**John von Neumann**

## John von Neumann 1903 –1957

**John von Neumann ( /vɒn ˈnɔɪmən/; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including mathematics (set theory,functional analysis, ergodic theory, geometry, numerical analysis, and many other mathematical fields), physics (quantum mechanics, hydrodynamics, and fluid dynamics), economics (game theory), computer science (linear programming,computer architecture, self-replicating machines, stochastic computing), andstatistics. He is generally regarded as one of the greatest mathematicians in modern history.**

The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians", while Peter Lax described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century, and Hans Bethe stated "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man". He was born in Budapest around the same time as Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913).

Von Neumann was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of theManhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor, and the digital computer. Von Neumann's mathematical analysis of the structure of self-replicationpreceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on theergodic theorem, Princeton, 1931–1932." Along with Teller and Stanisław Ulam, von Neumann worked out key steps in the nuclear physics involved inthermonuclear reactions and the hydrogen bomb.

Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital and later published in book form as The Computer and the Brain, gives an indication of the direction of his interests at the time of his death.

The mathematician Jean Dieudonné called von Neumann "the last of the great mathematicians", while Peter Lax described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century, and Hans Bethe stated "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man". He was born in Budapest around the same time as Theodore von Kármán (b. 1881), George de Hevesy (b. 1885), Leó Szilárd (b. 1898), Eugene Wigner (b. 1902), Edward Teller (b. 1908), and Paul Erdős (b. 1913).

Von Neumann was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of theManhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor, and the digital computer. Von Neumann's mathematical analysis of the structure of self-replicationpreceded the discovery of the structure of DNA. In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on theergodic theorem, Princeton, 1931–1932." Along with Teller and Stanisław Ulam, von Neumann worked out key steps in the nuclear physics involved inthermonuclear reactions and the hydrogen bomb.

Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital and later published in book form as The Computer and the Brain, gives an indication of the direction of his interests at the time of his death.

**Early life and education**

**Von Neumann was born Neumann János Lajos (Hungarian pronunciation: [ˈnojmɒn ˈjaːnoʃ ˈlɒjoʃ]; in Hungarian the family name comes first) in Budapest, Austro-Hungarian Empire, to wealthy Jewish parents. He was the eldest of three brothers. His father, Neumann Miksa (Max Neumann) was a banker, who held adoctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. His father (Mihály b. 1839) and grandfather (Márton) were both born in Ond (present day town of Szerencs?), Zemplén county, northern Hungary. John's mother was Kann Margit (Margaret Kann). In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian empire by Emperor Franz Josef. The Neumann family thus acquiring the hereditary title margittai, Neumann János became margittai Neumann János (John Neumann of Margitta), which he later changed to the German Johann von Neumann. János, nicknamed "Jancsi" (Johnny), was an extraordinary child prodigy in the areas of language, memorization, and mathematics. As a 6 year old, he could divide two 8-digit numbers in his head. By the age of 8, he was familiar with differential and integral calculus.**

**John entered the German-speaking Lutheran high school Fasori Evangelikus Gimnázium in Budapest in 1911. Although his father insisted he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Szegő subsequently visited the von Neumann house twice a week to tutor the child prodigy. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out with his father's stationery, 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, which superseded Georg Cantor's definition.**

**He received his Ph.D. in mathematics (with minors in experimental physics and chemistry) from Pázmány Péter University in Budapest at the age of 22. He simultaneously earned a diploma in chemical engineering from the ETH Zurich in Switzerland at his father's request, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavour than mathematics.**

**Geometry**

**Von Neumann founded 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, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.**

**Measure theory**

**In a series of famous papers, von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space." In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions (anticipating his later work, Mathematical formulation of quantum mechanics, on almost periodic functions).**

**In the 1936 paper on analytic measure theory, von Neumann used the Haar theorem in the solution of Hilbert's fifth problem in the case of compact groups.**

**Ergodic theory**

**Von Neumann made foundational contributions to ergodic theory, in a series of articles published in 1932, which have attained legendary status in mathematics. Of the 1932 papers on ergodic theory, Paul Halmos writes that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". By then von Neumann had already written his famous articles on operator theory, and the application of this work was instrumental in the von Neumann mean ergodic theorem.**

**Operator theory**

**Von Neumann introduced the study of rings of operators, through the von Neumann algebras. 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.**

**The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.**

**The direct integral was introduced in 1949 by John von Neumann. One of von Neumann's analyses was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.**

**Lattice theory**

**Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann worked on lattice theory between 1937 and 1939. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity."**

**Additionally, "In the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals of a suitable regular ring R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."**

**Mathematical formulation of quantum mechanics**

**Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics with his work Mathematische Grundlagen der Quantenmechanik.**

**After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a so-called Hilbert space, analogous to the 6N dimension (N is the number of particles, 3 general coordinate and 3 canonical momentum for each) phase space of classical mechanics but with infinitely many dimensions (corresponding to the infinitely many possible states of the system) instead: the traditional physical quantities (e.g., position and momentum) could therefore be represented as particular linear operators operating in these spaces. The physics of quantum mechanics was thereby reduced to the mathematics of the linear Hermitian operators on Hilbert spaces.**

**For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger, and culminated in his 1932 book Mathematische Grundlagen der Quantenmechanik.**

**Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs. non-determinism and in the book he presented a proof according to which quantum mechanics could not possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics. In 1966, a paper by John Bell was published, claiming that this proof contained a conceptual error and was therefore invalid (see the article on John Stewart Bell for more information). However, in 2010, Jeffrey Bub published an argument that Bell misconstrued von Neumann's proof, and that it is actually not flawed, after all. Regardless, the proof inaugurated a line of research that ultimately led, through the work of Bell in 1964 on Bell's theorem, and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics requires a notion of reality substantially different from that of classical physics.**

**In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, it never gained acceptance amongst the majority of physicists).**

**Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formalism of problems in quantum mechanics which underlies the majority of approaches and can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.**

**Mathematical statistics**

**Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.**

**Subsequently, John Denis Sargan and Alok Bhargava extended the results for testing if the errors on a regression model follow a Gaussianrandom walk (i.e. possess a unit root) against the alternative that they are a stationary first order autoregression.**