Book Review

John von Neumann: Selected Letters Reviewed by George Dyson

John von Neumann: Selected Letters “but develop the worst traits of pedantism and Miklós Rédei, editor inefficiency if I attempt to give a preliminary ac- AMS/LMS History of Mathematics, 2005 count of a subject which I do not have yet in what $59.00, 301 pages I can believe to be its final form.” His hesitancy to ISBN 0-8218-3776-1 discuss unfinished work in print is marvelously absent from his informal correspondence, leaving “If influence of a scientist is interpreted broadly a record both of the development of many of his enough to include impact on fields beyond science well-known results and the germs of many still- proper, then John von Neumann was probably the undeveloped avenues of research. He was fond of most influential mathematician who ever lived,” the postscript and with a simple “P.S.” would often states Miklós Rédei in John von Neumann: Selected Letters, introducing a convincing body of evidence, sketch out some completely new idea. much of it published for the first time. The 121 The present selection is rich in von Neumann’s letters (organized by correspondent) are lightly contributions to many fields of mathematics annotated, and despite the absence of an index (or (“every part of it except number theory and topol- even a chronological listing) this will be an endur- ogy", in the assessment of his colleague Eugene ing reference work. A detailed mathematical and Wigner) and despite inevitable omissions delivers biographical introduction occupies the first forty an impressive sampling of the breadth and influ- pages of the book. ence of von Neumann’s work. The book opens John von Neumann (1903–1957) roamed freely with a brief but eloquent foreword by Peter Lax, among all branches of science and mathematics, followed by a detailed and exceptionally informa- and, in his spare time, helped create several new tive introduction by Marina von Neumann Whit- fields in the gaps where disciplines did not yet man (only child of von Neumann, and an impor- exist. He was a prolific correspondent, delivering tant figure in her own right as an economist and penetrating commentary in handwritten letters advisor to four presidents of the United States). that were often composed in hotel rooms, aboard Rédei then presents a concise introductory survey ships, in airplanes, on trains, and in between of von Neumann’s mathematical contributions, meetings that kept him from ever settling down divided into the following subject areas: logic and to uninterrupted work. On every available scrap of foundations of mathematics, operator algebras, paper, he left a trail of ideas. There is an astonish- unbounded operators, quantum mechanics, quan- ing density of expression, due, in this reviewer’s tum logic, ergodic theory, computer science, and opinion, not only to von Neumann’s intellect but to game theory. the medium of handwriting as a process for distill- Subjects that are less well represented include: ing the essentials into a minimum of words. hydrodynamics (especially turbulence, and the Von Neumann’s correspondence provides a behavior of shock waves), statistics (especially much different picture from his published work. “I the “Monte Carlo” method and its repercussions), write rather freely and fast if a subject is ‘mature’ cellular automata, meteorology, and theoretical in my mind,” he explained to F. B. Silsbee (2 July 1945), apologizing for an undelivered manuscript, biology. Von Neumann’s manifold contributions to nuclear weapons design, weapons effects, George Dyson is a research associate at Western Washing- aerospace technology, military strategy, and the ton University. His email address is [email protected]. area once known as operations research, though

724 Notices of the AMS Volume 54, Number 6 mathematical in nature, are understandably largely Neumann was absent here. working as a The difficulty with von Neumann as a subject c o n s u l t a n t , is that he defied categorization from one minute s u p p o s e d l y to the next. “No matter which way you looked he on questions always seemed to belong somewhere else,” ex- of computer Whitman. Neumann plained Klari, his second wife who survived him modeling for John von Neumann: by six years. “The pure mathematicians claimed oil and gas that he had become a theoretical physicist; the exploration, theoretical physicists looked at him as a great but evidently Selected Letters help and advisor in applied mathematics; the ap- also on other plied mathematician was awed that such a pure things. “This

and ivory-towerish mathematician would show is the letter von Marina of permission with used Photograph Reviewed by George Dyson so much interest in his applied problems and, I that I prom- suspect, in certain government circles they may i s e d y o u , ” have thought of him as an experimental physicist, begins von John von Neumann, December 28, 1903– or even an engineer.” N e u m a n n , February 8, 1957. The unifying theme in von Neumann’s life and “ r e s t a t i n g work—abundantly demonstrated by his correspon- the proposal dence—was an insatiable urge to be absorbing that I made regarding the ‘La Salina Operations information and solving problems all the time. Problem’…that we discussed when we last met in “Johnny’s most characteristic trait was his bound- New York.” Von Neumann then precisely describes less curiosity about everything and anything, his the landscape: “This problem…deals with the op- compulsive ambition to know, to understand any erations of 18 tankers between La Salina and Las problem, no matter on what level,” wrote Klari after Piedras and Aruba. The comings and goings of the his death. “Anything that would tickle his curiosity tankers are described in statistical terms only, i.e., with a question mark, he could not leave alone; they are subject to fluctuations which depend on he would sulk, pout, and be generally impossible fortuitous events like weather, conditions in the ports of call, etc.” until, at least to his own satisfaction, he had found Then he becomes specific, demonstrating that the right answer.” he is not about to ignore the details: “Each one of To John von Neumann, the normal barriers be- the tankers has a separate and characteristic mean tween pure and applied mathematics did not exist. roundtrip time for its assigned run, which may be “A certain contact with the strivings and problems La Salina–Las Piedras–La Salina (for 6 tankers), or of the world that surrounds us is desirable and La Salina–Aruba–La Salina (for 12 tankers). The even necessary,” he wrote to J. Robert Oppen- times for the…actual trip of a specific tanker…may heimer (19 February 1948) defending his right, as be presumed to have a purely statistical distribu- a professor at the Institute for Advanced Study, to tion, to the extent to which it is due to weather be distracted (and remunerated) by outside work. and to similar factors. Under certain conditions, As explained by his collaborator Stanislaw Ulam, the captains or the crews of all ships heading for he had an ability, “perhaps somewhat rare among a certain port may desire to make port at a definite mathematicians, to commune with the physicists, moment. (You mentioned the effects of a good understand their language, and to transform it al- picture or amusement in port.)” most instantly into a mathematician’s schemes and expressions. Then, after following the problems as Suddenly, things don’t look so good: “My im- such, he could translate them back into expres- pression is that this problem…is one of consider- sions in common use among physicists.” able difficulty from the point of view of a strict And not only physicists. Von Neumann was able analytical-mathematical treatment. That is, I think to commune with anyone, translate their problems that it will be very difficult to derive complete (even problems that they did not recognize as formulas for the probabilities and means involved, problems) into the language of mathematics, find and to penetrate to the ultimate mean that is de- a solution, and translate this back into their lan- sired—namely, to the ‘mean turnover’ of the entire guage, bringing the results down to earth. fleet…as a function of the number of berths.” All the letters in this volume are historically or Now comes the light at the end of the tunnel: mathematically important, can be read on multiple “I would, therefore, suggest that the problem be levels, and are technically precise. To identify treated as a ‘statistical experiment’…The proce- favorites is impossible, but I will exercise the re- dure would have to be somewhat like this: Repre- viewer’s privilege by singling out one example that sent each tanker by some suitable form of record, shows the von Neumann mind at work. This letter e.g., by a punch card. Program calculations which was written on April 9, 1953, to T. V. Moore of the will develop the further history of this tanker, al- Standard Oil Development Company, for whom von ways deriving those quantities which depend on

June/July 2007 Notices of the AMS 725 chance…with for software engineers (to Marston Morse, 23 April the use of suit- 1952): “The difficulty is that most people who have able tables of been active in this field seem to believe that it is

Vonneumann. random num- easier to write a new code than to understand an bers…to control old one.” Either one of these insights are worth the the behavior of price of the book. the fictitious On the ever-recurring question—whether math- tankers on the ematicians should offer their services to the mili- punch card.” tary or not—von Neumann’s answer (to Saunders Of course, Mac Lane, 17 May 1948) is as pragmatic as it gets: we now see im- “The Army is a large and nonhomogeneous orga- mediately that nization. I think that we have a chance to do some Photograph used with permission of Nicholas Nicholas of permission with used Photograph von Neumann work which is useful, both to the Army and to the is describing mathematical community. We can do this in one Budapest, 1915, at age 11, with cousin a Monte Carlo particular sector of the Army where the authorities Katalin (Lily) Alcsuti. “She greatly admired, simulation, the have ‘seen the light’. I don’t think that we should but didn’t understand what John was now-ubiquitous be influenced too much by the inadequacies in writing,” explains von Neumanns’s brother method of sta- other sectors.” Nicholas Vonneumann. “He used such tistical approxi- It was generous, and perhaps risky, for the graphics as the letter sigma and so on.” mation that he, Notices to assign this review to someone who is Stan Ulam, Nick not a mathematician but has spent many years Metropolis, and others developed at Los Alamos tracking down von Neumann’s correspondence, for designing bombs. By replacing “neutron” with which could fill a shelf of volumes such as this. “tanker” and “cross section” with “number of This reviewer has assumed the technical accuracy berths”, von Neumann has captured the essence of of Miklós Rédei’s mathematical introduction, and the problem and explains to Standard Oil, in plain resisted the temptation to mention documents not English, exactly what he proposes to do: included in the book. There are two possible approaches to compiling In this scheme, then, each punch card a selection of von Neumann’s letters. One approach represents a tanker and has to be used is to select brief excerpts from as many letters to develop the tanker’s further history as possible. The other approach, taken by Miklós in its dependence on various chance Rédei, is to present a more limited number of let- events and their interaction with each ters in their entirety. This gives, in effect, a Monte other. It is thus possible to trace the Carlo sampling of the mind that produced the let- history of as many days of operation ters, based on the random walk of a limited subset as desired (say, a few years). I would of its unbounded store of ideas. The resulting only like to emphasize that the data statistical approximation is good. This approach handling and computing equipment is less likely to reflect the bias of the editor, and to process these punch cards at an ad- the result is one with which I believe the author of equately high speed exists, and that one the letters would agree. could in this way obtain good statistical averages in quite short times. Similarly delightful insights abound throughout this book, on subjects as diverse as the priority dispute between von Neumann and G. D. Birkhoff

over the ergodic theorem (handled with masterly Whitman. Neumann diplomacy) to von Neumann’s first impression of Alan Turing (“I think that he is quite promising”) and the attempt to win an overdue professorship for Kurt Gödel (“he may easily do more work in mathematics proper. In fact…his probability of doing some is no worse than that of most math- ematicians past 35”). Von Neumann gives advice on computer hard- von Marina of permission with used Photograph ware (to Lewis Strauss, 20 October 1945): “If we devote…several years to experimentation with Von Neumann in the Florida Everglades, such a machine, without a need for immediate January 1938, on his first American road applications, we shall be much better off at the trip with Klari von Neumann, who notes end of that period in every respect, including the that “between 1946 and 1955 we crossed applications.” And six years later he has advice the country twenty-eight times by car.”

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