John von Neumann

• 1. Introduction

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  John von Neumann once told a colleague that the second incompleteness theorem had struck a harder blow to mathematics than even its discoverer believed. He was born in Budapest in 1903 and died at age 53 in 1957, and in those years he reached further across human knowledge than almost any thinker of his time. He built the mathematical scaffolding of quantum physics. He founded game theory. He sketched the design behind the modern computer. He modeled the explosive lenses that compressed the plutonium core of the Fat Man bomb. How did one mind move so freely between pure abstraction and the machinery of war? What sort of person reads Borel's analysis at twelve, then later warns Congress that controlling the weather could be more dangerous than missiles? The answers run through 18-room apartments in Budapest, through the streets of Göttingen, and through the noisy parties where he did some of his best thinking.

• 2. The Budapest Prodigy

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  By age eight, according to a family legend, von Neumann already understood differential and integral calculus. He was the eldest of three brothers, tutored alongside his cousins by governesses in English, French, German and Italian, because his father believed languages beyond Hungarian were essential. He read Wilhelm Oncken's 46-volume world history series, and one room of the family apartment was converted into a library and reading room. His father, Max von Neumann, was a banker with a doctorate in law, and in February 1913 Emperor Franz Joseph elevated him to the Hungarian nobility. The family took the appellation Margittai and a coat of arms showing three marguerites, though they had no connection to the town it named. When von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1914, he met Eugene Wigner, a year ahead, who became his friend. At 15 he studied advanced calculus under Gábor Szegő, who, his wife recalled, came home from their first meeting with tears in his eyes. By 19 he had published two major papers, one giving the modern definition of ordinal numbers that superseded Georg Cantor's. He capped his school years by winning the Eötvös Prize, a national mathematics award.

• 3. Two Degrees At Once

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  Theodore von Kármán was once asked by von Neumann's father to talk his son out of mathematics. The father wanted John in industry, and they settled on chemical engineering as the safe path. So von Neumann took a two-year non-degree chemistry course at the University of Berlin, then passed the entrance exam to ETH Zurich in September 1923. He was also enrolled as a Ph.D. candidate in mathematics at the University of Budapest at the same time. In 1926 he graduated as a chemical engineer from ETH Zurich and, simultaneously, passed his doctoral examinations in mathematics summa cum laude, with minors in experimental physics and chemistry. A Rockefeller Foundation grant then sent him to the University of Göttingen to study under David Hilbert. Hermann Weyl remembered walking with von Neumann and Emmy Noether through the cold, rain-wet streets of Göttingen, arguing about hypercomplex number systems. His habilitation was completed on the 13th of December 1927, and he became the youngest Privatdozent in the history of the University of Berlin.

• 4. Loud Music And Late Nights

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  In Princeton, von Neumann drew complaints for playing extremely loud German march music while he worked. He did some of his best work in noisy, chaotic environments, and Churchill Eisenhart recalled that he could stay at parties until the early hours, then deliver a lecture at 8:30. His white clapboard house on Westcott Road was one of Princeton's largest private residences, and he always wore formal suits. He enjoyed Yiddish and off-color humor, and Stanisław Ulam, his closest friend in the United States, noted his hunger for earthy comedy alongside his taste for abstract wit. Von Neumann believed much of his thought happened intuitively, often going to sleep with a problem unsolved and waking with the answer. He was baptized Catholic in 1930, then married Marietta Kövesi, who had studied economics. Their daughter Marina, born in 1935, would become a professor. The couple divorced in November 1937, and the following year he married Klára Dán. He became a naturalized U.S. citizen in 1937 and tried to enlist as a lieutenant in the Army's Officers Reserve Corps, passing the exams but being rejected for his age. Wigner wrote that he supervised, in a casual sense, more work than any other modern mathematician.

• 5. Foundations And Their Limits

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  Russell's paradox, concerning the set of all sets that do not belong to themselves, had thrown the foundations of mathematics into crisis at the start of the 20th century. In his 1925 doctoral thesis, von Neumann offered two ways to exclude such self-membering sets: the axiom of foundation and the notion of class. To show the axiom added no contradictions, he introduced the method of inner models, which became an essential tool in set theory. His broader achievement was an elegant axiomatization of the ordinal and cardinal numbers and the first strict formulation of definition by transfinite induction. In September 1930, at the Second Conference on the Epistemology of the Exact Sciences, Kurt Gödel announced his first incompleteness theorem. Von Neumann suggested Gödel try transforming his results, then within a month wrote to him with a striking consequence: such systems cannot prove their own consistency. Gödel replied that he had already found this, now called the second incompleteness theorem, and von Neumann acknowledged his priority. The discovery changed von Neumann's views on mathematical rigor so deeply that he abandoned the foundations of mathematics for problems of application.

• 6. The Master Of Operators

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  Paul Halmos wrote that von Neumann's 1932 papers on ergodic theory alone would have guaranteed him mathematical immortality. That same fertile period saw him become the first to axiomatically define an abstract Hilbert space, as a complex vector space with a Hermitian scalar product, separable and complete. He developed the spectral theory of operators in seminal papers between 1929 and 1932, work that fed directly into his book Mathematical Foundations of Quantum Mechanics. In measure theory he argued that the problem was group-theoretic in character, showing that the existence of a measure depends on whether the transformation group of the space is solvable. His operator work led to his most profound invention in pure mathematics, the study of von Neumann algebras, originally called W*-algebras. With the partial collaboration of F. J. Murray, he produced six major papers between 1936 and 1940 that rank among the masterpieces of twentieth-century analysis. Garrett Birkhoff said his mind blazed over lattice theory like a meteor, and described pages of his razor-edged algebra written before breakfast, seated in a bathrobe at a living room writing-table.

• 7. Games, Economies, And Machines

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  The New York Times ran a front-page story on the book von Neumann wrote with Oskar Morgenstern, the 1944 Theory of Games and Economic Behavior. He had proved his minimax theorem back in 1928, showing that in zero-sum games of perfect information a pair of optimal strategies exists whose minimaxes are equal in value and opposite in sign. His model of an expanding economy, proved using a generalization of the Brouwer fixed-point theorem, was later called the greatest paper in mathematical economics by several authors. Paul Samuelson said that while many mathematicians built tools for economists, von Neumann was unique in contributing to economic theory itself. When George Dantzig described his work in a few minutes, an impatient von Neumann asked him to get to the point, then delivered an hourlong lecture conjecturing the equivalence between matrix games and linear programming. In computing, his incomplete First Draft of a Report on the EDVAC described a machine that stored data and program in the same address space, the basis of most modern digital designs. He invented the merge sort algorithm in 1945 and contributed to the Monte Carlo method. He once wrote that anyone using arithmetical methods to produce random digits is, of course, in a state of sin.

• 8. Bombs, Weather, And The End

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  In the late 1930s von Neumann became the leading authority on the mathematics of shaped charges, expertise that drew him into the Manhattan Project and frequent trips to Los Alamos in New Mexico. His principal contribution to the atomic bomb was the concept and design of the explosive lenses that compressed the plutonium core of the Fat Man weapon. After the war he founded a Meteorological Project at the Institute for Advanced Study in 1946, and by 1950 he and Jule Gregory Charney had written the world's first climate modeling software, running the first numerical weather forecasts on the ENIAC. In 1955 he observed that industry's burning of coal and oil may have warmed the world by about one degree Fahrenheit, and he warned Congress in 1956 that weather control could pose a bigger risk than ICBMs. That same decade he chaired Defense Department committees and was considered the nation's foremost expert on nuclear weaponry. In 1955 a mass was found near his collarbone, a cancer that may have come from radiation exposure at Los Alamos. As death neared he asked for a priest, but remained terrified and unable to accept it, telling his mother, "There probably has to be a God. Many things are easier to explain if there is than if there isn't." He died on the 8th of February 1957 at Walter Reed Army Medical Hospital and was buried at Princeton Cemetery, leaving behind a crater on the Moon named in his honor.