John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American

Holevo entropy

and conditional quantum entropy.

Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

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, wherein von Neumann and Birkhoff first proved that quantum mechanics requires a

Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American Theoretical physics, theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the History of the Teller–Ulam d ...

admitted that he "never could keep up with him". 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." Peter Lax wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics". Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...

described him in this way: "I have known a great many intelligent people in my life. I knew Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American Theoretical physics, theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the History of the Teller–Ulam d ...

have been among my closest friends; and quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...

, atomic,

Herman Goldstine
Herman Heine Goldstine (September 13, 1913 – June 16, 2004) was a mathematician and computer scientist, who worked as the director of the IAS machine at Princeton University's Institute for Advanced Study and helped to develop ENIAC, the ...

writes:
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. In his autobiography Stanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Teller ...

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.

zbMATH

an

Google Scholar

A list of his known works as of 1995 can be found i

The Neumann Compendium

available here

2018 edition: * 1937.

Continuous Geometry

', Israel Halperin, Halperin, I., Preface, Princeton Landmarks in Mathematics and Physics, Princeton University Press

online at archive.org

2016 edition: * 1937.

Continuous Geometries with a Transition Probability

', Israel Halperin, Halperin, I., Preface, Memoirs of the American Mathematical Society Vol. 34, No. 252, 1981 edition. * 1941.

Invariant Measures

'. American Mathematical Society. 1999 edition: * 1944. '' Theory of Games and Economic Behavior'', with Oskar Morgenstern, Morgenstern, O., Princeton University Press

online at archive.org**or here**

2007 edition: * 1950.

Functional Operators, Volume 1: Measures and Integrals

'. Annals of Mathematics Studies 21

online at archive.org

2016 edition: * 1951.

Functional Operators, Volume 2: The Geometry of Orthogonal Spaces

'. Annals of Mathematics Studies 22

online at archive.org

2016 edition * 1958. '' The Computer and the Brain'', Ray Kurzweil, Kurzweil, R. Preface, The Silliman Memorial Lectures Series, Yale University Press

online at archive.org**or here**

2012 edition: * 1966.

Theory of Self-Reproducing Automata

', Arthur Burks, Burks, A. W., Ed., University of Illinois Press.

On the introduction of transfinite numbers

(in German), ''Acta Szeged'', 1:199-208. * 1925

An axiomatization of set theory

(in German), ''J. f. Math.'', 154:219-240. * 1926

On the Prüfer theory of ideal numbers

(in German), ''Acta Szeged'', 2:193-227. * 1927

On Hilbert's proof theory

(in German), ''Math. Zschr.'', 26:1-46. * 1929

General eigenvalue theory of Hermitian functional operators

(in German), ''Math. Ann.'', 102:49-131. * 1932

Proof of the Quasi-Ergodic Hypothesis

''Proc. Nat. Acad. Sci.'', 18:70-82. * 1932

Physical Applications of the Ergodic Hypothesis

''Proc. Nat. Acad. Sci.'', 18:263-266. * 1932

On the operator method in classical mechanics

(in German), ''Ann. Math.'', 33:587-642. * 1934

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

On Rings of Operators

with F. J. Murray, ''Ann. Math.'', 37:116-229. * 1936

On an Algebraic Generalization of the Quantum Mechanical Formalism (Part I)

''Mat. Sborn.'', 1:415-484. * 1936

The Logic of Quantum Mechanics

with Garrett Birkhoff, G. Birkhoff, ''Ann. Math.'', 37:823-843. * 1936

Continuous Geometry

''Proc. Nat. Acad. Sci.'', 22:92-100. * 1936

Examples of Continuous Geometries

''Proc. Nat. Acad. Sci.'', 22:101-108. * 1936

On Regular Rings

''Proc. Nat. Acad. Sci.'', 22:707-713. * 1937

On Rings of Operators, II

with F. J. Murray, ''Trans. Amer. Math. Soc.'', 41:208-248. * 1937

Continuous Rings and Their Arithmetics

''Proc. Nat. Acad. Sci.'', 23:341-349. * 1938

On Infinite Direct Products

''Compos. Math.'', 6:1-77. * 1940

On Rings of Operators, III

''Ann. Math.'', 41:94-161. * 1942

Operator Methods in Classical Mechanics, II

with Paul Halmos, P. R. Halmos, ''Ann. Math.'', 43:332-350. * 1943

On Rings of Operators, IV

with F. J. Murray, ''Ann. Math.'', 44:716-808. * 1945

A Model of General Economic Equilibrium

''Rev. Econ. Studies'', 13:1-9. * 1945

''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

Numerical Inverting of Matrices of High Order

with Herman Goldstine, H. H. Goldstine, ''Bull. Amer. Math. Soc.'', 53:1021-1099. * 1948

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. 1–31

MR0045446

* 1949

On Rings of Operators. Reduction Theory

''Ann. Math.'', 50:401-485. * 1950

A Method for the Numerical Calculation of Hydrodynamic Shocks

with R. D. Richtmyer, ''J. Appl. Phys.'', 21:232-237. * 1950

Numerical Integration of the Barotropic Vorticity Equation

with Jule Gregory Charney, J. G. Charney and Ragnar Fjørtoft, R. Fjörtoft, ''Tellus'', 2:237-254. * 1951

A spectral theory for general operators of a unitary space

(in German), ''Math. Nachr.'', 4:258-281. * 1951

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

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

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.

The Mathematician

''The Works of the Mind''. ed. by R. B. Heywood, University of Chicago Press, 180–196. * 1951

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

The NORC and Problems in High Speed Computing

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

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.

Description

contents, incl. arrow-scrollable preview

&

review

* * * * * * ** ** * * * * * * * * * * * * * ** ** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ** ** * * * * * * * * * * * * * ** ** * * * * * * * * * * * * * * * * * * * * * * * *

available here

Journals * * * *

by Nelson H. F. Beebe *

von Neumann's profile

at Google Scholar

Oral History Project

- The Princeton Mathematics Community in the 1930s, contains many interviews that describe contact and anecdotes of von Neumann and others at the Princeton University and Institute for Advanced Study community.

Oral history interview with Alice R. Burks and Arthur W. Burks

Charles Babbage Institute, University of Minnesota, Minneapolis. Alice Burks and Arthur Burks describe

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.

zbMATH profile

Query for "von neumann"

on the digital repository of the Institute for Advanced Study.

Von Neumann vs. Dirac on Quantum Theory and Mathematical Rigor

– from ''Stanford Encyclopedia of Philosophy''

Quantum Logic and Probability Theory

- from ''Stanford Encyclopedia of Philosophy''

FBI files on John von Neumann released via FOI

Biographical video

by David Brailsford (John Dunford Professor Emeritus of computer science at the University of Nottingham)

A (very) Brief History of John von Neumann

video by YouTuber moderndaymath.

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).

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.

Greatest Mathematician Of The 20th Century

high-quality excerpt from above documentary whereEdward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American Theoretical physics, theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the History of the Teller–Ulam d ...

describes John von Neumann.
{{DEFAULTSORT:Neumann, John von
John von Neumann,
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Von Neumann family, John
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mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate caus ...

, computer scientist
A computer scientist is a person who is trained in the academic study of computer science.
Computer scientists typically work on the theoretical side of computation, as opposed to the computer hardware, hardware side on which computer engineer ...

, and polymath
A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...

. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences.
Von Neumann made major contributions to many fields, including mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

( foundations of mathematics, measure theory, functional analysis, ergodic theory, group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, lattice theory, representation theory, operator algebras, matrix theory, geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

, and numerical analysis), physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...

(quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...

, hydrodynamics, ballistics, nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...

and quantum statistical mechanics), economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...

(game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appl ...

and general equilibrium theory), computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , hardware and software. ...

(, linear programming, numerical meteorology, scientific computing
Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...

, self-replicating machines, stochastic computing), and statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

. 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 automata, the universal constructor and the digital computer
A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...

.
Von Neumann published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on special mathematical subjects or non-mathematical ones. His last work, an unfinished manuscript written while he was dying in hospital, 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
The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

, 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."
During World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

, von Neumann worked on the Manhattan Project
The Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...

with theoretical physicist Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American Theoretical physics, theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the History of the Teller–Ulam d ...

, mathematician Stanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Teller ...

and others, problem-solving key steps in the nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...

involved in thermonuclear
Thermonuclear fusion is the process of atomic nuclei combining or “fusing” using high temperatures to drive them close enough together for this to become possible. There are two forms of thermonuclear fusion: ''uncontrolled'', in which the re ...

reactions and the hydrogen bomb
A thermonuclear weapon, fusion weapon or hydrogen bomb (H bomb) is a second-generation nuclear weapon design. Its greater sophistication affords it vastly greater destructive power than first-generation nuclear bombs, a more compact size, a lowe ...

. He developed the mathematical models behind the explosive lens
An explosive lens—as used, for example, in nuclear weapons—is a highly specialized shaped charge. In general, it is a device composed of several explosive charges. These charges are arranged and formed with the intent to control the shape ...

es used in the implosion-type nuclear weapon
Nuclear weapon designs are physical, chemical, and engineering arrangements that cause the physics package of a nuclear weapon to detonate. There are three existing basic design types:
* pure fission weapons, the simplest and least technically ...

and coined the term "kiloton" (of TNT) as a measure of the explosive force generated. During this time and after the war, he consulted for a vast number of organizations including the Office of Scientific Research and Development
The Office of Scientific Research and Development (OSRD) was an agency of the United States federal government created to coordinate scientific research for military purposes during World War II. Arrangements were made for its creation during May 1 ...

, the Army's Ballistic Research Laboratory
The Ballistic Research Laboratory (BRL) was a leading United States Army, U.S. Army research establishment situated at Aberdeen Proving Ground, Maryland that specialized in ballistics (internal ballistics, interior, external ballistics, exterior, a ...

, the Armed Forces Special Weapons Project and the Oak Ridge National Laboratory
Oak Ridge National Laboratory (ORNL) is a U.S. multiprogram science and technology United States Department of Energy national laboratories, national laboratory sponsored by the United States Department of Energy, U.S. Department of Energy (DOE) ...

. At the peak of his influence in the 1950s he was the chair for a number of critical Defense Department committees including the Nuclear Weapons Panel of the Air Force
An air force – in the broadest sense – is the national military branch that primarily conducts aerial warfare. More specifically, it is the branch of a nation's armed services that is responsible for aerial warfare as distinct from an ar ...

Scientific Advisory Board and the ICBM Scientific Advisory Committee as well as a member of the influential Atomic Energy Commission. He played a key role alongside Bernard Schriever
Bernard Adolph Schriever (14 September 1910 – 20 June 2005), also known as Bennie Schriever, was a United States Air Force
The United States Air Force (USAF) is the Aerial warfare, air military branch, service branch of the United Sta ...

and Trevor Gardner in contributing to the design and development of the United States' first ICBM
An intercontinental ballistic missile (ICBM) is a ballistic missile with a range (aeronautics), range greater than , primarily designed for nuclear weapons delivery (delivering one or more Thermonuclear weapon, thermonuclear warheads). Conventi ...

programs. During this time he was considered the nation's foremost expert on nuclear weaponry and the leading defense scientist at the Pentagon
The Pentagon is the headquarters building of the United States Department of Defense. It was constructed on an accelerated schedule during World War II
World War II or the Second World War, often abbreviated as WWII or WW2, w ...

. As a Hungarian émigré, concerned that the Soviets would achieve nuclear superiority, he designed and promoted the policy of mutually assured destruction
Mutual assured destruction (MAD) is a doctrine of military strategy and national security, national security policy which posits that a full-scale use of nuclear weapons by an attacker on a nuclear-armed defender with Second strike, second-stri ...

to limit the arms race.
In honor of his achievements and contributions to the modern world, he was named in 1999 the ''Financial Times
The ''Financial Times'' (''FT'') is a British daily newspaper
A newspaper is a Periodical literature, periodical publication containing written News, information about current events and is often typed in black ink with a white or gray ba ...

'' 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.
Life and education

Family background

Von Neumann was born on December 28, 1903, to a wealthy, acculturated and non-observantJewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation
A nation is a community of people formed on the basis of a combination of shared features such as language, history, ethnicity, culture and/or ...

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.
Von Neumann was born in Budapest
Budapest (, ; ) is the capital and most populous city of Hungary. It is the ninth-largest city in the European Union
The European Union (EU) is a supranational union, supranational political union, political and economic union of ...

, Kingdom of Hungary
The Kingdom of Hungary was a monarchy in Central Europe that existed for nearly a millennium, from the Middle Ages into the 20th century. The Principality of Hungary emerged as a Christian kingdom upon the Coronation of the Hungarian monarch, c ...

, which was then part of the Austro-Hungarian Empire
Austria-Hungary, often referred to as the Austro-Hungarian Empire,, the Dual Monarchy, or Austria, was a constitutional monarchy and great power in Central Europe between 1867 and 1918. It was formed with the Austro-Hungarian Compromise ...

. 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). His father, Neumann Miksa (Max von Neumann, 1873–1928) 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 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); her parents were Jakab Kann and Katalin Meisels of the Meisels family. 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.
On February 20, 1913, Emperor Franz Joseph
Franz Joseph I or Francis Joseph I (german: Franz Joseph Karl, hu, Ferenc József Károly, 18 August 1830 – 21 November 1916) was Emperor of Austria, King of Hungary, and the other states of the Habsburg monarchy
The Habsburg monarchy (g ...

elevated John's father to the Hungarian nobility for his service to the Austro-Hungarian Empire. The Neumann family thus acquired the hereditary appellation ''Margittai'', meaning "of Margitta" (today Marghita, Romania
Romania ( ; ro, România ) is a country located at the crossroads of Central Europe, Central, Eastern Europe, Eastern, and Southeast Europe, Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, S ...

). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosen coat of arms
A coat of arms is a heraldry, heraldic communication design, visual design on an escutcheon (heraldry), escutcheon (i.e., shield), surcoat, or tabard (the latter two being outer garments). The coat of arms on an escutcheon forms the central ele ...

depicting three marguerites. Neumann János became margittai Neumann János (John Neumann de Margitta), which he later changed to the German Johann von Neumann.
Child prodigy

Von Neumann was achild prodigy
A child prodigy is defined in psychology research literature as a person under the age of ten who produces meaningful output in some domain at the level of an adult expert. The term is also applied more broadly to young people who are extraor ...

. When he was six years old, he could divide two eight-digit numbers in his head and could converse in Ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Greek Dark ...

. When the six-year-old von Neumann caught his mother staring aimlessly, he asked her, "What are you calculating?"
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 was essential, so the children were tutored in English, French, German and Italian. By the age of eight, von Neumann was familiar with differential and integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...

, and by twelve he had read and understood Borel's Théorie des Fonctions. But he was also particularly interested in history. He read his way through Wilhelm Oncken's 46-volume world history series (''General History in Monographs''). 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.
Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1914. Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...

was a year ahead of von Neumann at the Lutheran School and soon became his friend. 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. The Hungarian school system produced a generation noted for intellectual achievement, which included Theodore von Kármán
Theodore von Kármán ( hu, (Szőlőskislak, szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fie ...

(born 1881), George de Hevesy
George Charles de Hevesy (born György Bischitz; hu, Hevesy György Károly; german: Georg Karl von Hevesy; 1 August 1885 – 5 July 1966) was a Hungarian Radiochemistry, radiochemist and Nobel Prize in Chemistry laureate, recognized in 1943 ...

(born 1885), Michael Polanyi
Michael Polanyi (; hu, Polányi Mihály; 11 March 1891 – 22 February 1976) was a Hungarian-British polymath, who made important theoretical contributions to physical chemistry, economics, and philosophy. He argued that positivism supplies ...

(born 1891), Leó Szilárd (born 1898), Dennis Gabor
Dennis Gabor ( ; hu, Gábor Dénes, ; 5 June 1900 – 9 February 1979) was a Hungarian-British Electrical engineering, electrical engineer and physicist, most notable for inventing holography, for which he later received the 1971 Nobel Prize ...

(born 1900), Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...

(born 1902), Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American Theoretical physics, theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the History of the Teller–Ulam d ...

(born 1908), and Paul Erdős (born 1913). Collectively, they were sometimes known as " The Martians".
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. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. 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. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...

, which superseded Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, ...

's definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.
University studies

According to his friendTheodore von Kármán
Theodore von Kármán ( hu, (Szőlőskislak, szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fie ...

, 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. Von Neumann and his father decided that the best career path was to become a chemical engineer
In the field of engineering, a chemical engineer is a professional, equipped with the knowledge of chemical engineering, who works principally in the chemical industry to convert basic raw materials into a variety of products and deals with the ...

. 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
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin
Berlin ( , ) is the capital and largest city of Germany ...

, after which he sat for the entrance exam to the prestigious ETH Zurich
(colloquially)
, former_name = eidgenössische polytechnische Schule
, image = ETHZ.JPG
, image_size =
, established =
, type = Public
, budget = CHF 1.896 billion (2021)
, rector = Günther Dissertori
, president = Joël Mesot
, ...

, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

. For his thesis, he chose to produce an axiomatization of Cantor's set theory. 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),''The Collected Works of Eugene Paul Wigner: Historical, Philosophical, and Socio-Political Papers. Historical and Biographical Reflections and Syntheses'', By Eugene Paul Wigner, (Springer 2013), page 128 and passed his final examinations for his Ph.D. in mathematics simultaneously with his chemical engineering degree, of which Wigner wrote, "Evidently a Ph.D. thesis and examination did not constitute an appreciable effort." He then went to the University of Göttingen on a grant from the Rockefeller Foundation
The Rockefeller Foundation is an American private foundation and philanthropy, philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The second-oldest major philanthropic institution in America, aft ...

to study mathematics under David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

.
Career and private life

Von Neumann'shabilitation
Habilitation is the highest academic degree, university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, us ...

was completed on December 13, 1927, and he began to give lectures as a ''Privatdozent
''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualific ...

'' at the University of Berlin in 1928. He was the youngest person ever elected ''Privatdozent'' in the university's history in any subject. 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. In 1929, he briefly became a ''Privatdozent'' at the University of Hamburg
The University of Hamburg (german: link=no, Universität Hamburg, also referred to as UHH) is a public university, public research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ...

, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

as a visiting lecturer in mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

.
On New Year's Day 1930, von Neumann married Marietta Kövesi, who had studied economics at Budapest University. Von Neumann and Marietta had one child, a daughter, Marina
A marina (from Spanish , Portuguese and Italian : ''marina'', "coast" or "shore") is a dock or basin with moorings and supplies for yachts and small boats.
A marina differs from a port
A port is a maritime law, maritime ...

, born in 1935. As of 2021 Marina is a distinguished professor emerita of business administration and public policy at the University of Michigan
The University of Michigan (U-M, UMich, or Michigan) is a public university, public research university in Ann Arbor, Michigan. Founded in 1817 by an act of the old Michigan Territory as the History of the University of Michigan#The Catholepistemi ...

. The couple divorced on November 2, 1937. On November 17, 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest before the outbreak of World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

.
In 1930, before marrying Marietta, von Neumann was baptized into the Catholic Church
The Catholic Church, also known as the Roman Catholic Church, is the List of Christian denominations by number of members, largest Christian church, with 1.3 billion baptized Catholics Catholic Church by country, worldwide . It is am ...

. Von Neumann's father, Max, had died in 1929. None of the family had converted to Christianity while Max was alive, but all did afterward.
In 1933 Von Neumann was offered and accepted a life tenure professorship at the Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

in New Jersey, when that institution's plan to appoint Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

appeared to have failed. Von Neumann remained a mathematics professor there until his death, although he had announced his intention to resign and become a professor at large at the University of California, Los Angeles
The University of California, Los Angeles (UCLA) is a public university, public Land-grant university, land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a Normal school, teachers colle ...

. His mother, brothers and in-laws followed von Neumann to the United States in 1939. Von Neumann anglicized
Anglicisation is the process by which a place or person becomes influenced by Culture of England, English culture or Culture of the United Kingdom, British culture, or a process of cultural and/or linguistic change in which something non-English ...

his first name to John, keeping the German-aristocratic surname von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen
Naturalization (or naturalisation) is the legal act or process by which a non-citizen of a country may acquire citizenship
Citizenship is a "relationship between an individual and a state to which the individual owes allegiance and in turn ...

of the United States in 1937, and immediately tried to become a lieutenant
A lieutenant ( , ; abbreviated Lt., Lt, LT, Lieut and similar) is a commissioned officer
An officer is a person who holds a position of authority as a member of an Military, armed force or Uniformed services, uniformed service.
Broadly ...

in the United States Army's Officers Reserve Corps
The United States Army Reserve (USAR) is a Military reserve force, reserve force of the United States Army. Together, the Army Reserve and the Army National Guard constitute the Army element of the reserve components of the United States Armed F ...

. He passed the exams easily but was rejected because of his age. His prewar analysis of how France would stand up to Germany is often quoted: "Oh, France won't matter."
Klara and John von Neumann were socially active within the local academic community. His white clapboard house at 26 Westcott Road was one of Princeton's largest private residences. He always wore formal suits. He once wore a three-piece pinstripe while riding down the Grand Canyon
The Grand Canyon (, yuf-x-yav, Wi:kaʼi:la, , Southern Paiute language: Paxa’uipi, ) is a steep-sided canyon carved by the Colorado River in Arizona, United States. The Grand Canyon is long, up to wide and attains a depth of over a mi ...

astride a mule. 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.
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. He knew by heart much of the material in Gibbon's 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
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematics, mathematical research and scholarship, and serves the national and international community through its publicatio ...

, von Neumann would describe even the minutest details of the battles of the Civil War
A civil war or intrastate war is a war between organized groups within the same Sovereign state, state (or country).
The aim of one side may be to take control of the country or a region, to achieve independence for a region, or to change go ...

that occurred in the places they drove by. 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.
Von Neumann liked to eat and drink. His wife, Klara, said that he could count everything except calories. He enjoyed Yiddish
Yiddish (, or , ''yidish'' or ''idish'', , ; , ''Yidish-Taytsh'', ) is a West Germanic languages, West Germanic language historically spoken by Ashkenazi Jews. It originated during the 9th century in Central Europe, providing the nascent Ashke ...

and "off-color" humor (especially limericks). He was a non-smoker. In Princeton, he received complaints for regularly playing extremely loud German march music on his phonograph
A phonograph, in its later forms also called a gramophone (as a trademark since 1887, as a generic name in the UK since 1910) or since the 1940s called a record player, or more recently a turntable, is a device for the mechanical and analogu ...

, which distracted those in neighboring offices, including Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

, from their work. 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 its television playing loudly. 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.
Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Teller ...

. A later friend of Ulam's, Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians ...

, 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. 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. Ulam noted that von Neumann's way of thinking might not be visual, but more aural.
In February 1951 for the New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid d ...

he had his brain waves scanned while at rest and while thinking (along with Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

and Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...

). "They generally showed differences from the average".
Illness and death

In 1955, von Neumann was diagnosed with what was eitherbone
A bone is a rigid organ
Organ may refer to:
Biology
* Organ (biology), a part of an organism
Musical instruments
* Organ (music), a family of keyboard musical instruments characterized by sustained tone
** Electronic organ, an electronic ...

, pancreatic
The pancreas is an Organ (anatomy), organ of the digestive system and endocrine system of vertebrates. In humans, it is located in the abdominal cavity, abdomen behind the stomach and functions as a gland. The pancreas is a mixed or heterocrine ...

or prostate cancer
Prostate cancer is cancer of the prostate. Prostate cancer is the second most common cancerous tumor worldwide and is the fifth leading cause of cancer-related mortality among men. The prostate is a gland in the male reproductive system that surr ...

after he was examined by physicians following a fall, 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
Los Alamos National Laboratory (often shortened as Los Alamos and LANL) is one of the sixteen research and development Laboratory, laboratories of the United States Department of Energy National Laboratories, United States Department of Energy ...

. He was not able to accept the proximity of his own demise, and the shadow of impending death instilled great fear in him. He invited a Catholic priest, Father Anselm Strittmatter, O.S.B., to visit him for consultation. 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." "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." Father Strittmatter administered the last rites
The last rites, also known as the Commendation of the Dying, are the last prayers and ministrations given to an individual of Christian faith, when possible, dying, shortly before death. They may be administered to those Death row, awaiting ex ...

to him. Some of von Neumann's friends, such as Abraham Pais
Abraham Pais (; May 19, 1918 – July 28, 2000) was a Netherlands, Dutch-United States, American physicist and science historian. Pais earned his Ph.D. from University of Utrecht just prior to a Nazi ban on Jews, Jewish participation in Dutch u ...

and Oskar Morgenstern
Oskar Morgenstern (January 24, 1902 – July 26, 1977) was an Austrian-American economist
An economist is a professional and practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, ...

, said they had always believed him to be "completely agnostic". 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.
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''. 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. On his deathbed, his mental capabilities became a fraction of what they were before, causing him much anguish. At times Von Neumann even forgot the lines that his brother recited from ''Faust''. 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." He died on February 8, 1957, at the Walter Reed Army Medical Center
The Walter Reed Army Medical Center (WRAMC)known as Walter Reed General Hospital (WRGH) until 1951was the United States Army, U.S. Army's flagship medical center from 1909 to 2011. Located on in the Washington, D.C., District of Columbia, it se ...

in Washington, D.C.
)
, image_skyline =
, image_caption = Clockwise from top left: the Washington Monument
The Washington Monument is an obelisk shaped building within the National Mall in Washington, D.C., built to commemorate Geor ...

, 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
Mercer County is a county
A county is a geographic region of a country used for administrative or other purposesChambers Dictionary, L. Brookes (ed.), 2005, Chambers Harrap Publishers Ltd, Edinburgh in certain modern nations. The term is ...

.
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 UCLA
The University of California, Los Angeles (UCLA) is a public university, public Land-grant university, land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a Normal school, teachers colle ...

should he have recovered from his cancer.
Mathematics

Set theory

The axiomatization of mathematics, on the model ofEuclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

's '' Elements'', had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathemati ...

and Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...

, and in geometry, thanks to Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book ''Grundlagen der Geometrie'' (tr. ''The Foundations of Geometry'') as the foundation for a modern treatment of Euclidean geometry. Other well-known modern ax ...

. But at the beginning of the 20th century, efforts to base mathematics on naive set theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language
...

suffered a setback due to Russell's paradox
In mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...

(on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...

was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...

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 foundation'' and the notion of '' class.''
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 models'', which became an essential instrument in set theory.
The second approach to the problem of sets belonging to themselves took as its base the notion of 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.
Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the ordinal and cardinal number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...

s as well as the first strict formulation of principles of definitions by the transfinite induction
Transfinite induction is an extension of mathematical induction to well-order, well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of Zermelo–Fraenkel set theory, ZFC.
Induction by cases
...

".
Von Neumann paradox

Building on the work ofFelix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are c ...

, in 1924 Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Poland, Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an ...

and Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...

proved that given a solid ball
A ball is a round object (usually sphere, spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be ...

in 3‑dimensional space, there exists
In predicate logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-ord ...

a decomposition of the ball into a finite number of disjoint subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

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, 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
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...

(see below).
Proof theory

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 itsconsistency
In classical deductive logic, a consistent theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such proc ...

. 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 axioms
An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...

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
The operators in elementary arithmetic
Arithmetic () is an elementary part of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...

followed from Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have be ...

. Building on the work of Ackermann, von Neumann began attempting to prove (using the finistic methods of Hilbert's school) the consistency of first-order arithmetic. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on induction). He continued looking for a more general proof of the consistency of classical mathematics using methods from proof theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Jon Barwise, Barwise (1978) consists of four correspo ...

.
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
Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was named ...

, 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 expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.
Less than a month later, von Neumann, who had participated in the Conference, communicated to Gödel an interesting consequence of his theorem: that the usual axiomatic systems are unable to demonstrate their own consistency. Gödel had already discovered this consequence, now known as his second incompleteness theorem, and sent von Neumann a preprint of his article containing both theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything." However von Neumann's method of proof differed from Gödel's, as his used polynomials to explain consistency. With this discovery, von Neumann ceased work in mathematical logic
Mathematical logic is the study of formal logic within mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantiti ...

and foundations of mathematics and instead spent time on problems connected with applications.
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 ofdynamical systems
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...

with an invariant measure In mathematics, an invariant measure is a Measure (mathematics), measure that is preserved by some Function (mathematics), function. The function may be a geometric transformation. For examples, angle, circular angle is invariant under rotation, hyp ...

. Of the 1932 papers on ergodic theory, Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probabi ...

wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". By then von Neumann had already written his articles on operator theory, and the application of this work was instrumental in his mean ergodic theorem.
The theorem is about arbitrary one-parameter unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group (mathematics), group of Unitary matrix, unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linea ...

s $\backslash mathit\; \backslash to\; \backslash mathit$ and states that for every vector $\backslash phi$ in the Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

, $\backslash lim\_\; \backslash frac\; \backslash int\_^\; V\_t(\backslash phi)\; \backslash ,\; dt$ exists in the sense of the metric defined by the Hilbert norm and is a vector $\backslash psi$ which is such that $V\_t(\backslash psi)\; =\; \backslash psi$ for all $t$. 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 Boltzmann's ergodic hypothesis
In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of Microstate (statistical mechanics), microstates with the same energy is proportional to ...

. He also pointed out that ergodicity
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

had not yet been achieved and isolated this for future work.
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 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
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probabi ...

have significant applications in other areas of mathematics.
Measure theory

In measure theory, the "problem of measure" for an -dimensionalEuclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of ?" The work of Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are c ...

and Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Poland, Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an ...

had implied that the problem of measure has a positive solution if or 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
In mathematics, the automorphism group of an object ''X'' is the group (mathematics), group consisting of automorphisms of ''X'' under Function composition, composition of morphisms. For example, if ''X'' is a Dimension (vector space), finite-dime ...

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
In mathematics, a Euclidean group is the group of (Euclidean) isometry, isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transfo ...

is a solvable group
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

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." Around 1942 he told Dorothy Maharam how to prove that every complete
Complete may refer to:
Logic
* Completeness (logic)
* Complete theory, Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, ...

σ-finite measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-algebra, -algebra) and the me ...

has a multiplicative lifting, however he did not publish this proof and she later came up with a new one.
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. A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

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". He proved this in the positive, and in later papers with Stone
In geology, rock (or stone) is any naturally occurring solid mass or aggregate of minerals or mineraloid matter. It is categorized by the minerals included, its Chemical compound, chemical composition, and the way in which it is formed. Rocks ...

discussed various generalizations and algebraic aspects of this problem. He also proved by new methods the existence of 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
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...

s. He had to create entirely new techniques to apply this to locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact space, locally compact and Hausdorff space, Hausdorff. Locally compact groups are important because many examples of groups th ...

s. He also gave a new, ingenious proof for the Radon–Nikodym theorem. 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.
Topological groups

Using his previous work on measure theory von Neumann made several contributions to the theory oftopological group
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic functions to arbitrary groups. He continued this work with another paper in conjunction with Bochner that improved the theory of almost periodicity to include functions that took on elements of linear spaces as values rather than numbers. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis
Analysis (plural, : analyses) is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics a ...

in relation to these papers.
In a 1933 paper, he used the newly discovered Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This Measure (mathematics), measure was introduced by Alfré ...

in the solution of Hilbert's fifth problem for the case of compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...

s. The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of linear transformations
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

and found that closed subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...

s of a general linear group In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

are Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation a ...

s. This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem.
Functional analysis

Von Neumann was the first person to axiomatically define an abstractHilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

whereas it was previously defined as the Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although ...

. He defined it as a complex vector space with a Hermitian scalar product, with the corresponding norm being both separable and complete. In the same papers he also defined several other abstract inequalities such as the Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used Inequality (mathematics), inequalities in mathematics.
The inequality for sums was published by . The c ...

that were previously only defined for Euclidean spaces
Euclidean space is the fundamental space of geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. ...

. He continued with the development of the spectral theory In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mode ...

of operators in Hilbert space in 3 seminal papers between 1929 and 1932. This work cumulated in his Mathematical Foundations of Quantum Mechanics which among two other books by Stone
In geology, rock (or stone) is any naturally occurring solid mass or aggregate of minerals or mineraloid matter. It is categorized by the minerals included, its Chemical compound, chemical composition, and the way in which it is formed. Rocks ...

and Banach in the same year were the first monographs on Hilbert space theory. Previous work by others showed that a theory of weak topologies could not be obtained by using sequences
In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...

, and von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining locally convex spaces and topological vector spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...

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 Hausdorff from Euclidean to Hilbert spaces) such as boundness and total boundness are still used today. For twenty years von Neumann was considered the 'undisputed master' of this area. These developments were primarily prompted by needs in quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...

where von Neumann realized the need to extend the spectral theory of Hermitian operators from the bounded to the unbounded case. Other major achievements in these papers include a complete elucidation of spectral theory for normal operator
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...

s, the first abstract presentation of the trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...

of a positive operator, a generalisation of Riesz's presentation of Hilbert's spectral theorems at the time, and the discovery of Hermitian operators in a Hilbert space, as distinct from self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...

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 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 algebras.
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. 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.
With I. J. Schoenberg he wrote several items investigating translation invariant
In geometry, to translation (geometry), translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the Invariant ...

Hilbertian metrics
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...

on the real number line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...

which resulted in their complete classification. Their motivation lie in various questions related to embedding metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functio ...

s into Hilbert spaces.
With Pascual Jordan
Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matrix ...

he wrote a short paper giving the first derivation of a given norm from an inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

by means of the parallelogram identity. His trace inequality is a key result of matrix theory used in matrix approximation problems. 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). This paper leads naturally to the study of symmetric operator ideals and is the beginning point for modern studies of symmetric operator spaces.
Later with Robert Schatten he initiated the study of nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spa ...

s on Hilbert spaces, tensor products of Banach spaces, introduced and studied trace class In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a Trace (linear algebra), trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the tra ...

operators, their ideals, and their duality with compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps Bounded set, bounded subsets of X to relatively compact subsets of Y (subsets wit ...

s, and preduality with bounded operator
In functional analysis
Functional analysis is a branch of mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiati ...

s. The generalization of this topic to the study of nuclear operators on Banach spaces was among the first achievements of Alexander Grothendieck. Previously in 1937 von Neumann published several results in this area, for example giving 1-parameter scale of different cross norms on $\backslash textit\backslash ,\_2^n\backslash otimes\backslash textit\backslash ,\_2^n$ and proving several other results on what are now known as Schatten–von Neumann ideals.
Operator algebras

Von Neumann founded the study of rings of operators, through thevon Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of Bounded linear operator, bounded operators on a Hilbert space that is Closed set, closed in the weak operator topology and contains the identity operator. It is a special type ...

s (originally called W*-algebras). While his original ideas for rings of operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...

existed already in 1930, he did not begin studying them in depth until he met F. J. Murray several years later. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

that is closed in the weak operator topology and contains the identity operator
Identity may refer to:
* Identity document
* Identity (philosophy)
In philosophy, identity (from , "sameness") is the relation each thing bears only to itself. The notion of identity gives rise to List of unsolved problems in philosophy, ...

. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant. After elucidating the study of the commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

case, von Neumann embarked in 1936, with the partial collaboration of Murray, on the noncommutative case, the general study of 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". 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 factors. 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. 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. Another important result on polar decomposition
In mathematics, the polar decomposition of a square real number, real or complex number, complex matrix (mathematics), matrix A is a matrix decomposition, factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-d ...

was published in 1932. His work here lead on to the next major topic.
Continuous geometries & lattice theory

Between 1935 and 1937, von Neumann worked on lattice theory, the theory ofpartially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...

s in which every two elements have a greatest lower bound and a least upper bound. Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are co ...

described his work, "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann combined traditional projective geometry with modern algebra (linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

, ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ...

, lattice theory). Many previously geometric results could then be interpreted in the case of general modules over rings. His work laid the foundations for modern work in projective geometry.
His biggest contribution was founding the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

, where instead of the dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

of a subspace being in a discrete set $0,\; 1,\; ...,\; \backslash mathit$ it can be an element of the unit interval
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

$;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$axiom
An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...

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 non-negative integers), the dimensions of the elements of a continuous geometry can range continuously across the unit interval $;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of Bounded linear operator, bounded operators on a Hilbert space that is Closed set, closed in the weak operator topology and contains the identity operator. It is a special type ...

s with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.
In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of $\backslash mathit$ (continuous-dimensional projective geometry over an arbitrary division ring
In algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost a ...

$\backslash mathit\backslash ,$) in abstract language of lattice theory. Von Neumann provided an abstract exploration of dimension in completed complemented modular
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...

topological lattices (properties that arise in the lattices of subspaces of inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...

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 $n\; >\; 3$ every $\backslash mathit$-dimensional abstract projective geometry is isomorphic
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

to the subspace-lattice of an $\backslash mathit$-dimensional vector space
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

$V\_n(F)$ over a (unique) corresponding division ring $F$. This is known as the Veblen–Young theorem. Von Neumann extended this fundamental result in projective geometry to the continous dimensional case. This 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.
The theorem as described by Birkhoff: " the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice having a "basis" of pairwise perspective elements, is isomorphic with the lattice of all principal right-ideals of a suitable regular ring In commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry an ...

. 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."
This work required the creation of regular rings. A von Neumann regular ring is a ring where for every $a$, an element $x$ exists such that $axa\; =\; a$. These rings came from and have connections to his work on von Neumann algebras, as well as AW*-algebras and various kinds of C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution (mathematics), involution satisfying the properties of the Hermitian adjoint, adjoint. A particular case is t ...

s.
Many smaller technical results were proven during the creation and proof of the above theorems, particularly regarding 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 lattices.
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!"
Mathematical statistics

Von Neumann made fundamental contributions tomathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...

. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...

.
Subsequently, Denis Sargan and Alok Bhargava extended the results for testing if the errors on a regression model follow a Gaussian random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some Space (mathematics), mathematical space.
An elementary example of a random walk is the random walk on the integer n ...

(''i.e.'', possess a unit root
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...

) against the alternative that they are a stationary first order autoregression.
Other work in pure mathematics

In his early years von Neumann published several papers relating to set-theoretical real analysis and number theory. In a paper from 1925, he proved that for any dense sequence of points in $;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$algebraic number
An algebraic number is a number that is a root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plan ...

s, and their relation to p-adic number
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s. In 1928 he wrote a couple papers on set-theoretic analysis. The first dealt with partitioning an interval into countably many congruent subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...

s. It solved a problem of Hugo Steinhaus asking whether an interval is $\backslash aleph\_0$-divisible. Von Neumann proved that indeed that all intervals, half-open, open, or closed are $\backslash aleph\_0$-divisible by translations (i.e. that these intervals can be decomposed into $\backslash aleph\_0$ subsets that are congruent by translation). His next paper dealt with giving a constructive proof without the axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

that $2^$ algebraically independent reals exist. He proved that $A\_r\; =\; \backslash sum\_^\; \backslash frac$ are algebraically independent for $r\; >\; 0$. Consequently, there exists a perfect algebraically independent set of reals the size of the continuum. Other minor results from his early career include a proof of a maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic partial differential equation, elliptic ...

for the gradient of a minimizing function in the field of calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions
and functional (mathematics), functionals, to find maxima and minima of f ...

, specifically proving the following theorem: Let $u:\; \backslash mathbb^n\; \backslash rightarrow\; \backslash mathbb$ be a Lipschitz function with constant $K$, and $\backslash Omega$ an open and bounded set in $\backslash mathbb^n$. If $u$ is a minimum for $F$ in $Lip\_K(\backslash Omega)$, then $\backslash sup\_\; \backslash frac\; =\; \backslash sup\_\; \backslash frac$ and a small simplification of Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at University of Königsberg, Königsberg, University of Zürich, Zürich and University of Göttingen, Göttingen. He created and developed the ge ...

's theorem for linear forms in geometric number theory.
Later in his career together with Pascual Jordan
Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matrix ...

and Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...

he wrote a foundational paper classifying all finite-dimensional
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...

formally real Jordan algebras and discovering the Albert algebras. 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. 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.
Physics

Quantum mechanics

Von Neumann was the first to establish a rigorous mathematical framework forquantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...

, known as the Dirac–von Neumann axioms, in his widely influential 1932 work '' Mathematical Foundations of Quantum Mechanics''. 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
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...

acting on the Hilbert space associated with the quantum system.
The ''physics'' of quantum mechanics was thereby reduced to the ''mathematics'' of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of Inequality (mathematics), mathematical inequalities asserting a fundamental limit to the accuracy with which the values fo ...

, 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. When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...

and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?"
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. Hermann's work was largely ignored until after John S. Bell made essentially the same argument in 1966. 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. The validity of Bub's argument is, in turn, disputed. 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
Alain Aspect (; born 15 June 1947) is a French physicist noted for his experimental work on quantum entanglement.
Aspect was awarded the 2022 Nobel Prize in Physics, jointly with John Clauser and Anton Zeilinger, "for experiments with Quantum e ...

in 1982, to the demonstration that quantum physics either requires a ''notion of reality'' substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity.
In a chapter of ''The Mathematical Foundations of Quantum Mechanics'', von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex number, complex-valued probability amplitude, and the probabilities for the possible results of ...

. 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, the Von Neumann–Wigner interpretation never gained acceptance among the majority of physicists. The Von Neumann–Wigner interpretation has been summarized as follows:
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.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 interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations. Viewing von Neumann's work on quantum mechanics as a part of the fulfilment of Hilbert's sixth problem, noted mathematical physicist 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. 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).

Von Neumann entropy

Von Neumann entropy
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...

is extensively used in different forms (conditional entropy
In information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by t ...

, relative entropy
Relative may refer to:
General use
*Kinship and family, the principle binding the most basic social units society. If two people are connected by circumstances of birth, they are said to be ''relatives''
Philosophy
*Relativism, the concept that ...

, etc.) in the framework of quantum information theory
Quantum information is the information of the state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* The State (newspaper), ''The State'' (newsp ...

. Entanglement measures are based upon some quantity directly related to the von Neumann entropy. Given a statistical ensemble
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...

of quantum mechanical systems with the density matrix
In quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantu ...

$\backslash rho$, it is given by $S(\backslash rho)\; =\; -\backslash operatorname(\backslash rho\; \backslash ln\; \backslash rho).\; \backslash ,$ Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such aHolevo 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 theShannon entropy
Shannon may refer to:
People
* Shannon (given name)
* Shannon (surname)
* Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958)
* Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum Will ...

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.
Density matrix

The formalism of density operators and matrices was introduced by von Neumann in 1927 and independently, but less systematically byLev Landau
Lev Davidovich Landau (russian: Лев Дави́дович Ланда́у; 22 January 1908 – 1 April 1968) was a Soviet Union, Soviet-Azerbaijan, Azerbaijani physicist of Jewish descent who made fundamental contributions to many areas of th ...

and Felix Bloch
Felix Bloch (23 October 1905 – 10 September 1983) was a Swiss people, Swiss-United States, American physicist and Nobel physics laureate who worked mainly in the U.S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for Physics f ...

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 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.Quantum logic

Von Neumann first proposed a quantum logic in his 1932 treatise '' Mathematical Foundations of Quantum Mechanics'', where he noted that projections on apropositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

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'', 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 $(A\backslash land\; B)\backslash ne\; (B\backslash land\; A)$. It was also demonstrated that the laws of distribution of classical logic, $P\backslash lor(Q\backslash land\; R)=(P\backslash lor\; Q)\backslash land(P\backslash lor\; R)$ and $P\backslash land\; (Q\backslash lor\; R)=(P\backslash land\; Q)\backslash lor(P\backslash land\; R)$, 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 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 ''$A\; \backslash land\; (B\backslash lor\; C)=\; A\backslash land\; 1\; =\; A$'', while $(A\backslash land\; B)\backslash lor\; (A\backslash land\; C)=0\backslash lor\; 0=0$. As Hilary Putnam
Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions ...

writes, von Neumann replaced classical logic with a logic constructed in orthomodular lattices (isomorphic
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

to the lattices of subspaces of the Hilbert space of a given physical system).
Nevertheless, he was never satifisied 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 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.
Fluid dynamics

Von Neumann made fundamental contributions in the field offluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...

.
Von Neumann's contributions to fluid dynamics included his discovery of the classic flow solution to blast wave
In fluid dynamics, a blast wave is the increased pressure and flow resulting from the deposition of a large amount of energy in a small, very localised volume. The flow field can be approximated as a lead shock wave, followed by a self-similar sub ...

s, and the co-discovery (independently of Yakov Borisovich Zel'dovich and Werner Döring) of the ZND detonation model of explosives. During the 1930s, von Neumann became an authority on the mathematics of shaped charges
A shaped charge is an explosive charge shaped to form an explosively formed penetrator (EFP) to focus the effect of the explosive's energy. Different types of shaped charges are used for various purposes such as cutting and forming metal, init ...

.
Later with Robert D. Richtmyer, von Neumann developed an algorithm defining ''artificial viscosity
The viscosity of a fluid is a measure of its drag (physics), resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quant ...

'' that improved the understanding of shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propaga ...

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.
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
The Ballistic Research Laboratory (BRL) was a leading United States Army, U.S. Army research establishment situated at Aberdeen Proving Ground, Maryland that specialized in ballistics (internal ballistics, interior, external ballistics, exterior, a ...

, 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
Theodore von Kármán ( hu, (Szőlőskislak, szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fie ...

. After von Neumann had finished, von Kármán said "Well, Johnny, that's very interesting. Of course you realize Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiacontinuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis Cauchy was the first to fo ...

." It was evident from von Neumann's face, that he had been unaware of Lagrange's Mécanique analytique.
Other work in physics

While not as prolific in physics as he was in mathematics, he nevertheless made several other notable contributions to it. His pioneering papers withSubrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian Americans, Indian-American Theoretical physics, theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for ...

on the statistics of a fluctuating gravitational field
In physics, a gravitational field is a scientific model, model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational Field (physics), field is us ...

generated by randomly distributed star
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s were considered a ''tour de force''. In this paper they developed a theory of two-body relaxation and used the Holtsmark distribution to model the dynamics of stellar systems. He wrote several other unpublished manuscripts on topics in stellar structure
Stellar structure models describe the internal structure of a star in detail and make predictions about the luminosity, the stellar classification, color and the stellar evolution, future evolution of the star. Different classes and ages of stars ...

, some of which were included in Chandresekhar's other works. In some earlier work led by Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was long ...

von Neumann helped develop basic ideas involving spinors that would lead to Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics ...

's twistor theory
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of Theoretical physics, theoretical and mathematical physics. Penrose proposed that twistor space shoul ...

. Much of this was done in seminars conducted at the IAS during the 1930s. From this work he wrote a paper with A. H. Taub and Veblen extending the Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its Dirac equation#Covariant form and relativistic invariance, free form, or including Dirac equation#Comparison with the ...

to projective relativity, maintaining invariance with regards to coordinate, spin, and gauge transformations, as a part of early research into potential theories of quantum gravity
Quantum gravity (QG) is a field of theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, ...

in the 1930s. Additionally in the same time period he made several proposals to colleagues for dealing with the problems in the newly created quantum theory of fields and for quantizing spacetime
In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...

, however both his colleagues and he himself did not consider the ideas fruitful and he did not work on them further. Nevertheless, he maintained at least some interest in these ideas as he had as late as 1940 written a manuscript on the Dirac equation in De Sitter space
In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an n-sphere, ''n''-sphere (with i ...

.
Economics

Game theory

Von Neumann founded the field ofgame theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appl ...

as a mathematical discipline. He proved his minimax theorem
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality.
The first theorem in this sense is John von Neumann, von Neumann's minimax theorem from ...

in 1928. It establishes that in zero-sum game
Zero-sum game is a Mathematical model, mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, pla ...

s with perfect information
In economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behav ...

(i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies
Strategy (from Ancient Greek, Greek στρατηγία ''stratēgia'', "art of troop leader; office of general, command, generalship") is a general plan to achieve one or more long-term or overall goals under conditions of uncertainty. In the se ...

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.
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
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality.
The first theorem in this sense is John von Neumann, von Neumann's minimax theorem from ...

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
Oskar Morgenstern (January 24, 1902 – July 26, 1977) was an Austrian-American economist
An economist is a professional and practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, ...

. 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. The public interest in this work was such that ''The New York Times
''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid d ...

'' ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analysis, especially convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the Real number, reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set o ...

s and the topological
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

fixed-point theorem
In mathematics, a fixed-point theorem is a result saying that a function (mathematics), function ''F'' will have at least one fixed point (mathematics), fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that ...

, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.
Independently, Leonid Kantorovich
Leonid Vitalyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Soviet ...

's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector space
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

s to represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been the primary tools of mathematical economics ever since.
Mathematical economics

Von Neumann raised the intellectual and 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 theBrouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Egbertus Jan Brouwer, L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compactness, compact convex set to itself there is a po ...

. Von Neumann's model of an expanding economy considered the matrix pencil '' A − λB'' with nonnegative matrices A and B; von Neumann sought probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

vectors ''p'' and ''q'' and a positive number ''λ'' that would solve the complementarity equation
:$p^T\; (A\; -\; \backslash lambda\; B)\; q\; =\; 0$
along with two inequality systems expressing economic efficiency. In this model, the (transpose
In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among othe ...

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 rate of growth of the economy; the rate of growth equals the interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...

.
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. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities In mathematics a linear inequality is an inequality (mathematics), inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:. It shows the data which is not equal in graph form.
* greater than
* ...

, 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.
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
Nicholas Kaldor, Baron Kaldor (12 May 1908 – 30 September 1986), born Káldor Miklós, was a Cambridge economist in the post-war period. He developed the "compensation" criteria called Kaldor–Hicks efficiency for welfare comparisons (1939), d ...

, and they hit it off. Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras
Marie-Esprit-Léon Walras (; 16 December 1834 – 5 January 1910) was a French mathematical economist and Georgist
Georgism, also called in modern times Geoism, and known historically as the single tax movement, is an economic ideology holdi ...

. 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
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...

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.
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. 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 simplices, with the pivoting decision determined by a nonnegativeleast squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the Resi ...

subproblem with a convexity constraint ( projecting the zero-vector onto the convex hull
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field o ...

of the active simplex
In geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...

). Von Neumann's algorithm was the first interior point method of linear programming.
Computer science

Von Neumann was a founding figure incomputing
Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and development of both computer hardware , hardware and software. ...

. 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.
Von Neumann wrote the 23 pages 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. He also worked on the philosophy of artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animal cognition, animals and human intelligence, humans. Example tasks in ...

with Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical ...

when the latter visited Princeton in the 1930s.
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
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...

, which allowed solutions to complicated problems to be approximated using random numbers.
Von Neumann's algorithm for simulating a fair coin
In probability theory and statistics, a sequence of Independence (probability theory), independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is cal ...

with a biased coin is used in the "software whitening" stage of some hardware random number generator
In computing, a hardware random number generator (HRNG) or true random number generator (TRNG) is a device that random number generation, generates random numbers from a physical process, rather than by means of an algorithm. Such devices are o ...

s.
Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."
Von Neumann also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect.
While consulting for the Moore School of Electrical Engineering
The Moore School of Electrical Engineering at the University of Pennsylvania came into existence as a result of an endowment from Alfred Fitler Moore on June 4, 1923. It was granted to Penn's School of Electrical Engineering, located in the Towne ...

at the University of Pennsylvania
The University of Pennsylvania (also known as Penn or UPenn) is a Private university, private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest- ...

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
John William Mauchly (August 30, 1907 – January 8, 1980) was an American physicist who, along with J. Presper Eckert, designed ENIAC, the first general-purpose electronic digital computer, as well as EDVAC, BINAC and UNIVAC I, the first com ...

, described a computer architecture
In computer engineering, computer architecture is a description of the structure of a computer system made from component parts. It can sometimes be a high-level description that ignores details of the implementation. At a more detailed level, the ...

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
file:PaperTapes-5and8Hole.jpg, Five- and eight-hole punched paper tape
file:Harwell-dekatron-witch-10.jpg, Paper tape reader on the Harwell computer with a small piece of five-hole tape connected in a circle – creating a physical program loop
...

or plugboard
A plugboard or control panel (the term used depends on the application area) is an array of jack (connector), jacks or sockets (often called hubs) into which patch cords can be inserted to complete an electrical circuit. Control panels are som ...

. Although the single-memory, stored program architecture is commonly called von Neumann architecture
The von Neumann architecture — also known as the von Neumann model or Princeton architecture — is a computer architecture based on a 1945 description by John von Neumann, and by others, in the ''First Draft of a Report on the EDVAC''. The ...

as a result of von Neumann's paper, the architecture was based on the work of Eckert and Mauchly, inventors of the ENIAC
ENIAC (; Electronic Numerical Integrator and Computer) was the first Computer programming, programmable, Electronics, electronic, general-purpose digital computer, completed in 1945. There were other computers that had these features, but the ...

computer at the University of Pennsylvania.
Von Neumann consulted for the Army's Ballistic Research Laboratory
The Ballistic Research Laboratory (BRL) was a leading United States Army, U.S. Army research establishment situated at Aberdeen Proving Ground, Maryland that specialized in ballistics (internal ballistics, interior, external ballistics, exterior, a ...

, most notably on the ENIAC project, as a member of its Scientific Advisory Committee.
The electronics of the new ENIAC ran at one-sixth the speed, but this in no way degraded the ENIAC's performance, since it was still entirely I/O bound. Complicated programs could be developed and debugged in days rather than the weeks required for plugboarding the old ENIAC. Some of von Neumann's early computer programs have been preserved.
The next computer that von Neumann designed was the IAS machine
The IAS machine was the first electronic computer built at the Institute for Advanced Study (IAS) in Princeton, New Jersey. It is sometimes called the von Neumann machine, since the paper describing its design was edited by John von Neumann, a ...

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. Von Neumann recommended that the IBM 701
The IBM 701 Electronic Data Processing Machine, known as the Defense Calculator while in development, was IBM’s first commercial scientific computer and its first series production mainframe computer, which was announced to the public on May ...

, 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
The IBM 704 is a large digital computer, digital mainframe computer introduced by IBM in 1954. It was the first mass-produced computer with hardware for floating-point arithmetic. The IBM 704 ''Manual of operation'' states:
The type 704 Elec ...

.
Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953.
However, the theory could not be implemented until advances in computing of the 1960s. Around 1950 he was also among the first people to talk about the time complexity
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...

of computation
Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm).
Mechanical or electronic devices (or, History of computing hardware, historically, people) that perform computations are ...

s, which eventually evolved into the field of computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by ...

.
Herman Goldstine
Herman Heine Goldstine (September 13, 1913 – June 16, 2004) was a mathematician and computer scientist, who worked as the director of the IAS machine at Princeton University's Institute for Advanced Study and helped to develop ENIAC, the ...

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. As an example, he talked about 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 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-Rand
Sperry Corporation was a major American equipment and electronics company whose existence spanned more than seven decades of the 20th century. Sperry ceased to exist in 1986 following a prolonged takeover, hostile takeover bid engineered by ...

, which would result in the Stretch and 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. This view was held not just by him but by many other mathematicians and scientists of the time too.
Cellular automata, DNA and the universal constructor

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. Von Neumann created the field ofcellular automata
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

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
Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...

self-reproducing automaton. 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
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

.
Subsequently, the concept of the Von Neumann universal constructor
John von Neumann's universal constructor is a self-replicating machine in a cellular automaton (CA) environment. It was designed in the 1940s, without the use of a computer. The fundamental details of the machine were published in von Neumann's b ...

based on the von Neumann cellular automaton was fleshed out in his posthumously published lectures ''Theory of Self Reproducing Automata''.
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. Like Ulam's lattice network, von Neumann's cellular automata are two-dimensional, with his self-replicator implemented algorithmically. The result was a 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
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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.
Von Neumann addressed the evolutionary growth of complexity amongst his self-replicating machines. 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",''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 indicating it as a field of further study.
The cybernetics
Cybernetics is a wide-ranging field concerned with circular causality, such as feedback, in regulatory and purposive systems. Cybernetics is named after an example of circular causal feedback, that of steering a ship, where the helmsperson ma ...

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
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

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 moon
The Moon is Earth's only natural satellite. It is the List of natural satellites, fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth ( ...

or asteroid belt
The asteroid belt is a torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the ...

would be by using self-replicating spacecraft, taking advantage of their exponential growth
Exponential growth is a process that increases quantity over time. It occurs when the instantaneous Rate (mathematics)#Of change, rate of change (that is, the derivative) of a quantity with respect to time is proportionality (mathematics), propor ...

.
Von Neumann investigated the question of whether modelling evolution on a digital computer could solve the complexity problem in programming.
Beginning in 1949, von Neumann's design for a self-reproducing computer program is considered the world's first computer virus
A computer virus is a type of computer program that, when executed, replicates itself by modifying other computer programs and Code injection, inserting its own Computer language, code. If this replication succeeds, the affected areas are then s ...

, and he is considered to be the theoretical father of computer virology.
Scientific computing and numerical analysis

Considered to be possibly "the most influential researcher inscientific computing
Computational science, also known as scientific computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...

of all time", 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 stability analysis procedure that now bears his name, a scheme used to ensure that when linear partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...

are solved 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. His paper with Herman Goldstine
Herman Heine Goldstine (September 13, 1913 – June 16, 2004) was a mathematician and computer scientist, who worked as the director of the IAS machine at Princeton University's Institute for Advanced Study and helped to develop ENIAC, the ...

in 1947 was the first to describe backward error analysis, although only implicitly. He was also among the first researchers to write about the Jacobi method. During his time at Los Alamos, he was the first to consider how to solve various problems of gas dynamics
Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressibility, compressible, flows are usually treated as being incompressible flow, incom ...

numerically, writing several classified reports on the topic. However, he was frustrated by the lack of progress with analytic methods towards solving these problems, many of which were nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportionality (mathematics), proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, m ...

. As a result, he turned towards computational methods in order to break the deadlock. 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.
From his work at Los Alamos von Neumann realized that computation was not just a tool to brute force the solution to a problem numerically, but that computation could also provide insight for solving problems analytically too, 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 problems. 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 classical analysis methods. Titled "High-speed Computing Devices and Mathematical Analysis", he also described how wind tunnel
Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...

s, which at the time were being constructed at heavy cost, were actually analog computer
An analog computer or analogue computer is a type of Computation, computer that uses the continuous variation aspect of physical phenomena such as Electrical network, electrical, Mechanics, mechanical, or Hydraulics, hydraulic quantities (''a ...

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
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are co ...

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 US Navy
The United States Navy (USN) is the maritime service branch of the United States Armed Forces and one of the eight uniformed services of the United States. It is the largest and most powerful navy in the world, with the estimated tonnage ...

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 inverting matrices, random matrices
In probability theory and mathematical physics, a random matrix is a matrix (mathematics), matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can b ...

and automated relaxation methods for solving elliptic boundary value problem
In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an evolution problem. For example, the Dirichlet problem for the Laplacian gives the eventual distributi ...

s.
Weather systems and global warming

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. In 1946 von Neumann founded the "Meteorological Project" at the Institute for Advanced Study, securing funding for his project from the Weather Bureau along with theUS Air Force
The United States Air Force (USAF) is the air service branch of the United States Armed Forces
The United States Armed Forces are the Military, military forces of the United States. The armed forces consists of six Military branch, ...

and US Navy weather services.''Weather Architecture'' By Jonathan Hill (Routledge, 2013), page 216 With Carl-Gustaf Rossby
Carl-Gustaf Arvid Rossby ( 28 December 1898 – 19 August 1957) was a Swedish-born American meteorologist
A meteorologist is a scientist who studies and works in the field of meteorology
Meteorology is a branch of the atmospheric scie ...

, considered the leading theoretical meteorologist at the time, he gathered a twenty strong group of metereologists 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. 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 forecasts on the ENIAC computer that von Neumann had arranged to be used; von Neumann and his team published the results as ''Numerical Integration of the Barotropic Vorticity Equation''. Together they played a leading role in efforts to integrate sea-air exchanges of energy and moisture into the study of climate. Though primitive, news of the ENIAC forecasts quickly spread around the world and a number of parallel projects in other locations were initiated. In 1955 von Neumann, Charney and their collaborators convinced their funders to open up the Joint Numerical Weather Prediction Unit (JNWPU) in Suitland, Maryland
Suitland is an unincorporated area, unincorporated community and census designated place (CDP) in Prince George's County, Maryland, Prince George's County, Maryland, United States, approximately one mile (1.6 km) southeast of Washington, D.C. ...

which began routine real-time weather forecasting. 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." 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. 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.
His research into weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice cap
A polar ice cap or polar cap is a high-latitude region of a planet, dwarf planet, or natural satellite that is covered in ice.
There are no requirements with respect to size or composition for a body of ice to be termed a polar ice cap, nor a ...

s to enhance absorption of solar radiation (by reducing the albedo
Albedo (; ) is the measure of the diffuse reflection of sunlight, solar radiation out of the total solar radiation and measured on a scale from 0, corresponding to a black body that absorbs all incident radiation, to 1, corresponding to a body ...

), thereby inducing global warming
In common usage, climate change describes global warming—the ongoing increase in global average temperature—and its effects on Earth's climate system. Climate variability and change, Climate change in a broader sense also includes ...

. Von Neumann proposed a theory of global warming as a result of the activity of humans, noting that the Earth was only colder during the last glacial period, he wrote in 1955: "Carbon dioxide
Carbon dioxide ( chemical formula ) is a chemical compound made up of molecules that each have one carbon
Carbon () is a chemical element with the chemical symbol, symbol C and atomic number 6. It is nonmetallic and tetravalence, tetraval ...

released into the atmosphere by industry's burning of coal
Coal is a combustible black or brownish-black sedimentary rock, formed as stratum, rock strata called coal seams. Coal is mostly carbon with variable amounts of other Chemical element, elements, chiefly hydrogen, sulfur, oxygen, and nitrogen ...

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."''Engineering: Its Role and Function in Human Society''
edited by William H. Davenport, Daniel I. Rosenthal (Elsevier 2016), page 266 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." He also warned that weather and climate control could have military uses, telling Congress
A congress is a formal meeting of the Representative democracy, representatives of different countries, constituent states, organizations, trade unions, political party, political parties, or other groups. The term originated in Late Middle Eng ...

in 1956 that they could pose an even bigger risk than 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.
Technological singularity hypothesis

The first use of the concept of a singularity in the technological context is attributed to von Neumann, 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." This concept was fleshed out later in the book '' Future Shock'' byAlvin Toffler
Alvin Eugene Toffler (October 4, 1928 – June 27, 2016) was an American writer, futurist, and businessman known for his works discussing modern technologies, including the digital revolution and the Telecommunication, communication revolution, wi ...

.
Defense work

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 ofshaped charge
A shaped charge is an explosive charge shaped to form an explosively formed penetrator (EFP) to focus the effect of the explosive's energy. Different types of shaped charges are used for various purposes such as cutting and forming metal, init ...

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 Manhattan Project was a research and development undertaking during World War II that produced the first nuclear weapons. It was led by the United States with the support of the United Kingdom and Canada. From 1942 to 1946, the project w ...

. The involvement included frequent trips by train to the project's secret research facilities at the Los Alamos Laboratory
The Los Alamos Laboratory, also known as Project Y, was a secret laboratory established by the Manhattan Project and operated by the University of California during World War II. Its mission was to design and build the History of nuclear weapo ...

in a remote part of New Mexico.
Von Neumann made his principal contribution to the atomic bomb
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions ( thermonuclear bomb), producing a nuclear explosion. Both bom ...

in the concept and design of the explosive lenses that were needed to compress the plutonium
Plutonium is a radioactive decay, radioactive chemical element with the Symbol (chemistry), symbol Pu and atomic number 94. It is an actinide metal of silvery-gray appearance that tarnishes when exposed to air, and forms a dull coating plutoni ...

core of the Fat Man
"Fat Man" (also known as Mark III) is the codename for the type of nuclear bomb the United States Atomic bombings of Hiroshima and Nagasaki#Bombing of Nagasaki, detonated over the Japanese city of Nagasaki on 9 August 1945. It was the second ...

weapon that was later dropped on Nagasaki
is the capital and the largest Cities of Japan, city of Nagasaki Prefecture on the island of Kyushu in Japan.
It became the sole Nanban trade, port used for trade with the Portuguese and Dutch during the 16th through 19th centuries. The Hi ...

. 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".
When it turned out that there would not be enough uranium-235
Uranium-235 (235U or U-235) is an Isotopes of uranium, isotope of uranium making up about 0.72% of natural uranium. Unlike the predominant isotope uranium-238, it is fissile, i.e., it can sustain a nuclear chain reaction. It is the only fissile ...

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
Plutonium-239 (239Pu or Pu-239) is an isotopes of plutonium, isotope of plutonium. Plutonium-239 is the primary fissile isotope used for the production of nuclear weapons, although uranium-235 is also used for that purpose. Plutonium-239 is als ...

that was available from the Hanford Site
The Hanford Site is a decommissioned nuclear production complex operated by the United States federal government on the Columbia River in Benton County, Washington, Benton County in the U.S. state of Washington (state), Washington. The site h ...

. He established the design of the explosive lens
An explosive lens—as used, for example, in nuclear weapons—is a highly specialized shaped charge. In general, it is a device composed of several explosive charges. These charges are arranged and formed with the intent to control the shape ...

es required, but there remained concerns about "edge effects" and imperfections in the explosives. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, this was achieved by George Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945.
In a visit to Los Alamos in September 1944, von Neumann showed that the pressure increase from explosion shock wave reflection from solid objects was greater than previously believed if the angle of incidence of the shock wave was between 90° and some limiting angle. As a result, it was determined that the effectiveness of an atomic bomb would be enhanced with detonation some kilometers above the target, rather than at ground level.
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
is the capital of Hiroshima Prefecture in Japan. , the city had an estimated population of 1,199,391. The gross domestic product (GDP) in Greater Hiroshima, Hiroshima Urban Employment Area, was US$61.3 billion as of 2010. Kazumi Matsui ha ...

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. The cultural capital Kyoto
Kyoto (; Japanese language, Japanese: , ''Kyōto'' ), officially , is the capital city of Kyoto Prefecture in Japan. Located in the Kansai region on the island of Honshu, Kyoto forms a part of the Keihanshin, Keihanshin metropolitan area along wi ...

, which had been spared the bombing inflicted upon militarily significant cities, was von Neumann's first choice, a selection seconded by Manhattan Project leader General Leslie Groves
Lieutenant general (United States), Lieutenant General Leslie Richard Groves Jr. (17 August 1896 – 13 July 1970) was a United States Army Corps of Engineers Officer (armed forces), officer who oversaw the construction of the Pentagon and di ...

. However, this target was dismissed by Secretary of War
The secretary of war was a member of the President of the United States, U.S. president's United States Cabinet, Cabinet, beginning with George Washington's Presidency of George Washington, administration. A similar position, called either "Se ...

Henry L. Stimson.
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
The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God t ...

. The event was conducted as a test of the implosion method device, at the bombing range
A bombing range usually refers to a remote military aerial bombing and gunnery training range used by Military aircraft, combat aircraft to attack ground targets (air-to-ground bombing), or a remote area reserved for researching, developing, te ...

near Alamogordo Army Airfield, southeast of Socorro, New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to but Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the Atomic Age, nuclea ...

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. It was in von Neumann's 1944 papers that the expression "kilotons" appeared for the first time. After the war, Robert Oppenheimer
J. Robert Oppenheimer (; April 22, 1904 – February 18, 1967) was an American theoretical physics, theoretical physicist. A professor of physics at the University of California, Berkeley, Oppenheimer was the wartime head of the Los Alamos Lab ...

remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."
Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained the hydrogen bomb project. He collaborated with Klaus Fuchs
Klaus Emil Julius Fuchs (29 December 1911 – 28 January 1988) was a German theoretical physics, theoretical physicist and Atomic spies, atomic spy who supplied information from the American, British and Canadian Manhattan Project to the ...

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
Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is mani ...

. The Fuchs–von Neumann patent used radiation implosion, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design
A thermonuclear weapon, fusion weapon or hydrogen bomb (H bomb) is a second-generation nuclear weapon design. Its greater sophistication affords it vastly greater destructive power than first-generation nuclear bombs, a more compact size, a lowe ...

. Their work was, however, incorporated into the "George" shot of Operation Greenhouse
Operation Greenhouse was the fifth American nuclear test series, the second conducted in 1951 and the first to test principles that would lead to developing Teller-Ullam, thermonuclear weapons (''hydrogen bombs''). Conducted at the new Pacific ...

, which was instructive in testing out concepts that went into the final design. 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."
For his wartime services, von Neumann was awarded the Navy Distinguished Civilian Service Award in July 1946, and the Medal for Merit
The Medal for Merit was, during the period it was awarded, the highest civilian decoration of the United States. It was awarded by the President of the United States to civilians who "distinguished themselves by exceptionally meritorious conduct i ...

in October 1946.
Post war

In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group (WSEG), whose function was to advise theJoint Chiefs of Staff
The Joint Chiefs of Staff (JCS) is the body of the most senior uniformed leaders within the United States Department of Defense, that advises the president of the United States, the secretary of defense, the Homeland Security Council and ...

and the United States Secretary of Defense
The United States secretary of defense (SecDef) is the head of the United States Department of Defense, the United States federal executive departments, executive department of the United States Armed Forces, U.S. Armed Forces, and is a high r ...

on the development and use of new technologies. He also became an adviser to the Armed Forces Special Weapons Project (AFSWP), which was responsible for the military aspects on nuclear weapons. Over the following two years, he became a consultant to the Central Intelligence Agency
The Central Intelligence Agency (CIA ), known informally as the Agency and historically as the Company, is a civilian intelligence agency, foreign intelligence service of the federal government of the United States, officially tasked with gat ...

(CIA), a member of the influential General Advisory Committee of the Atomic Energy Commission, a consultant to the newly established Lawrence Livermore National Laboratory
Lawrence Livermore National Laboratory (LLNL) is a federal research facility in Livermore, California, United States. The lab was originally established as the University of California Radiation Laboratory, Livermore Branch in 1952 in response ...

, and a member of the Scientific Advisory Group of the United States Air Force
The United States Air Force (USAF) is the Aerial warfare, air military branch, service branch of the United States Armed Forces, and is one of the eight uniformed services of the United States. Originally created on 1 August 1907, as a part ...

among a host of other agencies. Beside the Coast Guard
A coast guard or coastguard is a maritime security organization of a particular country. The term embraces wide range of responsibilities in different countries, from being a heavily armed military force with customs and security duties t ...

, there was not a single US military or intelligence organization which von Neumann did not advise. During this time he became ''the'' "superstar" defense scientist at the Pentagon
The Pentagon is the headquarters building of the United States Department of Defense. It was constructed on an accelerated schedule during World War II
World War II or the Second World War, often abbreviated as WWII or WW2, w ...

. His authority was considered infalliable at the highest levels including the secretary of defense and Joint Chiefs of Staff. This applied not just to US government agencies. Supposedly, he was hired as a consultant to the RAND Corporation
The RAND Corporation (from the phrase "research and development") is an American nonprofit global policy think tank created in 1948 by Douglas Aircraft Company to offer research and analysis to the United States Armed Forces. It is finan ...

with the equivalent salary for an average full time analyst, yet his job was only to write down his thoughts each morning while shaving.
During several meetings of the advisory board of the US Air Force von Neumann and Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American Theoretical physics, theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the History of the Teller–Ulam d ...

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
Bernard Adolph Schriever (14 September 1910 – 20 June 2005), also known as Bennie Schriever, was a United States Air Force
The United States Air Force (USAF) is the Aerial warfare, air military branch, service branch of the United Sta ...

, 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. 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. 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
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...

advancements in both these areas and in strategic defenses against American bombers into reports prepared for the Department of Defense Department of Defence or Department of Defense may refer to:
Current departments of defence
* Department of Defence (Australia)
* Department of National Defence (Canada)
* Department of Defence (Ireland)
* Department of National Defense (Philippin ...

(DoD) in order to argue for the creation of ICBMs. Gardner on several occasions would bring von Neumann to the Pentagon in order to discuss with various senior officials his reports. Several design decisions in these reports such as inertial guidance mechanisms would form the basis for all ICBMs thereafter. By 1954 von Neumann was also regularly testifying to various 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 State Department
The United States Department of State (DOS), or State Department, is an United States federal executive departments, executive department of the Federal government of the United States, U.S. federal government responsible for the country's fore ...

and National Security Council
A national security council (NSC) is usually an executive branch governmental body responsible for coordinating policy on national security issues and advising chief executives on matters related to national security. An NSC is often headed by a na ...

(NSC).
However, this was not enough in order to have the ICBM program run at full throttle; they needed direct action by the President
President most commonly refers to:
*President (corporate title)
*President (education), a leader of a college or university
*President (government title)
President may also refer to:
Automobiles
* Nissan President, a 1966–2010 Japanese ful ...

. On July 28, 1955, Schriever, Gardner, and von Neumann had managed to arrange a direct meeting with President Eisenhower at the White House
The White House is the official residence and workplace of the president of the United States. It is located at 1600 Pennsylvania Avenue Northwest, Washington, D.C., NW in Washington, D.C., and has been the residence of every U.S. preside ...

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. 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
A vice president, also director in British English, is an Corporate officer, officer in government or business who is below the President (corporate title), president (chief executive officer) in rank. It can also refer to executive vice presid ...

Richard Nixon
Richard Milhous Nixon (January 9, 1913April 22, 1994) was the 37th president of the United States, serving from 1969 to 1974. A member of the Republican Party (United States), Republican Party, he previously served as a United States House ...

, Admiral
Admiral is one of the highest ranks in some navies. In the Commonwealth nations and the United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental ...

Arthur Radford, chairman of the Joint Chiefs of Staff
The chairman of the Joint Chiefs of Staff (CJCS) is the presiding officer of the United States Joint Chiefs of Staff (JCS). The chairman is the Chief of defence, highest-ranking and United States military seniority, most senior military offic ...

, the secretaries of State
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* The State (newspaper), ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, U ...

, Defense and Treasury
A treasury is either
*A government department related to finance and taxation, a Finance minister, finance ministry.
*A place or location where treasure, such as currency or precious items are kept. These can be State ownership, state or roya ...

, and the head of the CIA among others. The program officially belonged to Tommy Power as Commander in Chief of the Strategic Air Command
Strategic Air Command (SAC) was both a United States Department of Defense Specified Command and a United States Air Force (USAF) Major Command responsible for command and control of the strategic bomber and intercontinental ballistic missile c ...

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
Nuclear engineering is the branch of engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engi ...

to the intricacies of 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 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 A force structure is the combat
Combat (French language, French for ''fight'') is a purposeful violence, violent conflict meant to physically harm or kill the opposition. Combat may be armed (using weapons) or unarmed (Hand-to-hand combat, not u ...

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". 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. Evidence would later show that the Soviets indeed were already testing their own intermediate-range ballistic missile
An intermediate-range ballistic missile (IRBM) is a ballistic missile with a range (aeronautics), range of 3,000–5,500 km (1,864–3,418 miles), between a medium-range ballistic missile (MRBM) and an intercontinental ballistic missile ( ...

s at the time of the presentation to President Eisenhower at the White House. Von Neumann would continue to meet the President, including at his home in Gettysburg, Pennsylvania
Gettysburg (; non-locally ) is a borough
A borough is an administrative division
Administrative division, administrative unit,Article 3(1). country subdivision, administrative region, subnational entity, constituent state, as well as ma ...

, and other high-level government officials as a key advisor on ICBMs until his death.
Atomic Energy Commission

In 1955, von Neumann became a commissioner of the Atomic Energy Commission (AEC). He accepted this position and used it to further the production of compact hydrogen bombs suitable forintercontinental ballistic missile
An intercontinental ballistic missile (ICBM) is a ballistic missile with a range (aeronautics), range greater than , primarily designed for nuclear weapons delivery (delivering one or more Thermonuclear weapon, thermonuclear warheads). Conventi ...

(ICBM) delivery. He involved himself in correcting the severe shortage of tritium
Tritium ( or , ) or hydrogen-3 (symbol T or H) is a rare and radioactive isotope of hydrogen with half-life about 12 years. The nucleus of tritium (t, sometimes called a ''triton'') contains one proton and two neutrons, whereas the nucleu ...

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. 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
The Oppenheimer security hearing was a 1954 proceeding by the United States Atomic Energy Commission
The United States Atomic Energy Commission (AEC) was an agency of the United States government established after World War II by U.S ...

, at which he asserted that Oppenheimer was loyal, and praised him for his helpfulness once the program went ahead.
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
The SM-65 Atlas was the first operational intercontinental ballistic missile
An intercontinental ballistic missile (ICBM) is a ballistic missile with a range (aeronautics), range greater than , primarily designed for nuclear weapons de ...

passed its first fully functional test in 1959, two years after his death. The more advanced Titan rockets were deployed in 1962. Both had been proposed in the ICBM committees von Neumann chaired. 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.
Mutual assured destruction

Von Neumann is credited with developing the equilibrium strategy ofmutual assured destruction
Mutual assured destruction (MAD) is a doctrine of military strategy and national security, national security policy which posits that a full-scale use of nuclear weapons by an attacker on a nuclear-armed defender with Second strike, second-stri ...

(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. 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
ENIAC (; Electronic Numerical Integrator and Computer) was the first Computer programming, programmable, Electronics, electronic, general-purpose digital computer, completed in 1945. There were other computers that had these features, but the ...

by making it programmable and then wrote programs for it to do the H-bomb calculations (which further further 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.
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
The Sputnik crisis was a period of public fear and anxiety in Western Bloc, Western nations about the perceived technological gap between the United States and Soviet Union caused by the Soviets' launch of ''Sputnik 1'', the world's first arti ...

.
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
Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right politics, far-right Totalitarianism, totalitarian political ideology and practices associated with Adolf Hit ...

, Fascism
Fascism is a far-right, Authoritarianism, authoritarian, ultranationalism, ultra-nationalist political Political ideology, ideology and Political movement, movement,: "extreme militaristic nationalism, contempt for electoral democracy and pol ...

and Soviet Communism. During a Senate
A senate is a deliberative assembly, often the upper house
An upper house is one of two Debate chamber, chambers of a bicameralism, bicameral legislature, the other chamber being the lower house.''Bicameralism'' (1997) by George Tseb ...

committee hearing he described his political ideology as "violently anti-communist
Anti-communism is Political movement, political and Ideology, ideological opposition to communism. Organized anti-communism developed after the 1917 October Revolution in the Russian Empire, and it reached global dimensions during the Cold War, w ...

, and much more militaristic than the norm". He was quoted in 1950 remarking, "If you say why not bomb he Sovietstomorrow, I say, why not today? If you say today at five o'clock, I say why not one o'clock?"
On February 15, 1956, von Neumann was presented with the Medal of Freedom by President Dwight D. Eisenhower
Dwight David "Ike" Eisenhower (born David Dwight Eisenhower; ; October 14, 1890 – March 28, 1969) was an American military officer and statesman who served as the 34th president of the United States from 1953 to 1961. During World War II, ...

. His citation read:
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.
Consultancies

A list of consultancies given by various sources is as follows: While his appointment as full Atomic Energy Commissioner in late 1954 formally required he sever all his other consulting contracts, an exemption was made for von Neumann to continue working with several critical military committees after theAir Force
An air force – in the broadest sense – is the national military branch that primarily conducts aerial warfare. More specifically, it is the branch of a nation's armed services that is responsible for aerial warfare as distinct from an ar ...

and several key senators raised concerns.
Personality

Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians ...

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. Françoise Ulam described how she never saw von Neumann in anything but a formal suit and tie. 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, 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. 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. 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. 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. Von Neumann was not a quiet person either; he enjoyed going to and hosting parties several times a week, 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.
He was also known for always being happy to provide others with scientific and mathematical advice, even when the recipient did not later credit him, which he did on many occasions with mathematicians and scientists of all ability levels. 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. However, he did not particularly like it when he felt others were challenging him and his brilliance, being a very competitive person. A story went at the Aberdeen Proving Ground
Aberdeen Proving Ground (APG) (sometimes erroneously called Aberdeen Proving ''Grounds'') is a United States Army, U.S. Army facility located adjacent to Aberdeen, Maryland, Aberdeen, Harford County, Maryland, Harford County, Maryland, United Stat ...

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. Nevertheless, he would put in an effort to appear modest and did not like boasting or appearing in a self-effacing manner. Towards the end of his life on one occasion his wife 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.
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
Shorthand is an abbreviated symbolic writing method that increases speed and brevity of writing as compared to longhand, a more common method of writing a language. The process of writing in shorthand is called stenography, from the Greek ''s ...

. Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing strongly on technical ones. Herbert York
Herbert Frank York (24 November 1921 – 19 May 2009) was an American nuclear physicist of Mohawk origin.http://www.edge.org/conversation/nsa-the-decision-problem. The Decision Problem He held numerous research and administrative positions ...

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 Air Force
An air force – in the broadest sense – is the national military branch that primarily conducts aerial warfare. More specifically, it is the branch of a nation's armed services that is responsible for aerial warfare as distinct from an ar ...

long-range missile programs. 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, Ancient Latin and 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.
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. 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 advisery 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. As a child he was poor in athletics and thus did not make friends this way (but he did join in on class pranks).
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 on Neumannwanted 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.
Many people who had known von Neumann were puzzled by his relationship to the military and to power structures in general. He seemed to admire generals and admiral
Admiral is one of the highest ranks in some navies. In the Commonwealth nations and the United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country Continental ...

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. When hospitalized at the end of his life Ulam told him on one occasion he was on the same floor as president Dwight Eisenhower after the president suffered a heart attack, and von Neumann was greatly amused by this.
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. One of the many things he enjoyed reading was the precise and wonderful way Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...

historians such as Thucydides
Thucydides (; grc, , }; BC) was an Classical Athens, Athenian historian and general. His ''History of the Peloponnesian War'' recounts Peloponnesian War, the fifth-century BC war between Sparta and Athens until the year 411 BC. Thucydides has ...

and Herodotus
Herodotus ( ; grc, , }; BC) was an ancient Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided in ...

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.
Mathematical style

Rota, in describing von Neumann's relationship with his friendStanislaw Ulam
Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Teller ...

, wrote that von Neumann had "deep-seated and recurring self-doubts". As an example on one occasion he said in the future he would be forgotten while Gödel would be remembered with Pythagoras. 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 incompleteness theorems and Birkhoff's 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. However, according to Rota, von Neumann still had an "incomparably stronger technique" compared to his friend, despite describing Ulam as the more creative mathematician. 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, treated properties of objects as some logical consequence of an underlying fundamental physical assumption. 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."
Herman Goldstine
Herman Heine Goldstine (September 13, 1913 – June 16, 2004) was a mathematician and computer scientist, who worked as the director of the IAS machine at Princeton University's Institute for Advanced Study and helped to develop ENIAC, the ...

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. 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. 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.
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, and if he had a new one he would sometimes drop what he was doing to write them down.
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
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces ...

, and some time later Goldstine saw in a paper in the 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. 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. 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.
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.
Although he was commonly described as an analyst, he once classified himself an algebraist, and his style often displayed a mix of algebraic technique and set-theoretical intuition. 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, $\backslash phi\; (x)$ to $\backslash phi\; ((x))$. However, this process ended up being repeated several times, where the final result were equations such as $(\backslash psi((((a)))))^2\; =\; \backslash phi((((a))))$. The 1936 paper became known to students as "von Neumann's onion" 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. 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.
At times he could be ignorant of the standard mathematical literature, it would at times be easier 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 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.
After World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...

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, 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".
Recognition

Cognitive abilities

Nobel LaureateHans Bethe
Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American theoretical physics, theoretical physicist who made major contributions to nuclear physics, astrophysics, quantum electrodynamics, and solid-state physics, and who won ...

said "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man", and later Bethe wrote that " on Neumann'sbrain indicated a new species, an evolution beyond man". Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Kingdom of Hungary, Hungarian-born United States, American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probabi ...

states that "von Neumann's speed was awe-inspiring." Israel Halperin said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car." Claude Shannon
Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American people, American mathematician, electrical engineering, electrical engineer, and cryptography, cryptographer known as a "father of information theory".
As a 21-year-o ...

called him "the smartest person I’ve ever met", a common opinion.
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 theory of duality.
Lothar Wolfgang Nordheim
LotharHis name is sometimes misspelled as ''Lother''. Wolfgang Nordheim (November 7, 1899, Munich – October 5, 1985, La Jolla, California) was a Germans, German born Jewish American theoretical physicist. He was a pioneer in the applications o ...

described von Neumann as the "fastest mind I ever met", and Jacob Bronowski
Jacob Bronowski (18 January 1908 – 22 August 1974) was a Polish-British mathematician and philosopher. He was known to friends and professional colleagues alike by the nickname Bruno. He is best known for developing a humanistic approach to sc ...

wrote "He was the cleverest man I ever knew, without exception. He was a genius." George Pólya
George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical proble ...

, 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." Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the Atomic Age, nuclea ...

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!"
Max Planck
Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a Germans, German theoretical physicist whose discovery of quantum mechanics, energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial con ...

, Max von Laue
Max Theodor Felix von Laue (; 9 October 1879 – 24 April 1960) was a German physicist who received the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals.
In addition to his scientific endeavors with cont ...

, and Werner Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics
Quantum mechanics is a fundamental Scientific theory, theory in physics ...

. Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...

was my brother-in-Law; Leo Szilard
Leo Szilard (; hu, Szilárd Leó, pronounced ; born Leó Spitz; February 11, 1898 – May 30, 1964) was a Hungarian-German-American physicist and inventor. He conceived the nuclear chain reaction in 1933, patented the idea of a nuclear ...

and Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

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."
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:
Wigner told a similar story, only with a swallow instead of a fly, and says it was Max Born
Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a n ...

who posed the question to von Neumann in the 1920s.
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.
Accolades and anecdotes were not limited to those from the physical or mathematical sciences either, 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 Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

and Oppenheimer and Teller and—who's the mad genius from MIT? I don't mean 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." US President
The president of the United States (POTUS) is the head of state and head of government of the United States of America. The president directs the Federal government of the United States#Executive branch, executive branch of the Federal gove ...

Dwight D. Eisenhower
Dwight David "Ike" Eisenhower (born David Dwight Eisenhower; ; October 14, 1890 – March 28, 1969) was an American military officer and statesman who served as the 34th president of the United States from 1953 to 1961. During World War II, ...

considered him "the outstanding mathematician of the time". 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 missilery he was considered "the clearly dominant advisory figure" according to Herbert York
Herbert Frank York (24 November 1921 – 19 May 2009) was an American nuclear physicist of Mohawk origin.http://www.edge.org/conversation/nsa-the-decision-problem. The Decision Problem He held numerous research and administrative positions ...

whose opinions "everyone took very seriously".
Even for writer Arthur Koestler
Arthur Koestler, (, ; ; hu, Kösztler Artúr; 5 September 1905 – 1 March 1983) was a Hungarian-born author and journalist. Koestler was born in Budapest and, apart from his early school years, was educated in Austria. In 1931, Koestler join ...

, 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”)."
He is often given as an example that mathematicians could do great work in the physical sciences too, however 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 molecular
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...

, and nuclear physics
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...

, particle physics
Particle physics or high energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standa ...

, astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry
Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matt ...

, relativity, and physical and organic chemistry
Organic chemistry is a subdiscipline within chemistry involving the science, scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.Clay ...

. 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.
Eidetic memory

Von Neumann was also noted for hiseidetic memory
Eidetic memory ( ; more commonly called photographic memory or total recall) is the ability to recall an image from memory
Memory is the faculty of the mind by which data or information is Encoding (memory), encoded, stored, and retrieved ...

, particularly of the symbolic kind. 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''. James Glimm wrote: "he is regarded as one of the giants of modern mathematics". 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", while Peter Lax described him as possessing the "most scintillating intellect of this century". 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 -fold winner, for his work in physics, in particular, quantum mechanics". 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." 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.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
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher. He was a professor of mathematics at the Massachusetts Institute of Technology (MIT). A child prodigy, Wiener later became an early researcher i ...

had mastered Fourier transform
A Fourier transform (FT) is a mathematics, mathematical Integral transform, transform that decomposes function (mathematics), functions into frequency components, which are represented by the output of the transform as a function of frequency. Mo ...

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.
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 off what Wigner already knew. Using such circular paths he could make even the most difficult concepts easy. On another occasion he wrote, "Nobody knows all science, not even von Neumann did. But as for mathematics, he advanced every part of it except number theory and topology. That is, I think, something unique." 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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle.
If the axis of revolution ...

. While elementary even for modern graduate students this kind of work never crossed his path and thus he did not know it.
One time 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
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

, whom he thought was deeper and broader than von Neumann. Similarly 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.
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. 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.
In his biography of von Neumann, Salomon Bochner describes how much of von Neumann's works in pure mathematics involved finite and infinite dimensional vector space
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...

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.
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, 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
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mat ...

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.
Honors and awards

* The John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences (INFORMS, previously TIMS-ORSA) is awarded annually to an individual (or group) who have made fundamental and sustained contributions to theory in operations research and the management sciences. * The IEEE John von Neumann Medal is awarded annually by the Institute of Electrical and Electronics Engineers (IEEE) "for outstanding achievements in computer-related science and technology." * The John von Neumann Lecture is given annually at the Society for Industrial and Applied Mathematics (SIAM) by a researcher who has contributed to applied mathematics, and the chosen lecturer is also awarded a monetary prize. * The crater Von Neumann (crater), von Neumann on the Moon is named after him. * Asteroid 22824 von Neumann was named in his honor. * The John von Neumann Center in Plainsboro Township, New Jersey, was named in his honor. * The professional society of Hungarian computer scientists, John von Neumann Computer Society, was named after von Neumann. * 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. * 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. * John von Neumann University (:hu:Neumann János Egyetem) was established in Kecskemét, Hungary in 2016, as a successor to Kecskemét College. * 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 several articles were written about him and his work by friends and colleagues. * A large number of books have been dedicated to him from a wide variety of fields. * A large number of scientific papers have been dedicated to him from a wide variety of fields. * Many events have been dedicated to him from a wide variety of fields. * Twice List of International Congresses of Mathematicians Plenary and Invited Speakers, invited speaker at the International Congress of Mathematicians. A list of the following awards and honors was drawn from various biographic statements given by von Neumann. Awards: Co-Editorship: Honorary societies: Honorary doctorates: Honorary positions: Society memberships:Selected works

Collections of von Neumann's published works can be found ozbMATH

an

Google Scholar

A list of his known works as of 1995 can be found i

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 Pressavailable here

2018 edition: * 1937.

Continuous Geometry

', Israel Halperin, Halperin, I., Preface, Princeton Landmarks in Mathematics and Physics, Princeton University Press

online at archive.org

2016 edition: * 1937.

Continuous Geometries with a Transition Probability

', Israel Halperin, Halperin, I., Preface, Memoirs of the American Mathematical Society Vol. 34, No. 252, 1981 edition. * 1941.

Invariant Measures

'. American Mathematical Society. 1999 edition: * 1944. '' Theory of Games and Economic Behavior'', with Oskar Morgenstern, Morgenstern, O., Princeton University Press

online at archive.org

2007 edition: * 1950.

Functional Operators, Volume 1: Measures and Integrals

'. Annals of Mathematics Studies 21

online at archive.org

2016 edition: * 1951.

Functional Operators, Volume 2: The Geometry of Orthogonal Spaces

'. Annals of Mathematics Studies 22

online at archive.org

2016 edition * 1958. '' The Computer and the Brain'', Ray Kurzweil, Kurzweil, R. Preface, The Silliman Memorial Lectures Series, Yale University Press

online at archive.org

2012 edition: * 1966.

Theory of Self-Reproducing Automata

', Arthur Burks, Burks, A. W., Ed., University of Illinois Press.

Scholarly articles

* 1923On the introduction of transfinite numbers

(in German), ''Acta Szeged'', 1:199-208. * 1925

An axiomatization of set theory

(in German), ''J. f. Math.'', 154:219-240. * 1926

On the Prüfer theory of ideal numbers

(in German), ''Acta Szeged'', 2:193-227. * 1927

On Hilbert's proof theory

(in German), ''Math. Zschr.'', 26:1-46. * 1929

General eigenvalue theory of Hermitian functional operators

(in German), ''Math. Ann.'', 102:49-131. * 1932

Proof of the Quasi-Ergodic Hypothesis

''Proc. Nat. Acad. Sci.'', 18:70-82. * 1932

Physical Applications of the Ergodic Hypothesis

''Proc. Nat. Acad. Sci.'', 18:263-266. * 1932

On the operator method in classical mechanics

(in German), ''Ann. Math.'', 33:587-642. * 1934

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

On Rings of Operators

with F. J. Murray, ''Ann. Math.'', 37:116-229. * 1936

On an Algebraic Generalization of the Quantum Mechanical Formalism (Part I)

''Mat. Sborn.'', 1:415-484. * 1936

The Logic of Quantum Mechanics

with Garrett Birkhoff, G. Birkhoff, ''Ann. Math.'', 37:823-843. * 1936

Continuous Geometry

''Proc. Nat. Acad. Sci.'', 22:92-100. * 1936

Examples of Continuous Geometries

''Proc. Nat. Acad. Sci.'', 22:101-108. * 1936

On Regular Rings

''Proc. Nat. Acad. Sci.'', 22:707-713. * 1937

On Rings of Operators, II

with F. J. Murray, ''Trans. Amer. Math. Soc.'', 41:208-248. * 1937

Continuous Rings and Their Arithmetics

''Proc. Nat. Acad. Sci.'', 23:341-349. * 1938

On Infinite Direct Products

''Compos. Math.'', 6:1-77. * 1940

On Rings of Operators, III

''Ann. Math.'', 41:94-161. * 1942

Operator Methods in Classical Mechanics, II

with Paul Halmos, P. R. Halmos, ''Ann. Math.'', 43:332-350. * 1943

On Rings of Operators, IV

with F. J. Murray, ''Ann. Math.'', 44:716-808. * 1945

A Model of General Economic Equilibrium

''Rev. Econ. Studies'', 13:1-9. * 1945

''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

Numerical Inverting of Matrices of High Order

with Herman Goldstine, H. H. Goldstine, ''Bull. Amer. Math. Soc.'', 53:1021-1099. * 1948

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. 1–31

MR0045446

* 1949

On Rings of Operators. Reduction Theory

''Ann. Math.'', 50:401-485. * 1950

A Method for the Numerical Calculation of Hydrodynamic Shocks

with R. D. Richtmyer, ''J. Appl. Phys.'', 21:232-237. * 1950

Numerical Integration of the Barotropic Vorticity Equation

with Jule Gregory Charney, J. G. Charney and Ragnar Fjørtoft, R. Fjörtoft, ''Tellus'', 2:237-254. * 1951

A spectral theory for general operators of a unitary space

(in German), ''Math. Nachr.'', 4:258-281. * 1951

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

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

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

* 1947The Mathematician

''The Works of the Mind''. ed. by R. B. Heywood, University of Chicago Press, 180–196. * 1951

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

The NORC and Problems in High Speed Computing

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

Can We Survive Technology?

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. ** 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

* John von Neumann (sculpture), ''John von Neumann'' (sculpture), Eugene, Oregon * John von Neumann Award * q:John von Neumann, John von Neumann - Wikiquote * List of things named after John von Neumann * List of pioneers in computer science * Self-replicating spacecraft * Von Neumann–Bernays–Gödel set theory * Von Neumann algebra * * Von Neumann bicommutant theorem * Von Neumann conjecture *Von Neumann entropy
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...

* Von Neumann programming languages
* Von Neumann regular ring
* Von Neumann universal constructor
John von Neumann's universal constructor is a self-replicating machine in a cellular automaton (CA) environment. It was designed in the 1940s, without the use of a computer. The fundamental details of the machine were published in von Neumann's b ...

* Von Neumann universe
* Trace inequality#Von Neumann's trace inequality and related results, Von Neumann's trace inequality
* The Martians (scientists)
Notes

References

* * * * * * * * * * * * * ** ** ** ** ** ** ** ** ** ** ** ** ** * * * * * * * * *Description

contents, incl. arrow-scrollable preview

&

review

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Further reading

Books * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Popular periodicals * ''Good Housekeeping, Good Housekeeping Magazine'', September 1956, "Married to a Man Who Believes the Mind Can Move the World" * Video * ''John von Neumann, A Documentary'' (60 min.), Mathematical Association of Americaavailable here

Journals * * * *

External links

by Nelson H. F. Beebe *

von Neumann's profile

at Google Scholar

Oral History Project

- The Princeton Mathematics Community in the 1930s, contains many interviews that describe contact and anecdotes of von Neumann and others at the Princeton University and Institute for Advanced Study community.

Oral history interview with Alice R. Burks and Arthur W. Burks

Charles Babbage Institute, University of Minnesota, Minneapolis. Alice Burks and Arthur Burks describe

ENIAC
ENIAC (; Electronic Numerical Integrator and Computer) was the first Computer programming, programmable, Electronics, electronic, general-purpose digital computer, completed in 1945. There were other computers that had these features, but the ...

, EDVAC, and IAS machine, 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.

zbMATH profile

Query for "von neumann"

on the digital repository of the Institute for Advanced Study.

Von Neumann vs. Dirac on Quantum Theory and Mathematical Rigor

– from ''Stanford Encyclopedia of Philosophy''

Quantum Logic and Probability Theory

- from ''Stanford Encyclopedia of Philosophy''

FBI files on John von Neumann released via FOI

Biographical video

by David Brailsford (John Dunford Professor Emeritus of computer science at the University of Nottingham)

A (very) Brief History of John von Neumann

video by YouTuber moderndaymath.

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).

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.

Greatest Mathematician Of The 20th Century

high-quality excerpt from above documentary where