John von Neumann
Overview
Hungarian American
Hungarian Americans Hungarian are American citizens of Hungarian descent. The constant influx of Hungarian immigrants was marked by several waves of sharp increase.-History:...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and polymath
Polymath
A polymath is a person whose expertise spans a significant number of different subject areas. In less formal terms, a polymath may simply be someone who is very knowledgeable...
who made major contributions to a vast number of fields, including set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
and game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
, hydrodynamics, and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, as well as many other mathematical fields. He is generally regarded as one of the greatest mathematicians in modern history.
The mathematician Jean Dieudonné
Jean Dieudonné
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
called von Neumann "the last of the great mathematicians", while Peter Lax
Peter Lax
Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...
described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century, and Hans Bethe
Hans Bethe
Hans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...
stated "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man".
Discussions
Quotations
You should call it Entropy|entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.
Suggesting to Claude Elwood Shannon|Claude Shannon a name for his new uncertainty function, as quoted in Scientific American Vol. 225 No. 3, (1971), p. 180
Young man, in mathematics you don't understand things. You just get used to them.
Reply to Felix T. Smith who had said "I'm afraid I don't understand the method of characteristics." —as quoted in The Dancing Wu Li Masters|The Dancing Wu Li Masters: An Overview of the New Physics (1984) by Gary Zukav|Gary Zukav footnote in page 208.
You don't have to be responsible for the world that you're in.
Advice given by von Neumann to Richard Feynman as quoted in "Los Alamos from Below" in Surely You're Joking, Mr. Feynman!|Surely You're Joking, Mr. Feynman! (1985)
The goys have proven the following theorem...
Statement at the start of a classroom lecture, as quoted in 1,911 Best Things Anyone Ever Said (1988) by Robert Byrne
Truth is much too complicated to allow anything but approximations.
As quoted in Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise (1991) by Manfred Schroder
Encyclopedia
John von Neumann (December 28, 1903 – February 8, 1957) was a Hungarian-American
mathematician
and polymath
who made major contributions to a vast number of fields, including set theory
, functional analysis
, quantum mechanics
, ergodic theory
, geometry
, fluid dynamics
, economics
and game theory
, computer science
, numerical analysis
, hydrodynamics, and statistics
, as well as many other mathematical fields. He is generally regarded as one of the greatest mathematicians in modern history.
The mathematician Jean Dieudonné
called von Neumann "the last of the great mathematicians", while Peter Lax
described him as possessing the most "fearsome technical prowess" and "scintillating intellect" of the century, and Hans Bethe
stated "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man". Even in Budapest
, in the time that produced geniuses like Theodore von Kármán
(b. 1881), George de Hevesy
(b. 1885), Leó Szilárd
(b. 1898), Eugene Wigner (b. 1902), Edward Teller
(b. 1908), and Paul Erdős
(b. 1913), his brilliance stood out.
Von Neumann was a pioneer of the application of operator theory
to quantum mechanics
, in the development of functional analysis
, a principal 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
and the concepts of cellular automata, the universal constructor
, and the digital computer. Von Neumann's mathematical 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 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." Along with Teller and Stanisław Ulam, von Neumann worked out key steps in the nuclear physics
involved in thermonuclear reactions and the hydrogen bomb.
, Austro-Hungarian Empire, to wealthy Jewish parents. His father, Neumann Miksa (Max Neumann) was a banker, who held a doctorate in law
. He had moved to Budapest from Pécs
at the end of 1880s
. His mother was Kann Margit (Margaret Kann).
In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian empire
by Emperor Franz Josef. The Neumann family thus acquiring the hereditary title margittai, Neumann János became margittai Neumann János (John Neumann of Margitta), which he later changed to the German Johann von Neumann.
János, nicknamed "Jancsi" (Johnny), was a child prodigy
in the areas of language, memorization, and mathematics. By the age of six, he could exchange jokes in Classical Greek, memorize telephone directories on sight, and display prodigious mental calculation
abilities. As a 6 year old, he would astonish onlookers by instantly dividing two 8-digit numbers in his head, producing the answers to a decimal point. By the age of 8, he had attained mastery in calculus.
He entered the German-speaking Lutheran high school Fasori Evangelikus Gimnázium
in Budapest in 1911. Although his father insisted he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő
. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears.
Szegő subsequently visited the von Neumann house twice a week to tutor the child prodigy. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out with his father's stationary, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Cantor's.
He received his Ph.D.
in mathematics
(with minors in experimental physics
and chemistry
) from Pázmány Péter University in Budapest at the age of 22. He simultaneously earned a diploma in chemical engineering
from the ETH Zurich
in Switzerland at the behest of his father, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavour than mathematics.
Between 1926 and 1930, he taught as a Privatdozent
at the University of Berlin
, the youngest in its history. By the end of year 1927 Neumann had published twelve major papers in mathematics, and by the end of year 1929, thirty-two papers, at a rate of nearly one major paper per month.
In 1930, Von Neumann was invited to Princeton University
, New Jersey
, and, subsequently, was one of the first four people selected for the faculty of the Institute for Advanced Study
(two of the others being Albert Einstein
and Kurt Gödel
), where he remained a mathematics professor from its formation in 1933 until his death. His father, Max von Neumann had died in 1929. But his mother, and his brothers followed John to the United States. He anglicized his first name to John, keeping the Austrian-aristocratic surname of von Neumann.
In 1937, von Neumann became a naturalized citizen of the U.S. In 1938, he was awarded the Bôcher Memorial Prize
for his work in analysis.
Von Neumann married twice. He married Mariette Kövesi in 1930, just prior to emigrating to the United States. They had one daughter (von Neumann's only child), Marina
, who is now a distinguished professor of international trade and public policy at the University of Michigan
. The couple divorced in 1937. In 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II
. The von Neumanns were very active socially within the Princeton academic community.
In 1955, von Neumann was diagnosed with what was either bone or pancreatic cancer
. A von Neumann biographer Norman Macrae
has speculated: "It is plausible that in 1955 the then-fifty-one-year-old Johnny's cancer sprang from his attendance at the 1946 Bikini nuclear tests
." Von Neumann died a year and a half later. 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. This move shocked some of von Neumann's friends in view of his reputation as an agnostic.
Von Neumann, however, is reported to have said in explanation that Pascal
had a point, referring to Pascal's wager
. Father Strittmatter administered the last sacraments
to him. He died under military security lest he reveal military secrets while heavily medicated. von Neumann was buried at Princeton Cemetery
in Princeton
, Mercer County
, New Jersey
. On his death bed, he entertained his brother with a word for word memory of Goethe's Faust
.
Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital and later 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.
'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 geometry (thanks to David Hilbert
). At the beginning of the twentieth 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 and Fraenkel 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 belong 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
.
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.) 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 model
s) 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. Under the Zermelo/Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under 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 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 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: 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.)
. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry
, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebra
s with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor
.
In a number of von Neumann's papers, the methods of argument he employed are considered more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions (anticipating his later work on almost periodic functions).
In the 1936 paper on analytic measure theory, von Neumann used the Haar theorem in the solution of Hilbert's fifth problem
in the case of compact groups.
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 famous articles on operator theory
, and the application of this work was instrumental in the Von Neumann mean ergodic theorem.
s. A von Neumann algebra is a *-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 an algebra of symmetries.
The direct integral
was introduced in 1949 by John von Neumann in one of the final papers in the series On Rings of Operators. One of von Neumann's arguments was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.
: for example, the standard probability space
.
writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann worked on lattice theory between 1937-39. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity."
Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals
of a suitable regular ring
R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."
After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. 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 uncertainty principle
, 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 his 1932 book Mathematische Grundlagen der Quantenmechanik.
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs. non-determinism and in the book he presented a proof according to which quantum mechanics could not possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics. However, in 1966 it was discovered that this proof contained a conceptual error (see the article on John Stewart Bell
for more information). The proof nonetheless inaugurated a line of research that ultimately led, through the work of Bell in 1964 on Bell's Theorem
, and the experiments of Alain Aspect
in 1982, to the demonstration that quantum physics requires a notion of reality substantially different from that of classical physics.
In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem
. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, it never gained acceptance amongst the majority of physicists).
Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formalism of problems in quantum mechanics which underlies the majority of approaches and can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations
. It was also demonstrated that the laws of distribution of classical logic, 
and
, 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 in 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
, while 
.
Von Neumann proposes to replace classical logics, with a logic constructed in orthomodular lattices, (isomorphic to the lattice of subspaces of the Hilbert space of a given physical system).
as a mathematical discipline. Von Neumann's proved his minimax theorem in 1928. This theorem 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 (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy which will result in the minimization of his maximum loss.
Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Another result he proved during his German period was the nonexistence of a static equilibrium. An equilibrium can only exist in an expanding economy. Paul Samuelson
edited an anniversary volume dedicated to this short German paper in 1972 and stated in the introduction that von Neumann was the only mathematician ever to make a significant contribution to economic theory.
Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 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. In this book, von Neumann declared that economic theory needed to use functional analytic
methods, especially convex set
s and topological
fixed point theorem, rather than the traditional differential calculus
, because the maximum–operator did not preserve differentiable functions. Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting
and separating hyperplanes and convex set, and fixed-point theory—have been the primary tools of mathematical economics ever since. 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).
vector
s p and q and a positive number λ that would solve the complementarity
equation
along with two inequality systems expressing economic efficiency. In this model, the (transpose
d) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth
of the economy; the rate of growth equals the interest rate
. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.
Von Neumann's results have been viewed as a special case of linear programming
, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality
.
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
, in 1983 to Gérard Debreu
, and in 1994 to John Nash who used fixed point theorems to establish equilibria for noncooperative games and for bargaining problem
s in his Ph.D thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans
, Leonid Kantorovich
, Wassily Leontief
, Paul Samuelson
, Robert Dorfman
, Robert Solow
, and Leonid Hurwicz
.
, after George B. Dantzig described his work in a few minutes, when an impatient von Neumann asked him to get to the point. Then, Dantzig listened dumbfounded while von Neumann provided an hour lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of 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.
. In 1941, he derived the exact distribution of the ratio of mean square successive difference to the variance for normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression. Subsequently, John Denis Sargan and Alok Bhargava
extended the results for testing if the errors on a regression model follow a Gaussian random walk
(i.e. possess a unit root
) against the alternative that they are a stationary first order autoregression. Von Neumann's contributions to statistics have had a major impact on econometric methodology.
Beginning in the late 1930s, von Neumann developed an expertise in explosion
s—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
.
Von Neumann's principal contribution to the atomic bomb itself was in the concept and design of the explosive lenses
needed to compress the plutonium
core of the Trinity test
device and the "Fat Man
" weapon that was later dropped on Nagasaki. While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly" meaning compression. When it turned out that there would not be enough U235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the plutonium
-239 that was available from the Hanford site
. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, 5% was achieved by George Kistiakowsky
, and the construction of the Trinity bomb was completed in July 1944.
In a visit to Los Alamos in September 1944, von Neumann showed that the pressure increase from explosion shock wave reflection from solid objects was greater than previously believed if the angle of incidence of the shock wave was between 90° and some limiting angle. As a result, it was determined that the effectiveness of an atomic bomb would be enhanced with detonation some kilometers above the target, rather than at ground level.
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.
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.
After the war, Robert Oppenheimer
remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."
Von Neumann continued unperturbed in his work and became, along with Edward Teller
, one of those who sustained the hydrogen bomb project. He then collaborated with Klaus Fuchs
on further development of the bomb, and in 1946 the two filed a secret patent on "Improvement in Methods and Means for Utilizing Nuclear Energy", which outlined a scheme for using a fission bomb to compress fusion fuel to initiate a thermonuclear reaction. The Fuchs–von Neumann patent used radiation implosion
, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design. Their work was, however, incorporated into the "George" shot of Operation Greenhouse
, which was instructive in testing out concepts that went into the final design. The Fuchs–von Neumann work was passed on, by Fuchs, to the USSR as part of his nuclear espionage
, but it was not used in the Soviet's own, independent development of the Teller–Ulam design. The historian Jeremy Bernstein has pointed out that ironically, "John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made."
. Shortly before his death, when he was already quite ill, von Neumann headed the top secret von Neumann ICBM committee. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were indeed formidable, they could be overcome in time. The SM-65 Atlas passed its first fully functional test in 1959, two years after his death.
- or MANIAC).
, which allowed complicated problems to be approximated using random number
s. Because using lists of "truly" random numbers was extremely slow, von Neumann developed 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, and also noted that when it went awry it did so obviously, unlike methods which could be subtly incorrect.
While consulting for the Moore School of Electrical Engineering
at the University of Pennsylvania
on the EDVAC
project, von Neumann wrote an incomplete First Draft of a Report on the EDVAC
. The paper, which was widely distributed, described a computer
architecture in which the data and the program are both stored in the computer's memory in the same address space. This architecture is to this day the basis of modern computer design, unlike the earliest computers that were 'programmed' by altering the electronic circuitry. Although the single-memory, stored program architecture is commonly called von Neumann architecture
as a result of von Neumann's paper, the architecture's description was based on the work of J. Presper Eckert
and John William Mauchly, inventors of the ENIAC
at the University of Pennsylvania
.
Stochastic computing
was first introduced in a pioneering paper by von Neumann in 1953. However, the
theory could not be implemented until advances in computing of the 1960s.
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. Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire moon
or asteroid belt
would be by using self-replicating machines, taking advantage of their exponential growth
.
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.
His algorithm for simulating a fair coin
with a biased coin is used in the "software whitening" stage of some hardware random number generator
s.
s. It is possible that we would not understand much of astrophysics, and might not have highly developed jet and rocket engines without the work of von Neumann. A problem was that when computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock wave
s). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics.
Other well known contributions to fluid dynamics included the classic flow solution to blast wave
s, and the co-discovery of the ZND detonation model
of explosives.
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
, General Electric
, 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. Von Neumann entered government service (Manhattan Project) primarily because he felt that, if freedom and civilization were to survive, it would have to be because the U.S. would triumph over totalitarianism from the right (Nazism and Fascism) and totalitarianism from the left (Soviet Communism).
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
. During a Senate
committee hearing he described his political ideology as "violently anti-communist
, and much more militaristic than the norm". He was quoted in 1950 remarking, "If you say why not bomb [Russia] tomorrow, I say, why not today. If you say today at five o’clock, I say why not one o’clock?". As a result, he partly inspired the character of 'Doctor Strangelove' in Doctor Strangelove.
), thereby inducing global warming
.
.
Despite being a notoriously bad driver, he nonetheless enjoyed driving (frequently while reading a book) – occasioning numerous arrests as well as accidents.
When Cuthbert Hurd
hired him as a consultant to IBM, Hurd often quietly paid the fines for his traffic tickets. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish
and "off-color" humor
(especially limericks). At Princeton he received complaints for regularly playing extremely loud German marching music on his gramophone, which distracted those in neighbouring offices, including Einstein, from their work. Von Neumann's closest friend in America was the Polish mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota
writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend.
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
wrote that von Neumann effortlessly outdid anybody he ever met, and said "I never could keep up with him". Lothar Wolfgang Nordheim
described von Neumann as the "fastest mind I ever met", and Jacob Bronowski
wrote "He was the cleverest man I ever knew, without exception. He was a genius." George Pólya
, whose lectures at ETH Zurich
von Neumann attended as a student, said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem. He'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper." Halmos recounts a story told by Nicholas Metropolis
, concerning the speed of von Neumann's calculations, when somebody asked von Neumann to solve the famous fly puzzle:
Von Neumann had a photographic memory
. Herman Goldstine writes: "One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how The Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes."
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called von Neumann "the last of the great mathematicians", while Peter Lax
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stated "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man". Even in Budapest
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, in the time that produced geniuses like Theodore von Kármán
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(b. 1898), Eugene Wigner (b. 1902), Edward Teller
Edward Teller
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(b. 1908), and Paul Erdős
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(b. 1913), his brilliance stood out.
Von Neumann was a pioneer of the application of operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
to quantum mechanics
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, in the development of functional analysis
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, a principal member of the Manhattan Project
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and the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
(as one of the few originally appointed), and a key figure in the development of game theory
Game theory
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and the concepts of cellular automata, the universal constructor
Von Neumann universal constructor
John von Neumann's Universal Constructor is a self-replicating machine in a cellular automata environment. It was designed in the 1940s, without the use of a computer. The fundamental details of the machine were published in von Neumann's book Theory of Self-Reproducing Automata, completed in...
, and the digital computer. Von Neumann's mathematical analysis of the structure of self-replication
Self-replication
Self-replication is any behavior of a dynamical system that yields construction of an identical copy of that dynamical system. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and can be transmitted to offspring during reproduction...
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 Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932." Along with Teller and Stanisław Ulam, von Neumann worked out key steps in the nuclear physics
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
involved in thermonuclear reactions and the hydrogen bomb.
Biography
The eldest of three brothers, von Neumann was born Neumann János Lajos (ˈnojmɒn ˈjaːnoʃ ˈlɒjoʃ; in Hungarian the family name comes first) on December 28, 1903 in BudapestBudapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
, Austro-Hungarian Empire, to wealthy Jewish parents. His father, Neumann Miksa (Max Neumann) was a banker, who held a doctorate in law
Doctor of law
Doctor of Law or Doctor of Laws is a doctoral degree in law. The application of the term varies from country to country, and includes degrees such as the LL.D., Ph.D., J.D., J.S.D., and Dr. iur.-Argentina:...
. He had moved to Budapest from Pécs
Pécs
Pécs is the fifth largest city of Hungary, located on the slopes of the Mecsek mountains in the south-west of the country, close to its border with Croatia. It is the administrative and economical centre of Baranya county...
at the end of 1880s
1880s
The 1880s was the decade that spanned from January 1, 1880 to December 31, 1889. They occurred at the core period of the Second Industrial Revolution. Most Western countries experienced a large economic boom, due to the mass production of railroads and other more convenient methods of travel...
. His mother was Kann Margit (Margaret Kann).
In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian empire
Austria-Hungary
Austria-Hungary , more formally known as the Kingdoms and Lands Represented in the Imperial Council and the Lands of the Holy Hungarian Crown of Saint Stephen, was a constitutional monarchic union between the crowns of the Austrian Empire and the Kingdom of Hungary in...
by Emperor Franz Josef. The Neumann family thus acquiring the hereditary title margittai, Neumann János became margittai Neumann János (John Neumann of Margitta), which he later changed to the German Johann von Neumann.
János, nicknamed "Jancsi" (Johnny), was a child prodigy
Child prodigy
A child prodigy is someone who, at an early age, masters one or more skills far beyond his or her level of maturity. One criterion for classifying prodigies is: a prodigy is a child, typically younger than 18 years old, who is performing at the level of a highly trained adult in a very demanding...
in the areas of language, memorization, and mathematics. By the age of six, he could exchange jokes in Classical Greek, memorize telephone directories on sight, and display prodigious mental calculation
Mental calculation
Mental calculation comprises arithmetical calculations using only the human brain, with no help from calculators, computers, or pen and paper. People use mental calculation when computing tools are not available, when it is faster than other means of calculation , or in a competition context...
abilities. As a 6 year old, he would astonish onlookers by instantly dividing two 8-digit numbers in his head, producing the answers to a decimal point. By the age of 8, he had attained mastery in calculus.
He entered the German-speaking Lutheran high school Fasori Evangelikus Gimnázium
Fasori Gimnázium
Fasori Gimnázium , also known as Fasori Evangélikus Gimnázium , official name: Budapest-Fasori Evangélikus Gimnázium, is a famous secondary school in Budapest, Hungary...
in Budapest in 1911. Although his father insisted he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give him advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő
Gábor Szego
Gábor Szegő was a Hungarian mathematician. He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials.-Life:...
. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears.
Szegő subsequently visited the von Neumann house twice a week to tutor the child prodigy. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out with his father's stationary, are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Cantor's.
He received his Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
(with minors in experimental physics
Experimental physics
Within the field of physics, experimental physics is the category of disciplines and sub-disciplines concerned with the observation of physical phenomena in order to gather data about the universe...
and chemistry
Chemistry
Chemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
) from Pázmány Péter University in Budapest at the age of 22. He simultaneously earned a diploma in chemical engineering
Chemical engineering
Chemical engineering is the branch of engineering that deals with physical science , and life sciences with mathematics and economics, to the process of converting raw materials or chemicals into more useful or valuable forms...
from the ETH Zurich
ETH Zurich
The Swiss Federal Institute of Technology Zurich or ETH Zürich is an engineering, science, technology, mathematics and management university in the City of Zurich, Switzerland....
in Switzerland at the behest of his father, who wanted his son to follow him into industry and therefore invest his time in a more financially useful endeavour than mathematics.
Between 1926 and 1930, he taught as a Privatdozent
Privatdozent
Privatdozent or Private lecturer is a title conferred in some European university systems, especially in German-speaking countries, for someone who pursues an academic career and holds all formal qualifications to become a tenured university professor...
at the University of Berlin
Humboldt University of Berlin
The Humboldt University of Berlin is Berlin's oldest university, founded in 1810 as the University of Berlin by the liberal Prussian educational reformer and linguist Wilhelm von Humboldt, whose university model has strongly influenced other European and Western universities...
, the youngest in its history. By the end of year 1927 Neumann had published twelve major papers in mathematics, and by the end of year 1929, thirty-two papers, at a rate of nearly one major paper per month.
In 1930, Von Neumann was invited to Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, New Jersey
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, and, subsequently, was one of the first four people selected for the faculty of the Institute for Advanced Study
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(two of the others being Albert Einstein
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and Kurt Gödel
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), where he remained a mathematics professor from its formation in 1933 until his death. His father, Max von Neumann had died in 1929. But his mother, and his brothers followed John to the United States. He anglicized his first name to John, keeping the Austrian-aristocratic surname of von Neumann.
In 1937, von Neumann became a naturalized citizen of the U.S. In 1938, he was awarded the Bôcher Memorial Prize
Bôcher Memorial Prize
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for his work in analysis.
Von Neumann married twice. He married Mariette Kövesi in 1930, just prior to emigrating to the United States. They had one daughter (von Neumann's only child), Marina
Marina von Neumann Whitman
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, who is now a distinguished professor of international trade and public policy at the University of Michigan
University of Michigan
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. The couple divorced in 1937. In 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest prior to the outbreak of World War II
World War II
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. The von Neumanns were very active socially within the Princeton academic community.
Pancreatic cancer
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. A von Neumann biographer Norman Macrae
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has speculated: "It is plausible that in 1955 the then-fifty-one-year-old Johnny's cancer sprang from his attendance at the 1946 Bikini nuclear tests
Bikini Atoll
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." Von Neumann died a year and a half later. While at Walter Reed Hospital in Washington, D.C.
Washington, D.C.
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, he invited a Roman Catholic priest, Father Anselm Strittmatter, O.S.B.
Benedictine
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, to visit him for consultation. This move shocked some of von Neumann's friends in view of his reputation as an agnostic.
Von Neumann, however, is reported to have said in explanation that Pascal
Blaise Pascal
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had a point, referring to Pascal's wager
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. Father Strittmatter administered the last sacraments
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to him. He died under military security lest he reveal military secrets while heavily medicated. von Neumann was buried at Princeton Cemetery
Princeton Cemetery
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in Princeton
Princeton, New Jersey
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, Mercer County
Mercer County, New Jersey
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, New Jersey
New Jersey
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. On his death bed, he entertained his brother with a word for word memory of Goethe's Faust
Goethe's Faust
Johann Wolfgang von Goethe's Faust is a tragic play in two parts: and . Although written as a closet drama, it is the play with the largest audience numbers on German-language stages...
.
Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics. His last work, an unfinished manuscript written while in the hospital and later published in book form as The Computer and the Brain
The Computer and the Brain
The Computer and the Brain is an unfinished book by mathematician John von Neumann, begun shortly before his death. Von Neumann was an important figure in computer science, and the book discusses how the brain can be viewed as a computing machine...
, gives an indication of the direction of his interests at the time of his death.
Set theory
The axiomatization of mathematics, on the model of EuclidEuclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's Elements
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, had reached new levels of rigor and breadth at the end of the 19th century, particularly in arithmetic (thanks to the axiom schema
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano...
of Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
and Charles Sanders Peirce) and geometry (thanks to David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
). At the beginning of the twentieth century, efforts to base mathematics on naive
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
suffered a setback due to Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
(on the set of all sets that do not belong to themselves).
The problem of an adequate axiomatization of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
was resolved implicitly about twenty years later (by Ernst Zermelo
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
and Abraham Fraenkel). Zermelo and Fraenkel 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 belong 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
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
.
The axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
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.) 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 model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
s) 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. Under the Zermelo/Fraenkel approach, the axioms impede the construction of a set of all sets which do not belong to themselves. In contrast, under 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 1930 at the historic mathematical Congress of Königsberg
Königsberg
Königsberg was the capital of East Prussia from the Late Middle Ages until 1945 as well as the northernmost and easternmost German city with 286,666 inhabitants . Due to the multicultural society in and around the city, there are several local names for it...
, in which Kurt Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
announced his first theorem of incompleteness
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
: 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: 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.)
Geometry
Von Neumann founded the field of continuous geometryContinuous geometry
In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]...
. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Von Neumann was motivated by his discovery of von Neumann algebra
Von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
s with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor
Hyperfinite type II factor
In mathematics, there are up to isomorphism exactly two hyperfinite type II factors; one infinite and one finite. Murray and von Neumann proved that up to isomorphism there is a unique von Neumann algebra that is a factor of type II1 and also hyperfinite; it is called the hyperfinite type II1...
.
Measure theory
In a series of famous papers, Von Neumann made spectacular contributions to measure theory. The work of Banach had implied that the problem of measure has a positive solution if n = 1 or n = 2 and a negative solution in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character, and that, in particular, for the solvability of the problem of measure the ordinary algebraic concept of solvability of a group is relevant. Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."In a number of von Neumann's papers, the methods of argument he employed are considered more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions (anticipating his later work on almost periodic functions).
In the 1936 paper on analytic measure theory, von Neumann used the Haar theorem in the solution of Hilbert's fifth problem
Hilbert's fifth problem
Hilbert's fifth problem, is the fifth mathematical problem from the problem-list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics...
in the case of compact groups.
Ergodic theory
Von Neumann made foundational contributions to ergodic theory, in a series of articles published in 1932, which have attained legendary status in mathematics. Of the 1932 papers on ergodic theory, Paul HalmosPaul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
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 famous articles on operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, 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 algebraVon Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
s. A von Neumann algebra is a *-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
Von Neumann bicommutant theorem
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator...
shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
The direct integral
Direct integral
In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers...
was introduced in 1949 by John von Neumann in one of the final papers in the series On Rings of Operators. One of von Neumann's arguments was to reduce the classification of von Neumann algebras on separable Hilbert spaces to the classification of factors.
Probability Theory
Von Neumann's work on measure theory and operators led him to introduce a number of concepts in probability theoryProbability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
: for example, the standard probability space
Standard probability space
In probability theory, a standard probability space is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940...
.
Lattice theory
Garrett BirkhoffGarrett Birkhoff
Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
writes: "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann worked on lattice theory between 1937-39. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices: "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity."
Additionally, "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice L having a "basis" of n≥4 pairwise perspective elements, is isomorphic with the lattice ℛ(R) of all principal right-ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
of a suitable regular ring
Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...
R. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."
Mathematical formulation of quantum mechanics
Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics with his work Mathematische Grundlagen der Quantenmechanik.After having completed the axiomatization of set theory, von Neumann began to confront the axiomatization of quantum mechanics. He immediately realized, in 1926, that a quantum system could be considered as a point in a so-called Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, 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 uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
, 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 his 1932 book Mathematische Grundlagen der Quantenmechanik.
Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism vs. non-determinism and in the book he presented a proof according to which quantum mechanics could not possibly be derived by statistical approximation from a deterministic theory of the type used in classical mechanics. However, in 1966 it was discovered that this proof contained a conceptual error (see the article on John Stewart Bell
John Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...
for more information). The proof nonetheless inaugurated a line of research that ultimately led, through the work of Bell in 1964 on Bell's Theorem
Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
, and the experiments of Alain Aspect
Alain Aspect
Alain Aspect is a French physicist noted for his experimental work on quantum entanglement....
in 1982, to the demonstration that quantum physics requires a notion of reality substantially different from that of classical physics.
In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem
Measurement problem
The measurement problem in quantum mechanics is the unresolved problem of how wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer...
. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter (although this view was accepted by Eugene Wigner, it never gained acceptance amongst the majority of physicists).
Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formalism of problems in quantum mechanics which underlies the majority of approaches and can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations
Quantum logics
In a famous paper of 1936, the first work ever to introduce quantum logics, von Neumann 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 the new propositional calculus was demonstrated through several proofs. For example, 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. It was also demonstrated that the laws of distribution of classical logic, and, 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" northe 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 in 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
, while .Von Neumann proposes to replace classical logics, with a logic constructed in orthomodular lattices, (isomorphic to the lattice of subspaces of the Hilbert space of a given physical system).
Game theory
Von Neumann founded the field of game theoryGame theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
as a mathematical discipline. Von Neumann's proved his minimax theorem in 1928. This theorem establishes that in zero-sum games
Zero-sum
In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which a participant's gain of utility is exactly balanced by the losses of the utility of other participant. If the total gains of the participants are added up, and the total losses are...
with perfect information
Perfect information
In game theory, perfect information describes the situation when a player has available the same information to determine all of the possible games as would be available at the end of the game....
(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 (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy which will result in the minimization of his maximum loss.
Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Another result he proved during his German period was the nonexistence of a static equilibrium. An equilibrium can only exist in an expanding economy. Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...
edited an anniversary volume dedicated to this short German paper in 1972 and stated in the introduction that von Neumann was the only mathematician ever to make a significant contribution to economic theory.
Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior
Theory of Games and Economic Behavior
Theory of Games and Economic Behavior, published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinary research field of game theory...
(written with Oskar Morgenstern
Oskar Morgenstern
Oskar Morgenstern was a German-born Austrian-School economist. He, along with John von Neumann, helped found the mathematical field of game theory ....
). The public interest in this work was such that The New York Times
The New York Times
The New York Times is an American daily newspaper founded and continuously published in New York City since 1851. The New York Times has won 106 Pulitzer Prizes, the most of any news organization...
ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analytic
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
methods, especially convex set
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
s and topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
fixed point theorem, rather than the traditional differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
, because the maximum–operator did not preserve differentiable functions. Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting
Supporting hyperplane
Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set S in Euclidean space \mathbb R^n if it meets both of the following:...
and separating hyperplanes and convex set, and fixed-point theory—have been the primary tools of mathematical economics ever since. Von Neumann was also the inventor of the method of proof, used in game theory, known as backward induction
Backward induction
Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by first considering the last time a decision might be made and choosing what to do in any situation at that time. Using this...
(which he first published in 1944 in the book co-authored with Morgenstern, Theory of Games and Economic Behaviour).
Mathematical economics
Von Neumann raised the intellectual and mathematical level of economics in several stunning publications. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A − λB with nonnegative matrices A and B; von Neumann sought probabilityProbability vector
Stochastic vector redirects here. For the concept of a random vector, see Multivariate random variable.In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one....
vector
Generalized eigenvector
In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue λ is greater than its geometric multiplicity...
s p and q and a positive number λ that would solve the complementarity
Complementarity theory
A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing a function of two vector variables subject to certain requirements which include: that the inner product of the two variables must equal zero, i.e. = 0...
equation
- pT (A − λ B) q = 0,
along with two inequality systems expressing economic efficiency. In this model, the (transpose
Transpose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
d) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth
Economic growth
In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...
of the economy; the rate of growth equals the interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.
Von Neumann's results have been viewed as a special case of linear programming
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality
Dual problem
In constrained optimization, it is often possible to convert the primal problem to a dual form, which is termed a dual problem. Usually dual problem refers to the Lagrangian dual problem but other dual problems are used, for example, the Wolfe dual problem and the Fenchel dual problem...
.
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
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
, in 1983 to Gérard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...
, and in 1994 to John Nash who used fixed point theorems to establish equilibria for noncooperative games and for bargaining problem
Bargaining problem
The two person bargaining problem is a problem of understanding how two agents should cooperate when non-cooperation leads to Pareto-inefficient results. It is in essence an equilibrium selection problem; Many games have multiple equilibria with varying payoffs for each player, forcing the players...
s in his Ph.D thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans
Tjalling Koopmans
Tjalling Charles Koopmans was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences....
, Leonid Kantorovich
Leonid Kantorovich
Leonid Vitaliyevich Kantorovich was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources...
, Wassily Leontief
Wassily Leontief
Wassily Wassilyovich Leontief , was a Russian-American economist notable for his research on how changes in one economic sector may have an effect on other sectors. Leontief won the Nobel Committee's Nobel Memorial Prize in Economic Sciences in 1973, and three of his doctoral students have also...
, Paul Samuelson
Paul Samuelson
Paul Anthony Samuelson was an American economist, and the first American to win the Nobel Memorial Prize in Economic Sciences. The Swedish Royal Academies stated, when awarding the prize, that he "has done more than any other contemporary economist to raise the level of scientific analysis in...
, Robert Dorfman
Robert Dorfman
Robert Dorfman was emeritus professor of political economy at Harvard University. Dorfman made great contributions to the fields of economics, group testing and in the process of coding theory....
, Robert Solow
Robert Solow
Robert Merton Solow is an American economist particularly known for his work on the theory of economic growth that culminated in the exogenous growth model named after him...
, and Leonid Hurwicz
Leonid Hurwicz
Leonid "Leo" Hurwicz was a Russian-born American economist and mathematician. His nationality of origin was Polish. He was Jewish. He originated incentive compatibility and mechanism design, which show how desired outcomes are achieved in economics, social science and political science...
.
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 programmingLinear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
, after George B. Dantzig described his work in a few minutes, when an impatient von Neumann asked him to get to the point. Then, Dantzig listened dumbfounded while von Neumann provided an hour lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming. Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Gordan (1873) which was later popularized by Karmarkar's algorithm
Karmarkar's algorithm
Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems. It was the first reasonably efficient algorithm that solves these problems in polynomial time...
. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
subproblem with a convexity constraint (projecting the zero-vector onto the convex hull
Convex hull
In mathematics, the convex hull or convex envelope for a set of points X in a real vector space V is the minimal convex set containing X....
of the active simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
). Von Neumann's algorithm was the first interior-point method of linear programming.
Mathematical statistics
Von Neumann made fundamental contributions to mathematical statisticsMathematical statistics
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...
. In 1941, he derived the exact distribution of the ratio of mean square successive difference to the variance for normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression. Subsequently, John Denis Sargan and Alok Bhargava
Alok Bhargava
Alok Bhargava is an Indian econometrician. He studied mathematics at Delhi University and economics and econometrics at the London School of Economics. He is currently a full professor of economics at the University of Houston....
extended the results for testing if the errors on a regression model follow a Gaussian random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
(i.e. possess a unit root
Unit root
In time series models in econometrics , a unit root is a feature of processes that evolve through time that can cause problems in statistical inference if it is not adequately dealt with....
) against the alternative that they are a stationary first order autoregression. Von Neumann's contributions to statistics have had a major impact on econometric methodology.
Nuclear weapons
Explosion
An explosion is a rapid increase in volume and release of energy in an extreme manner, usually with the generation of high temperatures and the release of gases. An explosion creates a shock wave. If the shock wave is a supersonic detonation, then the source of the blast is called a "high explosive"...
s—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
Manhattan Project
The Manhattan Project was a research and development program, led by the United States with participation from the United Kingdom and Canada, that produced the first atomic bomb during World War II. From 1942 to 1946, the project was under the direction of Major General Leslie Groves of the US Army...
. The involvement included frequent trips by train to the project's secret research facilities in Los Alamos, New Mexico
Los Alamos National Laboratory
Los Alamos National Laboratory is a United States Department of Energy national laboratory, managed and operated by Los Alamos National Security , located in Los Alamos, New Mexico...
.
Von Neumann's principal contribution to the atomic bomb itself was in the concept and design of the explosive lenses
Nuclear weapon design
Nuclear weapon designs are physical, chemical, and engineering arrangements that cause the physics package of a nuclear weapon to detonate. There are three basic design types...
needed to compress the plutonium
Plutonium
Plutonium is a transuranic radioactive chemical element with the chemical symbol Pu and atomic number 94. It is an actinide metal of silvery-gray appearance that tarnishes when exposed to air, forming a dull coating when oxidized. The element normally exhibits six allotropes and four oxidation...
core of the Trinity test
Trinity test
Trinity was the code name of the first test of a nuclear weapon. This test was conducted by the United States Army on July 16, 1945, in the Jornada del Muerto desert about 35 miles southeast of Socorro, New Mexico, at the new White Sands Proving Ground, which incorporated the Alamogordo Bombing...
device and the "Fat Man
Fat Man
"Fat Man" is the codename for the atomic bomb that was detonated over Nagasaki, Japan, by the United States on August 9, 1945. It was the second of the only two nuclear weapons to be used in warfare to date , and its detonation caused the third man-made nuclear explosion. The name also refers more...
" weapon that was later dropped on Nagasaki. While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly" meaning compression. When it turned out that there would not be enough U235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with the plutonium
Plutonium
Plutonium is a transuranic radioactive chemical element with the chemical symbol Pu and atomic number 94. It is an actinide metal of silvery-gray appearance that tarnishes when exposed to air, forming a dull coating when oxidized. The element normally exhibits six allotropes and four oxidation...
-239 that was available from the Hanford site
Hanford Site
The Hanford Site is a mostly decommissioned nuclear production complex on the Columbia River in the U.S. state of Washington, operated by the United States federal government. The site has been known by many names, including Hanford Works, Hanford Engineer Works or HEW, Hanford Nuclear Reservation...
. His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. After a series of failed attempts with models, 5% was achieved by George Kistiakowsky
George Kistiakowsky
George Bogdan Kistiakowsky was a Ukrainian-American chemistry professor at Harvard who participated in the Manhattan Project and later served as President Eisenhower's Science Advisor...
, and the construction of the Trinity bomb was completed in July 1944.
In a visit to Los Alamos in September 1944, von Neumann showed that the pressure increase from explosion shock wave reflection from solid objects was greater than previously believed if the angle of incidence of the shock wave was between 90° and some limiting angle. As a result, it was determined that the effectiveness of an atomic bomb would be enhanced with detonation some kilometers above the target, rather than at ground level.
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
Hiroshima
is the capital of Hiroshima Prefecture, and the largest city in the Chūgoku region of western Honshu, the largest island of Japan. It became best known as the first city in history to be destroyed by a nuclear weapon when the United States Army Air Forces dropped an atomic bomb on it at 8:15 A.M...
and Nagasaki as the first targets of the atomic bomb
Atomic bombings of Hiroshima and Nagasaki
During the final stages of World War II in 1945, the United States conducted two atomic bombings against the cities of Hiroshima and Nagasaki in Japan, the first on August 6, 1945, and the second on August 9, 1945. These two events are the only use of nuclear weapons in war to date.For six months...
. 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
Kyoto
is a city in the central part of the island of Honshū, Japan. It has a population close to 1.5 million. Formerly the imperial capital of Japan, it is now the capital of Kyoto Prefecture, as well as a major part of the Osaka-Kobe-Kyoto metropolitan area.-History:...
, which had been spared the firebombing
Firebombing
Firebombing is a bombing technique designed to damage a target, generally an urban area, through the use of fire, caused by incendiary devices, rather than from the blast effect of large bombs....
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
Leslie Groves
Lieutenant General Leslie Richard Groves, Jr. was a United States Army Corps of Engineers officer who oversaw the construction of the Pentagon and directed the Manhattan Project that developed the atomic bomb during World War II. As the son of a United States Army chaplain, Groves lived at a...
. However, this target was dismissed by Secretary of War
United States Secretary of War
The Secretary of War was a member of the United States President's Cabinet, beginning with George Washington's administration. A similar position, called either "Secretary at War" or "Secretary of War," was appointed to serve the Congress of the Confederation under the Articles of Confederation...
Henry Stimson.
On July 16, 1945, with numerous other Los Alamos personnel, von Neumann was an eyewitness to the first atomic bomb blast
Trinity test
Trinity was the code name of the first test of a nuclear weapon. This test was conducted by the United States Army on July 16, 1945, in the Jornada del Muerto desert about 35 miles southeast of Socorro, New Mexico, at the new White Sands Proving Ground, which incorporated the Alamogordo Bombing...
, conducted as a test of the implosion method device, 35 miles (56 km) southeast of Socorro
Socorro, New Mexico
Socorro is a city in Socorro County in the U.S. state of New Mexico. It stands in the Rio Grande Valley at an elevation of . The population was 9,051 at the 2010 census...
, New Mexico
New Mexico
New Mexico is a state located in the southwest and western regions of the United States. New Mexico is also usually considered one of the Mountain States. With a population density of 16 per square mile, New Mexico is the sixth-most sparsely inhabited U.S...
. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5 kilotons
Ton
The ton is a unit of measure. It has a long history and has acquired a number of meanings and uses over the years. It is used principally as a unit of weight, and as a unit of volume. It can also be used as a measure of energy, for truck classification, or as a colloquial term.It is derived from...
of TNT
TNT equivalent
TNT equivalent is a method of quantifying the energy released in explosions. The ton of TNT is a unit of energy equal to 4.184 gigajoules, which is approximately the amount of energy released in the detonation of one ton of TNT...
, but Enrico Fermi
Enrico Fermi
Enrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
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.
After the war, Robert Oppenheimer
Robert Oppenheimer
Julius Robert Oppenheimer was an American theoretical physicist and professor of physics at the University of California, Berkeley. Along with Enrico Fermi, he is often called the "father of the atomic bomb" for his role in the Manhattan Project, the World War II project that developed the first...
remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it."
Von Neumann continued unperturbed in his work and became, along with Edward Teller
Edward Teller
Edward Teller was a Hungarian-American theoretical physicist, known colloquially as "the father of the hydrogen bomb," even though he did not care for the title. Teller made numerous contributions to nuclear and molecular physics, spectroscopy , and surface physics...
, one of those who sustained the hydrogen bomb project. He then collaborated with Klaus Fuchs
Klaus Fuchs
Klaus Emil Julius Fuchs was a German theoretical physicist and atomic spy who in 1950 was convicted of supplying information from the American, British and Canadian atomic bomb research to the USSR during and shortly after World War II...
on further development of the bomb, and in 1946 the two filed a secret patent on "Improvement in Methods and Means for Utilizing Nuclear Energy", which outlined a scheme for using a fission bomb to compress fusion fuel to initiate a thermonuclear reaction. The Fuchs–von Neumann patent used radiation implosion
Radiation implosion
The term radiation implosion describes the process behind a class of devices which use high levels of electromagnetic radiation to compress a target...
, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design. Their work was, however, incorporated into the "George" shot of Operation Greenhouse
Operation Greenhouse
Operation Greenhouse was the fifth American nuclear test series, the second conducted in 1951 and the first to test principles that would lead to developing thermonuclear weapons . Conducted at the new Pacific Proving Ground, all of the devices were mounted in large steel towers, to simulate air...
, which was instructive in testing out concepts that went into the final design. The Fuchs–von Neumann work was passed on, by Fuchs, to the USSR as part of his nuclear espionage
Nuclear espionage
Nuclear espionage is the purposeful giving of state secrets regarding nuclear weapons to other states without authorization . During the history of nuclear weapons there have been many cases of known nuclear espionage, and also many cases of suspected or alleged espionage...
, but it was not used in the Soviet's own, independent development of the Teller–Ulam design. The historian Jeremy Bernstein has pointed out that ironically, "John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made."
The ICBM Committee
In 1955, von Neumann became a commissioner of the United States Atomic Energy ProgramUnited States Atomic Energy Commission
The United States Atomic Energy Commission was an agency of the United States government established after World War II by Congress to foster and control the peace time development of atomic science and technology. President Harry S...
. Shortly before his death, when he was already quite ill, von Neumann headed the top secret von Neumann ICBM committee. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were indeed formidable, they could be overcome in time. The SM-65 Atlas passed its first fully functional test in 1959, two years after his death.
MAD
John von Neumann is credited with the strategy of Mutually assured destruction, providing the deliberately humorous acronym, MAD. (Other humorous acronyms coined by von Neumann include his computer, the Mathematical Analyzer, Numerical Integrator, and ComputerMANIAC I
The MANIAC was an early computer built under the direction of Nicholas Metropolis at the Los Alamos Scientific Laboratory...
- or MANIAC).
Computer science
Von Neumann was a founding figure in computer science. Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and Stanisław 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 methodMonte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
, which allowed complicated problems to be approximated using random number
Random number
Random number may refer to:* A number generated for or part of a set exhibiting statistical randomness.* A random sequence obtained from a stochastic process.* An algorithmically random sequence in algorithmic information theory....
s. Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middle-square method
Middle-square method
In mathematics, the middle-square method is a method of generating pseudorandom numbers. In practice it is not a good method, since its period is usually very short and it has some crippling weaknesses...
. 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 subtly incorrect.
While consulting for the Moore School of Electrical Engineering
Moore School of Electrical Engineering
The Moore School of Electrical Engineering at the University of Pennsylvania came into existence as a result of an endowment from Alfred Fitler Moore on June 4, 1923. It was granted to Penn's School of Electrical Engineering, located in the Towne Building...
at the University of Pennsylvania
University of Pennsylvania
The University of Pennsylvania is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Penn is the fourth-oldest institution of higher education in the United States,Penn is the fourth-oldest using the founding dates claimed by each institution...
on the EDVAC
EDVAC
EDVAC was one of the earliest electronic computers. Unlike its predecessor the ENIAC, it was binary rather than decimal, and was a stored program computer....
project, von Neumann wrote an incomplete First Draft of a Report on the EDVAC
First Draft of a Report on the EDVAC
The First Draft of a Report on the EDVAC was an incomplete 101-page document written by John von Neumann and distributed on June 30, 1945 by Herman Goldstine, security officer on the classified ENIAC project...
. The paper, which was widely distributed, described a computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
architecture in which the data and the program are both stored in the computer's memory in the same address space. This architecture is to this day the basis of modern computer design, unlike the earliest computers that were 'programmed' by altering the electronic circuitry. Although the single-memory, stored program architecture is commonly called von Neumann architecture
Von Neumann architecture
The term Von Neumann architecture, aka the Von Neumann model, derives from a computer architecture proposal by the mathematician and early computer scientist John von Neumann and others, dated June 30, 1945, entitled First Draft of a Report on the EDVAC...
as a result of von Neumann's paper, the architecture's description was based on the work of J. Presper Eckert
J. Presper Eckert
John Adam Presper "Pres" Eckert Jr. was an American electrical engineer and computer pioneer. With John Mauchly he invented the first general-purpose electronic digital computer , presented the first course in computing topics , founded the first commercial computer company , and...
and John William Mauchly, inventors of the ENIAC
ENIAC
ENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....
at the University of Pennsylvania
University of Pennsylvania
The University of Pennsylvania is a private, Ivy League university located in Philadelphia, Pennsylvania, United States. Penn is the fourth-oldest institution of higher education in the United States,Penn is the fourth-oldest using the founding dates claimed by each institution...
.
Stochastic computing
Stochastic computing
Stochastic computing is a collection of techniques that represent continuous values by streams of random bits. Complex computations can then be computed by simple bit-wise operations on the streams....
was first introduced in a pioneering paper by von Neumann in 1953. However, the
theory could not be implemented until advances in computing of the 1960s.
Von Neumann also created the field of cellular automata
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
without the aid of computers, constructing the first self-replicating
Self-replication
Self-replication is any behavior of a dynamical system that yields construction of an identical copy of that dynamical system. Biological cells, given suitable environments, reproduce by cell division. During cell division, DNA is replicated and can be transmitted to offspring during reproduction...
automata with pencil and graph paper. The concept of a universal constructor
Von Neumann universal constructor
John von Neumann's Universal Constructor is a self-replicating machine in a cellular automata environment. It was designed in the 1940s, without the use of a computer. The fundamental details of the machine were published in von Neumann's book Theory of Self-Reproducing Automata, completed in...
was fleshed out in his posthumous work Theory of Self Reproducing Automata. Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire moon
Natural satellite
A natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....
or asteroid belt
Asteroid belt
The asteroid belt is the region of the Solar System located roughly between the orbits of the planets Mars and Jupiter. It is occupied by numerous irregularly shaped bodies called asteroids or minor planets...
would be by using self-replicating machines, taking advantage of their exponential growth
Exponential growth
Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
.
Donald Knuth
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...
cites von Neumann as the inventor, in 1945, of the merge sort
Merge sort
Merge sort is an O comparison-based sorting algorithm. Most implementations produce a stable sort, meaning that the implementation preserves the input order of equal elements in the sorted output. It is a divide and conquer algorithm...
algorithm, in which the first and second halves of an array are each sorted recursively and then merged together.
His algorithm for simulating a fair coin
Fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin...
with a biased coin is used in the "software whitening" stage of some hardware random number generator
Hardware random number generator
In computing, a hardware random number generator is an apparatus that generates random numbers from a physical process. Such devices are often based on microscopic phenomena that generate a low-level, statistically random "noise" signal, such as thermal noise or the photoelectric effect or other...
s.
Fluid dynamics
Von Neumann made fundamental contributions in exploration of problems in numerical hydrodynamics. For example, with R. D. Richtmyer he developed an algorithm defining artificial viscosity that improved the understanding of shock waveShock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...
s. It is possible that we would not understand much of astrophysics, and might not have highly developed jet and rocket engines without the work of von Neumann. A problem was that when computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock wave
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...
s). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics.
Other well known contributions to fluid dynamics included the classic flow solution to blast wave
Blast wave
A blast wave in fluid dynamics is the pressure and flow resulting from the deposition of a large amount of energy in a small very localised volume. The flow field can be approximated as a lead shock wave, followed by a 'self-similar' subsonic flow field. In simpler terms, a blast wave is an area of...
s, and the co-discovery of the ZND detonation model
ZND detonation model
The ZND detonation model is a one-dimensional model for the process of detonation of an explosive. It was proposed during World War II independently by Y. B. Zel'dovich, John von Neumann, and Werner Döring, hence the name....
of explosives.
Politics and social affairs
Von Neumann obtained at the age of 29 one of the first five professorships at the new Institute for Advanced StudyInstitute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton, New Jersey
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
(another had gone to Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
). He was a frequent consultant for the Central Intelligence Agency
Central Intelligence Agency
The Central Intelligence Agency is a civilian intelligence agency of the United States government. It is an executive agency and reports directly to the Director of National Intelligence, responsible for providing national security intelligence assessment to senior United States policymakers...
, the United States Army
United States Army
The United States Army is the main branch of the United States Armed Forces responsible for land-based military operations. It is the largest and oldest established branch of the U.S. military, and is one of seven U.S. uniformed services...
, the RAND Corporation, Standard Oil
Standard Oil
Standard Oil was a predominant American integrated oil producing, transporting, refining, and marketing company. Established in 1870 as a corporation in Ohio, it was the largest oil refiner in the world and operated as a major company trust and was one of the world's first and largest multinational...
, General Electric
General Electric
General Electric Company , or GE, is an American multinational conglomerate corporation incorporated in Schenectady, New York and headquartered in Fairfield, Connecticut, United States...
, IBM
IBM
International Business Machines Corporation or IBM is an American multinational technology and consulting corporation headquartered in Armonk, New York, United States. IBM manufactures and sells computer hardware and software, and it offers infrastructure, hosting and consulting services in areas...
, 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. Von Neumann entered government service (Manhattan Project) primarily because he felt that, if freedom and civilization were to survive, it would have to be because the U.S. would triumph over totalitarianism from the right (Nazism and Fascism) and totalitarianism from the left (Soviet Communism).
As President of the Von Neumann Committee for Missiles, and later as a member of the United States Atomic Energy Commission
United States Atomic Energy Commission
The United States Atomic Energy Commission was an agency of the United States government established after World War II by Congress to foster and control the peace time development of atomic science and technology. President Harry S...
, 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
Mutual assured destruction
Mutual Assured Destruction, or mutually assured destruction , is a doctrine of military strategy and national security policy in which a full-scale use of high-yield weapons of mass destruction by two opposing sides would effectively result in the complete, utter and irrevocable annihilation of...
. During a Senate
United States Senate
The United States Senate is the upper house of the bicameral legislature of the United States, and together with the United States House of Representatives comprises the United States Congress. The composition and powers of the Senate are established in Article One of the U.S. Constitution. Each...
committee hearing he described his political ideology as "violently anti-communist
Communism
Communism is a social, political and economic ideology that aims at the establishment of a classless, moneyless, revolutionary and stateless socialist society structured upon common ownership of the means of production...
, and much more militaristic than the norm". He was quoted in 1950 remarking, "If you say why not bomb [Russia] tomorrow, I say, why not today. If you say today at five o’clock, I say why not one o’clock?". As a result, he partly inspired the character of 'Doctor Strangelove' in Doctor Strangelove.
Weather control
Von Neumann's team performed the world's first numerical weather forecasts on the ENIAC computer; von Neumann published the paper Numerical Integration of the Barotropic Vorticity Equation in 1950. Von Neumann's interest in weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice caps to enhance absorption of solar radiation (by reducing the albedoAlbedo
Albedo , or reflection coefficient, is the diffuse reflectivity or reflecting power of a surface. It is defined as the ratio of reflected radiation from the surface to incident radiation upon it...
), thereby inducing global warming
Global warming
Global warming refers to the rising average temperature of Earth's atmosphere and oceans and its projected continuation. In the last 100 years, Earth's average surface temperature increased by about with about two thirds of the increase occurring over just the last three decades...
.
Personality
Von Neumann had a wide range of cultural interests. Since the age of six, von Neumann had been fluent in Latin and ancient Greek, and he held a life-long passion for ancient history, being renowned for his prodigious historical knowledge. A professor of Byzantine history once reported that von Neumann had greater expertise on Byzantine history than he did. Von Neumann took great care over his clothing, and would always wear formal suits, once riding down the Grand Canyon astride a mule in a three-piece pin-stripe. He was extremely sociable and, during his first marriage, he enjoyed throwing large parties at his home in Princeton, occasionally twice a week. His white clapboard house at 26 Westcott Road was one of the largest in PrincetonPrinceton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
.
Despite being a notoriously bad driver, he nonetheless enjoyed driving (frequently while reading a book) – occasioning numerous arrests as well as accidents.
When Cuthbert Hurd
Cuthbert Hurd
Cuthbert Corwin Hurd was an American computer scientist and entrepreneur, who was instrumental in helping the International Business Machines Corporation develop its first general-purpose computers.-Life:...
hired him as a consultant to IBM, Hurd often quietly paid the fines for his traffic tickets. He believed that much of his mathematical thought occurred intuitively, and he would often go to sleep with a problem unsolved, and know the answer immediately upon waking up.
Von Neumann liked to eat and drink; his wife, Klara, said that he could count everything except calories. He enjoyed Yiddish
Jewish humor
Jewish humour is the long tradition of humour in Judaism dating back to the Torah and the Midrash from the ancient mid-east, but generally refers to the more recent stream of verbal, self-deprecating, crude, and often anecdotal humour originating in Eastern Europe and which took root in the United...
and "off-color" humor
Off-color humor
The term off-color humor is an Americanism used to describe jokes, prose, poems, black comedy, blue comedy, insult comedy, cringe comedy and skits that deal with topics that are considered to be in poor taste or overly vulgar by the prevailing morality of a culture...
(especially limericks). At Princeton he received complaints for regularly playing extremely loud German marching music on his gramophone, which distracted those in neighbouring offices, including Einstein, from their work. Von Neumann's closest friend in America was the Polish mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota
Gian-Carlo Rota
Gian-Carlo Rota was an Italian-born American mathematician and philosopher.-Life:Rota was born in Vigevano, Italy...
writes: "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in hospital, every time Ulam would visit he would come prepared with a new collection of jokes to cheer up his friend.
Cognitive and mnemonic abilities
Von Neumann's ability to instantaneously perform complex operations in his head stunned other mathematicians. Eugene Wigner wrote that, seeing von Neumann's mind at work, "one had the impression of a perfect instrument whose gears were machined to mesh accurately to a thousandth of an inch." Paul HalmosPaul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
states that "von Neumann's speed was awe-inspiring." Israel Halperin
Israel Halperin
Israel Halperin, was a Canadian mathematician and social activist.Born in Toronto, Ontario, the son of Russian immigrants Solomon Halperin and Fanny Lundy, Halperin attended Malvern Collegiate Institute, Victoria University in the University of Toronto, graduated from the University of Toronto in...
said: "Keeping up with him was... impossible. The feeling was you were on a tricycle chasing a racing car." Edward Teller
Edward Teller
Edward Teller was a Hungarian-American theoretical physicist, known colloquially as "the father of the hydrogen bomb," even though he did not care for the title. Teller made numerous contributions to nuclear and molecular physics, spectroscopy , and surface physics...
wrote that von Neumann effortlessly outdid anybody he ever met, and said "I never could keep up with him". Lothar Wolfgang Nordheim
Lothar Wolfgang Nordheim
Lothar Wolfgang Nordheim was a German-born Jewish theoretical physicist...
described von Neumann as the "fastest mind I ever met", and Jacob Bronowski
Jacob Bronowski
Jacob Bronowski was a Polish-Jewish British mathematician, biologist, historian of science, theatre author, poet and inventor...
wrote "He was the cleverest man I ever knew, without exception. He was a genius." George Pólya
George Pólya
George Pólya was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory...
, whose lectures at ETH Zurich
ETH Zurich
The Swiss Federal Institute of Technology Zurich or ETH Zürich is an engineering, science, technology, mathematics and management university in the City of Zurich, Switzerland....
von Neumann attended as a student, said "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem. He'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper." Halmos recounts a story told by Nicholas Metropolis
Nicholas Metropolis
Nicholas Constantine Metropolis was a Greek American physicist.-Work:Metropolis received his B.Sc. and Ph.D. degrees in physics at the University of Chicago...
, concerning the speed of von Neumann's calculations, when somebody asked von Neumann to solve the famous fly puzzle:
Von Neumann had a photographic memory
Eidetic memory
Eidetic , commonly referred to as photographic memory, is a medical term, popularly defined as the ability to recall images, sounds, or objects in memory with extreme precision and in abundant volume. The word eidetic, referring to extraordinarily detailed and vivid recall not limited to, but...
. Herman Goldstine writes: "One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how The Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes."
Honors
- The John von Neumann Theory PrizeJohn von Neumann Theory PrizeThe John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciencesis awarded annually to an individual who has made fundamental and sustained contributions to theory in operations research and the management sciences.The Prize named after mathematician John von...
of the Institute for Operations Research and the Management SciencesInstitute for Operations Research and the Management SciencesThe Institute for Operations Research and the Management Sciences is an international society for practitioners in the fields of 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 researchOperations researchOperations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...
and the management sciences. - The IEEE John von Neumann MedalIEEE John von Neumann MedalThe IEEE John von Neumann Medal was established by the IEEE Board of Directors in 1990 and may be presented annually "for outstanding achievements in computer-related science and technology." The achievements may be theoretical, technological, or entrepreneurial, and need not have been made...
is awarded annually by the IEEEInstitute of Electrical and Electronics EngineersThe Institute of Electrical and Electronics Engineers is a non-profit professional association headquartered in New York City that is dedicated to advancing technological innovation and excellence...
"for outstanding achievements in computer-related science and technology." - The John von Neumann Lecture is given annually at the Society for Industrial and Applied MathematicsSociety for Industrial and Applied MathematicsThe Society for Industrial and Applied Mathematics was founded by a small group of mathematicians from academia and industry who met in Philadelphia in 1951 to start an organization whose members would meet periodically to exchange ideas about the uses of mathematics in industry. This meeting led...
(SIAM) by a researcher who has contributed to applied mathematics, and the chosen lecturer is also awarded a monetary prize. - The crater Von NeumannVon Neumann (crater)Von Neumann is a lunar impact crater that lies on the far side of the Moon, in the northern hemisphere. It is nearly attached to the south-southeastern rim of the walled plain Campbell. The crater Ley is attached to the northeastern rim of Von Neumann, and is somewhat overlain by the outer rampart...
on the MoonMoonThe Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
is named after him. - The John von Neumann Computing Center in Princeton, New Jersey (40.348695°N 74.592251°W) was named in his honour.
- The professional society of Hungarian computer scientists, John von Neumann Computer SocietyJohn von Neumann Computer SocietyThe John von Neumann Computer Society is the central association for Hungarian researchers of Information communication technology and official partner of the International Federation for Information Processing founded in 1968....
, is named after John von Neumann. - On February 15, 1956, Neumann was presented with the Presidential Medal of FreedomPresidential Medal of FreedomThe Presidential Medal of Freedom is an award bestowed by the President of the United States and is—along with thecomparable Congressional Gold Medal bestowed by an act of U.S. Congress—the highest civilian award in the United States...
by President Dwight Eisenhower. - On May 4, 2005 the United States Postal ServiceUnited States Postal ServiceThe United States Postal Service is an independent agency of the United States government responsible for providing postal service in the United States...
issued the American Scientists commemorative postage stampPostage stampA postage stamp is a small piece of paper that is purchased and displayed on an item of mail as evidence of payment of postage. Typically, stamps are made from special paper, with a national designation and denomination on the face, and a gum adhesive on the reverse side...
series, a set of four 37-cent self-adhesive stamps in several configurations. The scientists depicted were John von Neumann, Barbara McClintockBarbara McClintockBarbara McClintock , the 1983 Nobel Laureate in Physiology or Medicine, was an American scientist and one of the world's most distinguished cytogeneticists. McClintock received her PhD in botany from Cornell University in 1927, where she was a leader in the development of maize cytogenetics...
, Josiah Willard GibbsJosiah Willard GibbsJosiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
, and Richard FeynmanRichard FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
. - The John von Neumann AwardJohn von Neumann AwardThe John von Neumann Award, named after John von Neumann is given annually by the Rajk László College for Advanced Studies , to an outstanding scholar in the exact social sciences, whose works have had substantial influence over a long period of time on the studies and intellectual activity of the...
of the Rajk László College for Advanced Studies was named in his honour, and has been given every year since 1995 to professors who have made an outstanding contribution to the exact social sciences and through their work have strongly influenced the professional development and thinking of the members of the college.
Selected works
- 1923. On the introduction of transfinite numbers, 346–54.
- 1925. An axiomatization of set theory, 393–413.
- 1932. Mathematical Foundations of Quantum Mechanics, Beyer, R. T., trans., Princeton Univ. Press. 1996 edition: ISBN 0-691-02893-1.
- 1944. Theory of Games and Economic Behavior, with Morgenstern, O., Princeton Univ. Press. 2007 edition: ISBN 978-0-691-13061-3.
- 1945. First Draft of a Report on the EDVAC TheFirstDraft.pdf
- 1963. Collected Works of John von Neumann, Taub, A. H., ed., Pergamon Press. ISBN 0080095666
- 1966. Theory of Self-Reproducing Automata, Burks, A. W., ed., Univ. of Illinois Press.
See also
PhD Students- Donald B. GilliesDonald B. GilliesDonald Bruce Gillies was a Canadian mathematician and computer scientist, known for his work in game theory, computer design, and minicomputer programming environments.- Education :...
, Ph.D. student - Israel HalperinIsrael HalperinIsrael Halperin, was a Canadian mathematician and social activist.Born in Toronto, Ontario, the son of Russian immigrants Solomon Halperin and Fanny Lundy, Halperin attended Malvern Collegiate Institute, Victoria University in the University of Toronto, graduated from the University of Toronto in...
, Ph.D. student
Further reading
- Aspray, William, 1990. John von Neumann and the Origins of Modern Computing.
- Chiara, Dalla, Maria Luisa and Giuntini, Roberto 1997, La Logica Quantistica in Boniolo, Giovani, ed., Filosofia della Fisica (Philosophy of Physics). Bruno Mondadori.
- Goldstine, Herman, 1980. The Computer from Pascal to von Neumann.
- Halmos, Paul R., 1985. I Want To Be A Mathematician Springer-Verlag
- Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903–1957). Teil 1: Lehrjahre eines jüdischen Mathematikers während der Zeit der Weimarer Republik. In: Informatik-Spektrum 29 (2), S. 133–141.
- Hashagen, Ulf, 2006: Johann Ludwig Neumann von Margitta (1903–1957). Teil 2: Ein Privatdozent auf dem Weg von Berlin nach Princeton. In: Informatik-Spektrum 29 (3), S. 227–236.
- Heims, Steve J., 1980. John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death MIT PressMIT PressThe MIT Press is a university press affiliated with the Massachusetts Institute of Technology in Cambridge, Massachusetts .-History:...
- Macrae, NormanNorman MacraeNorman Macrae CBE was a British economist, journalist and author, considered by some to have been one of the world's best forecasters when it came to economics and society...
, 1999. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More. Reprinted by the American Mathematical SocietyAmerican Mathematical SocietyThe American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
. - Poundstone, WilliamWilliam PoundstoneWilliam Poundstone is an American author, columnist, and skeptic. He has written a number of books including the Big Secrets series and a biography of Carl Sagan...
. Prisoner's Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb. 1992. - Redei, Miklos (ed.), 2005 John von Neumann: Selected Letters American Mathematical Society
- Ulam, Stanisław, 1983. Adventures of a Mathematician Scribner's
- Vonneuman, Nicholas A. John von Neumann as Seen by His Brother ISBN 0-9619681-0-9
- 1958, Bulletin of the American Mathematical Society 64.
- 1990. Proceedings of the American Mathematical Society Symposia in Pure Mathematics 50.
- John von Neumann 1903–1957, biographical memoir by S. Bochner, National Academy of SciencesUnited States National Academy of SciencesThe National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
, 1958
Popular periodicals
- Good Housekeeping Magazine, September 1956 Married to a Man Who Believes the Mind Can Move the World
- Life Magazine, February 25, 1957 Passing of a Great Mind
Video
- John von Neumann, A Documentary (60 min.), Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
External links
- von Neumann's contribution to economics — International Social Science Review
- Oral history interview with Alice R. Burks and Arthur W. Burks, Charles Babbage InstituteCharles Babbage InstituteThe Charles Babbage Institute is a research center at the University of Minnesota specializing in the history of information technology, particularly the history since 1935 of digital computing, programming/software, and computer networking....
, University of Minnesota, Minneapolis. Alice BurksAlice BurksAlice Rowe Burks is an American author of children's books and books about the history of electronic computers.Born Alice Rowe, she began her undergraduate degree at Oberlin College on a competitive mathematics scholarship and transferred to the University of Pennsylvania in Philadelphia where she...
and Arthur BurksArthur BurksArthur Walter Burks was an American mathematician who in the 1940s as a senior engineer on the project contributed to the design of the ENIAC, the first general-purpose electronic digital computer. Decades later, Burks and his wife Alice Burks outlined their case for the subject matter of the...
describe ENIACENIACENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....
, EDVACEDVACEDVAC was one of the earliest electronic computers. Unlike its predecessor the ENIAC, it was binary rather than decimal, and was a stored program computer....
, and IAS computers, and John von Neumann's contribution to the development of computers. - Oral history interview with Eugene P. Wigner, Charles Babbage InstituteCharles Babbage InstituteThe Charles Babbage Institute is a research center at the University of Minnesota specializing in the history of information technology, particularly the history since 1935 of digital computing, programming/software, and computer networking....
, University of Minnesota, Minneapolis. Wigner talks about his association with John von Neumann during their school years in Hungary, their graduate studies in Berlin, and their appointments to Princeton in 1930. Wigner discusses von Neumann's contributions to the theory of quantum mechanics, and von Neumann's interest in the application of theory to the atomic bomb project. - Oral history interview with Nicholas C. Metropolis, Charles Babbage InstituteCharles Babbage InstituteThe Charles Babbage Institute is a research center at the University of Minnesota specializing in the history of information technology, particularly the history since 1935 of digital computing, programming/software, and computer networking....
, University of Minnesota. Metropolis, the first director of computing services at Los Alamos National LaboratoryLos Alamos National LaboratoryLos Alamos National Laboratory is a United States Department of Energy national laboratory, managed and operated by Los Alamos National Security , located in Los Alamos, New Mexico...
, discusses John von Neumann's work in computing. Most of the interview concerns activity at Los Alamos: how von Neumann came to consult at the laboratory; his scientific contacts there, including Metropolis; von Neumann's first hands-on experience with punched card equipment; his contributions to shock-fitting and the implosion problem; interactions between, and comparisons of von Neumann and Enrico FermiEnrico FermiEnrico Fermi was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics...
; and the development of Monte Carlo methodMonte Carlo methodMonte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
s. Other topics include: the relationship between Alan TuringAlan TuringAlan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
and von Neumann; work on numerical methods for non-linear problems; and the ENIACENIACENIAC was the first general-purpose electronic computer. It was a Turing-complete digital computer capable of being reprogrammed to solve a full range of computing problems....
calculations done for Los Alamos. - Von Neumann vs. Dirac — from Stanford Encyclopedia of Philosophy.
- John von Neumann Postdoctoral Fellowship – Sandia National Laboratories
- Von Neumann's Universe, audio talk by George DysonGeorge Dyson (science historian)George Dyson is a scientific historian, the son of Freeman Dyson and Verena Huber-Dyson, brother of Esther Dyson, and the grandson of Sir George Dyson. He is the father of Lauren Dyson. When he was sixteen he went to live in British Columbia in Canada to pursue his interest in kayaking and...
- John von Neumann's 100th Birthday, article by Stephen WolframStephen WolframStephen Wolfram is a British scientist and the chief designer of the Mathematica software application and the Wolfram Alpha computational knowledge engine.- Biography :...
on Neumann's 100th birthday. - Annotated bibliography for John von Neumann from the Alsos Digital Library for Nuclear Issues
- Budapest Tech Polytechnical Institution – John von Neumann Faculty of Informatics
- John von Neumann speaking at the dedication of the NORD, December 2, 1954 (audio recording)
- The American Presidency Project
- John Von Neumann Memorial at Find A GraveFind A GraveFind a Grave is a commercial website providing free access and input to an online database of cemetery records. It was founded in 1998 as a DBA and incorporated in 2000.-History:...