John von Neumann

John von Neumann (/vɒn ˈnɔɪmən/; December 28, 1903 – February 8, 1957) was a Hungarian-American pure and applied mathematician and polymath. He made major contributions to a number of fields,[1] including mathematics (foundations of mathematics, functional analysis, ergodic theory, geometry, topology, and numerical analysis), physics (quantum mechanics, hydrodynamics, and fluid dynamics), economics (game theory), computer science (Von Neumann architecture, linear programming, self-replicating machines, stochastic computing), and statistics.[2] He was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a principal member of the Manhattan Project and the Institute for Advanced Study in Princeton (as one of the few originally appointed), and a key figure in the development of game theory[1][3] and the concepts of cellular automata,[1] the universal constructor, and the digital computer.

Von Neumann's mathematical analysis of the structure of self-replication preceded the discovery of the structure of DNA.[4] In a short list of facts about his life he submitted to the National Academy of Sciences, he stated "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932." Along with Hungarian-born American theoretical physicist Edward Teller and Polish mathematician Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb.

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

Early life and education[edit]

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.[5][6][7] He was the eldest of three brothers. His father, Neumann Miksa (Max Neumann) was a banker, who held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. Miksa's father (Mihály b. 1839)[8] and grandfather (Márton)[8] were both born in Ond (now part of the town of Szerencs), Zemplén county, northern Hungary. John's mother was Kann Margit (Margaret Kann).[9]

Her parents were Jakab Kann II (Pest (now Budapest) 1845–1928) and Katalin Meisels (Munkács, Kárpátalja c. 1854–1914). In 1913, his father was elevated to the nobility for his service to the 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.[10] By the age of 8, he was familiar with differential and integral calculus.[11]

Von Neumann was part of a Budapest generation noted for intellectual achievement: 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).[12]

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.[13]

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.[14] 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.[15]

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.[1] He simultaneously earned a diploma in chemical engineering from the ETH Zurich in Switzerland[1] at his father's request, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavour than mathematics.[N 1]

Career and abilities[edit]

Beginnings[edit]

Between 1926 and 1930, he taught as a Privatdozent at the University of Berlin, the youngest in its history.[N 2] By the end of 1927, von Neumann had published twelve major papers in mathematics, and by the end of 1929, thirty-two papers, at a rate of nearly one major paper per month.[17] Von Neumann's powers of speedy, massive memorization and recall allowed him to recite volumes of information, and even entire directories, with ease.[16]

In 1930, von Neumann was invited to Princeton University, New Jersey. In 1933, he was offered a position on the faculty of the Institute for Advanced Study when the institute's plan to appoint Hermann Weyl fell through; von Neumann remained a mathematics professor there until his death. His father, Max von Neumann had died in 1929. But his mother and his brothers followed John to the United States. He anglicized his first name to John, keeping the German-aristocratic surname of von Neumann. In 1937, von Neumann became a naturalized citizen of the U.S. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis.

Set theory[edit]

The axiomatization of mathematics, on the model of Euclid's Elements, had reached new levels of rigor and breadth at the end of the 19th century, particularly in arithmetic (thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce) and geometry (thanks to David Hilbert). At the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves).

The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later (by Ernst Zermelo and Abraham Fraenkel). Zermelo and Fraenkel provided Zermelo–Fraenkel set theory, a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics. But they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets: the axiom of foundation and the notion of class.

The axiom of foundation established that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel, in such a manner that if one set belongs to another then the first must necessarily come before the second in the succession (hence excluding the possibility of a set belonging to itself.) To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration (called the method of inner models) which later became an essential instrument in set theory.

The second approach to the problem took as its base the notion of class, and defines a set as a class which belongs to other classes, while a proper class is defined as a class which does not belong to other classes. Under the Zermelo/Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under the von Neumann approach, the class of all sets which do not belong to themselves can be constructed, but it is a proper class and not a set.

With this contribution of von Neumann, the axiomatic system of the theory of sets became fully satisfactory, and the next question was whether or not it was also definitive, and not subject to improvement. A strongly negative answer arrived in September 1930 at the historic mathematical Congress of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time.[18]

But von Neumann, who had participated at the Congress, confirmed his fame as an instantaneous thinker, and in less than a month was able to communicate to Gödel himself an interesting consequence of his theorem: namely that the usual axiomatic systems are unable to demonstrate their own consistency.[18] However, Gödel had already discovered this consequence (now known as his second incompleteness theorem), and sent von Neumann a preprint of his article containing both incompleteness theorems. Von Neumann acknowledged Gödel's priority in his next letter.[19]

Geometry[edit]

Von Neumann founded the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is an analogue of complex projective geometry, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.

Measure theory[edit]

In a series of famous papers, von Neumann made spectacular contributions to measure theory.[20] 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.[20][21]

Ergodic theory[edit]

Von Neumann made foundational contributions to ergodic theory, in a series of articles published in 1932.[22] 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".[20] 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.[23]

Operator theory[edit]

Von Neumann introduced the study of rings of operators, through the von Neumann algebras.[24] 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[edit]

Garrett Birkhoff writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor".[25] 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."[25]

Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal 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."[25]

Mathematical formulation of quantum mechanics[edit]

Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, with his 1932 work Mathematische Grundlagen der Quantenmechanik.

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.

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.[26] 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).[27]

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.

Quantum logic[edit]

In a famous paper of 1936, the first work ever to introduce quantum logics,[28] von Neumann first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work. But in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters which are polarized perpendicularly (e.g., one horizontally and the other vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession. But if the third filter is added in between the other two, the photons will indeed pass through. And this experimental fact is translatable into logic as the non-commutativity of conjunction . It was also demonstrated that the laws of distribution of classical logic,  and , are not valid for quantum theory. The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically determinate, while each of its members are necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., x and y) results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the x direction is positive." By the principle of indeterminacy, the value of the spin in the direction y will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of y is positive" nor the proposition "the spin in the direction of y is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of y is positive or the spin in the direction of y is negative" must be true for ɸ. In the case of distribution, it is therefore possible to have a situation in which , while .

Von Neumann proposes to replace classical logics, with a logic constructed in orthomodular lattices, (isomorphic to the lattice of subspaces of the Hilbert space of a given physical system).[29]

Game theory[edit]

Von Neumann founded the field of game theory as a mathematical discipline. Von Neumann proved his minimax theorem in 1928. This theorem establishes that in zero-sum games with perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy which will result in the minimization of his maximum loss.

Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Another result he proved during his German period was the nonexistence of a static equilibrium. An equilibrium can only exist in an expanding economy. Paul Samuelson edited an anniversary volume dedicated to this short German paper in 1972 and stated in the introduction that von Neumann was the only mathematician ever to make a significant contribution to economic theory.

Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior (written with Oskar Morgenstern). The public interest in this work was such that The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analytic methods, especially convex sets and topologicalfixed point theorem, rather than the traditional differential calculus, because the maximum–operator did not preserve differentiable functions.

Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairingsof real vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex set, and fixed-point theory—have been the primary tools of mathematical economics ever since.[30] Von Neumann was also the inventor of the method of proof, used in game theory, known as backward induction (which he first published in 1944 in the book co-authored with Morgenstern, Theory of Games and Economic Behaviour).[31]

Mathematical economics[edit]

Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization ofBrouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the matrix pencil  A − λB with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve thecomplementarity equation

pT (A − λ B) q = 0,

along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.[32][33][34]

Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices.[35] The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.[36][37][38] This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities,complementary slackness, and saddlepoint duality.

The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of Nobel prizes in 1972 to Kenneth Arrow, in 1983 to Gérard Debreu, and in 1994 to John Nash who used fixed point theorems to establish equilibria for noncooperative games and for bargaining problems in his Ph.D. thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans, Leonid Kantorovich, Wassily Leontief, Paul Samuelson, Robert Dorfman, Robert Solow, and Leonid Hurwicz.

Linear programming[edit]

Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming, after George B. Dantzig described his work in a few minutes, when an impatient von Neumann asked him to get to the point. Then, Dantzig listened dumbfounded while von Neumann provided an hour lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.[39]

Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873) which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior-point methodof linear programming.[39]

Mathematical statistics[edit]

Von 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.[40] This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic[41] for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.[41]

Subsequently, John Denis Sargan and Alok Bhargava[42] extended the results for testing if the errors on a regression model follow a Gaussian random walk (i.e. possess a unit root) against the alternative that they are a stationary first orderautoregression.