Life and Career

He was a pioneer of the application of operator theory 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, the universal constructor and the digital computer. He 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 in hospital, was later published in book form as The Computer and the Brain. His analysis of the structure of self-replication preceded 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 Gottingen 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, von Neumann worked on the Manhattan Project with theoretical physicist Edward Teller, mathematician Stanislaw Ulam and others, problem solving key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. He developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon, and coined the term "kiloton" (of TNT), as a measure of the explosive force generated. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, and consulted for a number of organizations, including the United States Air Force, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project, and the Lawrence Livermore National Laboratory. As a Hungarian émigré, concerned that the Soviets would achieve nuclear superiority, he designed and promoted the policy of mutually assured destruction to limit the arms race.

Illness and Death

In 1955, von Neumann was diagnosed with what was either bone, pancreatic or prostate cancer. 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", essentially saying that Pascal had a point, 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 Father Strittmatter administered the last rites to him. Some of von Neumann's friends (such as Abraham Pais and Oskar Morgenstern) 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. Von Neumann was on his deathbed when he entertained his brother by reciting by heart and word-for-word the first few lines of each page of Goethe's Faust.

He died at age 53 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 in Princeton, Mercer County, New Jersey.

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 of Richard Dedekind and Charles Sanders Peirce, and in geometry, thanks to Hilbert's axioms. But 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 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.

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 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 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 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 being equal to the bicommutant. Von Neumann embarked in 1936, with the partial collaboration of F.J. Murray, on 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 direct integral was later introduced in 1949 by John von Neumann.

Contributions to Computing

Von Neumann was a founding figure in computing. Von Neumann was the inventor,in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged. Von Neumann wrote the 23 pages long sorting program for the EDVAC in ink. On the first page, traces of the phrase "TOP SECRET", which was written in pencil and later erased, can still be seen. He also worked on the philosophy of artificial intelligence with Alan Turing when the latter visited Princeton in the 1930s.

Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and StanisLaw Ulam developed simulations on von Neumann's digital computers for the hydrodynamic computations. During this time he contributed to the development of the Monte Carlo method, which allowed solutions to complicated problems to be approximated using random numbers. Flow chart from von Neumann's "Planning and coding of problems for an electronic computing instrument," published in 1947. Von Neumann's algorithm for simulating a fair coin with a biased coin is used in the "software whitening" stage of some hardware random number generators. Because using lists of "truly" random numbers was extremely slow, von Neumann develope a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." Von Neumann also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect.

Cognitive Abilities

Nobel Laureate Hans Bethe said "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man", and later Bethe wrote that "[von Neumann's] brain indicated a new species, an evolution beyond man". Seeing von Neumann's mind at work, Eugene Wigner wrote, "one had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch." Paul Halmos states that "von Neumann's speed was awe-inspiring." Israel Halperin said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car." Edward Teller admitted that he "never could keep up with him". Teller also said "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us." Peter Lax wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics".