John von Neumann (; December 28, 1903 – February 8, 1957) was a Hungarian and American pure and applied mathematician, physicist, inventor and polymath. He made major contributions to a number of fields,^{[2]} including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and fluid dynamics), economics (game theory), computing (Von Neumann architecture, linear programming, selfreplicating machines, stochastic computing), and statistics.^{[3]} He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory^{[2]}^{[4]} and the concepts of cellular automata,^{[2]} the universal constructor, and the digital computer.
Von Neumann's mathematical analysis of the structure of selfreplication preceded the discovery of the structure of DNA.^{[5]} In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932." Along with Hungarianborn American theoretical physicist Edward Teller and Polish mathematician Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.
Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital and later published in book form as The Computer and the Brain, gives an indication of the direction of his interests at the time of his death.
Contents

Early life and education 1

Career and abilities 2

Beginnings 2.1

Set theory 2.2

Geometry 2.3

Measure theory 2.4

Ergodic theory 2.5

Operator theory 2.6

Lattice theory 2.7

Mathematical formulation of quantum mechanics 2.8

Quantum logic 2.9

Game theory 2.10

Mathematical economics 2.11

Linear programming 2.12

Mathematical statistics 2.13

Nuclear weapons 2.14

The Atomic Energy Committee 2.15

The ICBM Committee 2.16

Mutually assured destruction 2.17

Computing 2.18

Fluid dynamics 2.19

Politics and social affairs 2.20

On the eve of World War II 2.21

Greece and Rome 2.22

Weather systems 2.23

Cognitive abilities 2.24

Mastery of mathematics 2.25

Personal life 3

Later life 4

Honors 5

Info Park and Neumann János Street 5.1

Selected works 6

See also 7

Notes 8

Footnotes 8.1

Citations 8.2

References 9

Further reading 10

External links 11
Early life and education
Von Neumann was born Neumann János Lajos (Hungarian pronunciation: ; in Hungarian the family name comes first) in Budapest, AustroHungarian Empire, to wealthy Jewish parents.^{[6]}^{[7]}^{[8]} He was the eldest of three brothers. His father, Neumann Miksa (Max Neumann) was a banker, who held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. Miksa's father (Mihály b. 1839)^{[9]} and grandfather (Márton)^{[9]} were both born in Ond (now part of the town of Szerencs), Zemplén county, northern Hungary. John's mother was Kann Margit (Margaret Kann).^{[10]}
Her parents were Jakab Kann II (Pest (now Budapest) 1845–1928) and Katalin Meisels (Munkács, Kárpátalja c. 1854–1914). In 1913, his father was elevated to the nobility for his service to the AustroHungarian empire by Emperor Franz Josef. The Neumann family thus acquiring the hereditary title margittai, Neumann János became margittai Neumann János (John Neumann of Margitta), which he later changed to the German Johann von Neumann.
He was an extraordinary child prodigy in the areas of language, memorization, and mathematics. As a 6yearold, he could divide two 8digit numbers in his head.^{[11]} By the age of 8, he was familiar with differential and integral calculus.^{[12]}
Von Neumann was part of a Budapest generation noted for intellectual achievement: he was born in Budapest around the same time as Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913).^{[13]}
John entered the Germanspeaking Lutheran high school Fasori Evangelikus Gimnázium in Budapest in 1911. Although his father insisted he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears.^{[14]}
Szegő subsequently visited the von Neumann house twice a week to tutor the

.

von Neumann's contribution to economics—International Social Science Review

von Neumann's Scholar Google profile

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 computers, and John von Neumann's contribution to the development of computers.

Oral history interview with Eugene P. Wigner, Charles Babbage Institute, University of Minnesota, Minneapolis.

Oral history interview with Nicholas C. Metropolis, Charles Babbage Institute, University of Minnesota.

Von Neumann vs. Dirac — from Stanford Encyclopedia of Philosophy

John von Neumann Postdoctoral Fellowship at Sandia National Laboratories

Von Neumann's Universe, audio talk by George Dyson

John von Neumann's 100th Birthday, article by Stephen Wolfram on von Neumann's 100th birthday.

Annotated bibliography for John von Neumann from the Alsos Digital Library for Nuclear Issues

Budapest Tech Polytechnical Institution – John von Neumann Faculty of Informatics

John von Neumann speaking at the dedication of the NORD, December 2, 1954 (audio recording)

The American Presidency Project

Jewish.hu: Famous Hungarian Jews

John von Neumann (1903–1957).

John von Neumann at Find a Grave
External links

Video

Popular periodicals

Aspray, William, 1990. John von Neumann and the Origins of Modern Computing.

Chiara, Dalla, Maria Luisa and Giuntini, Roberto 1997, La Logica Quantistica in Boniolo, Giovani, ed., Filosofia della Fisica (Philosophy of Physics). Bruno Mondadori.

InformatikSpektrum 29 (2), S. 133–141.

Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903–1957). "Teil 2: Ein Privatdozent auf dem Weg von Berlin nach Princeton". In: InformatikSpektrum 29 (3), S. 227–236.

Heims, Steve J., 1980. John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death MIT Press

Poundstone, William. Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb. 1992.

Redei, Miklos (ed.), 2005 John von Neumann: Selected Letters American Mathematical Society

Ulam, Stanislaw, 1983. Adventures of a Mathematician Scribner's

Vonneuman, Nicholas A. John von Neumann as Seen by His Brother ISBN 0961968109

1958, Bulletin of the American Mathematical Society 64.

1990. Proceedings of the American Mathematical Society Symposia in Pure Mathematics 50.

John von Neumann 1903–1957, biographical memoir by S. Bochner, National Academy of Sciences, 1958
Further reading
This article is based on material taken from the Free Online Dictionary of Computing prior to 1 November 2008 and incorporated under the "relicensing" terms of the GFDL, version 1.3 or later.

Blair, Clay, Jr. Passing of a Great Mind, Life Magazine, 25 February 1957

Doran, Robert S.; John Von Neumann,

Goldstine, Herman The Computer from Pascal to von Neumann, Princeton University Press, 1980, ISBN 0691023670.

Heims, Steve J. (1980). John von Neumann and Norbert Wiener, from Mathematics to the Technologies of Life and Death. Cambridge, Massachusetts: MIT Press.

Herken, Gregg (2002). Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller.

Glimm, James; Impagliazzo, John; Singer, Isadore Manuel The Legacy of John von Neumann, American Mathematical Society 1990 ISBN 0821842196

Israel, Giorgio; Ana Millan Gasca (1995). The World as a Mathematical Game: John von Neumann, Twentieth Century Scientist.


Slater, Robert (1989). Portraits in Silicon. Cambridge, Mass.: MIT Press. pp. 23–33.
References

^ http://www.cambridgesystems.com/content/QFBenoitMandelbrotTribute.pdf

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f}

^ ^{a} ^{b} Glimm, p. vii

^ Nelson, David (2003). The Penguin Dictionary of Mathematics. London: Penguin. pp. 178–179.

^ ^{a} ^{b} Rocha, L.M. "Lecture Notes of I585Biologically Inspired Computing Course, Indiana University".

^ Doran, p. 1

^ Nathan Myhrvold, "John von Neumann". Time, March 21, 1999. Accessed September 5, 2010

^ Blair, p. 104

^ ^{a} ^{b} "Mihály Neumann".

^ MacRae, pp. 37–38

^ Poundstone, William, Prisoner's Dilemma, New York: Doubleday 1992

^ ^{a} ^{b} ^{c} ^{d} ^{e} Halmos, P.R. "The Legend of von Neumann". The American Mathematical Monthlyvolume= 80 (4–year=1973): 382–394.

^ Doran, p. 2

^ Glimm, p. 5

^ MacRae, p. 70

^ Nasar, Sylvia, A Beautiful Mind, London 2001, p. 81 ISBN 0743224574.

^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} Blair, pp. 89–104.

^ MacRae, p. 145

^ ^{a} ^{b} John von Neumann (2005). Miklós Rédei, ed. John von Neumann: Selected letters. History of Mathematics 27.

^ John von Neumann (2005). Miklós Rédei, ed. John von Neumann: Selected letters. History of Mathematics 27. "Many thanks for your letter and your reprint. As you have established the unprovability of consistency as a natural continuation and deepening of your earlier results, I clearly won't publish on this subject."

^ ^{a} ^{b} ^{c}

^ von Neumann, J. (1933). "Die Einfuhrung Analytischer Parameter in Topologischen Gruppen".

^ Two famous papers are below: von Neumann, John (1932). "Proof of the Quasiergodic Hypothesis". Proc Natl Acad Sci USA 18 (1): 70–82.

^ Michael C. Reed, Barry Simon, Methods of Modern Mathematical Physics, Volume 1: Functional Analysis, Academic Press; Revised edition (1980)

^ D.Petz and M.R. Redi, John von Neumann And The Theory Of Operator Algebras, in The Neumann compendium, World Scientific, 1995, pp. 163–181 ISBN 9810222017.

^ ^{a} ^{b} ^{c} Garrett Birkhoff (1958). "Von Neumann and lattice theory". Bull. Amer. Math. Soc. 64 (3): 50–56.

^ Bub, Jeffrey (2010). "Von Neumann's 'No Hidden Variables' Proof: A ReAppraisal".

^ von Neumann, John. (1932/1955). Mathematical Foundations of Quantum Mechanics. Princeton: Princeton University Press. Translated by Robert T. Beyer.

^ Dov M. Gabbay, John Woods, The Many Valued and Nonmonotonic Turn in Logic, Elsevier, 2007, pp. 205–217 ISBN 0444516239.

^ Philosophical Papers: Volume 3, Realism and Reason, Hilary Putnam, Cambridge University Press, 27 December 1985, p. 263

^



Green, Jerry; Heller, Walter P. (1981). "1 Mathematical analysis and convexity with applications to economics". In




^ John MacQuarrie. "Mathematics and Chess". School of Mathematics and Statistics,

^ John von Neumann, Documentary film.

^ For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem

A − λ I q = 0,
where the nonnegative matrix A must be square and where the Gerald L. Thompson in the 1950s and then by Stephen M. Robinson in the 1970s.

^ David Gale. The theory of linear economic models. McGraw–Hill, New York, 1960.

^

^ Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & Sons, 1998, ISBN 0471982326.

^

Rockafellar, R. Tyrrell (1967). Monotone processes of convex and concave type. Memoirs of the American Mathematical Society (77). Providence, R.I.: American Mathematical Society. pp. i+74.



^

^ Yinyu Ye. Chapter 9.1 "The von Neumann growth model", pp. 277–299 in Interior point algorithms: Theory and analysis. Wiley. 1997 ISBN 0471174203.

^ Contributions to von Neumann's Growth Model, Proceedings of a Conference Organized by the Institute for Advanced Studies Vienna, Austria, 6 and 7 July 1970, Prof. Dr. Gerhart Bruckmann and Prof. Dr. Wilhelm Weber (eds), ISBN 9783662227381 (Print) 9783662246672 (Online), Springer Verlag, September 21, 1971, doi:10.1007/9783662246672.

^ ^{a} ^{b} George B. Dantzig and Mukund N. Thapa. 2003. Linear Programming 2: Theory and Extensions. SpringerVerlag ISBN 1441931406.

^ von Neumann, John (1941). "Distribution of the ratio of the mean square successive difference to the variance".

^ ^{a} ^{b} Durbin, J., and Watson, G. S. (1950). "Testing for Serial Correlation in Least Squares Regression, I".

^ Sargan, J.D. and Bhargava, Alok (1983). "Testing residuals from least squares regression for being generated by the Gaussian random walk".

^ ^{a} ^{b} Lillian Hoddeson, with contributions from

^ Rhodes, Richard (1986). The Making of the Atomic Bomb. New York: Touchstone

^ Groves, Leslie (1962). Now It Can Be Told: The Story of the Manhattan Project. New York:

^ Herken, pp. 171, 374

^ ^{a} ^{b} Bernstein, Jeremy (2010). "John von Neumann and Klaus Fuchs: an Unlikely Collaboration". Physics in Perspective 12: 36.

^ John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life ans Death, Steve J. Heims, The MIT Press, Cambridge, Massachusetts, p.276, 1980

^ Pesavento, Umberto (1995). "An implementation of von Neumann's selfreproducing machine" (PDF).

^ Goldstine, pp. 167–178.

^ ^{a} ^{b} The name for the architecture is discussed in John W. Mauchly and the Development of the ENIAC Computer, part of the online ENIAC museum, in Robert Slater's computer history book, Portraits in Silicon (MIT Press, 1989), and in Nancy Stern's book From ENIAC to UNIVAC (Digital Press,1981).

^ Selected Papers on Computer Science (Center for the Study of Language and Information  Lecture Notes) by Donald E. Knuth (Nov 15, 2004)

^ John von Neumann: Selected Letters, edited by Miklós Rédei, letter to R.S. Burlington, p.73 et seq, Published jointly by The American Mathematics Society and The London Mathematical Society, 2005.

^ Turing's Cathedral, by George Dyson, 2012, ISBN 9781400075997, p. 26768, 287

^ von Neumann, J. (1963). "The Collected Works of John von Neumann". Macmillan.

^ Petrovic, R.; Siljak, D. (1962). "ACTES Proc. of 3rd Int. Analog Comp. Meeting".

^ Afuso, C. (1964). "Quart. Tech. Prog. Rept".

^ ^{a} ^{b} John von Neumann (1966). PDF reprint

^ Éric Filiol, Computer viruses: from theory to applications, Volume 1, Birkhäuser, 2005, pp. 19–38 ISBN 2287239391.

^

^ von Neumann, John (1951). "Various techniques used in connection with random digits".

^ Neumann, John von, "The Point Source Solution," John von Neumann. Collected Works, A. J. Taub (ed.), Vol. 6 [Elmsford, N.Y.: Pergamon Press, 1963], pp. 219–237

^ von Neumann, J. (April 1, 1942. PB 31090). "Theory of Detonation Waves. Progress Report to the National Defense Research Committee Div. B, OSRD549". In Taub, A. H. John von Neumann: Collected Works, 1903–1957 6. New York:

^ Mathematical Association of American documentary, especially comments by Morgenstern regarding this aspect of von Neumann's personality

^ ^{a} ^{b} "Conversation with Marina Whitman". Gray Watson (256.com). Retrieved 20110130.

^ Blair, p. 96.

^ John von Neumann: Selected Letters, Miklos Redei, editor, HMATH 27

^ John von Neumann: Selected Letters, Miklos Redei, editor, HMATH 27

^ Charney JG, Fjörtoft R and von Neumann J (1950). "Numerical Integration of the Barotropic Vorticity Equation".

^ MacRae, p. 332

^ Heims, pp. 236–247.

^ ^{a} ^{b} Goldstine, p. 171.

^ Eugene Wigner, Historical and Biographical Reflections and Syntheses, Springer 2002, p. 129 ISBN 3540572945.

^ Michael Kaplan, Ellen Kaplan, Chances are–: adventures in probability, Viking 2006

^ Darwin Among the Machines: the Evolution of Global Intelligence, Perseus Books, 1998, George Dyson, 77

^ John von Neumann, by Edward Teller, The Bulletin of the Atomic Scientists, April 1957, p. 150.

^ John von Neumann, a Documentary Film, Published in 1966 by the Mathematical Association of America

^ Jacob Bronowski, The Ascent of Man, BBC 1976, p. 433 ISBN 1849901155.

^ Miodrag Petković, Famous puzzles of great mathematicians, American Mathematical Soc., 2009, p. 157 ISBN 0821848143.

^ "Fly Puzzle (Two Trains Puzzle)". Mathworld.wolfram.com. 20140215. Retrieved 20140225.

^ Goldstine, p. 167.

^ Dictionary of Scientific Biography, ed. C. C. Gillispie, Scribners, 1981

^ Glimm, p. 7

^ Stan Ulam, Adventures of a Mathematician, p.96, University of California Press, 1991

^ Bochner, S. (1958). "John von Neumann; A Biographical Memoir" (PDF).

^

^ MacRae, pp. 170–171

^ Ed Regis. Who Got Einstein's Office?: Eccentricity and Genius at the Institute for Advanced Study. Perseus Books 1988 p. 103 ISBN 0671699237.

^ Nancy Stern (January 20, 1981). "An Interview with Cuthbert C. Hurd".

^ MacRae, p. 48

^ From Cardinals To Chaos: Reflections On The Life And Legacy Of Stanislaw Ulam, Necia Grant Cooper, Roger Eckhardt, Nancy Shera, CUP Archive, 1989, Chapter: "The Lost Cafe" by GianCarlo Rota, pp. 26–27 ISBN 0521367344.

^ While there is a general agreement that the initially discovered bone tumor was a secondary growth, sources differ as to the location of the primary cancer. While Macrae gives it as pancreatic, the Life magazine article says it was prostate.

^ MacRae, p. 231.

^ ^{a} ^{b} Macrae, Norman (1999). Story of Philosophy. Amer Mathematical Society; 2 edition.

^ Dies Irae, Stanzas 6—7.

^ John von Neumann at Find a Grave [1]

^ The question of whether or not von Neumann had formally converted to Catholicism upon his marriage to Mariette Kövesi (who was Catholic) is addressed in Halmos, P.R. "The Legend of von Neumann", The American Mathematical Monthly, Vol. 80, No. 4. (April 1973), pp. 382–394. He was baptised Roman Catholic, but certainly was not a practicing member of that religion after his divorce.

^ Norman MacRae (1992). John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More (2 ed.). American Mathematical Soc. p. 379.

^ Dransfield, Robert; Dransfield, Don (2003). Key Ideas in Economics.

^ Raymond George Ayoub (2005). Raymond George Ayoub, ed. Musings Of The Masters: An Anthology Of Mathematical Reflections. MAA. p. 170.

^ Marion Ledwig. "The Rationality of Faith", citing MacRae, p. 379.

^ Abraham Pais (2006). J. Robert Oppenheimer: A Life.

^ William Poundstone (1993). Prisoner's Dilemma.

^ Robert Dransfield; Don Dransfield (2003). Key Ideas in Economics. Nelson Thornes. p. 124.

^ William Poundstone (1993). Prisoner's Dilemma.

^ "Introducing the John von Neumann Computer Society". John von Neumann Computer Society. Retrieved 20080520.

^ ^{a} ^{b} John von Neumann at the Mathematics Genealogy Project. Accessed 20110305.

^ While Israel Halperin's thesis advisor is often listed as Salomon Bochner, this may be because "Professors at the university direct doctoral theses but those at the Institute do not. Unaware of this, in 1934 I asked von Neumann if he would direct my doctoral thesis. He replied Yes." (Israel Halperin, "The Extraordinary Inspiration of John von Neumann", Proceedings of Symposia in Pure Mathematics, vol. 50 (1990), pp. 15–17).
Citations

^ Life Magazine stated that he received both his undergraduate degree and his PhD at the age of 21.^{[17]}

^ While teaching as a professor he was frequently mistaken to be a student.
Notes

PhD Students
See also

1923. On the introduction of transfinite numbers, 346–54.

1925. An axiomatization of set theory, 393–413.

1932. Mathematical Foundations of Quantum Mechanics, Beyer, R. T., trans., Princeton Univ. Press. 1996 edition: ISBN 0691028931.

1944. Theory of Games and Economic Behavior, with Morgenstern, O., Princeton Univ. Press, online at archive.org. 2007 edition: ISBN 9780691130613.

1945. First Draft of a Report on the EDVAC TheFirstDraft.pdf

1963. Collected Works of John von Neumann, Taub, A. H., ed., Pergamon Press. ISBN 0080095666

1966. Theory of SelfReproducing Automata, Burks, A. W., ed., University of Illinois Press. ISBN 0598377980^{[61]}

von Neumann, John (1998) [1960]. Continuous geometry. Princeton Landmarks in Mathematics.

von Neumann, John (1981) [1937]. Halperin, Israel, ed. Continuous geometries with a transition probability. Memoirs of the American Mathematical Society 34 (252).
Selected works
Infopark is situated in the 11th district of Budapest, near the Buda side of Rákóczi bridge, in the university neighborhood, across the river from the National Theatre and the Palace of Arts. The streets bordering Infopark are Hevesy György Street, Boulevard of Hungarian Scientists, Street of Hungarian Nobel Prize Winners and Neumann János street.
Info Park and Neumann János Street

The John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences (INFORMS, previously TIMSORSA) is awarded annually to an individual (or group) who have made fundamental and sustained contributions to theory in operations research and the management sciences.

The IEEE John von Neumann Medal is awarded annually by the IEEE "for outstanding achievements in computerrelated science and technology."

The John von Neumann Lecture is given annually at the Society for Industrial and Applied Mathematics (SIAM) by a researcher who has contributed to applied mathematics, and the chosen lecturer is also awarded a monetary prize.

The crater von Neumann on the Moon is named after him.

The John von Neumann Center in Plainsboro Township, New Jersey () was named in his honour.

The professional society of Hungarian computer scientists, John von Neumann Computer Society, is named after John von Neumann.^{[109]}

On February 15, 1956, Neumann was presented with the Presidential Medal of Freedom by President Dwight Eisenhower.

On May 4, 2005 the United States Postal Service issued the American Scientists commemorative postage stamp series, a set of four 37cent selfadhesive stamps in several configurations designed by artist Victor Stabin. The scientists depicted were John von Neumann, Barbara McClintock, Josiah Willard Gibbs, and Richard Feynman.

The John von Neumann Award of the Rajk László College for Advanced Studies was named in his honour, and has been given every year since 1995 to professors who have made an outstanding contribution to the exact social sciences and through their work have strongly influenced the professional development and thinking of the members of the college.
Honors
While at Walter Reed, he invited a ^{[105]}^{[106]}^{[107]} Even after his conversion, Father Strittmatter recalled that von Neumann did not receive much peace or comfort from it as he still remained terrified of death.^{[108]}
Von Neumann died a year and a half after the diagnosis of cancer, at the Walter Reed Army Medical Center in Washington, D.C. under military security lest he reveal military secrets while heavily medicated. On his death bed, he entertained his brother with wordforword recitations of the first few lines of each page of Goethe's Faust.^{[17]} He was buried at Princeton Cemetery in Princeton, Mercer County, New Jersey.^{[99]}
John von Neumann held on to his exemplary knowledge of Latin and quoted to a deathbed visitor the declamation “Judex ergo cum sedebit,” and ends “Quid sum miser tunc dicturus? Quem patronum rogaturus, Cum vix iustus sit securus?” (When the judge His seat hath taken...What shall wretched I then plead? Who for me shall intercede when the righteous scarce is freed?)^{[97]}^{[98]}
His mother Margaret von Neumann was diagnosed as having cancer, and died within two weeks. John had eighteen months from diagnosis till death. In this period von Neumann returned to the Roman Catholic faith that had also been significant to his mother after the family’s conversion in 192930. There are those who say that he took instruction from the priest at the hospital mainly because the priest was an educated individual, whom von Neumann could talk to of classical Rome and Greece better than he could to the soldiers on guard. But John had earlier said to his mother, “There is probably a God. Many things are easier to explain if there is than if there isn’t."^{[97]}
In 1955, von Neumann was diagnosed with what was either bone or pancreatic cancer.^{[95]} A von Neumann biographer, Norman Macrae, has speculated that the cancer was caused by von Neumann's presence at the Operation Crossroads nuclear tests held in 1946 at Bikini Atoll.^{[96]}
Von Neumann's gravestone
Later life
Von Neumann's closest friend in the United States was the Polish mathematician Stanislaw Ulam. A later friend of Ulam's, GianCarlo Rota writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend.^{[94]}
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "offcolor" humor (especially limericks).^{[12]} At Princeton he received complaints for regularly playing extremely loud German marching music on his gramophone, which distracted those in neighbouring offices, including Einstein, from their work.^{[93]} Von Neumann did some of his best work blazingly fast in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its TV playing loudly.^{[17]}
Despite being a notoriously bad driver, he nonetheless enjoyed driving (frequently while reading a book)—occasioning numerous arrests as well as accidents. When Cuthbert Hurd hired him as a consultant to IBM, Hurd often quietly paid the fines for his traffic tickets.^{[92]} He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.^{[17]}
He was sociable and enjoyed throwing large parties at his home in Princeton,^{[17]} occasionally twice a week.^{[90]} His white clapboard house at 26 Westcott Road was one of the largest in Princeton.^{[91]}
Von Neumann took great care over his clothing, and would always wear formal suits, once riding down the Grand Canyon astride a mule in a threepiece pinstripe.^{[68]} Mathematician David Hilbert is reported to have asked at von Neumann's 1926 doctoral exam: "Pray, who is the candidate's tailor?" as he had never seen such beautiful evening clothes.^{[89]}
Von Neumann had a wide range of cultural interests. Since the age of six, von Neumann had been fluent in Latin and ancient Greek, and he held a lifelong passion for ancient history, being renowned for his prodigious historical knowledge. A professor of Byzantine history once said that von Neumann had greater expertise in Byzantine history than he did.^{[17]}
Von Neumann married twice. He married Mariette Kövesi in 1930, just prior to emigrating to the United States. Before his marriage he was also baptized a Catholic in 1930.^{[88]} They had one daughter (von Neumann's only child), Marina, who is now a distinguished professor of international trade and public policy at the University of Michigan. The couple divorced in 1937. In 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II. The von Neumanns were very active socially within the Princeton academic community.
Personal life

A facility with the symbolic manipulation of linear operators;

An intuitive feeling for the logical structure of any new mathematical theories;

An intuitive feeling for the combinatorial superstructure of new theories.
Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: "Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods." He went on to explain that the three methods were:^{[87]}
Mastery of mathematics
It has been said that von Neumann's intellect was absolutely unmatched. "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man", said Nobel Laureate Hans A. Bethe of Cornell University.^{[17]} "It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei in "Selected Letters." Glimm writes "he is regarded as one of the giants of modern mathematics".^{[3]} The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians",^{[85]} while Peter Lax described him as possessing the "most scintillating intellect of this century".^{[86]}
Von Neumann had a very strong eidetic memory, commonly called 'photographic' memory.^{[17]} Herman Goldstine writes: "One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how The Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes."^{[84]}
Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 mph. At the same time a fly that travels at a steady 15 mph starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover? The slow way to find the answer is to calculate what distance the fly covers on the first, northbound, leg of the trip, then on the second, southbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann, "All I did was sum the geometric series."^{[12]}
ETH Zurich von Neumann attended as a student, said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."^{[82]} 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:^{[83]}
Von Neumann's ability instantaneously to perform complex operations in his head stunned other mathematicians.^{[75]} Eugene Wigner wrote that, seeing von Neumann's mind at work, "one had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch."^{[76]} Paul Halmos states that "von Neumann's speed was aweinspiring."^{[12]} Israel Halperin said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car."^{[77]} Edward Teller wrote that von Neumann effortlessly outdid anybody he ever met,^{[78]} and said "I never could keep up with him".^{[79]} Teller also said "von Neumann would carry on a conversation with my 3 year old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us. Most people avoid thinking if they can, some of us are addicted to thinking, but von Neumann actually enjoyed thinking, maybe even to the exclusion of everything else."^{[80]}
Cognitive abilities
Von Neumann's team performed the world's first numerical weather forecasts on the ENIAC computer; von Neumann published the paper Numerical Integration of the Barotropic Vorticity Equation in 1950.^{[72]} Von Neumann's interest in weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice caps to enhance absorption of solar radiation (by reducing the albedo), thereby inducing global warming.^{[73]}^{[74]}
Weather systems

Yes, Europe has more culture than the US, just as Greece had more culture than Rome.

However, Europe is descended from Macedonia, who were barbarians, rather than Greece.

The Roman empire was purely defensive in its early days.

It became imperialistic only near the end.
He compared the US and Europe to Rome and Greece as follows:^{[71]}
Greece and Rome

France won't matter.

The US interest in the coming war is purely defensive and not imperialistic.

The US may develop imperialistic ambitions after the war.

It would not be profitable for the US to sell arms to combatants, because it would be on credit and such debts are never paid. (Roosevelt banned such sales in 1935.)
His shrewd prewar analysis is often quoted.^{[70]} His remarks included:
On the eve of World War II
As president of the von Neumann Committee for Missiles, and later as a member of the United States Atomic Energy Commission, from 1953 until his death in 1957, he was influential in setting U.S. scientific and military policy. Through his committee, he developed various scenarios of nuclear proliferation, the development of intercontinental and submarine missiles with atomic warheads, and the controversial strategic equilibrium called mutual assured destruction. During a Senate committee hearing he described his political ideology as "violently anticommunist, 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?".^{[69]}
Throughout his life von Neumann had a respect and admiration for business and government leaders; something which was often at variance with the inclinations of his scientific colleagues.^{[67]} Von Neumann entered government service (Manhattan Project) primarily because he felt that, if freedom and civilization were to survive, it would have to be because the U.S. would triumph over totalitarianism from the right (Nazism and Fascism) and totalitarianism from the left (Soviet Communism).^{[68]}
, and others.
IBM,
General Electric,
Standard Oil,
RAND Corporation, the
United States Army, the
Central Intelligence Agency). He was a frequent consultant for the
Albert Einstein (another had gone to
Princeton, New Jersey in
Institute for Advanced StudyVon Neumann obtained at the age of 29 one of the first five professorships at the new
John von Neumann at The Princeton Institute for Advanced Study (Left to right: Julian Bigelow, Herman Goldstine, J. Robert Oppenheimer, and John von Neumann).
Politics and social affairs
Other well known contributions to fluid dynamics included the classic flow solution to blast waves,^{[65]} and the codiscovery of the ZND detonation model of explosives.^{[66]}
A problem was that when computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics.
Von Neumann made fundamental contributions in exploration of problems in numerical hydrodynamics. For example, with R. D. Richtmyer he developed an algorithm defining artificial viscosity that improved the understanding of shock waves. It is possible that we would not understand much of astrophysics, and might not have highly developed jet and rocket engines without the work of von Neumann.
Fluid dynamics
His algorithm for simulating a fair coin with a biased coin^{[64]} is used in the "software whitening" stage of some hardware random number generators.
Donald Knuth cites von Neumann as the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged.^{[63]}
Beginning in 1949, Von Neumann's design for a selfreproducing computer program is considered the world's first computer virus, and he is considered to be the theoretical father of computer virology.^{[62]}
Von Neumann's rigorous mathematical analysis of the structure of selfreplication (of the semiotic relationship between constructor, description and that which is constructed), preceded the discovery of the structure of DNA.^{[5]}
Von Neumann also created the field of cellular automata without the aid of computers, constructing the first selfreplicating automata with pencil and graph paper. The concept of a universal constructor was fleshed out in his posthumous work Theory of Self Reproducing Automata.^{[61]} Von Neumann proved that the most effective way of performing largescale mining operations such as mining an entire moon or asteroid belt would be by using selfreplicating machines, taking advantage of their exponential growth.
Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953.^{[58]} However, the theory could not be implemented until advances in computing of the 1960s.^{[59]}^{[60]}
The next computer that von Neumann designed was the IAS machine at The Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the RCA Research Laboratory nearby. John von Neumann recommended that the IBM 701, nicknamed the defense computer include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successful IBM 704.^{[56]}^{[57]}
This architecture is to this day the basis of modern computer design, unlike the earliest computers that were 'programmed' by altering the electronic circuitry. Although the singlememory, stored program architecture is commonly called von Neumann architecture as a result of von Neumann's paper, the architecture's description was based on the work of J. Presper Eckert and John William Mauchly, inventors of the ENIAC at the University of Pennsylvania.^{[54]}
John von Neumann also consulted for the ENIAC project when ENIAC was being modified to contain stored programs. Since the modified ENIAC was fully functional by 1948 and the EDVAC wasn't delivered to Ballistics Research Laboratory until 1949, one could argue that ENIAC was the first computer to use a stored program in production runs. John von Neumann also designed the instruction set or op codes for the modified ENIAC, and he should be given credit for this. The electronics of the new ENIAC ran 6 times slower, but this in no way degraded the ENIAC's performance, since it was still entirely I/O bound. On the other hand, complicated programs could be developed and debugged in days rather than the weeks required for plugboarding the old ENIAC. Always mission driven, von Neumann had modified the ENIAC in order to get a head start on verifying the Hbomb calculations of Stan Ulam and C.J.Everett. Some of von Neumann's computer programs, the first programs written for a von Neumann machine, have been preserved.^{[55]}
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 William Mauchly, described a computer architecture in which the data and the program are both stored in the computer's memory in the same address space.^{[54]}
Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middlesquare method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, and also noted that when it went awry it did so obviously, unlike methods which could be subtly incorrect.
Von Neumann was a founding figure in computing.^{[53]} Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and Stanislaw Ulam developed simulations on von Neumann's digital computers for the hydrodynamic computations. During this time he contributed to the development of the Monte Carlo method, which allowed solutions to complicated problems to be approximated using random numbers. He was also involved in the design of the later IAS machine.
The first implementation of von Neumann's selfreproducing universal constructor.^{[52]} Three generations of machine are shown, the second has nearly finished constructing the third. The lines running to the right are the tapes of genetic instructions, which are copied along with the body of the machines. The machine shown runs in a 32state version of von Neumann's cellular automata environment.
Computing
John von Neumann is credited with the equilibrium strategy of mutually assured destruction, providing the deliberately humorous acronym, MAD. (Other humorous acronyms coined by von Neumann include his computer, the Mathematical Analyzer, Numerical Integrator, and Computer – or MANIAC).
Mutually assured destruction
In 1955, von Neumann became a commissioner of the United States Atomic Energy Program. Shortly before his death, when he was already quite ill, von Neumann headed the US government's top secret Intercontinental ballistic missile (ICBM) committee, and it would sometimes meet in his home. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were sizable, they could be overcome in time. The SM65 Atlas passed its first fully functional test in 1959, two years after his death. The feasibility of an ICBM owed as much to improved, smaller warheads as it did to developments in rocketry, and his understanding of the former made his advice invaluable.
The ICBM Committee
In 1954 von Neumann was invited to become a member of the Atomic Energy Committee. He accepted this powerful position and used it to further the production of compact Hbombs suitable for ICBM delivery. He involved himself personally 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 Hbombs 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 HBomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case.^{[51]}
The Atomic Energy Committee
The Fuchs–von Neumann work was passed on, by Fuchs, to the USSR as part of his nuclear espionage, but it was not used in the Soviets' own, independent development of the Teller–Ulam design. The historian Jeremy Bernstein has pointed out that ironically, "John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made."^{[50]}
Von Neumann continued unperturbed in his work and became, along with Operation Greenhouse, which was instructive in testing out concepts that went into the final design.^{[50]}
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."
On July 16, 1945, with numerous other Los Alamos personnel, von Neumann was an eyewitness to the first atomic bomb blast, code named Trinity, conducted as a test of the implosion method device, on the White Sands Proving Ground, 35 miles (56 km) southeast of Socorro, New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5 kilotons of TNT, but Enrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of tornup 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.^{[46]}
Beginning in the spring of 1945, along with four other scientists and various military personnel, von Neumann was included in the target selection committee responsible for choosing the Japanese cities of Hiroshima and Nagasaki as the first targets of the atomic bomb. Von Neumann oversaw computations related to the expected size of the bomb blasts, estimated death tolls, and the distance above the ground at which the bombs should be detonated for optimum shock wave propagation and thus maximum effect.^{[47]} The cultural capital Kyoto, which had been spared the firebombing inflicted upon militarily significant target cities like Tokyo in World War II, was von Neumann's first choice, a selection seconded by Manhattan Project leader General Leslie Groves. However, this target was dismissed by Secretary of War Henry Stimson.^{[48]}
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.^{[46]}
When it turned out that there would not be enough Trinity bomb was completed in July 1945.
Von Neumann's principal contribution to the atomic bomb itself was in the concept and design of the explosive lenses needed to compress the plutonium core of the Trinity test device and the "Fat Man" weapon that was later dropped on Nagasaki. While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly" (compression).
Beginning in the late 1930s, von Neumann developed an expertise in explosions—phenomena which are difficult to model mathematically. During this period von Neumann was the leading authority of the mathematics of shaped charges. This led him to a large number of military consultancies, primarily for the Navy, which in turn led to his involvement in the Manhattan Project. The involvement included frequent trips by train to the project's secret research facilities in Los Alamos, New Mexico.^{[2]}
Nuclear weapons
Subsequently, John Denis Sargan and Alok Bhargava^{[45]} 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.
Von Neumann made fundamental contributions to [44]
Mathematical statistics
Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873) which was later popularized by [42]
Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in [42]
Linear programming
The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of Nobel prizes in 1972 to Kenneth Arrow, in 1983 to Gérard Debreu, and in 1994 to John Nash who used fixed point theorems to establish equilibria for noncooperative games and for bargaining problems in his Ph.D. thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans, Leonid Kantorovich, Wassily Leontief, Paul Samuelson, Robert Dorfman, Robert Solow, and Leonid Hurwicz.
Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices.^{[37]} The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.^{[38]}^{[39]}^{[40]} This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixedpoint theorems, linear inequalities, complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself.^{[41]}
along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.^{[34]}^{[35]}^{[36]}

p^{T} (A − λ B) q = 0,
Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation
Mathematical economics
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.^{[33]}
[32]
Independently,
Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 The New York Times ran a frontpage story. In this book, von Neumann declared that economic theory needed to use functional analytic methods, especially convex sets and topological fixed point theorem, rather than the traditional differential calculus, because the maximum–operator did not preserve differentiable functions.
Von Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zerosum games with perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.
Game theory
Von Neumann proposes to replace classical logics, with a logic constructed in orthomodular lattices, (isomorphic to the lattice of subspaces of the Hilbert space of a given physical system).^{[30]}
In a famous paper of 1936, the first work ever to introduce quantum logics,^{[29]} von Neumann first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work. But in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters which are polarized perpendicularly (e.g., one horizontally and the other vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession. But if the third filter is added in between the other two, the photons will indeed pass through. And this experimental fact is translatable into logic as the noncommutativity of conjunction (A\land B)\ne (B\land A). It was also demonstrated that the laws of distribution of classical logic, P\lor(Q\land R)=(P\lor Q)\land(P\lor R) and P\land (Q\lor R)=(P\land Q)\lor(P\land R), are not valid for quantum theory. The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically determinate, while each of its members are necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semiintegral 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 A \land (B\lor C)= A\land 1 = A, while (A\land B)\lor (A\land C)=0\lor 0=0.
Quantum logic
Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formalism of problems in quantum mechanics which underlies the majority of approaches and can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.
In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the socalled measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, it never gained acceptance amongst the majority of physicists).^{[28]}
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs. nondeterminism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 1966, John S. Bell published a paper arguing that the proof contained a conceptual error and was therefore invalid (see the article on John Stewart Bell for more information). However, in 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a welldefined and important subset. Bub also suggests that von Neumann was aware of this limitation, and that von Neumann did not claim that his proof completely ruled out hidden variable theories.^{[27]} In any case, the proof inaugurated a line of research that ultimately led, through the work of Bell in 1964 on Bell's theorem, and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics either requires a notion of reality substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity.
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 noncommutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger.
After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a socalled Hilbert space, analogous to the 6N dimension (N is the number of particles, 3 general coordinate and 3 canonical momentum for each) phase space of classical mechanics but with infinitely many dimensions (corresponding to the infinitely many possible states of the system) instead: the traditional physical quantities (e.g., position and momentum) could therefore be represented as particular linear operators operating in these spaces. The physics of quantum mechanics was thereby reduced to the mathematics of the linear Hermitian operators on Hilbert spaces.
Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematische Grundlagen der Quantenmechanik.
Mathematical formulation of quantum mechanics
Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal ^{[26]}
Von Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity."^{[26]} Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor".^{[26]}
Lattice theory
The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.
The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
Von Neumann introduced the study of rings of operators, through the von Neumann algebras.^{[25]} A von Neumann algebra is a *algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator.
Operator theory
Von Neumann made foundational contributions to ergodic theory, in a series of articles published in 1932.^{[23]} Of the 1932 papers on ergodic theory, Paul Halmos writes that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality".^{[21]} By then von Neumann had already written his famous articles on operator theory, and the application of this work was instrumental in the von Neumann mean ergodic theorem.^{[24]}
Ergodic theory
In the 1936 paper on analytic measure theory, von Neumann used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups.^{[21]}^{[22]}
In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions (anticipating his later work, Mathematical formulation of quantum mechanics, on almost periodic functions).
In a series of famous papers, von Neumann made spectacular contributions to measure theory.^{[21]} The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially grouptheoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."
Measure theory
Von Neumann founded the field of continuous geometry. It followed his pathbreaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
Geometry
But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency.^{[19]} However, Gödel had already discovered this consequence (now known as his second incompleteness theorem), and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter.^{[20]}
With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time.^{[19]}
The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo/Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.
The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession (hence excluding the possibility of a set belonging to itself.) To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration (called the method of inner models) which later became an essential instrument in set theory.
The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later (by Ernst Zermelo and Abraham Fraenkel). Zermelo and Fraenkel provided Zermelo–Fraenkel set theory, a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets: the axiom of foundation and the notion of Class (set theory).
The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigor and breadth at the end of the 19th century, particularly in arithmetic (thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce) and geometry (thanks to David Hilbert). At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves).
Set theory
In 1930, von Neumann was invited to Princeton University, New Jersey. In 1933, he was offered a position on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through; von Neumann remained a mathematics professor there until his death. His mother and his brothers followed John to the United States, his father, Max Neumann, having died in 1929. He anglicized his first name to John, keeping the Germanaristocratic surname of von Neumann. In 1937, von Neumann became a naturalized citizen of the U.S. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.
Between 1926 and 1930, he taught as a Privatdozent at the University of Berlin, the youngest in its history.^{[N 2]} By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirtytwo papers, at a rate of nearly one major paper per month.^{[18]} Von Neumann's alleged powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease.^{[17]}
Beginnings
Career and abilities
He received his Ph.D. in mathematics (with minors in experimental physics and chemistry) from Pázmány Péter University in Budapest at the age of 22.^{[2]} He simultaneously earned a diploma in chemical engineering from the ETH Zurich in Switzerland^{[2]} at his father's request, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavour than mathematics.^{[N 1]}
[16]
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