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 interpreter 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 photographic 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 follow 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.
