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John von Neumann

2007 Schools Wikipedia Selection. Related subjects: Mathematicians

   CAPTION: John (Janos) von Neumann

   John von Neumann in the 1940s.
   John von Neumann in the 1940s.
         Born       December 28, 1903
                    Budapest, Austria-Hungary
         Died       February 8, 1957
                    Washington DC, USA
      Residence     USA
     Nationality    US - Hungarian
        Field       Mathematics
     Institution    Los Alamos
                    University of Berlin
                    Princeton University
      Alma Mater    University of Pázmány Péter
                    ETH Zurich
   Academic Advisor Leopold Fejer
   Notable Students Israel Halperin
      Known for     Game theory
                    von Neumann algebra
                    von Neumann architecture
                    Cellular automata
    Notable Prizes  Enrico Fermi Award 1956
       Religion     Roman Catholic

   John von Neumann (Neumann János Lajos Margittai) ( December 28, 1903 in
   Budapest, Austria-Hungary – February 8, 1957 in Washington DC, USA) was
   a Hungarian-born mathematician and polymath who made contributions to
   quantum physics, functional analysis, set theory, topology, economics,
   computer science, numerical analysis, hydrodynamics (of explosions),
   statistics and many other mathematical fields as one of history's
   outstanding mathematicians. Most notably, von Neumann was a pioneer of
   the modern digital computer and the application of operator theory to
   quantum mechanics (see von Neumann algebra), a member of the Manhattan
   Project and the Institute for Advanced Study at Princeton (as one of
   the few originally appointed — a group collectively referred to as the
   "demi-gods"), and the creator of game theory and the concept of
   cellular automata. Along with Edward Teller and Stanislaw Ulam, von
   Neumann worked out key steps in the nuclear physics involved in
   thermonuclear reactions and the hydrogen bomb.

Biography

   The eldest of three brothers, von Neumann was born Neumann János Lajos
   (Hungarian names have the family name first) in Budapest, 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 divide two 8-digit numbers in his head and converse with
   his father in ancient Greek. János was already very interested in math,
   the nature of numbers and the logic of the world around him. By age
   eight he had mastered calculus, and by twelve he was at the graduate
   level in mathematics reading such books as Emile Borel's " Theorie des
   Fonctions". His interests were not confined to mathematics, and
   accounts tell of him reading of all 44 volumes of the universal history
   by the age of 8. He could memorize pages on sight, a gift that later
   would surprise Nobel laureates. He loved to invent mechanical toys, and
   was an expert on the Civil War, the trial of Joan of Arc, and Byzantine
   history.

   He entered the Lutheran Gymnasium in Budapest in 1911. In 1913 his
   father purchased a title, and the Neumann family acquired the Hungarian
   mark of nobility Margittai, or the Austrian equivalent von. Neumann
   János therefore became János von Neumann, a name that he later changed
   to the German Johann von Neumann. After teaching as history's youngest
   Privatdozent of the University of Berlin from 1926 to 1930, he, his
   mother, and his brothers emigrated after Hitler's rise to power from
   Germany to the United States in the 1930s. Curiously, while he
   anglicized Johann to John , he kept the Austrian-aristocratic surname
   of von Neumann, whereas his brothers adopted surnames Vonneumann and
   Newman.

   Although von Neumann unfailingly dressed formally, he enjoyed throwing
   extravagant parties and driving hazardously (frequently while reading a
   book, and sometimes crashing into a tree or getting arrested). He was a
   profoundly committed 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 (for example: "bodily violence
   is a displeasure done with the intention of giving pleasure"), and
   insistently gaze 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.)

   He received his Ph.D. in mathematics (with minors in experimental
   physics and chemistry) from the University of Budapest at the age of
   23. He simultaneously earned his diploma in chemical engineering from
   the ETH Zurich in Switzerland at the behest of his father, who wanted
   his son to invest his time in a more financially viable endeavour than
   mathematics. Between 1926 and 1930 he was a private lecturer in Berlin,
   Germany.

   By age 25 he had published 10 major papers, and by 30, nearly 36.

   Von Neumann was invited to Princeton, New Jersey in 1930, and was one
   of four people selected for the first faculty of the Institute for
   Advanced Study (two of which were Albert Einstein and Kurt Gödel),
   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
   at Princeton. 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 US. 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 Kövesi, whom
   he married in 1930. When he proposed to her, he was incapable of
   expressing anything beyond "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 then he married his second wife, Klara Dan, in 1938. Von
   Neumann had one child, by his first marriage, daughter Marina. Marina
   is a distinguished professor of international trade and public policy
   at the University of Michigan.

   Von Neumann was diagnosed with bone cancer or pancreatic cancer in
   1957, possibly caused by exposure to radioactivity while observing
   A-bomb tests in the Pacific, and possibly in later work on nuclear
   weapons at Los Alamos, New Mexico. (Fellow nuclear pioneer Enrico Fermi
   had died of bone cancer in 1954.) Von Neumann died within a few months
   of the initial diagnosis, in excruciating pain. The cancer had also
   spread to his brain, cutting his thinking ability. As he lay dying in
   Walter Reed Hospital in Washington, D.C., he shocked his friends and
   acquaintances by asking to speak with a Roman Catholic priest. He died
   under military security lest he inadvertently reveal military secrets
   while heavily medicated.

   He had written 150 published papers in his life; 60 in pure
   mathematics, 20 in physics, and 60 in applied mathematics. When he
   died, he was developing a theory of the structure of the human brain.

   Von Neumann entertained notions which would now trouble many. His love
   for meteorological prediction led him to dreaming of manipulating the
   environment by spreading colorants on the polar ice caps in order to
   enhance absorption of solar radiation (by reducing the albedo) and
   thereby raise global temperatures. He also favored a preemptive nuclear
   attack on the USSR, believing that doing so could prevent it from
   obtaining the atomic 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 the twentieth century, 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 themselves, and therefore
   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 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 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 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 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 sets 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 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 cannot 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:
   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 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 had not been reduced, as Albert Einstein believed it
   must be 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 Schrödinger, 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 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 (e.g. 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 indeterminacy 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 and
   was based upon a strange type of function (the so-called Dirac delta
   function) which was harshly criticized by von Neumann.

   In any case, 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 could not 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.

   In a complementary work of 1936, von Neumann proved (along with Garrett
   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\land B)\ne
   (B\land A) . It was also demonstrated that the laws of distribution of
   classical logic, P\lor(Q\land R)=(P\lor Q)\land(P\lor R) and P\land
   (Q\lor R)=(P\land Q)\lor(P\land 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 there 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 in the x direction 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 \land (B\lor C)= A\land 1 = A , while (A\land
   B)\lor (A\land C)=0\lor 0=0 .

Economics

   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 Behaviour (written with Oskar Morgenstern). This resulted
   in such public attention that The New York Times did a front page
   story, the likes of which only Einstein had previously earned.

   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 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 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.

   Von Neumann was also the inventor of the method of proof, used in game
   theory, known as backward induction (first published in 1944 in the
   book co-authored with Morgenstern, Theory of Games and Economic
   Behaviour).

Armaments

   In 1937 von Neumann, having obtained his US citizenship, began to take
   an interest in applied mathematics. He became a top expert in
   explosives, and he committed himself to a large number of military
   consultancies, primarily for the Navy.

   One discovery was that large bombs are more devastating when detonated
   above the ground because of the force of shock waves. The most notable
   application of this occurred in August 1945, when nuclear weapons were
   detonated over Hiroshima and Nagasaki at the altitude calculated by von
   Neumann to produce the most damage.
   John von Neumann's wartime Los Alamos ID badge photo.
   John von Neumann's wartime Los Alamos ID badge photo.

   Von Neumann had been brought into the Manhattan Project to help design
   the explosive lenses needed to compress the plutonium core of the
   Trinity test device and the " Fat Man" weapon dropped on Nagasaki.

   Von Neumann was a member of the committee selecting potential targets.
   Von Neumann's first choice, the city of Kyoto, was dismissed by
   Secretary of War Henry Stimson.

   After the war, Robert Oppenheimer remarked that the physicists had
   "known sin" as a result of their development of the first atomic bombs.
   Von Neumann's rather cynical reply was that "sometimes someone
   confesses a sin in order to take credit for it." He continued
   unperturbed in this work and became, along with Edward Teller, one of
   the sustainers of hydrogen bomb project. Von Neumann had collaborated
   with spy Klaus Fuchs on hydrogen bomb development, and the two filed a
   secret patent on "Improvement in Methods and Means for Utilizing
   Nuclear Energy" in 1946, which outlined a scheme for using a fission
   bomb to compress fusion fuel to initiate a thermonuclear reaction.
   (Herken, pp. 171, 374). Though this was not the key to the hydrogen
   bomb — the Teller-Ulam design — it was judged to be in the right
   direction.

   Von Neumann's hydrogen bomb work was also 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 complicated problems to be approximated using random numbers.
   Because using lists of "truly" random numbers was extremely slow for
   the ENIAC, von Neumann developed a crude 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, and also noted that
   when it went awry it did so obviously, unlike methods which could be
   subtlely incorrect.

Computer science

   Von Neumann gave his name to the von Neumann architecture used in
   almost all computers, because of his publication of the concept. Some
   feel that this ignores the contributions of J. Presper Eckert, John
   William Mauchly and others who worked on the concept. Virtually every
   home computer, microcomputer, minicomputer and mainframe computer is a
   von Neumann machine. He also created the field of cellular automata
   without 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.
   Von Neumann proved that the most effective way large-scale mining
   operations such as mining an entire moon or asteroid belt could be
   accomplished is by using self-replicating machines, taking advantage of
   their exponential growth.

   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 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 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 would not 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 experienced a lightning-like academic career similar to the
   velocity of his own intellect, obtaining at the age of 29 one of the
   first five professorships at the new Institute for Advanced Study at
   Princeton (another had gone to Albert Einstein). He seemed compelled to
   seek other fields of interest, and he found this outlet in his
   collaboration with the American military-industrial complex. He was a
   frequent consultant for the CIA, the US Military, the RAND Corporation,
   Standard Oil, IBM, and others.

   During a Senate committee hearing he described his political ideology
   as "violently anti-communist, and much more militaristic than the
   norm." As President of the so-called Von Neumann Committee for Missiles
   at first, and as a member of the Commission 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 controversial strategic equilibrium called Mutually Assured
   Destruction. He was the mind behind the scientific aspects of the Cold
   War.

Honours

   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 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 professional society of Hungarian computer scientists, Neumann
   János Számítógéptudományi Társaság, 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.

Students

     * Donald B. Gillies, PhD student of John von Neumann.
     * John P. Mayberry, PhD student of John von Neumann.
     * John Forbes Nash Jr., student of John von Neumann.

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