The Fractalist Read online

  How to match eager candidates and limited-enrollment schools? Before the Revolution, well-paying offices were inherited, granted by sovereign pleasure, or purchased. By contrast, the grandes écoles recruit on merit. Their entrance exams were (and remain) the French counterpart of the cruel-by-design rites of passage practiced among the “savages.” To prepare for them, there were cramming programs like the one I took in Lyon at the Lycée du Parc.

  In the fall of 1944, back in Paris, I transferred as a resident student to the Lycée Louis-le-Grand—the crème de la crème, named by King Louis XIV himself. I sat in the class of Monsieur Pons, hardly ever speaking to him, but cramming by myself. The delayed exams began in December 1944 with a week of written Normale and one of written Carva, and ended in January 1945 with a week of oral Normale and one of oral Carva.

  At Normale, one mathematics test was so long the proctors called a brief break and fortified each candidate with a bowl of hot broth. Later, texts in several different languages were handed out, and we had to translate any two. To English, an obvious choice, I added Latin!

  By design, both exams were extremely difficult, sufficiently so to ensure that, typically, only the top man managed an average above 16/20. Rumor had it that, as of 1945, the all-time record had been set around 1885 by Jacques Hadamard, Grandfather’s senior guest at that 1930 dinner in Warsaw and later my grandfather of the mind.

  Unexpected Triumphs at Normale and Carva

  In January 1945, the week between the written and oral exams, I was racing across the Latin Quarter when my mathematics teacher, M. Pons, hailed me in the street, and we had our first and last private conversation. “Let’s talk about the big math problem at Polytechnique. I could not solve it in the time allowed, but examiners say that—in the whole of France—one student did solve it, and he is from my class. Could it be you?” “Well, I did solve the entire problem—including every optional question at the end.” “How did you manage? No human could resolve that triple integral in the time allowed!” “I saw that it is the volume of the sphere. But you must first change the given coordinates to the strange but intrinsic coordinates I thought the underlying geometry suggested.” “Oh!” And he walked away, repeating, “But of course, of course, of course!”

  When the exam ordeal ended, my grade was 19.75/20. Nobody ever received 20/20—ever! For this and other top mathematics marks, rumor appointed me the best math student in the country that year. Everyone seemed to know of my skimpy formal preparation, so I was credited with a feat that would be remembered for years to come.

  I was supposed to take those exams as practice for a serious second try. But that 19.75/20 was approached by some other very high marks, mostly in the additional math exams. Also, I wrote very good French and reasonable English and had high grades in freehand drawing. Somehow, subpar grades in “lesser” tests did not register, and I was widely believed to be number one. A major moment in my life!

  Plain and simple, not only had I survived the war, but in France I had it made for life. Of course, nothing could guarantee that I would mature into a great scientist—or a great anything. But either school could open every door and provided a kind of automatic lifelong insurance. All this was simply beyond belief. Only nine years since my move to France, only months since the liberation, and still officially residing in that slum of Belleville, I was in no way ready for such choices.

  As I look back, it seems to me that great opportunity in one way was wasted but in another was used in the best possible manner. For thirteen years, my suddenly acquired “capital” was not wisely invested and was ostensibly squandered over a period of slow maturation and wandering. Then I moved to the United States—where French credits had no value. However, there I managed to design a career to match my skills and tastes—one that lost out on all the French benefits earned by those exams but perfectly fit the dream I had conceived during the war.

  Reunion with Uncle Szolem

  One day when I was returning from a Normale exam to my quarters at the Lycée Louis-le-Grand, a man hailed me in the lobby. For a moment, I mistook him for Father. But he was younger and free from Father’s stigmata of perpetual deep worry. Sure enough, it was Szolem—back in Paris from the war years spent at Rice in Houston and then with the military in London.

  He noted I looked fit, rather than starved, and I told him why. We worried about those in Poland but reassured each other that in his own family and mine everybody was alive and well. His wife and son were about to return, and I put him in touch with Father.

  Then we moved on to the exams. “How far along are you now?” “I just took the big written math exam of Normale.” “How far did you get?” “All the way to the last question, and I can’t think of any bad mistake.” “Splendid, congratulations, splendid, splendid. You are a shoo-in. I am so glad. You are lucky. You will go to Normale and experience something marvelous that all my friends went through but I missed.”

  For many days, months, and years to follow, Szolem told me about the inner workings of the worlds of mathematics and science, which he knew well. He spoke often about his mentor, Hadamard, and his contemporaries. He described the colorful André Weil and the completely fictitious but increasingly influential group of Young Turks that Weil conceived, organized, led, and named Nicolas Bourbaki.

  All this was fascinating to me. Much was extremely attractive, but Weil and Bourbaki were positively repellent. Right after the war, I was wary of secret groups and charismatic leaders, and this leader’s taste was extremely far from mine. It will be seen shortly how this affected my life. Combining idealism and practicality, Szolem described very frankly both the system’s greatness and its warts—such as the pervasive patronage and the widespread inbreeding and nepotism facilitated by the fact that, even in mathematics, judgments of value are subjective.

  Family “War Council”

  It is my impression that among my fellow students who did well enough to have a choice between Normale and Polytechnique, very few agonized. Regular schooling identifies sensible ambitions, and my classmates had been preparing over much of their lives. By contrast, I was both underschooled and suddenly overadvised. Only months before, I had been desperately focused on staying alive. Now a marvelous long-term choice became available for me alone to make.

  A detail that became very important: entering Normale, students chose between mathematics or physics but could easily switch; Carva, to the contrary, allowed minimal advance planning, and opportunities were tightly restricted by the rank a person received at graduation.

  The high stakes terrified us all, and my parents did not trust my teachers. So a family “war council” was called to help: Szolem and a second cousin and close friend—the leading physical chemist, Michel Magat—met in February 1945 in our Belleville tenement.

  Uncle and Cousin were brilliant and forceful, politically engaged but unbelievably partisan and naïve, as it soon turned out. They battled against each other and Father for my future and my soul. Exact words are, of course, forgotten, but the message remains clear in my mind.

  UNCLE: Carva transforms bright students into soulless bureaucrats who can’t run anything properly. They won World War I, but lost World War II. Follow the path I took, and add one thing I missed. Go to Normale. No career brings the rewards of pure science. It gives you both freedom and insurance, because the alumni take care of their own. If you are unlucky and discover nothing important—but don’t worry, you will have no problems—you will become a high school teacher. No career comes closer in serving society, and you will be happy and proud of yourself.

  COUSIN: Inescapable social and political forces are about to abolish both schools. Carva is a bastion of obsolete ideas and old ways. They will teach you nothing, only make you feel you belong to the elite. Normale is just as bad. Consider the École Supérieure de Physique et de Chimie. It is supported by a city—not the state—and knows how to train people to become down-to-earth scientists.

  FATHER: Don’t listen to either
of them. Thriving as a scientist is a lottery. Szolem won a jackpot by being smart but also by coming to France at precisely the right time. But France, Europe, and much of the world are a total mess—no one can predict what will happen next. Cousin’s predictions are not serious. Besides, if the Russians help the Communists come to power here, you may be forced to pull up roots once again and move to a new country—Brazil, Argentina … who knows? Since we married, Mother and I were wiped out six times by events over which we had no control. Also, never forget something basic: professors are civil servants. Trouble may leave you somewhere—as it did Mother—with a worthless foreign certification. Keep away from state-certified fields and large national organizations. Education, health, and law are the plague. Go for broad engineering skills that every country will need under every political regime.

  Father, a skilled survivor, deeply admired scholarship and practiced it—but only to the extent that circumstances would allow. He strongly believed that a scholar’s happiness and independence hinged on a steady income largely independent of uncontrollable events. This attitude was forged by the chaos he had experienced. One hears the same advice today all over the media: don’t count on lifetime protection from one employer. Many years before, Father had given that very same advice to his twenty-year-old brother, Szolem.

  Years later, I realized that Father’s thinking had a far broader perspective. He was quite impressed by the work and misfortunes of a Portuguese Jewish philosopher born in Amsterdam, Benedict Spinoza (1632–77). Spinoza’s community shunned him, yet being a skilled lens grinder in tolerant Holland allowed him to think freely. His spiritual power stood in sharp contrast to his political powerlessness. In our family, achieving political power crossed no one’s mind.

  Similar family fights have occurred in the lives of two scientists I came to know. To help the biologist Jacques Monod decide between biology and music, his influential father appointed a committee. It reported that as a biologist he would match Pasteur and as a musician he would match Mozart. He chose biology and won a Nobel Prize.

  More important for me was the great mathematician John von Neumann, to be introduced later. Around 1920, Hungary, his motherland, was under a cloud of uncertainty far worse than Poland in 1920 and France in 1945. His rich father wanted him to play it safe and study chemical engineering, but agreed to hire a young Budapest professor named Michael Fekete to determine whether “Janos” should also be allowed to seek a Ph.D. in mathematics. The advice was that he should do both. He perfected an alloy whose composition is not expected to ever be encountered again.

  Much in my life is easily traced back to that family war council. In effect—in a most fruitful draw—Father and Uncle both won and earned my everlasting gratitude. Their respective influences did not just mix in my life—they simmered slowly under the blows and the heat of successive trials and errors, eventually yielding something quite distinct from each of them—a new alloy.

  A Die Cast One Day Is Retrieved the Next

  At first, Uncle’s academic position and personal authority prevailed, and I registered at Normale in extremely high spirits. I had every right to be proud of myself. I had survived the war, thanks mostly to help and luck, but also to fast thinking on my feet. Then—my acrobatic feat—I took this exam almost cold and came out near the top.

  On my first day at Normale, the deputy director for the sciences talked to me in the threshold of his office. We discussed my formal status as a foreign citizen who had passed the regular ENS exam and hoped to be naturalized. “There is no difficulty whatsoever,” he assured me. “As soon as your naturalization comes through, you will become a regular student. Till then, you will have to pay tuition and board. Your situation is rare but not unique.” One precedent he managed to recall was a philosopher then at the height of his fame, Henri Bergson (1859–1941), to whom—as he observed—“this initial complication did no harm.” We agreed that the precedent was flattering and promising.

  Unfortunately, as the first day went on, a good look around made me feel dreadful. “What am I doing here? This is absolutely the wrong place for me.” I finally faced a reality that Szolem had described to me—a reality I had previously disregarded. The Bourbaki cult was becoming dominant in pure mathematics, and Normale was about to be taken over. It was indeed the absolute worst place for a strong-willed person with already clearly defined tastes. I spent the day agonizing, could not imagine a good reason to stay, and went back home for the night.

  By the next day, I had yielded to Father, and returned to Normale to resign. Léon often reminisced about everybody’s surprise at my sudden change of mind. This key decision to switch schools—although it complicated the second stage of my life as a scientist—proved to be the right one and dominated my whole career.

  The decision was widely misunderstood and criticized, and some potential friends never forgave me. Szolem became upset and afraid, the way any fanatic, scientific purist fears new alloys. Even now, it is insinuated that I did something very wrong.

  The Weather and the Mood of the Day

  Individual decisions are randomly influenced by history in the making. In prosperous and happy times, the influence is very gradual, but not so on that day in the war-weary France of early 1945. The family war council was inevitably affected by the historical “microclimate.” The class of 1944 made choices in the middle of the abominable last winter of the war. How could this fail to matter? Only weeks had passed since an enemy counteroffensive in the Ardennes near Luxembourg created the scary Bastogne Bulge and threatened to push back the war’s end. Physically, Paris was nearly intact—but cold, bleak, and desolate, reeking of poverty and decay.

  Had I been a true believer in French mathematics à la André Weil, none of this would have been noticed. But I was not, and the mood of the day inevitably affected my decision. Sunny weather, good progress in the war, and a buoyant political situation might have made dwelling in the lay monastery of Normale acceptable. I shudder at the thought.

  Intergenerational Conflict Among Immigrants

  Around March 1945, Szolem resumed his chair at the Collège de France. At his first lecture, I was the only young person present, and he kept it at a level I could follow. The attendees proceeded to the cobblestone courtyard, mostly to exchange news of who had or had not survived the war.

  I recall clearly Szolem introducing me around and commenting on my scandalous choice in a tone appropriate for a funeral: “Having entered Normale, this boy has left on his second day and is about to enter Polytechnique.” He could not understand why anyone would look for mathematics different from his or that of Bourbaki.

  Michel Loève (1907–79), a Jew from Alexandria, Egypt, was there and spoke reassuringly: Polytechnique was of course second best, but fine, since I would study under Paul Lévy. That moment was my introduction to a great man and a major figure in the exciting field of probability theory. This encounter with Loève earned my gratitude. In due time, it would combine with other forces to steer my Keplerian dream toward the theory of chance.

  While Szolem had been anything but bland in his twenties, age and success had mellowed him. He was liberal on most issues—except those close to his heart. Since I wouldn’t follow in his footsteps, we had terrible fights. Until I pinned down what exactly I wanted to do, I kept losing—of course. He never understood my aspirations, continued to worry about my very bad taste and its inevitably horrible consequences, and felt to the end of his life that my intellectual gifts had been squandered.

  A major dynamic of our relationship was simply a classic reaction against a powerful father figure—in my case, not Father, but Szolem, twenty-five years older than I. This involved a theme favored in fiction and history: intergenerational conflict among immigrants. In our family, fleeing Poland put Szolem in the first generation; I stood on his shoulders and belonged to a freer second generation.

  Similar consequences result when a law subjecting a population to stringent restrictions is suddenly overturned.
Their natural reaction is to keep complying. Szolem’s youthful fling on the political and literary scene contradicts, but was transient. On the more important scene of mathematics, Szolem fit the first-generation stereotype by acting as a prudent conformist who promptly joined the soon-to-be-powerful Bourbaki.

  To the contrary, I fit to a tee the second-generation stereotype, which today’s France knows best through the children of immigrants from Africa. I never turned to political violence, yet became a nonconformist, a permanent questioner who managed to thrive without either joining an existing school or creating one for the few formal students I had. Therefore, seen from a distance, the path of Szolem’s scientific life seems straight as an arrow, while mine was … unquestionably fractal. But maturity brought out many similarities. It became important that we were both “ideological refugees” from utter abstraction. Sierpiński intellectual and political views made Uncle flee Poland, and Bourbaki made me leave Normale in 1945—and France in 1958.

  Two examples of sweet irony: Szolem loved and faithfully served through his life two topics of truly classical mathematics: the Taylor and the Fourier series. In the twentieth century, both developed into fields self-described as “fine” or “hard” mathematical analysis. They forgot their roots in physics, except for a massive contribution from another man who was to play an important role in my life, Norbert Wiener.

  After Szolem made me learn these topics, I flew away—but never jettisoned what I had learned. In Szolem’s theorems, the list of assumptions could take pages. The distinctions he enjoyed were elusive, and at his preferred level of complexity, no condition was both necessary and sufficient. The issues he tackled had a long pedigree within pure mathematics. This was for him a source of pride, but was for the younger me a source of aversion.