Everything about Von Neumann totally explained
John von Neumann (
December 28,
1903 –
February 8,
1957) was a
Hungarian-born American
mathematician who made major contributions to a vast range of fields including
set theory,
functional analysis,
quantum mechanics,
ergodic theory,
continuous geometry,
economics and
game theory,
computer science,
numerical analysis,
hydrodynamics (of explosions), and
statistics, as well as many other mathematical fields. He is generally regarded as one of the foremost mathematicians of the 20th century.
Most notably, von Neumann was a pioneer of the application of
operator theory to
quantum mechanics, a 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, possibly caused by exposure to
radiation during his witnessing of
atomic bomb tests. Von Neumann died a year and a half following the initial diagnosis, in great pain. While at
Walter Reed Hospital in
Washington, D.C., he invited a
Roman Catholic priest, Father Anselm Strittmatter,
O.S.B., to visit him for consultation (a move which shocked some of von Neumann's friends). The priest then administered to him the last
Sacraments. He died under military security lest he reveal military secrets while heavily medicated. John von Neumann was buried at
Princeton Cemetery in
Princeton,
Mercer County,
New Jersey.
Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, 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.
Logic and 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
Richard Dedekind and
Giuseppe Peano) and geometry (thanks to
David Hilbert). At the beginning of the twentieth century,
set theory, the new branch of mathematics discovered by
Georg Cantor, and thrown into crisis by
Bertrand Russell with the discovery of
his famous paradox (on the set of all sets which don't belong to themselves), hadn't yet been formalized.
The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later (by
Ernst Zermelo and
Abraham Fraenkel) by way of a series of principles which allowed for the construction of all sets used in the actual practice of mathematics, but which didn't 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 complementary 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 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.) In order to demonstrate that the addition of this new axiom to the others didn't 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 doesn't belong to other classes. Under the Zermelo/Fraenkel approach, the axioms impede the construction of a set of all sets which don't belong to themselves. In contrast, under the von Neumann approach, the class of all sets which don't belong to themselves can be constructed, but it's 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 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 can't prove every truth which is expressible in their language. This result was sufficiently innovative as to confound the majority of mathematicians of the time. 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. 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 1930s was quantum mechanics. QM found itself 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, its apparent non-determinism hadn't been reduced 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 Schrödinger, but there wasn't 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 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 (for example 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 famous
uncertainty principle of Heisenberg, 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, and culminated in the 1932 classic
The Mathematical Foundations of Quantum Mechanics. However, physicists generally ended up preferring another approach to that of von Neumann (which was considered elegant and satisfactory by mathematicians). This approach was formulated in 1930 by
Paul Dirac.
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs. non-determinism and in the book he demonstrated a theorem according to which quantum mechanics couldn't possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics. This demonstration contained a conceptual error, but it helped to inaugurate a line of research which, through the work of
John Stuart Bell in 1964 on
Bell's Theorem and the experiments of
Alain Aspect in 1982, demonstrated that quantum physics requires a
notion of reality substantially different from that of classical physics.
Economics and game theory
Up until the 1930s economics involved a great deal of mathematics and numbers, but almost all of this was either superficial or irrelevant. It was used, for the most part, to provide uselessly precise formulations and solutions to problems which were intrinsically
vague. Economics found itself
in a state similar to that of physics of the 17th century: still waiting for the development of an appropriate language in which to express and resolve its problems. While physics had found its language in the
infinitesimal calculus, von Neumann proposed the language of
game theory and a
general equilibrium theory for economics.
His first significant contribution was the
minimax theorem of 1928. This theorem establishes that in certain
zero sum games involving
perfect information (in which players know a priori the strategies of their opponents as well as their consequences), there exists one strategy which allows both players to minimize their maximum losses (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of the player's adversary and the maximum loss. The player 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
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, something which only
Einstein had previously elicited.
Von Neumann's second important contribution in this area was the solution, in 1937, of a problem first described by
Léon Walras in 1874, the existence of situations of equilibrium in mathematical models of market development based on supply and demand. He first recognized that such a model should be expressed through disequations and not equations, and then he found a solution to Walras' problem by applying a
fixed-point theorem derived from the work of
L. E. J. 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.
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).
Nuclear weapons
After obtaining U.S. citizenship, von Neumann took an interest in 1937 in applied mathematics, and then developed an expertise in explosions—phenomena which are difficult to model mathematically. 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 in
Los Alamos,
New Mexico
Beginning in the spring of 1945, along with four other scientists and various military personnel, von Neumann was included in the target selection committee responsible for choosing the
Japanese cities of
Hiroshima and
Nagasaki as the
first targets of the atomic bomb. Von Neumann oversaw computations related to the expected size of the bomb blasts, estimated death tolls, and the distance above the ground at which the bombs should be detonated for optimum shock wave propagation and thus maximum effect. The cultural capital
Kyoto, which had been spared the firebombing inflicted upon militarily significant target cities like
Tokyo in World War II, was von Neumann's first choice, a selection seconded by Manhattan Project leader General
Leslie Groves. However, this target was dismissed by
Secretary of War Henry Stimson, who had been impressed with the city during a visit while Governor General of the
Philippines.
On
July 16,
1945, with numerous other Los Alamos personnel, von Neumann was an eyewitness to the
first atomic bomb blast, conducted as a test of the implosion method device, 35 miles (56 km) southeast of
Socorro,
New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5
kilotons of
TNT, but
Enrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of torn-up paper as the shock wave passed his location and watching how far they scattered. The actual power of the explosion had been between 20 and 22 kilotons. With very few exceptions, all present-day home computers,
microcomputers,
minicomputers and
mainframe computers use this single-memory computer architecture.
Von Neumann also created the field of
cellular automata without the aid of computers, constructing the first
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. His algorithm for simulating a
fair coin with a biased coin is used in the "software whitening" stage of some
hardware random number generators.
He also engaged in exploration of problems in numerical
hydrodynamics. With
R. D. Richtmyer he developed an algorithm defining
artificial viscosity that improved the understanding of
shock waves. It is possible that we wouldn't understand much of astrophysics, and might not have highly developed jet and rocket engines without that work. The problem was that when computers solve hydrodynamic or aerodynamic problems, they try to put too many computational grid points 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 obtained at the age of 29 one of the first five professorships at the new
Institute for Advanced Study in
Princeton, New Jersey (another had gone to
Albert Einstein). He was a frequent consultant for the
Central Intelligence Agency, the
United States Army, the
RAND Corporation,
Standard Oil,
IBM, and others.
Throughout his life von Neumann had a respect and admiration for business and government leaders; something which was often at variance with the inclinations of his scientific colleagues. He enjoyed associating with persons in positions of power, and this led him into government service.
As President of the Von Neumann Committee for Missiles, and later as a member of the
United States Atomic Energy Commission, from 1953 until his death in 1957, he was influential in setting U.S. scientific and military policy. Through his committee, he developed various scenarios of nuclear proliferation, the development of intercontinental and submarine missiles with atomic warheads, and the controversial strategic equilibrium called
mutual assured destruction (aka the M.A.D. doctrine). During a Senate committee hearing he described his political ideology as "violently anti-communist, and much more militaristic than the norm".
Von Neumann's interest in meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice caps in order to enhance absorption of solar radiation (by reducing the
albedo), thereby raising global temperatures. He also favored a preemptive nuclear attack on the
USSR, believing that doing so could prevent it from obtaining the atomic bomb.
Personality
Although von Neumann invariably wore a conservative grey flannel business suit, he enjoyed throwing large parties at his home in Princeton, occasionally twice a week. Despite being a notoriously bad driver, he nonetheless enjoyed driving (frequently while reading a book) - occasioning numerous arrests as well as accidents. He once reported one of his car accidents in this way: "I was proceeding down the road. The trees on the right were passing me in orderly fashion at 60 miles per hour. Suddenly one of them stepped in my path." (The von Neumanns would return to Princeton at the beginning of each academic year with a new car.)
A committed hedonist, von Neumann liked to eat and drink heavily; his wife, Klara, said that he could count everything except calories. He enjoyed
yiddish and "off-color" humor (especially limericks) and could make very insensitive jokes (for example: "bodily violence is a displeasure done with the intention of giving pleasure"). Von Neumann persistently gazed at the legs of young women (so much so that female secretaries at Los Alamos often covered up the exposed undersides of their desks with cardboard).
Honors
The
John von Neumann Theory Prize of the
Institute for Operations Research and the Management Sciences (INFORMS, previously TIMS-ORSA) is awarded annually to an individual (or 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."
The John von Neumann Lecture is given annually at the
Society for Industrial and Applied Mathematics (SIAM) by a researcher who has contributed to applied mathematics, and the chosen lecturer is also awarded a monetary prize.
Von Neumann, a crater on
Earth's
Moon, is named after John von Neumann.
The John von Neumann Computing Center in Princeton, New Jersey was named in his honour.
The professional society of Hungarian computer scientists,
John von Neumann Computer Society, is named after John von Neumann.
On
May 4,
2005 the
United States Postal Service issued the
American Scientists commemorative
postage stamp series, a set of four 37-cent self-adhesive stamps in several configurations. The scientists depicted were John von Neumann,
Barbara McClintock,
Josiah Willard Gibbs, and
Richard Feynman.
The John von Neumann Award of the
Rajk László College for Advanced Studies was named in his honour, and is given every year from 1995 to professors, who had on outstanding contribution at the field of exact social sciences, and through their work they'd a heavy influence to the professional development and thinking of the members of the college.
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