Neumann, John von
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Wikipedia-Article "John von Neumann"
- A separate article covers Saint John Neumann, the American priest.
John von Neumann (Neumann János) (December 28, 1903 – February 8, 1957) was a Jewish Hungarian-born mathematician who made important contributions in quantum physics, functional analysis, set theory, computer science, economics and many other mathematical fields.
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Biography
The oldest of three children, von Neumann was born Neumann Margittai János Lajos (two surnames, two given names respectively) in Budapest, Austria-Hungary to Neumann Miksa (Max Neumann), a lawyer who worked in a bank, and Kann Margit (Margaret Kann). Growing up in a non-practising Jewish family, János, nicknamed "Jancsi", was an extraordinary prodigy. At the age of six, he could converse with his father in ancient Greek; at eight he was already knowledgable about the branch of mathematics called analysis; at ten he had read an entire historical encyclopedia. At the same age, when his mother once stared aimlessly in front of him, he asked, "What are you calculating?". It was said that he used to bring two books into the bathroom with him for fear of finishing one of them before having completed his bodily functions. János was already very interested in math, the nature of numbers and the logic of the world around him. He entered the Lutheran Gymnasium in 1911. In 1913, his father purchased a title, and János acquired the Austrian mark of nobility von, becoming János von Neumann—János was anglicised to John after he took up residence in the United States in the 1930s.
Although von Neumann unfailingly dressed formally, with suit and tie (even on the most improbable occasions such as while horseriding in the Grand Canyon, or during hiking trips in the mountains), he enjoyed throwing the most extravagant parties and driving hazardously (frequently while reading a book, and sometimes crashing into a tree or getting himself arrested as a consequence). He was a profoundly commited hedonist who liked to eat and drink heavily (it was said that he knew how to count everything, except calories), tell dirty stories and very insensitive jokes (e.g. "bodily violence is a displeasure done with the intention of giving pleasure"), and insistently gaze at the legs of young girls (so much so that the female secretaries at Los Alamos were often constrained to cover up the exposed parts of their desks with sheets of paper or cardboard.)
He received his Ph.D. in mathematics from the University of Budapest at the age of 23. He simultaneously learnt chemical engineering in Switzerland. Between 1926 and 1930 he was a private lecturer in Berlin, Germany.
Von Neumann was invited to Princeton University in 1930, and was one of four people selected for the first faculty of the Institute for Advanced Study, where he was a mathematics professor from its formation in 1933 until his death.
From 1936 to 1938 Alan Turing was a visitor at the Institute, where he completed a Ph.D. dissertation under the supervision of Alonzo Church. This visit occurred shortly after Turing's publication of his 1936 paper "On Computable Numbers with an Application to the Entscheidungsproblem" which involved the concepts of logical design and the universal machine. Von Neumann must have known of Turing's ideas but it is not clear whether he applied them to the design of the IAS machine ten years later.
In 1937, he became a naturalized citizen of the United States. In 1938 von Neumann was awarded the Bôcher Memorial Prize for his work in analysis.
Von Neumann was married twice. His first wife was Mariette Koevesi, whom he married in 1930. When he proposed to her, he was incapable of expressing anything beyond the very romantic-sounding phrase: "You and I might be able to have some fun together, seeing as how we both like to drink." Von Neumann agreed to convert to Catholicism to placate her family. The couple divorced in 1937 and Von Neumann married his second wife, Klara Dan, in 1938. Von Neumann had one child, a daughter Marina, from his first marriage.
He dreamed of environmental interventions for the purposes of climate control and manipulation, obtained for example by spreading artifical colorants on the polar ice caps in order to inhibit solar radiation and hence raise the global temperature, even for military purposes. As to the arms which already existed, he was favorable to the idea of a preventive nuclear attack on the USSR in order to prevent it from obtaining the A bomb.
Logic
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 Richard Dedekind and Giuseppe Peano) and geometry (thanks to David Hilbert). At the beginning of he twentieth century, however, set theory, the new branch of mathematics invented by Georg Cantor, and thrown into crisis by Bertrand Russell with the discovery of his famous paradox (on the set of all sets which do not belong to themselves), had not yet been formalized. Russell's paradox consisted in the observation that if the set x (of all sets which are not members of themselves) was a member of itself, then it must belong to the set of all sets which do not belong to themselves, and therefore cannot belong to itself; on the other hand, if the set x does not belong to itself, then it must belong to the set of all sets which do not belong to themslves, and therefore it must belong to itself.
The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later (thanks to Ernst Zermelo and Abraham Frankel) by way of a series of principles which allowed for the construction of all sets used in the actual practice of mathematics, but which did not explicitly exclude the possibility of the existence of sets which belong to themselves. In his doctoral thesis of 1925, von Neumann demonstrated how it was possible to exclude this possibility in two complemetary ways: 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 Frankel, 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.) In order 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 internal 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. While, on the Zermelo/Frankel approach, the axioms impede the construction of a set of all sets which do not belong to themselves, on the von Neumann approach, the class of all classes 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 of 1930 at the historical congress of Konigsberg, in which Kurt Gödel announced his famous 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 suffiently innovative as to confound the majority of mathematicians of the time. But von Neumann, who had participated at the Congress, confermed 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: the usual axiomatic systems are unable to demonstrate their own consistency. It is precisely this consequence which has attracted the most attention, even if Gödel originally considered it only a curiosity, and had derived it independently anyway (it is for this reason that the result is called Gödel's second theorem, without mention of von Neumann.)
Quantum Mechanics
At the International Congress of Mathematicians of 1900, David Hilbert presented his famous list of twenty-three problems considered central for the development of the mathematics of the new century: the sixth of these was the axiomatization of physical theories. Among the new physical theories of the century the only one which had yet to receive such a treatment by the end of the 1930's was quantum mecchanics. In fact, QM found itself, at this time, in a condition of foundational crisis similar to that of set theory at the beginning of the century, facing problems of both philosophical and technical natures: on the one hand, it's apparent non-determinism had not been reduced, as Albert Einstein believed it must have been in order to be satisfactory and complete, to an explanation of a deterministic form; on the other, there still existed two independent but equivalent heuristic formulations, the so-called matrix mechanical formulation due to Werner Heisenberg and the wave mechanical formulation due to Erwin Schrodinger, but there was not yet a single, unified satisfactory theoretical formulation.
After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of QM. He immediately realized, in 1926, that a quantum system could be considered as a point in a so-called Hilbert space, analogous to the common Euclidean space of elementary geometry but with infinitely many dimensions (corresponding to the infinitely many possible states of the system) instead of the usual three: 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 mecchanics was thereby reduced to the mathematics of the linear Hermitian operators on Hilbert spaces. For example, the famous indeterminacy principle of Heisenberg, according to which the determination of the position of a particle prevents the determination of its momentum and viceversa, 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 Schrodinger, and culminated in the 1932 classic The mathematical foundations of quantum mecchanics. However, physicists generally ended up preferring another approach to that of von Neumann (which was considered extremely elegant and satisfatory by mathematicians). This approach was formulated in 1930 by Paul Dirac and was based upon a strange type of function (the so-called delta of Dirac) which was harshly criticized by von Neumann.
In any case, von Neumann's abstract treatment permitted him to also confront the extremely hot-button foundational issue of determinism vs. non-determinism and in the book he demonstrated a theorem according to which quantum mechanics could not possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics. This demostration contained a conceptual error, but it helped to inaugurate a line of research which, through the work of John Bell in 1964 on Bell's Theorem and the experiments of Alain Aspect in 1982, eventually demonstrated that quantum physics does indeed require a notion of reality substantially different from that of classical physics.
In a complementary work of 1936, von Neumann proved (along with Garret Birkhoff) that quantum mechanics also requires a logic substantially different from the classical one. For example, light (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 (A & B) ǂ (B & A). It was also demonstrated that the laws of distribution of classical logic, P V (Q & R) = (P V Q) & (P V R) and P & (Q V R) = (P & Q) V (P & R), 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 their 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 x 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 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 A & (B V C) = A & 1 = A, while (A & B) V (A & C) = 0 V 0 = 0.
Economics
Up until the 1930s, the field of economics seemed to involve the utilization of a great deal of mathematics and numbers; but almost all of this was either superfical or irrelevant. It was used, for the most part, in order to provide uselessly precise formulations and solutions to problems which were, in fact, intrinsically vague. Economics found itself in a state similar to that of the physics of the XVIIth century: still waiting for the development of an appropriate language in which to express and resolve its problems. While physics, of course, had found its language in the infinitesimal calculus, von Neumann proposed the language of game theory and the theory of general equilibria for economics.
His first significant contribution was the minimax theorem of 1928. This theorem establishes that in certain so-called zero sum games (games in which the winnings of one player are equal and contrary to the losses of his opponent) involving perfect information (in which, that is, each player knows apriori both the strategies of their opponent as well as their consequences), there exists one strategy which allows both players to minimize their maximum losses (hence the name minimax). In particular, for every possible strategy of his own, a player must consider all the possible responses of his adversary and the maximum loss that he could derive. He then plays out the strategy which will result in the minimization of this maximum loss. Such a strategy, which minimizes the maximum loss, is called optimal for both players just in case their minimaxes are equal (in absolute value) and contrary (in sign). If the common value is zero, the game becomes pointless.
Von Neumann eventually improved and extended the minimax theorem to include games involving imperfect information and games with more than two players. This work culminated in the 1944 classic The theory of Games and Economic Behavior (written with Oskar Morgenstern).
Von Neumann's second important contribution in this area was the solution, in 1937, of a problem first described by Leon Walras in 1874: the existence of situations of equilibrium in mathematical models of market development based on supply an demand. He first recognized that such a model should be expressed through disequations (as is done today) and not equations (as had been the previous practice), and then he found a solution to Walras problem by applying a fixed-point theorem derived from the work of Luitzen Brouwer. 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 and, in 1983, to Gerard Debreu.
Armaments
In 1937 von Neumann, having recently obtained his US citizenship, began to take an interest in problems in applied mathematics. He rapidly became one of the top experts in the field of explosives, and he commited himself to a very large number of military consultancies, primarily for the Navy (it seems likely that he preferred socializing with admirals rather than generals because the former tended to enjoy drinking liquor while the latter preffered water.)
His most famous result in the field was the discovery that bombs of large dimension are more devastating if they detonate before touching the soil because of the additional force caused by waves of detonation (the media maintained more simply that von Neumann had discovered that it is better to miss a target than to hit it). The most famous (or infamous) application of this discovery occured on the 6th and 9th of August 1945, when the most powerful bombs in history detonated above the soils of Hiroshima and Nagasaki, at the precise altitude calculated by von Neumann himself in order that they would produce the most extensive damage possible.
But this was not the only contribution of von Neumann to atomic warfare. From a technical point of view, much more substantial was his work on the so-called lenses of implosion, the stratification of explosives around a mass of plutonium which permits for its compression to the point of initiating the chain reaction. From a political point of view, he was a member of the commitee whose job it was to select potential "targets". Von Neumann's first choice, the holy city of Kyoto, was dismissed out of hand by the Defense Secretary in person.
According to his own director of operations, Robert Oppenheimer, the atomic enterprise had transformed scientists into "destroyers of worlds". Von Neumann's rather cynical comment was that "sometimes someone confesses a sin in order to take credit for it." In any case, he continued unperturbed in this work, and eventually became, along with Edward Teller, one of the most convinced sustainers of the successive project of the construction of the hydrogen bomb (which was approved by Truman notwithstanding the contrary recommendation of the deliberately appointed scientific committee presided over by Oppenheimer, who had come to believe that scientists had already done more than enough harm to humanity.)
Computer Science
Von Neumann gave his name to the von Neumann architecture used in most non-parallel-processing computers, because of his publication of the concept, though many feel that this naming ignores the contribution of J. Presper Eckert and John William Mauchly who worked on the concept during their work on ENIAC. Virtually every commercially available home computer, microcomputer and supercomputer is a von Neumann machine. He created the field of cellular automata without computers, constructing the first examples of self-replicating automata with pencil and graph paper. The concept of a universal constructor was fleshed out in his posthumous work Theory of Self Reproducing Automata. The term "von Neumann machine" also refers to self-replicating machines. Von Neumann proved that the most effective way large-scale mining operations such as mining an entire moon or asteroid belt can be accomplished is through the use of self-replicating machines, to take advantage of the exponential growth of such mechanisms.
In addition to his work on architecture, he is credited with at least one contribution to the study of algorithms. Donald Knuth cites von Neumann as the inventor, in 1945, of the well known merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged together.
He also engaged in exploration of problems in the field of numerical hydrodynamics. With R. D. Richtmyer he developed an algorithm defining artificial viscosity, that proved essential to understanding many kinds of shock waves. It can fairly be said that we would not understand much of astrophysics, and might not even have highly developed jet and rocket engines, without that work. The problem to be solved was that when computers solve hydrodynamic or aerodynamic problems, they try to put too many computational gridpoints at regions of sharp discontinuity (shock waves). The artificial viscosity was a mathematical trick to slightly smooth the shock transition without sacrificing basic physics.
Politics and Social Affairs
Von Neumann had experienced a lightning-like academic career similar to the velocity of his own intellect, obtaining at the age of twenty-nine one of the first five professorships at the newly born Institute for Advanced Studies at Princeton (another had gone to Albert Einstein). He seemed compelled, therefore, to seek out other fields of interest in order to satisfy his ambitious personality, and he found this outlet in the collaboration (some might say, collaborationism) with the military, political and industrial complex. Through a frenetic activity of secret and proficuous relations of consultancy with various entities and organizations connected to the military/industrial complex, from the CIA, the US Military and the RAND Corporation to Standard Oil, IBM and others, he became something of a scientific prostitute for these agencies.
During a Senate committe hearing, he once described his political ideology as, in his own words "violently anti-communist, and much more militaristic than the norm." As President of the so-called Von Neumann Commitee for Missiles at first, and as a member of the restricted Commision for Atomic Energy later, starting from 1953 up until his death in 1957, he was the scientist with the most political power in the US. Through his committee, he developed various scenarios of nuclear proliferation, the development of intercontinental and submarine missiles with atomic warheads, and the highly controversial strategic equilibrium called Mutually Assured Destruction (MAD). In a word, he was the mind behind the scientific aspects of the Cold War which conditioned the Western world for forty years.
He died, tragically but perhaps ironically, of bone cancer contracted through exposure to the radiation of the atomic tests conducted at Bikini Atoll in 1946, tests whose security for observers he had so tenaciously defended so many years earlier.
Honors
The John von Neumann Theory Prize of the Institute for Operations Research and Management Science (INFORMS, previously TIMS-ORSA) is awarded annually to an individual (or sometimes group) who have made fundamental and sustained contributions to theory in operations research and the management sciences.
The IEEE John von Neumann Medal is awarded annually by the IEEE "for outstanding achievements in computer-related science and technology."
Von Neumann, a crater on Earth's Moon, is named after John von Neumann.
Along with American scientists Barbara McClintock, Josiah Willard Gibbs, and Richard Feynman, von Neumann was honored on a U.S. postage stamp in 2005. The set of self-adhesive 37-cent stamps was made available on May 4, 2005 in a pane of 20 stamps with five stamps for each.
References
- This article was originally based on material from the Free On-line Dictionary of Computing, which is licensed under the GFDL.
Heims, Michael. John von Neumann and Norbert Wiener, from Mathematics to the technologies of life and death. 1980. Macra, Norma. John von Neumann. 1992. Israel, Giorgio and Gasca, Ana Millan. The World as a Mathematical Game: John von Neumann, Twentieth Century Scientist. 1995
Further reading
- von Neumann, John and Arthur W. Burks. 1966. Theory of Self-Reproducing Automata, Univ. of Illinois Press, Urbana IL.
- von Neumann, John. 1932. "Mathematical Foundations of Quantum Mechanics", Princeton U. Press, Princeton NJ. Transl. by R.T. Beyer from the original German.
Students
- Donald B. Gillies, PhD student of John Von Neumann.
- John P. Mayberry, PhD student of John Von Neumann.
External links
- Biography at the MacTutor archive
- John von Neumann's contribution to economic science – By Maria Joao Cardoso De Pina Cabral, International Social Science Review, Fall-Winter 2003
- A Discussion of Artificial Viscosity
- Von Neuman's Universe, audio talk by George Dyson from ITConversations.com
- [1], article by Stephen Wolfram on Neumann's 100th birthday.