Odd Couple

The subjects of this biography, John von Neumann and Norbert Wiener, were two mathematicians of outstanding ability. Their careers had a few features in common: both were infant prodigies; both started in the most abstract fields of pure mathematics, and derived some of their inspiration from David Hilbert in Göttingen. Both later extended their interests to topics that had practical applications, and their work had indeed a major impact on practical matters. Both were of Jewish origin.

Yet their lives, their attitudes, and their personalities were completely different. The contrasts, and the thoughts they induce about the function of scientists, no doubt provided the impetus for Steve Heims’s embarking on this combined operation. It is unfortunate that from the very beginning he can be seen to be anything but neutral. While Wiener is described with great warmth, von Neumann is mentioned mostly in a tone of disapproval.

John von Neumann was Hungarian, the son of a banker who had established himself in Budapest society and had acquired a title of nobility, which the son translated into the German “von.” John graduated in mathematics in Budapest, and in chemical engineering in Zurich. He later worked in Göttingen, where his main interest was the foundations of mathematics. He followed Hilbert in a program to prove that the whole logical system of mathematics was free from contradiction. This was before Kurt Gödel proved that such a demonstration was impossible. He made many other important contributions to pure mathematics. In the mathematical circles of Göttingen, the late Twenties were also a time of great excitement over the new quantum mechanics, and von Neumann saw the challenge in this for a mathematician with a flexible mind. He wrote several important papers and a book about the mathematical foundations of quantum mechanics, and some of his ideas are still fresh today.

He had a deep understanding of the practical physical side of the problems, yet his approach is very much that of the mathematician—some of his arguments turned out to be based on assumptions which to the physicist were not necessary or not reasonable. The impact of his work on the development of quantum mechanics was therefore not as crucial as Heims suggests. By this time, however, he had impressed everyone with whom he came into contact by the brilliance and the phenomenal speed of his reasoning, the width of his knowledge, and his memory.

He had no resemblance to the traditional image of a mathematics professor—he was always immaculate, with expensive tastes, sociable and gregarious, always courteous and reasonable, and willing to listen to an opposing point of view. He seemed cold and lacking in emotions, yet he was propelled by tensions among which ambition was certainly a strong one. Heims repeatedly claims that some of these characteristics derived from his origin as a banker’s son in Budapest, but I have known many a banker’s son with very different tastes and attitudes.

Heims also muses about the fact that there were, in von Neumann’s generation, so many outstanding scientists from upper-middle-class backgrounds in Budapest. At least one other, the Nobel laureate Eugene Wigner, came from the same Gymnasium and the same class as von Neumann. The school, says Heims, cannot be responsible for this phenomenon, because “many schools of this type existed in Europe.” (One might have looked for the influence of one or a few outstanding teachers, rather than the “type” of school.) Instead Heims invokes the feeling of insecurity in Hungarian society as a driving force, quoting von Neumann on the necessity of “producing the unusual or facing extinction.” Such tensions might indeed keep a young man on his toes, and drive him to great effort, but they are hardly likely to produce such outstanding abilities as von Neumann’s or Wigner’s.

In Berlin, where von Neumann spent some time after Göttingen, he wrote his first paper on the theory of games. This was a study of the logical structure of the decisions made by a player in a game such as poker, in which the aim is to make moves that will give the greatest chance of financial gain and the least risk of loss. The rules resulting from this theory depend on considering what the adversary may do, and ensuring that the player will not be left at a disadvantage by the most skillful play of his opponent. This paper and later work by von Neumann, and his book with Oskar Morgenstern, The Theory of Games and Economic Behavior, form the basis of game theory as a branch of mathematics. It is an interesting discipline, in clarifying the logical nature of the problem of choice, but it has not yet led any gambler into making a fortune, because in practice the job of enumerating all the situations that may arise from any strategy of the opponent is prohibitive except in the simplest examples, such as a game of tick-tack-toe on a small field.

In 1933, with the advent of the Nazi regime, von Neumann moved to Princeton, where he became, and remained, a member of the new Institute for Advanced Study.

Norbert Wiener was American, the son of a Polish Jew with a very strong personality who became a professor at Harvard. The father attempted to mold his son’s personality according to his principles. Norbert entered college at the age of eleven, and obtained his PhD in mathematical philosophy at Harvard when he was eighteen. During a post-doctoral year at Cambridge, England, and Göttingen he came under the influence of Bertrand Russell, G.H. Hardy, and Hilbert. While these contacts led him into very abstract fields of mathematics, he soon discovered the pleasure to be derived from applying mathematical tools to problems of practical interest. The first of these applications concerned Brownian motion, the irregular motion of tiny particles of dust immersed in a liquid, caused by the buffeting they receive from the motion of the molecules of the liquid. He found that the concept of Lebesgue integration, a highly esoteric mathematical method, was appropriate for handling this problem. Other abstract ideas that he picked up, such as functional integration—which he pioneered—proved unexpectedly to have very practical applications in statistical mechanics and quantum theory. One suspects that, without foreseeing these precise applications, he knew intuitively that such methods were likely to prove applicable.

Wiener was very much the eccentric professor: absent-minded, unsure of himself in spite of his early recognition, given to moods. Heims reports many anecdotes of his generosity to students, his concern—sometimes perhaps quixotic—with ethical principles, as when he resigned from the National Academy of Sciences in protest against its official powers and conformity with establishment views.

Wiener was not as great a wizard in mathematical reasoning as von Neumann, but he was more inventive. His best-known achievement is the foundation of a branch of science he called “cybernetics,” which describes the functioning of control systems. Whether in the operation of a thermostat that keeps a constant temperature in a house or in a laboratory device, or in the coordination of the eye, brain, and hand of the driver who keeps his car following the road, we continuously meet examples of control mechanisms in which information received about a deviation from the intended object (temperature, or position and direction of the car) is fed back to cause changes which counteract the deviation. This principle of “negative feedback” (described not too clearly by Heims on pages 216-217) is one of the essential ingredients of cybernetics, which explores the effectiveness and stability of such systems. Wiener’s work did much to clarify the foundations of this field, which today has many applications through the growing use of automation. Attempts were also made by Wiener and others to use similar ideas in economics, and in psychology, in order to understand the functioning of the brain.

In some of these more general discussions von Neumann and Wiener, who got to know each other well, were collaborating. Indeed von Neumann’s game theory also had some application to economics and to the problem of the brain. Heims devotes much effort to comparing and confronting the two approaches, but in my view he exaggerates the contradictions between them; they differ less in their fundamental assumptions than in their emphasis on different aspects. Neumann’s approach seems deterministic, in that all possible moves in a game are to be foreseen and listed, whereas Wiener’s treatment is statistical, allowing for unpredictable external influences. Yet Neumann’s method must also allow for the unknown choices of the opponent in the game, and he also realized that in practice it is not usually feasible to have command of all possible situations; he is quoted as having stressed the impossibility of mechanizing decision making.

Another difference is that in the theory of games there is an obvious single criterion: a move is good if it leads to an increase in the expectation of financial gain, and bad otherwise. Heims makes much of this single motive of greed. He does not seem to note that the theory applies equally directly to a situation in which other factors of merit apply, as long as one can define clearly the relative importance to be attached to each. There is a problem when there is no such clear principle to resolve the conflict between, say, moral and economic requirements. The same problem arises in Wiener’s approach. A control mechanism can take account of diverse desiderata, but only if a decision is made about the weight to be given to each.

The lives of these two mathematicians, like those of many others, were profoundly affected by the Second World War, which Heims refers to as the “Watershed.” He is fond of citing negative statements, about things which did not happen, with an air of surprise. Thus he records that von Neumann’s and Wiener’s attitudes toward formal logic, probabilities, time and process, limits and errors, did not undergo any abrupt changes. What did change were their activities. Von Neumann, in particular, became involved as a consultant in many military research projects including the US Army Manhattan District—the atom bomb project. He became a frequent visitor to Los Alamos, where the weapon design work was carried out, and his help with the many intricate mathematical problems arising from it soon became invaluable. He was more than a mathematical trouble-shooter; with his clear mind and quick perception of issues he became increasingly useful on committees and in general administrative problems.

As a result he remained a trusted adviser to many government and private organizations after the war. This included serving on committes of the AEC and later on the commission itself. He had been interested in the early ideas about the hydrogen bomb, pressed by Edward Teller at Los Alamos, and he later favored the speedy completion of the hydrogen bomb program, and participated in its theoretical work. Regarding the political implications of all this, he was a hawk, favoring increases in the development of modern weapons, opposing disarmament, and regarding the Soviet Union as unavoidably the enemy of the United States. He is quoted as advocating preventive nuclear war against the Soviet Union, and while this is hearsay (“a number of…friends and associates”) it is believable.

At Los Alamos he had become interested in the use of mechanical punchcard machines as aids to computing, and soon he was involved in the design of electronic computers, making important contributions to their development. High-speed computing was necessary for the work on the hydrogen bomb, but I do not believe Heims is right in seeing this as the major motivation for von Neumann’s work on computers. He would have been fascinated by the challenge of the computing problem in any case.

His work on nuclear weapons and his views on weapons policy (with which indeed many of his colleagues disagree) are the basis of Heims’s hostility. Heims does not like nuclear weapons (who does?), but surely there is no need to distort the facts relating to them. Historians of World War II will raise their eyebrows at his assertion that “Truman’s deliberate tactic was to prevent Japan’s surrender until the Americans had a chance to drop the atomic bomb”; radio biologists may be surprised by the statement that von Neumann “had increased his own chances of getting cancer by personally attending nuclear weapons tests and by staying at Los Alamos for long periods.”

The reader senses the book’s attitude to von Neumann as early as his school days. After mentioning the hated communist regime of Bela Kun (no doubt an important cause of the strong anticommunist and anti-Soviet emotions of so many Hungarian emigrés), Heims writes that “the available evidence indicates that [von Neumann] was not at all attracted to idealistic leftish regimes [another of the typical negative statements] and that his opinion was asked about how one should deal with his less reliable classmates.” The evidence does not seem to tell us what was the nature of his answer, if any.

Von Neumann was fond of logical reasoning, and this is exaggerated by Heims into the statement that he tried to resolve all problems of life by formal logic. “This confidence in the power of logic, which after all is mechanical, to resolve the problems of human life is merely another form of the optimism and faith in technology that von Neumann inherited from his early years during Hungary’s economic take-off.” The same inclination to see connections between abstract epistemological considerations and practical matters shows up in a statement referring to von Neumann’s theory of measurement in quantum mechanics, where he uses the phrase that the measuring instrument is merely an extension of the observer:

During and after the war, when working with computers, von Neumann regarded these machines as extensions, supplements to his mental powers…. The continuity of this theme makes it possible to speculate that he also regarded nuclear weapons as means of self-aggrandizement; as extensions of himself.

What an impressive non sequitur! But it would take too long to summarize the chain of reasoning that leads to the obscure conclusion that “…von Neumann’s interest in the bomb as an unusual tool he had helped produce reflected a desire for a long life, but focused it on the political scene in a tribal, parochial way.”

Much is read into von Neumann’s cooperation with the military and industrial establishment: “He seemed more like a talented nineteenth-century banker to some Central European court, aspiring to aristocracy, than a mid-twentieth-century American scientist accustomed to egalitarian and democratic values. Perhaps this was the mark that centuries of persecution of European Jewry had left on John von Neumann.” He is blamed for the use of game theory by American strategy analysts, including those in the Rand Corporation. Without following up the many references, one has the impression that such conclusions are exaggerated. One even suspects there may be a confusion in Heims’s mind between the theory of games and war games, the latter being an exercise that has gained in popularity since the availability of computers.

In criticizing the application of game theory to economics, Heims stresses that it rests on the traditional assumption of the scarcity of resources, and quotes Walter Weisskopf, who, he says, shows that this is an attitude characteristic of modern industrial society. However, the words he quotes from Weisskopf actually say, “There is…a sense in which the scarcity principle is universally valid because it is rooted in the conditions under which human beings exist.”

The heading of one chapter “Von Neumann: Only Human, in Spite of Himself,” adds to the flavor of the von Neumann part of the biography. But this chapter contains what is perhaps the best and one of the simplest sentences about von Neumann in the book: “Johnny von Neumann, who knew how to live so fully, did not know how to die.”

Wiener had also been called in to help with military projects during the war, including work on directing anti-aircraft fire, which fitted in with his then developing ideas about control systems. He received the news about nuclear weapons and their use on Japan with horror and shock. His immediate reaction was to refuse from then on to provide information for military purposes. “I do not expect to publish any further work of mine which may do damage in the hands of irresponsible militarists.” In many lectures, articles, and books he stressed the view that scientists carry a responsibility for the way in which their discoveries or inventions are used; he said, in Heims’s paraphrase: “If an inventor regards his government as irresponsible in its military policies, then it is his moral responsibility to withhold his ideas on weaponry from that government.”

Heims clearly approves of Wiener’s attitude, and the parts of the book about him picture him as an attractive, though complicated, human being. This presentation is weakened only by the frequent cross-references to the deplorable attitudes of von Neumann, and by the tendency to seek motivations and connections between the mathematical concepts in Wiener’s work and his attitudes to other matters. Heims makes a strange remark about Wiener’s mathematical theories, “in which he perhaps incorporated his own image.” Much is made of the pain he felt when confronting an unsolved problem, but this experience is surely common in mathematics, indeed in most creative activities. These, however, are minor points in a generally very acceptable presentation of Wiener’s thoughts on general issues and his way of life.

In the epilogue Heims comes out as opposed to science in general. He refers to the “naïve but conventional conviction…that scientific knowledge, confined to the accepted methodology of science, has some kind of absolute character as ‘truth,”‘ and adds, “even as the only kind of truth acceptable.” In this way he runs together a moderate and a provocatively extreme statement, and by quoting refutations of the latter, gives the impression of having disposed of the former. He quotes Einstein as opposing the university training of physicists, and recalls that the early members of the Royal Society in the seventeenth century were largely amateurs, without seeming to be aware of the quite exceptional ability of Einstein to teach himself modern physics, or of the shortage in our days of “gentlemen of leisure” who could command the complexities of modern science, let alone the necessary equipment.

Comments in this spirit are scattered throughout the book. “Rarely does a serious mathematician follow his muse as a freelance artist might, wholly outside the establishment. Mathematics is distinctly useful to the state and to industrial enterprises, and directly or indirectly they supply the mathematician’s salary. Thus aesthetic pleasure and worldly practicality dovetail all too neatly” (my emphasis).

Heims’s closing sentiment is that our civilization let its mode of existence “be determined by science and technology” but was awakened to realize that “what dominated it, after all, was people—play and affections, politics and passions, pleasures and pains.” So science and technology did not dominate after all. May they still plead “Not Guilty”?

Some of the fifteen illustrations are excellent and include characteristic pictures of the subjects, many of them shown in assorted company and in not very interesting poses. One on page 393, has no visible connection with either of the mathematicians. It is labeled “Quantum Theorists.” Assuming it is meant to typify the species, I feel honored, as the photographer, by its selection.

Reading John von Neumann and Norbert Wiener takes time, not only because of its 414 pages of text and 115 pages of notes, but because so much of it, like some of the passages quoted above, expresses such far-fetched “explanations” and imagined connections that one casts about for some more plausible meaning. The convoluted style also holds up the reading. Some sentences have to be read at least twice before one can see what is intended. Typically: “I believe that generally a mathematician’s predilection for mathematics related to empirics and applications expresses an impulse to have an impact on a wider world than that of the mathematical cognoscenti.” Or, “…whereas von Neumann sought to explicate primarily through formal logical structure, Wiener sought an intellectually comprehensive synthesis as a context within which to examine a few concrete topics with mathematical rigor.” A certain insensitivity to the sound of language is exemplified by: “Von Neumann’s and Wiener’s social theories…contained ideas about the nature of human nature, of what a human being is.”

If, in studying his material, the author had picked up a little of von Neumann’s clarity of thought and a little of Wiener’s kindness, this would have been a better book.