John von neumann biography summary

Hungarian and American mathematician and physicist (1903–1957)

The native form of this personal name is Neumann János Lajos. This article uses Western name order when mentioning individuals.

John von Neumann (von NOY-mən; Hungarian: Neumann János Lajos[ˈnɒjmɒnˈjaːnoʃˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including cellular automata, the universal constructor and the digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA.

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

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

Life and education

Family background

Von Neumann was born in Budapest, Kingdom of Hungary (then part of the Austro-Hungarian Empire),^[13][14] on December 28, 1903, to a wealthy, non-observant Jewish family. His birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names are equivalent to John Louis in English.

He was the eldest of three brothers; his two younger siblings were Mihály (Michael) and Miklós (Nicholas). His father Neumann Miksa (Max von Neumann) was a banker and held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. Miksa's father and grandfather were born in Ond (now part of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (Margaret Kann); her parents were Kann Jákab and Meisels Katalin 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 elevated John's father to the Hungarian nobility for his service to the Austro-Hungarian Empire.^[22] The Neumann family thus acquired the hereditary appellation Margittai, meaning "of Margitta" (today Marghita, Romania). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosen coat of arms depicting three 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 who at six years old could divide two eight-digit numbers in his head^[24] and converse in Ancient Greek.^[26] He, 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 age eight, von Neumann was familiar with differential and integral calculus, and by twelve he had read Borel'sLa Théorie des Fonctions. He was also interested in history, reading Wilhelm Oncken's 46-volume world history series Allgemeine Geschichte in Einzeldarstellungen (General History in Monographs). One of the rooms in the apartment was converted into a library and reading room.

Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1914.Eugene Wigner was a year ahead of von Neumann at the school and soon became his friend.

Although von Neumann's father insisted that he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give von Neumann advanced instruction. At 15, he began to study advanced calculus under the analyst Gábor Szegő. On their first meeting, Szegő was so astounded by von Neumann's mathematical talent and speed that, as recalled by his wife, he came back home with tears in his eyes.^[33] By 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition.^[34] At the conclusion of his education at the gymnasium, he applied for and won the Eötvös Prize, a national award for mathematics.

University studies

According to his friend Theodore von Kármán, von Neumann's father wanted John to follow him into industry, and asked von Kármán to persuade his son not to take mathematics.^[36] Von Neumann and his father decided that the best career path was chemical engineering. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year, non-degree course in chemistry at the University of Berlin, after which he sat for the entrance exam to ETH Zurich, which he passed in September 1923. Simultaneously von Neumann entered Pázmány Péter University, then known as the University of Budapest, as a Ph.D. candidate in mathematics.^[39] For his thesis, he produced an axiomatization of Cantor's set theory.^[40] In 1926, he graduated as a chemical engineer from ETH Zurich and simultaneously passed his final examinations summa cum laude for his Ph.D. in mathematics (with minors in experimental physics and chemistry) at the University of Budapest.^[42]

He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.Hermann Weyl remembers how in the winter of 1926–1927 von Neumann, Emmy Noether, and he would walk through "the cold, wet, rain-wet streets of Göttingen" after class discussing hypercomplex number systems and their representations.^[45]

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.^[46] He was the youngest person elected Privatdozent in the university's history.^[47] He began writing nearly one major mathematics paper per month. In 1929, he briefly became a Privatdozent at the University of Hamburg, where the prospects of becoming a tenured professor were better, then in October of that year moved to Princeton University as a visiting lecturer in mathematical physics.

Von Neumann was baptized a Catholic in 1930. Shortly afterward, he married Marietta Kövesi, who had studied economics at Budapest University. Von Neumann and Marietta had a daughter, Marina, born in 1935; she would become a professor.^[52] The couple divorced on November 2, 1937.^[53] On November 17, 1938, von Neumann married Klára Dán.

In 1933 Von Neumann accepted a tenured professorship at the Institute for Advanced Study in New Jersey, when that institution's plan to appoint Hermann Weyl appeared to have failed. His mother, brothers and in-laws followed von Neumann to the United States in 1939. Von Neumann anglicized his name to John, keeping the German-aristocratic surname von Neumann. Von Neumann became a naturalized U.S. citizen in 1937, and immediately tried to become a lieutenant in the U.S. Army's Officers Reserve Corps. He passed the exams but was rejected because of his age.

Klára and John von Neumann were socially active within the local academic community. His white clapboard house on Westcott Road was one of Princeton's largest private residences.^[60] He always wore formal suits.^[61] He enjoyed Yiddish and "off-color" humor. In Princeton, he received complaints for playing extremely loud German march music; Von Neumann did some of his best work in noisy, chaotic environments. According to Churchill Eisenhart, von Neumann could attend parties until the early hours of the morning and then deliver a lecture at 8:30.^[64]

He was known for always being happy to provide others of all ability levels with scientific and mathematical advice.^[4][66] Wigner wrote that he perhaps supervised more work (in a casual sense) than any other modern mathematician. His daughter wrote that he was very concerned with his legacy in two aspects: his life and the durability of his intellectual contributions to the world.^[68]

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

He also maintained his knowledge of languages learnt in his youth. He knew Hungarian, French, German and English fluently, and maintained a conversational level of Italian, Yiddish, Latin and Ancient Greek. His Spanish was less perfect. He had a passion for and encyclopedic knowledge of ancient history, and he enjoyed reading Ancient Greek historians in the original Greek. Ulam suspected they may have shaped his views on how future events could play out and how human nature and society worked in general.

Von Neumann's closest friend in the United States was the mathematician Stanisław Ulam.^[76] 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. Ulam recalled, "Quite independently of his liking for abstract wit, he had a strong appreciation (one might say almost a hunger) for the more earthy type of comedy and humor".

Illness and death

In 1955, a mass was found near von Neumann's collarbone, which turned out to be cancer originating in the skeleton, pancreas or prostate. (While there is general agreement that the tumor had metastasised, sources differ on the location of the primary cancer.)^[79][80] The malignancy may have been caused by exposure to radiation at Los Alamos National Laboratory. As death neared he asked for a priest, though the priest later recalled that von Neumann found little comfort in receiving the last rites – he remained terrified of death and unable to accept it.^[82][85] Of his religious views, Von Neumann reportedly said, "So long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end," referring to Pascal's wager. He confided to his mother, "There probably has to be a God. Many things are easier to explain if there is than if there isn't."^[86][87]

He died Roman Catholic on February 8, 1957, at Walter Reed Army Medical Hospital and was buried at Princeton Cemetery.^[89]

Mathematics

Set theory

See also: Von Neumann–Bernays–Gödel set theory

At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his 1925 doctoral thesis, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.^[91]

The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the Zermelo–Fraenkel principles. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced the method of inner models, which became an essential demonstration instrument in set theory.^[91]

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.^[91]

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 numbers as well as the first strict formulation of principles of definitions by the transfinite induction".

Von Neumann paradox

Main article: Von Neumann paradox

Building on the Hausdorff paradox of Felix Hausdorff (1914), Stefan Banach and Alfred Tarski in 1924 showed how to subdivide a three-dimensional ball into disjoint sets, then translate and rotate these sets to form two identical copies of the same ball; this is the Banach–Tarski paradox. They also proved that a two-dimensional disk has no such paradoxical decomposition. But in 1929,^[93] von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preserving affine transformations instead of translations and rotations. The result depended on finding free groups of affine transformations, an important technique extended later by von Neumann in his work on measure theory.

Proof theory

See also: Hilbert's program

With the contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its consistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger axioms that could be used to prove a broader class of theorems.^[95]

By 1927, von Neumann was involving himself in discussions in Göttingen on whether elementary arithmetic followed from Peano axioms.^[96] Building on the work of Ackermann, he 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).^[97] 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 Second Conference on the Epistemology of the Exact Sciences, 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. At the conference, von Neumann suggested to Gödel that he should try to transform his results for undecidable propositions about integers.

Less than a month later, von Neumann communicated to Gödel an interesting consequence of his theorem: the usual axiomatic systems are unable to demonstrate their own consistency. Gödel replied that he had already discovered this consequence, now known as his second incompleteness theorem, and that he would send a preprint of his article containing both results, which never appeared.^[101] Von Neumann acknowledged Gödel's priority in his next letter. However, von Neumann's method of proof differed from Gödel's, and he was also of the opinion that the second incompleteness theorem had dealt a much stronger blow to Hilbert's program than Gödel thought it did.^[106] With this discovery, which drastically changed his views on mathematical rigor, von Neumann ceased research in the foundations of mathematics and metamathematics and instead spent time on problems connected with applications.

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.^[108] Of the 1932 papers on ergodic theory, Paul Halmos wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". 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-parameterunitary groups 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'sergodic hypothesis. He also pointed out that ergodicity had not yet been achieved and isolated this for future work.^[111]

Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodic measure preserving actions of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with Paul Halmos have significant applications in other areas of mathematics.^[111][112]

Measure theory

See also: Lifting theory

In measure theory, the "problem of measure" for an n-dimensional Euclidean spaceRⁿ may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of Rⁿ?" The work of Felix Hausdorff and Stefan Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution (because of the Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of the transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space." Around 1942 he told Dorothy Maharam how to prove that every completeσ-finitemeasure space has a multiplicative lifting; 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 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 discussed various generalizations and algebraic aspects of this problem.^[118] 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 groups. He had to create entirely new techniques to apply this to locally compact groups. He also gave a new, ingenious proof for the Radon–Nikodym theorem.^[120] 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.^[122][123]

Topological groups

Using his previous work on measure theory, von Neumann made several contributions to the theory of topological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic functions to arbitrary groups.^[124] 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.^[125] In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis in relation to these papers.^[126]

In a 1933 paper, he used the newly discovered Haar measure in the solution of Hilbert's fifth problem for the case of compact groups.^[128] 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 and found that closed subgroups of a general linear group are Lie groups.^[129] This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem.

Functional analysis

Main article: Operator theory

See also: Spectral theorem

Von Neumann was the first to axiomatically define an abstract Hilbert space. 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 proved the general form of the Cauchy–Schwarz inequality that had previously been known only in specific examples. He continued with the development of the spectral theory of operators in Hilbert space in three seminal papers between 1929 and 1932. This work cumulated in his Mathematical Foundations of Quantum Mechanics which alongside two other books by Stone 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. 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 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 where von Neumann realized the need to extend the spectral theory of Hermitian operators from the bounded to the unbounded case.^[136] Other major achievements in these papers include a complete elucidation of spectral theory for normal operators, the first abstract presentation of the trace of a positive operator,^[137] 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 operators, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. He wrote a paper detailing how the usage of 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 work on spectral theory and providing a new way of working through the geometric content by the use of direct integrals of Hilbert spaces.^[136] Like in his work on measure theory he proved several theorems that he did not find time to publish. He told Nachman Aronszajn and K. T. Smith that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the invariant subspace problem.

With I. J. Schoenberg he wrote several items investigating translation invariant Hilbertian metrics on the real number line which resulted in their complete classification. Their motivation lie in various questions related to embedding metric spaces into Hilbert spaces.^[141][142]

With Pascual Jordan he wrote a short paper giving the first derivation of a given norm from an inner product 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).^[145][147] 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 operators on Hilbert spaces,^[149][150]tensor products of Banach spaces, introduced and studied trace class operators, their ideals, and their duality with compact operators, and preduality with bounded operators. 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

Main article: Von Neumann algebra

See also: Direct integral

Von Neumann founded the study of rings of operators, through the von Neumann algebras (originally called W*-algebras). While his original ideas for rings of operators existed already in 1930, he did not begin studying them in depth until he met F. J. Murray several years later.^[158] 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.^[159] The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant.^[160] After elucidating the study of the commutative algebra 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"; they collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification of factors. In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949.^[164] 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.^[166] Another important result on polar decomposition was published in 1932.

Lattice theory

Main article: Continuous geometry

See also: Complemented lattice § Orthomodular lattices

Between 1935 and 1937, von Neumann worked on lattice theory, the theory of partially ordered sets in which every two elements have a greatest lower bound and a least upper bound. As Garrett Birkhoff wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann combined traditional projective geometry with modern algebra (linear algebra, ring theory, lattice theory). Many previously geometric results could then be interpreted in the case of general modules over rings. His work laid the foundations for some of the modern work in projective geometry.^[169]

His biggest contribution was founding the field of continuous geometry.^[170] It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry, where instead of the dimension of a subspace being in a discrete set it can be an element of the unit interval