radiation|radiation]] at [[Los Alamos National Laboratory]].{{sfn|Jacobsen|2015|p=62}} As death neared he asked for a priest, though the priest later recalled that von Neumann found little comfort in receiving the [[last rites]]{{snd}}he remained terrified of death and unable to accept it.<ref>{{cite book|last=Poundstone |first=William |author-link=William Poundstone |title=Prisoner's Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb|year=1993|publisher=Random House Digital |isbn=978-0-385-41580-4 |page=194 |ref=none}}</ref>{{sfn|Halmos|1973|pp=383, 394}}{{sfn|Jacobsen|2015|p=63}}<ref>{{cite book |last=Read |first=Colin |title=The Portfolio Theorists: von Neumann, Savage, Arrow and Markowitz |series=Great Minds in Finance |url=https://books.google.com/books?id=YZd9DAAAQBAJ&pg=PA65 |access-date=September 29, 2017 |year=2012 |publisher=Palgrave Macmillan |isbn=978-0230274143 |page=65 |quote=When von Neumann realised he was incurably ill his logic forced him to realise that he would cease to exist... [a] fate which appeared to him unavoidable but unacceptable.}}</ref> Of his religious views, Von Neumann reportedly said, "So long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end," referring to [[Pascal's wager]]. He confided to his mother, "There probably has to be a God. Many things are easier to explain if there is than if there isn't."<ref>{{harvnb|Macrae|1992|page=379}}"</ref><ref>{{cite book |last=Ayoub |first=Raymond George |title=Musings Of The Masters: An Anthology Of Mathematical Reflections |year=2004 |publisher=MAA |location=Washington, D.C. |isbn=978-0-88385-549-2 |oclc=56537093 |page=170}}</ref>

He died Roman Catholic{{sfn|Bochner|1958|p=446}} on February 8, 1957, at [[Walter Reed Army Medical Hospital]] and was buried at [[Princeton Cemetery]].{{sfn|Macrae|1992|p=380}}<ref>{{Cite web|url=https://nassauchurch.org/about/princetoncemetery/|title = Nassau Presbyterian Church}}</ref>

== Mathematics ==

=== Set theory ===
{{See also|Von Neumann–Bernays–Gödel set theory}}
[[File:NBG Evolution svg.svg|thumb|upright=1.5|History of approaches that led to NBG set theory]]
At the beginning of the 20th century, efforts to base mathematics on [[naive set theory]] suffered a setback due to [[Russell's paradox]] (on the set of all sets that do not belong to themselves).{{sfn|Macrae|1992|pp=104–105}} The problem of an adequate axiomatization of [[set theory]] was resolved implicitly about twenty years later by [[Ernst Zermelo]] and [[Abraham Fraenkel]]. [[Zermelo–Fraenkel set theory]] provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his 1925 doctoral thesis, von Neumann demonstrated two techniques to exclude such sets—the ''[[axiom of regularity|axiom of foundation]]'' and the notion of ''[[Class (set theory)|class]].''<ref name=vanheijenoort>{{cite book |last=Van Heijenoort |first=Jean |author-link=Jean van Heijenoort |year=1967 |title=From Frege to Gödel: a Source Book in Mathematical Logic, 1879–1931 |url=https://archive.org/details/fromfregetogodel0000vanh |url-access=registration |location=Cambridge, Massachusetts |publisher=Harvard University Press |isbn=978-0-674-32450-3|oclc=523838 }}</ref>

The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the Zermelo–Fraenkel principles. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes 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 the ''method of [[inner model]]s'', which became an essential demonstration instrument in set theory.{{r|vanheijenoort}}

The second approach to the problem of sets belonging to themselves took as its base the notion of [[Class (set theory)|class]], and defines a set as a class that belongs to other classes, while a ''proper class'' is defined as a class that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is a ''proper class'', not a set.{{r|vanheijenoort}}

Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the [[Ordinal number|ordinal]] and [[cardinal number]]s as well as the first strict formulation of principles of definitions by the [[transfinite induction]]".{{sfn|Murawski|2010|p=196}}

====Von Neumann paradox====
{{main|Von Neumann paradox}}
Building on the [[Hausdorff paradox]] of [[Felix Hausdorff]] (1914), [[Stefan Banach]] and [[Alfred Tarski]] in 1924 showed how to subdivide a three-dimensional [[ball (mathematics)|ball]] into [[disjoint sets]], then translate and rotate these sets to form two identical copies of the same ball; this is the [[Banach–Tarski paradox]]. They also proved that a two-dimensional disk has no such paradoxical decomposition. But in 1929,<ref>{{citation | first=J. | last=von Neumann | author-link=John von Neumann | url=http://matwbn.icm.edu.pl/ksiazki/fm/fm13/fm1316.pdf | title=Zur allgemeinen Theorie des Masses |trans-title=On the general theory of mass |language=de | journal=[[Fundamenta Mathematicae]] | volume=13 | pages=73–116 | year=1929 | doi=10.4064/fm-13-1-73-116 | doi-access=free }}</ref> von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preserving [[affine transformation]]s instead of translations and rotations. The result depended on finding [[free group]]s of affine transformations, an important technique extended later by von Neumann in [[#Measure theory|his work on measure theory]].{{sfn|Ulam|1958|pages=14–15}}

=== Proof theory ===
{{See also|Hilbert's program}}

With the contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its [[consistency]]. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger [[Axiom#Mathematical logic|axioms]] that could be used to prove a broader class of theorems.<ref>{{cite encyclopedia |last=Von Plato |first=Jan |title=The Development of Proof Theory |encyclopedia=The Stanford Encyclopedia of Philosophy |year=2018 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/proof-theory-development/ |publisher=Stanford University |edition=Winter 2018 |access-date=2023-09-25 }}</ref>

By 1927, von Neumann was involving himself in discussions in Göttingen on whether [[elementary arithmetic]] followed from [[Peano axioms]].<ref>{{cite journal |last1=van der Waerden |first1=B. L. |author-link1=Bartel Leendert van der Waerden |title=On the sources of my book Moderne algebra |journal=Historia Mathematica |date=1975 |volume=2 |issue=1 |pages=31–40 |doi=10.1016/0315-0860(75)90034-8 |doi-access=free }}</ref> Building on the work of [[Wilhelm Ackermann|Ackermann]], he began attempting to prove (using the [[Finitism|finistic]] methods of [[Hilbert's program|Hilbert's school]]) the consistency of [[Peano axioms#Peano arithmetic as first-order theory|first-order arithmetic]]. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on [[Mathematical induction|induction]]).<ref>{{cite journal |last1=Neumann |first1=J. v. |title=Zur Hilbertschen Beweistheorie |journal=Mathematische Zeitschrift |date=1927 |volume=24 |pages=1–46 |language=German |doi=10.1007/BF01475439 |s2cid=122617390 |url=https://eudml.org/doc/167910}}</ref> He continued looking for a more general proof of the consistency of classical mathematics using methods from [[proof theory]].{{sfn|Murawski|2010|pp=204-206}}

A strongly negative answer to whether it was definitive arrived in September 1930 at the [[Second Conference on the Epistemology of the Exact Sciences]], 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 expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.{{sfn|Rédei|2005|p=123}} At the conference, von Neumann suggested to Gödel that he should try to transform his results for undecidable propositions about integers.{{sfn|von Plato|2018|p=4080}}

Less than a month later, von Neumann communicated to Gödel an interesting consequence of his theorem: the usual axiomatic systems are unable to demonstrate their own consistency.{{sfn|Rédei|2005|p=123}} Gödel replied that he had already discovered this consequence, now known as his [[second incompleteness theorem]], and that he would send a preprint of his article containing both results, which never appeared.<ref>{{cite book |last=Dawson |first=John W. Jr. |author-link=John W. Dawson, Jr. |year=1997 |title=Logical Dilemmas: The Life and Work of Kurt Gödel |location=Wellesley, Massachusetts |publisher=A. K. Peters |isbn=978-1-56881-256-4 |page=70}}</ref>{{sfn|von Plato|2018|pp=4083-4088}}{{sfn|von Plato|2020|pp=24-28}} Von Neumann acknowledged Gödel's priority in his next letter.{{sfn|Rédei|2005|p=124}} However, von Neumann's method of proof differed from Gödel's, and he was also of the opinion that the second incompleteness theorem had dealt a much stronger blow to Hilbert's program than Gödel thought it did.{{sfn|von Plato|2020|p=22}}<ref>{{cite book |last1=Sieg |first1=Wilfried |title=Hilbert's Programs and Beyond |date=2013 |publisher=Oxford University Press |isbn=978-0195372229 |url=https://books.google.com/books?id=4lDrwqo-8TkC&pg=PA149 |page=149}}</ref> With this discovery, which drastically changed his views on mathematical rigor, von Neumann ceased research in the [[foundations of mathematics]] and [[metamathematics]] and instead spent time on problems connected with applications.{{sfn|Murawski|2010|p=209}}

=== Ergodic theory ===
In a series of papers published in 1932, von Neumann made foundational contributions to [[ergodic theory]], a branch of mathematics that involves the states of [[dynamical systems]] with an [[invariant measure]].<ref>{{cite journal|author-link=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|language=German}} {{pb}}
Two of the papers are: {{pb}}
{{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 |doi-access=free |bibcode-access=free }} {{pb}}
{{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|doi-access=free}}.</ref> Of the 1932 papers on ergodic theory, [[Paul Halmos]] wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality".{{sfn|Halmos|1958|p=93}} By then von Neumann had already written his articles on [[operator theory]], and the application of this work was instrumental in his [[Ergodic theory#Mean ergodic theorem|mean ergodic theorem]].{{sfn|Halmos|1958|p=91}}

The theorem is about arbitrary [[One-parameter group|one-parameter]] [[unitary group]]s <math>\mathit{t} \to \mathit{V_t}</math> and states that for every vector <math>\phi</math> in the [[Hilbert space]], <math display=inline>\lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} V_t(\phi) \, dt</math> exists in the sense of the metric defined by the Hilbert norm and is a vector <math>\psi</math> which is such that <math>V_t(\psi) = \psi</math> for all <math>t</math>. This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating to [[Ludwig Boltzmann|Boltzmann's]] [[ergodic hypothesis]]. He also pointed out that [[ergodicity]] had not yet been achieved and isolated this for future work.<ref name=mackey1990>{{harvc |last1=Mackey |first1=George W. |author-link=George Mackey |year=1990 |chapter=Von Neumann and the Early Days of Ergodic Theory |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=27–30}}</ref>

Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodic [[Measure-preserving dynamical system|measure preserving actions]] of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with [[Paul Halmos]] have significant applications in other areas of mathematics.{{r|mackey1990}}<ref>{{harvc |last1=Ornstein |first1=Donald S. |author-link=Donald Samuel Ornstein |year=1990 |chapter=Von Neumann and Ergodic Theory |in1=Glimm |in2=Impagliazzo |in3=Singer |page=39}}</ref>

=== Measure theory ===
{{See also|Lifting theory}}

In [[Measure (mathematics)|measure theory]], the "problem of measure" for an {{mvar|n}}-dimensional [[Euclidean space]] {{math|'''R'''<sup>''n''</sup>}} may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of {{math|'''R'''<sup>''n''</sup>}}?"{{sfn|Halmos|1958|p=86}} The work of [[Felix Hausdorff]] and [[Stefan Banach]] had implied that the problem of measure has a positive solution if {{math|1=''n'' = 1}} or {{math|1=''n'' = 2}} and a negative solution (because of the [[Banach–Tarski paradox]]) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of the [[transformation group]] of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the [[Euclidean group]] is a [[solvable group]] for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."{{sfn|Halmos|1958|p=87}} Around 1942 he told [[Dorothy Maharam]] how to prove that every [[Complete measure|complete]] [[σ-finite measure|σ-finite]] [[measure space]] has a multiplicative lifting; he did not publish this proof and she later came up with a new one.{{sfn|Pietsch|2007|p=168}}

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.{{sfn|Halmos|1958|p=88}} A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of [[Alfréd Haar|Haar]] regarding whether there existed an [[Algebra over a field|algebra]] of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions".{{sfn|Dieudonné|2008}} He proved this in the positive, and in later papers with [[Marshall Harvey Stone|Stone]] discussed various generalizations and algebraic aspects of this problem.<ref>{{cite book |last1=Ionescu-Tulcea |first1=Alexandra |author1-link=Alexandra Bellow |last2=Ionescu-Tulcea |first2=Cassius |author2-link=Cassius Ionescu-Tulcea |title=Topics in the Theory of Lifting |date=1969 |publisher=Springer-Verlag Berlin Heidelberg |isbn=978-3-642-88509-9 |page=V |url=https://www.springer.com/gp/book/9783642885099}}</ref> He also proved by new methods the existence of [[Disintegration theorem|disintegrations]] for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for [[compact group]]s.{{sfn|Dieudonné|2008}} He had to create entirely new techniques to apply this to [[locally compact group]]s.{{sfn|Halmos|1958|p=89}} He also gave a new, ingenious proof for the [[Radon–Nikodym theorem]].<ref>{{cite journal |last1=Neumann |first1=J. v. |title=On Rings of Operators. III. |journal=Annals of Mathematics |date=1940 |volume=41 |issue=1 |pages=94–161 |doi=10.2307/1968823 |jstor=1968823 |url=https://www.jstor.org/stable/1968823}}</ref> His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.{{sfn|Halmos|1958|p=90}}<ref>{{cite book |last1=Neumann |first1=John von |title=Functional Operators, Volume 1: Measures and Integrals |date=1950 |publisher=Princeton University Press |isbn=9780691079660 |url=https://press.princeton.edu/books/paperback/9780691079660/functional-operators-am-21-volume-1}}</ref><ref>{{cite book |last1=von Neumann |first1=John |title=Invariant Measures |date=1999 |publisher=American Mathematical Society |isbn=978-0-8218-0912-9 |url=https://bookstore.ams.org/inmeas}}</ref>

=== Topological groups ===
Using his previous work on measure theory, von Neumann made several contributions to the theory of [[topological group]]s, beginning with a paper on almost periodic functions on groups, where von Neumann extended [[Harald Bohr|Bohr's]] theory of [[almost periodic function]]s to arbitrary [[Group (mathematics)|groups]].<ref>{{cite journal |last1=von Neumann |first1=John |title=Almost Periodic Functions in a Group. I. |journal=Transactions of the American Mathematical Society |date=1934 |volume=36 |issue=3 |pages=445–492 |doi=10.2307/1989792 |jstor=1989792 |url=https://www.jstor.org/stable/1989792}}</ref> He continued this work with another paper in conjunction with [[Salomon Bochner|Bochner]] that improved the theory of almost [[Periodic function|periodicity]] to include [[Function (mathematics)|functions]] that took on elements of [[Vector space|linear spaces]] as values rather than numbers.<ref>{{cite journal |last1=von Neumann |first1=John |last2=Bochner |first2=Salomon |title=Almost Periodic Functions in Groups, II. |journal=Transactions of the American Mathematical Society |date=1935 |volume=37 |issue=1 |pages=21–50 |doi=10.2307/1989694 |jstor=1989694 |url=https://www.jstor.org/stable/1989694}}</ref> In 1938, he was awarded the [[Bôcher Memorial Prize]] for his work in [[Mathematical analysis|analysis]] in relation to these papers.<ref>{{cite web |url=https://www.ams.org/profession/prizes-awards/pabrowse?purl=bocher-prize#year=1938 |title=AMS Bôcher Prize |publisher=AMS |date=January 5, 2016 |access-date=2018-01-12}}</ref>{{sfn|Bochner|1958|p=440}}

In a 1933 paper, he used the newly discovered [[Haar measure]] in the solution of [[Hilbert's fifth problem]] for the case of [[compact group]]s.<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=170–190|doi=10.2307/1968347|jstor=1968347|language=German}}</ref> The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of [[Linear map|linear transformations]] and found that closed [[subgroup]]s of a general [[linear group]] are [[Lie group]]s.<ref>{{cite journal |last1=v. Neumann |first1=J. |title=Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen |journal=Mathematische Zeitschrift |date=1929 |volume=30 |issue=1 |pages=3–42 |doi=10.1007/BF01187749 |s2cid=122565679 |language=German}}</ref> This was later extended by [[Élie Cartan|Cartan]] to arbitrary Lie groups in the form of the [[closed-subgroup theorem]].{{sfn|Bochner|1958|p=441}}{{sfn|Dieudonné|2008}}

=== Functional analysis ===
{{Main|Operator theory}}{{See also|Spectral theorem}}
Von Neumann was the first to axiomatically define an abstract [[Hilbert space]]. He defined it as a [[Vector space|complex vector space]] with a [[Inner product space|Hermitian scalar product]], with the corresponding [[Norm (mathematics)|norm]] being both separable and complete. In the same papers he also proved the general form of the [[Cauchy–Schwarz inequality]] that had previously been known only in specific examples.{{sfn|Pietsch|2007|p=11}} He continued with the development of the [[spectral theory]] of operators in Hilbert space in three seminal papers between 1929 and 1932.{{sfn|Dieudonné|1981|p=172}} This work cumulated in his ''[[Mathematical Foundations of Quantum Mechanics]]'' which alongside two other books by [[Marshall Harvey Stone|Stone]] and [[Stefan Banach|Banach]] in the same year were the first monographs on Hilbert space theory.{{sfn|Pietsch|2007|p=14}} Previous work by others showed that a theory of [[Weak topology|weak topologies]] could not be obtained by using [[Weak convergence (Hilbert space)|sequences]]. Von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining [[Locally convex topological vector space|locally convex spaces]] and [[topological vector spaces]] for the first time. In addition several other topological properties he defined at the time (he was among the first mathematicians to apply new topological ideas from [[Felix Hausdorff|Hausdorff]] from Euclidean to Hilbert spaces){{sfn|Dieudonné|1981|pp=211,218}} such as [[Bounded set (topological vector space)|boundness]] and [[Totally bounded space|total boundness]] are still used today.{{sfn|Pietsch|2007|pp=58,65-66}} For twenty years von Neumann was considered the 'undisputed master' of this area.{{sfn|Dieudonné|2008}} These developments were primarily prompted by needs in [[quantum mechanics]] where von Neumann realized the need to extend [[Self-adjoint operator#Spectral theorem|the spectral theory of Hermitian operators]] from the bounded to the [[Unbounded operator|unbounded]] case.<ref name=steen>{{cite journal |last1=Steen |first1=L. A. |author-link1=Lynn Steen |title=Highlights in the History of Spectral Theory |journal=The American Mathematical Monthly |date=April 1973 |volume=80 |issue=4 |pages=359–381, esp. 370–373 |doi=10.1080/00029890.1973.11993292 |jstor=2319079 |url=https://doi.org/10.2307/2319079}}</ref> Other major achievements in these papers include a complete elucidation of spectral theory for [[normal operator]]s, the first abstract presentation of the [[Trace (linear algebra)|trace]] of a [[Positive operator (Hilbert space)|positive operator]],<ref>{{cite journal |last1=Pietsch |first1=Albrecht |author-link=:de:Albrecht Pietsch |title=Traces of operators and their history |journal=Acta et Commentationes Universitatis Tartuensis de Mathematica |date=2014 |volume=18 |issue=1 |pages=51–64 |doi=10.12697/ACUTM.2014.18.06 |url=https://acutm.math.ut.ee/index.php/acutm/article/download/ACUTM.2014.18.06/22|doi-access=free }}</ref>{{sfn|Lord|Sukochev|Zanin|2012|p=1}} a generalisation of [[Frigyes Riesz|Riesz]]'s presentation of [[David Hilbert|Hilbert]]'s spectral theorems at the time, and the discovery of [[Self-adjoint operator#Definitions|Hermitian operators]] in a Hilbert space, as distinct from [[self-adjoint operator]]s, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. He wrote a paper detailing how the usage of [[Matrix (mathematics)#Infinite matrices|infinite matrices]], common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory lead to his most profound invention in pure mathematics, the study of von Neumann algebras and in general of [[operator algebra]]s.{{sfn|Dieudonné|1981|pp=175–176, 178–179, 181, 183}}

His later work on rings of operators lead to him revisiting his work on spectral theory and providing a new way of working through the geometric content by the use of direct integrals of Hilbert spaces.{{r|steen}} Like in his work on measure theory he proved several theorems that he did not find time to publish. He told [[Nachman Aronszajn]] and K. T. Smith that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the [[invariant subspace problem]].{{sfn|Pietsch|2007|p=202}}

With [[Isaac Jacob Schoenberg|I. J. Schoenberg]] he wrote several items investigating [[Translational symmetry|translation invariant]] Hilbertian [[Metric (mathematics)|metrics]] on the [[Number line|real number line]] which resulted in their complete classification. Their motivation lie in various questions related to embedding [[metric space]]s into Hilbert spaces.<ref>{{cite arXiv |last1=Kar |first1=Purushottam |last2=Karnick |first2=Harish |title=On Translation Invariant Kernels and Screw Functions |date=2013 |page=2 |class=math.FA |eprint=1302.4343}}</ref><ref>{{cite journal |last1=Alpay |first1=Daniel |last2=Levanony |first2=David |title=On the Reproducing Kernel Hilbert Spaces Associated with the Fractional and Bi-Fractional Brownian Motions |journal=Potential Analysis |date=2008 |volume=28 |issue=2 |pages=163–184 |doi=10.1007/s11118-007-9070-4 |arxiv=0705.2863 |s2cid=15895847 }}</ref>

With [[Pascual Jordan]] he wrote a short paper giving the first derivation of a given norm from an [[inner product space|inner product]] by means of the [[Parallelogram law#The parallelogram law in inner product spaces|parallelogram identity]].{{sfn|Horn|Johnson|2013|p=320}} His [[Trace inequality#Von Neumann's trace inequality and related results|trace inequality]] is a key result of matrix theory used in matrix approximation problems.{{sfn|Horn|Johnson|2013|p=458}} He also first presented the idea that the dual of a pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as  symmetric absolute norms).<ref>{{cite book |last1=Horn |first1=Roger A. |author1-link=Roger Horn |last2=Johnson |first2=Charles R. |author2-link=Charles Royal Johnson |title=Topics in Matrix Analysis |date=1991 |publisher=Cambridge University Press |isbn=0-521-30587-X |url=https://www.cambridge.org/core/books/topics-in-matrix-analysis/B988495A235F1C3406EA484A2C477B03 |page=139}}</ref>{{sfn|Horn|Johnson|2013|p=335}}<ref>{{cite book |last1=Bhatia |first1=Rajendra |title=Matrix Analysis |series=Graduate Texts in Mathematics |date=1997 |volume=169 |publisher=Springer |location=New York |isbn=978-1-4612-0653-8 |page=109 |doi=10.1007/978-1-4612-0653-8 |url=https://link.springer.com/book/10.1007/978-1-4612-0653-8}}</ref> This paper leads naturally to the study of symmetric [[operator ideal]]s and is the beginning point for modern studies of symmetric [[operator space]]s.{{sfn|Lord|Sukochev|Zanin|2021|p=73}}

Later with [[Robert Schatten]] he initiated the study of [[nuclear operator]]s on Hilbert spaces,<ref>{{cite journal |last1=Prochnoa |first1=Joscha |last2=Strzelecki |first2=Michał |title=Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings |journal=Journal of Approximation Theory |date=2022 |volume=277 |page=105736 |doi=10.1016/j.jat.2022.105736 |arxiv=2103.13050 |s2cid=232335769 }}</ref><ref>{{cite web |url=https://encyclopediaofmath.org/wiki/Nuclear_operator |archive-url=https://web.archive.org/web/20210623153701/https://encyclopediaofmath.org/wiki/Nuclear_operator |archive-date=2021-06-23 |title=Nuclear operator |access-date=August 7, 2022 |publisher=Encyclopedia of Mathematics}}</ref> [[Topological tensor product#Cross norms and tensor products of Banach spaces|tensor products of Banach spaces]],{{sfn|Pietsch|2007|p=372}} introduced and studied [[trace class]] operators,{{sfn|Pietsch|2014|p=54}} their [[Ideal (ring theory)|ideals]], and their [[Duality (mathematics)|duality]] with [[compact operator]]s, and [[predual]]ity with [[bounded operator]]s.{{sfn|Lord|Sukochev|Zanin|2012|p=73}} The generalization of this topic to the study of [[Nuclear operators between Banach spaces|nuclear operators on Banach spaces]] was among the first achievements of [[Alexander Grothendieck]].{{sfn|Lord|Sukochev|Zanin|2021|p=26}}{{sfn|Pietsch|2007|p=272}} Previously in 1937 von Neumann published several results in this area, for example giving 1-parameter scale of different cross norms on <math>\textit{l}\,_2^n\otimes\textit{l}\,_2^n</math> and proving several other results on what are now known as Schatten–von Neumann ideals.{{sfn|Pietsch|2007|pp=272,338}}

=== Operator algebras ===
{{Main|Von Neumann algebra}}{{See also|Direct integral}}
Von Neumann founded the study of rings of operators, through the [[von Neumann algebra]]s (originally called W*-algebras). While his original ideas for [[Noncommutative ring|rings]] of [[Linear map|operators]] existed already in 1930, he did not begin studying them in depth until he met [[Francis Joseph Murray|F. J. Murray]] several years later.{{sfn|Pietsch|2007|p=140}}<ref>{{harvc |last1=Murray |first1=Francis J. |author-link=Francis Joseph Murray |year=1990 |chapter=The Rings of Operators Papers |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=57–59}}</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 function|identity operator]].<ref>{{harvc |last1=Petz |first1=D. |author-link1=Dénes Petz |last2=Rédei |first2=M. R. |contribution=John von Neumann And The Theory Of Operator Algebras |in1=Bródy |in2=Vámos |year=1995 |pages=163–181}}</ref> The [[von Neumann bicommutant theorem]] shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the [[bicommutant]].<ref>{{cite web |url=https://www.princeton.edu/~hhalvors/restricted/jones.pdf |title=Von Neumann Algebras |access-date=January 6, 2016 |publisher=Princeton University }}</ref> After elucidating the study of the [[Commutative ring|commutative algebra]] case, von Neumann embarked in 1936, with the partial collaboration of Murray, on the [[Noncommutative ring|noncommutative]] case, the general study of [[von Neumann algebra|factors]] classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 "rank among the masterpieces of analysis in the twentieth century";{{sfn|Dieudonné|2008|p=90}} they collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification of [[Von Neumann algebra#Factors|factors]].{{sfn|Pietsch|2007|pp=151}} In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949.{{sfn|Pietsch|2007|p=146}}<ref>{{cite web |url=https://www.math.ucla.edu/~brh6/DirectIntegral.pdf |archive-url=https://web.archive.org/web/20150702001911/http://www.math.ucla.edu/~brh6/DirectIntegral.pdf |archive-date=2015-07-02 |title=Direct Integrals of Hilbert Spaces and von Neumann Algebras |access-date=January 6, 2016 |publisher=University of California at Los Angeles }}</ref> Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly write out.{{sfn|Segal|1965}}<ref>{{harvc |last1=Kadison |first1=Richard V. |author-link=Richard Kadison |year=1990 |chapter=Operator Algebras - An Overview |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=65,71,74}}</ref> Another important result on [[polar decomposition]] was published in 1932.{{sfn|Pietsch|2007|p=148}}

=== Lattice theory ===
{{Main|Continuous geometry}}{{See also|Complemented lattice#Orthomodular lattices}}

Between 1935 and 1937, von Neumann worked on [[Lattice (order)|lattice theory]], the theory of [[partially ordered set]]s in which every two elements have a greatest lower bound and a least upper bound. As [[Garrett Birkhoff]] wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor".{{sfn|Birkhoff|1958|p=50}} Von Neumann combined traditional projective geometry with modern algebra ([[linear algebra]], [[Ring (mathematics)|ring theory]], lattice theory). Many previously geometric results could then be interpreted in the case of general [[Module (mathematics)|modules]] over rings. His work laid the foundations for some of the modern work in projective geometry.<ref name=lashkhi1995>{{cite journal |last=Lashkhi |first=A. A. |title=General geometric lattices and projective geometry of modules |journal=Journal of Mathematical Sciences |date=1995 |volume=74 |issue=3 |pages=1044–1077 |doi=10.1007/BF02362832 |s2cid=120897087 |doi-access=free }}</ref>

His biggest contribution was founding the field of [[continuous geometry]].<ref>{{cite journal | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Examples of continuous geometries | jstor=86391 | doi=10.1073/pnas.22.2.101 | jfm=62.0648.03 | year=1936 | journal=Proc. Natl. Acad. Sci. USA | volume=22 | issue=2 | pages=101–108 | pmid=16588050 | pmc=1076713| bibcode=1936PNAS...22..101N | doi-access=free}} {{pb}}
{{cite journal | last1=von Neumann | first1=John | author1-link=John von Neumann | title=Continuous geometry | journal=Proceedings of the National Academy of Sciences of the United States of America| orig-year=1960 | url=https://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| volume=22| issue=2| pages=92–100| doi=10.1073/pnas.22.2.92| pmid=16588062| pmc=1076712| doi-access=free}} {{pb}}
{{cite book | last1=von Neumann | first1=John | author1-link=John von Neumann | editor1-last=Taub | editor1-first=A. H. | title=Collected works. Vol. IV: Continuous geometry and other topics | url=https://books.google.com/books?id=HOTXAAAAMAAJ | publisher=Pergamon Press | location=Oxford | mr=0157874 | year=1962}} {{pb}}
{{cite journal | last1=von Neumann | first1=John| author1-link=John von Neumann | editor1-last=Halperin | editor1-first=Israel | title=Continuous geometries with a transition probability | orig-year=1937 | url=https://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=978-0-8218-2252-4 | doi=10.1090/memo/0252}}</ref> It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex [[projective geometry]], where instead of the [[Dimension (vector space)|dimension]] of a [[Linear subspace|subspace]] being in a discrete set <math>0, 1, ..., \mathit{n}</math> it can be an element of the [[unit interval]] <math>[0,1]</math>. Earlier, [[Karl Menger|Menger]] and Birkhoff had axiomatized [[Complex projective space|complex projective geometry]] in terms of the properties of its [[Linear subspace#Lattice of subspaces|lattice of linear subspaces]]. Von Neumann, following his work on rings of operators, weakened those [[axiom]]s to describe a broader class of lattices, the continuous geometries.

While the dimensions of the subspaces of projective geometries are a discrete set (the [[Natural number|non-negative integers]]), the dimensions of the elements of a continuous geometry can range continuously across the unit interval <math>[0,1]</math>. 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 [[Von Neumann algebra#Projections|projections]] of the [[hyperfinite type II factor]].{{sfn|Macrae|1992|p=140}}<ref>{{cite journal|first=John |last=von Neumann|doi=10.1007/BF01782352|title= Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren|language=de |journal= [[Mathematische Annalen]]|volume=102 |issue=1 |year=1930|pages= 370–427|bibcode=1930MatAn.102..685E|s2cid=121141866}}. The original paper on von Neumann algebras.</ref>

In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of <math>\mathit{CG(F)}</math> (continuous-dimensional projective geometry over an arbitrary [[division ring]] <math>\mathit{F}\,</math>) in abstract language of lattice theory.{{sfn|Birkhoff|1958|pp=50-51}} Von Neumann provided an abstract exploration of dimension in completed [[Complemented lattice|complemented]] [[Modular lattice|modular]] topological lattices (properties that arise in the [[Linear subspace#Lattice of subspaces|lattices of subspaces]] of [[inner product space]]s): <blockquote>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.</blockquote>

For any integer <math>n > 3</math> every <math>\mathit{n}</math>-dimensional abstract projective geometry is [[Isomorphism|isomorphic]] to the subspace-lattice of an <math>\mathit{n}</math>-dimensional [[vector space]] <math>V_n(F)</math> over a (unique) corresponding  division ring <math>F</math>. This is known as the [[Veblen–Young theorem]]. Von Neumann extended this fundamental result in projective geometry to the continuous dimensional case.{{sfn|Birkhoff|1958|p=51}} This [[Continuous geometry#Coordinatization theorem|coordinatization theorem]] stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques.{{r|lashkhi1995}}<ref>{{cite journal |last1=Wehrung |first1=Friedrich |title=Von Neumann coordinatization is not first-order |journal=Journal of Mathematical Logic |date=2006 |volume=6 |issue=1 |pages=1–24 |doi=10.1142/S0219061306000499 |arxiv=math/0409250 |s2cid=39438451 }}</ref> Birkhoff described this theorem as follows: <blockquote>Any complemented modular lattice {{mvar|L}} having a "basis" of {{math|''n'' ≥ 4}} pairwise perspective elements, is isomorphic with the lattice {{math|ℛ(''R'')}} of all principal [[Ideal (ring theory)|right-ideals]] of a suitable [[Von Neumann regular ring|regular ring]] {{mvar|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.{{sfn|Birkhoff|1958|p=52}}</blockquote>

This work required the creation of [[Von Neumann regular ring|regular rings]].<ref>{{cite book |last1=Goodearl |first1=Ken R. |title=Von Neumann Regular Rings |date=1979 |publisher=Pitman Publishing |isbn=0-273-08400-3 |page=ix}}</ref> A von Neumann regular ring is a [[Ring (mathematics)|ring]] where for every <math>a</math>, an element <math>x</math> exists such that <math>axa = a</math>.{{sfn|Birkhoff|1958|p=52}} These rings came from and have connections to his work on von Neumann algebras, as well as [[AW*-algebra]]s and various kinds of [[C*-algebra]]s.<ref>{{cite journal |last1=Goodearl |first1=Ken R. |title=Von Neumann regular rings: connections with functional analysis |journal=Bulletin of the American Mathematical Society |date=1981 |volume=4 |issue=2 |pages=125–134 |doi=10.1090/S0273-0979-1981-14865-5 |doi-access=free }}</ref>

Many smaller technical results were proven during the creation and proof of the above theorems, particularly regarding [[Distributive property|distributivity]] (such as infinite distributivity), von Neumann developing them as needed. He also developed a theory of valuations in lattices, and shared in developing the general theory of [[metric lattice]]s.{{sfn|Birkhoff|1958|pp=52-53}}

Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two-year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians. A final contribution in 1940 was for a joint seminar he conducted with Birkhoff at the Institute for Advanced Study on the subject where he developed a theory of σ-complete lattice ordered rings. He never wrote up the work for publication.{{sfn|Birkhoff|1958|pp=55-56}}

=== 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 [[Normal distribution|normally]] distributed variables.<ref>{{cite journal|last=von Neumann |first=John|year=1941|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|doi-access=free}}</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 |last1=Durbin |first1=J. |last2=Watson |first2=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 |jstor=2332391}}</ref> for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order [[Autoregressive model|autoregression]].<ref name="jstor.org"/>

Subsequently, [[Denis Sargan]] and [[Alok Bhargava]] extended the results for testing whether 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.<ref>{{cite journal |last1=Sargan |first1=J.D. |last2=Bhargava |first2=Alok |year=1983 |jstor=1912252 |title=Testing residuals from least squares regression for being generated by the Gaussian random walk|journal=[[Econometrica]]|volume=51|issue=1 |pages=153–174|doi=10.2307/1912252}}</ref>

=== Other work ===
In his early years, von Neumann published several papers related to set-theoretical real analysis and number theory.<ref>{{cite journal |last1=Rédei |first1=László |author-link1=László Rédei |title=Neumann János munkássága az algebrában és számelméletben |journal=Matematikai Lapok |date=1959 |volume=10 |pages=226–230 |language=Hungarian |url=http://real-j.mtak.hu/id/eprint/9389}}</ref> In a paper from 1925, he proved that for any dense sequence of points in <math>[0,1]</math>, there existed a rearrangement of those points that is [[Equidistributed sequence|uniformly distributed]].<ref>{{cite journal |last1=von Neumann |first1=J. |title=Egyenletesen sürü szämsorozatok (Gleichmässig dichte Zahlenfolgen) |journal=Mat. Fiz. Lapok |date=1925 |volume=32 |pages=32–40 |url=http://real-j.mtak.hu/7301/}}</ref><ref>{{cite journal |last1=Carbone |first1=Ingrid |last2=Volcic |first2=Aljosa |title=A von Neumann theorem for uniformly distributed sequences of partitions |journal=Rend. Circ. Mat. Palermo |date=2011 |volume=60 |issue=1–2 |pages=83–88 |doi=10.1007/s12215-011-0030-x|arxiv=0901.2531 |s2cid=7270857 }}</ref><ref>{{cite journal |last1=Niederreiter |first1=Harald |author1-link=Harald Niederreiter |title=Rearrangement theorems for sequences |journal=Astérisque |date=1975 |volume=24-25 |pages=243–261 |url=http://www.numdam.org/item/?id=AST_1975__24-25__243_0}}</ref> In 1926 his sole publication was on [[Heinz Prüfer|Prüfer's]] theory of [[Ideal number|ideal algebraic numbers]] where he found a new way of constructing them, thus extending Prüfer's theory to the [[Field (mathematics)|field]] of all [[algebraic number]]s, and clarified their relation to [[p-adic number]]s.<ref>{{cite journal |last1=von Neumann |first1=J. |title=Zur Prüferschen Theorie der idealen Zahlen |journal=Acta Szeged |date=1926 |volume=2 |pages=193–227 |url=http://acta.bibl.u-szeged.hu/13323/ |jfm=52.0151.02}}</ref>{{sfn|Ulam|1958|pp=9-10}}<ref>{{cite book |last1=Narkiewicz |first1=Wladyslaw |title=Elementary and Analytic Theory of Algebraic Numbers |series=Springer Monographs in Mathematics |date=2004 |publisher=Springer |isbn=978-3-662-07001-7  |doi=10.1007/978-3-662-07001-7 |edition=3rd |page=120}} {{pb}} {{cite book |last1=Narkiewicz |first1=Władysław |title=The Story of Algebraic Numbers in the First Half of the 20th Century: From Hilbert to Tate |series=Springer Monographs in Mathematics |date=2018 |publisher=Springer |doi=10.1007/978-3-030-03754-3 |isbn=978-3-030-03754-3 |page=144}}</ref><ref>{{cite journal |last=van Dantzig |first=D. |author-link=David van Dantzig |title=Nombres universels ou p-adiques avec une introduction sur l'algèbre topologique |journal=Annales scientifiques de l'École Normale Supérieure |date=1936 |volume=53 |pages=282–283 |language=fr |doi=10.24033/asens.858 |url=https://eudml.org/doc/81525|doi-access=free }}</ref><ref>{{cite book |last1=Warner |first1=Seth |title=Topological Rings |date=1993 |publisher=North-Hollywood |isbn=9780080872896 |url=https://www.elsevier.com/books/topological-rings/warner/978-0-444-89446-5 |page=428}}</ref> 
In 1928 he published two additional papers continuing with these themes. The first dealt with [[Partition of a set|partitioning]] an [[Interval (mathematics)|interval]] into [[Countable set|countably]] many [[Congruence relation|congruent]] [[subset]]s. It solved a problem of [[Hugo Steinhaus]] asking whether an interval is <math>\aleph_0</math>-divisible. Von Neumann proved that indeed that all intervals, half-open, open, or closed are <math>\aleph_0</math>-divisible by translations (i.e. that these intervals can be decomposed into <math>\aleph_0</math> subsets that are congruent by translation).<ref>{{cite journal |last1=von Neumann |first1=J. |title=Die Zerlegung eines Intervalles in abzählbar viele kongruente Teilmengen |journal=Fundamenta Mathematicae |date=1928 |volume=11 |issue=1 |pages=230–238 |doi=10.4064/fm-11-1-230-238 |url=https://eudml.org/doc/211437 |jfm=54.0096.03|doi-access=free }}</ref>{{sfn|Wagon|Tomkowicz|2016|p=73}}{{sfn|Dyson|2013|p=156}}<ref>{{cite journal |last1=Harzheim |first1=Egbert |title=A Construction of Subsets of the Reals which have a Similarity Decomposition |journal=Order |date=2008 |volume=25 |issue=2 |pages=79–83 |doi=10.1007/s11083-008-9079-3|s2cid=45005704 }}</ref> His next paper dealt with giving a [[constructive proof]] without the [[axiom of choice]] that <math>2^{\aleph_0}</math> [[Algebraic independence|algebraically independent]] [[Real number|reals]] exist. He proved that <math>A_r = \textstyle\sum_{n=0}^{\infty} 2^{2^{[nr]}}\! \big/ \, 2^{2^{n^2}}</math> are algebraically independent for <math>r > 0</math>. Consequently, there exists a perfect algebraically independent set of reals the size of the [[Continuum (set theory)|continuum]].<ref>{{cite journal |last1=von Neumann |first1=J. |title=Ein System algebraisch unabhängiger Zahlen |journal=Mathematische Annalen |date=1928 |volume=99 |pages=134–141 |doi=10.1007/BF01459089 |url=https://eudml.org/doc/159249 |jfm=54.0096.02|s2cid=119788605 }}</ref><ref>{{cite journal |last1=Kuiper |first1=F. |last2=Popken |first2=Jan |title=On the So-Called von Neumann-Numbers |journal=Indagationes Mathematicae (Proceedings) |date=1962 |volume=65 |pages=385–390 |doi=10.1016/S1385-7258(62)50037-1|doi-access=free }}</ref><ref>{{cite journal |last1=Mycielski |first1=Jan |author1-link=Jan Mycielski |title=Independent sets in topological algebras |journal=Fundamenta Mathematicae |date=1964 |volume=55 |issue=2 |pages=139–147 |doi=10.4064/fm-55-2-139-147 |url=https://eudml.org/doc/213780 |doi-access=free }}</ref>{{sfn|Wagon|Tomkowicz|2016|p=114}} Other minor results from his early career include a proof of a [[maximum principle]] for the gradient of a minimizing function in the field of [[calculus of variations]],<!-- specifically proving the following theorem: Let <math>u: \mathbb{R}^n \rightarrow \mathbb{R}</math> be a [[Lipschitz continuity|Lipschitz function]] with constant <math>K</math>, and <math>\Omega</math> an open and bounded set in <math>\mathbb{R}^n</math>. If <math>u</math> is a minimum for <math>F</math> in <math>Lip_K(\Omega)</math>, then <math>\sup_{x \in \Omega, y \in \delta\Omega} \frac{|u(x) - u(y)|}{|x - y|} = \sup_{x \neq y \in \Omega} \frac{|u(x) - u(y)|}{|x - y|}</math> (unnecessary detail for a minor result) --><ref>{{cite journal |last1=von Neumann |first1=J. |title=Über einen Hilfssatz der Variationsrechnung |journal=Abhandlungen Hamburg |date=1930 |volume=8 |pages=28–31 |url=https://abhandlungen.math.uni-hamburg.de/en/archiv.php?vol=8 |jfm=56.0440.04}}</ref><ref>{{cite journal |last=Miranda |first=Mario |title=Maximum principles and minimal surfaces |journal=Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |date=1997 |volume=4, 25 |issue=3–4 |pages=667–681 |url=http://www.numdam.org/item/ASNSP_1997_4_25_3-4_667_0/}}</ref><ref>{{cite book |last1=Gilbarg |first1=David |last2=Trudinger |first2=Neil S. |author1-link=David Gilbarg |author2-link=Neil Trudinger |title=Elliptic Partial Differential Equations of Second Order |date=2001 |publisher=Springer |doi=10.1007/978-3-642-61798-0 |isbn=978-3-642-61798-0 |edition=2 |url=https://link.springer.com/book/10.1007/978-3-642-61798-0 |page=316}}</ref><ref>{{cite book |last1=Ladyzhenskaya |first1=Olga A. |author1-link=Olga Ladyzhenskaya |last2=Ural'tseva |first2=Nina N. |author2-link=Nina Uraltseva |title=Linear and Quasilinear Elliptic Equations |date=1968 |publisher=Academic Press |isbn=978-1483253329 |pages= 14, 243}}</ref> and a small simplification of [[Hermann Minkowski]]'s theorem for linear forms in [[Geometry of numbers|geometric number theory]].<ref>{{cite journal |last1=von Neumann |first1=J. |title=Zum Beweise des Minkowskischen Stazes über Linearformen |journal=Mathematische Zeitschrift |date=1929 |volume=30 |pages=1–2 |doi=10.1007/BF01187748 |url=https://eudml.org/doc/168103 |jfm=55.0065.04|s2cid=123066944 }}</ref><ref>{{cite book |last=Koksma |first=J. F. |author-link=Jurjen Ferdinand Koksma |title=Diophantische Approximationen |date=1936 |publisher=Springer |doi=10.1007/978-3-642-65618-7 |isbn=978-3-642-65618-7 |language=German |url=https://link.springer.com/book/10.1007/978-3-642-65618-7 |page=15}}</ref>{{sfn|Ulam|1958|pp=10,23}}
Later in his career together with [[Pascual Jordan]] and [[Eugene Wigner]] he wrote a foundational paper classifying all [[Dimension (vector space)|finite-dimensional]] [[Jordan algebra#Formally real Jordan algebras|formally real Jordan algebras]] and discovering the [[Albert algebra]]s while attempting to look for a better [[Mathematical formulation of quantum mechanics|mathematical formalism for quantum theory]].<ref>{{cite web |last1=Baez |first1=John |author-link1=John C. Baez |title=State-Observable Duality (Part 2) |url=https://golem.ph.utexas.edu/category/2010/11/stateobservable_duality_part_2.html |website=The n-Category Café |access-date=20 August 2022}}</ref><ref>{{cite book |last1=McCrimmon |first1=Kevin |author-link1=Kevin McCrimmon |title=A Taste of Jordan Algebras |series=Universitext |date=2004 |publisher=Springer |location=New York |doi=10.1007/b97489 |isbn=978-0-387-21796-3 |url=https://link.springer.com/book/10.1007/b97489 |page=68}}</ref> In 1936 he attempted to further the program of replacing the axioms of his previous Hilbert space program with those of Jordan algebras<ref>{{cite journal |last=Rédei |first=Miklós |title=Why John von Neumann did not Like the Hilbert Space formalism of quantum mechanics (and what he liked instead) |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |date=1996 |volume=27 |issue=4 |pages=493–510 |doi=10.1016/S1355-2198(96)00017-2 |bibcode=1996SHPMP..27..493R |url=https://www.sciencedirect.com/science/article/abs/pii/S1355219896000172 }}</ref> in a paper investigating the infinite-dimensional case; he planned to write at least one further paper on the topic but never did.<ref>{{cite journal |last1=Wang |first1=Shuzhou |last2=Wang |first2=Zhenhua |title=Operator means in JB-algebras |journal=Reports on Mathematical Physics |date=2021 |volume=88 |issue=3 |page=383 |doi=10.1016/S0034-4877(21)00087-2 |arxiv=2012.13127 |bibcode=2021RpMP...88..383W |s2cid=229371549 }}</ref> Nevertheless, these axioms formed the basis for further investigations of algebraic quantum mechanics started by [[Irving Segal]].<ref>{{cite book |last=Landsman |first=Nicolaas P. |year=2009 |chapter=Algebraic Quantum Mechanics |editor-last1=Greenberger |editor-first1=Daniel |editor-link1=Daniel Greenberger |editor-last2=Hentschel |editor-first2=Klaus |editor-link2=Klaus Hentschel |editor-last3=Weinert |editor-first3=Friedel |title=Compendium of Quantum Physics: Concepts, Experiments, History and Philosophy |publisher=Springer |isbn=978-3-540-70626-7 |doi=10.1007/978-3-540-70626-7 |pages=6–7}}</ref><ref>{{cite encyclopedia |last1=Kronz |first1=Fred |last2=Lupher |first2=Tracy |year=2021 |title=Quantum Theory and Mathematical Rigor |url=https://plato.stanford.edu/entries/qt-nvd/ |edition=Winter 2021 |editor-last=Zalta |editor-first=Edward N. |encyclopedia=Stanford Encyclopedia of Philosophy |publisher=Stanford University |access-date=2022-12-21}}</ref>

==Physics==
=== Quantum mechanics ===
{{See also|Quantum mutual information|Measurement in quantum mechanics|Wave function collapse}}
Von Neumann was the first to establish a rigorous mathematical framework for [[quantum mechanics]], known as the [[Dirac–von Neumann axioms]], in his influential 1932 work ''[[Mathematical Foundations of Quantum Mechanics]]''.<ref name=VanHove>{{cite journal|author-link=Léon Van Hove|last1=Van Hove|first1=Léon|title=Von Neumann's Contributions to Quantum Theory|journal=[[Bulletin of the American Mathematical Society]]|date=1958|volume=64|issue=3|url=http://projecteuclid.org/euclid.bams/1183522374 |pages=95–99|doi=10.1090/s0002-9904-1958-10206-2|doi-access=free}}</ref> After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as [[linear operators]] acting on the Hilbert space associated with the quantum system.{{sfn|Macrae|1992|pp=139–141}}

The ''physics'' of quantum mechanics was thereby reduced to the ''mathematics'' of Hilbert spaces and linear operators acting on them. For example, the [[uncertainty principle]], according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the ''non-commutativity'' of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger.{{sfn|Macrae|1992|pp=139–141}}

Von Neumann's abstract treatment permitted him to confront the foundational issue of determinism versus non-determinism, and in the book he presented a [[Mathematical Foundations of Quantum Mechanics#No hidden variables proof|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 1935, [[Grete Hermann]] published a paper arguing that the proof contained a conceptual error and was therefore invalid.<ref>{{cite journal|first=Grete |last=Hermann |author-link=Grete Hermann |title=Die naturphilosophischen Grundlagen der Quantenmechanik |journal =[[Naturwissenschaften]] |volume=23 |number=42 |pages=718&ndash;721 |year=1935 |doi=10.1007/BF01491142|bibcode=1935NW.....23..718H |s2cid=40898258 }} English translation in {{cite book|last=Hermann |first=Grete |title=Grete Hermann &mdash; Between physics and philosophy |editor1-first=Elise |editor1-last=Crull |editor2-first=Guido |editor2-last=Bacciagaluppi |publisher=Springer |year=2016 |pages=239&ndash;278}}</ref> Hermann's work was largely ignored until after [[John S. Bell]] made essentially the same argument in 1966.<ref>{{cite journal|first=John S. |last=Bell |author-link=John S. Bell |title=On the problem of hidden variables in quantum mechanics |journal=[[Reviews of Modern Physics]] |volume=38 |issue=3 |pages=447&ndash;452 |doi=10.1103/RevModPhys.38.447|bibcode=1966RvMP...38..447B |year=1966 |osti=1444158 }}</ref> In 2010, [[Jeffrey Bub]] argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all [[hidden variable theories]], does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation and did not claim that his proof completely ruled out hidden variable theories.<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|s2cid=118595119}}</ref> The validity of Bub's argument is, in turn, disputed. [[Gleason's theorem]] of 1957 provided an argument against hidden variables along the lines of von Neumann's, but founded on assumptions seen as better motivated and more physically meaningful.<ref>{{cite journal|title=Homer nodded: von Neumann's surprising oversight |journal=[[Foundations of Physics]] |volume=48 |issue=9 |pages=1007–1020 |year=2018 |arxiv=1805.10311 |last1=Mermin |first1=N. David |last2=Schack |first2=Rüdiger |author-link1=N. David Mermin|doi=10.1007/s10701-018-0197-5 |bibcode=2018FoPh...48.1007M |s2cid=118951033 }}</ref><ref>{{cite journal|last=Peres |first=Asher |author-link=Asher Peres |date=1992 |doi=10.1016/0375-9601(92)91005-C |title=An experimental test for Gleason's theorem |journal=[[Physics Letters A]] |volume=163 |issue=4 |pages=243–245|bibcode=1992PhLA..163..243P }}</ref>

Von Neumann's proof inaugurated a line of research that ultimately led, through [[Bell's theorem]] and the [[Aspect's experiment|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 [[quantum nonlocality|nonlocality]] in apparent violation of special relativity.<ref>{{cite journal |journal=Studies in History and Philosophy of Modern Physics |volume=37 |issue=4 |year=2006 |pages=577–616 |title=Philosophy enters the optics laboratory: Bell's theorem and its first experimental tests (1965–1982) |first=Olival Jr. |last=Freire |author-link=Olival Freire Jr. |doi=10.1016/j.shpsb.2005.12.003 |bibcode=2006SHPMP..37..577F |arxiv=physics/0508180 |s2cid=13503517 }}</ref>

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. He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer. In other words, while the line between observer and observed could be drawn in different places, the theory only makes sense if an observer exists somewhere.<ref>{{cite journal|last=Stacey |first=B. C. |title=Von Neumann was not a Quantum Bayesian |journal=Philosophical Transactions of the Royal Society A |year=2016 |volume=374 |issue=2068 |pages=20150235 |doi=10.1098/rsta.2015.0235 |pmid=27091166 |arxiv=1412.2409 |bibcode=2016RSPTA.37450235S|s2cid=16829387 }}</ref> Although the idea of consciousness causing collapse was accepted by Eugene Wigner,<ref name="Wigner">{{cite journal |doi=10.1119/1.1973829 |volume=35 |issue=12| pages=1169–1170 |last1=Wigner |first1=Eugene |author-link=Eugene Wigner |last2=Margenau |first2=Henry |author2-link=Henry Margenau |title=Remarks on the Mind Body Question, in Symmetries and Reflections, Scientific Essays |journal=[[American Journal of Physics]] |date=December 1967 |bibcode=1967AmJPh..35.1169W}}</ref> the [[Von Neumann–Wigner interpretation]] never gained acceptance among the majority of physicists.<ref>{{cite journal |first1=M. |last1=Schlosshauer |first2=J.|last2= Koer |first3=A. |last3=Zeilinger |author3-link=Anton Zeilinger |title=A Snapshot of Foundational Attitudes Toward Quantum Mechanics |year=2013 |pages=222–230 |volume=44 |issue=3 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |arxiv=1301.1069 |doi=10.1016/j.shpsb.2013.04.004|bibcode=2013SHPMP..44..222S |s2cid=55537196 }}</ref>

Though theories of quantum mechanics continue to evolve, a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches can be traced back to the mathematical formalisms and techniques first used by von Neumann. 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.<ref name=VanHove/>

Viewing von Neumann's work on quantum mechanics as a part of the fulfilment of [[Hilbert's sixth problem]], mathematical physicist [[Arthur Wightman]] said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. With his 1932 book, quantum mechanics became a mature theory in the sense it had a precise mathematical form, which allowed for clear answers to conceptual problems.<ref>{{cite book |last1=Wightman |first1=A. S. |author1-link=Arthur Wightman |chapter=Hilbert's Sixth Problem: Mathematical Treatment of the Axioms of Physics |editor1-last=Browder |editor1-first=Felix E. |editor1-link=Felix Browder |title=Mathematical Developments Arising from Hilbert Problems |date=1976 |publisher=American Mathematical Society |isbn=978-0821814284 |url=https://bookstore.ams.org/pspum-28 |pages=157–158}}</ref> Nevertheless, von Neumann in his later years felt he had failed in this aspect of his scientific work as despite all the mathematics he developed, he did not find a satisfactory mathematical framework for quantum theory as a whole.{{sfn|Kac|Rota|Schwartz|2008|p=168}}{{sfn|Rédei|2005|pp=21,151–152,194}}

==== Von Neumann entropy ====
{{Main|Von Neumann entropy}}
[[Von Neumann entropy]] is extensively used in different forms ([[conditional entropy]], [[relative entropy]], etc.) in the framework of [[quantum information theory]].<ref>{{cite book |last1=Nielsen |first1=Michael A. |author1-link=Michael Nielsen |last2=Chuang |first2=Isaac |author2-link=Isaac Chuang |title=Quantum computation and quantum information |year=2001 |publisher=Cambridge University Press |isbn=978-0-521-63503-5 |pages=700 |edition=reprinted}}</ref> Entanglement measures are based upon some quantity directly related to the von Neumann entropy. Given a [[statistical ensemble]] of quantum mechanical systems with the [[density matrix]] <math>\rho</math>, it is given by <math> S(\rho) = -\operatorname{Tr}(\rho \ln \rho). \, </math> Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy<ref>{{cite web | url=http://www.mi.ras.ru/~holevo/eindex.html | title=Alexandr S. Holevo }}</ref> and [[conditional quantum entropy]]. Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy, a cornerstone in the former's development; the [[Entropy (information theory)|Shannon entropy]] applies to classical information theory.<ref>{{cite book |title=Quantum Information Theory |author-link=Mark Wilde |first=Mark M. |last=Wilde |publisher=Cambridge University Press |date=2013 |page=252}}</ref>

==== Density matrix ====
{{main|Density matrix}}
The formalism of [[density matrix|density operators and matrices]] was introduced by von Neumann<ref>{{cite journal | last = von Neumann | first = John  | year = 1927 | author-link = John von Neumann  |title=Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik | journal = Göttinger Nachrichten | volume = 1 |pages= 245–272 |url=https://eudml.org/doc/59230 |language=de}}</ref> in 1927 and independently, but less systematically by [[Lev Landau]]<ref name="PT">{{Citation| title=Density functional theory | last1=Schlüter |first1=Michael |author2-link=Lu Jeu Sham |first2=Lu Jeu |last2=Sham | journal=Physics Today | year=1982 | volume=35 | pages=36–43 | doi=10.1063/1.2914933 | issue=2 | bibcode=1982PhT....35b..36S | s2cid=126232754 }}</ref> and [[Felix Bloch]]<ref name="Ugo">{{Citation  | title=Density matrices as polarization vectors | first=Ugo |last=Fano | author-link=Ugo Fano | journal=[[:it:Rendiconti Lincei|Rendiconti Lincei]] |date=June 1995 | volume=6 | issue=2 | pages=123–130 | doi=10.1007/BF03001661| s2cid=128081459 }}</ref> in 1927 and 1946 respectively. The density matrix allows the representation of probabilistic mixtures of quantum states ([[Quantum state#Mixed states of wave functions|mixed states]]) in contrast to [[wavefunction]]s, which can only represent [[pure state]]s.<ref name=Hall2013pp419-440>{{cite book |doi=10.1007/978-1-4614-7116-5_19 |chapter=Systems and Subsystems, Multiple Particles |title=Quantum Theory for Mathematicians |volume=267 |pages=419–440 |series=Graduate Texts in Mathematics |year=2013 |last1=Hall |first1=Brian C. |isbn=978-1-4614-7115-8 }}</ref>

==== Von Neumann measurement scheme ====
The [[Measurement in quantum mechanics#History of the measurement concept|von Neumann measurement scheme]], the ancestor of quantum [[decoherence]] theory, represents measurements projectively by taking into account the measuring apparatus which is also treated as a quantum object. The 'projective measurement' scheme introduced by von Neumann led to the development of quantum decoherence theories.<ref>{{Cite book|last1=Giulini |first1=Domenico|title=Decoherence and the Appearance of a Classical World in Quantum Theory|date=1996|publisher=Springer Berlin Heidelberg|last2=Joos |first2=Erich |last3=Kiefer |first3=Claus |last4=Kupsch |first4=Joachim |last5=Stamatescu |first5=Ion-Olimpiu |author6-link=H. Dieter Zeh |last6=Zeh |first6=H. Dieter |isbn=978-3-662-03263-3 |location=Berlin, Heidelberg|oclc=851393174}}</ref><ref>{{cite encyclopedia |last=Bacciagaluppi |first=Guido |title=The Role of Decoherence in Quantum Mechanics |date=2020 |url=https://plato.stanford.edu/entries/qm-decoherence/ |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |edition=Fall 2020 |publisher=Stanford University |access-date=2023-09-25}}</ref>

==== Quantum logic ====
{{Main|Quantum logic}}
Von Neumann first proposed a quantum logic in his 1932 treatise ''[[Mathematical Foundations of Quantum Mechanics]]'', where he noted that projections on a [[Hilbert space]] can be viewed as propositions about physical observables. The field of quantum logic was subsequently inaugurated in a 1936 paper by von Neumann and Garrett Birkhoff, the first to introduce quantum logics,<ref>{{cite book |last1=Gabbay |first1=Dov M. |author-link1=Dov Gabbay |last2=Woods |first2=John |author2-link=John Woods (logician) |title=The Many Valued and Nonmonotonic Turn in Logic |chapter-url=https://books.google.com/books?id=3TNj1ZkP3qEC&pg=PA205|year=2007|publisher=Elsevier|isbn=978-0-08-054939-2|pages=205–2017|chapter=The History of Quantum Logic}}</ref> wherein von Neumann and Birkhoff first proved that quantum mechanics requires a [[propositional calculus]] substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., horizontally and vertically), and therefore, ''[[A fortiori argument|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 ''between'' the other two, the photons will indeed pass through. 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) = {}</math><math>(P\lor Q)\land(P\lor R)</math> and <math>P\land (Q\lor R) ={}</math><math>(P\land Q)\lor(P\land R)</math>, are not valid for quantum theory.<ref name="quantum logic"/>

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 is necessarily indeterminate. Consequently, the [[distributive law]] of classical logic must be replaced with a weaker condition.<ref name="quantum logic">{{cite journal |title=The Logic of Quantum Mechanics |first1=Garrett |last1=Birkhoff |author-link=Garrett Birkhoff |first2=John |last2=von Neumann |journal=Annals of Mathematics |volume=37 |issue=4 |date=October 1936 |pages=823–843 |doi=10.2307/1968621 |jstor=1968621 }}</ref> Instead of a distributive lattice, propositions about a quantum system form an [[orthomodular lattice]] isomorphic to the lattice of subspaces of the Hilbert space associated with that system.<ref name="Putnam1985">{{cite book|last=Putnam|first=Hilary |author-link=Hilary Putnam|title=Philosophical Papers |volume=3: Realism and Reason|url=https://books.google.com/books?id=HAjfSA3ir3kC|year=1985|publisher=Cambridge University Press|isbn=978-0-521-31394-0|page=263}}</ref>

Nevertheless, he was never satisfied with his work on quantum logic. He intended it to be a joint synthesis of formal logic and probability theory and when he attempted to write up a paper for the Henry Joseph Lecture he gave at the [[Philosophical Society of Washington|Washington Philosophical Society]] in 1945 he found that he could not, especially given that he was busy with war work at the time. During his address at the 1954 [[International Congress of Mathematicians]] he gave this issue as one of the unsolved problems that future mathematicians could work on.{{sfn|Rédei|2005|pp=30-32}}{{sfn|Rédei|Stöltzner|2001|pp=53,153–154,168–169}}

=== Fluid dynamics ===
Von Neumann made fundamental contributions in the field of [[fluid dynamics]], including the classic flow solution to [[blast wave]]s,<ref>{{harvc |last1=von Neumann |first1=John |year=1963 |chapter=The Point Source Solution |in=Taub |pages=219–237}}</ref> and the co-discovery (independently by [[Yakov Borisovich Zel'dovich]] and [[Werner Döring]]) of the [[ZND detonation model]] of explosives.<ref>{{harvc |last1=von Neumann |first1=John |year=1963 |chapter=Theory of Detonation Waves. Progress Report to the National Defense Research Committee Div. B, OSRD-549 |orig-year=1942 |url=https://web.archive.org/web/20160610184551/http://www.geocities.jp/nomonomo2007/ReadingPhysics/von_Neumann/Detonation.pdf |access-date=June 10, 2016 |in=Taub |pages=205–218}}</ref> During the 1930s, von Neumann became an authority on the mathematics of [[shaped charges]].<ref>{{cite book |title=Ballistics: Theory and Design of Guns and Ammunition |edition=2nd |first1=Donald E. |last1=Carlucci |first2=Sidney S. |last2=Jacobson |publisher=CRC Press |date=26 August 2013 |page=523}}</ref>

Later with [[Robert D. Richtmyer]], von Neumann developed an algorithm defining ''artificial [[viscosity]]'' that improved the understanding of [[shock wave]]s. When computers solved hydrodynamic or aerodynamic problems, they put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics.<ref>{{cite journal |title=A Method for the Numerical Calculation of Hydrodynamic Shocks |last1=von Neumann |first1=J. |last2=Richtmyer |first2=R. D. |author-link2=Robert D. Richtmyer |journal=Journal of Applied Physics |volume=21 |issue=3 |pages=232–237 |date=March 1950 |doi=10.1063/1.1699639 |bibcode=1950JAP....21..232V }}</ref>

Von Neumann soon applied computer modelling to the field, developing software for his ballistics research. During World War II, he approached R. H. Kent, the director of the US Army's [[Ballistic Research Laboratory]], with a computer program for calculating a one-dimensional model of 100 molecules to simulate a shock wave. Von Neumann gave a seminar on his program to an audience which included his friend [[Theodore von Kármán]]. After von Neumann had finished, von Kármán said "Of course you realize [[Joseph-Louis Lagrange|Lagrange]] also used digital models to simulate [[continuum mechanics]]." Von Neumann had been unaware of Lagrange's {{lang|fr|[[Mécanique analytique]]}}.<ref name=mhr1980>{{cite book |editor1-last=Metropolis |editor1-first=Nicholas |editor1-link=Nicholas Metropolis |editor2-first=J. |editor2-last=Howlett |editor2-link=Jack Howlett |editor3-first=Gian-Carlo |editor3-last=Rota |editor3-link=Gian-Carlo Rota |title=A History of Computing in the Twentieth Century  |publisher=Elsevier |year=1980 |doi=10.1016/C2009-0-22029-0 |isbn=978-1-4832-9668-5 |pages=24–25}}</ref>

=== Other work ===
[[File:Neumann János emléktáblája szülőháza falán (Budapest V. kerület, Báthory u 26.).jpg|thumb|Von Neumann's memorial plaque on the wall of his birthplace in Budapest, 5th district Báthory u. 26.]]
While not as prolific in physics as he was in mathematics, he nevertheless made several other notable contributions. His pioneering papers with [[Subrahmanyan Chandrasekhar]] on the statistics of a fluctuating [[gravitational field]] generated by [[Probability distribution|randomly distributed]] [[star]]s were considered a ''tour de force''.<ref>{{cite journal |last1=Binney |first1=James |author1-link=James Binney |title=The stellar-dynamical oeuvre |journal=Journal of Astrophysics and Astronomy |date=1996 |volume=17 |issue=3–4 |pages=81–93 |doi=10.1007/BF02702298 |bibcode=1996JApA...17...81B |s2cid=56126751 |url=https://link.springer.com/article/10.1007/BF02702298}}</ref> In this paper they developed a theory of two-body relaxation<ref>{{cite journal |last1=Benacquista |first1=Matthew J. |last2=Downing |first2=Jonathan M. B. |title=Relativistic Binaries in Globular Clusters |journal=Living Reviews in Relativity |date=2013 |volume=16 |issue=1 |page=4 |doi=10.12942/lrr-2013-4 |doi-access=free |pmid=28179843|pmc=5255893 |arxiv=1110.4423 |bibcode=2013LRR....16....4B }}</ref> and used the [[Holtsmark distribution]] to model<ref>{{cite book |last1=Uchaikin |first1=Vladimir V. |last2=Zolotarev |first2=Vladimir M. |title=Chance and Stability: Stable Distributions and their Applications |date=1999 |publisher=De Gruyter |doi=10.1515/9783110935974 |isbn=9783110631159 |pages=xviii, 281, 424 }}</ref> the [[Stellar dynamics|dynamics of stellar systems]].<ref>{{cite journal |last1=Silva |first1=J. M. |last2=Lima |first2=J. A. S. |last3=de Souza |first3=R. E. |last4=Del Popolo |first4=A. |last5=Le Delliou |first5=Morgan |last6=Lee |first6=Xi-Guo |title=Chandrasekhar's dynamical friction and non-extensive statistics |journal=[[Journal of Cosmology and Astroparticle Physics]] |date=2016 |volume=2016 |issue=5 |page=21 |doi=10.1088/1475-7516/2016/05/021|arxiv=1604.02034 |bibcode=2016JCAP...05..021S |hdl=11449/173002 |s2cid=118462043 }}</ref> He wrote several other unpublished manuscripts on topics in [[stellar structure]], some of which were included in Chandrasekhar's other works.{{sfn|Taub|1963|pp=172–176}}<ref>{{cite journal |last=Bonolis |first=Luisa |title=Stellar structure and compact objects before 1940: Towards relativistic astrophysics |journal=The European Physical Journal H |date=2017 |volume=42 |issue=2 |pages=311–393, esp. pp. 351, 361 |doi=10.1140/epjh/e2017-80014-4 |arxiv=1703.09991 |bibcode=2017EPJH...42..311B |doi-access=free }}</ref> In earlier work led by [[Oswald Veblen]] von Neumann helped develop basic ideas involving [[spinor]]s that would lead to [[Roger Penrose]]'s [[twistor theory]].<ref>{{cite journal |last1=Trautman |first1=Andrzej |author1-link=Andrzej Trautman |last2=Trautman |first2=Krzysztof |title=Generalized pure spinors |journal=Journal of Geometry and Physics |date=1994 |volume=15 |issue=1 |pages=1–22 |doi=10.1016/0393-0440(94)90045-0|bibcode=1994JGP....15....1T }}</ref><ref>{{cite journal |last1=Forstnerič |first1=Franc |title=The Calabi–Yau Property of Superminimal Surfaces in Self-Dual Einstein Four-Manifolds |journal=The Journal of Geometric Analysis |date=2021 |volume=31 |issue=5 |pages=4754–4780 |doi=10.1007/s12220-020-00455-6 |arxiv=2004.03536 |s2cid=215238355 }}</ref> Much of this was done in seminars conducted at the [[Institute for Advanced Study|IAS]] during the 1930s.<ref>{{harvc |last1=Segal |first1=Irving E. |author-link=Irving Segal |year=1990 |chapter=The Mathematical Implications of Fundamental Physical Principles |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=162–163}}</ref> From this work he wrote a paper with [[Abraham H. Taub|A. H. Taub]] and Veblen extending the [[Dirac equation]] to [[Projective geometry|projective]] relativity, with a key focus on maintaining [[Invariant (physics)|invariance]] with regards to coordinate, [[Spin (physics)|spin]], and [[Gauge theory|gauge]] transformations, as a part of early research into potential theories of [[quantum gravity]] in the 1930s.{{sfn|Rickles|2020|p=89}} In the same time period he made several proposals to colleagues for dealing with the problems in the newly created [[quantum field theory]] and for [[Quantization (physics)|quantizing]] spacetime; however, both his colleagues and he did not consider the ideas fruitful and did not pursue them.{{sfn|Rédei|2005|pp=21–22}}{{sfn|Rédei|Stöltzner|2001|pp=222–224}}{{sfn|Rickles|2020|pp=202-203}} Nevertheless, he maintained at least some interest, in 1940 writing a manuscript on the Dirac equation in [[de Sitter space]].{{sfn|Taub|1963|p=177}}

== Economics ==
=== Game theory ===
Von Neumann founded the field of [[game theory]] as a mathematical discipline.<ref name="KuhnTucker">{{cite journal|last1=Kuhn|first1= H. W.|author-link=Harold W. Kuhn|last2=Tucker|first2= A. W.|author-link2=Albert W. Tucker|title=John von Neumann's work in the theory of games and mathematical economics|journal=Bull. Amer. Math. Soc.|year=1958|volume=64 (Part 2)|issue=3|pages=100–122|mr=0096572|doi=10.1090/s0002-9904-1958-10209-8|citeseerx= 10.1.1.320.2987}}</ref> He proved his [[Minimax#Minimax theorem|minimax theorem]] in 1928. It establishes that in [[zero-sum game]]s 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 [[Strategy (game theory)|strategies]] for both players that allows each to minimize their maximum losses.<ref name="Game Theory">{{cite journal |last=von Neumann |first=J |title=Zur Theorie der Gesellschaftsspiele |language=de |journal=[[Mathematische Annalen]] |volume=100 |pages=295–320 |doi=10.1007/bf01448847|year=1928 |s2cid=122961988 }}</ref> Such strategies are called ''optimal''. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). He 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.<ref>{{Cite news |last=Lissner |first=Will |date=1946-03-10 |title=Mathematical Theory of Poker Is Applied to Business Problems; GAMING STRATEGY USED IN ECONOMICS Big Potentialities Seen Strategies Analyzed Practical Use in Games |language=en-US |newspaper=The New York Times |url=https://www.nytimes.com/1946/03/10/archives/mathematical-theory-of-poker-is-applied-to-business-problems-gaming.html |access-date=2020-07-25 |issn=0362-4331 }}</ref> In this book, von Neumann declared that economic theory needed to use [[functional analysis]], especially [[convex set]]s and the [[topology|topological]] [[fixed-point theorem]], rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.<ref name="KuhnTucker"/>

Von Neumann's functional-analytic techniques—the use of [[Dual space|duality pairings]] of real [[vector space]]s to represent prices and quantities, the use of [[Supporting hyperplane|supporting]] and [[Hyperplane separation theorem|separating hyperplanes]] and convex sets, and fixed-point theory—have been primary tools of mathematical economics ever since.<ref>{{cite book |last=Blume |first=Lawrence E. |author-link=Lawrence E. Blume |contribution=Convexity |year=2008 |title=The New Palgrave Dictionary of Economics |pages=225–226 |editor1-last=Durlauf |editor1-first=Steven N. |editor1-link=Steven N. Durlauf  |editor2-last=Blume |editor2-first=Lawrence E. |publisher=Palgrave Macmillan |location=New York |edition=2nd |url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508|doi=10.1057/9780230226203.0315 |isbn=978-0-333-78676-5}}</ref>

=== Mathematical economics ===
Von Neumann raised the [[Mathematical economics|mathematical level of economics]] in several influential publications. For his model of an expanding economy, he proved the existence and uniqueness of an equilibrium using his generalization of the [[Brouwer fixed-point theorem]].<ref name=KuhnTucker/> Von Neumann's model of an expanding economy considered the [[Eigendecomposition of a matrix#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 <math>p^T (A - \lambda B) q = 0</math> 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 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]].<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) [[Eigenvalues and eigenvectors|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 [[D. G. 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]], Morgenstern, and [[Gerald L. Thompson]] in the 1950s and then by Stephen M. Robinson in the 1970s.</ref><ref>{{cite book|last1=Morgenstern|first1=Oskar|author-link1=Oskar Morgenstern|last2=Thompson|first2=Gerald L.|author-link2=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|isbn=978-0-669-00089-4|url-access=registration|url=https://archive.org/details/mathematicaltheo0000morg |pages=xviii, 277}}</ref>

Von Neumann's results have been viewed as a special case of [[linear programming]], where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists.<ref>{{cite book |last=Rockafellar |first=R. T. |author-link=R. Tyrrell Rockafellar |title=Convex analysis |publisher=Princeton University Press |year=1970 |isbn=978-0-691-08069-7 |oclc=64619 |pages=i, 74}} {{pb}} {{cite conference |last=Rockafellar |first=R. T. |author-link=R. Tyrrell Rockafellar |chapter=Convex Algebra and Duality in Dynamic Models of production |title=Mathematical Models in Economics |conference=Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972 |editor1-first=Josef |editor1-last=Loz |editor2-first=Maria |editor2-last=Loz |publisher=Elsevier North-Holland Publishing and Polish Academy of Sciences |location=Amsterdam |year=1974 |oclc=839117596 |pages=351–378}}</ref><ref>{{cite book |last=Ye |first=Yinyu |author-link=Yinyu Ye |year=1997 |url=https://books.google.com/books?id=RQZd7ru8cmMC&pg=PA277 |contribution=The von Neumann growth model |title=Interior point algorithms: Theory and analysis |publisher=Wiley |location=New York |isbn=978-0-471-17420-2 |oclc=36746523 |pages= 277–299}}</ref> This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, [[Linear inequality|linear inequalities]], [[Linear programming#Complementary slackness|complementary slackness]], and [[Duality (optimization)|saddlepoint duality]].{{sfn|Dore|Chakravarty|Goodwin|1989|p=xi}} 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>{{cite book|editor1-last=Bruckmann|editor1-first=Gerhart|editor2-last=Weber|editor2-first=Wilhelm|date=September 21, 1971|doi=10.1007/978-3-662-24667-2|title=Contributions to von Neumann's Growth Model|series=Proceedings of a Conference Organized by the Institute for Advanced Studies Vienna, Austria, July 6 and 7, 1970|publisher=Springer–Verlag|isbn=978-3-662-22738-1}}</ref> The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of [[Nobel prize]]s 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 [[non-cooperative 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]].{{sfn|Dore|Chakravarty|Goodwin|1989|p=234}}

Von Neumann's interest in the topic began while he was lecturing at Berlin in 1928 and 1929. He spent his summers in Budapest, as did the economist [[Nicholas Kaldor]]; Kaldor recommended that von Neumann read a book by the mathematical economist [[Léon Walras]]. Von Neumann noticed that Walras's [[General Equilibrium Theory]] and [[Walras's law]], which led to systems of simultaneous linear equations, could produce the absurd result that profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced his paper.{{sfn|Macrae|1992|pp=250–253}}

=== 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 when [[George Dantzig]] described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.<ref>{{cite book | last=Dantzig | first=G. B. | author-link=George Dantzig | year=1983 | contribution=Reminiscences about the origins of linear programming. | title=Mathematical Programming The State of the Art: Bonn 1982 | editor1-last=Bachem | editor1-first=A. | editor2-last=Grötschel |editor2-first=M. | editor3-last=Korte | editor3-first=B. | location=Berlin, New York | publisher=Springer-Verlag | pages=78–86 | isbn=0387120823 | oclc=9556834}}</ref>

Later, von Neumann suggested a new method of [[linear programming]], using the homogeneous linear system of [[Paul Gordan]] (1873), which was later popularized by [[Karmarkar's algorithm]]. Von Neumann's method used a pivoting algorithm between [[simplex|simplices]], with the pivoting decision determined by a nonnegative [[least squares]] subproblem with a convexity constraint ([[Projection (linear algebra)#Orthogonal projections|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">{{cite book | last1 = Dantzig | first1 = George |author1-link=George Dantzig |last2=Thapa |first2=Mukund N. | title = Linear Programming : 2: Theory and Extensions | publisher = [[Springer Science+Business Media|Springer-Verlag]]| location = New York, NY | year = 2003| isbn = 978-1-4419-3140-5}}</ref>

== Computer science ==
Von Neumann was a founding figure in [[computing]],{{sfn|Goldstine|1980|pp=167–178}} with significant contributions to computing hardware design, to [[theoretical computer science]], to [[scientific computing]], and to the [[philosophy of computer science]].

===Hardware===
[[File:Living Large -- Argonne's First Computer (8056998342).jpg|thumb|The [[AVIDAC]] computer was partially based on the architecture of the [[IAS machine]] developed by Von Neumann.]]
Von Neumann consulted for the Army's [[Ballistic Research Laboratory]], most notably on the [[ENIAC]] project,{{sfn|Macrae|1992|pp=279–283}} as a member of its Scientific Advisory Committee.<ref>
{{cite web
 |title       = BRL's Scientific Advisory Committee, 1940
 |publisher   = U.S. Army Research Laboratory
 |url         = http://ftp.arl.army.mil/~mike/comphist/40sac/index.html
 |access-date  = 2018-01-12
}}
</ref> Although the single-memory, stored-program architecture is commonly called [[von Neumann architecture]], the architecture was based on the work of [[J. Presper Eckert]] and [[John Mauchly]], inventors of ENIAC and its successor, [[EDVAC]].
While consulting for the EDVAC project at the [[University of Pennsylvania]], von Neumann wrote an incomplete ''[[First Draft of a Report on the EDVAC]]''. The paper, whose premature distribution nullified the patent claims of Eckert and Mauchly, described a computer that stored both its data and its program in the same address space, unlike the earliest computers which stored their programs separately on [[paper tape]] or [[plugboard]]s. This architecture became the basis of most modern computer designs.<ref name="ENIAC museum">{{cite web |url=http://www.library.upenn.edu/exhibits/rbm/mauchly/jwm9.html |title=John W. Mauchly and the Development of the ENIAC Computer |publisher=University of Pennsylvania |access-date=January 27, 2017 |archive-url=https://web.archive.org/web/20070416112324/http://www.library.upenn.edu/exhibits/rbm/mauchly/jwm9.html |archive-date=April 16, 2007 |url-status=dead }}</ref>

Next, von Neumann designed the [[IAS machine]] at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the [[Sarnoff Corporation|RCA Research Laboratory]] nearby. Von Neumann recommended that the [[IBM 701]], nicknamed ''the defense computer'', include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successful [[IBM 704]].{{sfn|Rédei|2005|p=73}}{{sfn|Dyson|2012|pp=267–268, 287}}

=== Algorithms ===
[[File:Flow chart of Planning and coding of problems for an electronic computing instrument, 1947.jpg|thumb|[[Flow chart]] from von Neumann's "Planning and coding of problems for an electronic computing instrument", published in 1947]]
Von Neumann was 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|author-link=Donald Knuth|year=1998|title=The Art of Computer Programming: Volume 3 Sorting and Searching|isbn=978-0-201-89685-5|publisher=Addison-Wesley|location=Boston |page=159}}</ref><ref name="Papers J. Neumann Computing Computer Theory. 1987, pp. 89-95">
{{cite book
 | last= Knuth
 | first= Donald E.
 | author-link= Donald Knuth
 | title= Papers of John von Neumann on computing and computer theory
 | editor1-last= Aspray
 | editor1-first= W.
 | editor2-last= Burks
 | editor2-first= A.
 | publisher= MIT Press
 | year= 1987
 | location= Cambridge
 | isbn= 978-0-262-22030-9
 | pages= [https://archive.org/details/papersofjohnvonn00vonn/page/89 89–95]
 | chapter= Von Neumann's First Computer Program
 | chapter-url= https://books.google.com/books?id=lvgmAAAAMAAJ
 | url= https://archive.org/details/papersofjohnvonn00vonn/page/89
 }}
</ref>

As part of Von Neumann's hydrogen bomb work, he and Stanisław Ulam developed simulations for hydrodynamic computations. He also contributed to the development of the [[Monte Carlo method]], which used [[Algorithmically random sequence|random numbers]] to approximate the solutions to complicated problems.{{sfn|Macrae|1992|pp=334–335}}

Von Neumann's algorithm for simulating a [[fair coin]] with a biased coin is used in the "software whitening" stage of some [[hardware random number generator]]s.<ref name=nbsams/> Because obtaining "truly" random numbers was impractical, von Neumann developed a form of [[pseudorandomness]], using the [[middle-square method]]. He justified this crude method as faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."<ref name="nbsams">
{{cite journal
 | last= Von Neumann
 | first= John
 | title= Various techniques used in connection with random digits
 | journal= National Bureau of Standards Applied Mathematics Series
 | year= 1951
 | volume= 12
 | pages= 36–38
 | url= https://babel.hathitrust.org/cgi/pt?id=osu.32435030295547&view=image&seq=48
}}
</ref> He also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect.<ref name="nbsams"/>

[[Stochastic computing]] was introduced by von Neumann in 1953,<ref>{{harvc |last1=von Neumann |first1=J. |year=1995 |contribution=Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components |in1=Bródy |in2=Vámos |pages= 567–616}}</ref> but 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
 | book-title= ACTES Proc. of 3rd Int. Analog Comp. Meeting
}}
</ref><ref>
{{citation
 | last= Afuso
 | first= C.
 | title= Quart. Tech. Prog. Rept
 | location= Illinois |publisher=[[Department of Computer Science, University of Illinois at Urbana-Champaign]]
 | year= 1964
}}
</ref> Around 1950 he was also among the first to talk about the [[time complexity]] of [[computation]]s, which eventually evolved into the field of [[computational complexity theory]].<ref>{{cite book |last1=Chaitin |first1=Gregory J. |author-link1=Gregory Chaitin |title=Conversations with a Mathematician: Math, Art, Science and the Limits of Reason |date=2002 |publisher=Springer |location=London |isbn=978-1-4471-0185-7 |doi=10.1007/978-1-4471-0185-7 |url=https://link.springer.com/book/10.1007/978-1-4471-0185-7 |page=28}}</ref>

=== Cellular automata, DNA and the universal constructor ===
{{See also|von Neumann cellular automaton|von Neumann universal constructor|von Neumann neighborhood|von Neumann Probe}}
[[File:Nobili Pesavento 2reps.png|right|thumb|upright=1.8|The first implementation of von Neumann's self-reproducing universal constructor.<ref name=Pesavento1995>{{cite journal|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|url=http://dragonfly.tam.cornell.edu/~pesavent/pesavento_self_reproducing_machine.pdf|archive-url=https://web.archive.org/web/20070621164824/http://dragonfly.tam.cornell.edu/~pesavent/pesavento_self_reproducing_machine.pdf |archive-date=June 21, 2007 |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.]]
[[File:VonNeumann CA demo.gif|right|frame|A simple configuration in von Neumann's cellular automaton. A binary signal is passed repeatedly around the blue wire loop, using excited and quiescent ''ordinary transmission states''. A confluent cell duplicates the signal onto a length of red wire consisting of ''special transmission states''. The signal passes down this wire and constructs a new cell at the end. This particular signal (1011) codes for an east-directed special transmission state, thus extending the red wire by one cell each time. During construction, the new cell passes through several sensitised states, directed by the binary sequence.]]
Von Neumann's mathematical analysis of the structure of [[self-replication]] preceded the discovery of the structure of DNA.<ref>{{cite book |last=Rocha |first=L.M. |author-link=Luis M. Rocha |title=Lecture Notes of I-585-Biologically Inspired Computing Course, Indiana University |contribution=Von Neumann and Natural Selection |year=2015 |url=http://informatics.indiana.edu/rocha/i-bic/pdfs/ibic_lecnotes_c6.pdf |access-date=February 6, 2016  |archive-url=https://web.archive.org/web/20150907173700/http://www.informatics.indiana.edu/rocha/i-bic/pdfs/ibic_lecnotes_c6.pdf |archive-date=September 7, 2015 |url-status=dead |pages=25–27}}</ref> Ulam and von Neumann are also generally credited with creating the field of [[cellular automata]], beginning in the 1940s, as a simplified mathematical model of biological systems.<ref name="John von Neumann Cellular Automata">{{cite web |editor-last=Damerow |editor-first=Julia |publisher=Arizona State University. School of Life Sciences. Center for Biology and Society.| title=John von Neumann's Cellular Automata | website=Embryo Project Encyclopedia | date=2010-06-14 | url=https://embryo.asu.edu/pages/john-von-neumanns-cellular-automata | access-date=2024-01-14}}</ref>

In lectures in 1948 and 1949, von Neumann proposed a [[kinematic]] self-reproducing automaton.<ref>{{cite book| last=von Neumann| first=John| year=1966| title=The Theory of Self-reproducing Automata| url=https://archive.org/details/theoryofselfrepr00vonn_0| editor= A. Burks| publisher=Univ. of Illinois Press| location= Urbana, IL| isbn=978-0-598-37798-2}}</ref><ref>{{cite web|url=http://www.MolecularAssembler.com/KSRM/2.1.htm |title=2.1 Von Neumann's Contributions |publisher=Molecularassembler.com |access-date=2009-09-16}}</ref> By 1952, he was treating the problem more abstractly. He designed an elaborate 2D [[cellular automaton]] that would automatically make a copy of its initial configuration of cells.<ref>{{cite web|url=http://www.MolecularAssembler.com/KSRM/2.1.3.htm |title=2.1.3 The Cellular Automaton (CA) Model of Machine Replication |publisher=Molecularassembler.com |access-date=2009-09-16}}</ref>  The [[Von Neumann universal constructor]] based on the [[von Neumann cellular automaton]] was fleshed out in his posthumous ''Theory of Self Reproducing Automata''.<ref name=TSRA>
{{cite book
 | year= 1966
 | last= von Neumann
 | first= John
 | editor= [[Arthur W. Burks]]
 | title= Theory of Self-Reproducing Automata
 | publisher= [[University of Illinois Press]]
 | place= Urbana and London
 | isbn= 978-0-598-37798-2
 | url= http://www.history-computer.com/Library/VonNeumann1.pdf
}}
</ref>
The [[von Neumann neighborhood]], in which each cell in a two-dimensional grid has the four orthogonally adjacent grid cells as neighbors, continues to be used for other cellular automata.<ref>{{cite book|title=Cellular Automata Machines: A New Environment for Modeling|first1=Tommaso|last1=Toffoli|author1-link=Tommaso Toffoli|first2=Norman|last2=Margolus|author2-link=Norman Margolus|year=1987|publisher=MIT Press|page=60}}.</ref>

=== Scientific computing and numerical analysis ===
Considered to be possibly "the most influential researcher in [[Computational science|scientific computing]] of all time",{{sfn|Gustafsson|2018|p=91}} von Neumann made several contributions to the field, both technically and administratively. He developed the [[Von Neumann stability analysis]] procedure,{{sfn|Gustafsson|2018|pp=101-102}} still commonly used to avoid errors from building up in [[Numerical methods for partial differential equations|numerical methods for linear partial differential equations]].{{sfn|Gustafsson|2018|p=235}} His paper with [[Herman Goldstine]] in 1947 was the first to describe [[Error analysis (mathematics)#Backward error analysis|backward error analysis]], although implicitly.{{sfn|Brezinski|Wuytack|2001|p=27}} He was also one of the first to write about the [[Jacobi method]].{{sfn|Brezinski|Wuytack|2001|p=216}} At Los Alamos, he wrote several classified reports on solving problems of [[Compressible flow|gas dynamics]] numerically. However, he was frustrated by the lack of progress with [[Mathematical analysis|analytic methods]] for these [[Nonlinear partial differential equation|nonlinear]] problems. As a result, he turned towards computational methods.{{sfn|Gustafsson|2018|pp=112-113}} Under his influence Los Alamos became the leader in computational science during the 1950s and early 1960s.<ref>{{cite interview |last=Lax |first=Peter D. |subject-link=Peter Lax|interviewer1=Martin Raussen|interviewer2=Christian Skau|title=Interview with Peter D. Lax |date=2005 |publisher=[[Notices of the American Mathematical Society]] |location=Oslo |url=https://www.ams.org/notices/200602/comm-lax.pdf |page=223}}</ref>

From this work von Neumann realized that computation was not just a tool to [[Proof by exhaustion|brute force]] the solution to a problem numerically, but could also provide insight for solving problems analytically,<ref>{{cite book |last1=Ulam |first1=Stanisław M. |author-link1=Stanisław Ulam |editor1-last=Reynolds |editor1-first=Mark C. |editor2-last=Rota |editor2-first=Gian-Carlo |editor-link2=Gian-Carlo Rota |title=Science, Computers, and People: From the Tree of Mathematics |date=1986 |publisher=Birkhäuser |location=Boston |doi=10.1007/978-1-4615-9819-0 |isbn=978-1-4615-9819-0 |url=https://link.springer.com/book/10.1007/978-1-4615-9819-0 |page=224}}</ref> and that there was an enormous variety of scientific and engineering problems towards which computers would be useful, most significant of which were [[Nonlinear system|nonlinear problems]].<ref>{{cite book |last=Hersh |first=Reuben |author-link=Reuben Hersh |title=Peter Lax, Mathematician: An Illustrated Memoir |publisher=American Mathematical Society |date=2015 |isbn=978-1-4704-2043-7 |url=https://bookstore.ams.org/mbk-88 |page=170}}</ref> In June 1945 at the First Canadian Mathematical Congress he gave his first talk on general ideas of how to solve problems, particularly of fluid dynamics numerically.{{r|mhr1980}} He also described how [[wind tunnel]]s were actually [[analog computer]]s, and how digital computers would replace them and bring a new era of fluid dynamics. [[Garrett Birkhoff]] described it as "an unforgettable sales pitch". He expanded this talk with Goldstine into the manuscript "On the Principles of Large Scale Computing Machines" and used it to promote the support of scientific computing. His papers also developed the concepts of [[Invertible matrix|inverting matrices]], [[Random matrix|random matrices]] and automated [[Relaxation (iterative method)|relaxation methods]] for solving [[elliptic boundary value problem]]s.<ref>{{cite book |last1=Birkhoff |first1=Garrett |author-link=Garrett Birkhoff |year=1990 |chapter=Fluid dynamics, reactor computations, and surface representation |editor-last1=Nash |editor-first1=Stephen G.  |publisher=Association for Computing Machinery |isbn=978-0-201-50814-7 |title=A history of scientific computing |doi=10.1145/87252.88072 |pages=64–69}}</ref>

=== Weather systems and global warming ===
{{See also|History of numerical weather prediction|History of climate change science#Increasing concern, 1950s–1960s}}
As part of his research into possible applications of computers, von Neumann became interested in weather prediction, noting similarities between the problems in the field and those he had worked on during the Manhattan Project.{{sfn|Edwards|2010|p=115}} In 1946 von Neumann founded the "Meteorological Project" at the Institute for Advanced Study, securing funding for his project from [[National Weather Service|the Weather Bureau]], the [[557th Weather Wing|US Air Force]] and US Navy weather services.<ref name="Jonathan Hill 2013 page 216">''Weather Architecture'' By Jonathan Hill (Routledge, 2013), page 216</ref> With [[Carl-Gustaf Rossby]], considered the leading theoretical meteorologist at the time, he gathered a group of twenty meteorologists to work on various problems in the field. However, given his other postwar work he was not able to devote enough time to proper leadership of the project and little was accomplished.

This changed when a young [[Jule Gregory Charney]] took up co-leadership of the project from Rossby.{{sfn|Edwards|2010|pp=117–118}} By 1950 von Neumann and Charney wrote the world's first climate modelling software, and used it to perform the world's first numerical [[Weather forecasting|weather forecasts]] on the ENIAC computer that von Neumann had arranged to be used;<ref name="Jonathan Hill 2013 page 216"/> von Neumann and his team published the results as ''Numerical Integration of the Barotropic Vorticity Equation''.<ref>
{{Cite journal
 | last1= Charney
 | first1= J. G.
 | last2= Fjörtoft
 | first2= R.
 | last3= Neumann
 | first3= J.
 | title= Numerical Integration of the Barotropic Vorticity Equation
 | journal= Tellus
 | year= 1950
 | volume= 2
 | issue= 4
 | pages= 237–254
 | doi= 10.3402/TELLUSA.V2I4.8607
 | bibcode= 1950Tell....2..237C
 | doi-access= free
 }}
</ref> Together they played a leading role in efforts to integrate sea-air exchanges of energy and moisture into the study of climate.<ref>[[Bruce Gilchrist|Gilchrist, Bruce]], {{cite web|url=http://www.columbia.edu/cu/epic/gilchrist_3.07.06.pdf |title=Remembering Some Early Computers, 1948–1960 |accessdate=December 12, 2006 |url-status=dead |archive-url=https://web.archive.org/web/20061212200023/http://www.columbia.edu/cu/epic/gilchrist_3.07.06.pdf |archive-date=December 12, 2006 }}, ''Columbia University EPIC'', 2006, pp.7-9. (archived 2006) Contains some autobiographical material on Gilchrist's use of the IAS computer beginning in 1952.</ref> Though primitive, news of the ENIAC forecasts quickly spread around the world and a number of parallel projects in other locations were initiated.{{sfn|Edwards|2010|p=126}}

In 1955 von Neumann, Charney and their collaborators convinced their funders to open the Joint Numerical Weather Prediction Unit (JNWPU) in [[Suitland, Maryland]], which began routine real-time weather forecasting.{{sfn|Edwards|2010|p=130}} Next up, von Neumann proposed a research program for climate modeling: <blockquote>The approach is to first try short-range forecasts, then long-range forecasts of those properties of the circulation that can perpetuate themselves over arbitrarily long periods of time, and only finally to attempt forecast for medium-long time periods which are too long to treat by simple hydrodynamic theory and too short to treat by the general principle of equilibrium theory.<ref>''Intraseasonal Variability in the Atmosphere-Ocean Climate System'', By William K.-M. Lau, Duane E. Waliser (Springer 2011), page V</ref></blockquote> Positive results of [[Norman A. Phillips]] in 1955 prompted immediate reaction and von Neumann organized a conference at Princeton on "Application of Numerical Integration Techniques to the Problem of the General Circulation". Once again he strategically organized the program as a predictive one to ensure continued support from the Weather Bureau and the military, leading to the creation of the General Circulation Research Section (now the [[Geophysical Fluid Dynamics Laboratory]]) next to the JNWPU.{{sfn|Edwards|2010|pp=152–153}} He continued work both on technical issues of modelling and in ensuring continuing funding for these projects.{{sfn|Edwards|2010|pp=153,161,189–190}}

During the late 19th century, [[Svante Arrhenius]] suggested that human activity could cause [[global warming]] by adding [[carbon dioxide]] to the atmosphere.<ref>{{cite web|url=https://history.aip.org/climate/co2.htm |website=The Discovery of Global Warming |title=The Carbon Dioxide Greenhouse Effect |publisher=[[American Institute of Physics]] |date=May 2023 |access-date=9 October 2023}}</ref> In 1955, von Neumann observed that this may already have begun: "Carbon dioxide released into the atmosphere by industry's burning of [[coal]] and oil – more than half of it during the last generation – may have changed the atmosphere's composition sufficiently to account for a general warming of the world by about one degree Fahrenheit."{{sfn|Macrae|1992|p=16}}<ref name="William H. Davenport 2016 page 266">''Engineering: Its Role and Function in Human Society''
edited by William H. Davenport, Daniel I. Rosenthal (Elsevier 2016), page 266</ref> His research into weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the [[polar ice cap]]s to enhance absorption of solar radiation (by reducing the [[albedo]]).{{sfn|Macrae|1992|p=332}}{{sfn|Heims|1980|pp=236–247}}{{sfn|Macrae|1992|p=332}}{{sfn|Heims|1980|pp=236–247}}  However, he urged caution in any program of atmosphere modification: <blockquote>What ''could'' be done, of course, is no index to what ''should'' be done... In fact, to evaluate the ultimate consequences of either a general cooling or a general heating would be a complex matter. Changes would affect the level of the seas, and hence the habitability of the continental coastal shelves; the evaporation of the seas, and hence general precipitation and glaciation levels; and so on... But there is little doubt that one ''could'' carry out the necessary analyses needed to predict the results, intervene on any desired scale, and ultimately achieve rather fantastic results.<ref name="William H. Davenport 2016 page 266"/></blockquote> He also warned that weather and climate control could have military uses, telling [[United States Congress|Congress]] in 1956 that they could pose an even bigger risk than [[Intercontinental ballistic missile|ICBMs]].{{sfn|Edwards|2010|pp=189–191}}

===Technological singularity hypothesis===
{{See also|Technological singularity}}
{{Quote box
| width = 390px
| align = right
| quote = "The technology that is now developing and that will dominate the next decades is in conflict with traditional, and, in the main, momentarily still valid, geographical and political units and concepts. This is a maturing crisis of technology... The most hopeful answer is that the human species has been subjected to similar tests before and it seems to have a congenital ability to come through, after varying amounts of trouble."
| source = —von Neumann, 1955<ref name="William H. Davenport 2016 page 266"/>
}}
The first use of the concept of a [[Wiktionary:singularity|singularity]] in the technological context is attributed to von Neumann,<ref>''The Technological Singularity'' by [[Murray Shanahan]], (MIT Press, 2015), page 233</ref> who according to Ulam discussed the "ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue."<ref name="chalmers">{{Cite journal|last=Chalmers|first=David|author-link=David Chalmers|date=2010|title=The singularity: a philosophical analysis|journal=Journal of Consciousness Studies|volume=17|issue=9–10|pages=7–65}}</ref> This concept was later fleshed out in the 1970 book ''[[Future Shock]]'' by [[Alvin Toffler]].

== Defense work ==
[[File:Vonneumann-john r.jpg|thumb|Von Neumann's wartime [[Los Alamos National Laboratory|Los Alamos]] ID badge photo]]

=== Manhattan Project ===
Beginning in the late 1930s, von Neumann developed an expertise in explosions—phenomena that are difficult to model mathematically. During this period, he was the leading authority of the mathematics of [[shaped charge]]s, leading him to a large number of military consultancies and consequently his involvement in the [[Manhattan Project]]. The involvement included frequent trips to the project's secret research facilities at the [[Los Alamos Laboratory]] in New Mexico.<ref name="NYT"/>

Von Neumann made his principal contribution to the [[Nuclear weapon|atomic bomb]] in the concept and design of the [[nuclear weapon design|explosive lenses]] that were needed to compress the [[plutonium]] core of the [[Fat Man]] weapon that was later dropped on [[Nagasaki]].{{sfn|Jacobsen|2015|loc=Ch. 3}} While von Neumann did not originate the "[[Nuclear weapon design#Implosion-type 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".{{sfn|Hoddeson|Henriksen|Meade|Westfall|1993|pp=130–133, 157–159}}

When it turned out that there would not be enough [[uranium-235]] to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the [[plutonium-239]] that was available from the [[Hanford Site]].{{sfn|Hoddeson|Henriksen|Meade|Westfall|1993|pp=239–245}} He established the design of the [[explosive lens]]es required, but there remained concerns about "edge effects" and imperfections in the explosives.{{sfn|Hoddeson|Henriksen|Meade|Westfall|1993|p=295}} His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry.<ref>{{cite web |url=http://nuclearweaponarchive.org/Nwfaq/Nfaq8.html |title=Section 8.0 The First Nuclear Weapons |publisher=Nuclear Weapons Frequently Asked Questions  |first=Carey |last=Sublette |access-date=January 8, 2016}}</ref> After a series of failed attempts with models, this was achieved by [[George Kistiakowsky]], and the construction of the Trinity bomb was completed in July 1945.{{sfn|Hoddeson|Henriksen|Meade|Westfall|1993|pp=320–327}}

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.{{sfn|Macrae|1992|p=209}}{{sfn|Hoddeson|Henriksen|Meade|Westfall|1993|p=184}}

[[File:Implosion bomb animated.gif|thumb|left|Implosion mechanism]]
Von Neumann was included in the target selection committee that was responsible for choosing the Japanese cities of [[Hiroshima]] and 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. The cultural capital [[Kyoto]] was von Neumann's first choice,{{sfn|Macrae|1992|pp=242–245}} 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 L. Stimson]].<ref>{{cite book |last=Groves|first=Leslie |author-link=Leslie Groves |title=Now it Can be Told: The Story of the Manhattan Project |url=https://archive.org/details/nowitcanbetolds00grov|url-access=registration|location=New York |publisher=Harper & Row |year=1962 |isbn=978-0-306-70738-4|oclc=537684 |pages=268–276 }}</ref>

On July 16, 1945, von Neumann and numerous other Manhattan Project personnel were eyewitnesses to the first test of an atomic bomb detonation, which was code-named [[Trinity (nuclear test)|Trinity]]. The event was conducted as a test of the implosion method device, at the [[Alamogordo Bombing Range]] in New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to {{convert|5|ktonTNT|lk=on}} 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.{{sfn|Hoddeson|Henriksen|Meade|Westfall|1993|pp=371–372}} It was in von Neumann's 1944 papers that the expression "kilotons" appeared for the first time.{{sfn|Macrae|1992|p=205}}

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 collaborated with [[Klaus Fuchs]] on further development of the bomb, and in 1946 the two filed a secret patent outlining a scheme for using a fission bomb to compress fusion fuel to initiate [[nuclear fusion]].<ref>{{cite book | last=Herken |first=Gregg |author-link=Gregg Herken | title=Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller |publisher=Holt | location=New York |year=2002 | isbn=978-0-8050-6589-3| oclc=48941348 |pages=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 [[History of the Teller–Ulam design|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|issue=1|pages=36–50|year=2010|doi=10.1007/s00016-009-0001-1|bibcode = 2010PhP....12...36B|s2cid=121790196}}</ref> The Fuchs–von Neumann work was passed on to the Soviet Union by Fuchs 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"/>

For his wartime services, von Neumann was awarded the [[Navy Distinguished Civilian Service Award]] in July 1946, and the [[Medal for Merit]] in October 1946.{{sfn|Macrae|1992|p=208}}

=== Post-war work===
In 1950, von Neumann became a consultant to the [[Weapons Systems Evaluation Group]],{{sfn|Macrae|1992|pp=350–351}} whose function was to advise the [[Joint Chiefs of Staff]] and the [[United States Secretary of Defense]] on the development and use of new technologies.<ref>{{cite news|title=Weapons' Values to be Appraised|url=https://news.google.com/newspapers?id=wBIzAAAAIBAJ&pg=7379%2C6398588|date=December 15, 1948|newspaper=[[Spokane Daily Chronicle]]|access-date=January 8, 2015 }}</ref> He also became an adviser to the [[Armed Forces Special Weapons Project]], which was responsible for the military aspects on [[nuclear weapon]]s.{{sfn|Macrae|1992|pp=350–351}} Over the following two years, he became a consultant across the US government.{{sfn|Sheehan|2010|p=182}} This included the [[Central Intelligence Agency]] (CIA), a member of the influential General Advisory Committee of the [[United States Atomic Energy Commission|Atomic Energy Commission]], a consultant to the newly established [[Lawrence Livermore National Laboratory]], and a member of the [[Scientific Advisory Group]] of the [[United States Air Force]]{{sfn|Macrae|1992|pp=350–351}} During this time he became a "superstar" defense scientist at [[the Pentagon]]. His authority was considered infallible at the highest levels of the US government and military.{{sfn|Jacobsen|2015|p=40}}

During several meetings of the advisory board of the US Air Force, von Neumann and [[Edward Teller]] predicted that by 1960 the US would be able to build a hydrogen bomb light enough to fit on top of a rocket. In 1953 [[Bernard Schriever]], who was present at the meeting, paid a personal visit to von Neumann at Princeton to confirm this possibility.{{sfn|Sheehan|2010|pp=178–179}} Schriever enlisted [[Trevor Gardner]], who in turn visited von Neumann several weeks later to fully understand the future possibilities before beginning his campaign for such a weapon in Washington.{{sfn|Sheehan|2010|p=199}} Now either chairing or serving on several boards dealing with strategic missiles and nuclear weaponry, von Neumann was able to inject several crucial arguments regarding potential [[Soviet Union|Soviet]] advancements in both these areas and in strategic defenses against American bombers into government reports to argue for the creation of [[Intercontinental ballistic missile|ICBMs]].{{sfn|Sheehan|2010|pp=217,219-220}} Gardner on several occasions brought von Neumann to meetings with the US Department of Defense to discuss with various senior officials his reports.{{sfn|Sheehan|2010|p=221}} Several design decisions in these reports such as inertial guidance mechanisms would form the basis for all ICBMs thereafter.{{sfn|Sheehan|2010|p=259}} By 1954, von Neumann was also regularly testifying to various [[United States Congress|Congressional]] military subcommittees to ensure continued support for the ICBM program.{{sfn|Sheehan|2010|pp=273,276-278}}

However, this was not enough. To have the ICBM program run at full throttle they needed direct action by the President of the United States.{{sfn|Sheehan|2010|pp=275,278}} They convinced [[Dwight D. Eisenhower|President Eisenhower]] in a direct meeting in July 1955, which resulted in a presidential directive on September 13, 1955. It stated that "there would be the gravest repercussions on the national security and on the cohesion of the free world" if the Soviet Union developed the ICBM before the US and therefore designated the ICBM project "a research and development program of the highest priority above all others." The Secretary of Defense was ordered to commence the project with "maximum urgency".{{sfn|Sheehan|2010|pp=287–299}} Evidence would later show that the Soviets indeed were already testing their own [[intermediate-range ballistic missile]]s at the time.{{sfn|Sheehan|2010|p=311}} Von Neumann would continue to meet the President, including at his home in [[Gettysburg, Pennsylvania]], and other high-level government officials as a key advisor on ICBMs until his death.{{sfn|Aspray|1990|p=250}}

=== Atomic Energy Commission ===
In 1955, von Neumann became a commissioner of the [[United States Atomic Energy Commission|Atomic Energy Commission]] (AEC), which at the time was the highest official position available to scientists in the government.{{sfn|Heims|1980|p=275}} (While his appointment formally required that he sever all his other consulting contracts,{{sfn|Aspray|1990|pp=244-245}} an exemption was made for von Neumann to continue working with several critical military committees after the [[United States Air Force|Air Force]] and several key [[United States Senate|senators]] raised concerns.{{sfn|Aspray|1990|p=250}}) He used this position to further the production of compact hydrogen bombs suitable for [[intercontinental ballistic missile]] (ICBM) delivery. He involved himself in correcting the severe shortage of [[tritium]] and [[lithium 6]] needed for these weapons, and he argued against settling for the intermediate-range missiles that the Army wanted. He was adamant that H-bombs delivered deep into enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile would not be a problem with an H-bomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case.{{sfn|Heims|1980|p=276}}{{sfn|Macrae|1992|pp=367–369}} While [[Lewis Strauss]] was away in the second half of 1955 von Neumann took over as acting chairman of the commission.{{sfn|Heims|1980|p=282}}

In his final years before his death from cancer, von Neumann headed the United States government's top-secret ICBM committee, which would sometimes meet in his home. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were sizable, they could be overcome. The [[SM-65 Atlas]] passed its first fully functional test in 1959, two years after his death.{{sfn|Macrae|1992|pp=359–365}} The more advanced [[Titan (rocket family)|Titan]] rockets were deployed in 1962. Both had been proposed in the ICBM committees von Neumann chaired.{{sfn|Aspray|1990|p=250}} The feasibility of the ICBMs owed as much to improved, smaller warheads that did not have guidance or heat resistance issues as it did to developments in rocketry, and his understanding of the former made his advice invaluable.{{sfn|Macrae|1992|pp=359–365}}{{sfn|Aspray|1990|p=250}}

Von Neumann entered government service primarily because he felt that, if freedom and civilization were to survive, it would have to be because the United States would triumph over totalitarianism from [[Nazism]], [[Fascism]] and [[Ideology of the Communist Party of the Soviet Union|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) |access-date=January 30, 2011 |ref=whiteman |url-status=dead |archive-url=https://web.archive.org/web/20110428125353/http://256.com/gray/docs/misc/conversation_with_marina_whitman.shtml |archive-date=April 28, 2011 }}</ref> 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".{{sfn|Blair|1957|p=96}}{{sfn|Pais|2006|p=109}}

== Personality ==

=== Work habits ===
[[Herman Goldstine]] commented on von Neumann's ability to intuit hidden errors and remember old material perfectly.{{sfn|Goldstine|1985|pp=9–10}}{{sfn|Albers|Alexanderson|2008|p=81}} When he had difficulties he would not labor on; instead, he would go home and sleep on it and come back later with a solution.{{sfn|Goldstine|1985|p=16}} This style, 'taking the path of least resistance', sometimes meant that he could go off on tangents. It also meant that if the difficulty was great from the very beginning, he would simply switch to another problem, not trying to find weak spots from which he could break through.{{sfn|Ulam|1976|p=78}} At times he could be ignorant of the standard mathematical literature, finding it easier to rederive basic information he needed rather than chase references.{{sfn|Halmos|1973|pp=387-388}}

After [[World War II]] began, he became extremely busy with both academic and military commitments. His habit of not writing up talks or publishing results worsened.<ref name=lax1990>{{harvc |last1=Lax |first1=Peter D. |author-link=Peter Lax |year=1990 |chapter=Remembering John von Neumann |in1=Glimm |in2=Impagliazzo |in3=Singer |page=6}}</ref> He did not find it easy to discuss a topic formally in writing unless it was already mature in his mind; if it was not, he would, in his own words, "develop the worst traits of pedantism and inefficiency".{{sfn|Rédei|Stöltzner|2001|p=168}}

=== Mathematical range ===

The mathematician [[Jean Dieudonné]] said that von Neumann "may have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions".{{sfn|Dieudonné|2008|p=90}} According to Dieudonné, his specific genius was in analysis and "combinatorics", with combinatorics being understood in a very wide sense that described his ability to organize and axiomize complex works that previously seemed to have little connection with mathematics. His style in analysis followed the German school, based on foundations in [[linear algebra]] and [[general topology]]. While von Neumann had an encyclopedic background, his range in pure mathematics was not as wide as [[Henri Poincaré|Poincaré]], [[David Hilbert|Hilbert]] or even [[Hermann Weyl|Weyl]]: von Neumann never did significant work in [[number theory]], [[algebraic topology]], [[algebraic geometry]] or [[differential geometry]]. However, in applied mathematics his work equalled that of [[Carl Friedrich Gauss|Gauss]], [[Augustin-Louis Cauchy|Cauchy]] or [[Henri Poincaré|Poincaré]].{{sfn|Dieudonné|2008}}

According to Wigner, "Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique."{{sfn|Dyson|1998|p=77}} Halmos noted that while von Neumann knew lots of mathematics, the most notable gaps were in algebraic topology and number theory; he recalled an incident where von Neumann failed to recognize the topological definition of a [[torus]].{{sfn|Halmos|1973|p=389}} Von Neumann admitted to Herman Goldstine that he had no facility at all in topology and he was never comfortable with it, with Goldstine later bringing this up when comparing him to [[Hermann Weyl]], who he thought was deeper and broader.{{sfn|Goldstine|1985|p=16}}

In his biography of von Neumann, [[Salomon Bochner]] wrote that much of von Neumann's works in pure mathematics involved finite and infinite dimensional [[vector space]]s, which at the time, covered much of the total area of mathematics. However he pointed out this still did not cover an important part of the mathematical landscape, in particular, anything that involved geometry "in the global sense", topics such as [[topology]], [[differential geometry]] and [[Hodge theory|harmonic integrals]], [[algebraic geometry]] and other such fields. Von Neumann rarely worked in these fields and, as Bochner saw it, had little affinity for them.{{sfn|Bochner|1958|p=441}}

In one of von Neumann's last articles, he lamented that pure mathematicians could no longer attain deep knowledge of even a fraction of the field.{{sfn|Ulam|1958|p=8}} In the early 1940s, Ulam had concocted for him a doctoral-style examination to find weaknesses in his knowledge; von Neumann was unable to answer satisfactorily a question each in differential geometry, number theory, and algebra. They concluded that doctoral exams might have "little permanent meaning". However, when Weyl turned down an offer to write a history of mathematics of the 20th century, arguing that no one person could do it, Ulam thought von Neumann could have aspired to do so.{{sfn|Ulam|1976|p=291}}

=== Preferred problem-solving techniques ===
Ulam remarked that most mathematicians could master one technique that they then used repeatedly, whereas von Neumann had mastered three:
# A facility with the symbolic manipulation of linear operators;
# An intuitive feeling for the logical structure of any new mathematical theory;
# An intuitive feeling for the combinatorial superstructure of new theories.{{sfn|Ulam|1976|p=96}}

Although he was commonly described as an analyst, he once classified himself an algebraist,<ref>{{cite interview |last=Halperin |first=Israel |subject-link=Israel Halperin|interviewer=[[Albert W. Tucker|Albert Tucker]] |title=Interview Transcript #18 - Oral History Project |date=1984 |publisher=Princeton Mathematics Department |url=https://web.math.princeton.edu/oral-history/c17.pdf |access-date=4 April 2022 |page=12}}</ref> and his style often displayed a mix of algebraic technique and set-theoretical intuition.{{sfn|Ulam|1958|p=9}} He loved obsessive detail and had no issues with excess repetition or overly explicit notation. An example of this was a paper of his on rings of operators, where he extended the normal functional notation, <math>\phi (x)</math> to <math>\phi ((x))</math>. However, this process ended up being repeated several times, where the final result were equations such as <math>(\psi((((a)))))^2 = \phi((((a))))</math>. The 1936 paper became known to students as "von Neumann's onion"<ref name=segal1990>{{harvc |last1=Segal |first1=Irving E. |author-link=Irving Segal |year=1990 |chapter=The Mathematical Implications of Fundamental Physical Principles |in1=Glimm |in2=Impagliazzo |in3=Singer |pages=154–156}}</ref> because the equations "needed to be peeled before they could be digested". Overall, although his writings were clear and powerful, they were not clean or elegant.{{sfn|Halmos|1973|p=388}} Although powerful technically, his primary concern was more with the clear and viable formation of fundamental issues and questions of science rather than just the solution of mathematical puzzles.{{r|segal1990}}

According to Ulam, von Neumann surprised physicists by doing dimensional estimates and algebraic computations in his head with fluency Ulam likened to [[blindfold chess]]. His impression was that von Neumann analyzed physical situations by abstract logical deduction rather than concrete visualization.{{sfn|Ulam|1958|p=38}}

=== Lecture style ===

Goldstine compared his lectures to being on glass, smooth and lucid. By comparison, Goldstine thought his scientific articles were written in a much harsher manner, and with much less insight.{{sfn|Goldstine|1985|p=7}} [[Paul Halmos|Halmos]] described his lectures as "dazzling", with his speech clear, rapid, precise and all encompassing. Like Goldstine, he also described how everything seemed "so easy and natural" in lectures but puzzling on later reflection.{{sfn|Halmos|1973|pp=387-388}} He was a quick speaker: [[Banesh Hoffmann]] found it very difficult to take notes, even in [[shorthand]],<ref>{{cite interview |last=Hoffmann |first=Banesh |subject-link=Banesh Hoffmann|interviewer=[[Albert W. Tucker|Albert Tucker]] |title=Interview Transcript #20 - Oral History Project |date=1984 |publisher=Princeton Mathematics Department |url=https://web.math.princeton.edu/oral-history/c19.pdf |access-date=4 April 2022 |page=4}}</ref> and [[Albert W. Tucker|Albert Tucker]] said that people often had to ask von Neumann questions to slow him down so they could think through the ideas he was presenting. Von Neumann knew about this and was grateful for his audience telling him when he was going too quickly.{{sfn|Tucker|1984|p=4}} Although he did spend time preparing for lectures, he rarely used notes, instead jotting down points of what he would discuss and for how long.{{sfn|Halmos|1973|pp=387-388}}

=== Eidetic memory ===
Von Neumann was also noted for his [[eidetic memory]], particularly of the symbolic kind. [[Herman Goldstine]] writes:

{{blockquote|One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how ''[[A Tale of Two Cities]]'' started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.{{sfn|Goldstine|1980|pp=167}}}}

Von Neumann was reportedly able to memorize the pages of telephone directories. He entertained friends by asking them to randomly call out page numbers; he then recited the names, addresses and numbers therein.{{sfn|Blair|1957|p=90}}<ref>[https://www.ias.edu/von-neumann John von Neumann: Life, Work, and Legacy] Institute of Advanced Study, Princeton</ref>  [[Stanisław Ulam]] believed that von Neumann's memory was auditory rather than visual.{{sfn|Ulam|1976|pp=147–148}}

=== Mathematical quickness ===

Von Neumann's mathematical fluency, calculation speed, and general problem-solving ability were widely noted by his peers. [[Paul Halmos]] called his speed "awe-inspiring."{{sfn|Halmos|1973|p=386}} [[Lothar Wolfgang Nordheim]] described him as the "fastest mind I ever met".{{sfn|Goldstine|1980|pp=171}} [[Enrico Fermi]] told physicist [[Herbert L. Anderson]]: "You know, Herb, Johnny can do calculations in his head ten times as fast as I can! And I can do them ten times as fast as you can, Herb, so you can see how impressive Johnny is!"<ref>''Fermi Remembered'', [[James Cronin|James W. Cronin]], University of Chicago Press (2004), page 236</ref> [[Edward Teller]] admitted that he "never could keep up with him",<ref>{{cite journal |title=John von Neumann|first=Edward |last=Teller |author-link=Edward Teller |journal=Bulletin of the Atomic Scientists |date=April 1957 |volume=13 |issue=4 |pages=150–151 |doi=10.1080/00963402.1957.11457538 |bibcode=1957BuAtS..13d.150T }}</ref> and [[Israel Halperin]] described trying to keep up as like riding a "tricycle chasing a racing car."<ref>Kaplan, Michael and Kaplan, Ellen (2006) ''Chances are–: adventures in probability''. Viking.</ref>

He had an unusual ability to solve novel problems quickly. [[George Pólya]], whose lectures at [[ETH Zürich]] von Neumann attended as a student, said, "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."<ref>{{cite book |last=Petković |first=Miodrag |author-link=Miodrag Petković |year=2009 |title=Famous puzzles of great mathematicians |url=https://archive.org/details/famouspuzzlesgre00mpet/page/n175 |url-access=limited |publisher=American Mathematical Society |page=157 |isbn=978-0-8218-4814-2}}</ref> When [[George Dantzig]] brought von Neumann an unsolved problem in linear programming "as I would to an ordinary mortal", on which there had been no published literature, he was astonished when von Neumann said "Oh, that!", before offhandedly giving a lecture of over an hour, explaining how to solve the problem using the hitherto unconceived [[Linear programming#Duality|theory of duality]].<ref>{{cite book |last=Mirowski |first=Philip |author-link=Philip Mirowski |title=Machine Dreams: Economics Becomes a Cyborg Science |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-77283-9 |oclc=45636899 |url-access=limited |url=https://archive.org/details/machinedreamseco0000miro |page=258}}</ref>

A story about von Neumann's encounter with the famous [[fly puzzle]]<ref>{{cite web|url=http://mathworld.wolfram.com/TwoTrainsPuzzle.html |title=Fly Puzzle (Two Trains Puzzle) |publisher=Wolfram MathWorld |date=February 15, 2014 |access-date=February 25, 2014}}</ref> has entered [[Mathematical folklore#Stories, sayings and jokes|mathematical folklore]]. In this puzzle, two bicycles begin 20 miles apart, and each travels toward the other at 10 miles per hour until they collide; meanwhile, a fly travels continuously back and forth between the bicycles at 15 miles per hour until it is squashed in the collision. The questioner asks how far the fly traveled in total; the "trick" for a quick answer is to realize that the fly's individual transits do not matter, only that it has been traveling at 15 miles per hour for one hour. As [[Eugene Wigner]] tells it,<ref>{{cite web|title=John von Neumann – A Documentary|url=https://www.youtube.com/watch?v=vQp70uqsBV4&t=1020s|publisher=The Mathematical Association of America|access-date=26 August 2022|at=17m00s – 19m11s|date=1966}}</ref> [[Max Born]] posed the riddle to von Neumann. The other scientists to whom he had posed it had laboriously computed the distance, so when von Neumann was immediately ready with the correct answer of 15 miles, Born observed that he must have guessed the trick. "What trick?" von Neumann replied. "All I did was sum the [[geometric series]]."{{sfn|Halmos|1973|pp=386-387}}

=== Self-doubts ===

[[Gian-Carlo Rota|Rota]] wrote that von Neumann had "deep-seated and recurring self-doubts".{{sfn|Rota|1997|p=71}} [[John L. Kelley]] reminisced in 1989 that "Johnny von Neumann has said that he will be forgotten while [[Kurt Gödel]] is remembered with [[Pythagoras]], but the rest of us viewed Johnny with awe."<ref>{{cite book |last1=Kelley |first1=J. L. |author-link1=John L. Kelley |year=1989 |chapter=Once Over Lightly |editor-last1=Duren |editor-first1=Peter |editor-link1=Peter Duren |title=A Century of Mathematics in America: Part III |publisher=American Mathematical Society |isbn=0-8218-0136-8 |page=478|url = http://at.yorku.ca/t/o/p/d/03.htm}}</ref> Ulam suggests that some of his self-doubts with regard for his own creativity may have come from the fact he had not discovered several important ideas that others had, even though he was more than capable of doing so, giving the [[Gödel's incompleteness theorems|incompleteness theorems]] and [[George David Birkhoff|Birkhoff's]] [[Ergodic theory#Ergodic theorems|pointwise ergodic theorem]] as examples. Von Neumann had a virtuosity in following complicated reasoning and had supreme insights, yet he perhaps felt he did not have the gift for seemingly irrational proofs and theorems or intuitive insights. Ulam describes how during one of his stays at Princeton while von Neumann was working on rings of operators, continuous geometries and quantum logic he felt that von Neumann was not convinced of the importance of his work, and only when finding some ingenious technical trick or new approach did he take some pleasure in it.{{sfn|Ulam|1976|pp=76-77}} However, according to Rota, von Neumann still had an "incomparably stronger technique" compared to his friend, despite describing Ulam as the more creative mathematician.{{sfn|Rota|1997|p=71}}

== Legacy ==
===Accolades===
Nobel Laureate [[Hans Bethe]] said "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man".{{sfn|Blair|1957|p=90}} [[Edward Teller]] observed "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."<ref>{{cite web|last1=Nowak|first1=Amram|title=John Von Neumann a documentary|url=https://www.youtube.com/watch?v=gn3U9G0gfu4|publisher=Mathematical Association of America, Committee on Educational Media|language=en|date=1 January 1966|oclc=177660043}}, DVD version (2013) {{oclc|897933992}}.</ref> [[Peter Lax]] wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics".{{r|lax1990}} [[Eugene Wigner]] said, "He understood mathematical problems not only in their initial aspect, but in their full complexity."{{sfn|Szanton|1992|p=58}} [[Claude Shannon]] called him "the smartest person I've ever met", a common opinion.<ref>{{cite book |last1=Soni |first1=Jimmy |author-link1=Jimmy Soni |last2=Goodman |first2=Rob |title=A Mind at Play: How Claude Shannon Invented the Information Age |date=2017 |publisher=Simon & Schuster |isbn=978-1476766683 |page=76}}</ref> [[Jacob Bronowski]] wrote "He was the cleverest man I ever knew, without exception. He was a genius."<ref>{{cite book |last=Bronowski |first=Jacob |author-link=Jacob Bronowski |year=1974 |title=The Ascent of Man |location=Boston |publisher=Little, Brown |page=433}}</ref> Due to his wide reaching influence and contributions to many fields, von Neumann is widely considered a [[polymath]].<ref>{{Cite web |title=John von Neumann |url=https://ahf.nuclearmuseum.org/ahf/profile/john-von-neumann/ |access-date=2024-11-06 |website=[[Atomic Heritage Foundation]] |language=en-US}}</ref><ref>{{Cite web |last=Robinson |first=Andrew |date=2021-12-18 |title=Brilliant polymath, troubled person: how John von Neumann shaped our world |url=https://physicsworld.com/a/brilliant-polymath-troubled-person-how-john-von-neumann-shaped-our-world/ |access-date=2024-11-06 |website=Physics World |language=en-GB}}</ref><ref>{{Cite book |last=Adami |first=Christoph |title=The Evolution of Biological Information: How Evolution Creates Complexity, from Viruses to Brains |date=2024 |publisher=Princeton University Press |isbn=978-0-691-24114-2 |location=Princeton |pages=189–190 |language=en}}</ref>

Wigner noted the extraordinary mind that von Neumann had, and he described von Neumann as having a mind faster than anyone he knew, stating that:{{sfn|Szanton|1992|p=58}}
{{blockquote|I have known a great many intelligent people in my life. I knew [[Max Planck]], [[Max von Laue]], and [[Werner Heisenberg]]. [[Paul Dirac]] was my brother-in-law; [[Leo Szilard]] and Edward Teller have been among my closest friends; and [[Albert Einstein]] was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me.}}"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]].{{sfn|Rédei|2005|p=7}} Peter Lax commented that von Neumann would have won a [[Nobel Prize in Economics]] had he lived longer, and that "if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too."{{sfn|Rédei|2005|p=xiii}} Rota writes that "he was the first to have a vision of the boundless possibilities of computing, and he had the resolve to gather the considerable intellectual and engineering resources that led to the construction of the first large computer" and consequently that "No other mathematician in this century has had as deep and lasting an influence on the course of civilization."{{sfn|Rota|1997|p=70}} He is widely regarded as one of the greatest and most influential mathematicians and scientists of the 20th century.<ref>
{{harvnb|Ulam|1976|p=4}}; {{harvnb|Kac|Rota|Schwartz|2008|p=206}}; {{harvnb|Albers|Alexanderson|2008|p=168}}; {{harvnb|Szanton|1992|p=51}} {{pb}} {{cite book |last1=Rhodes |first1=Richard |author-link1=Richard Rhodes |title=Dark Sun: The Making of the Hydrogen Bomb |date=1995 |publisher=Simon & Schuster |location=New York |isbn=0-684-80400-X |url=https://www.simonandschuster.com/books/Dark-Sun/Richard-Rhodes/9780684824147 |page=250}} {{pb}} {{cite journal | url=https://www.worldscientific.com/worldscibooks/10.1142/5982 | doi=10.1142/5982 | title=Modeling and Computations in Dynamical Systems | journal=World Scientific Series on Nonlinear Science Series B | date=March 2006 | last1=Doedel | first1=Eusebius J. | last2=Domokos | first2=Gábor | last3=Kevrekidis | first3=Ioannis G. | volume=13 | isbn=978-981-256-596-9 }}</ref>

[[Neurophysiology|Neurophysiologist]] [[Leon Harmon]] described him in a similar manner, calling him the only "true genius" he had ever met: "von Neumann's mind was all-encompassing. He could solve problems in any domain. ... And his mind was always working, always restless."<ref>{{cite book |last=McCorduck |first=Pamela |author-link=Pamela McCorduck |title=Machines Who Think: A Personal Inquiry into the History and Prospects of Artificial Intelligence |date=2004 |publisher=Routledge |isbn=978-1568812052 |edition=2nd |page=81}}</ref> While consulting for non-academic projects von Neumann's combination of outstanding scientific ability and practicality gave him a high credibility with military officers, engineers, and industrialists that no other scientist could match. In [[Nuclear weapon|nuclear missilery]] he was considered "the clearly dominant advisory figure" according to [[Herbert York]].{{sfn|York|1971|p=85}} Economist [[Nicholas Kaldor]] said he was "unquestionably the nearest thing to a genius I have ever encountered."{{sfn|Dore|Chakravarty|Goodwin|1989|p=xi}} Likewise, [[Paul Samuelson]] wrote, "We economists are grateful for von Neumann's genius. It is not for us to calculate whether he was a [[Carl Friedrich Gauss|Gauss]], or a [[Henri Poincaré|Poincaré]], or a [[David Hilbert|Hilbert]]. He was the incomparable Johnny von Neumann. He darted briefly into our domain and it has never been the same since."{{sfn|Dore|Chakravarty|Goodwin|1989|p=121}}

=== Honors and awards ===
{{main|List of things named after John von Neumann|List of awards and honors received by John von Neumann}}
[[File:Von Neumann crater 5103 h2 h3.jpg|thumb|right|The von Neumann crater, on the far side of the Moon]]

Events and awards named in recognition of von Neumann include the annual [[John von Neumann Theory Prize]] of the [[Institute for Operations Research and the Management Sciences]],<ref>{{cite web |url=https://www.informs.org/Recognize-Excellence/INFORMS-Prizes-Awards/John-von-Neumann-Theory-Prize |title=John von Neumann Theory Prize |publisher=[[Institute for Operations Research and the Management Sciences]] |accessdate=May 17, 2016 |url-status=dead |archive-url=https://web.archive.org/web/20160513155431/https://www.informs.org/Recognize-Excellence/INFORMS-Prizes-Awards/John-von-Neumann-Theory-Prize |archive-date=May 13, 2016 }}</ref> [[IEEE John von Neumann Medal]],<ref>{{cite web |url=https://corporate-awards.ieee.org/award/ieee-john-von-neumann-medal/ |publisher=[[Institute of Electrical and Electronics Engineers]] |title= IEEE John von Neumann Medal |website= IEEE Awards |access-date=2024-07-30 }}</ref> and the [[John von Neumann Prize]] of the [[Society for Industrial and Applied Mathematics]].<ref>{{cite web |url=https://www.siam.org/prizes/sponsored/vonneumann.php |publisher=[[Society for Industrial and Applied Mathematics]]|title=The John von Neumann Lecture  |access-date=May 17, 2016 }}</ref> Both the crater [[Von Neumann (crater)|von Neumann]] on the [[Moon]]<ref>{{cite web |url=http://planetarynames.wr.usgs.gov/Feature/6442?__fsk=1809478007 |title=Von Neumann |publisher=[[United States Geological Survey]] |access-date=May 17, 2016 }}</ref> and the asteroid [[22824 von Neumann]] are named in his honor.<ref>{{cite web |url=https://ssd.jpl.nasa.gov/sbdb.cgi?ID=a0022824 |title=22824 von Neumann (1999 RP38) |publisher=[[Jet Propulsion Laboratory]] |access-date=February 13, 2018 }}</ref><ref>{{cite web |url=https://minorplanetcenter.net/db_search/show_object?utf8=%E2%9C%93&object_id=22824 |title=(22824) von Neumann = 1999 RP38 = 1998 HR2 |publisher=[[Minor Planet Center]] |access-date=February 13, 2018 }}</ref>

Von Neumann received awards including the [[Medal for Merit]] in 1947, the [[Medal of Freedom (1945)|Medal of Freedom]] in 1956,<ref>{{cite web |url=https://www.presidency.ucsb.edu/documents/citation-accompanying-medal-freedom-presented-dr-john-von-neumann |title=Dwight D. Eisenhower: Citation Accompanying Medal of Freedom Presented to Dr. John von Neumann |publisher= The American Presidency Project }}</ref> and the [[Enrico Fermi Award]] also in 1956. He was elected a member of multiple honorary societies, including the [[American Academy of Arts and Sciences]] and the [[National Academy of Sciences]], and he held eight honorary doctorates.{{sfn|Aspray|1990|pp=246-247}}{{sfn|Ulam|1958|pp=41-42}}<ref>{{cite web |title=Von Neumann, John, 1903–1957 |url=https://history.aip.org/phn/11610032.html |website=Physics History Network |publisher=American Institute of Physics |accessdate=October 12, 2023 }}</ref> On May 4, 2005, the [[United States Postal Service]] issued the ''American Scientists'' commemorative postage stamp series, designed by artist [[Victor Stabin]]. The scientists depicted were von Neumann, [[Barbara McClintock]], [[Josiah Willard Gibbs]], and [[Richard Feynman]].<ref>{{cite web |url=http://arago.si.edu/category_2046196.html |archive-url=https://web.archive.org/web/20160202230912/http://arago.si.edu/category_2046196.html |archivedate=February 2, 2016 |url-status=dead |title=American Scientists Issue |website=Arago: People, Postage & the Post |publisher=[[National Postal Museum]] |access-date=August 2, 2022}}</ref>

{{ill|John von Neumann University|hu|Neumann János Egyetem}} was established in [[Kecskemét]], Hungary in 2016, as a successor to Kecskemét College.<ref>{{Cite web|url=https://nje.hu/|title=Neumann János Egyetem|website=Neumann János Egyetem}}</ref>

== Selected works ==
{{main|List of scientific publications by John von Neumann}}
Von Neumann's first published paper was ''On the position of zeroes of certain minimum polynomials'', co-authored with [[Michael Fekete]] and published when von Neumann was 18. At 19, his solo paper ''On the introduction of transfinite numbers'' was published.{{sfn|Dyson|2013|p=154}} He expanded his second solo paper, ''An axiomatization of set theory'', to create his PhD thesis.{{sfn|Dyson|2013|p=155}} His first book, ''Mathematical Foundations of Quantum Mechanics'', was published in 1932.{{sfn|Dyson|2013|p=157}} Following this, von Neumann switched from publishing in German to publishing in English, and his publications became more selective and expanded beyond pure mathematics. His 1942 ''Theory of Detonation Waves'' contributed to military research,{{sfn|Dyson|2013|p=158}} his work on computing began with the unpublished 1946 ''On the principles of large scale computing machines'', and his publications on weather prediction began with the 1950 ''Numerical integration of the barotropic vorticity equation''.{{sfn|Dyson|2013|p=159}} Alongside his later papers were informal essays targeted at colleagues and the general public, such as his 1947 ''The Mathematician'',<ref>{{cite book |last=von Neumann |first=John |editor-last=Heywood |editor-first=Robert B. |title=The Works of the Mind |chapter=The Mathematician |year=1947 |publisher=University of Chicago Press |oclc=752682744 }}</ref> described as a "farewell to pure mathematics", and his 1955 ''Can we survive technology?'', which considered a bleak future including nuclear warfare and deliberate climate change.{{sfn|Dyson|2013|pp=159–160}} His complete works have been compiled into a six-volume set.{{sfn|Dyson|2013|p=154}}

== See also ==
* [[List of pioneers in computer science]]
* [[Teapot Committee]]
* ''[[The MANIAC (book)|The MANIAC]]'', 2023 book about von Neumann
* {{langx|de|[[:de:Abenteuer eines Mathematikers|Abenteuer eines Mathematikers]]}} (English title: ''Adventures of a Mathematician''), biopic about Stanislaw Ulam also features John von Neumann.

== Notes ==
{{reflist|25em}}

== References ==
{{refbegin|30em}}
* {{cite book |editor-last1=Albers |editor-first1=Donald J. |editor-last2=Alexanderson |editor-first2=Gerald L. |editor-link2=Gerald L. Alexanderson |title=Mathematical People: Profiles and Interviews |publisher=CRC Press |date=2008 |doi=10.1201/b10585 |edition=2 |isbn=978-1568813400 |url=https://www.taylorfrancis.com/books/mono/10.1201/b10585/mathematical-people-gerald-alexanderson-donald-albers}}
* {{cite book |last=Aspray |first=William |year=1990 |title=John von Neumann and the Origins of Modern Computing |location=Cambridge, Massachusetts |publisher=MIT Press |url=https://mitpress.mit.edu/9780262518857/john-von-neumann-and-the-origins-of-modern-computing/ |isbn=978-0262518857|oclc=21524368 |bibcode=1990jvno.book.....A}}
* {{cite book |last=Bhattacharya |first=Ananyo |year=2022 |title=The Man from the Future: The Visionary Life of John von Neumann |publisher=W. W. Norton & Company |isbn=978-1324003991 |url=https://wwnorton.com/books/the-man-from-the-future}}
* {{cite journal |last1=Birkhoff |first1=Garret |author-link=Garrett Birkhoff |title=Von Neumann and lattice theory |journal=Bulletin of the American Mathematical Society |year=1958 |volume=64 |issue=3, Part 2 |pages=50–56 |doi=10.1090/S0002-9904-1958-10192-5 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-64/issue-3.P2/Von-Neumann-and-lattice-theory/bams/1183522370.full|doi-access=free }}
* {{cite magazine|last=Blair |first=Clay Jr. |author-link=Clay Blair |url=https://books.google.com/books?id=rEEEAAAAMBAJ&pg=PA89 |title=Passing of a Great Mind|magazine=[[Life (magazine)|Life]] |pages=89–104 |date=February 25, 1957}}
* {{cite web |last=Bochner |first=S. |author-link=Salomon Bochner |title=John von Neumann 1903–1957: A Biographical Memoir |url=https://www.nasonline.org/wp-content/uploads/2024/06/von-neumann-john.pdf |publisher=[[National Academy of Sciences]] |year=1958 |access-date=January 16, 2025}}
* {{cite book |last1=Brezinski |first1=C. |last2=Wuytack |first2=L. |title=Numerical Analysis: Historical Developments in the 20th Century |publisher=Elsevier |date=2001 |doi=10.1016/C2009-0-10776-6 |isbn=9780444598585}}
* {{cite book |editor-last1=Bródy |editor-first1=F. |editor-last2=Vámos |editor-first2=Tibor |title=The Neumann Compendium |series=World Scientific Series in 20th Century Mathematics |year=1995 |volume=1 |publisher=World Scientific Publishing Company |location=Singapore |isbn=978-981-02-2201-7 |oclc=32013468 |url=https://archive.org/details/neumanncompendiu00neum_179 |url-access=limited |doi=10.1142/2692}}
* {{cite book |last1=Dieudonné |first1=Jean |author1-link=Jean Dieudonné |title=History of Functional Analysis |date=1981 |publisher=North-Hollywood Publishing Company |isbn=978-0444861481}}
* {{cite book |last=Dieudonné |first=J. |author-link=Jean Dieudonné |contribution=Von Neumann, Johann (or John) |edition=7th |title=Complete Dictionary of Scientific Biography |volume=14 |location=Detroit |publisher=Charles Scribner's Sons |year=2008 |pages=88–92 Gale Virtual Reference Library |editor-last=Gillispie |editor-first=C. C. |editor-link=Charles Coulston Gillispie |url=https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/von-neumann-johann-or-john |isbn=978-0-684-31559-1|oclc=187313311  }}
* {{cite book |editor1-last=Dore |editor1-first=Mohammed |editor2-last=Chakravarty |editor2-first=Sukhamoy |editor2-link=Sukhamoy Chakraborty |editor3-last=Goodwin |editor3-first=Richard |editor3-link=Richard M. Goodwin |title=John von Neumann and Modern Economics |location=Oxford |publisher=Clarendon|year=1989|isbn=978-0-19-828554-0|oclc=18520691|url=https://archive.org/details/johnvonneumannmo0000unse}}
* {{cite book |last=Dyson |first=George |author-link=George Dyson (science historian) |title=Darwin among the machines the evolution of global intelligence |publisher=Perseus Books |year=1998 |location=Cambridge, Massachusetts |isbn=978-0-7382-0030-9 |oclc=757400572  |url=https://archive.org/details/darwinamongmachi00dyso }}
* {{cite book |last=Dyson |first=George |author-link=George Dyson (science historian) |year=2012 |title=Turing's Cathedral: the Origins of the Digital Universe |publisher=Pantheon Books |location=New York |isbn=978-0-375-42277-5 |oclc=745979775  |url-access=registration |url=https://archive.org/details/turingscathedral0000dyso_n1l6|quote=“I am thinking of something much more important than bombs. I am thinking about computers”}}
* {{cite journal |last1=Dyson |first1=Freeman |author1-link=Freeman Dyson |title=A Walk through Johnny von Neumann's Garden |journal=Notices of the AMS |date=2013 |volume=60 |issue=2 |pages=154–161 |doi=10.1090/noti942|doi-access=free }}
* {{cite book |last1=Edwards |first1=Paul N. |title=A Vast Machine: Computer Models, Climate Data, and the Politics of Global Warming |date=2010 |publisher=The MIT Press |isbn=978-0-262-01392-5 |url=https://mitpress.mit.edu/books/vast-machine}}
* {{cite book|editor-last1=Glimm |editor-first1=James |editor1-link=James Glimm |editor-last2=Impagliazzo |editor-first2=John |editor-last3=Singer |editor-first3=Isadore Manuel |editor3-link=Isadore Singer |title=The Legacy of John von Neumann |publisher=American Mathematical Society |year=1990 |url=https://bookstore.ams.org/pspum-50 |isbn=978-0-8218-4219-5}}
* {{cite book|last=Goldstine |first=Herman |author-link=Herman Goldstine |title=The Computer from Pascal to von Neumann|publisher=Princeton University Press|year=1980|isbn=978-0-691-02367-0}}
* {{cite interview |last=Goldstine |first=Herman |subject-link=Herman Goldstine|interviewer1=[[Albert W. Tucker|Albert Tucker]]|interviewer2=Frederik Nebeker|title=Interview Transcript #15 - Oral History Project |date=1985 |publisher=Princeton Mathematics Department |location=Maryland |url=https://web.math.princeton.edu/oral-history/c14.pdf |access-date=3 April 2022}}
* {{cite book |last=Gustafsson |first=Bertil |author-link=Bertil Gustafsson |title=Scientific Computing: A Historical Perspective |series=Texts in Computational Science and Engineering |publisher=Springer |date=2018 |volume=17 |doi=10.1007/978-3-319-69847-2 |isbn=978-3-319-69847-2 |url=https://link.springer.com/book/10.1007/978-3-319-69847-2}}
* {{cite journal |last=Halmos |first=Paul R. |author-link=Paul Halmos |title=Von Neumann on measure and ergodic theory |journal=Bulletin of the American Mathematical Society |date=1958 |volume=64 |issue=3, Part 2 |pages=86–94 |doi=10.1090/S0002-9904-1958-10203-7 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-64/issue-3.P2/Von-Neumann-on-measure-and-ergodic-theory/bams/1183522373.full|doi-access=free }}
* {{cite journal |last=Halmos |first=Paul |author-link=Paul Halmos |title=The Legend of John Von Neumann |journal=The American Mathematical Monthly |date=1973 |volume=80 |issue=4 |pages=382–394 |doi=10.1080/00029890.1973.11993293 |url=https://www.tandfonline.com/toc/uamm20/80/4}}
* {{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=978-0-262-08105-4|url=https://archive.org/details/johnvonneumannno00heim}}
* {{cite book |last1=Hoddeson |first1=Lillian|author-link=Lillian Hoddeson|first2=Paul W. |last2=Henriksen |first3=Roger A. |last3=Meade |first4=Catherine L. |last4=Westfall|author4-link= Catherine Westfall |title=Critical Assembly: A Technical History of Los Alamos During the Oppenheimer Years, 1943–1945 |location=New York |publisher=Cambridge University Press |year=1993 |isbn=978-0-521-44132-2 |oclc=26764320  |url-access=registration |url=https://archive.org/details/criticalassembly0000unse }}
* {{cite book |last1=Horn |first1=Roger A. |author1-link=Roger Horn |last2=Johnson |first2=Charles R. |author2-link=Charles Royal Johnson |title=Matrix Analysis |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-83940-2 |edition=2 |url=https://www.cambridge.org/9780521548236}}
* {{cite book |last=Jacobsen |first=Annie |author-link=Annie Jacobsen |year=2015 |title=The Pentagon's Brain: An Uncensored History Of DARPA, America's Top Secret Military Research Agency |publisher=Little, Brown and Company |isbn=978-0316371667 |oclc=1037806913 |url=https://www.littlebrown.com/titles/annie-jacobsen/the-pentagons-brain/9780316371650 }}
* {{cite book |last1=Kac |first1=Mark |author-link1=Mark Kac |last2=Rota |first2=Gian-Carlo |author-link2=Gian-Carlo Rota |last3=Schwartz |first3=Jacob T. |author-link3=Jacob T. Schwartz |title=Discrete Thoughts: Essays on Mathematics, Science and Philosophy |date=2008 |publisher=Birkhäuser |location=Boston |doi=10.1007/978-0-8176-4775-9 |isbn=978-0-8176-4775-9 |edition=2 |url=https://link.springer.com/book/10.1007/978-0-8176-4775-9}}
* {{cite book |last1=Lord |first1=Steven |last2=Sukochev |first2=Fedor |last3=Zanin |first3=Dmitriy |title=Singular Traces: Theory and Applications |date=2012 |publisher=De Gruyter |doi=10.1515/9783110262551 |isbn=9783110262551 |edition=1 |url=https://www.degruyter.com/document/doi/10.1515/9783110262551/html}}
* {{cite book |last1=Lord |first1=Steven |last2=Sukochev |first2=Fedor |last3=Zanin |first3=Dmitriy |title=Singular Traces Volume 1: Theory |date=2021 |publisher=De Gruyter |doi=10.1515/9783110378054 |isbn=9783110378054 |s2cid=242485577 |edition=2 |url=https://www.degruyter.com/document/doi/10.1515/9783110378054/html}}
* {{cite book |last=Macrae |first=Norman |author-link=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=978-0-679-41308-0}}  [https://books.google.com/books?id=LfTup9S2suQC Description], [https://books.google.com/books?id=iF2mDwAAQBAJ contents, incl. arrow-scrollable preview], &  [https://www.kirkusreviews.com/book-reviews/norman-macrae/john-von-neumann/ review].
* {{cite book |last1=Murawski |first1=Roman |title=Essays in the Philosophy and History of Logic and Mathematics |date=2010 |publisher=Rodopi |location=Amsterdam |isbn=978-90-420-3091-6 |chapter=John Von Neumann and Hilbert's School |pages=195–209 |url=https://brill.com/view/book/9789042030916/B9789042030916-s015.xml |doi=10.1163/9789042030916_015}}
* {{cite book |last1=Pais |first1=Abraham |author-link1=Abraham Pais |title=The Genius of Science: A Portrait Gallery: A Portrait Gallery of Twentieth-Century Physicists |date=2000 |publisher=Oxford University Press |isbn=978-0198506140}}
* {{cite book |last=Pais |first=Abraham |author-link=Abraham Pais |title=J. Robert Oppenheimer: A Life |year=2006 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-516673-6 |oclc=475574884  |url=https://archive.org/details/jrobertoppenheim00pais_0 }}
* {{cite book |last=Pietsch |first=Albrecht |author-link=:de:Albrecht Pietsch |title=History of Banach Spaces and Linear Operators |date=2007 |publisher=Birkhäuser |location=Boston |doi=10.1007/978-0-8176-4596-0 |isbn=978-0-8176-4596-0 |url=https://link.springer.com/book/10.1007/978-0-8176-4596-0}}
* {{cite journal |last=Rédei |first=Miklós |title=Unsolved Problems in Mathematics |journal=[[The Mathematical Intelligencer]] |volume=21 |pages=7–12 |date=1999 |doi=10.1007/BF03025331 |s2cid=117002529 |url=http://phil.elte.hu/redei/cikkek/intel.pdf}}
* {{cite book |editor1-last=Rédei |editor1-first=Miklós |editor2-last=Stöltzner |editor2-first=Michael |title=John von Neumann and the Foundations of Quantum Physics |publisher=Springer |year=2001 |doi=10.1007/978-94-017-2012-0 |isbn=978-0792368120 |url=https://link.springer.com/book/10.1007/978-94-017-2012-0}}
* {{cite book |editor-last=Rédei |editor-first=Miklós |year=2005 |title=John von Neumann: Selected Letters |publisher=American Mathematical Society |location=Providence, Rhode Island |series=History of Mathematics |volume=27 |isbn=978-0-8218-3776-4 |oclc=60651134 |url=https://bookstore.ams.org/hmath-27}}
* {{cite book |last=Rickles |first=Dean |author-link=Dean Rickles |title=Covered with Deep Mist: The Development of Quantum Gravity 1916–1956 |date=2020 |publisher=Oxford University Press |doi=10.1093/oso/9780199602957.001.0001 |isbn=9780199602957 |url=https://doi.org/10.1093/oso/9780199602957.001.0001}}
* {{cite book |last=Rota |first=Gian-Carlo |author-link=Gian-Carlo Rota |editor1-first=Fabrizio |editor1-last=Palombi |title=Indiscrete Thoughts |date=1997 |publisher=Birkhäuser |location=Boston, MA |doi=10.1007/978-0-8176-4781-0 |isbn=978-0-8176-4781-0 |url=https://link.springer.com/book/10.1007/978-0-8176-4781-0}}
* {{cite book |last1=Schneider |first1=G. Michael |first2=Judith |last2=Gersting |first3=Bo |last3=Brinkman |author2-link=Judith Gersting |title=Invitation to Computer Science |location=Boston |publisher=Cengage Learning |year=2015 |isbn=978-1-305-07577-1|oclc=889643614 }}
* {{cite journal |last1=Segal |first1=Irving |author-link1=Irving Segal |title=Algebraic Integration Theory |journal=Bulletin of the American Mathematical Society |date=1965 |volume=71 |issue=3 |pages=419–489 |doi=10.1090/S0002-9904-1965-11284-8 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-71/issue-3.P1/Algebraic-integration-theory/bams/1183526903.full|doi-access=free }}
* {{cite book|last=Sheehan |first=Neil |author-link=Neil Sheehan |title=A Fiery Peace in a Cold War: Bernard Schriever and the Ultimate Weapon |publisher=Vintage |year=2010 |url=https://archive.org/details/fierypeaceincold00shee |isbn=978-0679745495}}
* {{cite book |last1=Szanton |first1=Andrew |author-link1=Andrew Szanton |title=The Recollections of Eugene P. Wigner: as told to Andrew Szanton |date=1992 |publisher=Springer |edition=1 |isbn=978-1-4899-6313-0 |doi=10.1007/978-1-4899-6313-0 |url=https://link.springer.com/book/10.1007/978-1-4899-6313-0}}
* {{cite book |editor-last1=Taub |editor-first1=A. H. |editor-link1=Abraham H. Taub |title=John von Neumann Collected Works Volume VI: Theory of Games, Astrophysics, Hydrodynamics and Meteorology |year=1963 |publisher=Pergamon Press |location=New York |isbn=978-0-08-009566-0 |oclc=493423386}}
* {{cite interview |last=Tucker |first=Albert |subject-link=Albert W. Tucker|interviewer=William Aspray |title=Interview Transcript #34 - Oral History Project |date=1984 |publisher=Princeton Mathematics Department |url=https://web.math.princeton.edu/oral-history/c32.pdf |location=Princeton |access-date=4 April 2022}}
* {{cite journal |last1=Ulam |first1=Stanisław |author1-link=Stanisław Ulam |title=John von Neumann 1903–1957 |journal=Bull. Amer. Math. Soc. |date=1958 |volume=64 |issue=3 |pages=1–49 |doi=10.1090/S0002-9904-1958-10189-5 |url=https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10189-5/S0002-9904-1958-10189-5.pdf}}
* {{cite book |last=Ulam |first=Stanisław |author-link=Stanislaw Ulam |title=Adventures of a Mathematician |location=New York |publisher=Charles Scribner's Sons |year=1976 |isbn=0-684-14391-7}}
* {{cite journal |last1=von Plato |first1=Jan |title=In search of the sources of incompleteness |journal=Proceedings of the International Congress of Mathematicians 2018 |date=2018 |volume=3 |pages=4075–4092 |doi=10.1142/9789813272880_0212 |isbn=978-981-327-287-3 |s2cid=203463751 |url=https://eta.impa.br/dl/209.pdf}}
* {{cite book |last1=von Plato |first1=Jan |title=Can Mathematics Be Proved Consistent? |series=Sources and Studies in the History of Mathematics and Physical Sciences |date=2020 |publisher=Springer International Publishing |isbn=978-3-030-50876-0 |doi=10.1007/978-3-030-50876-0 |s2cid=226522427 |url=https://www.springer.com/gp/book/9783030508753}}
* {{cite book |last1=Wagon |first1=Stan |last2=Tomkowicz |first2=Grzegorz |author1-link=Stan Wagon |title=The Banach–Tarski Paradox |date=2016 |publisher=Cambridge University Press |isbn=9781316572870 |edition=2 |url=https://www.cambridge.org/9781316572870}}
* {{cite book |last1=York |first1=Herbert |author1-link=Herbert York |title=Race to Oblivion: A Participant's View of the Arms Race |date=1971 |publisher=Simon and Schuster |location=New York |isbn=978-0671209315 |url=https://archive.org/details/racetooblivionpa0000york}}
{{refend}}

== Further reading ==
{{refbegin|30em}}
''' Books'''
* {{cite book |last1=Erickson |first1=Paul |title=The World the Game Theorists Made |date=2015 |publisher=University of Chicago Press |isbn=9780226097176 |ref=none}}
* {{cite book |last=Halmos |first=Paul R. |author-link=Paul Halmos |year=1985 |title=I Want To Be A Mathematician: an Automathography |publisher=Springer-Verlag |location=New York |doi=10.1007/978-1-4612-1084-9 |isbn=978-0-387-96078-4 |oclc=11497873 |ref=none}}
* {{cite book |last1=Hargittai |first1=Balazs |last2=Hargittai |first2=Istvan |title=Wisdom of the Martians of Science: In Their Own Words with Commentaries |date=2015 |publisher=World Scientific |doi=10.1142/9809 |isbn=978-9814723817 |ref=none}}
* {{cite book |last=Hargittai |first=Istvan |title=Martians of Science: Five Physicists Who Changed the Twentieth Century |date=2008 |publisher=Oxford University Press |isbn=978-0195365566 |doi=10.1093/acprof:oso/9780195178456.001.0001 |ref=none}}
* {{cite book |editor-last=Horvath |editor-first=Janos |editor-link=John Horvath (mathematician) |title=A Panorama of Hungarian Mathematics in the Twentieth Century I |series=Bolyai Society Mathematical Studies |publisher=Springer |year=2006 |volume=14 |doi=10.1007/978-3-540-30721-1 |isbn=978-3540289456 |ref=none}}
* {{cite book |last1=Krehl |first1=Peter O. K. |title=History of Shock Waves, Explosions and Impact: A Chronological and Biographical Reference |date=2009 |publisher=Springer |doi=10.1007/978-3-540-30421-0 |isbn=978-3-540-30421-0 |ref=none}}
* {{cite book |last=Leonard |first=Robert |title=Von Neumann, Morgenstern, and the Creation of Game Theory: From Chess to Social Science, 1900–1960 |publisher=Cambridge University Press |year=2010 |doi=10.1017/CBO9780511778278 |isbn=978-1107609266 |ref=none}}
* {{cite book|last=Poundstone |first=William |author-link=William Poundstone |title=Prisoner's Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb|year=1993|publisher=Random House Digital |isbn=978-0-385-41580-4 |ref=none}}
* {{cite book |last=Purrington |first=Robert D. |title=The Heroic Age: The Creation of Quantum Mechanics, 1925–1940 |publisher=Oxford University Press |year=2018 |isbn=978-0190655174 |ref=none}}
* {{cite book|last=Slater|first=Robert|author-link=Robert Slater|title=Portraits in Silicon |url=https://archive.org/details/portraitsinsilic00slat|url-access=registration|year=1989|publisher=MIT Press|location=Cambridge, Massachusetts |isbn=978-0-262-19262-0|oclc=15630421 |ref=none}}
* {{cite book |last=von Neumann Whitman |first=Marina |author-link=Marina von Neumann Whitman |year=2012 |title=The Martian's Daughter—A Memoir |publisher=University of Michigan Press |location=Anne Arbor |isbn=978-0-472-03564-9 |oclc=844308382 |url=https://muse.jhu.edu/book/17623 |url-access=subscription |ref=none}}
* {{cite book |last=Vonneuman |first=Nicholas A. |year=1987 |title=John von Neumann as Seen by His Brother |location=Meadowbrook, Pennsylvania |publisher=N.A. Vonneuman |isbn=978-0-9619681-0-6|oclc=17547196 |url=https://itf.njszt.hu/324rtr4/uploads/2019/07/neumann_02.pdf |ref=none}}
* {{cite book |editor1-last=Weintraub |editor1-first=E. Roy |editor-link1=E. Roy Weintraub |title=Towards a History of Game Theory |series=History of Political Economy |date=1992 |volume=24 (Supplement) |url=https://read.dukeupress.edu/hope/issue/24/Supplement  |url-access=subscription |issn=0018-2702 |ref=none}}

'''Popular periodicals'''
* {{cite magazine|magazine=[[Good Housekeeping|Good Housekeeping Magazine]]|date=September 1956|title=Married to a Man Who Believes the Mind Can Move the World|type=Interview with Klari von Neumann|pages=80–81, 282–292|last=Grafton|first=Samuel}}

'''Journals'''
* {{cite journal |last1=Formica |first1=Giambattista |title=John von Neumann's Discovery of the 2nd Incompleteness Theorem |journal=History and Philosophy of Logic |date=2022 |volume=44 |pages=66–90 |doi=10.1080/01445340.2022.2137324 |s2cid=256699234 |url=https://www.tandfonline.com/doi/full/10.1080/01445340.2022.2137324}}
* {{cite journal |last1=Smithies |first1=F. |author-link=Frank Smithies |title=John von Neumann |journal=Journal of the London Mathematical Society |date=1959 |volume=s1-34 |issue=3 |pages=373–384 |doi=10.1112/jlms/s1-34.3.373 |ref=none}}
* {{cite journal |last1=Carvajalino |first1=Juan|title=Unlocking the Mystery of the Origins of John von Neumann's Growth Model |journal=History of Political Economy |date=2021 |volume=53 |issue=4 |pages=595–631 |doi=10.1215/00182702-9308883 |url=https://read.dukeupress.edu/hope/article/53/4/595/173872/Unlocking-the-Mystery-of-the-Origins-of-John-von?searchresult=1}}
* {{cite journal |last1=Carvajalino |first1=Juan|title=Where Did John von Neumann's Mathematical Economics Come From? |journal=History of Political Economy |date=2022 |volume=54 |issue=5 |pages=823–858 |doi=10.1215/00182702-10005718 |url=https://read.dukeupress.edu/hope/article/54/5/823/313473/Where-Did-John-von-Neumann-s-Mathematical?searchresult=1}}
* {{cite journal |last1=Boldyrev|first1=Ivan|title=The Frame for the Not-Yet Existent: How American, European, and Soviet Scholars Jointly Shaped Modern Mathematical Economics |journal=History of Political Economy |date=2024 |volume=56 |issue=3 |pages=467–488 |doi=10.1215/00182702-11156216 |url=https://read.dukeupress.edu/hope/article/56/3/467/386155/The-Frame-for-the-Not-Yet-Existent-How-American}}
{{refend}}

== External links ==
{{sister project links|c=Category:John von Neumann|n=no|b=no|v=no|voy=no|m=no|mw=no|wikt=no|s=no|species=no}}
* [https://www.netlib.org/bibnet/authors/v/von-neumann-john.html A more or less complete bibliography of publications of John von Neumann] by Nelson H. F. Beebe
* {{MacTutor|id=Von_Neumann}}
* [https://scholar.google.com.au/citations?user=6kEXBa0AAAAJ von Neumann's profile] at [[Google Scholar]]
* [https://www.math.princeton.edu/about/oral-history Oral History Project] - The Princeton Mathematics Community in the 1930s, contains many interviews that describe contact and anecdotes of von Neumann and others at the Princeton University and Institute for Advanced Study community.
* Oral history interviews (from the [[Charles Babbage Institute]], [[University of Minnesota]]) with: [http://purl.umn.edu/107206 Alice R. Burks and Arthur W. Burks]; [http://purl.umn.edu/107714 Eugene P. Wigner]; and [http://purl.umn.edu/107493 Nicholas C. Metropolis].
* [https://zbmath.org/authors/?q=Neumann%2C+John zbMATH profile]
* [https://albert.ias.edu/search?von%20neumann=&query=von%20neumann Query for "von neumann"] on the digital repository of the Institute for Advanced Study.
* [https://plato.stanford.edu/entries/qt-nvd/ Von Neumann vs. Dirac on Quantum Theory and Mathematical Rigor] – from ''[[Stanford Encyclopedia of Philosophy]]''
* [https://plato.stanford.edu/entries/qt-quantlog/ Quantum Logic and Probability Theory] - from ''Stanford Encyclopedia of Philosophy''
* [https://documents2.theblackvault.com/documents/fbifiles/scientists/johnvonneumann-fbi1.pdf FBI files on John von Neumann released via FOI]
* [https://www.youtube.com/watch?v=Ml3-kVYLNr8 Biographical video] by David Brailsford (John Dunford Professor Emeritus of computer science at the University of Nottingham)
* [https://www.youtube.com/watch?v=RuwnywSpl40 John von Neumann: Prophet of the 21st Century] 2013 [[Arte]] documentary on John von Neumann and his influence in the modern world (in German and French with English subtitles).
* [https://www.youtube.com/watch?v=vQp70uqsBV4 John von Neumann - A Documentary] 1966 detailed documentary by the [[Mathematical Association of America]] containing remarks by several of his colleagues including Ulam, Wigner, Halmos, Morgenstern, Bethe, Goldstine, Strauss and Teller.

{{Manhattan Project}}
{{Set theory}}
{{Game theory}}
{{Timelines of computing}}
{{AMS Presidents|state=collapsed}}
{{Authority control}}

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