Examine individual changes

'{{short description|Hungarian-American mathematician and physicist (1903–1957)}} {{Good article}} {{Use mdy dates|date=March 2015}} {{Hungarian name|Neumann János Lajos}} {{Infobox officeholder | name = John von Neumann | image = JohnvonNeumann-LosAlamos.gif | image_size = | caption = von Neumann in the 1940s | office = Member of the [[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 (1867–1918)|Kingdom of Hungary]], [[Austria-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]] | 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]]{{sfn|Israel|Gasca|2009|p=14}}|[[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."{{sfn|Halperin|1990|p=16}}}}|[[Friederich Ignaz Mautner|Friederich Mautner]]<ref name=genealogy>{{MathGenealogy |id=53213}}. Retrieved March 17, 2015.</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|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 = {{ubl| [[Abelian von Neumann algebra]]| [[Affiliated operator]]| [[Almost periodic function#Almost periodic functions on a locally compact abelian group|Almost periodic functions on a locally compact abelian group]]| [[Projections onto convex sets|Alternating projection method]]| [[Amenable group]]| [[Arithmetic logic unit]]| [[History of artificial life|Artificial life]]| [[viscosity|Artificial viscosity]]| [[Axiom of regularity]]| [[Axiom of limitation of size]]| [[Backward induction]]| [[Birkhoff algorithm|Birkhoff-von-Neumann algorithm]]| [[Doubly stochastic matrix#Birkhoff–von Neumann theorem|Birkhoff–von Neumann theorem]]| [[Blast wave]] (fluid dynamics)| [[Taylor–von Neumann–Sedov blast wave]]| [[Bounded set (topological vector space)]]| [[Calkin correspondence]]| [[Carry-save adder]]| [[Cellular automaton|Cellular automata]]| [[Class (set theory)]]| [[Closed-subgroup theorem#Proof|Closed-subgroup theorem for matrix groups]]| [[Computer virus]]| [[Commutation theorem for traces]]| [[Continuous geometry]]| [[Coupling constant]]| [[Cumulative hierarchy]]| [[Quantum decoherence|Decoherence theory (quantum mechanics)]]}} {{collapsible list|title=+93 more|[[Density matrix]]| [[Dirac–von Neumann axioms]]| [[Direct integral]]| [[Doubly stochastic matrix]]| [[Duality (optimization)|Duality Theorem]]| [[Durbin–Watson statistic]]| [[EDVAC]]| [[Enveloping von Neumann algebra]]| [[Ergodic theory]]| [[nuclear weapon design|Explosive lenses]]| [[Flowchart]]| [[Finite von Neumann algebra]]| [[Game theory]]| [[Haar measure#A construction using mean values of functions|Haar measure]]| [[Hilbert's fifth problem]]| [[Hyperfinite type II factor]]| [[IAS machine]]| [[Inner model]]| [[Inner model theory]]| [[Interior point method]]| [[John von Neumann (sculpture)]]| [[Koopman–von Neumann classical mechanics]]| [[Lattice (order)|Lattice theory]]| [[Lifting theory]]| [[Mathematical Foundations of Quantum Mechanics]]| [[Merge sort]]| [[Middle-square method]]| [[Minimax theorem]]| [[Monte Carlo method]]| [[Mutual assured destruction]]| [[Noncommutative harmonic analysis]]| [[Normal-form game]]| [[Operation Greenhouse]]| [[Operator theory]]| [[Operator topologies]]| [[Pointless topology]]| [[Polarization identity]]| [[Project Rover]]| [[Pseudorandomness]]| [[Pseudorandom number generator]]| [[Quantum logic]]| [[Quantum mutual information]]| [[Quantum statistical mechanics]]| [[Radiation implosion]]| [[Rank ring]]| [[Schatten norm|Schatten–von-Neumann norm]]| [[Self-replication]]| [[Semantic neural network]]| [[Hardware random number generator#Software whitening|Software whitening]]| [[Sorted array]]| [[Spectral set]]| [[Spectral theory]]| [[Standard probability space]]| [[Stochastic computing]]| [[Stone–von Neumann theorem]]| [[Subfactor]]| [[Teapot Committee]]| [[Technological singularity]]| [[The Computer and the Brain]]| [[Theory of Games and Economic Behavior]]| [[Ultrastrong topology]]| [[Von Neumann algebra]]| [[Von Neumann architecture]]| [[Von Neumann bicommutant theorem]]| [[Bornological space#Bornology of a topological vector space|von Neumann bornology]]| [[Von Neumann cardinal assignment]]| [[Von Neumann cellular automaton]]| [[Von Neumann–Wigner interpretation]]| [[Von Neumann measurement scheme]]| [[Von Neumann ordinals]]| [[Von Neumann universal constructor]]| [[Von Neumann entropy]]| [[Density matrix#The Von Neumann equation for time evolution|Von Neumann equation]]| [[Von Neumann neighborhood]]| [[Von Neumann paradox]]| [[Self-replicating spacecraft#Von Neumann probes|Von Neumann probes]]| [[Von Neumann programming languages]]| [[Von Neumann regular ring]]| [[Von Neumann–Bernays–Gödel set theory]]| [[Von Neumann universe]]| [[Spectral theorem#Spectral theorem|Von Neumann spectral theorem]]| [[Von Neumann conjecture]]| [[Von Neumann's inequality]]| [[Von Neumann's theorem]]| [[Trace inequality#Von Neumann's trace inequality and related results|Von Neumann's trace inequality]]| [[Von Neumann stability analysis]]| [[Randomness extractor#Von Neumann extractor|Von Neumann extractor]]| [[Ergodic theory#Mean ergodic theorem|Von Neumann ergodic theorem]]| [[Von Neumann–Morgenstern utility theorem]]| [[Weak topology|Weak topology on Hilbert spaces]]| [[Weyl–von Neumann theorem]]| [[Wold–von Neumann decomposition]]| [[ZND detonation model]]}} | author_abbrev_bot = | author_abbrev_zoo = | influences = | influenced = {{ubl|[[Maurice Pryce]]|[[John Williams Calkin|John W. Calkin]]|[[Jule Gregory Charney|Jule Charney]]|[[Herman Goldstine|Herman H. Goldstine]]|[[Oskar Morgenstern]]|[[Francis Joseph Murray|Francis J. Murray]]|[[Robert Schatten]]|[[Irving Segal|Irving E. Segal]]|[[Abraham H. Taub]]|[[Stanislaw Ulam]]{{sfn|Halmos|1973|p=388}}|[[John G. Kemeny]]{{sfn|Albers|Alexanderson|2008|p=169}}|[[Hugh Everett III]]{{sfn|Byrne|2010|pp=56,73}}}} | 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 = [[file:Neumann de Margitta Arms 1913.png|100px|Coat of Arms<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>]] }} | resting_place = [[Princeton Cemetery]] }} '''John von Neumann''' ({{IPAc-en|v|ɒ|n|_|ˈ|n|ɔɪ|m|ən}} {{respell|von|_|NOY|mən}}; {{lang-hu|Neumann János Lajos}} {{IPA-hu|ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ|}}; December 28, 1903&nbsp;– February&nbsp;8, 1957) was a Hungarian-American [[mathematician]], [[physicist]], [[computer scientist]], [[engineer]] and [[polymath]]. He was regarded as having perhaps the widest coverage of any mathematician of his time{{sfn|Rédei|1999|p=7}} and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics".{{sfn|Dieudonné|2008|p=90}}{{sfn|Burke|2020|pp=157}} He integrated [[Basic research|pure]] and [[Applied science#Applied research|applied sciences]]. Von Neumann made major contributions to many fields, including [[mathematics]] ([[mathematical logic]], [[Measure (mathematics)|measure theory]], [[functional analysis]], [[ergodic theory]], [[group theory]], [[Lattice (order)|lattice theory]], [[representation theory]], [[operator algebra]]s, [[Matrix (mathematics)|matrix theory]], [[geometry]], and [[numerical analysis]]), [[physics]] ([[quantum mechanics]], [[Fluid dynamics|hydrodynamics]] & [[ballistics]], [[nuclear physics]] and [[quantum statistical mechanics]]), [[economics]] ([[game theory]] and [[general equilibrium theory]]), [[computing]] ([[Von Neumann architecture]], [[linear programming]], [[Numerical weather prediction|numerical meteorology]], [[scientific computing]], [[Von Neumann universal constructor|self-replicating machines]], [[stochastic computing]]), and [[statistics]]. He was a pioneer of the application of [[operator theory]] to quantum mechanics in the development of functional analysis, and a key figure in the development of game theory and the concepts of [[Cellular automaton|cellular automata]], the [[Von Neumann universal constructor|universal constructor]] and the [[Computer|digital computer]]. Von Neumann published over 150 papers: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on special mathematical subjects or non-mathematical subjects.{{sfn|Doran|Kadison|2004|p=8}} His last work, an unfinished manuscript written while he was dying, was later published in book form as ''[[The Computer and the Brain]]''. His analysis of the structure of [[self-replication]] preceded the discovery of the structure of [[DNA]]. In a shortlist of facts about his life he submitted to the [[National Academy of Sciences]], he wrote, "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."<ref>{{cite journal | last = Zund | first = Joseph D. | doi = 10.1006/hmat.2001.2338 | issue = 2 | journal = Historia Mathematica | mr = 1896971 | pages = 138–156 | title = George David Birkhoff and John von Neumann: a question of priority and the ergodic theorems, 1931–1932 | volume = 29 | year = 2002| doi-access = free }} See page 151.</ref> During [[World War II]], von Neumann worked on the [[Manhattan Project]] with theoretical physicist [[Edward Teller]], mathematician [[Stanislaw Ulam]] and others, problem-solving key steps in the [[nuclear physics]] involved in [[Thermonuclear fusion|thermonuclear]] reactions and the [[Thermonuclear weapon|hydrogen bomb]]. He developed the mathematical models behind the [[explosive lens]]es used in the [[implosion-type nuclear weapon]] and coined the term "kiloton" (of [[TNT equivalent|TNT]]) as a measure of the explosive force generated.{{sfn|Macrae|1992}} During this time and after the war, he consulted for a vast number of 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 was the chair for a number of critical [[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 contributing to the design and development of the United States' first [[Intercontinental ballistic missile|ICBM]] programs.{{sfn|Sheehan|2010}} During this time he was considered the nation's foremost expert on [[nuclear weapon]]ry and the leading defense scientist at [[the Pentagon]]. As a Hungarian émigré, concerned that the Soviets would achieve nuclear superiority, he designed and promoted the policy of [[mutually assured destruction]] to limit the arms race.{{sfn|Jacobsen|2015|pp=40,53}} In honor of his achievements and contributions to the modern world, he was named in 1999 the ''[[Financial Times]]'' [[Financial Times Person of the Year|Person of the Century]], as a representative of the century's characteristic ideal that the power of the mind could shape the physical world, and of the "intellectual brilliance and human savagery" that defined the 20th century.<ref>{{cite web |author1=Andrew Hill |title=Person of the Year: Past winners |url=https://www.ft.com/content/b7744df2-9f76-11e5-beba-5e33e2b79e46 |website=[[Financial Times]] |access-date=23 April 2022 |archive-url=https://web.archive.org/web/20190918031543/https://www.ft.com/content/b7744df2-9f76-11e5-beba-5e33e2b79e46 |archive-date=18 September 2019 |date=14 December 2015}}</ref><ref>{{cite web |author1=Peter Martin |title=VON NEUMANN: Architect of the computer age |url=https://www.math.uwaterloo.ca/~hwolkowi//neumann1.html |website=[[Financial Times]] |access-date=23 April 2022 |archive-url=https://web.archive.org/web/20150919175755/https://www.math.uwaterloo.ca/~hwolkowi//neumann1.html |archive-date=19 September 2015 |date=24 December 1999}}</ref><ref>{{cite web |title=Every FT Person of the Year since 1970 |url=https://ig.ft.com/sites/person-of-the-year/ |website=[[Financial Times]] |access-date=23 April 2022 |archive-url=https://web.archive.org/web/20211212160517/https://ig.ft.com/sites/person-of-the-year/ |archive-date=12 December 2021 |date=12 December 2017}}</ref> == Life and education == === Family background === Von Neumann was born in [[Budapest]], [[Kingdom of Hungary]] (which was then part of the [[Austria-Hungary|Austro-Hungarian Empire]]){{sfn|Doran|Kadison|2004|p=1}}<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, acculturated and non-observant [[Jewish]] family. His Hungarian 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 (English: Michael von Neumann; 1907–1989) and Miklós (Nicholas von Neumann, 1911–2011).{{sfn|Dyson|1998|p=xxi}} His father, Neumann Miksa (Max von Neumann, 1873–1928) was a banker, who held a [[doctor of law|doctorate in law]]. He had moved to Budapest from [[Pécs]] at the end of the 1880s.{{sfn|Macrae|1992|pp=38–42}} Miksa's father and grandfather were both born in Ond (now part of the town of [[Szerencs]]), [[Zemplén County]], northern Hungary. John's mother was Kann Margit (English: Margaret Kann);{{sfn|Macrae|1992|pp=37–38}} her parents were Jakab Kann and Katalin Meisels 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"/> 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]]. When he was six years old, he could divide two eight-digit numbers in his head{{sfn|Henderson|2007|p=30}}{{sfn|Schneider|Gersting|Brinkman|2015|p=28}} and could converse in [[Ancient Greek]]. When the six-year-old von Neumann caught his mother staring aimlessly, he asked her, "What are you calculating?"{{sfn|Mitchell|2009|p=124}} When they were young, von Neumann, 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 the age of eight, von Neumann was familiar with [[differential calculus|differential]] and [[integral calculus]], and by twelve he had read and understood [[Émile Borel|Borel's]] Théorie des Fonctions.{{sfn|Halmos|1973|p=383}} But he was also particularly interested in history. He read his way through [[Wilhelm Oncken]]'s 46-volume world history series {{lang|de|Allgemeine Geschichte in Einzeldarstellungen}} (''General History in Monographs'').{{sfn|Blair|1957|p=90}} A copy was contained in a private library Max purchased. One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor.{{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 Lutheran School and soon became his friend.{{sfn |Macrae |1992 |pp=70–71}} This was one of the best schools in Budapest and was part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one [[Gymnasium (school)|gymnasium]]. The [[Education in Hungary|Hungarian school]] system produced a generation noted for intellectual achievement, many of which were Jews or of Jewish descent, which included [[Theodore von Kármán]] (born 1881), [[George de Hevesy]] (born 1885), [[Michael Polanyi]] (born 1891), [[Leó Szilárd]] (born 1898), [[Dennis Gabor]] (born 1900), [[Eugene Wigner]] (born 1902), [[Edward Teller]] (born 1908), and [[Paul Erdős]] (born 1913).{{sfn|Doran|Kadison|2004|p=3}} Collectively, they were sometimes known as "[[The Martians (scientists)|The Martians]]".{{sfn|Macrae|1992|pp=32–33}} Although von Neumann's father insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give von Neumann advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst [[Gábor Szegő]].{{sfn|Macrae|1992|pp=70–71}} On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears.{{sfn|Lax|1990|p=5}} Some of von Neumann's instant solutions to the problems that Szegő posed in calculus are sketched out on his father's stationery and are still on display at the von Neumann archive in Budapest.{{sfn|Macrae|1992|pp=70–71}} As for his other subjects, he received a grade of A for all barring B's in geometrical drawing, writing and music, and a C for physical education.{{sfn|Pais|2000|p=186}} By the age of 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.{{sfn|Nasar|2001|p=81}} At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize 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 thereby invest his time in a more financially useful endeavor than mathematics. In fact, his father asked von Kármán to persuade his son not to take mathematics as his major.<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 to become a [[chemical engineer]]. 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 the prestigious [[ETH Zurich]],{{sfn|Macrae|1992|pp=85–87}} which he passed in September 1923.{{sfn|Macrae|1992|p=97}} At the same time, von Neumann also entered [[Eötvös Loránd University|Pázmány Péter University]] in Budapest,<ref name="NYT">{{cite web|author-link=Ed Regis (author)|last=Regis |first=Ed|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=February 4, 2008}}</ref> as a [[Doctor of Philosophy|Ph.D.]] candidate in [[mathematics]]. For his thesis, he chose to produce 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}} He graduated as a [[chemical engineer]] from ETH Zurich in 1926 (although Wigner says that von Neumann was never very attached to the subject of [[chemistry]]), and passed his final examinations with ''[[summa cum laude]]'' for his Ph.D. in mathematics (with minors in [[experimental physics]] and chemistry) simultaneously with his chemical engineering degree, of which Wigner wrote, "Evidently a Ph.D. thesis and examination did not constitute an appreciable effort."{{sfn|Wigner|2001|p=128}}{{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]], in his obituary of [[Emmy Noether]], remembers how in the winter of 1926-1927 von Neumann, Noether and himself would take walks after his classes through "the cold, wet, rain-wet streets of Göttingen" where they discussed [[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 ever elected ''Privatdozent'' in the university's history in any subject.{{sfn|Dimand|Dimand|2002|p=129}} By the end of 1927, von Neumann had published 12 major papers in mathematics, and by the end of 1929, 32, a rate of nearly one major 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}} but in October of that year a better offer presented itself when he was invited to [[Princeton University]] as a visiting lecturer in [[mathematical physics]].{{sfn|Macrae|1992|pp=155–157}} On New Year's Day 1930, von Neumann married Marietta Kövesi, who had studied economics at Budapest University.{{sfn|Macrae|1992|pp=155–157}} Von Neumann and Marietta had one child, a daughter, [[Marina von Neumann Whitman|Marina]], born in 1935. As of 2021 Marina is a distinguished professor emerita of business administration and public policy at the [[University of Michigan]].<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|Klara Dan]], whom he had met during his last trips back to Budapest before the outbreak of [[World War II]].{{sfn|Heims|1980|p=178}}{{sfn|Macrae|1992|pp=170–174}} In 1930, before marrying Marietta, von Neumann was baptized into the [[Catholic Church]].{{sfn|Bochner|1958|p=446}} Von Neumann's father, Max, had died in 1929. None of the family had [[Conversion to Christianity|converted to Christianity]] while Max was alive, but all did afterward.{{sfn |Macrae |1992 |pp=43, 157}} In 1933 Von Neumann was offered and accepted a life tenure 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 first name to John, keeping the German-aristocratic surname [[von]] Neumann. His brothers changed theirs to "Neumann" and "Vonneumann".{{sfn|Macrae|1992|pp=57–58}} Von Neumann became a [[Naturalization#United States|naturalized citizen]] of the United States in 1937, and immediately tried to become a [[lieutenant]] in the United States Army's [[Officers Reserve Corps]]. He passed the exams easily but was rejected because of his age.{{sfn|Macrae|1992|pp=190–195}} His prewar analysis of how France would stand up to Germany is often quoted: "Oh, France won't matter."{{sfn|Ulam|1976|p=70}} Klara and John von Neumann were socially active within the local academic community.{{sfn|Macrae|1992|pp=170–171}} His white [[clapboard]] house at 26 Westcott Road was one of Princeton's largest private residences.{{sfn|Regis|1987|p=103}} He always wore formal suits. He once wore a three-piece pinstripe while riding down the [[Grand Canyon]] astride a mule.<ref name="whitman"/> Hilbert is reported to have asked, "Pray, who is the candidate's tailor?" at von Neumann's 1926 doctoral exam, as he had never seen such beautiful evening clothes.<ref>{{cite news |url=https://www.nytimes.com/2012/05/06/books/review/turings-cathedral-by-george-dyson.html |title=Unleashing the Power |date=May 4, 2012 |newspaper=[[The New York Times]] |last=Poundstone |first=William |author-link=William Poundstone }}</ref> Von Neumann held a lifelong passion for ancient history and was renowned for his historical knowledge. A professor of [[Byzantine history]] at Princeton once said that von Neumann had greater expertise in Byzantine history than he did.{{sfn|Blair|1957|p=93}} He knew by heart much of the material in [[Edward Gibbon|Gibbon's]] [[The History of the Decline and Fall of the Roman Empire|Decline and Fall]] and after dinner liked to engage in various historical discussions. Ulam noted that one time while driving south to a meeting of the [[American Mathematical Society]], von Neumann would describe even the minutest details of the battles of the [[American Civil War|Civil War]] that occurred in the places they drove by.{{sfn|Ulam|1958|p=6}} This kind of travel where he could be in a car and talk for hours on topics ranging from mathematics to literature without interruption was something he enjoyed very much.{{sfn|Ulam|1976|pp=100-101}} Von Neumann liked to eat and drink. His wife, Klara, said that he could count everything except calories. He enjoyed [[Jewish humour|Yiddish]] and [[Off-color humor|"off-color"]] humor (especially [[Limerick (poetry)|limericks]]).{{sfn|Halmos|1973|p=383}} He was a non-smoker.{{sfn|Macrae|1992|p=150}} In Princeton, he received complaints for regularly playing extremely loud German [[March (music)|march music]] on his [[phonograph]], which distracted those in neighboring offices, including [[Albert Einstein]], from their work.{{sfn|Macrae|1992|p=48}} Von Neumann did some of his best work in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with his wife's phonograph playing loudly.{{sfn|Blair|1957|p=94}} Despite being a notoriously bad driver, he enjoyed driving—frequently while reading a book—occasioning numerous arrests as well as accidents. When [[Cuthbert Hurd]] hired him as a consultant to [[IBM]], Hurd often quietly paid the fines for his traffic tickets.<ref name="cch">{{cite web|title=An Interview with Cuthbert C. Hurd|author=Stern, Nancy |publisher= [[Charles Babbage Institute]], University of Minnesota|date= January 20, 1981|url=https://conservancy.umn.edu/bitstream/handle/11299/107368/oh076cch.pdf?sequence=1&isAllowed=y|access-date=June 3, 2010 }}</ref> Von Neumann's closest friend in the United States was mathematician [[Stanislaw Ulam]]. A later friend of Ulam's, [[Gian-Carlo Rota]], wrote, "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in the hospital, every time Ulam visited, he came prepared with a new collection of jokes to cheer him up.{{sfn|Rota|1989|pp=26–27}} 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}} In February 1951 for [[the New York Times]] he had his brain waves scanned while at rest and while thinking (along with [[Albert Einstein]] and [[Norbert Wiener]]). "They generally showed differences from the average" was the conclusion.{{sfn|Pais|1994|p=228}} === Illness and death === [[File:John von neumann tomb 2004.jpg|thumb|right|Von Neumann's gravestone]] In 1955, von Neumann was diagnosed with what was either [[Bone tumor|bone]], [[pancreatic cancer|pancreatic]] or [[prostate cancer]]<ref>While there is a general agreement that the initially discovered bone tumour was a secondary growth, sources differ as to the location of the primary cancer. While Macrae gives it as pancreatic, the ''Life'' magazine article says it was prostate. 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> after he was examined by physicians following a fall, where they discovered a mass growing near his collarbone. The cancer was possibly caused by his radiation exposure during his time in [[Los Alamos National Laboratory]].{{sfn|Jacobsen|2015|p=62}} He was not able to accept the proximity of his own demise, and the shadow of impending death instilled great fear in him.<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> He invited a Catholic priest, Father Anselm Strittmatter, [[Order of Saint Benedict|O.S.B.]], to visit him for consultation.{{sfn|Halmos|1973|p=394}}{{sfn|Jacobsen|2015|p=63}}<ref>At the time of Von Neumann's death, Fr. Strittmatter was on leave of absence from his Washington, D.C. monastery, [[Saint Anselm's Abbey (Washington, D.C.)|Saint Anselm's Priory, which became Saint Anselm's Abbey in 1961]].</ref> 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 had earlier 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>{{harvnb|Ayoub|2004|p=170}} "On the other hand, von Neumann, giving in to Pascal's wager on his death bed, received extreme unction."</ref> Father Strittmatter administered the [[last rites]] to him.{{sfn|Halmos|1973|p=383}} Some of von Neumann's friends, such as [[Abraham Pais]] and [[Oskar Morgenstern]], said they had always believed him to be "completely agnostic".<ref name=Thornes2003>{{harvnb|Dransfield|Dransfield|2003|p=124}} "He was brought up in a Hungary in which anti-Semitism was commonplace, but the family were not overly religious, and for most of his adult years von Neumann held agnostic beliefs."</ref><ref>{{harvnb|Pais|2006|p=109}} "He had been completely agnostic for as long as I had known him. As far as I could see this act did not agree with the attitudes and thoughts he had harbored for nearly all his life."</ref> Of this deathbed conversion, Morgenstern told Heims, "He was of course completely agnostic all his life, and then he suddenly turned Catholic—it doesn't agree with anything whatsoever in his attitude, outlook and thinking when he was healthy." Father Strittmatter recalled that even after his conversion, von Neumann did not receive much peace or comfort from it, as he still remained terrified of death.{{sfn|Poundstone|1993|p=194}} On his deathbed he entertained his brother by reciting by heart and word-for-word the first few lines of each page of [[Goethe's Faust|Goethe's ''Faust'']]. For example, it is recorded that one day his brother Mike read ''Faust'' to him, and when Mike paused to turn the pages, Von Neumann recited from memory the first few lines of the following page.{{sfn|Blair|1957|p=104}} On his deathbed, his mental capabilities became a fraction of what they were before, causing him much anguish. At times he even forgot the lines that his brother recited from ''Faust''.{{sfn|Jacobsen|2015|p=63}} Meanwhile, [[Clay Blair]] remarked that von Neumann did not give up research until his death: "It was characteristic of the impatient, witty and incalculably brilliant John von Neumann that although he went on working for others until he could do no more, his own treatise on the workings of the brain—the work he thought would be his crowning achievement in his own name—was left unfinished."{{sfn|Blair|1957|p=104}} He died on February 8, 1957, at the [[Walter Reed Army Medical Center]] in [[Washington, D.C.]], under military security lest he reveal military secrets while heavily medicated. He was buried at [[Princeton Cemetery]] of Nassau Presbyterian Church in Princeton, [[Mercer County, New Jersey]].{{sfn|Macrae|1992|p=380}}<ref>{{Cite web|url=https://nassauchurch.org/about/princetoncemetery/|title = Nassau Presbyterian Church}}</ref> Ulam reflected on his death in his autobiography, originally intended to be a book on von Neumann, saying that he died so prematurely, "seeing the promised land but hardly entering it". His published work on [[automata]] and the [[brain]] contained only the barest sketches of what he planned to think about, and although he had a great fascination with them, many of the significant discoveries and advancements in [[molecular biology]] and [[computing]] were made only after he died before he could make any further contributions to them. On his deathbed he was still unsure of whether he had done enough important work in his life. Although he never lived to see it, he had also accepted an appointment as professor at large at the [[University of California, Los Angeles]] should he have recovered from his cancer.{{sfn|Ulam|1976|pp=242-243}} == Mathematics == === Set theory === {{See also|Von Neumann–Bernays–Gödel set theory}} [[File:NBG Evolution svg.svg|thumb|300px|History of approaches that led to NBG set theory]] The axiomatization of mathematics, on the model of [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the [[Peano axioms|axiom schema]] of [[Richard Dedekind]] and [[Charles Sanders Peirce]], and in geometry, thanks to [[Hilbert's axioms]].{{sfn|Van Heijenoort|1967|pp=393–394}} But 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 doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the ''[[axiom of regularity|axiom of foundation]]'' and the notion of ''[[Class (set theory)|class]].''{{sfn|Van Heijenoort|1967|pp=393–394}} 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 principles of Zermelo and Fraenkel. 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 a method of demonstration called the ''method of [[inner model]]s'', which became an essential instrument in set theory.{{sfn|Van Heijenoort|1967|pp=393–394}} 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.{{sfn|Van Heijenoort|1967|pp=393–394}} 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 work of [[Felix Hausdorff]], in 1924 [[Stefan Banach]] and [[Alfred Tarski]] proved that given a solid [[ball (mathematics)|ball]] in 3‑dimensional space, [[existence theorem|there exists]] a decomposition of the ball into a finite number of [[Disjoint sets|disjoint]] [[subset]]s that can be reassembled together in a different way to yield two identical copies of the original ball. Banach and Tarski proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But in a 1929 paper,<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 | journal=[[Fundamenta Mathematicae]] | volume=13 | pages=73–116 | year=1929 | doi=10.4064/fm-13-1-73-116 | doi-access=free }}</ref> von Neumann proved that paradoxical decompositions could use a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations contains such subgroups, and this opens the possibility of performing paradoxical decompositions using these subgroups. The class of groups von Neumann isolated in his work on Banach–Tarski decompositions was very important in many areas of mathematics, including von Neumann's own later work in [[measure theory]] (see below). === Proof theory === {{See also|Hilbert's program}} With the aforementioned 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. By 1925 he was involving himself in discussions with others in Göttingen on whether [[elementary arithmetic]] followed from [[Peano axioms]].{{sfn|van der Waerden|1975|p=34}} Building on the work of [[Wilhelm Ackermann|Ackermann]], von Neumann began attempting to prove (using the [[Finitism|finistic]] methods of [[Hilbert's program|Hilbert's school]]) the consistency of [[Peano axioms#First-order theory of arithmetic|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 historic [[Second Conference on the Epistemology of the Exact Sciences]] of [[Königsberg]], in which [[Kurt Gödel]] announced his [[Gödel's incompleteness theorems|first theorem of incompleteness]]: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.{{sfn|Rédei|2005|p=123}} Von Neumann, who participated in the conference, pulled Gödel for conversation afterwards and the pair had a discussion on the topic, where von Neumann suggested that Gödel 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: that the usual axiomatic systems are unable to demonstrate their own consistency.{{sfn|Rédei|2005|p=123}} Gödel replied, saying 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.{{sfn|Dawson|1997|p=70}}{{sfn|von Plato|2018|pp=4083-4086}} Von Neumann acknowledged Gödel's priority in his next letter.{{sfn|Rédei|2005|p=124}} He never thought much of "the American system of claiming personal priority for everything."{{sfn|Macrae|1992|p=182}} 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 (effectively ending it) than Gödel thought it did.{{sfn|von Plato|2020|p=22}}{{sfn|Sieg|2012|p=149}} 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}} However, Gödel's promised exposition of the second incompleteness theorem never appeared.{{sfn|von Plato|2018|p=4088}}{{sfn|von Plato|2020|pp=24-28}} === 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>Two of the papers are: * {{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}}. * {{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}}. {{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}}</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>\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.{{sfn|Mackey|1990|p=27}} Later in the year he published another long and influential paper that began the systematic study of ergodicity. In this paper 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.{{sfn|Mackey|1990|pp=28-30}}{{sfn|Ornstein|1990|p=39}} === Measure theory === {{See also|Lifting theory}} In [[Measure (mathematics)|measure theory]], the "problem of measure" for an {{mvar|n}}-dimensional [[Euclidean space]] {{math|'''R'''^''n''}} may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of {{math|'''R'''^''n''}}?"{{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, however 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".<ref name="Dieudonne21">{{cite web |last1=Dieudonné |first1=Jean |title=Von Neumann, Johann (or John) |url=https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/von-neumann-johann-or-john |website=Encyclopedia.com |publisher=Complete Dictionary of Scientific Biography |access-date=7 August 2021}}</ref> 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.<ref name="Dieudonne21" /> 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}}<ref name="Dieudonne21" /> === Functional analysis === {{Main|Operator theory}}{{See also|Spectral theorem}} Von Neumann was the first person to axiomatically define an abstract [[Hilbert space]] whereas it was previously defined as the [[Lp 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 defined several other abstract inequalities such as the [[Cauchy–Schwarz inequality]] that were previously only defined for [[Euclidean spaces]].{{sfn|Pietsch|2007|p=11}} He continued with the development of the [[spectral theory]] of operators in Hilbert space in 3 seminal papers between 1929 and 1932.{{sfn|Dieudonné|1981|p=172}} This work cumulated in his [[Mathematical Foundations of Quantum Mechanics]] which among 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]], and 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.<ref name="Dieudonne21" /> 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.{{sfn|Steen|1973|p=370}} 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]],{{sfn|Pietsch|2014|p=54}}{{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. In addition 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 earlier work on spectral theory and providing a new way of working through the geometric content of the spectral theory by the use of direct integrals of Hilbert spaces.{{sfn|Steen|1973|p=373}} Like in his work on measure theory he proved several theorems that he did not find time to publish. [[Nachman Aronszajn]] and K. T. Smith were told by him 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).{{sfn|Horn|Johnson|2013|p=335}}{{sfn|Horn|Johnson|1991|p=139}}<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}}{{sfn|Murray|1990|pp=57-59}} 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]].{{sfn|Petz|Rédei|1995|pp=163–181}} 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}} The nearly 500 pages that the papers span 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}}{{sfn|Kadison|1990|pp=70,86}} In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors yet 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}}{{sfn|Kadison|1990|pp=65,71,74}} Another important result on [[polar decomposition]] was published in 1932.{{sfn|Pietsch|2007|p=148}} His work here lead on to the next major topic. === 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. [[Garrett Birkhoff]] described his work, "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 modern work in projective geometry.{{sfn|Lashkhi|1995|p=1044}} His biggest contribution was founding the field of [[continuous geometry]].<ref>*{{Citation | last1=Neumann | first1=John von| author1-link=John von Neumann | title=Examples of continuous geometries | jstor=86391 | doi=10.1073/pnas.22.2.101 | jfm=62.0648.03 | year=1936b | journal=Proc. Natl. Acad. Sci. USA | volume=22 | issue=2 | pages=101–108 | pmid=16588050 | pmc=1076713| bibcode=1936PNAS...22..101N | doi-access=free}} *{{Citation | last1=Neumann | first1=John von| 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}} *{{Citation | last1=Neumann | first1=John von| 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}} *{{Citation | last1=Neumann | first1=John von| 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): "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." 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]] is a deep and important result that stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques.{{sfn|Lashkhi|1995|pp=1045-1046}}{{sfn|Wehrung|2006|p=1}} The theorem as described by Birkhoff: "[I]n the general case, von Neumann proved the following basic representation theorem. 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}} This work required the creation of [[Von Neumann regular ring|regular rings]].{{sfn|Goodearl|1979|p=ix}} 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.{{sfn|Goodearl|1981}} 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 and afterwards became busy with war work and his interests moved to computers. He finished his article by saying, "One wonders what would have been the effect on lattice theory, if von Neumann's intense two-year preoccupation with lattice theory had continued for twenty years!"{{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|author=von Neumann, 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|author=Durbin, J.|author2= Watson, G. S.|year=1950|title=Testing for Serial Correlation in Least Squares Regression, I|journal=[[Biometrika]]|volume=37|pages=409–428|pmid=14801065|issue=3–4|doi=10.2307/2332391|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 if the errors on a regression model follow a Gaussian [[random walk]] (''i.e.'', possess a [[unit root]]) against the alternative that they are a stationary first order autoregression.<ref>{{cite journal|author=Sargan, J.D. |author2=Bhargava, Alok |year=1983|jstor=1912252 |title=Testing residuals from least squares regression for being generated by the Gaussian random walk|journal=[[Econometrica]]|volume=51|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.{{sfn|Rédei|1959|p=226}} 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>{{sfn|Carbone|Volcic|2011|p=84}}{{sfn|Niederreiter|1975|p=243}} 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}}{{sfn|Narkiewicz|2004|p=120}}{{sfn|Narkiewicz|2018|p=144}}{{sfn|van Dantzig|1936|pp=282-283}}{{sfn|Warner|1993|p=428}} In 1928 he a couple 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}}{{sfn|Harzheim|2008|p=83}} 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} \tfrac{2^{2^{[nr]}}}{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>{{sfn|Kuiper|Popken|1962|p=385}}{{sfn|Mycielski|1964|p=139}}{{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>{{sfn|Miranda|1997|pp=667-668}}{{sfn|Gilbarg|Trudinger|2001|p=316}}{{sfn|Ladyzhenskaya|Ural'tseva|1968|pp=14, 243}} 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>{{sfn|Koksma|1936|p=15}}{{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>{{sfn|McCrimmon|2004|p=68}} A couple years later in 1936 he wrote another paper by himself in an attempt to further the program of replacing the axioms of his previous Hilbert space program with those of Jordan algebras.{{sfn|Rédei|1996|p=494}} In this paper he investigated the infinite-dimensional case and planned to write at least one further paper on the topic however this paper never came to fruition.<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]].{{sfn|Landsman|2009|p=6-7}}<ref>{{cite web |last1=Kronz |first1=Fred |last2=Lupher |first2=Tracy |title=Quantum Theory and Mathematical Rigor |url=https://plato.stanford.edu/entries/qt-nvd/ |website=Stanford Encyclopedia of Philosophy |publisher=Stanford University |access-date=21 December 2022}}</ref> ==Physics== === Quantum mechanics === {{See also|von Neumann entropy|Quantum mutual information|Measurement in quantum mechanics#von Neumann measurement scheme|label 4 = von Neumann measurement scheme|Wave function collapse}} {{Quantum mechanics}} Von Neumann was the first to establish a rigorous mathematical framework for [[quantum mechanics]], known as the [[Dirac–von Neumann axioms]], in his widely 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}} When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a [[self-adjoint operator]] and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?"{{sfn|Macrae|1992|p=142}} Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 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.<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> In any case, [[Gleason's theorem]] of 1957 fills the gaps in von Neumann's approach. Von Neumann's proof inaugurated a line of research that ultimately led, through [[Bell's theorem]] and the experiments of [[Alain Aspect]] in 1982, to the demonstration that quantum physics either requires a ''notion of reality'' substantially different from that of classical physics, or must include [[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. Although this view 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> The Von Neumann–Wigner interpretation has been summarized as follows:<ref name=Schreiber/> <blockquote> The rules of quantum mechanics are correct but there is only one system which may be treated with quantum mechanics, namely the entire material world. There exist external observers which cannot be treated within quantum mechanics, namely human (and perhaps animal) ''minds'', which perform measurements on the brain causing wave function collapse.<ref name=Schreiber>{{Cite arXiv |eprint=quant-ph/9501014 |last1=Schreiber |first1=Zvi |author-link=Zvi Schreiber |title=The Nine Lives of Schroedinger's Cat |year=1995}}</ref> </blockquote> Though theories of quantum mechanics continue to evolve, there is a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches that can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about [[Interpretations of quantum mechanics|interpretation of the theory]], and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.<ref name=VanHove/> Viewing von Neumann's work on quantum mechanics as a part of the fulfilment of [[Hilbert's sixth problem]], noted mathematical physicist [[Arthur Wightman|A. S. Wightman]] said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. In the publication of 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.{{sfn|Wightman|1976|pp=157-158}} 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 (operator theory, von Neumann algebras, continuous geometries, etc.), he did not find a satisfactory mathematical framework for quantum theory as a whole (including quantum field theory).{{sfn|Kac|Rota|Schwartz|2008|p=168}}{{sfn|Rédei|2005|pp=21,151-152,194}} ==== Von Neumann entropy ==== {{Main|Von Neumann entropy}} {{Statistical mechanics}} [[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 Univ. Press |location=Cambridge [u.a.]|isbn=978-0-521-63503-5 |pages=700 |edition=Repr.}}</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 [http://www.mi.ras.ru/~holevo/eindex.html Holevo entropy] and [[conditional quantum entropy]]. ==== Quantum mutual information ==== Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy. The von Neumann entropy is the cornerstone in the development of quantum information theory, while the [[Entropy (information theory)|Shannon entropy]] applies to classical information theory. This is considered a historical anomaly, as Shannon entropy might have been expected to be discovered before von Neumann entropy, given the latter's more widespread application to quantum information theory. But von Neumann discovered von Neumann entropy first, and applied it to questions of statistical physics. Decades later, Shannon developed an information-theoretic formula for use in classical information theory, and asked von Neumann what to call it. Von Neumann said to call it Shannon entropy, as it was a special case of von Neumann entropy.<ref>Quantum Information Theory, By [[Mark Wilde|Mark M. Wilde]], (Cambridge University Press 2013), page 252</ref> ==== Density matrix ==== {{main|Density matrix}} The formalism of [[density matrix|density operators and matrices]] was introduced by von Neumann<ref>{{Citation | 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=German}}</ref> in 1927 and independently, but less systematically by [[Lev Landau]]<ref name="PT">{{Citation| title=Density functional theory | author=Schlüter, Michael and [[Lu Jeu 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 | author=Ugo Fano | author-link=Ugo Fano | journal=[[:it:Rendiconti Lincei|Rendiconti Lincei (it)]] |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 is an alternative way to represent the state of a quantum system, which could otherwise be represented using the wavefunction. The density matrix allows the solution of certain time-dependent problems in quantum mechanics. ==== 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|last=Giulini, Domenico.|title=Decoherence and the Appearance of a Classical World in Quantum Theory|date=1996|publisher=Springer Berlin Heidelberg|others=Joos, Erich., Kiefer, Claus., Kupsch, Joachim., Stamatescu, Ion-Olimpiu., [[H. Dieter Zeh|Zeh, H. Dieter]].|isbn=978-3-662-03263-3|location=Berlin, Heidelberg|oclc=851393174}}</ref><ref>{{Citation|last=Bacciagaluppi|first=Guido|title=The Role of Decoherence in Quantum Mechanics|date=2020|url=https://plato.stanford.edu/archives/sum2020/entries/qm-decoherence/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2020|publisher=Metaphysics Research Lab, Stanford University|access-date=2020-07-25}}</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 famous paper of 1936 by von Neumann and Garrett Birkhoff, the first work ever 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)=(P\lor Q)\land(P\lor R)</math> and <math>P\land (Q\lor R)=(P\land Q)\lor(P\land R)</math>, are not valid for quantum theory.<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. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (spin angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., ''x'' and ''y'') results in a pair of incompatible quantities. Suppose that the state '''ɸ''' of a certain electron verifies the proposition "the spin of the electron in the ''x'' direction is positive." By the principle of indeterminacy, the value of the spin in the direction ''y'' will be completely indeterminate for '''ɸ'''. Hence, '''ɸ''' can verify neither the proposition "the spin in the direction of ''y'' is positive" nor the proposition "the spin in the direction of ''y'' is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of ''y'' is positive or the spin in the direction of ''y'' is negative" must be true for '''ɸ'''. In the case of distribution, it is therefore possible to have a situation in which ''<math>A \land (B\lor C)= A\land 1 = A</math>'', while <math>(A\land B)\lor (A\land C)=0\lor 0=0</math>.<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> As [[Hilary Putnam]] writes, "The logic von Neumann proposes is based on [[orthomodular lattice]]s (the ones of physical interest are isomorphic to the lattice of subspaces of the Hilbert space of the system being talked about)."<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. He just could not make himself write something he did not fully understand to his satisfaction. 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]]. Von Neumann's contributions to fluid dynamics included his discovery of the classic flow solution to [[blast wave]]s,{{sfn|von Neumann|1963a|pp=219–237}} and the co-discovery (independently of [[Yakov Borisovich Zel'dovich]] and [[Werner Döring]]) of the [[ZND detonation model]] of explosives.{{sfn|von Neumann|1963b|pp=205–218}} During the 1930s, von Neumann became an authority on the mathematics of [[shaped charges]].<ref>''Ballistics: Theory and Design of Guns and Ammunition'', Second Edition By Donald E. Carlucci, Sidney S. Jacobson, (CRC Press, 26 Aug 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 tried to put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of ''artificial viscosity'' smoothed the shock transition without sacrificing basic physics.<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 WW2, he arrived one day at the office of R.H. Kent, the Director of the US Army's [[Ballistic Research Laboratory]], with a computer program he had created for calculating a one-dimensional model of 100 molecules to simulate a shock wave. Von Neumann then gave a seminar on his computer program to an audience which included his friend [[Theodore von Kármán]]. After von Neumann had finished, von Kármán said "Well, Johnny, that's very interesting. Of course you realize [[Joseph-Louis Lagrange|Lagrange]] also used digital models to simulate [[continuum mechanics]]." It was evident from von Neumann's face, that he had been unaware of Lagrange's [[Mécanique analytique]].{{sfn|Metropolis|Howlett|Rota|1980|p=24}} === 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 to it. 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''.{{sfn|Binney|1996|pp=84-87}} 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 |pmid=28179843|pmc=5255893 |arxiv=1110.4423 |bibcode=2013LRR....16....4B }}</ref> and used the [[Holtsmark distribution]] to model {{sfn|Uchaikin|Zolotarev|1999|pp=xviii, 281, 424}} 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 Chandresekhar's other works.{{sfn|Taub|1963|pp=172-176}}{{sfn|Bonolis|2017|pp=41, 51}} In some 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.{{sfn|Segal|1990|pp=162-163}} From this work he wrote a paper with [[Abraham H. Taub|A. H. Taub]] and Veblen extending the [[Dirac equation]] to [[Projective geometry|projective]] [[General relativity|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}} Additionally in the same time period he made several proposals to colleagues for dealing with the problems in the newly created [[quantum field theory|quantum theory of fields]] and for [[Quantization (physics)|quantizing]] [[spacetime]], however both his colleagues and he himself did not consider the ideas fruitful and he did not work on them further.{{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 these ideas as he had in as late as 1940 written 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 his maximum losses. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.<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, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). 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]]. Morgenstern wrote a paper on game theory and thought he would show it to von Neumann because of his interest in the subject. He read it and said to Morgenstern that he should put more in it. This was repeated a couple of times, and then von Neumann became a coauthor and the paper became 100 pages long. Then it became a book.{{Citation needed|date=October 2022}} 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|work=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"/> Independently, [[Leonid Kantorovich]]'s functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and [[Riesz space|vector lattices]]. 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 the primary tools of mathematical economics ever since.{{sfn|Blume|2008}} === Mathematical economics === Von Neumann raised the intellectual and [[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>{{sfn|Morgenstern|Thompson|1976|pages=xviii, 277}} 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 with interests in computational economics.{{sfn|Rockafellar|1970|pp=i, 74}}{{sfn|Rockafellar|1974|pp=351–378}}{{sfn|Ye|1997|pp=277–299}} 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]]. These can all have been said to have been unquestionably influenced by von Neumann's work.{{sfn|Dore|Chakravarty|Goodwin|1989|p=234}} Von Neumann's famous 9-page paper started life as a talk at Princeton and then became a paper in German that was eventually translated into English. His interest in economics that led to that paper began while he was lecturing at Berlin in 1928 and 1929. He spent his summers back home in Budapest, as did the economist [[Nicholas Kaldor]], and they hit it off. Kaldor recommended that von Neumann read a book by the mathematical economist [[Léon Walras]]. Von Neumann found some faults in the book and corrected them–for example, replacing equations by inequalities. He 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 the 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}} 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.{{sfn|Knuth|1998|p=159}}<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> Von Neumann wrote the 23-page-long sorting program for the [[EDVAC]] in ink. On the first page, traces of the phrase "TOP SECRET", which was written in pencil and later erased, can still be seen.<ref name="Papers J. Neumann Computing Computer Theory. 1987, pp. 89-95"/> He also worked on the philosophy of [[artificial intelligence]] with [[Alan Turing]] when the latter visited Princeton in the 1930s.{{sfn|Macrae|1992|pp=183–184}} Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and Stanisław Ulam developed simulations on von Neumann's digital computers for the hydrodynamic computations. During this time he contributed to the development of the [[Monte Carlo method]], which allowed solutions to complicated problems to be approximated using [[Algorithmically random sequence|random numbers]].{{sfn|Macrae|1992|pp=334–335}} [[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'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="digits"> {{cite journal | last= von Neumann | first= John | title= Various techniques used in connection with random digits | journal= National Bureau of Standards Applied Math Series | year= 1951 | volume= 12 | page= 36 | url= https://babel.hathitrust.org/cgi/pt?id=osu.32435030295547&view=image&seq=48 }} </ref> Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making [[Pseudorandomness|pseudorandom numbers]], using the [[middle-square method]]. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, 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> Von Neumann also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect.<ref name="nbsams"/> While consulting for the [[Moore School of Electrical Engineering]] at the [[University of Pennsylvania]] on the EDVAC project, von Neumann wrote an incomplete ''[[First Draft of a Report on the EDVAC]]''. The paper, whose premature distribution nullified the patent claims of EDVAC designers [[J. Presper Eckert]] and [[John Mauchly]], described a [[computer architecture]] in which the data and the program are both stored in the computer's memory in the same address space. This architecture is the basis of most modern computer designs, unlike the earliest computers that were "programmed" using a separate memory device such as a [[paper tape]] or [[plugboard]]. Although the single-memory, stored program architecture is commonly called [[von Neumann architecture]] as a result of von Neumann's paper, the architecture was based on the work of Eckert and Mauchly, inventors of the [[ENIAC]] computer at the University of Pennsylvania.<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> 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> The electronics of the new ENIAC ran at one-sixth the speed, but this in no way degraded the ENIAC's performance, since it was still entirely [[I/O bound]]. Complicated programs could be developed and [[Debugging|debugged]] in days rather than the weeks required for plugboarding the old ENIAC. Some of von Neumann's early computer programs have been preserved.<ref> {{cite book | last= Knuth | first= Donald E. | author-link= Donald Knuth | title= Selected papers on computer science (Center for the Study of Language and Information – Lecture Notes) | publisher= CSLI Publications Cambridge University Press | location= Stanford, Calif. Cambridge, Mass. | year= 1996 | isbn= 978-1-881526-91-9 }} </ref> The next computer that von Neumann designed was the [[IAS machine]] at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the [[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}} [[Stochastic computing]] was first introduced in a pioneering paper by von Neumann in 1953.{{sfn|von Neumann|1995|pp=567-616}} However, the theory could not be implemented until advances in computing of the 1960s.<ref> {{cite conference | last1= Petrovic | first1= R. | last2= Siljak | first2= D. | title= Multiplication by means of coincidence | year= 1962 | book-title= ACTES Proc. of 3rd Int. Analog Comp. Meeting }} </ref><ref> {{citation | last= Afuso | first= C. | title= Quart. Tech. Prog. Rept | location= [[Department of Computer Science, University of Illinois at Urbana-Champaign]], Illinois | year= 1964 }} </ref> Around 1950 he was also among the first people to talk about the [[time complexity]] of [[computation]]s, which eventually evolved into the field of [[computational complexity theory]].{{sfn|Chaitin|2002|p=28}} [[Herman Goldstine]] once described how he felt that even in comparison to all his technical achievements in computer science, it was the fact that he was held in such high esteem, had such a reputation, that the digital computer was accepted so quickly and worked on by others.{{sfn|Chaitin|2002|p=6}} As an example, he talked about [[Thomas J. Watson Jr.|Tom Watson, Jr.'s]] meetings with von Neumann at the Institute for Advanced Study, whom he had come to see after having heard of von Neumann's work and wanting to know what was happening for himself personally. [[IBM]], which Watson Jr. later became CEO and president of, would play an enormous role in the forthcoming computer industry. The second example was that once von Neumann was elected Commissioner of the [[United States Atomic Energy Commission|Atomic Energy Commission]], he would exert great influence over the commission's laboratories to promote the use of computers and to spur competition between IBM and [[Sperry Corporation|Sperry-Rand]], which would result in the [[IBM 7030 Stretch|Stretch]] and [[UNIVAC LARC|LARC]] computers that lead to further developments in the field. Goldstine also notes how von Neumann's expository style when speaking about technical subjects, particularly to non-technical audiences, was very attractive.{{sfn|Goldstine|1990|p=14}} This view was held not just by him but by many other mathematicians and scientists of the time too.{{sfn|Kac|Rota|Schwartz|2008|p=206}} === 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>{{Citation|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 rigorous mathematical analysis of the structure of [[self-replication]] (of the semiotic relationship between constructor, description and that which is constructed), preceded the discovery of the structure of DNA.{{sfnb|Rocha|2015|pp=25–27}} Von Neumann created the field of [[cellular automaton|cellular automata]] without the aid of computers, constructing the first self-replicating automata with pencil and graph paper. The detailed proposal for a physical non-biological self-replicating system was first put forward in lectures von Neumann delivered in 1948 and 1949, when he first only 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> While qualitatively sound, von Neumann was evidently dissatisfied with this model of a self-replicator due to the difficulty of analyzing it with mathematical rigor. He went on to instead develop a more abstract model self-replicator based on his original concept of [[cellular automata]].<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> Subsequently, the concept of the [[Von Neumann universal constructor]] based on the [[von Neumann cellular automaton]] was fleshed out in his posthumously published lectures ''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 | type= PDF }} </ref> Ulam and von Neumann created a method for calculating liquid motion in the 1950s. The driving concept of the method was to consider a liquid as a group of discrete units and calculate the motion of each based on its neighbors' behaviors.<ref name=bialynick-8>{{Harvnb|Białynicki-Birula|Białynicki-Birula|2004|p=8}}</ref> Like Ulam's lattice network, [[Von Neumann cellular automata|von Neumann's cellular automata]] are two-dimensional, with his self-replicator implemented algorithmically. The result was a [[Von Neumann universal constructor|universal copier and constructor]] working within a cellular automaton with a small neighborhood (only those cells that touch are neighbors; for von Neumann's cellular automata, only [[orthogonal]] cells), and with 29 states per cell. Von Neumann gave an existence proof that a particular pattern would make infinite copies of itself within the given cellular universe by designing a 200,000 cell configuration that could do so. {{quote box | width = 390px | align = right | quote = [T]here exists a critical size below which the process of synthesis is degenerative, but above which the phenomenon of synthesis, if properly arranged, can become explosive, in other words, where syntheses of automata can proceed in such a manner that each automaton will produce other automata which are more complex and of higher potentialities than itself. | source = —von Neumann, 1948<ref name="TSRA"/> }} Von Neumann addressed the evolutionary growth of complexity amongst his self-replicating machines.<ref name=McMullin2000>{{Citation|journal=Artificial Life|last=McMullin|first=B.|year=2000|title=John von Neumann and the Evolutionary Growth of Complexity: Looking Backwards, Looking Forwards...|volume=6|issue=4|pages=347–361|url=http://www.eeng.dcu.ie/~alife/bmcm-alj-2000/|doi=10.1162/106454600300103674|pmid=11348586|s2cid=5454783}}</ref> His "proof-of-principle" designs showed how it is logically possible, by using a general purpose programmable ("universal") constructor, to exhibit an indefinitely large class of self-replicators, spanning a wide range of complexity, interconnected by a network of potential mutational pathways, including pathways from the most simple to the most complex. This is an important result, as prior to that it might have been conjectured that there is a fundamental logical barrier to the existence of such pathways; in which case, biological organisms, which do support such pathways, could not be "machines", as conventionally understood. Von Neumann considers the potential for conflict between his self-reproducing machines, stating that "our models lead to such conflict situations",<ref name="Autonomous Systems 1992 page 236">''Toward a Practice of Autonomous Systems: Proceedings of the First European Conference on Artificial Life'', Francisco J. Varela, Paul Bourgine, (MIT Press 1992), page 236</ref> indicating it as a field of further study.<ref name="TSRA"/>{{rp|147}} The [[cybernetics]] movement highlighted the question of what it takes for self-reproduction to occur autonomously, and in 1952, John von Neumann designed an elaborate 2D [[cellular automaton]] that would automatically make a copy of its initial configuration of cells. 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. Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire [[natural satellite|moon]] or [[asteroid belt]] would be by using [[self-replicating spacecraft]], taking advantage of their [[exponential growth]].<ref> {{cite journal | title= A Self-Reproducing Interstellar Probe | first= Robert A., Jr. | last= Freitas | author-link= Robert Freitas | journal= Journal of the British Interplanetary Society | volume= 33 | pages= 251–264 | year= 1980 | bibcode= 1980JBIS...33..251F | url= http://www.rfreitas.com/Astro/ReproJBISJuly1980.htm | access-date= January 9, 2015 }} </ref> Von Neumann investigated the question of whether modelling evolution on a digital computer could solve the complexity problem in programming.<ref name="Autonomous Systems 1992 page 236"/> Beginning in 1949, von Neumann's design for a self-reproducing computer program is considered the world's first [[computer virus]], and he is considered to be the theoretical father of computer virology.{{sfn|Filiol|2005|pp=19–38}} === Scientific computing and numerical analysis === {{Computational physics}} 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 on the technical side and on the administrative side. He was one of the key developers of the [[Von Neumann stability analysis|stability analysis]] procedure that now bears his name,{{sfn|Gustafsson|2018|pp=101-102}} a scheme used to ensure that when linear [[partial differential equation]] are solved [[Numerical analysis|numerically]], the errors at each time step of the calculation do not build up. This scheme is still the mostly commonly used technique for stability analysis today.{{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 only implicitly.{{sfn|Brezinski|Wuytack|2001|p=27}} He was also among the first researchers to write about the [[Jacobi method]].{{sfn|Brezinski|Wuytack|2001|p=216}} During his time at Los Alamos, he was the first to consider how to solve various problems of [[Compressible flow|gas dynamics]] numerically, writing several classified reports on the topic. However, he was frustrated by the lack of progress with [[Mathematical analysis|analytic methods]] towards solving these problems, many of which were [[Nonlinear partial differential equation|nonlinear]]. As a result, he turned towards computational methods in order to break the deadlock.{{sfn|Gustafsson|2018|pp=112-113}} While von Neumann only occasionally worked there as a consultant, under his influence Los Alamos became the undisputed leader in computational science during the 1950s and early 1960s.{{sfn|Lax|2005|p=223}} From his work at Los Alamos von Neumann realized that computation was not just a tool to [[Proof by exhaustion|brute force]] the solution to a problem numerically, but that computation could also provide insight for solving problems analytically too,{{sfn|Ulam|1986|p=224}} through heuristic hints, 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]].{{sfn|Hersh|2015|p=170}} 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, which would defeat the current stalemate there was when trying to solve them by [[Mathematical analysis|classical analysis]] methods.{{sfn|Metropolis|Howlett|Rota|1980|pp=24-25}} Titled "High-speed Computing Devices and Mathematical Analysis", he also described how [[wind tunnel]]s, which at the time were being constructed at heavy cost, were actually [[analog computer]]s, and how digital computers, which he was developing, would replace them and dawn a new era of fluid dynamics. He was given a very warm reception, with [[Garrett Birkhoff]] describing it as "an unforgettable sales pitch". Instead of publishing this talk in the proceedings of the congress, he expanded on it with Goldstine into the manuscript "On the Principles of Large Scale Computing Machines", which he would present to the [[United States Navy|US Navy]] and other audiences in the hopes of drumming up their support for scientific computing using digital computers. In his papers, many in conjunction with others, he 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.{{sfn|Birkhoff|1990|pp=64-69}} === Weather systems and global warming === {{See also|History of numerical weather prediction}} As part of his research into possible applications of computers, von Neumann became interested in weather prediction, noting the similarities between the problems in the field and previous problems he had worked on during the Manhattan Project, both of which involved nonlinear fluid dynamics.{{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]] along with 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 twenty strong group of meteorologists who began to work on various problems in the field. However, as other postwar work took up considerable portions of his time he was not able to devote enough of it to proper leadership of the project and little was done during this time period. However 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 |access-date=2006-12-12 |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 up 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: "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> 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 in order to ensure continued support from the Weather Bureau and the military, leading to the creation of the General Circulation Research Section (now known as the [[Geophysical Fluid Dynamics Laboratory]]) next to the JNWPU in Suitland, Maryland.{{sfn|Edward|2010|pp=152-153}} He continued work both on technical issues of modelling and in ensuring continuing funding for these projects, which, like many others, were enormously helped by von Neumann's unwavering support to legitimize them.{{sfn|Edwards|2010|pp=153,161,189-190}} 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}} thereby inducing [[global warming]].{{sfn|Macrae|1992|p=332}}{{sfn|Heims|1980|pp=236–247}} Von Neumann proposed a theory of global warming as a result of the activity of humans, noting that the Earth was only {{convert|6|F-change|C-change}} colder during the [[last glacial period]], he wrote in 1955: "[[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> However, von Neumann urged a degree of caution in any program of intentional human weather manufacturing: "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"/> 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]]. Although he died the next year, this continuous advocacy ensured that during the Cold War there would be continued interest and funding for research.{{sfn|Edwards|2010|pp=189-191}} {{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"/> }} ===Technological singularity hypothesis=== {{See also|Technological singularity}} 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 fleshed out later in the book ''[[Future Shock]]'' by [[Alvin Toffler]]. == Defense work == [[File:John von Neumann ID badge.png|thumb|upright=1.15|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, von Neumann was the leading authority of the mathematics of [[shaped charge]]s. This led him to a large number of military consultancies, primarily for the Navy, which in turn led to his involvement in the [[Manhattan Project]]. The involvement included frequent trips by train to the project's secret research facilities at the [[Los Alamos Laboratory]] in a remote part of 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]]. In addition, he helped determine which Japanese cities the atomic bombs would be used on, and was also the singular person responsible for the complex calculations needed to determine at which height the bombs would be detonated over Hiroshima and Nagasaki to achieve the maximum kill rate. He determined that height to be 1,800 feet.{{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, four other scientists, and various military personnel were 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 and thus maximum effect. The cultural capital [[Kyoto]], which had been spared the [[Air raids on Japan|bombing inflicted upon militarily significant cities]], 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]].{{sfn|Groves|1962|pp=268–276}} 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|bombing range]] near [[Alamogordo Army Airfield]], {{convert|35|mi}} southeast of [[Socorro, 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}} After the war, [[Robert Oppenheimer]] remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."{{sfn|Macrae|1992|p=245}} 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 on "Improvement in Methods and Means for Utilizing Nuclear Energy", which outlined a scheme for using a fission bomb to compress fusion fuel to initiate [[nuclear fusion]].{{sfn|Herken|2002|pages=171, 374}} 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 === In 1950, von Neumann became a consultant to the [[Weapons Systems Evaluation Group]] (WSEG),{{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]] (AFSWP), which was responsible for the military aspects on [[nuclear weapon]]s. Over the following two years, he became a consultant to 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}} among a host of other agencies. Beside the [[United States Coast Guard|Coast Guard]], there was not a single US military or intelligence organization which von Neumann did not advise.{{sfn|Sheehan|2010|p=182}} During this time he became ''the'' "superstar" defense scientist at [[the Pentagon]]. His authority was considered infallible at the highest levels including the [[United States Secretary of Defense|secretary of defense]] and Joint Chiefs of Staff.{{sfn|Jacobsen|2015|p=40}} This applied not just to US government agencies. Supposedly, he was hired as a consultant to the [[RAND Corporation]] with the equivalent salary for an average full time analyst, yet his job was only to write down his thoughts each morning while shaving.{{sfn|Jacobsen|2015|p=43}} 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, one not only powerful but light enough too to fit on top of a rocket. In 1953 [[Bernard Schriever]], who was present at the meeting with Teller and von Neumann, paid a personal visit to von Neumann at Princeton in order to confirm this possibility.{{sfn|Sheehan|2010|pp=178-179}} Schriever would then enlist [[Trevor Gardner]], who in turn would also personally visit von Neumann several weeks later in order 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 reports prepared for the [[United States Department of Defense|Department of Defense]] (DoD) in order to argue for the creation of [[Intercontinental ballistic missile|ICBMs]].{{sfn|Sheehan|2010|pp=217,219-220}} Gardner on several occasions would bring von Neumann to the Pentagon in order 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, which would later expand to include senior officials from all over the US government including those from the [[United States Department of State|State Department]] and [[United States National Security Council|National Security Council]] (NSC).{{sfn|Sheehan|2010|pp=273,276-278}} However, this was not enough. In order to have the ICBM program run at full throttle they needed direct action by the [[President of the United States|President]].{{sfn|Sheehan|2010|pp=275,278}} On July 28, 1955, Schriever, Gardner, and von Neumann had managed to arrange a direct meeting with [[Dwight D. Eisenhower|President Eisenhower]] at the [[White House]] in order to relay their concerns. While the other two would focus on the introduction and conclusion, von Neumann would present the technical meat of the argument. White House staff had told them all three presentations could take up a maximum of half an hour and could only include "straightforward and factual" information, with no attempts to "sell" to the President their specific needs. [[Dillon Anderson]], who was head of the NSC staff, was skeptical of the wide-ranging solutions that the trio posed as they could downgrade attention given to other defense projects. [[Thomas S. Power|General Tommy Power]], who was there with them that day, did not think there was enough time to get a subject of such importance across given the restrictions however the three thought they could compress their arguments enough to do so. At 10:00 AM their meeting was set to begin. They were to address not only President Eisenhower, but a whole host of the top civilian and military leaders of the country including [[Vice President of the United States|Vice President]] [[Richard Nixon]], [[Admiral (United States)|Admiral]] [[Arthur W. Radford|Arthur Radford]], [[chairman of the Joint Chiefs of Staff]], the secretaries of [[United States Secretary of State|State]], Defense and [[United States Secretary of the Treasury|Treasury]], and the [[Director of the Central Intelligence Agency|head of the CIA]] among others. The program officially belonged to Tommy Power as [[List of commanders-in-chief of the Strategic Air Command|Commander in Chief of the]] [[Strategic Air Command]] yet he was considered a lesser figure. Gardner began by describing the strategic consequences of ICBMs and briefly what the other two presenters would say. Von Neumann then began his speech, with no notes as he often did, speaking as the nation's preeminent scientist in matters of nuclear weaponry. He discussed technical matters, from the base [[nuclear engineering]] to the intricacies of [[missile guidance|missile targeting]]. Within these discussions, he once again mixed warnings that there were no known defenses against such weapons, and the fifteen minutes of warning that would be provided with the available [[Radar|radar system technology]] was all so little. One of the participants at the meeting, Vince Ford, was keeping track of the faces of all those listening to try to see if anyone was confused or lost. He saw no one, and thought that von Neumann had "knocked the ball out of the park." Now it was General Schriever's turn to speak. However, it was already 11:05 AM, and the meeting was supposed to finish five minutes before. Von Neumann had spoken for much longer than was originally planned however there was no restlessness or desire from anyone to leave; everyone was paying close attention to the speakers. Schriever spoke on how to realize the technology physically, in terms of manpower and what organizations were working on it, and the strategic plans for how to complete the project in the fastest way if it would be approved. He smartly attributed all the work being done to the recommendations of the earlier [[Teapot Committee]] that von Neumann chaired, and hence capitalized on the credibility of all the distinguished scientists that served on it too. The early restriction by Anderson also no longer mattered as much, as his proposed solutions were no longer his own but the solutions proposed in the final report of the Teapot Committee. Once Schriever had finished speaking on the cost of the project he wrapped up and Eisenhower thanked all three men for their presentations. Of them he said, "This has been most impressive, most impressive! There is no question this weapon will have a profound impact on all aspects of human life, not only in the United States but in every corner of the globe—military, sociological, political.” Immediately he asked Admiral Radford to find out what effect the long-range missiles would have on the [[force structure]] and report back to him. The others in attendance likewise thanked each of the men and left despite being more than an hour overtime. Nixon and the head of the CIA stayed and questioned why this had not been done earlier and what was the hold up. Later that day, the trio would once again repeat their briefings to the NSC Planning Board. The board would then physically write the directive for the President to sign. However the board was mostly made up of DoD staffers who did not believe in the project as strongly as Gardner or Schriever. Luckily, by now Vice President Nixon had been won over, and when he chaired a full NSC meeting that would decide the issue on September 8, he personally invited von Neumann to give another presentation. The result was NSC Action No. 1433, a presidential directive signed by Eisenhower 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 America did 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}} From the first time Schriever heard the presentation of von Neumann and Teller to the signing of the presidential directive the trio had moved heaven and earth in order to make the ICBM program a reality.{{sfn|Sheehan|2010|pp=391,408-409}} Evidence would later show that the Soviets indeed were already testing their own [[intermediate-range ballistic missile]]s at the time of the presentation to President Eisenhower at the White House.{{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}} He accepted this position and used it 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 compact weapons, and he argued against settling for the intermediate-range missiles that the Army wanted. He was adamant that H-bombs delivered into the heart of enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile wouldn't be a problem with an H-bomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case.{{sfn|Heims|1980|p=276}}{{sfn|Macrae|1992|pp=367–369}} Despite his disagreement with Oppenheimer over the need for a crash program to develop the hydrogen bomb, he testified on the latter's behalf at the 1954 [[Oppenheimer security hearing]], at which he asserted that Oppenheimer was loyal, and praised him for his helpfulness once the program went ahead.{{sfn|Halmos|1973|p=391}} 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 in time. 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}} === Mutual assured destruction === [[File:Redwing Mohawk 002.jpg|thumb|right|[[Operation Redwing]] nuclear test in July 1956]] Von Neumann is credited with developing the equilibrium strategy of [[mutual assured destruction]] (MAD). He also "moved heaven and earth" to bring MAD about. His goal was to quickly develop ICBMs and the compact hydrogen bombs that they could deliver to the USSR, and he knew the Soviets were doing similar work because the [[CIA]] interviewed German rocket scientists who were allowed to return to Germany, and von Neumann had planted a dozen technical people in the CIA. The Soviets considered that bombers would soon be vulnerable, and they shared von Neumann's view that an H-bomb in an ICBM was the [[ne plus ultra]] of weapons; they believed that whoever had superiority in these weapons would take over the world, without necessarily using them.{{sfn|Macrae|1992|pp=362–363}} He was afraid of a "missile gap" and took several more steps to achieve his goal of keeping up with the Soviets: *He modified the [[ENIAC]] by making it programmable and then wrote programs for it to do the H-bomb calculations (which furthered the feasibility of the Teller-Ulam design). *Under the aegis of the AEC he promoted the development of a compact H-bomb which could fit in an ICBM. *He personally interceded to speed up the production of lithium-6 and tritium needed for the compact bombs. *He caused several separate missile projects to be started, because he felt that competition combined with collaboration got the best results.{{sfn|Heims|1980|pp=258–260}} Von Neumann's assessment that the Soviets had a lead in missile technology, considered pessimistic at the time, was soon proven correct in the [[Sputnik crisis]].{{sfn|Macrae|1992|pp=362–364}} 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". He was quoted in 1950 remarking, "If you say why not bomb [the Soviets] tomorrow, I say, why not today? If you say today at five o'clock, I say why not one o'clock?"{{sfn|Blair|1957|p=96}}{{sfn|Pais|2006|p=109}} On February 15, 1956, von Neumann was presented with the [[Medal of Freedom (1945)|Medal of Freedom]] by President [[Dwight D. Eisenhower]]. His citation read: {{blockquote|Dr. von Neumann, in a series of scientific study projects of major national significance, has materially increased the scientific progress of this country in the armaments field. Through his work on various highly classified missions performed outside the continental limits of the United States in conjunction with critically important international programs, Dr. von Neumann has resolved some of the most difficult technical problems of national defense.<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>}} Even when dying of cancer, von Neumann continued his work while he still could. [[Lewis Strauss]], who at the time was chairman of the AEC and a close friend, described some of his final memories of von Neumann in his memoir. {{blockquote|Until the last, he continued to be a member of the Commission and chairman of an important advisory committee to the Defense Department. On one dramatic occasion, I was present at a meeting at Walter Reed Hospital, where, gathered around Johnny's bedside were the Secretary of Defense and his deputies, the Secretaries of the three Armed Services, and all the military Chiefs of Staff. The central figure was a young man who but a few years before had come to the United States as an immigrant.{{sfn|Strauss|1962|p=236}}}} === Consultancies === A list of consultancies given by various sources is as follows:{{sfn|Aspray|1990|pp=245-246}}{{sfn|Ulam|1958|p=42}}{{sfn|AIP Profile|2022}}{{sfn|Sheehan|2010}}{{sfn|Booß-Bavnbek|Høyrup|2003}}{{sfn|Steingart|2013}}{{sfn|Christensen|2017}} {{columns-list| * 1940-1957 Member, [[Ballistic Research Laboratory#Advisors and consultants|Scientific Advisory Committee]], Ballistics Research Labs, U.S. Army, Aberdeen Proving Ground, MD * 1940-1942 Chief Consultant, Ballistics, War Preparedness Committee, American Mathematical Society/[[Mathematical Association of America]] * 1941-1946 Consultant, [[National Defense Research Committee]]/[[Office of Scientific Research and Development]] * 1941-1955 Member, [[Bureau of Ordnance|Bureau of Ordnancy]], U.S. Navy, Washington, DC * 1942-19?? Consultant, Anti-Submarine Warfare Operations Research Group, U.S. Navy, [[MIT Radiation Laboratory]] * 1943-1955 Consultant, [[Los Alamos National Laboratory|Los Alamos Scientific Lab]], Los Alamos, NM * 1946-1947 Member, Special Committee on Applied Mathematics, [[American Mathematical Society]] * 1946-19?? Chairman, Advisory Committee to the [[Office of Naval Research]] Mathematics Branch, [[National Academy of Sciences]] * 1946-1952 Chairman, Committee on High-Speed Computing, [[National Academies of Sciences, Engineering, and Medicine#Division on Engineering and Physical Sciences (DEPS)|Physical Sciences Division]], National Research Council * 1947-1955 Consultant, [[Naval Ordnance Laboratory|Naval Ordnancy Laboratory]], Silver Spring, MD * 1948-1955 Consultant, [[RAND Corporation]] * 1949-1953 Member, [[Under Secretary of Defense for Research and Engineering#History|Research and Development Board]], Department of Defense, Washington, DC * 1949-1954 Consultant, [[Oak Ridge National Laboratory]], Oak Ridge, TN * 1950-1955 Member, [[Armed Forces Special Weapons Project]], Department of Defense, Washington, DC * 1950-1955 Consultant, [[Weapons Systems Evaluation Group]], Department of Defense, Washington, DC * 1951-1953 Member, Special Cryptologic Advisory Group, [[National Security Agency#World War II and its aftermath|Armed Forces Security Agency]], Washington, DC * 1951-1957 Member, [[United States Air Force Scientific Advisory Board|Scientific Advisory Board]], U.S. Air Force, Washington, DC * 1951-1955 Consultant, [[IBM|IBM Corporation]] * 1952-1954 Member, General Advisory Committee, [[United States Atomic Energy Commission|U.S. Atomic Energy Commission]], Washington, DC * 1952-1955 [[Central Intelligence Agency]], Washington, DC * 1952-195? Consultant, Committee on Research and Training in Applied Mathematics, [[National Academies of Sciences, Engineering, and Medicine#Program units|National Research Council]] * 1952-1955 [[Lawrence Berkeley National Laboratory#History|Radiation Laboratory]], University of California, Livermore, CA * 1953-1954 Chairman, [[Teapot Committee|Strategic Missiles Evaluation Committee]], U.S. Air Force * 1953-1955 [[Sandia National Laboratories|Sandia Corporation]], Albuquerque, NM * 1953-1955 [[TRW Inc.|Ramo-Wooldridge Corporation]], Inglewood, CA * 1953-1955 Member, [[National Security Agency]] Scientific Advisory Board, Washington, DC * 1953-1957 Chairman, Nuclear Weapons Panel, Scientific Advisory Board, [[United States Air Force|U.S. Air Force]], Washington, DC * 1953-1957 Member, Technical Advisory Panel on Atomic Energy, [[United States Department of Defense|Department of Defense]], Washington, DC * 1954-1957 Chairman, ICBM Scientific Advisory Committee, Department of Defense, Washington, DC * 1955 Ad Hoc Panel on University Computing Facilities, [[National Science Foundation]], Washington, DC * 1955-1957 Commissioner, U.S. Atomic Energy Commission, Washington, DC }} <!-- Anti-Submarine Warfare Operations Research Group (ASWORG) is not fully ascertained, von Neumann worked with several US navy research groups however they split and conjoined in several ways at various times and is it not clear of the distinctions between the various groups. In this case whether the ASWORG work was included under his Bureau of Ordnancy consult or whether it was separate isn't known, they worked in separate locations on similar topics. See (Booß-Bavnbek & Høyrup, 2003) and their reference https://link.springer.com/chapter/10.1007/978-94-017-3596-4_24 on page 332 for more details. I have limited the listed consultancies to those related to the military or academics, however von Neumann also consulted for several purely commercial companies such as Standard Oil that I have not added as they seemed to be more "per project" rather than on a yearly basis and thus did not fit the list format. --> While his appointment as full Atomic Energy Commissioner in late 1954 formally required 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}} == Personality == [[Gian-Carlo Rota]] wrote in his famously controversial book, ''Indiscrete Thoughts'', that von Neumann was a lonely man who had trouble relating to others except on a strictly formal level.{{sfn|Rota|1997|p=71}} [[Françoise Aron Ulam|Françoise Ulam]] described how she never saw von Neumann in anything but a formal suit and tie.{{sfn|Ulam|1990|p=9}} His daughter wrote in her memoirs that she believed her father was motivated by two key convictions, one, that every person had the responsibility to make full use of their intellectual capacity, and two, that there is a critical importance of an environment of political freedom in order to pursue the first conviction. She added that he "enjoyed the good life, liked to live well, and counted a number of celebrities among his friends and colleagues". He was also very concerned with his legacy, in two aspects, the first being the durability of his intellectual contributions to the world,{{sfn|von Neumann Whitman|1990|p=2}} and secondly the life of his daughter. His brother, Nicholas noted that John tended to take a statistical view of the world, and that characterized many of his views.{{sfn|von Neumann Whitman|1990|p=3}} His encyclopedic knowledge of history did not help him in this point of view, nor did his work in game theory. He often liked to discuss the future in world events and politics and compare them with events in the past, predicting in 1936 that war would break out in Europe and that the French army was weak and would not matter in any conflict.{{sfn|Ulam|1976|pp=69-70,80}} On the other hand, Stan Ulam described his warmth this way, "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". He delighted in gossip and dirty jokes. Conversations with friends on scientific topics could go on for hours without respite, never being a shortage of things to discuss, even when leaving von Neumann's specialty in mathematics.{{sfn|Ulam|1958|pp=4-6}} He would mix in casual jokes, anecdotes and observations of people into his conversations, which allowed him to release any tension or wariness if there were disagreements, especially on questions of politics.{{sfn|Szanton|1992|p=122}} Von Neumann was not a quiet person either; he enjoyed going to and hosting parties several times a week,{{sfn|Ulam|1976|pp=67,69-70}} [[Churchill Eisenhart]] recalls in an interview that von Neumann could attend parties until the early hours of the morning, then the next day right at 8:30 could be there on time and deliver clear, lucid lectures. Graduate students would try to copy von Neumann in his ways; however, they did not have any success.{{sfn|Eisenhart|1984|p=7}} He was also known for always being happy to provide others with scientific and mathematical advice,{{sfn|Halperin|1990|p=16}} even when the recipient did not later credit him, which he did on many occasions with mathematicians and scientists of all ability levels.{{sfn|Goldstine|1985|p=7}}{{sfn|DeGroot|1989|p=592}} Wigner wrote that he perhaps supervised more work (in a casual sense) than any other modern mathematician. Collected works of colleagues at Princeton are full of references to hints or results from casual conversations with him.{{sfn|Szanton|1992|p=227}}<ref>As an example, see the collected works of [[Shizuo Kakutani]] https://books.google.com/books?id=O-msNu5vcXEC&q=neumann</ref> However, he did not particularly like it when he felt others were challenging him and his brilliance, being a very competitive person.{{sfn|Goldstine|1985|p=10}}{{sfn|Tucker|1984|p=8}} A story went at the [[Aberdeen Proving Ground]] how a young scientist had pre-prepared a complicated expression with solutions for several cases. When von Neumann came to visit, he asked him to evaluate them, and for each case would give his already calculated answer just before Johnny did. By the time they came to the third case it was too much for Johnny and he was upset until the joker confessed.{{sfn|Halmos|1973|p=386}} Nevertheless, he would put in an effort to appear modest and did not like boasting or appearing in a self-effacing manner.{{sfn|Ulam|1976|p=207}} Towards the end of his life on one occasion his wife [[Klára Dán von Neumann|Klari]] chided him for his great self-confidence and pride in his intellectual achievements. He replied only to say that on the contrary he was full of admiration for the great wonders of nature compared to which all we do is puny and insignificant.{{sfn|Ulam|1976|p=244}} In addition to his speed in mathematics, he was also a quick speaker, with [[Banesh Hoffmann]] noting that it made it very difficult to take notes, even in [[shorthand]].{{sfn|Hoffmann|1984|p=4}} Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing strongly 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}} He also maintained his knowledge of languages he learnt in his youth, becoming somewhat of a linguist. He knew Hungarian, French, German and English fluently, and maintained at least a conversational level of Italian, Yiddish, Latin and Ancient Greek. His Spanish was less perfect, but once on a trip to Mexico he tried to create his own "neo-Castilian" mix of English and Spanish.{{sfn|Ulam|1958|pp=5-6}} Even from a young age he was somewhat emotionally distant, and some women felt that he was lacking curiosity in subjective and personal feelings. Despite this the person he was confided to most was his mother.{{sfn|Szanton|1992|pp=57,180}} Ulam felt he did not devote enough time to ordinary family affairs and that in some conversations with him Johnny was shy about such topics. The fact he was constantly working on all kinds of intellectual, academic and advisory affairs probably meant he could not be a very attentive husband. This may show in the fact his personal life was not so smooth compared to his working one. Friendship wise he felt most at ease with those of similar background, third or fourth generation wealthy Jews like himself, and was quite conscious of his position in society.{{sfn|Ulam|1976|pp=78-80}} As a child he was poor in athletics and thus did not make friends this way (but he did join in on class pranks).{{sfn|Szanton|1992|p=57}} In general, he did not disagree with people. If someone was inclined to think or do things in a certain way he would not try to contradict or dissuade them. His manner was just to go along, even when asked for advice. Ulam said he had an innocent little trick that he used where he would suggest to someone that something he [von Neumann] wanted done had in fact originated from that person in order to get them to do it. Nevertheless, he held firm on scientific matters he believed in.{{sfn|Ulam|1976|p=80}}{{sfn|Szanton|1992|p=130}} 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}} He seemed to admire [[General officer|generals]] and [[admiral]]s and more generally those who wielded power in society. Ulam suspected that he had a hidden admiration for people or organizations that could influence the thoughts and decision making of others. During committee meetings he was not a particularly strong debater and as a whole preferred to avoid controversy and yield to those more forceful in their approaches.{{sfn|Ulam|1976|pp=231-232}} When hospitalized at the end of his life Ulam told him on one occasion he was on the same floor as [[Dwight D. Eisenhower|president Dwight Eisenhower]] after the president suffered a heart attack, and von Neumann was greatly amused by this.{{sfn|Ulam|1976|p=243}} As a whole he was overwhelmingly, universally, curious. Compared to other mathematicians or scientists of the time he had a broader view of the world and more 'common sense' outside of academics. Mathematics and the sciences, history, literature, and politics were all major interests of his. In particular his knowledge of ancient history was encyclopedic and at the level of a professional historian.{{sfn|Szanton|1992|p=277}} One of the many things he enjoyed reading was the precise and wonderful way [[Ancient Greece|Greek]] historians such as [[Thucydides]] and [[Herodotus]] wrote, which he could of course read in the original language. Ulam suspected these 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}} === Mathematical style === [[Gian-Carlo Rota|Rota]], in describing von Neumann's relationship with his friend [[Stanislaw Ulam]], wrote that von Neumann had "deep-seated and recurring self-doubts".{{sfn|Rota|1997|p=71}} As an example on one occasion he said in the future he would be forgotten while [[Kurt Gödel|Gödel]] would be remembered with Pythagoras.{{sfn|Kelley|1989|p=478}} Ulam suggests that some of his self-doubts with regard for his own creativity may have come from the fact he had not himself 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. Johnny had a virtuosity in following complicated reasoning and had supreme insights, yet he perhaps felt he did have the gift for seemingly irrational proofs and theorems or intuitive insights that came from nowhere. 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 Johnny was not convinced of the importance of his work, and only when finding some ingenious technical trick or new approach that he took some pleasure from his work that satiated his concerns.{{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}} Ulam, in his obituary of von Neumann, described how he was adept in dimensional estimates and did algebraical or numerical computations in his head without the need for pencil and paper, often impressing physicists who needed the help of physical utensils. His impression of the way von Neumann thought was that he did not visualise things physically; instead, he thought abstractly, treating properties of objects as some logical consequence of an underlying fundamental physical assumption.{{sfn|Ulam|1958|p=38}} [[Albert W. Tucker|Albert Tucker]] described von Neumann's overall interest in things as problem oriented, not even that, but as he "would deal with the point that came up as a thing by itself."{{sfn|Tucker|1984|p=4}} [[Herman Goldstine]] compared his lectures to being on glass, smooth and lucid. You would sit down and listen to them and not even feel the need to write down notes because everything was so clear and obvious; however, once one would come home and try understand the subject, you would suddenly realize it was not so easy. By comparison, Goldstine thought his scientific articles were written in a much harsher manner, and with much less insight.{{sfn|Goldstine|1985|p=7}} Another person who attended his lectures, Albert Tucker, described his lecturing as "terribly quick" and said that people often had to ask von Neumann questions in order to slow him down so they could think through the ideas he was going through. Even if his presentation was clear they would still be thinking of the previous idea when von Neumann moved on to the next one. Von Neumann knew about this and was grateful for the assistance of his audience in telling him when he was going too quickly.{{sfn|Tucker|1984|p=4}} [[Paul Halmos|Halmos]] described his lectures as "dazzling", with his speech clear, rapid, precise and all encompassing. He would cover all approaches to the subject he was speaking on and relate them to each other. Like Goldstine, he also described how everything seemed "so easy and natural" in lectures and a puzzled feeling once one tried to think over it at home.{{sfn|Halmos|1973|pp=387-388}} His work habits were rather methodical. After waking up and having breakfast at the [[Nassau Club]], he would visit the Institute for Advanced Study and begin work for the day. He would continue working for the entirety of the day, including after going home at five. Even if he was entertaining guests or hosting a party, he could still spend some time in his work room working away, still following the conversation in the other room where guests were. Although he went to bed at a reasonable time, he would awaken late in the night two or three in the morning by which time his brain had thought through problems he had in the previous day and begin working again and writing things down. He placed great importance on writing down ideas he had in detail,{{sfn|Goldstine|1985|pp=8-9}}{{sfn|Ulam|1986|p=xv}} and if he had a new one he would sometimes drop what he was doing to write it down.{{sfn|Ulam|1976|p=78}} Goldstine also writes of many quirks of intuition von Neumann had. One such quirk was that one time von Neumann had asked to review an old paper he had not published because he believed there was an error there yet he could not find it. After Goldstine found it, he exclaimed, "Damn it, of course. There is some instinct that kept me from publishing that paper and it must have been a realization that I had a mistake somewhere in it, but I just never knew where it was." Another one was his ability to lecture off old material many years after he had originally given it. Goldstine's example was based on material von Neumann had written in German but was now lecturing on in English, with Goldstine noting that the lecture was almost word for word, symbol for symbol the same. A final example Goldstine writes about was that one time von Neumann had difficulty proving something related to the bounds of [[Eigenvalues and eigenvectors|eigenvalues]], and some time later Goldstine saw in a paper in the [[Mathematical Reviews|Math Reviews]] where someone had proved a related theorem and described the theorem to von Neumann, who was then able to come to the blackboard and write down a proof. Goldstine says that just knowing that a proof was possible allowed von Neumann to see how to write it down even when previously he had difficulty.{{sfn|Goldstine|1985|pp=9-10}}{{sfn|Albers|Alexanderson|2008|p=81}} Likewise when he had difficulties he would not labor on and struggle with them as soon as he found them; 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 working on tangents simply because he saw how to do so. It also meant that while he could crush any small obstacles in his path while solving a problem, if the difficulty was great from the very beginning, he would simply switch to another problem. He would not labor on them or try to find weak spots from which he could break through.{{sfn|Ulam|1976|p=78}} Von Neumann was asked to write an essay for the layman describing what mathematics is. He explained that mathematics straddles the world between the empirical and logical, arguing that geometry was originally empirical, but Euclid constructed a logical, deductive theory. However, he argued that there is always the danger of straying too far from the real world and becoming irrelevant sophistry.<ref>{{cite web |url=http://www-history.mcs.st-and.ac.uk/Extras/Von_Neumann_Part_1.html |title=Von Neumann: The Mathematician |publisher=MacTutor History of Mathematics Archive |access-date=December 16, 2016 }}</ref><ref>{{cite web |url=http://www-history.mcs.st-and.ac.uk/Extras/Von_Neumann_Part_2.html |title=Von Neumann: The Mathematician, Part 2 |publisher=MacTutor History of Mathematics Archive |access-date=December 16, 2016 }}</ref>{{sfn|von Neumann|1947|pp=180–196}} Although he was commonly described as an analyst, he once classified himself an algebraist,{{sfn|Halperin|1984|p=12}} 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"{{sfn|Segal|1990|p=156}} 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. Von Neumann always saw the bigger picture and the trees never concealed the forest for him.{{sfn|Halmos|1973|p=388}} Although powerful technically, his primary concern seemed to be more with the clear and viable formation of fundamental issues and questions of science rather than just the solution of mathematical puzzles.{{sfn|Segal|1990|p=154}} At times he could be ignorant of the standard mathematical literature. It would at times be easier for him to rederive basic information he needed rather than chase references. He did not 'write down' to a specific audience, but rather he wrote it exactly as he saw it. Although he did spend time preparing for lectures, often it was just before he was about to present them, and he rarely used notes, instead jotting down points of what he would discuss and how long he would spend on it.{{sfn|Halmos|1973|pp=387-388}} After [[World War II]] began, he increasingly became extremely busy with a multitude of both academic and military commitments. He already had a bad habit of not writing up talks or publishing results he found,{{sfn|Lax|1990|p=6}} which only worsened. Another factor was that he did not find it easy to discuss a topic formally in writing to others unless it was already mature in his mind. If it was, he could talk freely and without hesitation, but if it was not, he would, in his own words, "develop the worst traits of pedantism and inefficiency".{{sfn|Rédei|Stöltzner|2001|p=168}} == Recognition == === Cognitive abilities === 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}} and later Bethe wrote that "[von Neumann's] brain indicated a new species, an evolution beyond man".{{sfn|Macrae|1992|p=backcover}} [[Paul Halmos]] states that "von Neumann's speed was awe-inspiring."{{sfn|Halmos|1973|p=386}} [[Israel Halperin]] said: "Keeping up with him was&nbsp;... impossible. The feeling was you were on a tricycle chasing a racing car."<ref>Kaplan, Michael and Kaplan, Ellen (2006) ''Chances are–: adventures in probability''. Viking.</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> Teller also said "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."<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".{{sfn|Lax|1990|p=6}} [[Claude Shannon]] called him "the smartest person I’ve ever met", a common opinion.{{sfn|Soni|Goodman|2017|p=76}} 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]].{{sfn|Mirowski|2002|p=258}} [[Lothar Wolfgang Nordheim]] described von Neumann as the "fastest mind I ever met",{{sfn|Goldstine|1980|pp=171}} and [[Jacob Bronowski]] wrote "He was the cleverest man I ever knew, without exception. He was a genius."{{sfn|Bronowski|1974|p=433}} [[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."{{sfn|Petković|2009|p=157}} [[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> Nobel Laureate [[Eugene Wigner]] described him in this way: "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. You saw immediately the quickness and power of von Neumann's mind. He understood mathematical problems not only in their initial aspect, but in their full complexity. Swiftly, effortlessly, he delved deeply into the details of the most complex scientific problem. He retained it all. His mind seemed a perfect instrument, with gears machined to mesh accurately to one thousandth of an inch."{{sfn|Szanton|1992|p=58}} Halmos recounts a story told by [[Nicholas Metropolis]], concerning the speed of von Neumann's calculations, when somebody asked von Neumann to solve the famous fly puzzle:<ref>{{cite web|url=http://mathworld.wolfram.com/TwoTrainsPuzzle.html |title=Fly Puzzle (Two Trains Puzzle) |publisher=Mathworld.wolfram.com |date=February 15, 2014 |access-date=February 25, 2014}}</ref> {{Blockquote|Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 mph. At the same time a fly that travels at a steady 15 mph starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover? The slow way to find the answer is to calculate what distance the fly covers on the first, southbound, leg of the trip, then on the second, northbound, leg, then on the third, etc., etc., and, finally, to sum the [[Series (mathematics)|infinite series]] so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann, "All I did was sum the [[geometric series]]."{{sfn|Halmos|1973|pp=386-387}}}} Wigner told a similar story, only with a swallow instead of a fly, and says it was [[Max Born]] who posed the question to von Neumann in the 1920s.<ref>{{cite web|title=John von Neumann – A Documentary|url=https://www.youtube.com/watch?v=gn3U9G0gfu4&t=1006|publisher=The Mathematical Association of America|access-date=26 August 2022|pages=16m46s–19m04s|date=1966}}</ref> Similarly, when the first computers he was helping develop were completed, simple tests like "what is the lowest power of 2 that has the number 7 in the fourth position from the end?" were conducted to ensure their accuracy. For modern computers this would take only a fraction of a second but for the first computers Johnny would race against them in calculation, and win.{{sfn|Halmos|1973|p=386}} Accolades and anecdotes were not limited to those from the physical or mathematical sciences either, [[Neurophysiology|neurophysiologist]] [[Leon Harmon]], described him in a similar manner, "Von Neumann was a true genius, the only one I've ever known. I've met [[Albert Einstein|Einstein]] and [[J. Robert Oppenheimer|Oppenheimer]] and [[Edward Teller|Teller]] and—who's [[Norbert Wiener|the mad genius from MIT]]? I don't mean [[Warren Sturgis McCulloch|McCulloch]], but a mathematician. Anyway, a whole bunch of those other guys. Von Neumann was the only genius I ever met. The others were supersmart .... And great prima donnas. But von Neumann's mind was all-encompassing. He could solve problems in any domain. ... And his mind was always working, always restless."{{sfn|McCorduck|2004|p=81}} [[President of the United States|US President]] [[Dwight D. Eisenhower]] considered him "the outstanding mathematician of the time".<ref>{{cite book |last1=Eisenhower |first1=Dwight D. |author1-link=Dwight D. Eisenhower |title=The White House Years: Waging Peace 1956-1961 |date=1965 |publisher=Doubleday & Company |location=New York |page=686}}</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, industrialists and scientists that no other scientist could match. In [[Nuclear weapon|nuclear missilery]] he was considered "the clearly dominant advisory figure" according to [[Herbert York]] whose opinions "everyone took very seriously".{{sfn|York|1971|p=85}} [[Nicholas Kaldor]] said he was "unquestionably the nearest thing to a genius I have ever encountered."{{sfn|Dore|Chakravarty|Goodwin|1989|p=xi}} [[Paul Samuelson]], "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}} Even for writer [[Arthur Koestler]], who was not an academic, von Neumann was "one of the few people for whom Koestler entertained not only respect but reverence, and he shared Koestler's Central European addiction to abstruse philosophical discussions, political debate, and dirty jokes. The two of them derived considerable pleasure from discussing the state of American civilization (was it in crisis or simply at the stage of adolescence?), the likely future of Europe (would there be war?), free will versus determinism, and the definition of pregnancy (“the uterus taking seriously what was pointed at it in fun”)."{{sfn|Scammell|2009|p=879}} He is often given as an example that mathematicians could do great work in the physical sciences too, however [[Robert D. Richtmyer|R. D. Richtmyer]] describes how during von Neumann's time at Los Alamos he functioned not as a mathematician applying his art to physics problems, but rather entirely as a physicist in the mind and thought (except faster). He describes him as a first-rate physicist who knew [[quantum mechanics]], [[Atomic physics|atomic]], [[Molecular physics|molecular]], and [[nuclear physics]], [[particle physics]], [[astrophysics]], [[general relativity|relativity]], and [[Physical chemistry|physical]] and [[organic chemistry]]. As such any mathematician who does not possess the same talent as von Neumann should not be fooled into thinking physics is easy just because they study mathematics.{{sfn|Hecker|Rota|2000|pp=235-236}} === 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> In his autobiography [[Stanislaw Ulam]] writes that Johnny's memory was auditory rather than visual. He did not have to any extent an intuitive 'common sense' for guessing what may happen in a given physical situation.{{sfn|Ulam|1976|pp=147-148}} == Legacy == "It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei in ''John von Neumann: Selected Letters''.{{sfn|Rédei|2005|p=7}} [[James Glimm]] wrote: "he is regarded as one of the giants of modern mathematics".{{sfn|Glimm|Impagliazzo|Singer|1990|p=vii}} 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}} while [[Peter Lax]] described him as possessing the "most scintillating intellect of this century".{{sfn|Lax|1990|p=7}} In the foreword of Miklós Rédei's ''Selected Letters'', Peter Lax wrote, "To gain a measure of von Neumann's achievements, consider that had he lived a normal span of years, he would certainly have been a recipient of a Nobel Prize in economics. And if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too. So the writer of these letters should be thought of as a triple Nobel laureate or, possibly, a {{frac|3|1|2}}-fold winner, for his work in physics, in particular, quantum mechanics".{{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 believed in the power of mathematical reasoning to influence modern civilization, an idea which expressed itself through his life work. He is widely regarded as one of the greatest and most influential mathematicians and scientists of the 20th century.{{sfn|Glimm|Impagliazzo|Singer|1990|p=vii}}{{sfn|Ulam|1976|p=4}}{{sfn|Rhodes|1995|p=250}}{{sfn|Kac|Rota|Schwartz|2008|p=206}}{{sfn|Albers|Alexanderson|2008|p=168}}{{sfn|Szanton|1992|p=51}}<ref>{{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> === Mastery of mathematics === Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: "Most mathematicians know one method. For example, [[Norbert Wiener]] had mastered [[Fourier transform]]s. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods." He went on to explain that the three methods were: # 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}} As an example of the last point Eugene Wigner described how once he did not understand a mathematical theorem and asked von Neumann for help. Von Neumann would ask Wigner whether he knew several other different but related theorems and then he would then explain the problematic theorem based on what Wigner already knew. Using such circular paths he could make even the most difficult concepts easy.{{sfn|Szanton|1992|pp=129-130}} On another occasion he wrote, "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}} Likewise Halmos noted that while von Neumann knew lots of mathematics, the most notable gaps were in algebraic topology and number theory, describing a story of how von Neumann once was walking by and saw something on the blackboard he didn't understand. Upon asking Halmos told him it was just the usual identification for a [[torus]]. While elementary even for modern graduate students this kind of work never crossed his path and thus he did not know it.{{sfn|Halmos|1973|p=389}} On another occasion he 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]], whom he thought was deeper and broader than von Neumann.{{sfn|Goldstine|1985|p=16}} Similarly [[Albert W. Tucker|Albert Tucker]] said he never saw von Neumann work on anything he would call "topological" and described how once von Neumann was giving a proof of a topological theorem, which he thought, while ingenious, was the kind of proof an analyst would give rather than someone who worked on [[combinatorial topology]].{{sfn|Tucker|1984|p=4}} 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 in one way or another, 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. In these fields he said von Neumann worked on rarely, and had very little affinity for it in his thinking.{{sfn|Bochner|1958|p=441}} Likewise [[Jean Dieudonné]] noted in his biographical article that while he 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]]. 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 a priori that previously seemed to have little connection with mathematics. His style in analysis was not of the traditional English or French schools but rather of the German one, where analysis is based extensively on foundations in [[linear algebra]] and [[general topology]]. As with Bochner, he noted von Neumann never did significant work in [[number theory]], [[algebraic topology]], algebraic geometry or differential geometry. However, for his limits in pure mathematics he made up for in applied mathematics, where his work certainly equalled that of legendary mathematicians such as [[Carl Friedrich Gauss|Gauss]], [[Augustin-Louis Cauchy|Cauchy]] or [[Henri Poincaré|Poincaré]]. Dieudonné notes that during the 1930s when von Neumann's work in pure mathematics was at its peak, there was hardly an important area he didn't have at least passing acquaintance with.<ref name="Dieudonne21" /> Towards the end of his life he deplored to Ulam the fact that it no longer felt possible for anyone to have more than passing knowledge of one-third of the field of pure mathematics.{{sfn|Ulam|1958|p=8}} In fact in the early 1940s Ulam himself concocted for him at his suggestion a doctoral style examination in various fields in order to find weaknesses in his knowledge. He did find them, with von Neumann being unable to answer satisfactorily a question each in differential geometry, number theory, and algebra. "This may also tend to show that doctoral exams have little permanent meaning" was their conclusion. However while 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 Johnny could have aspired to do so.{{sfn|Ulam|1976|p=291}} === Honors and awards === [[File:Von Neumann crater 5103 h2 h3.jpg|thumb|right|240px|The von Neumann crater, on the far side of the Moon]] * The [[John von Neumann Theory Prize]] of the [[Institute for Operations Research and the Management Sciences]] (INFORMS, previously TIMS-ORSA) is awarded annually to an individual (or group) who have made fundamental and sustained contributions to theory in [[operations research]] and the management sciences.<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]] |access-date=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 |df=mdy-all }}</ref> * The [[IEEE John von Neumann Medal]] is awarded annually by the [[Institute of Electrical and Electronics Engineers]] (IEEE) "for outstanding achievements in computer-related science and technology."<ref>{{cite news |url=http://www.ieee.org/about/awards/medals/vonneumann.html |publisher= [[Institute of Electrical and Electronics Engineers]]|title= IEEE John von Neumann Medal |newspaper= IEEE Awards|access-date=May 17, 2016 }}</ref> * The [[John von Neumann Lecture]] is given annually at the [[Society for Industrial and Applied Mathematics]] (SIAM) by a researcher who has contributed to applied mathematics, and the chosen lecturer is also awarded a monetary prize.<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> * The crater [[Von Neumann (crater)|von Neumann]] on the [[Moon]] is named after him.<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> * Asteroid [[22824 von Neumann]] was 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> * The [[John von Neumann Center]] in [[Plainsboro Township, New Jersey]], was named in his honor.<ref>{{cite news |url=http://www.the-scientist.com/?articles.view/articleNo/10771/title/NSF-Supercomputer-Program-Looks-Beyond-Princeton-Recall/ |title=NSF Supercomputer Program Looks Beyond Princeton Recall |newspaper=[[The Scientist (magazine)|The Scientist]] |first=Christopher |last=Anderson |date=November 27, 1989 |access-date=May 17, 2016 }}</ref> * The professional society of Hungarian computer scientists, [[John von Neumann Computer Society]], was named after von Neumann.<ref>{{cite web|title=Introducing the John von Neumann Computer Society |publisher=[[John von Neumann Computer Society]] |url=http://www.njszt.hu/neumann/neumann.head.page?nodeid=210 |access-date=May 20, 2008 |url-status=dead |archive-url=https://web.archive.org/web/20080429192308/http://www.njszt.hu/neumann/neumann.head.page?nodeid=210 |archive-date=April 29, 2008 }}</ref><ref>{{cite web | url=https://njszt.hu/en | title=NJSZT - John von Neumann Computer Society }}</ref> * The [[John von Neumann Environmental Research Institute of the Pacific]] in [[Colombia]] is named in his honor. * On May 4, 2005, the [[United States Postal Service]] issued the ''American Scientists'' commemorative postage stamp series, a set of four 37-cent self-adhesive stamps in several configurations designed by artist [[Victor Stabin]]. The scientists depicted were 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 |archive-date=2016-02-02 |url-status=dead |title=American Scientists Issue |website=Arago: People, Postage & the Post |publisher=[[National Postal Museum]] |access-date=August 2, 2022}}</ref> * The [[John von Neumann Award]] of the [[Rajk László College for Advanced Studies]] was named in his honor, and has been given every year since 1995 to professors who have made an outstanding contribution to the exact social sciences and through their work have strongly influenced the professional development and thinking of the members of the college.<ref>{{cite web |url=http://rajk.eu/en/dijaink/ |title=John von Neumann Award |publisher=díjaink – Rajk |access-date=May 17, 2016 |archive-url=https://web.archive.org/web/20141215020145/http://rajk.eu/en/dijaink/ |archive-date=December 15, 2014 |url-status=dead }}</ref> * 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> * The May 1958 issue of the [[Bulletin of the American Mathematical Society]] was dedicated as a memorial volume (in an act without precedent) to von Neumann and eight articles were written about him and his work by friends and colleagues.<ref>https://www.ams.org/journals/bull/1958-64-03<br />https://archive.org/details/bulletinof64pt3.2oxtomiss<br />https://archive.org/details/johnvonneumann190000unse</ref>{{sfn|Halperin|1990|p=15}} In addition, obituaries were written in several other journals, including the ''Journal of the [[London Mathematical Society]]'',<ref>{{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 |url=https://academic.oup.com/jlms/article-abstract/s1-34/3/373/847975}}</ref> ''[[National Academy of Sciences]]'',{{sfn|Bochner|1958}} ''[[Bulletin of the Atomic Scientists]]'',<ref>{{cite journal |last1=Teller |first1=Edward |title=John Von Neumann |journal=Bulletin of the Atomic Scientists |date=1957 |volume=13 |issue=4 |pages=150–151 |doi=10.1080/00963402.1957.11457538 |bibcode=1957BuAtS..13d.150T |url=https://www.tandfonline.com/doi/abs/10.1080/00963402.1957.11457538}}</ref> ''Matematikai Lapok'',<ref>{{cite journal |title=NEUMANN JÁNOS |journal=Matematikai Lapok |date=1957 |volume=8 |url=http://real-j.mtak.hu/9387/}}</ref> ''[[Physics Today]]'',<ref>{{cite journal |title=John von Neumann |journal=Physics Today |date=1957 |volume=10 |issue=4 |page=58 |doi=10.1063/1.3060351 |url=https://physicstoday.scitation.org/doi/10.1063/1.3060351}}</ref> ''[[Science (journal)|Science]]'',<ref>{{cite journal |last1=Goldstine |first1=Herman H. |last2=Wigner |first2=Eugene |title=Scientific Work of J. von Neumann |journal=Science |date=1957 |volume=125 |issue=3250 |pages=683–684 |doi=10.1126/science.125.3250.683 |pmid=17796247 |bibcode=1957Sci...125..683. |url=https://www.science.org/doi/10.1126/science.125.3250.683}}</ref> ''[[Mathematics of Computation]]''<ref>{{cite journal |last1=Tompkins |first1=Charles Brown |title=John von Neumann, 1903–1957 |journal=Mathematics of Computation |date=1957 |volume=11 |issue=58 |pages=127–128 |doi=10.1090/S0025-5718-1957-0086002-9 |url=https://www.ams.org/journals/mcom/1957-11-058/S0025-5718-1957-0086002-9/}}</ref> and ''[[The Economic Journal]]''.<ref>{{cite journal |last1=Morgenstern |first1=Oskar |title=John von Neumann 1903-57 |journal=The Economic Journal |date=1958 |volume=68 |issue=269 |pages=170–174 |doi=10.1093/ej/68.269.170 |url=https://academic.oup.com/ej/article-abstract/68/269/170/5248833}}</ref> * A large number of books have been dedicated to him from a wide variety of fields.{{sfn|Halmos|1985}}<ref>Further Examples:<br />https://books.google.com/books?id=6CnqCAAAQBAJ&q=neumann<br />https://www.worldscientific.com/worldscibooks/10.1142/5982<br />https://books.google.com/books?id=uqDDAgAAQBAJ&q=neumann<br />https://books.google.com/books?id=J8XG8Y5Ou5QC&q=neumann<br />https://books.google.com/books?id=9lSVFzsTGWsC&q=neumann<br />https://books.google.com/books?id=uWQnAQAAIAAJ&q=neumann<br />https://books.google.com/books?id=fAvgBAAAQBAJ&q=neumann<br />https://books.google.com/books?id=uqDDAgAAQBAJ&q=neumann</ref> * A large number of scientific papers have been dedicated to him from a wide variety of fields.{{sfn|Glimm|Impagliazzo|Singer|1990}}{{sfn|Doran|Kadison|2004}}<ref>Further Examples:<br />https://www.osti.gov/biblio/836372<br />https://iopscience.iop.org/article/10.1070/PU2004v047n12ABEH001896<br />https://ccse.lbl.gov/Publications/sepp/differential/LBNL-51940.pdf<br />https://www.osti.gov/biblio/839224<br />http://www.math.uni.wroc.pl/~pms/files/33.2/Article/33.2.24.pdf<br />https://ccse.lbl.gov/Publications/sepp/leastSquares/LBNL-52434.pdf<br />https://doi.org/10.1016/S0927-5452(04)80004-2<br />https://zbmath.org/?q=bi%3Avon-neumann.john</ref> * Many events have been dedicated to him from a wide variety of fields.<ref>Examples:<br />https://www.ieeesmc.org/publications/enewsletter/report-on-the-ieee-systems-man-and-cybernetics-2016-international-conference<br />https://math.bme.hu/~szasz/lp.html<br />https://golem.ph.utexas.edu/category/2007/09/deep_beauty_understanding_the.html<br />https://bookstore.ams.org/pspum-59<br />https://www.rockefeller.edu/events-and-lectures/25596-joshua-lederberg-john-von-neumann-symposium-towards-quantitative-biology<br />https://www.ams.org/meetings/amsconf/symposia/symposia<br />https://archive.siam.org/news/news.php?id=318<br />https://archive.siam.org/news/news.php?id=36<br />http://hdl.handle.net/2117/344825<br />http://conf.uni-obuda.hu/Neumann/CFP.htm</ref> * Twice [[List of International Congresses of Mathematicians Plenary and Invited Speakers|invited speaker]] at the [[International Congress of Mathematicians]].<ref>{{cite web |last1=O'Connor |first1=J. J. |last2=Robertson |first2=E. F. |author-link2=Edmund F. Robertson |title=International Congresses of Mathematicians |url=https://mathshistory.st-andrews.ac.uk/Honours/ICM/ |website=[[MacTutor History of Mathematics archive|MacTutor History of Mathematics]] |publisher=University of St Andrews, Scotland |access-date=15 September 2022}}</ref> A list of the following awards and honors was drawn from various biographic statements given by von Neumann.{{sfn|Aspray|1990|pp=246-247}}{{sfn|Ulam|1958|pp=41-42}}{{sfn|AIP Profile|2022}} Awards: {{columns-list| * 1926 [[Rockefeller Foundation|Rockefeller]] Fellowship * 1937 [[Bôcher Memorial Prize]], American Mathematical Society * 1947 [[Medal for Merit]] (Presidential Award) * 1947 [[Navy Distinguished Civilian Service Award|Distinguished Civilian Service Award]], U.S. Navy * 1955 Science Award, [[Air & Space Forces Association|Air Force Association]]<ref>{{cite web |title=List of Theodore von Karman Award Recipients |url=https://secure.afa.org/AboutUs/recipients.asp |website=Air & Space Forces Association |access-date=20 December 2022 |archive-url=https://web.archive.org/web/20220328074812/https://secure.afa.org/AboutUs/recipients.asp |archive-date=28 March 2022}}</ref> * 1956 [[Medal of Freedom (1945)|Medal of Freedom]] (Presidential Award) * 1956 Albert Einstein Commemorative Award * 1956 [[Enrico Fermi Award]], U.S. Atomic Energy Commission * 1957 [[American Meteorological Society]] [[Carl-Gustaf Rossby Research Medal|Award for Extraordinary Scientific Accomplishment]] * 1997 [[United States Air Force|Air Force]] Space and Missile Pioneers Award (Posthumous)<ref>{{cite web |title=Air Force Space and Missile Pioneers |url=https://www.afspc.af.mil/About-Us/Heritage/spaceandmissilepioneers/ |website=Air Force Space Command |access-date=20 December 2022 |archive-url=https://web.archive.org/web/20220908172458/https://www.afspc.af.mil/About-Us/Heritage/spaceandmissilepioneers/ |archive-date=8 September 2022}}</ref> * 1997 Hall of Fame, Space Command Headquarters, [[Peterson Space Force Base|Peterson Air Force Base]] (Posthumous){{sfn|Sheehan|2010|p=464}} }} Co-Editorship: {{columns-list| * 1933-1957 [[Annals of Mathematics]] * 1935-1957 [[Compositio Mathematica]] * 1947 [[Journal of Applied Physics]]<ref>{{cite web |title=December 5, 1946 |url=https://www.aip.org/content/december-5-1946 |website=American Institute of Physics |access-date=20 December 2022 |archive-url=https://web.archive.org/web/20221220125458/https://www.aip.org/content/december-5-1946 |archive-date=20 December 2022}}</ref> }} Honorary societies: {{columns-list| * Academia Nacional de Ciencias Exactas, Lima, Peru * [[Accademia dei Lincei|Academia Nazionale dei Lincei]], Rome, Italy * [[American Academy of Arts and Sciences]] * [[American Philosophical Society]] * [[Istituto Lombardo Accademia di Scienze e Lettere]], Milan, Italy * [[National Academy of Sciences]] * [[Royal Netherlands Academy of Arts and Sciences|Royal Netherlands Academy of Sciences and Letters]], Amsterdam, Netherlands }} Honorary doctorates: {{columns-list| * 1947 [[Princeton University]] * 1949 [[University of Pennsylvania]] * 1949 [[Harvard University]] * 1952 [[Istanbul University|University of Istanbul]] * 1952 [[Case Western Reserve University|Case Institute of Technology]] * 1952 University of Maryland * 1953 [[Technical University of Munich|Polytechnics Institut]], Munich * 1954 [[Columbia University]] }} Honorary positions: {{columns-list| * 1937 [[American Mathematical Society]] Colloquium Lecturer * 1944 [[Josiah Willard Gibbs Lectureship|Gibbs Lecturer]], American Mathematical Society * 1951-1953 President, American Mathematical Society * 1953 Vanuxem Lecturer, Princeton University * 1950-1957 Member, Board of Advisors, [[University of Los Andes (Colombia)|Universidad de Los Andes]], Colombia }} Society memberships: {{columns-list| * American Mathematical Society * [[American Physical Society]] * [[Econometric Society]] * [[International Statistical Institute]], The Hague, Netherlands * [[Sigma Xi]] }} == Selected works == Collections of von Neumann's published works can be found on [https://zbmath.org/authors/?s=0&q=Neumann%2C+John zbMATH] and [https://scholar.google.com.au/citations?user=6kEXBa0AAAAJ Google Scholar]. A list of his known works as of 1995 can be found in [https://www.worldscientific.com/doi/epdf/10.1142/9789812831088_bmatter The Neumann Compendium]. === Books authored / coauthored === * 1932. ''[[Mathematical Foundations of Quantum Mechanics|Mathematical Foundations of Quantum Mechanics: New Edition]]'', Wheeler, N. A., Ed., [[Robert T. Beyer|Beyer, R. T.]], Trans., Princeton University Press, [https://doi.org/10.1515/9781400889921 available here]. 2018 edition: {{isbn|9780691178561}} * 1937. ''[https://www.degruyter.com/document/doi/10.1515/9781400883950 Continuous Geometry]'', [[Israel Halperin|Halperin, I.]], Preface, Princeton Landmarks in Mathematics and Physics, Princeton University Press, [https://archive.org/details/continuousgeomet0000vonn online at archive.org]. 2016 edition: {{isbn|9781400883950}} {{mr|0120174}} * 1937. ''[https://bookstore.ams.org/memo-34-252 Continuous Geometries with a Transition Probability]'', [[Israel Halperin|Halperin, I.]], Preface, [[Memoirs of the American Mathematical Society]] Vol. 34, No. 252, 1981 edition. {{isbn|978-1-4704-0659-2}} {{mr|634656}} * 1941. ''[https://bookstore.ams.org/inmeas Invariant Measures]''. American Mathematical Society. 1999 edition: {{isbn|978-0-8218-0912-9}} * 1944. ''[[Theory of Games and Economic Behavior]]'', with [[Oskar Morgenstern|Morgenstern, O.]], Princeton University Press, [https://archive.org/details/in.ernet.dli.2015.215284 online at archive.org] [https://www.degruyter.com/document/doi/10.1515/9781400829460 or here]. 2007 edition: {{isbn|9781400829460}} * 1950. ''[https://www.degruyter.com/document/doi/10.1515/9781400881895 Functional Operators, Volume 1: Measures and Integrals]''. Annals of Mathematics Studies 21, [https://archive.org/details/dli.ernet.5825/ online at archive.org]. 2016 edition: {{isbn|9781400881895}} * 1951. ''[https://www.degruyter.com/document/doi/10.1515/9781400882250 Functional Operators, Volume 2: The Geometry of Orthogonal Spaces]''. Annals of Mathematics Studies 22, [https://archive.org/details/dli.ernet.5826 online at archive.org]. 2016 edition {{isbn|9781400882250}} * 1958. ''[[The Computer and the Brain]]'', [[Ray Kurzweil|Kurzweil, R.]] Preface, The [[Silliman Memorial Lectures]] Series, Yale University Press, [https://archive.org/details/computerbrain0000vonn online at archive.org] [https://www.degruyter.com/document/doi/10.12987/9780300188080 or here]. 2012 edition: {{isbn|9780300181111}} * 1966. ''[https://archive.org/details/theoryofselfrepr00vonn_0 Theory of Self-Reproducing Automata]'', [[Arthur Burks|Burks, A. W.]], Ed., University of Illinois Press. {{isbn|0-598-37798-0}} === Scholarly articles === * 1923. [https://zbmath.org/?q=an%3A49.0139.02 On the introduction of transfinite numbers], (in German), ''Acta Szeged'', '''1''':199-208. * 1925. [https://zbmath.org/?q=an%3A51.0163.04 An axiomatization of set theory], (in German), ''J. f. Math.'', '''154''':219-240. * 1926. [https://zbmath.org/?q=an%3A52.0151.02 On the Prüfer theory of ideal numbers], (in German), ''Acta Szeged'', '''2''':193-227. * 1927. [https://zbmath.org/?q=an%3A53.0041.02 On Hilbert's proof theory], (in German), ''Math. Zschr.'', '''26''':1-46. * 1929. [https://zbmath.org/?q=an%3A55.0824.02 General eigenvalue theory of Hermitian functional operators], (in German), ''Math. Ann.'', '''102''':49-131. * 1932. [https://zbmath.org/?q=an%3A0004.31004 Proof of the Quasi-Ergodic Hypothesis], ''Proc. Nat. Acad. Sci.'', '''18''':70-82. * 1932. [https://zbmath.org/?q=an%3A58.1272.01 Physical Applications of the Ergodic Hypothesis], ''Proc. Nat. Acad. Sci.'', '''18''':263-266. * 1932. [https://zbmath.org/?q=an%3A58.1270.04 On the operator method in classical mechanics], (in German), ''Ann. Math.'', '''33''':587-642. * 1934. [https://zbmath.org/?q=an%3A60.0902.02 On an Algebraic Generalization of the Quantum Mechanical Formalism], with [[Pascual Jordan|P. Jordan]] and [[Eugene Wigner|E. Wigner]], ''Ann. Math.'', '''35''':29-64. * 1936. [https://zbmath.org/?q=an%3A62.0449.03 On Rings of Operators], with [[Francis Joseph Murray|F. J. Murray]], ''Ann. Math.'', '''37''':116-229. * 1936. [https://zbmath.org/?q=an%3A62.0447.11 On an Algebraic Generalization of the Quantum Mechanical Formalism (Part I)], ''Mat. Sborn.'', '''1''':415-484. * 1936. [https://zbmath.org/?q=an%3A62.1061.04 The Logic of Quantum Mechanics], with [[Garrett Birkhoff|G. Birkhoff]], ''Ann. Math.'', '''37''':823-843. * 1936. [https://zbmath.org/?q=an%3A0014.22307 Continuous Geometry], ''Proc. Nat. Acad. Sci.'', '''22''':92-100. * 1936. [https://zbmath.org/?q=an:62.0648.03 Examples of Continuous Geometries], ''Proc. Nat. Acad. Sci.'', '''22''':101-108. * 1936. [https://zbmath.org/?q=an%3A0015.38802 On Regular Rings], ''Proc. Nat. Acad. Sci.'', '''22''':707-713. * 1937. [https://zbmath.org/?q=an%3A63.1008.03 On Rings of Operators, II], with [[Francis Joseph Murray|F. J. Murray]], ''Trans. Amer. Math. Soc.'', '''41''':208-248. * 1937. [https://zbmath.org/?q=an%3A0017.14804 Continuous Rings and Their Arithmetics], ''Proc. Nat. Acad. Sci.'', '''23''':341-349. * 1938. [https://zbmath.org/?q=an%3A64.0377.01 On Infinite Direct Products], ''Compos. Math.'', '''6''':1-77. * 1940. [https://zbmath.org/?q=an%3A66.0547.01 On Rings of Operators, III], ''Ann. Math.'', '''41''':94-161. * 1942. [https://zbmath.org/?q=an%3A0063.01888 Operator Methods in Classical Mechanics, II], with [[Paul Halmos|P. R. Halmos]], ''Ann. Math.'', '''43''':332-350. * 1943. [https://zbmath.org/?q=an%3A0060.26903 On Rings of Operators, IV], with [[Francis Joseph Murray|F. J. Murray]], ''Ann. Math.'', '''44''':716-808. * 1945. [https://academic.oup.com/restud/article-abstract/13/1/1/1543455 A Model of General Economic Equilibrium], ''Rev. Econ. Studies'', '''13''':1-9. * 1945. [https://web.archive.org/web/20110503181603/http://systemcomputing.org/turing%20award/Maurice_1967/TheFirstDraft.pdf ''First Draft of a Report on the EDVAC''], Report prepared for the U.S. Army Ordnance Department and the University of Pennsylvania, under Contract W670-ORD-4926, June 30, ''Summary Report No. 2'', ed. by [[J. Presper Eckert|J. P. Eckert]], [[John Mauchly|J. W. Mauchly]] and S. R. Warren, July 10. [The typescript original of this report has been re-edited by M. D. Godrey: ''IEEE Ann. Hist. Comp.'', Vol 15, No. 4, 1993, 27-75]. * 1947. [https://zbmath.org/?q=an%3A0031.31402 Numerical Inverting of Matrices of High Order], with [[Herman Goldstine|H. H. Goldstine]], ''Bull. Amer. Math. Soc.'', '''53''':1021-1099. * 1948. [https://www.vordenker.de/ggphilosophy/jvn_the-general-and-logical-theory-of-automata.pdf The General and Logical Theory of Automata], in ''Cerebral Mechanisms in Behavior - The Hixon Symposium'', [[Lloyd A. Jeffress|Jeffress, L.A.]] ed., John Wiley & Sons, New York, N. Y, 1951, pp.&nbsp;1–31, [https://www.ams.org/mathscinet-getitem?mr=0045446 MR0045446]. * 1949. [https://zbmath.org/?q=an%3A0034.06102 On Rings of Operators. Reduction Theory], ''Ann. Math.'', '''50''':401-485. * 1950. [https://zbmath.org/?q=an%3A0037.12002 A Method for the Numerical Calculation of Hydrodynamic Shocks], with [[Robert D. Richtmyer|R. D. Richtmyer]], ''J. Appl. Phys.'', '''21''':232-237. * 1950. [https://onlinelibrary.wiley.com/doi/10.1111/j.2153-3490.1950.tb00336.x Numerical Integration of the Barotropic Vorticity Equation], with [[Jule Gregory Charney|J. G. Charney]] and [[Ragnar Fjørtoft|R. Fjörtoft]], ''Tellus'', '''2''':237-254. * 1951. [https://zbmath.org/?q=an%3A0042.12301 A spectral theory for general operators of a unitary space], (in German), ''Math. Nachr.'', '''4''':258-281. * 1951. [https://zbmath.org/?q=an%3A1178.01069 Discussion on the Existence and Uniqueness or Multiplicity of Solutions of the Aerodynamical Equations], Chapter 10 of ''Problems of Cosmical Aerodynamics'', Proceedings of the Symposium on the Motion of Gaseous Masses of Cosmical Dimensions held in Paris, August 16–19, 1949. * 1951. [http://fsaad.mit.edu/randomseed/citing-a-rare-paper-by-von-neumann-various-techniques-used-in-connection-with-random-digits Various Techniques Used in Connection with Random Digits], Chapter 13 of "Proceedings of Symposium on 'Monte Carlo Method'", held June–July 1949 in Los Angeles, Summary written by [[George Forsythe|G. E. Forsynthe]]. * 1956. [https://www.degruyter.com/document/doi/10.1515/9781400882618-003 Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components], January 1952, Calif. Inst. of Tech., Lecture notes taken by R. S. Pierce and revised by the author, ''Automata Studies'', ed. by [[Claude Shannon|C. E. Shannon]] and [[John McCarthy (computer scientist)|J. McCarthy]], Princeton University Press, 43–98. === Popular articles === * 1947. [https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_Part_1/ The Mathematician], ''The Works of the Mind''. ed. by R. B. Heywood, University of Chicago Press, 180–196. * 1951. [https://longstreet.typepad.com/thesciencebookstore/2011/10/history-of-the-future-john-von-neumanns-crystal-ball-the-computer-1949.html The Future of High-Speed Computing], Digest of an address at the IBM Seminar on Scientific Computation, November 1949, ''Proc. Comp. Sem.'', IBM, 13. * 1954. The Role of Mathematics in the Sciences and in Society. Address at ''4th Conference of Association of Princeton Graduate Alumni'', June, 16–29. * 1954. [https://ieeexplore.ieee.org/document/4640689 The NORC and Problems in High Speed Computing], [https://www.youtube.com/watch?v=0L4aLmmjOCI Address] on the occasion of the first public showing of the [[IBM Naval Ordnance Research Calculator]], December 2. * 1955. Method in the Physical Sciences, ''The Unity of Knowledge'', ed. by L. Leary, Doubleday, 157–164. * 1955. [http://geosci.uchicago.edu/~kite/doc/von_Neumann_1955.pdf Can We Survive Technology?] [https://fortune.com/2013/01/13/can-we-survive-technology/ Fortune], June. * 1955. Impact of Atomic Energy on the Physical and Chemical Sciences, Speech at M.I.T. Alumni Day Symposium, June 13, Summary, Tech. Rev. 15–17. * 1955. Defense in Atomic War, Paper delivered at a symposium in honor of Dr. R. H. Kent, December 7, 1955, ''The Scientific Bases of Weapons'', Journ. Am. Ordnance Assoc., 21–23. * 1956. The Impact of Recent Developments in Science on the Economy and on Economics, Partial text of a talk at the National Planning Assoc., Washington, D.C., December 12, 1955, ''Looking Ahead'', '''4''':11. === Collected works === * 1963. ''John von Neumann Collected Works (6 Volume Set)'', [[Abraham H. Taub|Taub, A. H.]], editor, Pergamon Press Ltd. {{isbn|9780080095660}} ** 1961. ''Volume I: Logic, Theory of Sets and Quantum Mechanics'' ** 1961. ''Volume II: Operators, Ergodic Theory and Almost Periodic Functions in a Group'' ** 1961. ''Volume III: Rings of Operators'' ** 1962. ''Volume IV: Continuous Geometry and other topics'' ** 1963. ''Volume V: Design of Computers, Theory of Automata and Numerical Analysis'' ** 1963. ''Volume VI: Theory of Games, Astrophysics, Hydrodynamics and Meteorology'' == See also == {{Portal|Biography|Economics|History of Science|Hungary|Mathematics|Nuclear technology|Physics|Systems science|Technology|World War II}} {{cols|colwidth=26em}} * [[John von Neumann (sculpture)|''John von Neumann'' (sculpture)]], Eugene, Oregon * [[John von Neumann Award]] * [[List of things named after John von Neumann]] * [[List of pioneers in computer science]] * [[Mathematical formulation of quantum mechanics]] * [[Measurement in quantum mechanics]] * [[Self-replicating spacecraft]] * [[Von Neumann–Bernays–Gödel set theory]] * [[Von Neumann algebra]] * [[Von Neumann architecture]] * [[Von Neumann bicommutant theorem]] * [[Von Neumann cellular automaton]] * [[Von Neumann conjecture]] * [[Von Neumann entropy]] * [[Von Neumann programming languages]] * [[Von Neumann regular ring]] * [[Von Neumann's theorem]] * [[Von Neumann universal constructor]] * [[Von Neumann universe]] * [[Von Neumann–Morgenstern utility theorem]] * [[Teapot Committee]] * [[Trace inequality#Von Neumann's trace inequality and related results|Von Neumann's trace inequality]] * [[The Martians (scientists)]] {{colend}} == Notes == {{Reflist|20em}} == References == {{refbegin|30em}} * {{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 }} * {{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 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|editor-link=Steven N. Durlauf |editor-last=Durlauf |editor-first=Steven N. |editor2-last=Blume |editor2-first=Lawrence E.|publisher=Palgrave Macmillan|location=New York|edition=Second|url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508|doi=10.1057/9780230226203.0315|isbn=978-0-333-78676-5}} * {{cite book |last1=Białynicki-Birula |first1=Iwo |last2=Białynicki-Birula |first2=Iwona |title=Modeling Reality: How Computers Mirror Life. |date=2004 |publisher=Oxford University Press |location=Oxford |isbn=9780198531005}} * {{cite book |last=Bronowski |first=Jacob |author-link=Jacob Bronowski |year=1974 |title=The Ascent of Man |location=Boston |publisher=Little, Brown }} * {{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 }} * {{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 |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}} * {{cite book |last1=Dransfield |first1=Robert |first2=Don |last2=Dransfield |title=Key Ideas in Economics |year=2003 |publisher=Nelson Thornes |location=Cheltenham |isbn=978-0-7487-7081-6|oclc=52395899 }} * {{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 book |last=Filiol |first=Éric |year=2005 |title=Computer viruses: from theory to applications, Volume 1 |location=New York |publisher=Springer|pages=9–38 |isbn=978-2-287-23939-7|oclc=224779290 }} * {{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}} ** {{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}} ** {{harvc |last1=Lax |first1=Peter D. |author-link=Peter Lax |year=1990 |chapter=Remembering John von Neumann |in1=Glimm |in2=Impagliazzo |in3=Singer}} ** {{harvc |last1=Ulam |first1=Françoise |author-link=Françoise Aron Ulam |year=1990 |chapter=Nonmathematical Reminiscences about Johnny von Neumann |in1=Glimm |in2=Impagliazzo |in3=Singer}} ** {{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}} ** {{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}} ** {{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}} ** {{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}} ** {{harvc |last1=Kadison |first1=Richard V. |author-link=Richard Kadison |year=1990 |chapter=Operator Algebras - An Overview |in1=Glimm |in2=Impagliazzo |in3=Singer}} ** {{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}} ** {{harvc |last1=Glimm |first1=James |author-link=James Glimm |year=1990 |chapter=Scientific Computing: Von Neumann's Vision, Today's Realities, and the Promise of the Future |in1=Glimm |in2=Impagliazzo |in3=Singer}} ** {{harvc |last1=Cowan |first1=Jack D. |year=1990 |chapter=Von Neumann and Neural Networks |in1=Glimm |in2=Impagliazzo |in3=Singer}} ** {{harvc |last1=Aspray |first1=William |year=1990 |anchor-year=1990a |chapter=The Origins of John von Neumann's Theory of Automata |in1=Glimm |in2=Impagliazzo |in3=Singer}} ** {{harvc |last1=Pippenger |first1=Nicholas |author-link=Nick Pippenger |year=1990 |chapter=Developments in "The Synthesis of Reliable Organisms from Unreliable Components" |in1=Glimm |in2=Impagliazzo |in3=Singer}} * {{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 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 }} * {{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 |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 }} * {{cite book | last1 = Herken |first1 = Gregg |author1-link=Gregg Herken | title = Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller |publisher = Holt Paperbacks | location = New York, New York |year = 2002 | isbn = 978-0-8050-6589-3| oclc = 48941348}} * {{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 |editor-last=Kent |editor-first=Allen |editor2-first=James G. |editor2-last=Williams |title=Encyclopedia of Computer Science and Technology, Volume=30, Supplement 15 |location=New York |publisher=Dekker |year=1994 |isbn=978-0-8247-2283-8|oclc=832033269 }} * {{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}} * {{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|last=Mirowski|first=Philip|author-link=Philip Mirowski|title=Machine Dreams: Economics Becomes a Cyborg Science|publisher=Cambridge University Press|year=2002|location=New York|isbn=978-0-521-77283-9|oclc=45636899|url-access=registration|url=https://archive.org/details/machinedreamseco0000miro}} * {{cite book |last=Mitchell |first=Melanie |author-link=Melanie Mitchell |title=Complexity: A Guided Tour |location=New York |publisher=Oxford University Press |year=2009 |isbn=978-0-19-512441-5|oclc=216938473}} * {{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}} * {{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 }} * {{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 |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}} ** {{harvc |last1=Petz |first1=D. |author-link1=Dénes Petz |last2=Rédei |first2=M. R. |year=1995 |contribution=John von Neumann And The Theory Of Operator Algebras |in1=Bródy |in2=Vámos}} ** {{harvc |last1=von Neumann |first1=J. |year=1995 |contribution=Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components |in1=Bródy |in2=Vámos}} * {{cite book |last=Petković |first=Miodrag |author-link=Miodrag Petković |year=2009 |title=Famous puzzles of great mathematicians |url=https://archive.org/details/famouspuzzlesgre00mpet |url-access=limited |publisher=American Mathematical Society |page=[https://archive.org/details/famouspuzzlesgre00mpet/page/n175 157] |isbn=978-0-8218-4814-2}} * {{cite book|last=Poundstone |first=William |author-link=William Poundstone |title=Prisoner's Dilemma |year=1993 |publisher=Random House Digital|isbn=978-0-385-41580-4}} * {{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 |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 }} * {{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 }} *{{cite book|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 (Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972)|editor-first=Josef |editor-last=Loz |editor2-first=Maria |editor2-last=Loz |publisher=Elsevier North-Holland Publishing and Polish Academy of Sciences (PAN) |location=Amsterdam |year=1974 |oclc=839117596 }} * {{cite book|last=Rockafellar|first=R. T.|author-link=R. Tyrrell Rockafellar |title=Convex analysis |publisher=Princeton University Press |location=Princeton, New Jersey |year=1970 |isbn=978-0-691-08069-7|oclc=64619 }} * {{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 |location=Cambridge |publisher=Cambridge University Press |isbn=978-0-521-36734-9|oclc=18290810 }} * {{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 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 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 }} * {{cite book |last = von Neumann|first=John|editor-last=Heywood |editor-first=Robert B. |title=The Works of the Mind: The Mathematician |year=1947 |publisher=University of Chicago Press |location=Chicago |oclc=752682744 }} * {{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}} ** {{harvc |last1=von Neumann |first1=John |year=1963 |anchor-year=1963a |chapter=The Point Source Solution |in=Taub}} ** {{harvc |last1=von Neumann |first1=John |year=1963 |anchor-year=1963b |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}} * {{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 |last1=Wigner |first1=Eugene Paul |first2=Jagdish |last2=Mehra |first3=A. S. |last3=Wightman |author1-link=Eugene Wigner |author2-link=Jagdish Mehra |author3-link=Arthur Wightman |year=1995 |title=Volume 7, Part B, Philosophical Reflections and Syntheses |location=Berlin |publisher=Springer |isbn=978-3-540-63372-3 }} * {{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 |pages=277–299 |title=Interior point algorithms: Theory and analysis |publisher=Wiley |location=New York |isbn=978-0-471-17420-2|oclc=36746523 }} * {{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 journal |last1=Ulam |first1=Stanislaw |author1-link=Stanislaw 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 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}} * {{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 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}} * {{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 book|last=Rockafellar|first=R. Tyrrell|author-link=R. Tyrrell Rockafellar|title=Monotone Processes of Convex and Concave Type|series=Memoirs of the American Mathematical Society|publisher=American Mathematical Society|location=Providence, Rhode Island|issue=77|isbn=978-0-8218-1277-8|year=1967|oclc=1318941}} * {{cite book |last1=Burke |first1=Peter |author1-link=Peter Burke (historian) |title=The Polymath: A Cultural History from Leonardo da Vinci to Susan Sontag |date=2020 |publisher=Yale University Press |isbn=978-0300250022}} * {{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}} * {{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 web |title=Von Neumann, John, 1903-1957 |url=https://history.aip.org/phn/11610032.html |website=American Institute of Physics |access-date=25 April 2022 |archive-url=https://web.archive.org/web/20211106194620/https://history.aip.org/phn/11610032.html |archive-date=6 November 2021 |ref={{harvid|AIP Profile|2022}}}} * {{cite book |last1=Strauss |first1=Lewis L. |author1-link=Lewis Strauss |title=Men and Decisions |date=1962 |publisher=Doubleday |url=https://archive.org/details/mendecisions00stra |isbn=978-1299121713}} * {{cite journal |last1=Halmos |first1=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 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}} * {{cite book |last1=McCorduck |first1=Pamela |author1-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=2}} * {{cite book |last1=Scammell |first1=Michael |author1-link=Michael Scammell |title=Koestler: The Literary and Political Odyssey of a Twentieth-Century Skeptic |date=2009 |publisher=Random House |isbn=978-0394576305}} * {{cite web |first=S. |last=Bochner |author-link=Salomon Bochner |title=John von Neumann 1903-1957: A Biographical Memoir |url=http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/von-neumann-john.pdf |publisher=[[National Academy of Sciences]] |year=1958 |access-date=August 16, 2015}} * {{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 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 }} * {{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}} * {{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 |page=120 |doi=10.1007/978-3-662-07001-7 |edition=3 |url=https://link.springer.com/book/10.1007/978-3-662-07001-7}} * {{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}} * {{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 |url=https://link.springer.com/book/10.1007/978-3-030-03754-3}} * {{cite journal |last1=van Dantzig |first1=D. |author1-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=275–307 |language=French |doi=10.24033/asens.858 |url=https://eudml.org/doc/81525|doi-access=free }} * {{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 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 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 }} * {{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 }} * {{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 }} * {{cite journal |last1=Miranda |first1=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/}} * {{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}} * {{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}} * {{cite book |last1=Koksma |first1=J. F. |author1-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}} * {{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}} * {{cite book |last1=Rickles |first1=Dean |author1-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 journal |last1=Bonolis |first1=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 |doi=10.1140/epjh/e2017-80014-4 |arxiv=1703.09991 |bibcode=2017EPJH...42..311B |s2cid=119550393 |url=https://link.springer.com/article/10.1140/epjh/e2017-80014-4}} * {{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 |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 |url=https://www.degruyter.com/document/doi/10.1515/9783110935974/html?lang=en}} * {{cite book |editor-last1=Hecker |editor-first1=Siegfried S. |editor1-link=Siegfried S. Hecker |editor-last2=Rota |editor-first2=Gian-Carlo |editor2-link=Gian-Carlo Rota |title=Essays on the Future: In Honor of Nick Metropolis |date=2000 |publisher=Springer |location=New York |isbn=978-1-4612-6898-7 |doi=10.1007/978-1-4612-0777-1 |url=https://link.springer.com/book/10.1007/978-1-4612-0777-1}} * {{cite book|last=Jacobsen|first=Annie|author-link=Annie Jacobsen|title=The Pentagon's Brain: An Uncensored History Of DARPA, America's Top Secret Military Research Agency|publisher=Little, Brown and Company|date=2015|isbn=978-0316371667|oclc=1037806913|url=https://www.littlebrown.com/titles/annie-jacobsen/the-pentagons-brain/9780316371650}} * {{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 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}} * {{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}} * {{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 |title=A History of Computing in the Twentieth Century |editor1-first=Nicholas |editor1-last=Metropolis |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 |publisher=Elsevier |year=1980 |doi=10.1016/C2009-0-22029-0 |isbn=978-1-4832-9668-5}} * {{cite book |editor-last1=Nash |editor-first1=Stephen G. |title=A History of Scientific Computing |date=1990 |publisher=Association for Computing Machinery |url=https://dl.acm.org/doi/book/10.1145/87252 |doi=10.1145/87252 |isbn=978-0-201-50814-7}} ** {{harvc |last1=Goldstine |first1=Herman H. |author-link=Herman Goldstine |year=1990 |chapter=Remembrance of things past |doi=10.1145/87252.88059 |url=https://dl.acm.org/doi/10.1145/87252.88059 |in=Nash}} ** {{harvc |last1=Birkhoff |first1=Garrett |author-link=Garrett Birkhoff |year=1990 |chapter=Fluid dynamics, reactor computations, and surface representation |doi=10.1145/87252.88072 |url=https://dl.acm.org/doi/10.1145/87252.88072 |in=Nash}} * {{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 |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 |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}} * {{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 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 }} * {{cite book |last1=Pietsch |first1=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 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 |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 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}} * {{cite book |last1=Booß-Bavnbek |first1=Bernhelm |last2=Høyrup |first2=Jens |title=Mathematics and War |date=2003 |publisher=Birkhäuser |location=Basel |doi=10.1007/978-3-0348-8093-0 |isbn=978-3-0348-8093-0 |edition=1 |url=https://link.springer.com/book/10.1007/978-3-0348-8093-0}} * {{cite thesis |last=Steingart |first=Alma |date=2013 |title=Conditional Inequalities: American Pure and Applied Mathematics, 1940-1975 |type=PhD |publisher=Massachusetts Institute of Technology |url=https://core.ac.uk/download/pdf/19879029.pdf}} * {{cite book |editor-last1=Duren |editor-first1=Peter |editor-link1=Peter Duren |title=A Century of Mathematics in America: Part III |date=1989 |publisher=American Mathematical Society |isbn=0-8218-0136-8}} ** {{harvc |last1=Kelley |first1=J. L. |author-link1=John L. Kelley |year=1989 |chapter=Once Over Lightly |in=Duren}} ** {{harvc |last1=DeGroot |first1=Morris H. |author-link1=Morris H. DeGroot |year=1989 |chapter=A Conversation with David Blackwell |in=Duren}} * {{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 |page=373 |doi=10.1080/00029890.1973.11993292 |jstor=2319079 |url=https://doi.org/10.2307/2319079}} * {{cite journal |last1=Christensen |first1=Chris |title=The evolving relationship between mathematics and cryptology, 1951–1952: SCAG and the beginnings of SCAMP and NSASAB |journal=Cryptologia |date=2017 |volume=41 |issue=4 |pages=329–387 |doi=10.1080/01611194.2016.1219788 |s2cid=41317076 |url=https://www.tandfonline.com/doi/full/10.1080/01611194.2016.1219788}} * {{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}} * {{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}} * {{cite journal |last1=Rédei |first1=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}} * {{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}} * {{cite book |last1=Rhodes |first1=Richard |author-link1=Richard Rhodes |title=The Making of the Atomic Bomb |date=1986 |publisher=Simon & Schuster |isbn=978-0671657192 |url=https://www.simonandschuster.com/books/The-Making-of-the-Atomic-Bomb/Richard-Rhodes/9781451677614}} * {{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=Ulam |first1=Stanislaw M. |author-link1=Stanislaw 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}} * {{cite book |last1=Goodearl |first1=Ken R. |title=Von Neumann Regular Rings |date=1979 |publisher=Pitman Publishing |isbn=0-273-08400-3 |edition=1}} * {{cite journal |last1=Lashkhi |first1=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 }} * {{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 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 }} * {{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 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-4/issue-2/Von-Neumann-regular-rings-connections-with-functional-analysis/bams/1183547995.full|doi-access=free }} * {{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 journal |last1=Halmos |first1=Paul R. |author-link1=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 |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 }} * {{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 |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}} * {{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 |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}} * {{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 |url=https://link.springer.com/book/10.1007/978-1-4612-1084-9}} * {{cite book |last1=Byrne |first1=Peter |title=The Many Worlds of Hugh Everett III: Multiple Universes, Mutual Assured Destruction, and the Meltdown of a Nuclear Family |date=2010 |publisher=Oxford University Press |isbn=978-0199552276 |edition=1}} * {{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 |edition=1}} * {{cite book |last1=Heukelom |first1=Floris |title=Behavioral Economics: A History |date=2014 |publisher=Cambridge University Press |isbn=9781139600224 |url=https://www.cambridge.org/core/books/behavioral-economics/E0A9410BBA25772ECAF4C0C3BABB9709 |doi=10.1017/CBO9781139600224}} * {{cite book |last1=Giocoli |first1=Nicola |title=Modeling Rational Agents: From Interwar Economics to Early Modern Game Theory |date=2003 |publisher=Edward Elgar Pub |isbn=978-1-84064-868-3 |url=https://www.e-elgar.com/shop/gbp/modeling-rational-agents-9781840648683.html}} * {{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}} * {{cite book |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 |date=2001 |publisher=Springer |location=Berlin, Heidelberg |doi=10.1007/978-3-662-07791-7 |isbn=978-3-662-07791-7 |edition=1 |url=https://link.springer.com/book/10.1007/978-3-662-07791-7}} ** {{harvc |last1=Wigner |first1=Eugene |author-link=Eugene Wigner |year=2001 |chapter=John von Neumann (1903–1957) |in1=Mehra}} * {{cite book |last1=Pais |first1=Abraham |author-link1=Abraham Pais |title=Einstein Lived Here |date=1994 |publisher=Oxford University Press |isbn=0-19-853994-0}} * {{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 |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 |date=2009 |publisher=Springer |isbn=978-3-540-70626-7 |url=https://link.springer.com/book/10.1007/978-3-540-70626-7 |doi=10.1007/978-3-540-70626-7}} ** {{harvc |last1=Landsman |first1=Nicolaas P. |year=2009 |chapter=Algebraic Quantum Mechanics |in1=Greenberger |in2=Hentschel |in3=Weinert}} * {{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 |last1=Sieg |first1=Wilfried |title=Hilbert's Programs and Beyond |date=2013 |publisher=Oxford University Press |isbn=978-0195372229 |url=https://global.oup.com/academic/product/hilberts-programs-and-beyond-9780195372229}} * {{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=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=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}} {{refend}} == Further reading == {{refbegin|30em}} ''' Books''' * {{cite book |editor-last=Rédei |editor-first=Miklós |year=2005 |title=John von Neumann: Selected Letters |publisher=American Mathematical Society |location=Providence, Rhode Island |isbn=978-0-8218-3776-4|oclc=60651134 |url=https://bookstore.ams.org/hmath-27 |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 |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 |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 |ref=none}} * {{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 |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 |isbn=978-1107609266 |url=https://www.cambridge.org/core/books/von-neumann-morgenstern-and-the-creation-of-game-theory/FFF74A9B0D845E0BED16E91AFE4ED052 |ref=none}} * {{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 |ref=none}} * {{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 |date=1999 |publisher=American Mathematical Society |isbn=978-0821826768 |edition=2 |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|url=https://www.penguinrandomhouse.com/books/133168/prisoners-dilemma-by-william-poundstone|isbn=978-0-385-41580-4 |ref=none}} * {{cite book |last=Aspray |first=William |year=1990 |title=John von Neumann and the Origins of Modern Computing |location=Cambridge, Massachusetts |publisher=MIT Press |isbn=978-0-262-01121-1|oclc=21524368 |bibcode=1990jvno.book.....A |url=https://mitpress.mit.edu/books/john-von-neumann-and-origins-modern-computing |ref=none}} * {{cite book |last=Ulam |first=Stanisław |author-link=Stanislaw Ulam |title=Adventures of a Mathematician |publisher=University of California Press |year=1991 |isbn=978-0520071544 |url=https://www.ucpress.edu/book/9780520071544/adventures-of-a-mathematician |ref=none}} * {{cite book|last=Goldstine |first=Herman |author-link=Herman Goldstine |title=The Computer from Pascal to von Neumann|publisher=Princeton University Press|year= 1980|doi=10.1515/9781400820139 |isbn=978-0-691-02367-0 |url=https://www.degruyter.com/document/doi/10.1515/9781400820139/html?lang=en |ref=none}} * {{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|ref=none}} * {{cite book |last1=Erickson |first1=Paul |title=The World the Game Theorists Made |date=2015 |publisher=University of Chicago Press |isbn=9780226097176 |url=https://press.uchicago.edu/ucp/books/book/chicago/enwiki/w/bo17588566.html |ref=none}} * {{cite book|last=Mirowski|first=Philip|author-link=Philip Mirowski|title=Machine Dreams: Economics Becomes a Cyborg Science|publisher=Cambridge University Press|year=2002|location=New York|isbn=978-0-521-77283-9|oclc=45636899|url-access=registration|url=https://archive.org/details/machinedreamseco0000miro|ref=none}} * {{cite journal |editor1-last=Weintraub |editor1-first=E. Roy |editor-link1=E. Roy Weintraub |title=Towards a History of Game Theory |journal=History of Political Economy |date=1992 |volume=24 |issue=Special Issue |url=https://read.dukeupress.edu/hope/issue/24/Supplement |issn=0018-2702 |ref=none}} * {{cite book |editor1-first=Mohammed |editor1-last=Dore |editor2-first=Sukhamoy |editor2-last=Chakravarty |editor2-link=Sukhamoy Chakraborty |editor3-first=Richard |editor3-last=Goodwin |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 |ref=none}} * {{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 |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 |last1=Sieg |first1=Wilfried |title=Hilbert's Programs and Beyond |date=2013 |publisher=Oxford University Press |isbn=978-0195372229 |url=https://global.oup.com/academic/product/hilberts-programs-and-beyond-9780195372229 |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 |url=https://link.springer.com/book/10.1007/978-3-540-30721-1 |ref=none}} * {{cite book |last1=Pietsch |first1=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 |ref=none}} * {{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 |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 |url=https://link.springer.com/book/10.1007/978-3-540-30421-0 |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 |url=https://www.worldscientific.com/worldscibooks/10.1142/9809 |ref=none}} * {{cite book |last1=Hargittai |first1=Istvan |title=Martians of Science: Five Physicists Who Changed the Twentieth Century |date=2008 |publisher=Oxford University Press |isbn=978-0195365566 |url=https://oxford.universitypressscholarship.com/view/10.1093/acprof:oso/9780195178456.001.0001/acprof-9780195178456 |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 |url=https://global.oup.com/academic/product/the-heroic-age-9780190655174 |isbn=978-0190655174 |ref=none}} * {{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 |ref=none}} * {{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 |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 |url=https://link.springer.com/book/10.1007/978-1-4612-1084-9 |ref=none}} * {{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 |doi=10.1142/2692 |isbn=978-981-02-2201-7 |oclc=32013468 |url=https://www.worldscientific.com/worldscibooks/10.1142/2692 |ref=none}} '''Popular periodicals''' * ''[[Good Housekeeping|Good Housekeeping Magazine]]'', September 1956, "Married to a Man Who Believes the Mind Can Move the World" * {{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 |ref=none}} '''Video''' * ''John von Neumann, A Documentary'' (60 min.), [[Mathematical Association of America]], [https://www.youtube.com/watch?v=gn3U9G0gfu4 available here] '''Journals''' * {{cite journal |last1=Ulam |first1=Stanislaw |author-link=Stanislaw 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/ |ref=none|doi-access=free }} * {{cite journal |last1=Halmos |first1=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 |ref=none}} * {{cite web |first=S. |last=Bochner |author-link=Salomon Bochner |title=John von Neumann 1903-1957: A Biographical Memoir |url=http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/von-neumann-john.pdf |publisher=[[National Academy of Sciences]] |year=1958 |access-date=August 16, 2015 |ref=none}} * {{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 |url=https://academic.oup.com/jlms/article-abstract/s1-34/3/373/847975 |ref=none}} {{refend}} == External links == {{sisterlinks|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. * [http://purl.umn.edu/107206 Oral history interview with Alice R. Burks and Arthur W. Burks], [[Charles Babbage Institute]], [[University of Minnesota]], Minneapolis. [[Alice Burks]] and [[Arthur Burks]] describe [[ENIAC]], [[EDVAC]], and [[IAS machine|IAS]] computers, and John von Neumann's contribution to the development of computers. * [http://purl.umn.edu/107714 Oral history interview with Eugene P. Wigner], [[Charles Babbage Institute]], University of Minnesota, Minneapolis. * [http://purl.umn.edu/107493 Oral history interview with Nicholas C. Metropolis], [[Charles Babbage Institute]], University of Minnesota. * [https://zbmath.org/authors/?q=Neumann%2C+John zbMATH profile] * [https://albert.ias.edu/discover?query=von+neumann 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=QhBvuW-kCbM A (very) Brief History of John von Neumann] video by [[YouTuber]] moderndaymath. * [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. * [https://www.youtube.com/watch?v=Oh31I1F2vds Greatest Mathematician Of The 20th Century] high-quality excerpt from above documentary where [[Edward Teller]] describes John von Neumann. {{Manhattan Project}} {{Set theory}} {{Game theory}} {{Timelines of computing}} {{AMS Presidents|state=collapsed}} {{Authority control}} {{DEFAULTSORT:Neumann, John von}} [[Category:John von Neumann| ]] [[Category:1903 births]] [[Category:1957 deaths]] [[Category:20th-century American mathematicians]] [[Category:20th-century American physicists]] [[Category:Algebraists]] [[Category:American anti-communists]] [[Category:American computer scientists]] [[Category:American nuclear physicists]] [[Category:American operations researchers]] [[Category:American people of Hungarian-Jewish descent]] [[Category:American Roman Catholics]] [[Category:American systems scientists]] [[Category:Austrian nobility]] [[Category:Mathematicians from Austria-Hungary]] [[Category:Ballistics experts]] [[Category:Burials at Princeton Cemetery]] [[Category:Deaths from cancer in Washington, D.C.]] [[Category:Carl-Gustaf Rossby Research Medal recipients]] [[Category:Cellular automatists]] [[Category:Computer designers]] [[Category:Computer scientists]] [[Category:Converts to Roman Catholicism from Judaism]] [[Category:Cyberneticists]] [[Category:Elected Members of the International Statistical Institute]] [[Category:Enrico Fermi Award recipients]] [[Category:ETH Zurich alumni]] [[Category:Fasori Gimnázium alumni]] [[Category:Fellows of the American Physical Society]] [[Category:Fellows of the Econometric Society]] [[Category:Fluid dynamicists]] [[Category:Functional analysts]] [[Category:Game theorists]] [[Category:Hungarian anti-communists]] [[Category:Hungarian computer scientists]] [[Category:Hungarian emigrants to the United States]] [[Category:Hungarian inventors]] [[Category:Hungarian Jews]] [[Category:20th-century Hungarian mathematicians]] [[Category:20th-century Hungarian physicists]] [[Category:Hungarian nobility]] [[Category:Hungarian nuclear physicists]] [[Category:Hungarian Roman Catholics]] [[Category:Institute for Advanced Study faculty]] [[Category:Jewish American scientists]] [[Category:Jewish anti-communists]] [[Category:Jewish physicists]] [[Category:Lattice theorists]] [[Category:Manhattan Project people]] [[Category:Mathematical economists]] [[Category:Mathematical physicists]] [[Category:Mathematicians from Budapest]] [[Category:Measure theorists]] [[Category:Medal for Merit recipients]] [[Category:Members of the American Philosophical Society]] [[Category:Members of the Lincean Academy]] [[Category:Members of the Royal Netherlands Academy of Arts and Sciences]] [[Category:Members of the United States National Academy of Sciences]] [[Category:Mental calculators]] [[Category:Monte Carlo methodologists]] [[Category:Naturalized citizens of the United States]] [[Category:Numerical analysts]] [[Category:Oak Ridge National Laboratory people]] [[Category:Operations researchers]] [[Category:Operator theorists]] [[Category:People from Pest, Hungary]] [[Category:Presidents of the American Mathematical Society]] [[Category:Princeton University faculty]] [[Category:Probability theorists]] [[Category:Quantum physicists]] [[Category:RAND Corporation people]] [[Category:Recipients of the Medal of Freedom]] [[Category:Researchers of artificial life]] [[Category:Set theorists]] [[Category:Theoretical physicists]] [[Category:Academic staff of the University of Göttingen]] [[Category:Von Neumann family|John]] [[Category:Yiddish-speaking people]]'