John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American
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, structure, space, models, and change.
History
On ...
,
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 cau ...
,
computer scientist,
engineer
Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the limit ...
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
Pure may refer to:
Computing
* A pure function
* A pure virtual function
* PureSystems, a family of computer systems introduced by IBM in 2012
* Pure Software, a company founded in 1991 by Reed Hastings to support the Purify tool
* Pure-FTPd, F ...
and
applied sciences.
Von Neumann made major contributions to many fields, including
mathematics (
foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
,
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
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
,
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
,
operator algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of ...
s,
matrix theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begi ...
,
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 ...
, and
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
),
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
(
quantum mechanics
Quantum mechanics is a fundamental 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 quantum chemistr ...
,
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, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
(
game theory and
general equilibrium theory),
computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, ...
(
Von Neumann architecture,
linear programming,
numerical meteorology,
scientific computing,
self-replicating machines,
stochastic computing), and
statistics. He was a pioneer of the application of
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
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
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 ...
, the
universal constructor and the
digital computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These program ...
.
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
Self-replication is any behavior of a dynamical system that yields construction of an identical or similar copy of itself. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and c ...
preceded the discovery of the structure of
DNA. In a shortlist of facts about his life he submitted to the
National Academy of Sciences, he wrote, "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."
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 vast majority of the world's countries—including all of the great powers—forming two opposing ...
, 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 physicist who is known colloquially as "the father of the hydrogen bomb" (see the Teller–Ulam design), although he did not care for ...
, mathematician
Stanislaw Ulam 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 reactions and the
hydrogen bomb. 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
Trinitrotoluene (), more commonly known as TNT, more specifically 2,4,6-trinitrotoluene, and by its preferred IUPAC name 2-methyl-1,3,5-trinitrobenzene, is a chemical compound with the formula C6H2(NO2)3CH3. TNT is occasionally used as a reagen ...
) 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 U.S. Army research establishment situated at Aberdeen Proving Ground, Maryland that specialized in ballistics ( interior, exterior, and terminal) as well as vulnerability and lethality analys ...
, the
Armed Forces Special Weapons Project
The Armed Forces Special Weapons Project (AFSWP) was a United States military agency responsible for those aspects of nuclear weapons remaining under military control after the Manhattan Project was succeeded by the Atomic Energy Commission on ...
and the
Oak Ridge National Laboratory. 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 ...
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 and
Trevor Gardner in contributing to the design and development of the United States' first
ICBM 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. As a symbol of the U.S. military, the phrase ''The Pentagon'' is often used as a meton ...
. 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 policy which posits that a full-scale use of nuclear weapons by an attacker on a nuclear-armed defender with second-strike capabilities would cause the ...
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 printed in broadsheet and published digitally that focuses on business and economic current affairs. Based in London, England, the paper is owned by a Japanese holding company, Ni ...
''
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-observant
Jewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...
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 by population within city limits and the second-largest city on the Danube river; the city has an estimated population ...
,
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 first king Stephen ...
, which was then part of the
Austro-Hungarian Empire.
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
Pécs ( , ; hr, Pečuh; german: Fünfkirchen, ; also known by other #Name, alternative names) is List of cities and towns of Hungary#Largest cities in Hungary, the fifth largest city in Hungary, on the slopes of the Mecsek mountains in the countr ...
at the end of the 1880s. Miksa's father and grandfather were both born in Ond (now part of the town of
Szerencs
Szerencs is a town in Borsod-Abaúj-Zemplén county, Northern Hungary. It lies away from Miskolc, and away from Budapest. It has about 9,100 inhabitants.
History
Szerencs grew into a town where the Great Plain and the Zemplén mountains meet. ...
),
Zemplén County
Zemplén ( hu, Zemplén, sk, Zemplín, german: Semplin, Semmlin, la, Zemplinum) was an administrative county (comitatus) of the Kingdom of Hungary. The northern part of its territory is now situated in eastern Slovakia ( Zemplín region), while ...
, 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 from 2 December 1848 until his ...
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
Marghita (; hu, Margitta ; yi, מארגארעטין ''Margaretin'') is a city in Bihor County, Romania. It administers two villages, Cheț (''Magyarkéc'') and Ghenetea (''Genyéte'').
Geography
Marghita is located in the northern part of the c ...
,
Romania
Romania ( ; ro, România ) is a country located at the crossroads of Central, Eastern, and Southeastern Europe. It borders Bulgaria to the south, Ukraine to the north, Hungary to the west, Serbia to the southwest, Moldova to the east, and ...
). 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 heraldic visual design on an escutcheon (i.e., shield), surcoat, or tabard (the latter two being outer garments). The coat of arms on an escutcheon forms the central element of the full heraldic achievement, which in its ...
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 a
child prodigy. 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 ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...
. 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
Italian(s) may refer to:
* Anything of, from, or related to the people of Italy over the centuries
** Italians, an ethnic group or simply a citizen of the Italian Republic or Italian Kingdom
** Italian language, a Romance language
*** Regional Ita ...
. 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 co ...
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, ( 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 fields of aeronaut ...
(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 radiochemist and Nobel Prize in Chemistry laureate, recognized in 1943 for his key rol ...
(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
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 ...
(born 1898),
Dennis Gabor
Dennis Gabor ( ; hu, Gábor Dénes, ; 5 June 1900 – 9 February 1979) was a Hungarian-British electrical engineer and physicist, most notable for inventing holography, for which he later received the 1971 Nobel Prize in Physics. He obtained ...
(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 co ...
(born 1902),
Edward Teller
Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American theoretical physicist who is known colloquially as "the father of the hydrogen bomb" (see the Teller–Ulam design), although he did not care for ...
(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 ...
, which superseded
Georg Cantor'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 friend
Theodore von Kármán
Theodore von Kármán ( hu, ( 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 fields of aeronaut ...
, 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. 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. It was established by Frederick William III on the initiative ...
, after which he sat for the entrance exam to the prestigious
ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered
Pázmány Péter University in Budapest,
as a
Ph.D.
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
candidate in
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
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded ...
on a grant from the
Rockefeller Foundation to study mathematics under
David Hilbert.
Career and private life
Von Neumann's
habilitation was completed on December 13, 1927, and he began to give lectures as a ''
Privatdozent'' at the University of Berlin in 1928. 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 research university in Hamburg, Germany. It was founded on 28 March 1919 by combining the previous General Lecture System ('' Allgemeines Vo ...
, 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 research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
as a visiting lecturer in
mathematical physics
Mathematical physics refers to the development of 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 the developme ...
.
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, born in 1935. As of 2021 Marina is a distinguished professor emerita of business administration and public policy at the
University of Michigan
, mottoeng = "Arts, Knowledge, Truth"
, former_names = Catholepistemiad, or University of Michigania (1817–1821)
, budget = $10.3 billion (2021)
, endowment = $17 billion (2021)As o ...
. 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 vast majority of the world's countries—including all of the great powers—forming two opposing ...
.
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 largest Christian church, with 1.3 billion baptized Catholics worldwide . It is among the world's oldest and largest international institutions, and has played a ...
. 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 schola ...
in New Jersey, when that institution's plan to appoint
Hermann Weyl 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. 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 English culture or British culture, or a process of cultural and/or linguistic change in which something non-English becomes English. It can also refer to the influenc ...
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 or nationality of that country. It may be done automatically by a statute, i.e., without any effort on the part of the in ...
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 rank in the armed forces of many nations.
The meaning of lieutenant differs in different militaries (see comparative military ranks), but it is often ...
in the United States Army's
Officers Reserve Corps
The United States Army Reserve (USAR) is a 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 Forces.
Since July 2020, ...
. 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
Clapboard (), also called bevel siding, lap siding, and weatherboard, with regional variation in the definition of these terms, is wooden siding of a building in the form of horizontal boards, often overlapping.
''Clapboard'' in modern Americ ...
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 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
This history of the Byzantine Empire covers the history of the Eastern Roman Empire from late antiquity until the Fall of Constantinople in 1453 AD. Several events from the 4th to 6th centuries mark the transitional period during which the Rom ...
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
''Decline and Fall'' is a novel by the English author Evelyn Waugh, first published in 1928. It was Waugh's first published novel; an earlier attempt, titled '' The Temple at Thatch'', was destroyed by Waugh while still in manuscript form. '' ...
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 mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, von Neumann would describe even the minutest details of the battles of the
Civil War 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 and
"off-color" humor (especially
limericks
A limerick ( ) is a form of verse, usually humorous and frequently rude, in five-line, predominantly trimeter with a strict rhyme scheme of AABBA, in which the first, second and fifth line rhyme, while the third and fourth lines are shorter and ...
). He was a non-smoker. In Princeton, he received complaints for regularly playing extremely loud German
march music on his
phonograph, which distracted those in neighboring offices, including
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
, 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. A later friend of Ulam's,
Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, proba ...
, 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 physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
and
Norbert Wiener). "They generally showed differences from the average".
Illness and death
In 1955, von Neumann was diagnosed with what was either
bone
A bone is a rigid organ that constitutes part of the skeleton in most vertebrate animals. Bones protect the various other organs of the body, produce red and white blood cells, store minerals, provide structure and support for the body, ...
,
pancreatic
The pancreas is an organ of the digestive system and endocrine system of vertebrates. In humans, it is located in the abdomen behind the stomach and functions as a gland. The pancreas is a mixed or heterocrine gland, i.e. it has both an endocr ...
or
prostate cancer 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 laboratories of the United States Department of Energy (DOE), located a short distance northwest of Santa Fe, New Mexico, ...
. 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, shortly before death. They may be administered to those awaiting execution, mortall ...
to him. Some of von Neumann's friends, such as
Abraham Pais
Abraham Pais (; May 19, 1918 – July 28, 2000) was a Dutch-American physicist and science historian. Pais earned his Ph.D. from University of Utrecht just prior to a Nazi ban on Jewish participation in Dutch universities during World War II. ...
and
Oskar Morgenstern
Oskar Morgenstern (January 24, 1902 – July 26, 1977) was an Austrian-American economist. In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory as applied to the social sciences and strategic decis ...
, 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 in Washington, D.C., under military security lest he reveal military secrets while heavily medicated. He was buried at Princeton Cemetery of Nassau Presbyterian Church in Princeton, Mercer County, New Jersey.
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 land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the California ...
should he have recovered from his cancer.
Mathematics
Set theory
The axiomatization of mathematics, on the model of Euclid'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 In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
Formal definition
An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...
of Richard Dedekind and Charles Sanders Peirce, 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 suffered a setback due to Russell's paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
(on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel
Abraham Fraenkel ( he, אברהם הלוי (אדולף) פרנקל; February 17, 1891 – October 15, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. ...
. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the ''axiom of foundation
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the ax ...
'' 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, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
s as well as the first strict formulation of principles of definitions by the transfinite induction".
Von Neumann paradox
Building on the work of Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
, in 1924 Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a 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 origina ...
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 in 3‑dimensional space, there exists
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
a decomposition of the ball into a finite number of disjoint subsets 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 (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 its consistency
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
. 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 that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
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 are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type ...
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 mathematician Giuseppe Peano. These axioms have been used nearly ...
. 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.
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 The Second Conference on the Epistemology of the Exact Sciences (german: 2. Tagung für Erkenntnislehre der exakten Wissenschaften in Königsberg) was held on 5–7 September 1930 in Königsberg, then located in East Prussia. It was at this conferen ...
of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth 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 and foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
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 of dynamical systems with an invariant measure. Of the 1932 papers on ergodic theory, Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
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
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
, 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 of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
s and states that for every vector in the Hilbert space, exists in the sense of the metric defined by the Hilbert norm and is a vector which is such that for all . 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. He also pointed out that ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
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 Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator ...
have significant applications in other areas of mathematics.
Measure theory
In measure theory, the "problem of measure" for an -dimensional Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, an ...
and Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a 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 origina ...
had implied that the problem of measure has a positive solution if or and a negative solution (because of the Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
) 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 consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the gr ...
of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
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 σ-finite measure space 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 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 mathematics.
Elementary ...
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 topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
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 and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
s. He also gave a new, ingenious proof for the Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurab ...
. 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 of topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Haral ...
s 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
The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). It is awarded every three years (formerly every five year ...
for his work in analysis in relation to these papers.
In a 1933 paper, he used the newly discovered Haar measure in the solution of Hilbert's fifth problem for the case of compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
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 mapping V \to W between two vector spaces that pre ...
and found that closed subgroup
In group theory, a branch of mathematics, given a 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, ''H'' is a subgroup ...
s of a general linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a f ...
are Lie groups. 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 abstract Hilbert space whereas it was previously defined as the Lp space. He defined it as a complex vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 that were previously only defined for Euclidean spaces
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
. He continued with the development of the spectral theory of operators in Hilbert space in 3 seminal papers between 1929 and 1932. This work cumulated in his Mathematical Foundations of Quantum Mechanics
The book ''Mathematical Foundations of Quantum Mechanics'' (1932) by John von Neumann is an important early work in the development of quantum theory.
Publication history
The book was originally published in German in 1932 by Julius Springer, un ...
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 objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called t ...
, and von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining locally convex spaces
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological spa ...
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 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 quantum chemistr ...
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, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''.
Normal opera ...
s, the first abstract presentation of the trace of a positive operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \l ...
, a generalisation of Riesz's presentation of 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 ...
's spectral theorems at the time, and the discovery of Hermitian operators
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 its ...
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 its ...
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 algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study of ...
s.
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
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many vari ...
.
With I. J. Schoenberg he wrote several items investigating translation 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 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 number to a poin ...
which resulted in their complete classification. Their motivation lie in various questions related to embedding metric spaces 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 matri ...
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 pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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 spac ...
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 may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace ...
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 subsets of X to relatively compact subsets of Y (subsets with compact c ...
s, and preduality with bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector ...
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 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 the von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algebra ...
s (originally called W*-algebras). While his original ideas for rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
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:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
of operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
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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 that is closed in the weak operator topology and contains the identity operator
Identity may refer to:
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* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film) ...
. The von Neumann bicommutant theorem
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection bet ...
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 ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
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 or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive se ...
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
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
, the theory of partially ordered sets 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. He is best known for his work in lattice theory.
The mathematician George Birkhoff (1884–1944) was his father.
Life
The son of the mathematician Ge ...
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, ring theory
In algebra, ring theory is the study of 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 of rings, their r ...
, 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, ...
, where instead of the dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of a subspace being in a discrete set it can be an element of the unit interval