But this was not all! Every regular Carva student received a working civil servant’s starting pay as pocket money. This helps answer the question I am often asked by U.S. parents or teachers: “How come twenty-year-old students in France are so much better in math?” Part of the answer: “Because they are, in effect, bribed.”
This pocket money was denied to me. Bless their hearts, the elected class representatives (one of them had met me in Lyon) intervened. They were called caissiers because they were trusted to manage a cash box to which all the students contributed—and to which were added various windfalls. They thought that, to preserve collegiality, I should receive a comparable benefit from the student council budget.
This suggestion was voted on and defeated. Some classmates explained their nay as a matter of high principle (“You signed no obligation and therefore are owed no compensation”); other excuses sounded more like low politics (“I like you, but my friend so-and-so has objections, and I will vote as he did”).
Bless their hearts—again—the caissiers had a higher idea of school solidarity. Their responsibilities included contributing to the neighborhood charities. The rue Mouffetard, next door, was not yet the spruced-up baby Disneyland of today, but an ancient slum. In effect, as few classmates knew, I was handled as a neighborhood charity case and granted that “benefit” anyhow.
How Did My Carva Classmates Fare?
Did all that competition pay off? Not really. Graduation rank actually predicted future performance very poorly. Yet many of my classmates played key roles in rebuilding France after the war. They faced weak competition because our immediate elders had led largely disrupted lives, were not fluent in English, and suffered other handicaps.
Low exit rank guaranteed comfort but not always a grand life—with two notable exceptions. Jean-Claude Simon (1923–2000), a roommate, gave no thought to class ranking—except that he tried to be ranked last, while managing not to flunk out. Having inherited a banking job that he found unbearably boring, he was rich enough to purchase freedom. Almost from scratch, he started a second career in electronics and did well—first in research and then in senior management. He then had a third career as a university professor of computer science. After retiring, he built an imaginative and successful start-up that managed automated signature recognition on checks under a certain amount within installment plans. He was fun, an interesting man, and a good friend.
(Illustration Credit 9.3)
Another classmate, Valéry Giscard d’Estaing, stood out in school by wearing a blue uniform different from our khakis—and later by being elected president of France. I first saw him when he entered the twelve-man casert to which I was assigned in 1945–46. “I am looking for Simon. Do you know where he is?” “Not the foggiest.” “Tell him I stopped by.” When Simon came back, I did tell him, and asked who was this remarkably self-assured man wearing a different uniform. “Oh, you haven’t yet met your classmate Valéry Giscard d’Estaing?” He explained the uniform and continued, “I’ve known him since high school. He kept telling everyone that he will be a député [national representative] by thirty, minister of finance by forty, president of the republic by fifty, and president of Europe by sixty. How stupid can you get?” Everyone present laughed in unison. Of course, my own ambitions may have been even wilder, but involved no schedule—and were not made public.
Amazingly, Giscard’s first three youthful goals were indeed achieved—ahead of schedule. The final goal, becoming president of Europe before sixty, was missed. He remained in the public eye, as author of a European constitution. Put to the vote in France and the Netherlands, it lost. Will his dream ever be reached?
When Giscard was president of France, Jean-Claude Simon had to hand him a report he had edited. The French second-person pronoun has a familiar form, tu, which Simon planned to use. It is the unbreakable rule between Carva classmates and alumni from classes less than seven years apart, and he had known Giscard for years. But at the fateful moment, his mouth disobeyed his brain and uttered, “Monsieur le Président, vous …” He was crestfallen, and remained so each time he retold the story. I did not see Giscard closely again until our school’s 1994 bicentennial in New York. He gave a masterful speech and we chatted, but I minded Simon’s experience and kept away from the minefields.
Professors Leprince-Ringuet and Platrier
Students did not attend Carva for quality teaching, but rather for useful classmates and good jobs. Being a foreign student who didn’t have to cram suited me very well and, if anything, increased my wish to excel. As a result, I received a very fine education in a broad mathematical sciences program, one that straddled the U.S. bachelor’s level of the day—definitely above what I would need at the next stage of life, as a graduate student at Caltech.
My professor of physics, Louis Leprince-Ringuet (1901–2000), was a man of great charm, ambition, and energy. Fully committed to reviving experimental physics in France after its many years at a standstill, he was investigating high energies using the best tool of the day—cosmic rays. The observations were made at the Pic du Midi Observatory, in the Pyrénées near the Spanish border, and analyzed in Paris. Very popular—nicknamed Le Petit Prince after the best seller by Antoine de Saint-Exupéry—he was actively recruiting for his lab. I rushed to join his team as a part-time apprentice.
From my inherited love of gadgets and my training as a toolmaker in wartime Périgueux, I could visualize instantly—in three space dimensions plus time—the complicated instruments that the team was designing. But the rhythm of experimentation was too slow for me, and while my Keplerian plans had not yet coalesced, I was definitely bound to become a theorist of some kind.
The lecture notes of Leprince-Ringuet were uneven. On topics close to his heart, they were up-to-date, but hastily edited. Otherwise, he kept close to the notes of a Carva predecessor who had borrowed right and left. The ways of fate being inscrutable, the mysterious Carva notes made me pay special attention to thermodynamics. Even so, I didn’t get it. So when I went on to Caltech in 1947, this was a course I would not miss (and thermodynamics has inspired much of my research). The Carva course had been just good enough to mystify me—and just bad enough to leave me hungry.
The Chair of Mechanics had once been held by a classmate of Jacques Hadamard, Paul Painlevé (1863–1933). After he lost the creative touch, he went into national politics—serving briefly as prime minister of France in the middle of World War I! Since the incomparable Lazare Carnot (1753–1823), I can’t think of a better example of a scholar-warrior. Incidentally, his son Sadi Carnot (1796–1832) founded thermodynamics.
Painlevé continued to teach whenever he could. When he could not, his stand-in was the little-known Charles Platrier. The course and course notes changed slowly from Painlevé to Platrier, and in many small steps. Painlevé was Wilbur Wright’s first passenger after Orville Wright’s accident—qualifying him as a very early airplane enthusiast. The course notes Platrier prepared for my class were supplemented by many additional readings. One of them was hilarious. It contained Painlevé’s pre-Wright proof that—granted certain “natural” mathematical assumptions—airplanes could not possibly fly! This proof deserves to be republished as a warning to scientists that a theory can be killed by an assumption that looks mathematically “natural” but was not chosen by nature.
Professors Julia and Lévy
Our pure mathematics teachers Gaston Julia and Paul Lévy differed in innumerable ways. When I was their student, the Paris mathematical world respected neither, and these two men and Szolem had no love for one another. This did not matter to me, and they all influenced me profoundly.
The terms “Julia set” and “Lévy process” drew blank stares when I introduced them. Today, fractalists use them every day. I was also first to use the Lévy stable processes in science, and named them Lévy flights. Although some cynics attribute to Julia or Lévy ideas that I originated, I am delighted that this terminology has taken root.
&nb
sp; Those who closely relate to their teachers are expected to fall into a rut, and when the teachers are not fashionable, that rut is bound to be a dead end. But Julia and Lévy differed too much from each other to lead me into a single rut. Plus, all generally valid rules suffer from deviant exceptions, and I went on to prove that a person profoundly rooted in classics may very well be a successful, yet troublemaking, maverick.
Each fall Julia taught differential geometry at Carva, and each spring he was a senior professor at the Sorbonne. One course was intermediate, and the other was advanced. Double-dipping was legal, convenient, and widely practiced. A by-product is that the faculties of different institutions were not as separate as in the United States.
In 1917, Julia published his 199-page Mémoire sur l’itération des fonctions rationnelles. This masterpiece received the Grand Prix from the Académie des Sciences. Its topic—iteration of rational functions led to a parallel investigation by Pierre Fatou and was fashionable for a brief time. But it was filled with special examples and narrowly valid results. Bourbaki thought it was too concrete, and it fell into thirty years of scorn and neglect.
To his credit, Szolem always praised the Julia-Fatou theory, and suggested I pick it up as a Ph.D. topic. I failed to move it an inch. Who could have imagined that, thirty years later, I would revive that field with new questions that fired it with enthusiasm and brought it well-deserved glory.
Nearing sixty, Lévy was still viewed as a brilliant oddball of the first magnitude, but was “molting” into a great man in probability theory, arguably the greatest probabilist of all time. But Lévy’s way of doing probability theory was too intuitive for some and too strange for others. As a result he was a loner, never to be an insider. His self-directed boldness and insight cost him much in his career and early recognition, but I found his independence admirable. I felt ready to pay the same price.
10
Pasadena: Student at Caltech During a Golden Age, 1947–49
IN 1947, THIS WOULD-BE Kepler of complexity had reached another fork in the road. As I had hoped, Carva had granted me two years to think, and the future promised considerable freedom of choice. I learned a great deal, matured, and became very French. But freedom of choice was a negative asset; it set me on a wide sea without sufficient guidance.
I wanted to stay far away from organized physics and mathematics, and to find different, fun ways to apply my growing knowledge and gift for shape. I wanted to feel the excitement of being the first to find a degree of order in some real, concrete, and complex area where everyone else saw a lawless mess. Of bringing to a field the element of rational mathematical structure that Kepler had brought to physics several centuries before. But that Keplerian dream remained stuck in a holding pattern. I was aware that the next step after Carva was going to be hard.
Admiral Brard Recommends Caltech
In the real world of Paris and Carva in 1947, the most obvious person to ask for advice was neither Szolem nor Paul Lévy, but rather the professor of applied mathematics, Roger Brard (1907–77). A naval engineer, he held the rank of admiral and headed a large bassin des carènes—the lovely old-fashioned term for water tunnel. He had no office at Carva, so we met in his car. I still recall the make: Matford. A sign of the times, there were so few cars in town that he always found a parking space near the school.
In the 1930s, when the lovely SS Normandie, touted by Popular Mechanics as the latest “giant of the sea,” took a trial cruise, a resonance was revealed between the hull and the propellers; Brard helped with the diagnosis and the cure. Although his numerous papers in probability theory are no longer quoted, Carva viewed him as very practical (contrary to Paul Lévy) and put him in charge of all topics in applied mathematics.
Ambitious Carva students focused solely on their graduation rank had no need for advisers. But I had a desperate need for someone with broad down-to-earth experience to help me carve a path. Brard was friendly and, to my surprise, made himself available.
With little hesitation, he made two suggestions. First, the right field for me was fluid mechanics. Second, I should go to Caltech—in Pasadena, a suburb of Los Angeles, and study under the illustrious Theodore von Kármán. Kármán was a magician who knew precisely how to find the proper mathematics to deal with great complexity. Kármán worked in aeronautics, but Brard thought that he would be open-minded.
Szolem warned me against Brard’s advice. To do well teaching fluid mechanics in Paris, it was absolutely necessary to find an appropriate and reliable local patron, establish proper credentials, and only then go to Caltech. But I was restless, and none of the possible patrons in Paris claimed the magician’s skills that Brard credited to Kármán.
Father viewed Caltech as an excellent idea. He had already encouraged Léon to go into aeronautics. Only later did his enthusiasm cool when he saw how close the aircraft industry was to the state.
Truth is, Father and I agreed on a plan of action but for very different reasons. I viewed aeronautics not as my final field of work but as the best available path toward reaching my Keplerian dream. So I applied to Caltech with a letter of recommendation from my Carva physics professor, Louis Leprince-Ringuet. I was accepted and spent two years there. For my travel, I got a generous stipend from Carva—arranged by Professor Brard, who had gone far beyond giving advice.
I wondered whether Father remembered that he had wanted to send Szolem to Berlin to study engineering. In both cases, engineering involved the technology that had won the previous war: from the 1920s to the 1940s, it had changed from chemical to aeronautical.
Father could not possibly have heard of the advice received from the fathers of three famous Hungarians: the mathematician John von Neumann, who will play a large role in this story, and his contemporaries, the physicists Edward Teller and Eugene Wigner. Their fathers—far more prosperous and worldly than mine—had also insisted that their sons study chemical engineering. So they did, with profound historical consequences when they worked for the U.S. government during World War II.
Welcome to Los Angeles
In terms of direct preparation for a career, I pretty much wasted my two years at Caltech, though some courses came in handy later. However, my time there gave me the chance to refine my Keplerian dream. I am very lucky to have gone.
The only crossing I could book was from Southampton to New York on the SS Queen Elizabeth. It had recently been converted from a troop transport back to a luxury liner—though my tiny shared cabin on the lowest deck was grim.
I managed a sightseeing stop in London and reached New York dangerously close to the beginning of classes at Caltech. Someone paid for an air ticket, and I flew to Los Angeles. The limousine from Manhattan to the airport stopped next to an opening in a big wire fence. Right on the other side stood a gleaming silver plane, an early model Lockheed Constellation of Trans World Airlines. Its four propeller-driven engines could not reach Los Angeles without a stop halfway at the TWA hub in St. Louis. Tickets were checked by one of several employees idling at the gate, and off we went. This was my first acquaintance with what is now New York’s La Guardia Airport.
My first impressions of California involved smog and the Bible. For days after I arrived, my eyes hurt uncontrollably. I recalled that a man to whom I had been recommended by friends was an ophthalmologist. He lived west of downtown, much too far for social interaction, but I called him for professional help. He gave me an appointment at no cost, but on the way I missed the big red streetcar along the Arroyo Seco. Desperate, I hitchhiked. I was soon picked up by a two-door sedan with a young driver and an older passenger. I sat in the back. Once the car started, the passenger turned to look at me and inquired, “Are you saved?” Wondering whether my ears were also affected by Los Angeles, I failed to respond. The car swerved left and right, and the passenger turned toward me again. “My son is a safe driver, but accidents often happen on the Arroyo. Think again, are you saved?” At that point, the car stopped on the side of the road, and the passen
ger moved to the backseat next to me. He opened his Bible and read a passage. “Does not this story read the same in English as in French? Don’t you agree that this proves the existence of God? Think again, are you saved?” At that point, I got out and they drove down a side road, leaving me stranded. A streetcar finally arrived, but I was very late for my appointment. A message taped on the door informed me that the doctor could not wait. My response informed him that I was very sorry but had been delayed by a preacher.
When I finally saw the ophthalmologist, he greeted me and asked, “Was that preacher any good?” His medical diagnosis was that my eyes were fine except for being overly sensitive to the smog. “What is smog?” “Oh, you had not been told? It is smoke mixed with fog. It’s part of the weather here. Some of your friends at Caltech are working on it. Ask them.” I did.
Another very different surprise met me at the first room I rented. The landlady and her friends spoke German to each other! Their ancestors left for America after Prussia defeated its liberals in 1848.
American Academia in Rapid Transition
When I checked in, the admissions office at Caltech told me, “The yearly tuition is six hundred dollars. We have not mentioned it because you won’t have to pay anything. It has been taken care of by a benefactor of the institute who is interested in international cooperation. He lives close by, in San Marino. Perhaps you should send him a note of thanks.”
Mea maxima culpa: I didn’t. Worse, I forgot the benefactor’s name. As you may recall, not only were tuition, room, and board free at Carva, but students actually received a stipend, even those who bragged of millionaire parents. The very existence of tuition at Caltech seemed abnormal to me. My feeling of guilt did not dissipate until my sons went to college. I could afford the tuition, and I paid it instead of pressing them to seek scholarships. So my debt to Caltech was repaid to Yale and Harvard.
The Fractalist Page 12