Guo, Wen
University of Helsinki
Feb 2016
John von Neumann
Introduction
John von Neumann should be one of the most prolific scientists, mathematician. In
all his life, he produced seventy-five papers. His contribution cover quite board fields,
including mathematics, physics, computing, and economics. No doubt he was genius.
Just try to find some clues of how could he have done major contributions to so many
areas.
Early Age
John von Neumann was born on December 28, 1903, in Budapest, Hungary. His
father was a successful lawyer and banker. His maternal grandfather had four
daughters and gave each daughter a spacious apartment in the house he owned. Each
family in that house employed at least on German and one French governess. That
made the house actually a private education institution for John, his cousins and
siblings.
John received his early education in this extensive big family's premises
until ten years old. He displayed a great deal of intellectual endowments at very early
age, among them, language talents, photographic memory, and rapid mental
1 arithmetic ability. Their family members kept a tradition to discuss recent events,
science, arts, philosophy regularly and esteemed local intellectuals were invited to the
house occasionally. John became very interested in history and science. He devoured
a forty-four volume general history from their family library and was said to remember
it for the rest of his life.
When John was ten, he was admitted to Lutheran Gimnázium (secondary school). At
that time the secondary school was an elite education in Hungary. Lutheran
Gimnázium, as one of the best, maintained a very high standard, it had a broad
curriculum, only admit a selected group of students, hired well educated teachers who
are highly respected. In Lutheran Gimnázium, László Rátz tutored him mathematics.
He also involved Michael Fekete from the University of Budapest to tutor John
mathematics at home. This arrangement lasted 8 years long and result in John's first
publication with Fekete on Deutschen Math Journal. László Rátz was not any
mediocre teacher. He himself had studied in Budapest, Berlin, and Strasbourg and
took a important part in improving mathematics teaching and had awarded in his
devotion. He also tutored Nobel Prize winner Eugene Wigner.
During 1867–1918 Budapest was the fastest-growing city in Europe. Electric railroad,
telephone, radio was introduced. Influx of nearly one million inhabitants bring along
educational and cultural advancements. Especially, Jews' integration into Hungary
society along with their high regard for learning was a key factor in the prosperity of
the state.
1920s in Europe
In 1921, John was enrolled into University of Budapest to study math. However, he
spent most of college time in Berlin, where he had the opportunity to hear the lecture
by Nobel Prize winner Fritz Haber, Albert Einstein. He also had regular contact with
scientists or mathematicians like Denis Babor, Leo Szilard, and Eugene Wigner. He
only attended end period of each term in Budapest to take examination.
In 1923, John started his studying in Chemical engineering at ETH in Zurich under
the influence of his father. Which seems benefit him in his period of computer
2 research. At the same time, he was still doing mathematical researches. He graduated
with a chemical engineering degree from ETH in 1925 and get Ph.D. in mathematics
from the University of Budapest in 1926.
In 1927 He was appointed as a Privatdozent, a university lecturer or teacher in
German-speaking countries, by the University of Berlin. At the age 24, he was the
youngest Privatdozent in the university's history.
Set theory. At this period, David Hilbert, in his famous lecture at the International
Congress of Mathematicians, Paris, 1900, listed 23 unsolved problems, trying to enlist
mathematicians to restore the foundations of mathematics. As a result, number theory
and set theory axiomatization is quite popular among young minds.
Von Neumann works at this period focused on axiomatization. They mainly involved
construction of ordinal numbers from abstract set theory, continuing algebraically
construction of ideal numbers as infinite and continuous domain. Instead of taking
Russell’s approach to restrict the logical foundation to prevent the appearance of
paradoxes, von Neumann apply axiom of foundation and notion of class to prevent
Russell's paradoxes, which later evolved into von Neumann–Bernays–Gödel set theory.
Quantum mechanics.
In 1925, John started to work with Hilbert and Lothar
Nordheim. He generalized Nordheim's mathematical account of the Jordan_Dirac
transformation basing on the extension of Hilbert spaces, gave a mathematically
rigorous foundation to Quantum mechanics. Later, he developed measurement theory
and quantum statistics, where a finitely additive measurement is treated as subsets of
groups.
Von Neumann himself regarded analysis of properties of linear operators and an
algebraical study of rings of operators in infinite-dimensional spaces as one of his
most important contribution to mathematics in addition to his contribution to
quantum mechanics.
After Gödel published his result of famous incompleteness theory, von Neumann
stopped his pursuit of establishing a rigid foundation for mathematics. However, in
later conversations, von Neumann implied that Gödel's discovery should lead to a new
3 approach to the understanding of the role formalism in mathematics instead of nailing
the coffin of the subject. [1]
1930s in Princeton
In an effort to modernize American mathematics, Princeton professor Oswald Veblen
traveled Europe hunting for talent. In the fall of 1929, Von Neumann was offered a
position there.
Before leaving for Princeton, von Neumann went to Budapest and
married Mariette Kövesi.
To von Neumann, the Americans were “sane people, less
formal and traditional than the Europeans and a bit more commercial. But a great
deal more sensible too.”( Andrew :134).
Ergodic theorem. Von Neumann continued his work on operator theory and
quantum mechanics. At the same time produce a paper on ergodic theorem, which
studies dynamic systems with invariant measure. Von Neumann formulated and
proved the mean ergodic theorem for unitary operator in Hilbert space. This was the
last of his three most important contributions in mathematics.
In 1933, Adolf Hitler became Germany’s chancellor. As the political situation
deteriorated for Jews in Berlin, Von Neumann would like to secure a permanent
position in American. Fortunately, Institute for Advanced Study was established in
Princeton and he became one of its lifetime professors at the age of 30.
Game theory, Economics.
Von Neumann expanded his work to economics. He
developed a mathematical mode of equilibrium under a uniformly expanding economy
with unlimited resources and fixed return to scale of production. He extended his
research on minimax theorem of 1928, which was originally under the assumption of
perfect information in zero-sum games, to suitable in imperfect information condition
and involving more than two players in the game. The book co-authored with Oskar
Morgenstern, Theory of Games and Economic Behavior, was published in 1946.
It
covered conditions like monopolies, oligopolies, and free trade, gave detailed
mathematic formalization of the problem in question. It was believed to thrust the
study and application of Game theory. Von Neumann had not taken any active part in
game theory research after 1944. However, US Military explored the theory application
4 both in aerial combat and planning problems extensively. Although results seemed
unsatisfactory, the extensive pooling of the resources makes study in linear
programming, network flow analysis, and inventory control flourished. (Giorgio:134)
1940s Design Computer
Von Neumann became very much involved in defense research after the Pearl Harbor
attack in December 1941. He served as a member of National Defense Research
Council from 1942. He and several scientists worked on detonation waves research.
He mainly focused on means to concentrate and direct physical effect of a detonation
concerning high explosives.
From 1943 he started to work at Aberdeen on
aerodynamics. September 1943, he was enlisted as a consultant to Los Alamos on the
atomic bomb.
In 1943, von Neumann immersed himself in computational
mathematics, in a letter he wrote:
I think that I see clearly that the best course for me at present is to concentrate on
Ordnance work, and the Gas Dynamical matters connected therewith. I think that I have
learned here a good deal of experimental physics, particularly of the Gas Dynamical
variety, and that I shall return a better and impurer man. I have also developed an
obscene interest in computational techniques.
A month later, von Neumann had written his first program for a calculating machine.
And through his contact with John Todd, mathematician and the organizer of the
Admiralty Computing Service, John knew about automate dynamical calculations.
Later on, von Neumann told Todd that he "received in that period a decisive impulse
which determined my interest in computing machines."( William:28)
Manhattan Project. In design of atomic bomb, von Neumann contributed to design
of implosion detonation, which generate a spherical shock wave to compress
fissionable core to a critical state. To get an effective design, implosion simulation was
carried out by an IBM accounting equipment. On each run, machine read a punched
card and printed out a set of intermediate value and then a new card need to be
punched with these values as input for next step. At that time, even the state-of-art
could not fully meet the requirement of the project.
5 EDVAC, IAS mahine. From June 1943, work on ENIAC, the first electronic generalpurpose computer, began in secret at the University of Pennsylvania's Moore School of
Electrical Engineering, under the code name "Project PX". Unfortunately, von
Neumann only heard about it in summer of 1943. Before he got there, principle design
was already over, and the project is on test run. Although von Neumann missed the
opportunity to participate in the design, he did embark on taking part in next project,
later named EDVAC. Unlike its predecessor, EDVAC was binary rather than decimal
and stored program computer. Though it's controversial to assign individual credit on
whole design, von Neumann "has contributed to discussions on the logical controls of
the EDVAC, has proposed certain instruction codes, and has tested these proposed
systems by writing out the coded instructions for specific problems."[5] And he also
discussed the relative merits of the binary, octal, and decimal systems. Of course, the
most important contribution of von Neumann was his "First Draft of a Report on the
EDVAC" in 1945.
It presented the first written description of the stored-program
concept and explained how a stored-program computer processes information. In the
report, von Neumann described abstractly the composing part, its function and their
interactions without delve into the detail of physical characteristics of each part.
EDVAC was delivered in August 1949. At the same period, von Neumann directed the
construction of IAS machine in Princeton. Von Neumann and his colleges divided
problems in the design into two classes: Logical design described in the "Preliminary
Discussion for an Electronic Computing Instrument" report of 1946. The design
continues to be the basis of most computers today; the design of a system for
instructing the machine to carry out specified tasks, addressed in a three-volume
report by Goldstine and von Neumann entitled "Planning and Coding of Problems for
an Electronic Computing Instrument," which appeared in 1947 and 1948. The report
wrote about general principles of coding, described system of flow diagramming, gave a
number of program examples for typical scientific and engineering problems, and
discussed the use of libraries of subroutines. A lengthy introduction discussed the
general principles of coding and flow diagramming and outlined the complexities and
subtleties of programming. It wrote, as computer need to jump through instruction
sequence and instruction could be modified, it was not a static translation. Moreover it
contended program had to be viewed as a logical problem and only a new branch of
formal logics can describe them. [William:68] IAS machine started to work in 1952.
6 Numerical Analysis, Simulation, Random Numbers.
The development of
computer revived the numerical analysis to finding approximate solution in
mathematics. When designing the computer, von Neumann realized accumulation
round-off errors could change the way we find the approximation. For instance,
approximation methods seemed to be more suitable than elimination in solving linear
systems as it is more stable hence demands less precision to be carried through. The
new method needed to be developed to accommodate the intrinsic characteristic of
computers.
Von Neumann also found another way to exploit the computer power.
Normally, partial differential equations governing fluid flow are analytically intractable
and impractical to investigate. Under certain conditions, the complexity of problem
could make it numerical unsolvable. By simulate the fluid with a line of beads
connected by spring in a simple kinetic he approximated the hydrodynamic
phenomena.
Furthermore, Stan Ulam and von Neumann invented Monte Carlo
method. The method is to evaluate a deterministic model using random numbers as
input. Idea behind it is to repeat mental experiments in a system until you get some
ideas of its properties you care about. With computer you can run experiment
repeatedly at nearly no cost. First Monte Carlo calculation is to simulate chain
reaction from some initial states. It soon found its way to solve deterministic problem
and applications including PDEs solution, multiple integrals, and matrix calculation.
Von Neumann also look into the method to generate random value, devised a simple
pseudo-random generator and Neumann's rejection technique to get numbers conform
to distribution other than a rectangular one.
1950s
In 1951–1952, von Neumann became consultant to the CIA, a member of the
General Advisory Committee (GAC) to the Atomic Energy Commission (AEC) and
several additional appointments. In November 1952, Eisenhower was elected president;
Lewis Strauss became chairman of the AEC. He and other big players regarded von
Neumann “as America’s quickest-thinking scientific genius.” Von Neumann started to
have influence over Washington. He favored a high-priority program for the
development of the super-bomb. In March 1955, von Neumann accepted full-time job
as an AEC commissioner. Although von Neumann was an advocator of preventative
7 war against Soviet Union, he advocated neutral within the country. He wrote letters
urging that “the best men should always be chosen for any scientific post or grant,
irrespective of past or even present communist leanings.” (István:158)
Von Neumann decided probabilistic theory of automata, complexity and self
replication in automata, and comparisons between the computer and the brain as his
current research interests. And he hoped automata research would shade light on
human nervous system and helped the design of computer.
In 1955, von Neumann was diagnosed with cancer. In January 1956, President
Eisenhower awarded the Medal of Freedom to him. He died at age 53 on February 8,
1957.
Comments
Eugene Wigner was a keen observer of von Neumann, Wigner writes: “Jancsi was
somewhat retiring. He participated in the pranks of the class, but a bit half-heartedly,
just enough to avoid unpopularity.” Eugene Wigner said von Neumann was the one
from whom he learned most. When he asked von Neumann to explain Von Neumann
would then explain what Wigner wanted to know using knowledge that Wigner already
possessed, even if he had had to skirt around the things that Wigner was not familiar
with in a laborious way. Wigner “learned more mathematics from him than from
anyone else.”
Wigner found “the accuracy of his logic to be the most decisive
character” of von Neumann’s mind.( István:221)
8 References
[1]Stanislaw Ulam. JOHN VON NEUMANN. 1903-1957. in John von Neumann's death on
February 8, 1957
[2]Andrew Szanton. The Recollections Of Eugene P. Wigner. Basic Books, 2003
[3]Giorgio Israel & Ana Millán Gasca. The World as a Mathematical Game. Birkhäuser Verlag AG,
2009
[4]Letter, 21 May 1943, Oswald Veblen Papers, Library of Congress
[5] Automatic High-Speed Computing: A Progress Report on the EDVAC. 30 September 1945, Report
on Contract No. W-670-ORD-4926, Supp. 4, HSRC.Congress
[6]István Hargittai. The Martians of Science: Five Physicists Who Changed the Twentieth Century.
Oxford University Press, 2006
[7]William Aspray. John von Neumann and the Origins of Modern Computing. The MIT Press, 1990
[8]Norman MacRae. John von Neumann: The Scientific Genius Who Pioneered the Modern Computer,
Game Theory, Nuclear Deterrence, and Much More. Amer Mathematical Society,1999
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