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certain, however, that he was instrumental in demolishing misguided proposals that would have resulted in considerable waste of time and funds. It is all but certain that the seed idea that finally worked was his own. At any rate, the ensuing loud dispute with Teller over the priority of the invention brought him wide publicity. (The patent application for the device was jointly submitted by Teller and Ulam.) The Democrats soon saw their advantage in adopting Ulam as a bulwark against the Republicans, who had Teller on their side. He was invited to sit in on important Washington committees and later became a darling of the Kennedy era.

At last some of the glitter of his Polish youth had come back, if not in the form of tangible wealth, at least in the guise of public recognition.

he late forties and fifties were the high point of Stan Ulam's life. His personality thrived. His conversation, always lively, became all the more witty and engaging. The better part of his day was spent telling jokes and funny stories and inventing one interesting mathematical idea after another, like a wheel of fortune that never stopped. The joke was the literary form he most appreciated. He would come up with anecdotes, ideas, and stories on any subject of his acquaintance, however little his competence. He so liked to dominate a conversation that some of his colleagues began to take pains to avoid him. Now he had to win every argument. When he felt he was on the losing side, he would abruptly change the subject, but not before seeing the bottom of the other person's position and summarizing it with irritating accuracy. Considering how fast it all happened, it is remarkable how seldom he misunderstood. Mathematicians felt put down, and Ulam's ways alienated him from the guild. He retaliated by claim

ing not to be a "professional" mathematician and by going into rambling tirades against the myopia of much contemporary mathematics.

The free rein Ulam gave to his fantasy fed on one of his latent weaknesseshis wishful thinking. He became an artist at self-deception. He would go to great lengths to avoid facing the unpleasant realities of daily life. When anyone close to him became ill, he would seize on every straw to pretend that nothing was really wrong. When absolutely forced to face an unpleasant fact, he would drop into a chair and fall into a silent and wide-eyed panic.

His severest critics were those close to him who felt excluded from his private world, who stood outside the mighty fortress of mathematics. His daughter would browbeat him and cut him to pieces at regular intervals, incredulous of her father's achievements. He took her criticisms in silence, and was fond of quoting one of James Thurber's lovely generalizations: "Generals are afraid of their daughters."

Despite the comfort of the Los Alamos Laboratory (in the fifties and sixties Ulam was one of two research advisors to the Director of the laboratory), Stan could find no peace there. Since his return in 1946, he had, unbelievable as it may sound, lived out of a suitcase. He owned beautiful homes in Los Alamos and Boulder, but he thought of himself as permanently on the road. (Significantly, his ashes are now in Montparnasse Cemetery in Paris.) The Scottish Café was gone forever, and he was a

passenger on an imaginary ship, who survived on momentary thrills designed to get him through the day. He surrounded himself with traveling companions who were fun to be with and to talk to. He went to great lengths to avoid being alone. When he was, only the lure of mathematics could draw his mind away from the clamor of his memories.

I will always treasure the image of Stan Ulam sitting in his study in Santa Fe early in the morning, rapt in thought, scribbling formulas in drafts that would probably fill a couple of postage stamps.

he traits of Stan Ulam's personality that became dominant in his later years were laziness, gen

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erosity, considerateness, and most of all, depth of thought.

Those who knew Stan and did not know what to make of him covered up the mixture of envy and resentment they felt toward him by pronouncing him lazy. He was in fact lazy, in the dictionary sense of the word. In the thirties he would take a taxi to Harvard from his apartment in Boston to avoid tackling the petty decisions that a ride on the subway required. In Los Alamos there is a spot on a pathway up the Jémez Mountains that is called Ulam's Landing. It is as far as Stan ever went on a hike before turning back. More often, he would watch the hikers with binoculars from the porch of his house, while sipping gin and tonics and talking to his friends.

Like all words denoting human conditions, laziness, taken by itself, is neutral. It is a catchall that conceals a tension of opposites. Fata ducunt, non trahunt. Ulam turned his laziness into elegance in mathematics and into grand seigneur behavior in his life. He had to give all of his thinking an epigrammatic twist of elegant definitiveness. His failing became an imperious demand to get to

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the heart of things with a minimum of jargon.

He had a number of abrupt conversation stoppers that he used to get rid of bores. One of them was a question designed to stop some long tirade: "What is this compared to E = mc2?" When I first heard it (undoubtedly it was being used to stop me), I thought it a sign of conceit. But I was wrong. He would wake up in the middle of the night and compare his own work, too, to E = mc2, and he developed ulcers from these worries. In truth, his apparent conceit was a way of concealing from others, and most of all from himself, the aging of his brain. On rare occasions he felt overwhelmed by guilt at his inability to concentrate, which he viewed as avoidance of "serious" work. He looked at me, his intense blue-green eyes popping and slightly twitching (they were the eyes of a prophet, like Madame Blavatsky's), his mask about to come down, and asked, "Isn't it true that I am a charlatan?" I proceeded to set his mind to rest by giving him, as a sedative, varied examples of flaming charlatans taken from scientists we both knew (both with and without Nobel Prizes). But soon his gnawing doubts would start all over again. He knew he would remain to the end a Yehudi Menuhin who never practiced.

His generosity was curiously linked to his laziness. A generous action is often impulsive and calls for little foresight. Its opposite requires the careful advance planning that Stan loathed. He fancied himself a grand seigneur of bottomless means, and in matters of money he was apt to practice the art of self-deception. In his penurious years he went to great lengths to conceal his shaky financial condition. He always lived as the spirit moved him, sometimes beyond his means. He carried on his person bundles of fifty and one-hundred dollar bills, partly from a remnant of the refugee mentality, partly

to impress whomever he met during his travels.

He was also too much of a grand seigneur to insist on his priority for the many new ideas he contributed to science. His nonchalance as to the fate and success of his work has unjustly lowered his standing as a scientist. When he saw one of his ideas circulating without credit, he remarked, "Why should they remember me? No one quotes Newton or Einstein in the bibliographies of their papers."

His way of expressing himself lent itself to his being exploited. He would speak in sibylline pronouncements that seemed to make little sense. Those of his listeners who decided to pursue his proposals (and often ended up writing dozens of research papers on them) felt they had spent enough of an effort in figuring out what Stan really meant to reward themselves by claiming full credit.

A seed idea is the last thing we want to acknowledge, all the more so when it originates from a native intelligence seemingly blessed with inexhaustible luck. After we silently appropriate it, we will soon enough figure out a way to obliterate all memory of its source. In a last-ditch effort to salvage our pride, we will also manage to find fault with the person to whom we are indebted. Stan Ulam's weaknesses were all too apparent and made him more vulnerable than most. But the strength of his thinking more than made up for what he lost to the pettiness of others.

Stan once showed me in five minutes the central idea of the theory of

continued fractions and thereby saved me much work. Once I bragged to him about some computations I had done on the speed of convergence in the central limit theorem, and he showed me how to derive the same result by an elegant argument with ordinary square roots.

Stan did his best work in fields where no one dared to tread, where he would be sure of having the first shot, free from all fear of having been anticipated. He used to brag about being lucky. But the source of his luck was his boundless intellectual courage, which let him see an interesting possibility where everyone else could see only a blur.

He refused to write down some of his best ideas. He thought he would find some day the time and the help he needed to work them out. But he was misjudging the time he had left. His best problems will survive only if his students ever write them down.

Two of them have struck me. In the nineteenth century mathematicians could not conceive of a surface unless it was defined by specific equations. After a tortuous period of abstraction, the point-set topologists in this century arrived at the abstract notion of a topological space, which renders in precise terms our intuitive grasp of the notion of extension. Ulam proposed going through a similar process of refinement on Maxwell's equations to arrive at an abstract structure for electromagnetic theory free of algebraic irrelevancies.

The second problem bore on ergodic dynamical systems. Poincaré, and several others after him, taught us that in such a system every state is visited infinitely often, given a sufficiently long time. In practice, however, the recurrence times are so large that one cannot observe successive visits, and the practical import of ergodicity is nil. This paradox became strikingly evident after the Fermi-Pasta-Ulam computer sim

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ulations of coupled nonlinear oscillators. (These were written up in one of Fermi's last papers. It is rumored that Fermi considered this to have been his most important discovery.) In these nonlinear systems the initial state is visited several times before another set of available states is even approached. After observing this phenomenon, Ulam guessed that in some ergodic systems the phase space ought to be measuretheoretically represented by two or more big blobs connected by thin tubes. He wanted to express his guess in terms of ergodic theory. I wish we knew how.

Stan's fascination with physics led him to formulate mathematical thoughts that had a background of physics, but they invariably bore the unmistakable ring of mathematics. (He once started to draft a long paper that was to be titled "Physics for Mathematicians.") One of the most striking is his proposal for the reconstruction of the cgs system (distance, mass, and time) on the basis of a random walk. Another, which Dan Mauldin has recently proved true, is the existence of a limiting energy distribution for systems in which energy is redistributed through particle collisions.

Stan Ulam's best work is a game played in the farthest reaches of abstraction, where the cares of the world cannot intrude: in set theory, in measure theory, and in the foundations of mathematics. He used to refer to his volume of collected papers as a slim volume of poems. It is just that.

As a mathematician, his name is most likely to survive for his two problem books, which will remain bedside books for young mathematicians eager to make their mark by solving at least one of them. He also wanted to be remembered for those of his insights that found substantial practical applications, such as the Monte Carlo method, for which he will share the credit with Metropolis and von Neumann, and the bomb, for

which he will be remembered alongside Teller.

Only in the last years of his life did his thinking take a decisively speculative turn. He always professed to dislike philosophical discussions, and he excoriated ponderous treatises in philosophy. He thought them in bad taste, "Germanic" (one of his words of reprobation). Nonetheless, he had an instinctive grasp of philosophical issues, which he refused to express in words. When forced to take a philosophical stand, he would claim to agree with the naive scientism of H. G. Wells and with the positivism of the Vienna Circle (the reigning philosophy of his time), but in his actual thinking he was closer to the phenomenology of Husserl and Heidegger. His knowledge of philosophy suffered from his habit of scanning without reading. He seldom read a book from top to bottom; more often he would handle it long enough to pick out the main point, sometimes after correcting a few misprints, and then literally toss it away. I once set up a little test of his understanding of existentialism, by way of teasing him. I gave him a collection of poems written by Trakl, the first existential poet in German. Stan read them all and was visibly moved. I will always regret not being able to hold his attention long enough for him to get the basic idea of Husserl's phenomenology. He would have liked it.

Those of us who were close to him at the end of his life (Bednarek, Beyer, Everett, Mauldin, Metropolis, Mycielski, Stein, and I, to name a few) were

drawn to him by a fascination that went beyond the glitter of new ideas of arresting beauty, beyond the trenchant remarks that laid bare the hidden weakness of some well-known theory, beyond the endless repertoire of amusing anecdotes. The fascination of Stan Ulam's personality rested in his supreme self-confidence. His self-confidence was not the complacency of success. It rested on the realization that the outcome of all undertakings, no matter how exalted, will be ultimate failure. From this unshakeable conviction he drew his strength.

This conviction of his, of course, was kept silent. What we heard from him instead were rambling tirades against mathematicians and scientists who took themselves too seriously. He would tear to shreds some of the physics that goes on today, which is nothing but poor man's mathematics, poorly learned and poorly dressed up in a phoney physical language. But his faith in a few men whom he considered great remained unshaken: Einstein, Fermi, Brouwer, President Truman.

Thinking back and recalling the ideas, insights, analogies, nuances of style that I drew from my association with him for twenty-one years, I am at a loss to tell where Ulam ends and where I really begin. Perhaps this is one way he chose to survive.

He could not bear to see unhappiness among his friends, and he went to any lengths to cheer us up when we were down. One day, we were driving towards the Jémez Mountains, along the stretch of straight road that starts right after the last site of the laboratory. I felt depressed, and drove silently, looking straight ahead. I could feel his almost physical discomfort at my unhappiness. He tried telling some funny stories, but they didn't work. After a minute of silence, he deployed another tactic. He

knew I had been interested in finding out just how much physics he really knew, and that I had unsuccessfully tried to quiz him. Now he launched on a description of the Planck distribution (which he knew I didn't know) and its role in statistical mechanics. I turned around, surprised at the thoroughness of his knowledge, and he smiled. But a few minutes later he again fell silent, and the gloom started all over. After a pause that was undoubtedly longer than he could bear, he blurted out: "You are not the best mathematician I have ever met, because von Neumann was a better one. You are not the best Italian I have ever met, because Fermi was a better one. But you are the best psychologist I have ever met." This time I smiled. It was his way of acknowledging our friendship. He knew that I could see through his weaknesses, through his laziness, through his inability to do any prolonged stint of work. He knew that I discounted those weaknesses, and that I saw, beyond them, the best of his person. That he appreciated.

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o other period of civilization has been so dependent on hypocrisy for survival as the belle époque, the Victorian Age. It has bequeathed us a heritage of lies that we are now charged with erasing, like a huge national debt: the image of the hero as the fair-haired boy, and the sharp partition of all people into "good guys" and "bad guys." These false illusions must now make way for biographies in which ambiguity, duplicity, and the tension of opposites are seen as the fundamental forces that drive every person.

The prejudice that the scientist, as a seeker of the truth, is immune from the passions of the world and is capable of doing no wrong, a prejudice propagated for over a century by bigoted biographers, has done harm. One shudders to guess how many talented young

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minds have been discouraged from a career in science by reading such unrealistic portrayals of the scientist as a saint. Moreover the presumption that "good" behavior (as interpreted by the biographer) is a prerequisite for success in science betrays a lack of faith in science. Lastly, one should tell the truth, even when such a truth belies our ideas of how things ought to be.

Stan Ulam was lazy, he talked too much, he was hopelessly self-centered (though not egotistical), he had an overpowering personality. But he bequeathed us a view that bears the imprint of depth and elegance, one that enriches our lives and will enrich the lives of those who come after us. For this he will always be remembered.

Gian-Carlo Rota is a Professor of Applied Mathematics and Philosophy at the Massachusetts Institute of Technology. He has served as a consultant to the Laboratory for over twenty years.

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any people thought of Stan as lazy. In one way of speaking they were right: he never was observed cutting the lawn or washing dishes, and he engaged only extremely rarely in the tedious, straightforwardalbeit demanding-business of actually carrying through long and complicated and step-wise dependent numerical calculations.

But in a more important way of speaking they were wrong. He was never asleep. He was constantly alert to the strange ways in which people applied and immersed themselves in immediate and detailed problems. He did not consider such efforts to be undeserving of credit-indeed he recognized them as essential to any practical outcome-but for the most part they were outside the range of his direct interest and personal style. He thought, and thought constantly, in a qualitative and unconstrained way. In this fashion he made a number of exceedingly

Alamos Science Special Issue 1987

important observations bearing on the weapons program.

He was among the first to realize how valuable the Monte Carlo method would be when electronic computers should make extensive applications of the method feasible. Even before that time, in connection with a particular aspect of the Super Program that was essentially impregnable against analytic efforts, he and C. J. Everett applied the method, by hand, in a highly schematic but still enormously time-demanding manner. Stan was not fired by any desire that the hydrogen bomb should add a new dimension to the already intolerable capabilities of the atomic bomb-indeed he hoped that it might be possible to show that a thermonuclear weapon was impractical. And actually his and Everett's work strongly indicated impracticality for the particular pattern envisaged at that time (1950). In the meantime, others had prepared a much more elaborate treatment of the

same central problem to be handled on electronic computers. The point had been indicated, and was probably right, and further detailed examination was in proper hands. It was time for him to drop the problem.

Stan subsequently turned his attention to the unique conditions-temperatures, pressures, and energy densities that are established in the near neighborhood of a fission explosion, and he asked himself about the possible ways one might apply these to realize unprecedented effects. In discussing these speculations with Edward Teller, a completely new approach to obtaining a thermonuclear explosion came in sight. Very quickly the whole program of the Los Alamos Laboratory was redirected to the problems involved in this new approach. From the theoretical point of view, the new effort immediately required intensive and extensive calculations that could be carried out only on the most capable computing machines then available. This was, of course, done, and Stan followed the results with interestbut again, the conduct of this work was not his style. He turned his own personal attention to more long-standing questions having to do, for example, with random processes, nuclear propulsion, mathematical models for biological processes, patterns of growth, and so on. At the time he retired from the staff of Los Alamos (1967), and to a very large extent since then, the weaponsrelated activities of the Laboratory were directed mainly to realizing modifications, improvements, or refinements of devices of the sort presaged in the Ulam-Teller proposal of 1951. ■

Carson Mark first came to Los Alamos in May 1945 as a Canadian collaborator on the Manhattan Project. In 1946 he became a member of the permanent staff of the laboratory that arose from the wartime project and was head of its Theoretical Division from 1947 until he retired in 1973.

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