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JOHN VON NEUMANN |
John "Johnny" von Neumann (/vɒn ˈnɔɪmən/; Hungarian: Neumann János Lajos, pronounced [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer, and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time[13] and was said to be "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics".[14][15] He integrated pure and applied sciences.
Von Neumann made major contributions to many fields, including mathematics (foundations of mathematics, measure theory, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, matrix theory, geometry, and numerical analysis), physics (quantum mechanics, hydrodynamics, ballistics, nuclear physics, and quantum statistical mechanics), economics (game theory and general equilibrium theory), computing (Von Neumann architecture, linear programming, numerical meteorology, scientific computing, self-replicating machines, stochastic computing), and statistics.
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.
Von Neumann 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.[16] His last work, an unfinished manuscript written while he was dying 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 shortlist of facts about his life he submitted to the National Academy of Sciences, he wrote, "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."[17]
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.[18]
During this time and after the war, he consulted for a vast number of organizations including the Office of Scientific Research and Development, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project, and the Oak Ridge National Laboratory.[19] At the peak of his influence in the 1950s, he was the chairman of a number of critical U.S. Department of Defense committees including the Nuclear Weapons Panel of the Air Force Scientific Advisory Board and the ICBM Scientific Advisory Committee as well as a member of the influential U.S. Atomic Energy Commission.
He played a key role alongside Bernard Schriever and Trevor Gardner in contributing to the design and development of the United States' first ICBM programs.[20] During this time he was considered the nation's foremost expert on nuclear weaponry and the leading defense scientist at the Pentagon.[21] 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.[22]
In honor of his achievements and contributions to the modern world, he was named in 1999 the Financial Times Person of the Century, as a representative of the century's characteristic ideal that the power of the mind could shape the physical world, and of the "intellectual brilliance and human savagery" that defined the 20th century.[23][24][25]
Life and Education: Family background
Von Neumann's birthplace is at 16 Báthory Street, Budapest. Since 1968, it has housed the John von Neumann Computer Society.
Von Neumann was born on December 28, 1903, to a wealthy, acculturated and non-observant Jewish family. His Hungarian birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names are equivalent to John Louis in English.
Child prodigy
Von Neumann was a child prodigy. When he was six years old, he could divide two eight-digit numbers in his head[36][37] and could converse in Ancient Greek. When the six-year-old von Neumann caught his mother staring aimlessly, he asked her, "What are you calculating?"[38]
When they were young, von Neumann, 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.[39]
By the age of eight, von Neumann was familiar with differential and integral calculus, and by twelve he had read and understood Borel's Théorie des Fonctions.[40] But he was also particularly interested in history. He read his way through Wilhelm Oncken's 46-volume world history series Allgemeine Geschichte in Einzeldarstellungen (General History in Monographs).[41] A copy was contained in a private library Max purchased. One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor.[42]
University studies
According to his friend Theodore von Kármán, von Neumann's father wanted John to follow him into the industry and thereby invest his time in a more financially useful endeavor than mathematics. In fact, his father asked von Kármán to persuade his son not to take mathematics as his major.[50] Von Neumann and his father decided that the best career path was to become a chemical engineer.
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 the prestigious ETH Zurich,[51] which he passed in September 1923.[52]
At the same time, von Neumann also entered Pázmány Péter University in Budapest,[53] as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory.[54][55] He graduated as a chemical engineer from ETH Zurich in 1926 (although Wigner says that von Neumann was never very attached to the subject of chemistry),[56] and passed his final examinations for his Ph.D. in mathematics simultaneously with his chemical engineering degree, which Wigner wrote, "Evidently a Ph.D. thesis and examination did not constitute an appreciable effort."[56] He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.[57]
Mathematics: Set theory
See also: Von Neumann–Bernays–Gödel set theory
History of approaches that led to NBG set theory
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 in geometry, thanks to Hilbert's axioms.[102] 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).[103]
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 doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.[102]....
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.[133]
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.[134] In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis in relation to these papers.[135][136]
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.[137] 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.[138] This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem.[139][126]....
Other work in pure mathematics
In his early years, von Neumann published several papers relating to set-theoretical real analysis and number theory.[194] In a paper from 1925, he proved that for any dense sequence of points in {\displaystyle [0,1]}[0,1], there existed a rearrangement of those points that is uniformly distributed.[195][196][197] In 1926 his sole publication was on Prüfer's theory of ideal algebraic numbers where von Neumann found a new way of constructing them, extending Prüfer's theory to the field of all algebraic numbers, and their relation to p-adic numbers.
...
QUANTUM MECHANICS:
Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, in his widely influential 1932 work Mathematical Foundations of Quantum Mechanics.[225] After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics.
He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system.[226]....
Quantum logic Main article: Quantum logic
Von Neumann first proposed quantum logic in his 1932 treatise Mathematical Foundations of Quantum Mechanics, where he noted that projections on a Hilbert space can be viewed as propositions about physical observables.
The field of quantum logic was subsequently inaugurated, in a famous paper of 1936 by von Neumann and Garrett Birkhoff, the first work ever to introduce quantum logic,[246] wherein von Neumann and Birkhoff 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 a new propositional calculus was demonstrated through several proofs.----
Economics: Game theory
Von Neumann founded the field of game theory as a mathematical discipline.[270] He proved his minimax theorem in 1928. It 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. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.[...
Linear programming
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 when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.[281]
Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul 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 method of linear programming......
Computer science
Von Neumann was a founding figure in computing.[283] 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.[284][285] 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.[285] He also worked on the philosophy of artificial intelligence with Alan Turing when the latter visited Princeton in the 1930s.[286]...
CELLULAR AUTOMATA:
Von Neumann created the field of cellular automata without the aid of computers, constructing the first self-replicating automata with pencil and graph paper.
The detailed proposal for a physical non-biological self-replicating system was first put forward in lectures von Neumann delivered in 1948 and 1949 when he first only proposed a kinematic self-reproducing automaton.[305][306] While qualitatively sound, von Neumann was evidently dissatisfied with this model of a self-replicator due to the difficulty of analyzing it with mathematical rigor. He went on to instead develop a more abstract model self-replicator based on his original concept of cellular automata.[307]...
Manhattan Project
Beginning in the late 1930s, von Neumann developed an expertise in explosions—phenomena that are difficult to model mathematically. During this period, von Neumann was the leading authority on the mathematics of shaped charges. This led him to a large number of military consultancies, primarily for the Navy, which in turn led to his involvement in the Manhattan Project. The involvement included frequent trips by train to the project's secret research facilities at the Los Alamos Laboratory in a remote part of New Mexico.[53]....
Atomic Energy Commission
In 1955, von Neumann became a commissioner of the Atomic Energy Commission (AEC). He accepted this position and used it to further the production of compact hydrogen bombs suitable for intercontinental ballistic missile (ICBM) delivery. He involved himself in correcting the severe shortage of tritium and lithium 6 needed for these compact weapons, and he argued against settling for the intermediate-range missiles that the Army wanted. He was adamant that H-bombs delivered into the heart of enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile wouldn't be a problem with an H-bomb....
Personality
Gian-Carlo Rota wrote in his famously controversial book, Indiscrete Thoughts, that von Neumann was a lonely man who had trouble relating to others except on a strictly formal level.[390] Françoise Ulam described how she never saw von Neumann in anything but a formal suit and tie.[391] His daughter wrote in her memoirs that she believed her father was motivated by two key convictions, one, that every person had the responsibility to make full use of their intellectual capacity, and two, that there is a critical importance of an environment of political freedom in order to pursue the first conviction......
Selected works
Collections of von Neumann's published works can be found on zbMATH and Google Scholar. A list of his known works as of 1995 can be found in The Neumann Compendium.
https://en.wikipedia.org/wiki/John_von_Neumann
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