The Fractalist

by Benoit Mandelbrot

  Except in cases of extraordinary longevity, friendship with an older colleague is generally brief. Thus, it was a rare privilege that my friendship with the physicist László Tisza (1907–2009) lasted far longer than usual. He obliged by being born on 7/7/07 and on the next 7/7/07 provided me with my only chance so far to talk to someone in the process of turning a hundred years old.

  The man was short, slight, retiring, and soft-spoken. Upon meeting him, I was told that he had been a well-known and productive researcher—in fact, had come close to fame by almost explaining a curious phenomenon called superfluidity of very low-temperature helium. However—as was added immediately—serious mistakes in that work had to be corrected by his onetime adviser, a star physicist named Lev Landau (1908–68). In truth, Tisza had made no mistake and deserved the credit he did not receive. Tisza was victimized by Landau, but lived long enough for this to be recognized. Instead of clamoring for full credit, he nominated Landau for a prize for this work.

  Tisza and I interacted intensely for a few years after a symposium on information theory held at MIT in the summer of 1956. The paper I presented there described an axiomatic for statistical thermodynamics that developed from the second half of my Ph.D. thesis. Asked to comment on my advance text, Tisza praised it handsomely and described himself, on this occasion, as being my student! Given the age difference, his words were a rarity—balm on my heart.

  Tisza was a hugely helpful professor. I was delighted to trigger an early celebration of his centennial. A large room was filled, a few people came from far away, and the mood was warm and altogether cheerful. His life had produced little needless sound and fury addressed to outsiders, and much reflection for his friends and his own pleasure. It extended late and added at least one solid brick to the permanent edifice of physics. Many mysteries remain open, but long live diversity. I was very moved.

  Effects of Prosperity on the Sciences

  Why did I find RLE so attractive? Because it was close to Wiener, but mostly because of its ambition to be the kind of place described earlier. Nearly isolated in Paris, I was eager for a more open and varied environment to live in and to help me decide whether to continue in the direction of my Ph.D. thesis or move on.

  I admired this great incubator of imaginative science and engineering and was disappointed that the format I had known did not last. But there were many reasons why it could not. Like my Ph.D. topic, RLE’s seemingly perfect timing had arisen not from brilliant long-range planning but from a postwar period of trust in the benevolent power of science and a buildup of expectations that relied on many outside factors. The spiritual health of RLE greatly depended on the financial health of the communications industry. As technology moved on, the role of incubator passed to computers—and so to different institutions, such as IBM Research.

  Early on, outside pressures imposed a certain degree of breadth and cohesion on university departments to create overlap. But when mathematics and physics became suddenly rich—when a rising tide lifted all boats—they indulged in a sort of “ethnic cleansing” and restricted their scope down to very pure, or core, topics.

  In sharp and most fortunate contrast, communications, and later computers, were atypical. Each chose to interpret their role in very broad fashion. Sadly, RLE’s miraculous mix of old and new academic technology and science is only remembered by a few old men.

  16

  Princeton: John von Neumann’s Last Postdoc, 1953–54

  “I MUST PROTEST! This is the worst lecture I ever heard. Not only do I see no relation to the title, but what we have heard makes absolutely no sense at all!”

  We were in Princeton at the Institute for Advanced Study (IAS), and a luminary named Otto Neugebauer (1899–1990), a mathematician who had made himself a famous historian of Babylonian astronomy, was commenting on a lecture I had just finished.

  I stood frozen with gaping mouth as the physicist J. Robert Oppenheimer, father of the atom bomb, sprung up. “May I respond, Otto? If Dr. Mandelbrot will allow, I would like to make a few comments. The title listed in the announcement of this lecture was tentative and should have been changed. But I had the privilege of hearing about his work. I am impressed, but also fear he may not have given full justice to his striking results. I would like to sketch what I remember.”

  The audience became transfixed, being unexpectedly treated to one of the “Oppie talks” for which he was famous. In a few flawless sentences one could print as they were spoken, he was able to summarize every seminar he attended and made the speaker see—often for the first time fully—what had been accomplished and should have been shared with the audience.

  As he sat down, the mathematician John von Neumann, father of the computer, stood up. “I invited Dr. Mandelbrot to spend the year here, and we have had very interesting conversations. If he allows me, I would like to sketch some points that Oppie did not mention.” The transfixed audience was then treated to a “Johnny talk”—equally compelling, and delivered with a strong Hungarian accent. The meeting went from abysmally low to unforgettably high and concluded in triumph.

  Am I describing a nightmare? No, but I wish I was. Having left MIT, I was spending the year 1953–54 at IAS as the last postdoctoral fellow that von Neumann sponsored. That lecture came about one day during a chat with Oppie on the commuter train.

  John von Neumann

  Many pure mathematicians I knew well—like Szolem or Paul Lévy—were not attuned to other fields. John von Neumann (1903–57) was a man of many trades—all sought after—and a known master of each. He continually stunned the mathematical sciences by zeroing in on problems acknowledged as the most challenging of the day, and with his speed, intellectual flexibility, and unsurpassed power, he arrived at solutions that encountered instant acclaim. He did not seem to consciously search for any single holy grail or Golden Fleece of the mind beyond his readiness to tackle many diverse investigations. From the most abstract foundations of the purest mathematics to strategic advice to U.S. presidents, moved by insatiable curiosity and aided by personal wealth, von Neumann let his fancy run free. As soon as he heard a field had become hot, he made himself an expert with a competitive edge and identified several key issues he could solve.

  Von Neumann had a “normal” childhood. So did other Hungarians in that celebrated age. Cohorts Eugene Wigner (1902–95) and Edward Teller (1908–2003) also achieved high fame in the United States and a substantial—though less flamboyant—level of versatility in combining abstract skills with interest in applications. The glittering culture to which they all belonged vanished after the Hapsburg double monarchy collapsed in 1918 and Hungary lost half of its historical lands. Thus, their development was thoroughly disturbed by an external element. Von Neumann started in the 1920s with a fundamental Ph.D. thesis in logic, specifically, abstract set theory. Next he did two great pieces of work, which I knew well. He first formalized the foundations of quantum mechanics. Before his work, two approaches had been in competition. In appearance, they were very different, but he showed them to yield identical results. Later, he “invented” the theory of games, which he meant to provide a foundation for economics. Then—still very young—he proceeded to other works that made him famous as a pure mathematician.

  By the time I met him, he had long left pure for applied mathematics. Fascinated with weather predictions, he had become convinced that theoretical meteorology would remain primitive until the underlying mathematical equations could be solved numerically. To solve them, he had reinvented himself as an entrepreneur in an entirely untried form of engineering, and closely supervised a team building one of the first electronic computers—from scratch.

  Inherited wealth saved him from ever working in a garret (figuratively or otherwise) or fleeing for his life—though for the hundred days of the Bolshevik dictatorship of Béla Kun, his family prudently left his native Budapest. Concluding that he would never achieve a professorship in Europe, von Neumann moved to professorships in Princeton, long before Hitle
r’s rise to power, first at the university, then at the IAS—the most desirable of all academic institutions. He also became a highly paid consultant.

  In truth, I disdained the nature of his interests and the fact that, while multiple unrelated interests made us fellow throwbacks, he was the precise opposite of a self-motivated solo scientist. As I already mentioned, the “hot” specialties that attracted him were overflowing with skilled competitors, and he was a formidable visiting expert who did not threaten his hosts. At that stage in life, I did not seek competition, but craved variety. He filled me with admiration, awe, and the desire to emulate the sheer vastness of his pursuits. I was also hoping to gain hints about how he managed.

  Von Neumann’s diverse interests continue to thrive separately from one another. The nearest thing to a proper centennial celebration was held in his native Hungary. Von Neumann was lucky that his country of birth—a very small nation—finds continuing solace in the greatness of its sons who went away and achieved fame abroad, therefore absolving (or enjoying) their idiosyncrasies.

  Warren Weaver Saves the Day, More Than Once

  Naturally, I had sent von Neumann a copy of my Ph.D. thesis. He sent word back that I should come see him—any day, even on a Saturday morning. While at MIT, I took a few days off to pay him a visit.

  Very well dressed compared to some other academics, he looked like a banker. We talked and he asked if I could visit for a year. I said that I would love to, but when? It was late May, and I assumed that everything was settled for the next academic year. He responded that the Rockefeller Foundation in New York could easily solve this problem. On the coming Monday, I should see one of the great movers and shakers of scientific policy during World War II, Warren Weaver. Von Neumann would leave a message on my behalf, and everything would be settled in no time.

  On the forty-ninth floor of 49 West Forty-ninth Street, in New York City, the receptionist waved me toward Weaver’s secretary, who waved me into his office. On my way out after a brief but very nice chat, I asked for an application. None was necessary, I was told. Everything had been arranged. No major turn in my entire life proceeded more smoothly.

  Over the years, I saw Weaver every so often. He always bubbled with new projects. At one time, he was committed to helping launch mathematical biology, wanted me to take a lead, and offered substantial funding. But I felt—correctly, it seems—that the field was not yet ripe enough for me to abandon my other activities.

  My last encounter with Weaver, in 1968, was very different from the first, but equally unforgettable. By then, I was working at IBM. I had just started reporting to an individual who made my life difficult. The IBM policy at the time was never to fire anybody, but this new supervisor could easily hound me out by assigning some project that I would simply hate.

  Fearing that the end was coming, I went to see Weaver, who was then at the Sloan Foundation. He revealed that years earlier “Johnny” (then dying of cancer) had asked him to keep an eye on me—he saw that my chosen path was dangerous and I might need help. So Weaver offered me a two-year fellowship as a visiting professor at a university of my choice. He also suggested that this money could find other uses, so I should first try to settle my differences at IBM.

  Observing my surprise at these revelations, Weaver disclosed other significant facts. Von Neumann had long been unhappy at the Institute. Many mathematicians resented him for leaving “real” mathematics for computers. Mathematicians and physicists detested his well-known hawkish military views. In a way, as long as he was a pure scientist among pure scientists, he could impress the “natives.” When he moved on to engineering and politics, the tolerance ended. As I found out, during the year I was at the IAS, he had accepted a position at UCLA—less prestigious than Princeton, but also presumed to be less stressful. He died too soon to find out.

  I was so relieved at Weaver’s offer that I did not question him further. How did my case come up between them? What other untold details of his story were lurking? Ignorance was bliss.

  Back at work, the storm that I had feared soon dissipated, but I am grateful it led me to witness this extraordinary offer of help from beyond the grave. Von Neumann was not exactly a warm person, but (maverick to maverick?) he understood me.

  A Commuter Ride with J. Robert Oppenheimer

  One day, having boarded the train from Princeton to New York, I was quite pleased when J. Robert Oppenheimer sat down next to me. After scanning the newspaper, he turned to me. “Have you not just arrived from MIT? Please tell me about your work.” That work was my Ph.D. thesis. Delighted, I proceeded to sketch it. He got my point instantly, confirming the observation by the physicist Hans Bethe that “Oppie” could often understand an entire problem after he heard a single sentence, and the observation by the physicist Robert Wilson that in his presence, I became more intelligent, more vocal, more intense, more prescient, more poetic myself.

  I had hesitated to insist on the role of thermodynamics in the context of a social science—a topic other physicists tended to scorn. To the contrary, surprised and impressed, he told me, “Everybody tries to apply thermodynamics to social science problems but fails; you have actually achieved something.”

  He was especially thrilled to hear that my story of the Zipf-Mandelbrot law of word frequencies involved the notion of temperature of discourse. This fundamental exponent is usually greater than 1, but in certain special cases is smaller. In the theory of heat analogy, this meant that the temperature could be less than zero! A fact I thought had no counterpart in physics. Oppie interrupted in a very excited tone. “Indeed, it used to have no counterpart, but let me tell you about physicist Norman Ramsey at Harvard. His very recent work involves problems in which a negative temperature is not only unavoidable but very important.”

  Oppie ended by asking for my help. “I have been trying to organize evening lectures for ‘the historians and the ladies’ but find too few suitable speakers. Would you be the first?” I took a deep breath and agreed.

  Ordeal by Fire: The Lecture and a Good Recovery

  For days after Oppie’s secretary fixed a date, I sweated to write a talk totally devoid of formulas and long words I might not enunciate clearly.

  On the day of the lecture, I was in the room ahead of time and—to my horror—watched several Institute giants join the audience. Oppenheimer came in. “You need not come; you have heard everything I have to say!” “Not necessarily, and I want to be present.” Then von Neumann came in. “You need not come; you have heard everything I have to say!” “Perhaps, but the discussion may be interesting. Besides, I am the chairman.”

  I trembled in fear throughout my lecture, watching famous people in the audience fall asleep and then snore. After forty-five minutes of agony, I called it a day.

  Von Neumann stood up. “Any questions or comments?” Two friends commented and questioned me dutifully. As the gruesome experience was about to end, another man stood up. That’s when Otto Neugebauer proceeded with the blast reported in this chapter’s first lines. Everyone was wide awake.

  Through the night that followed, I was profoundly happy, but a question nagged me. I trusted that my worth and pitiful misery contributed to the obvious enjoyment Oppie and Johnny had both found in defending me. But could there be another reason? An answer came shortly after, when The New York Times publicized the gist of the celebrated trial in which von Neumann testified against Oppenheimer. Both wanted to go out on that night, and in very quiet Princeton, mine was the only show.

  The next day, I visited Neugebauer in his office. He was very apologetic. “Please do forgive my outburst.” “To the contrary, I come to thank you. Without your outburst, my two skilled lieutenants would not have been motivated to stand up to defend my work.” The ambience became very pleasant, and he gave a demonstration of his astonishing craft. His research dealt with tablets that used the same cuneiform alphabet for a mixture of two different languages that once coexisted in Mesopotamia: Akkadian, which was Semitic, and S
umerian, of unknown origin. Therefore, each tablet could have either of two meanings, and identifying the proper one was a very difficult task.

  To be so well treated by Oppenheimer was a high compliment. An institution that had Oppie in residence was automatically the center of living theoretical physics, a highly active field at that time. Hence, the competition in physics was ferocious, and the level of the junior members extremely high. Computers were so new that they were not yet rated, and the staff on von Neumann’s project was not even considered academic.

  The Name-droppers’ Nirvana

  I had visited numerous palaces, museums, and state monuments, but IAS was the first place I lived and worked where I was surrounded by elegance and gentility. Carva was a barracks, RLE’s Building 20 prided itself on its decrepitude, and MIT’s corridors suffered traffic jams between classes. By contrast, IAS seemed an oasis of motionless meditation; it even boasted of noiseless light switches, which were new to me.

  One exception occurred every weekday at teatime. Practically everybody attended, except for the likes of Einstein and von Neumann. Clearly one era was passing and a new one was coming in, so between the great men and us low-ranked beginners, there was hardly any “middle.” Most careers—including mine, for a long time—were doomed never to rise higher than this year at IAS. Thus, what should have been a marvelous experience was in many ways no fun at all.

  Throughout the short IAS term, I had great fun, but I also worked diligently on many topics and obtained wide-ranging results. I presented everything I had done at a meeting organized by the Brooklyn Polytechnic Institute. When it came time for publication, logic and concern with a future career should have suggested “retailing” that work through several separate papers. Instead, I wrote a single long and involved “memory dump.” I doubt that anybody ever heard of that meeting’s utterly obscure Proceedings. For example, how long did it take for “normal” research to duplicate my findings? It was years before a formula I self-effacingly called Szilard’s inequality really came out as the McMillan inequality of coding theory. Other formulas took decades to be duplicated. To my delight, a long and tedious calculation carried out in that paper proved its mettle by starring in a far more widely interesting context … in 1995.