The Fractalist

by Benoit Mandelbrot

  I benefited—for life—from meeting IAS graduate student Henry P. McKean, who went on to a brilliant career. His thesis topic was pure mathematical esoterica. Puzzled by some complications and difficulties, I requested and received very useful coaching. The lesson was filed in my memory, and something called the Hausdorff-Besicovitch dimension of the values of a Lévy stable processes became essential to fractal geometry—which led to that dimension becoming well known.

  17

  Paris, 1954–55

  VON NEUMANN WAS LEAVING PRINCETON for Washington, so I could not remain for the usual second year of a postdoc. Nearing thirty, I felt ready for a regular job. In the United States, a quick check yielded nothing I liked. France had no teaching job either, but excellent insurance.

  Supported by the National Center for Scientific Research

  Indeed, a research position granted by the CNRS had prudently been maintained. When I came back from Caltech, being drafted put me on unpaid leave. Later, Philips passed unnoticed, and that leave continued automatically when I was at MIT and Princeton.

  I met the big CNRS boss in person and heard that not only a paying job was waiting, but also a promotion to maître de recherches, the third of four ranks from the bottom up. The CNRS was famously bureaucratic, and my official letter was anything but welcoming, mostly listing all kinds of prohibited activities. To the contrary, other recent Ph.D.’s received all-too-final letters of termination. Incidentally, later beneficiaries of CNRS largesse lived in a less generous system that is now further threatened. Tenure was immediate and ironclad, but promotion was glacial at best, and some remained at the lowest rank until retirement.

  My rank of junior research professor was expected to last until a teaching position became open. To increase my chances, I volunteered to teach pro bono publico. Searching for a format one could not confuse with that of a “real” course, I settled on a misnamed groupe de recherche that resided in my briefcase and consisted of lectures on information theory that I gave and published. In research, I kept running around.

  France happened to be abuzz with great political theater, thanks to its prime minister, Pierre Mendès-France (1907–82). Later, his mathematician son, Michel, told me about the origin of this name—a Portuguese ancestor named Mendes married a young lady named Francia. Fleeing the Inquisition, they moved to Bordeaux, where “Francia” became “France.” His nickname, PMF, was a straight copy of Franklin Delano Roosevelt’s FDR. Mendès had been an unusually young subcabinet minister before the war, then a wartime pilot, and next a minister of de Gaulle in London. Among the many French prime ministers between de Gaulle in 1945 and de Gaulle in 1958, he was rated best and remains most fondly remembered. But he was an incorrigible maverick and could never give full measure of his talents. Indefatigable, he was ridiculed by opponents for keeping a glass and a bottle of milk on his desk. In France? Yes, his constituency in Normandy produced milk, not wine. Besides, he was effective in fighting cheap alcohol.

  Normal conditions would never have allowed him to become premier. But—shortly after a story told earlier, of the Carva contingent marching to honor the Vietnamese leader Ho Chi Min—the situation had become quite abnormal. Unthinking French governments had rushed into a war in Vietnam. It was doing very badly and encountering every problem Mendès had predicted. Ultimately, the fathers of the war—his unforgiving enemies—asked him to clean up their mess. This was the path followed by the old Ottoman Empire: whenever it had to abandon some territory, it just so happened that its foreign minister was not a Turk but a Greek. PMF delivered—helped by smoke and mirrors—then rushed to give up French protectorates in Tunisia and Morocco. At that point he was overthrown, and the political situation resumed the course that soon returned de Gaulle to power.

  Close to home, Mendès took every opportunity to promote the sciences and bemoan their weakened state in France. He was widely heard, which may be why—before Britain or Germany—a wild and uncontrolled enrollment surge hit the French universities and the academic market flipped from puny to wide open. In a short time, this would have a major impact on me.

  Paul Lévy

  Getting to know Paul Lévy was one of my few academic accomplishments in 1954–55. He never had a formal disciple, I never had a formal teacher, and I never thought of becoming his clone or shadow. Yet much of probability theory has long consisted of filling logical gaps in his works, and in a real, though indirect, fashion, he was the teacher of several members of his family, and also mine.

  He documented his life, thoughts, and opinions at length in a book well worth reading because of his lack of any attempt to appear better or worse than he was. The best passages are splendid. In particular, he describes in touching terms both his fear of being a mere survivor of the last century, and his feeling of being a mathematician unlike all the others. This feeling was widely shared. I recall John von Neumann saying in 1954, “I think I understand how every other mathematician operates, but Lévy is like a visitor from a strange planet. His own private methods of arriving at the truth leave me ill at ease.”

  When Lévy died in 1971, I lobbied for a memorial at Polytechnique, but very few people came. However, the centennial in 1986 was a different story. By then, Lévy’s mistakes and idiosyncrasies were forgotten and forgiven, and a large meeting was organized by pure mathematicians. (A Polytechnique building came to be called Lévy.) Late in the process, I was invited, discreetly informed of strong opposition to my participation, and advised to avoid the shrillest opponents. Sadly, I wondered whether Lévy himself would have been invited and—if so—would have felt comfortable. I did not.

  Lévy was the least flashy person on earth, so how to explain the profound influence his work and manner had on me and on many other scientists? Herein lies a familiar and always surprising story concerning the very nature of probability theory.

  One half of the story is part of the mystery the great mathematical physicist Eugene Wigner called the unreasonable effectiveness of mathematics in the natural sciences. A symmetric mystery should never be forgotten: the unreasonable effectiveness of the sciences in mathematics. Together these mysteries acknowledge that human thinking is unified within itself (and even with feeling), not in a trendy New Age fashion but very fundamentally.

  Georg Cantor claimed that the essence of mathematics lies in its freedom. But mathematicians do not pick problems from thin air for the pleasure of solving them. To the contrary, a mark of greatness resides in the ability to identify the most interesting problem in the framework of what is already known. And the highest level of the label “interesting” is invariably accompanied by a restrictive label, such as “in mathematics” or “in physics.” My admiration for Lévy’s “mathematical taste” increases each time his mark is revealed on yet another tool I need when tackling a problem in science that he could not conceivably have had in mind.

  What a contrast with the period around 1960! Then Lévy stability was viewed as a specialized and uninteresting concept. It received at most a page in textbooks, with the exception of one by Boris Gnedenko and Andrei Kolmogorov. The English translation expresses the hope that Lévy stable limits will also receive diverse applications in time … in, say, the field of statistical physics. But no actual application was either described or referenced—until my work.

  Lévy’s minicourses—I attended several—have marked my whole life. Not a charismatic lecturer, he looked frail and withdrawn. The auditors were few, and I recall (wrongly, I hope) having often been alone. I also watched Lévy closely at the weekly seminar on probability. One speaker began by describing a problem on the blackboard, then faced Lévy squarely and invited him to guess the answer. The guess was correct. But how reliably could Lévy proceed beyond guesses? A book by Kiyosi Ito and Henry P. McKean is pointedly dedicated to Lévy, whose work has been our spur and admiration. It includes this comment: The difficult point of this proof is the jump between [two equations on that page]; although the meaning is clear, the complete justifica
tion escapes us.

  Andrei Kolmogorov

  A giant of my teachers’ generation, the polymath Andrei Nikolaevich Kolmogorov (1903–87) lived in Moscow. Had it been possible, he would have joined Wiener and von Neumann in influencing my intellectual growth directly, but the Iron Curtain was then an insurmountable barrier. Like Lévy, he was celebrated for work in pure mathematics. He also thought about many aspects of the real world, including the structure of Russian poetry. In the 1930s, he obtained results in genetics that became textbook material. But he antagonized the notorious Trofim Lysenko, a quack favored by Stalin who destroyed genetics in Russia, and fell into disfavor. He reemerged with a pathbreaking paper on turbulence that we studied at Caltech and that was to have a direct influence on my research.

  To everybody’s delight and surprise, political maneuvers allowed Kolmogorov to spend the spring of 1958 in Paris. At a packed and unforgettable colloquium, he outlined the results of the work of two of his students, who both went on to great fame: Vladimir Arnold and Yakov Sinai. Arnold’s results added tangibly to a major issue I would contribute to over the years—the distinction between objects of different dimensions. The first square-filling curve, demonstrated in 1890 by Giuseppe Peano (1858–1932), showed that a continuous motion can visit every point in a square. “Intuition” claimed that one- and two-dimensional objects did not mix; hence, in 1890, a plane-filling curve was called monstrous. That scandal lasted until fractal geometry transformed that monster into an intuitive tool. Arnold’s results revealed to us in 1958 by Kolmogorov showed that every continuous function of a point in the plane can be expressed by combining ten functions of a point on the line. There must be a catch somewhere! Yes! Those ten functions have to be special fractals—long before the word was coined.

  Kolmogorov had coauthored a textbook featuring an obscure mathematical object universally regarded as a mere toy, one that I later called Lévy stable probability distributions. The only real-world application in the literature was quite isolated and did not lend itself to development. But I was going to change that toy into an essential tool in economics. However, I was concerned about a sentence in that textbook. So I went to see Kolmogorov. My results obviously surprised him, and he praised them warmly. Then I asked for references to the precursors claimed in that textbook. He changed the subject. My suspicion that those references never existed was confirmed.

  Wiener’s nonobvious motivations are described in his memoir. John von Neumann seemed to seek the hottest topics of the day. What about Kolmogorov’s motivations? A 1962 talk he gave in Marseille on turbulence was raw, and he never followed it with a piece true to his high standards. So when Russians close to Kolmogorov came west, I inquired about the motivations of that paper on turbulence. Again, I received no answer.

  I still think the issue is important from the viewpoint of the unity of mathematics and continue to hope that a well-informed and bolder soul will educate us. I would also welcome the story of how an orphan from a small ethnic enclave of Russia rose to such a level of admiration and respect.

  18

  Wooing and Marrying Aliette, 1955

  CHRONOLOGICALLY AFTER MOTHER, the most important woman in my life has been Aliette. Of course. We met in October 1950, shortly after my release from the air force. We did not rush to join our lives, marrying after five years of acquaintance. So, no matter how you count, our golden anniversary has passed. Being ill defined is a feature common to all important concepts.

  Following a custom on its way out, my parents had not married until Father had settled down as a reasonable provider. My own slowness in settling down was acutely on my mind, but as I watched friends and my kid brother marry, it was obvious that I should rather wait.

  My activities in 1954–55 left me plenty of time to woo Aliette, the second cousin of my Caltech classmate Leon Trilling. He had a traveling fellowship for 1950–51, decided to use it in Paris, and asked me to find an apartment he could rent. This I did, and was invited to the housewarming, where I met members of his family who had survived the war in France. This included his first cousin and her daughter, Aliette Kagan. Aliette was eighteen, a recent high school graduate registered as a student of law. She eventually changed to biology. I also met her brothers. Years later, the older received the Wolf Prize for chemistry.

  The Trilling family had been prominent in Białystok, a Polish city halfway from Warsaw to Wilno. Family legend claims that Czar Peter the Great (1672–1725) brought their ancestor to Russia when he invited diverse experts from Holland to Westernize his empire. Therefore, they were merchants of the First Guild and could live or travel where they wanted in the empire. Typical of Russian upper classes, their first language was French. By 1939, Aliette’s branch had moved to France, where everyone adjusted well.

  I had confessed to my future wife that I had a very demanding mistress I did not intend to abandon. She did not mind. That mistress was—and is—science. Throughout, my wife has been extraordinarily supportive. Without her willingness to let me gamble my life—and hers and our children’s—the odd career I undertook would have been unthinkable.

  When we were getting to know each other, our evenings out were almost entirely musical. Soon after we married, we attended the most unforgettable opera performance of our lives: Mozart’s Don Giovanni in the Monte Carlo Opera House, a miniature of the Paris Opera designed by the same architect, Charles Garnier. The marriage of Prince Rainier of Monaco to a famous actress called for the ne plus ultra of gala performances, which filled the opera’s first rows with recognizable faces and fanciful evening dresses. We happened to be driving by and—on a wild whim—stopped to inquire. We were told that if we promised to keep quiet and invisible, we could be seated in the last row. The seats were cheap enough, and that last row was the thirteenth—close enough. The greatest singers of the day and a small but top-notch orchestra—nirvana!

  Honeymoon at the Divine La Boverie

  Aliette and I could not forget Geneva. Visiting in search of a place to rent, I had noticed a beautifully written newspaper ad, a poem in praise of a house in what seemed to be a distant suburb. I telephoned to ask for directions to Satigny and was instructed to just wait in front of the railroad station and watch for a car one could not miss, an Alfa Romeo 1800. In no time, a man with a small boy arrived. He introduced himself as Marbot, and the boy as his grandson, and took me to his estate.

  The divine La Boverie consisted of a huge park with a manor house subdivided into apartments, a “farm,” and a plain little apartment for rent above the garage. The estate dated to the eighteenth century—the name suggests that at one time oxen were kept there—and I was soon told that, to make the design last for eternity, the views were framed by sequoia trees, then a recent novelty in Europe. The same architect also featured sequoias in the big Parc de la Grange in Geneva.

  Here are pictures of us in the Alps in April 1955, shortly after becoming engaged and on November 5, 1955, at our wedding celebration.

  (Illustration Credit 18.1)

  A vineyard covered the sunny hill across the road. Together with the wheat fields, it had belonged to the bishops of Geneva, later overthrown by Calvin. Both lands had been said to be under the bishops’ mandement, an ecclesiastic term with the same root as “command.” Therefore, to this day, these lands continue to be called pays du Mandement. The high-sounding wine brand Perle du Mandement was plonk mostly used to dilute the far better Swiss wines from cantons farther east.

  A deed was signed and—to show my fiancée that she was going to live in a grand place—Marbot gave me an aerial photo taken by an enterprising pilot. The view was as divine from the air as from the ground. Compared to the small apartment we would have had to settle for in Paris or downtown Geneva, this was a winner. Not like my parents’ apartment, with windows on an alley, or a house like in Piranesi’s Carceri.

  We spent a two-year honeymoon there, and it is where Aliette brought Laurent, our older son, home from the maternity. It evokes a flood of
memories. An immense lawn had to be rented to a farmer to plant wheat. That field hid a proper cherry orchard with fruit ranging from small and tart to big and plummy—making dessert into a glorious many-course banquet.

  Our apartment had a well-framed open view across the Rhône, with the Mont Salève to the south in full glory; in rare clear weather, we even saw the Aiguille du Midi in the high Alps near Mont Blanc. From the manor house, the view to the west went straight through Bellegarde, where the Rhône cuts across the Jura Mountains.

  (Illustration Credit 18.2)

  La Boverie, in the cool and discreet style of Calvinist Geneva, spoiled us for life. Each new shelter we considered had—in its own way—to match our first. Our future house in New York instantly attracted us with its oak tree reminiscent of “our oak” in Geneva. And later, when our younger son, Didier, insisted on a dog, we found for him a brindled boxer—dark brown with a black mouth. Magnificent thoroughbreds demand proper names, and we chose Bruno Boccanegra de la Boverie.