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John von Neumann

One of the most creative and outstanding
mathematicians of the 20th century. Made
major contributions to game theory, utility
theory, design of stored-program computers,
numerical analysis, Monte Carlo simulation.

Born: December 28, 1903, Budapest, Hungary.

Died: February 8, 1957, Washington, DC,
USA.

Education: Diploma, Chemical Engineering,
Technische Hochschule,
(1925); Doctorate, Mathematics, University of
Budapest (1926).

Key positions: Visiting Lecturer, Princeton
University (1930–1933); Professor of Mathematics,
Institute for Advanced Study, Princeton
(1933–1957); Scientific Advisory Committee,
US Army Ballistic Research Laboratory,
Aberdeen Proving Ground (1940–1957); Consultant,
US Navy Bureau of Ordnance (1941–
1955); Consultant, Los Alamos Scientific Laboratory
(1943–1955); Member, US Armed
Forces Special Weapons Project, Washington,
DC (1950–1955); Scientific Advisory Board,
US Air Force, Washington, DC (1951–1957);
Member, General Advisory Committee, US
Atomic Energy Commission, Washington, DC
(1952–1954); Commissioner, US Atomic Energy
Commission (1955–1957).

Key positions: Visiting Lecturer, Princeton
University (1930–1933); Professor of Mathematics,
Institute for Advanced Study, Princeton
(1933–1957); Scientific Advisory Committee,
US Army Ballistic Research Laboratory,
Aberdeen Proving Ground (1940–1957); Consultant,
US Navy Bureau of Ordnance (1941–
1955); Consultant, Los Alamos Scientific Laboratory
(1943–1955); Member, US Armed
Forces Special Weapons Project, Washington,
DC (1950–1955); Scientific Advisory Board,
US Air Force, Washington, DC (1951–1957);
Member, General Advisory Committee, US
Atomic Energy Commission, Washington, DC
(1952–1954); Commissioner, US Atomic Energy
Commission (1955–1957).

 

On the death of his friend John von Neumann, the physicist Stanislaw Ulam wrote (Ulam
1958): ‘‘. . . the world of mathematics lost a most original, penetrating, and versatile mind. Science
suffered the loss of a universal intellect and a unique inter preter of mathematics , who could bring
the latest (and develop latent) applications of its methods to bear on problems of physics,
astronomy, biology, and the new technology.’’ Ulam goes on to discuss von Neumann’s
contributions to set theory and algebra, theory of functions, measure theory, topology,
continuous groups, Hilbert space theory, operator theory, theory of lattices, continuous
geometry, theoretical physics, quantum theory, statistical mechanics, hydrodynamics, systems
of linear equations and matrix inversion, game theory, economics, theory and practice of
electronic computers, Monte Carlo method, theory of automata, probabilistic logic, and nuclear
energy and nuclear weapons. Ulam mentions ‘‘operational research,’’ more or less in passing, as
stemming from von Neumann’s and Morgenstern’s ‘‘classical treatise’’ Theory of Games and
Economic Behavior (1994). But, from an operational research (OR) perspective, von Neumann’s
influence far exceeds a passing glance, as evidenced by the establishment of the prestigious von
Neumann Theory Prize by the Operations Research Society of America (ORSA) and the Institute
of Management Sciences (TIMS) in 1974, and now sponsored by the Institute of Operations
Research and the Management Sciences (INFORMS). In what follows, we recount briefly von
Neumann’s early years and, with respect to his impact on OR, describe his contributions to game
theory, numerical analysis, and Monte Carlo simulation.

John ( Jansci, Johnny) von Neumann was born on December 28, 1903 in Budapest, Hungary to
Margaret (Kann) and Max Neumann. Max was a successful banker who obtained the appellation
Neumann de Margitta (title of nobility) in 1913 from the Emperor Franz Josef. This was later
changed by Johnny to the Germanized version von Neumann.

As a young child, Johnny exhibited a photo graphic memory and a remarkable ability
in mathematics. Soon after starting his formal education at the Budapest Lutheran Gymnasium,
his mathematics teacher recognized that he was a child prodigy and recommended that he
be tutored by a University of Budapest professor, Michael Fekete. A special mathematics
program was initiated and by the time Johnny left the Gymnasium, he and Fekete
had written and published a joint paper that extended a theorem in analysis (1922). In 1921,
von Neumann enrolled in the mathematics program at the University of Budapest, but did
not take any classes. He also registered at the University of Berlin where he studied chemistry
through 1923. He then moved to Zurich and enrolled in the Technische
Hochschule, where he received an undergraduate degree in chemical engineering in 1925. During
his time in Berlin and Zurich, he would return to Budapest at the end of each semester so he could
take (and pass) the exams at the University. He was thus able to receive his doctorate in
mathematics in 1926.

Von Neumann was appointed Privat Dozent (assistant professor) at the University of Berlin
where he remained from 1927 to 1929. He then held the same title at the University of Hamburg
through 1930. After spending a semester in 1929 lecturing on quantum mechanics at Princeton
University, he was offered a position there as Visiting Professor, which he accepted and held from
1930 to 1933. When the Princeton Institute for Advanced Study was opened in 1933, Johnny was
appointed as a Professor in Mathematics, a position he held until his death in 1957. He was the
youngest member of the Institute when he joined the newly formed illustrious staff of Albert
Einstein, Marston Morse, Oswald Veblen, and Hermann Weyl.

The origins of modern game theory can be traced to the work of Ernst Zermelo and
Borel, but it was von Neumann who set the stage for what was to fol low by his 1928 Minimax
Theorem paper. In it he proved the existence of optimal randomized mixed strategies for any twoperson,
zero-sum game, as well as the existence of a unique value for the game, the minimax value.
There was a long hiatus before von Neumann’s next game theory publication in 1944. This came
about because of his friendship with the Princeton University economist Oskar Morgenstern who
introduced him to the competitive problems inherent in economic activities. Together they wrote
the seminal book Theory of Games and Economic Behavior. Besides establishing the theory of
games in a rigorous fashion, this book also set the stage for the development of modern utility
theory by giving it an axiomatic base that leads to an existence theorem of a real-valued utility
function (the theorem is proved in the 1947 second edition).

Looking at von Neumann’s game theory mathematical results in terms of matrix and linear
relationships, one can see how and why von Neumann reacted to George Dantzig’s description of
the newly formulated linear -programming model when they first met in 1947. Here is that story as
told by Dantzig (1982, p. 45):

On October 3, 1947, I visited him (von Neumann) for the first time at the Institute for Advanced Study
at Princeton. I remember trying to describe to von Neumann, as I would to an ordinary mortal, the Air
Force problem. I began with the formulation of the linear programming model in terms of activities
and items, etc. Von Neumann did something which I believe was uncharacteristic of him. ‘‘Get to the
point,’’ he said impatiently. Having at times a somewhat low kindling-point, I said to myself ‘‘O.K., if
he wants a quicky, then that’s what he will get.’’ In under one minute I slapped the geometric and
algebraic version of the problem on the blackboard. Von Neumann stood up and said ‘‘Oh that!’’ Then
for the next hour and a half, he proceeded to give me a lecture on the mathematical theory of linear
programs.

At one point seeing me sitting there with my eyes popping and my mouth open (after I had searched
the literature and found nothing), von Neumann said: ‘‘I don’t want you to think I am pulling all this
out of my sleeve at the spur of the moment like a magician. I have just recently completed a book with
Oscar [sic]Morgenstern on the theory of games. What I am doing is conjecturing that the two problems
are equivalent . The theory that I am outlining for your problem is an analogue to the one we have
developed for games.’’ Thus I learned about Farkas’ Lemma, and about duality for the first time.

Thus, in the 1940s, we have the almost simultaneous appearance of modern game theory and the
field of linear programming. Both areas have become mainstays of OR theory and its application.
But they did not evolve from the exigencies of World War II military operations, as did the origin
and practice of early OR. (How this came to be is a story yet to be told.) What is really fascinating
and beautiful about these two areas is that, although they were developed independently in two
very different research environments, they are intimately related as it can be shown that they solve
the same mathematical problem.

Von Neumann is considered to be the originator of modern numerical analysis and a key
contributor to the development and application of Monte Carlo simulation. He is also the first
one to consider and describe an electronic computer in terms of a logical structure that included
the stored-program concept and how such a computer processes information. These three areas –
numerical analysis, Monte Carlo simulation, stored-program computers – have had major
impacts on the development of OR methods and their application.

Many techniques used in OR require the inversion of matrices (the solution of linear systems),
e.g., linear programming and least squares . In the early 1940s, an acceptable matrix inverse was
rather difficult to determine because of the matrix size, accuracy in computation, and the human
effort required to calculate it. In addition to von Neumann, the computer giants John Atanasoff,
Herman Goldstine, and Alan Turing recognized that a stable means of solving aX =b was of great
importance. It was the ‘‘. . . absolutely fundamental problem in numerical analysis: how best to
solve a large system of linear equations’’ (Goldstine, 1972, p. 289).

By the early 1940s, Atanasoff had designed a new ‘‘computing machine for the solution of linear
algebraic equations’’ that applied Gaussian elimination. He noted:

The solution of general systems of linear equations with a number of unknowns greater than ten is not
often attempted. But this is precisely what is needed to make approximate methods more effective in the
solution of practical problems (Goldstine, 1972, p. 124).

The question at that time was whether or not numerical procedures for solving large-scale
systems could be developed that would produce accurate solutions. In 1943, a heuristic analysis
by the statistician Harold Hotelling indicated that Gaussian elimination was unstable; to achieve
a five- digit accuracy for a 100*100 system approximately 65 digits would be needed! This
caused Von Neumann and his associates, Valentine Bargmann and Deane Montgomery,
to consider iterative procedures in their 1946 report ‘‘Solution of linear systems of high order .’’
But von Neumann and Goldstine, figuring that Gauss was too skilled a ‘‘computer’’ to be
caught in such an accuracy problem, decided to pursue the matter. In their seminal paper
(1947), ‘‘Numerical inverting of matrices of high order,’’ they concluded that Gaussian
elimination was ‘‘very good indeed provided the original problem was not ‘ill-conditioned’;
in other words, the procedure was stable.’’ This paper helped to set the future of
modern numerical analysis. According to Goldstine, he and von Neumann were so involved
with matrix inversion that Mrs. von Neumann named their newly acquired Irish Setter puppy
Inverse.

During and after World War II, von Neumann was involved with the theory and design of
nuclear weapons being developed at the Los Alamos Laboratory. He was a recognized expert in
quantum theory and hydrodynamics, and, most importantly, he had the rare ability to work with
physicists and extract their problems into a mathematical form that could then be subjected to
analysis and calculation. One of his associates at Los Alamos was the physicist Stanislaw Ulam.
They were both involved in the difficult numerical computations of neutron diffusion and nuclear
explosions in general. Ulam traced the birth of the Monte Carlo method to a question that
occurred to him while he was playing solitaire during his convalescence from a brain operation in
January 1946. Ulam’s unpublished account notes:

. . . the question was what are the chances that a Canfield solitaire laid out with 52 cards will come out
successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I
wondered whether a more practical method . . . might not be to lay it out say one hundred times and
simply observe and count the number of successful plays. This was already possible to envisage with the
beginning of the new era of fast computers, and I immediately thought of problems of neutron
diffusion . . . Later . . . [in 1946, I] described the idea to John von Neumann and we began to plan actual
calculations (Eckhardt, 1989, p. 131).

In a letter to Robert Richtmyer (March 11, 1947), theoretical division leader at Los Alamos,
von Neumann wrote about the ‘‘possibility of using statistical methods to solve neutron diffusion
and multiplication problems , in accordance with the principle suggested by Stan Ulam.’’ The
name Monte Carlo was coined by the Los Alamos theoretical physicist Nicholas Metropolis, the
leader of the group that solved the first computer-based Monte Carlo analysis – the simulation of
chain reactions, done on the Aberdeen Proving Ground ENIAC in 1947.

The application of Monte Carlo techniques requires a source of random numbers, either by
physical means (counting radiation hits on a Geiger counter) or by arithmetic processes using a
generating function. For the latter, von Neumann proposed the middle-square procedure in which
the calculations are done by squaring an initial n-digit number (seed) and extracting the middle n
digits for the next number in the sequence. Von Neumann recognized that this would yield a
pseudo-random sequence, at best; the middle-square method is now out of favor as the sequence
can be short, degenerate to a zero, or continuously repeat. Von Neumann cautioned users of
arithmetic schemes with the statement (von Neumann, 1951, p. 36):

Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.
For, as has been pointed out several times, there is no such thing as a random number – there are only
methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.

John von Neumann’s contributions to mathematics, physics, economics, and computers have
helped to define many of the major scientific contributions of the 20th century. His intellect and
talents were wide ranging. As measured by his total output, von Neumann’s contribution to
operations research is just a subset, but what an important subset! Just imagine the scope and
breadth and impact that OR would now enjoy if John von Neumann had provided us with his
wisdom in a more active and direct manner. He would have filled most of the rooms that adjoin
the IFORS Hall of Fame.

Saul I. Gass

Selected original works

Bargmann, V., Montgomery, D., von Neumann, J., 1946. Solution of Linear Systems of High Order. Bureau of
Ordnance, Department of the Navy, Washington, DC.
Fekete, M., von Neumann, J., 1922. Uber die Lage der Nulstellen gewisser Minimumpolynome. Jahresbericht der
deutschen Mathematiker-Vereinigung 31, 128–138.
von Neumann, J., 1928. Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100, 295–320. Translated by
Bargmann, S., 1959. On the theory of games of strategy. In Tucker A. W. and Luce R. D. (eds) Contributions to the
Theory of Games, Vol. IV. Annals of Mathematics Studies 40, Princeton University Press, Princeton, NJ, pp. 13–42.
von Neumann, J., 1946. Principles of large-scale computing machines. Paper delivered by von Neumann on May 15,
1946 at a meeting of the Mathematical Computing Advisory Panel, Office of Research and Inventions Department
of the Navy, Washington, DC, reprinted in Annals of the History of Computing 3 (3), 1981, 262–273.
von Neumann, J., 1951. Various techniques used in connection with random digits. Summarized by George E.
Forsythe. Journal of Research of the National Bureau of Standards, Applied Mathematics Series 3, 36–38.
Von Neumann, J., Goldstine, H.H., 1947. Numerical inverting of matrices of high order. Bulletin of the American
Mathematical Society 53, 1021–1099.
Von Neumann, J., Morgenstern, O., 1944. Theory of Games and Economic Behavior, (1st edition). Princeton University
Press, Princeton, NJ (2nd edition, 1947, 3rd edition, 1953).

Biographical material

Aspray, W., 1990. John von Neumann and the Origins of Modern Computing. The MIT Press, Cambridge, MA.
Dantzig, G.B., 1982. Reminiscences about the origins of linear programming. Operations Research Letters 1 (2), 43–48.
also see Dantzig, G.B., 2002. Linear programming. Operations Research 50, 42–47.
Eckhardt, R., 1989. S. Ulam, J. von Neumann, and the Monte Carlo method. In Cooper, N.G. (ed.). From Cardinals to
Chaos: Reflections on the Life and Legacy of Stanislaw Ulam. Cambridge University Press, New York, pp. 131–137.
Goldstine, H.M., 1972. The Computer from Pascal to von Neumann. Princeton University Press, Princeton, NJ.
Heims, S.J., 1980. John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death.
MIT Press, Cambridge, MA.
Kuhn, H.W., Tucker, A.W., 1958. John von Neumann’s work in the theory of games and mathematical economics.
Bulletin of the American Mathematical Society 64, 100–122.
Metropolis, N., Ulam, S., 1949. The Monte Carlo method. Journal of the American Statistical Association 44B, 335–341.
Poundstone, W., 1992. Prisoner’s Dilemma. Doubleday, New York, NY.
Ulam, S., 1958. John von Neumann. Bulletin of the American Mathematical Society 64, 1–49.

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