The Consummate Gamesman: John von Neumann

June 15, 2009

Book Review
Philip J. Davis

The World as a Mathematical Game: John von Neumann and Twentieth Century Science. By Giorgio Israel and Ana Millán Gasca, Birkhäuser, Basel, 2009, 207 pages, $119.

Why are there multiple biographies of one person? Abraham Lincoln, to name an extreme example, has probably been the subject of at least 10,000. Reasons appear to be as numerous as biographies. The emergence of new information can be the motivation for a new biography; individual points of view and perceptions of significance can alter with the passage of time. A biographer might place special emphasis on a particular aspect of the subject's life, or might target a certain audience or perhaps speakers of a certain language. Different treatments might be written for specialists and for non-specialists. A biographer might pursue a political or social agenda. Biographies vary in tone: straightforward vs. satiric prose; hagiographic vs. calm, reasoned appraisal.

John von Neumann, while not up there with Lincoln, has been the inspiration for an immense amount of material, primary and secondary. There are the Collected Works, in six volumes; also available are surveys, conference proceedings, analyses, letters, quotes, anecdotes, descriptions of collaborations, legends, evaluations, attributions and misattributions, not to mention an unanalyzed Nachlass, and of course, discussions of "what happened afterward," i.e., his legacy. The Library of Congress has a von Neumann archive. Such a plethora is an encouragement to biographers.

John von Neumann, 1903–1957, born Neumann János Lájos in Budapest, Hungary, was the closest thing to a universal genius that the 20th century can claim. His vita shows contributions to game theory and economics, cellular automata, atomic energy, the theory of quantum mechanics, science and society, operator theory, measure theory, von Neumann algebras, numerical analysis and numerical methods, axiomatics, proof theory, computer design, automata theory, computational strategies, meteorology. All this and more.

Von Neumann was a dominant, iconic, and almost mythic 20th-century scientist, one of the group of remarkable Hungarian scientists whose large numbers constitute the "Hungarian miracle" that was described so well by Reuben Hersh and Vera John-Steiner in the Mathematical Intelligencer (Vol. 15, 1993). A phenomenon of nature, von Neumann might be called in medieval language the stupor et dominus scientiae mundi, the wonder of the scientific world.

Von Neumann was a guru, a consultant, an adviser to those whose fingers were on the buttons of power. Although an inspiration to the scientific world, he was anathema to those who rejected his politically right-wing, hawkish cold war stance that, in the opinion of some, actively promoted the nuclear arms race. This stance was elicited, perhaps, by the long, often tragic and ambiguous history of Hungary, and the insane inhuman brutalities of the Nazi years. Were those who survived and those who did not a matter of mere probabilities?

Von Neumann was one of the foremost enemies of narrow specialization. This is clear from both his publications and his scientific activities, and he states as much explicitly in his frequently anthologized essay "The Mathematician" (1947). And yet, without in any way tarnishing von Neumann's accomplishments, it must be said that his additions to the mathematical corpus are not commensurate with those of earlier geniuses, e.g., Euler or even Riemann. The ponderous, 2000-page CRC Concise Encyclopaedia of Mathematics lists only one item under his name: von Neumann algebra. A better measure (in these days of bean counting) might be the half million or so references you pick up on entering his name in a search engine.

Von Neumann grew up in a highly cultured and intellectual Budapest family, where conversation over frequent family lunches and dinners touched on the important ideas and events of the day. Although he wrote on the relation between mathematics, science, and society, I don't see that he bridged the "two cultures" gap in the sense of C.P. Snow's phrase.

Two colossi bestrode the scientific world of the first half of the 20th century: John von Neumann and Norbert Wiener (1894–1964). Differing in background, experience, temperament, and outlook, differing in their production of theorematic mathematics, they were friendly enemies who promoted different views of mathematical computation and experimentation: the continuous/analog versus the discrete/digital. Wiener was the analog man; von Neumann went for the digital. The digital won the battle; ac-cording to an argument that goes back to ancient times (natura non facit saltum), however, reality cannot be described by the digital alone. As Wiener remarked of von Neumann's theory of games,

"von Neumann's picture of the player as a completely intelligent, completely ruthless person is an abstraction and a perversion of the facts."

Marcus Aurelius, the "Philosopher Emperor," wrote that "all is ephemeral; fame and the famous as well." Measured by name recognition alone, even the greatest fame, with few exceptions, inevitably decays. I wonder what reactions the name "John von Neumann" would elicit among today's graduate students. His brother, Nicholas A. Vonneuman, in the informal, charming, and revealing John von Neumann as Seen by his Brother (Revised Edition, 1992), points out that John was aware to a much larger extent than most people of the evanescence of fame and concurred with the sentiment ex-pressed in Baron Joseph Eötvös's Last Will and Testament:

"If my name survives, then the victory of my ideas, rather than a marble statue, should become my memorial."

In von Neumann's case, his works and ideas have entered mainstream mathematical and scientific theory and practice in many ways, to be joined there by the contributions of others made over a period of 5000 years. When pundits assert that "such and such is not a zero-sum game," we know that von Neumann's concept from The Theory of Games and Economic Behavior is flourishing on Main Street.

In a description of the celebration of the 100th birthday of John von Neumann (SIAM News, May 2003), I wrote that

"There are hundreds of metaphors for life: Life is a vale of tears, a dream, a joke. In As You Like It, Shakespeare says that life is a stage. What was von Neumann's metaphor? From a few indications I've gathered, it was that life is a game. Sensitive to the double-edged sword of knowledge and the idiocy of mankind, von Neumann's main legacy might be the deepening of the ancient dilemma of Prometheus."

The meaninglessness of human events asserted in many quarters denies the lines of the poet Alexander Pope, who wrote that "All nature is but art, unknown to thee; All chance, direction, which thou canst not see." This divergence leads to an argument that rages today, in the year of the 200th anniversary of the birth of Darwin, and the probabilistic view of games developed in von Neumann and Morgenstern's book led me, rightly or wrongly, to the metaphor I selected for von Neumann.

Ana Millán Gasca and Giorgio Israel, in their Il Mondo Come GIOCO MATEMATICO: La vita e le idee di John von Neumann (Italian edition, 1995; Spanish edition, 2001), picked up on this characterization and have given us now a revised and enlarged English-language version with the title The World as a Mathematical Game: John von Neumann and Twentieth Century Science.

Giorgio Israel, a professor of mathematics in the Faculty of Mathematical, Physical and Natural Sciences at the University of Rome "La Sapienza," is a knowledgeable, facile, and prolific writer on all imaginable intellectual topics, as well as a media personality. Coining a phrase, I should call him "Omni-fluent."

Ana Millán Gasca, a historian of science and technology, is an associate professor of the history and epistemology of science at the University L'Aquila; she has written numerous books, a number of them collaborations with Israel. Their names have been familiar to me since 2005, when I reviewed their Biology of Numbers: Correspondence of Vito Volterra on Mathematical Biology in these pages (SIAM News, October 2005).

In view of the numerous major biographies of von Neumann that have appeared, including those of Steve J. Heims, Norman Macrae, and William Aspray, and the innumerable lesser treatments, what special insight have Israel and Gasca brought to our understanding of their subject? Everything mentioned in this review can be found in their book's intense and detailed descriptions of the cultural and scientific background of the times, and of the genesis of ideas that accompany and extend beyond von Neumann's work, as well as in the explications of his work in terms that should be comprehensible to the laity. Israel and Gasca's book is an impressive accomplishment and a valuable contribution to and resource for burgeoning von Neumann-iana.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at [email protected].

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