The Fractalist

by Benoit Mandelbrot

  Our first car was basic: a Citroën 2CV, the fabled Deux Chevaux, which our friend Mark Kac called the Platonic essence of a car. Of the innumerable cars I owned, that alone deserves mention. Rolling down the canvas roof made it into a roadster; it could never be called dirty because at that time it only came in one color: dried mud.

  “2CV” would seem to stand for the puny power of a two-horse team (CV meaning cheval-vapeur, or horsepower). But it doesn’t. As soon as cars appeared, the government imposed an excise tax meant to increase with the engine’s power. But horsepower was subject to argument and fraud. Hence, the law froze the relation between engine displacement and power that had prevailed when a “normal” engine of about 400 cc could produce 2 CV. The regular model had 375 cc, and our “luxury” model had 425 cc—less than most motorcycles.

  André Citroën was a highly educated, very sophisticated, and daring innovator, both in design and advertising. He tamed front-wheel drive for mass production, and his brilliant engineers rethought every part from scratch so that even some key parts could be duplicated, if needed, in a home garage. The result was quirky in the extreme. One day, our car stopped dead in the Alps. I opened the hood and was mystified by a small part covered with grime. Once cleaned, it turned out to be a lever, which I played with just in case—and identified as an auxiliary fuel pump! A few pumping stops got us back home, and I rushed to the garage, which was housed in an old smithy. The proprietor told me that one part of the fuel pump was underdesigned, but there was no need to order and wait for a spare. On the spot, he machined a replacement from a chunk of steel scrap picked from a big barrel.

  19

  In Geneva with Jean Piaget, Mark Kac, and Willy Feller, 1955–57

  MY POSTDOCS AT MIT AND PRINCETON had been carefully laid out, and the research position that followed in Paris was planned to support me while I was waiting for an academic opening. Unlike those periods of the grand tour, the 1955–57 stage of my “career”—in Geneva, Switzerland—was completely unplanned.

  Jean Piaget

  In 1955, the Institute of Statistics of the University of Paris was squeezed into a few rooms in the Institut Henri Poincaré, part of a small campus on what is now the rue Pierre et Marie Curie. One day when I had to stop by on some administrative business, a spry but elderly looking gentleman breezed in and asked the secretary where and when he could find me. Having found me quite easily, he introduced himself as Jean Piaget (1896–1980). He was pleased to hear that I was aware of his fame in trying to bring rationality to child psychology. He had long been a professor in Geneva, and at this point he also taught a day each week in Paris, commuting by overnight sleeper train.

  We sat down to chat, and he described the notion that the nature of knowledge could be inferred from the way knowledge was acquired in early childhood—something he had studied all his life and called genetic epistemology. He had a Rockefeller Foundation grant to establish an interdisciplinary center that he was sure would move at lightning speed—if he had help from a suitable mathematician in residence. He was looking for someone whose work showed open-mindedness, was impressed by my work in linguistics, and wanted me to be that mathematician. I was thunderstruck. Only a few days before, Aliette and I had decided to marry, and we were both keen to live neither too close nor too far from our mothers in Paris but did not know how we would be able to do that. Piaget’s sudden offer provided a timely, surprising, and most elegant solution. Our “negotiation” was brief. Yes, I could be an assistant professor at the university, but most important for him, I should be very active in a weekly get-together of all the participants and a broad symposium at the end of the year, with immediate publication of our results.

  A truly ambitious program but—at my stage in life—a godsend! Geneva was close enough to Paris for me to keep an eye on openings in France. Piaget seemed like an interesting person, and working with social scientists looked challenging—and might help me land a job. I accepted, and Piaget attended my wedding party.

  From a home up in the hills, Piaget biked to office and classroom, downhill or uphill, sunshine or rain. Therefore, his face was weathered, but he was young in spirit and years. His Ph.D.—earned when he was twenty—concerned mountain snails and familiarized him with the scientific practices of zoology. He promptly changed fields and set out on a lifelong effort to extend proper scientific principles to human behavior.

  His first books on children’s intelligence were based on his observations of his own babies and written when he was in his early twenties. Not resting on his laurels, he was always at work on papers, reports, or a book. Early in the school year, he asked me to look at his current book and handed me a chapter. I found it interesting but asked him to explain a few lines in greater detail. Piaget apologized and obliged: in no time obscure lines became obscure whole pages. It soon became clear that—until that moment—he had never heard the words “I do not quite understand. Please explain.”

  Before founding his Center for Genetic Epistemology, Piaget had achieved international fame while leading a completely sheltered and very austere life. He had mostly interacted with either students of education—awed and in a hurry to be certified—or confirmed schoolteachers who would never dare contradict him.

  While Piaget could be vague or wrong, he was not a phony, and I always perceived in him an element of genius. Due to extreme isolation before the 1950s, his scientific talent had never been honed by competition. His ambition was boundless, with no inkling of the deep truth I had learned from John von Neumann: that a scientist shows mettle by identifying problems that are neither too easy nor too difficult. Science is best at giving credit for thinking big, but not too big. I worked hard, but sparked no miracle.

  I admired Piaget’s ambition to become the Kepler of psychology—but not his expectation that, with my help, a year or two would suffice. His center continued for years, and reportedly my successors worked out better than I had.

  Mark Kac

  By extraordinary good chance, my years in Geneva had a plus: close acquaintance with two other visitors who happened to be the most active probabilists on the west side of the Iron Curtain. In 1955–56, Mark Kac (1914–84), a Cornell mathematician, was in residence. He had no built-in group of associates, and there was no obstacle to our becoming close. The 1956–57 visitors were Willy Feller and Joseph Doob (1910–2004), whom I knew less well.

  Mark Kac was quick-witted—always the life of the party. His storytelling skills were well above the typical mathematician’s, and he was tireless in advocating greater harmony between mathematics and science.

  His personal style, likes, and dislikes did not in the least match his dry-as-bones articles. He had been deeply influenced by his teacher and spiritual father, Hugo Steinhaus (1887–1972), a mathematician who had trained in Vienna around 1900, at a time when it was a major intellectual center. His ideal was not too far from what Hadamard had accomplished, Szolem had spurned, and I was hoping to achieve: a harmonious alloy of mathematics and science.

  But life was dangerous in Poland, and the first order of the day was to find a way out. A fellowship to Cornell eluded him in 1937. He was bitterly disappointed, but—as he later gleefully told anybody who cared to listen—fate had been on his side. If successful, he would have returned to Poland in 1938—and likely perished in the war.

  The appointment did go through for 1938, but the letter from Cornell was adamant: under no circumstances could it be renewed. He had hit a wall and was ready for anything. He watched the content and style of the conventional mathematics that were in favor at Cornell. Then, contradicting the openness inherited from Steinhaus, he simply morphed—for the duration—into a follower of fashion. Conversions under duress were very common during the Depression—as I knew from the case of Father.

  Our turbulent childhoods made us react very differently. He gained high respect for order and fear of anarchy. One day when we were chatting after a lecture, another attendee came up and expressed delight at seeing t
wo mavericks together. Smiling as usual, Kac responded: “Benoit is a true maverick, but I am not one in the least. I am a staunch conservative who tries to act intelligently.”

  Years later, he influenced my life by firmly telling me that, instead of more papers that looked unrelated, I must write a book. So I did; my first was in 1975, in French. He reviewed the later English version of 1977 favorably. But in private, he expressed fears that I would open the gates to a flood of nonsense—fears I shared but had to face.

  William Feller

  I first met the mathematician William Feller (1906–70) in Paris; I next saw him in Princeton in 1953–54, then in Geneva in 1956–57, and later—repeatedly—when he consulted at IBM. He deserves a few words here not because he became a role model for me—he definitely did not—but because he was expected to become one.

  Let me begin with a quote from a tribute to Feller by Joseph Doob:

  Those who knew him personally remember Feller best for his gusto, the pleasure with which he met life, and the excitement with which he drew on his endless fund of anecdotes about life and its absurdities, particularly the absurdities involving mathematics and mathematicians. To listen to him lecture was a unique experience, for no one else could lecture with such intense excitement.

  Feller had been a prodigy at the University of Göttingen and earned his Ph.D. at age twenty. Feller’s paternal grandfather was Jewish, so he had to leave Germany. The Depression brought him to Stockholm, under Harald Cramér (1893–1985). Cramér loved pure mathematics but owed his funding to strict Swedish regulation of the insurance industry and had to satisfy his benefactors. So did Feller.

  When in Sweden, and later as a colleague of Mark Kac, Feller became an effective teacher of probability theory, and his marvelous textbook was beloved by many scientists who trusted that mathematics was of genuine use in the sciences. But astonishingly, Feller went out of his way to pooh-pooh this trust. In a published interview, he described as fraudulent the idea that the famed bell curve of mathematical errors ever represents anything real. He even denied it had any role in what is called thermal noise, where it is a pillar of excellent theory and unquestioned practice.

  Probability saved his career and made him rich, but it was never a true love. It was a stopgap until he could return to purer mathematics by leaving Cornell for Princeton—then in a golden age.

  My 1962 pioneering work on the price of cotton and other commodities had been my first Keplerian jackpot. Soon it was pushed away in horror. Its detractors included Feller. When I submitted an early paper on prices, IBM asked him to comment. He flattered me by praising a technical angle, but he proclaimed that what I did had nothing to do with the real world. This made my IBM manager very unhappy, and to save myself, I had to exhibit Feller’s infamous article about the bell curve and thermal noise.

  My work on cotton prices was followed by work on the ebb and flow of the Nile. The brilliant Harold Edwin Hurst (1880–1978) had discovered a relationship that everyone characterized as a deep riddle. Feller credited Hurst in a paper, but immediately proceeded to tackle a related topic that he could handle, one that led to new mathematics, yet was distinctly traditional, while Hurst’s was not.

  After the Hurst-Mandelbrot theory had solved the empirical riddle, I asked Feller to stop by my IBM office during one of his visits. To be frank, I set him up. He began by restating his belief that Hurst’s riddle could be resolved in a way his paper had suggested. I ventured that he did not think this riddle had much bite. He conceded, with a smile. Only at that point did I reveal my solution and its consequences—both theoretical and practical. He got the point and became uncharacteristically subdued. Never did he accept me, but at least he ceased to be an albatross.

  20

  An Underachieving and Restless Maverick Pulls Up Shallow Roots, 1957–58

  THE SUMMER OF 1957 was scheduled to mark the end of the grand tour apprenticeship. My postdoctoral experiences in Cambridge, Princeton, and Geneva had been absolutely crucial to my personal and scientific development. Unfortunately, my various enterprises up to 1957 had not gone very far to further my aging but still vibrant Keplerian dream. The start of the academic year 1957–58 was supposed to be the beginning of “real” working life as a French academic in Lille and Paris.

  Returning to Paris from Geneva in the fall of 1957, I could not help but think back to the fall of 1944, shortly after Paris was liberated. Then, despite my grossly curtailed preparation, I was on the way to shine at the tough entrance exams at the tiny École Normale Supérieure and the École Polytechnique. I was the academic star of the year, and Uncle Szolem—being impressed—was doing his best to recruit me for pure mathematics.

  Marvelous Surprises!

  The academic year 1957–58 represented a development I had completely given up on. I landed a teaching job with ironclad tenure at the University of Lille, plus a lovely moonlighting slot at Carva and other attractive prospects in Paris. When I received my Ph.D. in 1952, French universities had few openings, and I had not made the short list for an academic slot. But in 1956, when enrollments ballooned, teachers were suddenly in great demand.

  Scarcity was such that my nominal adviser, Georges Darmois, remembered I was available. He telephoned to Geneva, asking me to come back and fill a vacancy. I was already committed for another year but for 1957 gladly agreed to become a soon-to-be-tenured junior professor of mathematics. I chose the University of Lille—only two hours north by train from my home in Paris. Also, ten years after graduating from the École Polytechnique, I was invited—practically begged!—to come back “home,” as a junior professor of mathematical analysis, untenured and on short contract.

  So the husband of Aliette, the father of baby Laurent, and the new owner of a very nice apartment in Paris close to the beautiful Parc Monsouris was now also a university professor. Drawing two half-time salaries from the National Treasury was a privileged but fairly common practice. In Lille, my teaching largely fit in two successive midweek days, with only one night at a hotel.

  Teaching at Lille

  Officially, the state felt obligated to provide housing to every civil servant, but all they offered me was a mean worker’s cottage in a distant suburb of Lille. I took one look and decided to fend for myself. Anyhow, Aliette and I were ready to live again in Paris—the usual attractions being enhanced by two grandmothers waiting to provide for our baby son. Therefore, the most desirable provincial university became Lille.

  We lived south of the Latin Quarter. Every conceivable way to the Gare du Nord crossed the old Les Halles area of midtown, but—compared to what it is today—traffic in Paris was a sweet dream and my trusty Citroën 2CV always found free street parking a short walk from the station.

  In other words, I was joining the ranks of the part-time turboprofs, with little social life in Lille. The locals never invited us and criticized us for being absentees and less available to the students.

  Substantial welfare-state perks were paid only once, and one salary was withheld to repay the zero-interest loan for my apartment. Therefore, on top of job security, my financial situation in France was satisfactory. In addition, Darmois was close to retirement. Musical chairs would open a junior position in Paris. Candidates were few, and—amazingly—my chances of being chosen and saved from a commute looked excellent.

  Even this was not all! A marvelous and completely unexpected additional “escape route” soon opened thanks to a celebrated historian. Fernand Braudel (1902–85) was best known for The Mediterranean, a sweeping masterpiece written from memory when he was a war prisoner during World War II. I had read with fascination his description of the 1571 naval battle of Lepanto, where Spain had prevented Turkey from taking over the whole Mediterranean—and beyond. Braudel’s group of historians, the Annales school, wielded considerable academic power and felt, at that point, that the wave of the future was quantitative history. They held an overly enthusiastic interpretation of the Zipf-Mandelbrot law of linguistics and of my effe
ctiveness in Geneva with the psychologist Jean Piaget. So they invited me to set up an ambitious research group in Paris, west of the Luxembourg Gardens.

  So by the fall of 1957, I was a beginning assistant professor at the University of Lille. I was not much noticed by the powerful French pure mathematics establishment, which I had spurned in 1945. A career as a disaffected civil servant with axes to grind would have been pleasant enough. But safety was not my goal, so that very thought made me shudder.

  The Glitter Wears Off and My Plans Change

  The marvelous surprise of the previous fall had worn off fast, and I ended the year 1957 in a very unsettled mood. I saw no compatibility between a university position in France and my still-burning wild ambition and dreams.

  For the experienced survivor I was by 1958, the omens for an intellectually satisfying career in French or U.S. academia looked grim. Not to mention that teaching—even in a university—is a hard profession—one had better start practicing much earlier than I did. Moreover, in May, heavy political clouds burst with the return of Charles de Gaulle to power. No one could predict that he was to feed the French universities and later leave them to the self-destructive devices of the system that went up in flames in the events that shook the general’s rule in the May 1968 riots.

  I escaped with minimal agonizing. That year changed character, and a summer job sent me instead to IBM Research in the United States. My midlife crisis led me to forgo ironclad French tenure for an unknown position in the United States. Would this position last? In many ways, my timing was perfect, and my career bloomed beyond my wildest dreams. Had I not chosen a very risky path? Indeed. In fact, I allowed risk to increase enormously. Instead of joining any existing community of scientists, I went my own way and kept moving into topics that were not part of any field or establishment.