The Fractalist Read online

  Writing this memoir earlier might have made my professional life a bit easier. But the delay has been fruitful. It has rubbed out some less important details, and my life’s course has become clearer, even to me.

  In this memoir, exact quotes are in italics. Conversations I recall vividly are in roman type between quotation marks. There are no footnotes and few references.

  1

  Roots: Of Flesh and the Mind

  IN PEACEFUL AND PROSPEROUS COUNTRIES, the children of landowners, bakers, or bankers have an easy option: to follow the family traditions and trade. But I was born in Poland and my family came from Lithuania—neither peaceful nor prosperous. As observed by a writer native to that part of Europe, Woe to the poet born in an interesting piece of geography in a violent time.

  My ancestors’ main inheritable property consisted of well-worn books. Indeed, over many generations, the family tradition has been to forsake greed and to worship works of the mind.

  To be a scientist, a thinker, an inventor, was viewed as a higher calling, as something near divine. A scientist or creative mind was described as “immense.” For the young people in our household and my friends it was an incredible, extraordinary privilege to be allowed to think and devote one’s life to science. Money counted for little and one did not [seek] a path to riches or a career. Not at all! To the contrary, one wanted to sacrifice oneself for science.

  The author of these words was my uncle Szolem (1899–1983). In very different ways, he and I both made this sacrifice. He became a well-known mainstream mathematician. His words may sound naïve, even corny. They strike me as describing an extraordinary alloy of Jewish and Russian traditions, in many ways a high point in man’s readiness to face insoluble issues. His world would not have known the meaning of the word “corny” and that motivated many forms of heroism … and of destruction.

  On gifted youngsters, this environment bestowed absolutely no feeling of entitlement and offered no encouragement by flattery. Not only did it not provide shelter from the tragic reality of life, but it imposed a heavy burden: to shine or at least to try to become scholars of some sort—but not without leaving time for a family and fun.

  How did I react? In one word, I obliged. But contrary to Uncle and influenced by World War II in France, I took a path that nobody I knew of had previously attempted.

  Momentous Dinner Party

  Too often, an event’s importance is not recognized until it is too late for proper recording. Also, many plays or retellings of history contrive an early scene that recalls the past and brings the main actors together. In my life, such a scene actually happened and was properly recorded in June 1930, in the family home where I was born on November 20, 1924.

  This extraordinary event brought several of the most important people in my life together at the same table. Through those present, mathematical ideas rooted in the late nineteenth century would have a greater and more direct influence on my life and work than would a twentieth-century invention, the computer.

  The stage was the dining room of our family apartment, at Ulica Muranowska 14, in the Warsaw ghetto. Across from a pocket-size park, it overlooked the shell of a large building that was abandoned during construction and was probably still not finished when World War II flattened that whole area, that whole world.

  A professional photographer hired for the occasion produced an instant family heirloom, admired and commented upon for as long as I can remember. It documented much of my family history and the fact that I grew up in what may be called a house of mathematics.

  Every person at that dinner deeply affected either my blood or my spirit. At different times, they have been examples to follow, inflexible spurs, or a stern panel of judges. Being a maverick weakly rooted in his present, I have also found them a lasting source of comfort. Let me introduce these actors, then return to the main ones at greater leisure.

  (Illustration Credit 1.1)

  The hostess and only woman in this photograph was Aunt Helena Loterman. Of Father’s four sisters (including two dentists), she was the third and the only one willing and able to keep a home. Hence, in a community in which the women were formally educated before the men—and were expected to work outside the home if they could—Helena remained a contented full-time childless housewife. She was also the unflappable caregiver for the eighty-year-old white-bearded patriarch in this photo, her father and my grandfather, Szlomo, who died five years after this event.

  Grandfather’s only language was Yiddish—my only language was Polish—so we could not communicate. But his influence on our family was profound. He was born in a sizable old city in the Russian Empire that a cruel history had endowed with many alternative names. He called it Vilna. Poles write it “Wilno” and pronounce it “Vilno.” Reclaiming an older name, Vilnius, it is again the capital of independent Lithuania, today a small Baltic state, but once a powerful grand duchy that extended to the Black Sea and became linked with Poland. Napoléon Bonaparte, on his ill-fated journey to conquer Moscow in 1812, called it the Jerusalem of the North. My ancestors on both sides had lived there for five centuries, practicing an intellectual variant of Judaism, somewhat Calvinist as opposed to the Baptist-like Hasidic variants that arose farther south in Ukraine. Economic opportunity brought Szlomo to booming Warsaw, where Father was born.

  The few families sharing our name—under variant spellings—may or may not be related. But it is indeed a proper Ashkenazi construction, and John Hersey gave it to a hero in his book on the World War II Warsaw ghetto.

  Reportedly, all of Father’s male ancestors belonged to the caste of priests and were men of great learning, some even famous within Jewry. Following tradition, each made sure that his preferred daughter married his preferred disciple; that is how Grandfather’s teacher became my great-grandfather.

  As proclaimed by their distinctive styles of dress, a sharp break had occurred between the generations. Grandfather and other elderly relatives belonged to a ghetto where religion was paramount. Their children belonged to an altogether different world, where religion mattered far less. We never felt rich, but Grandfather’s household seemed comfortable and sometimes had one or even two peasant servants. How did he manage before his many children could support him? I never could help wondering. Ostensibly, by purchasing yeast wholesale and retailing it to regular customers. Szolem, his youngest son, made the deliveries when he was a boy. But there was more. That community strove to support its learned men. Grandfather had been a respected and beloved adviser. For the more prosperous men praying with him, commercial dealings were a nice cover to make him comfortable and put him in a position to hold court—ensuring access to his valued advice.

  Leaning on the back of Grandfather’s chair is my cousin Leon (circa 1900–70), then an editor at the most important Polish-language Jewish daily. He kept us all in touch with what was really happening. He was to escape the war in Poland by being deported to eastern Siberia, then moved back west again. We met several times after the war. His wife, Maria Bar, was a leading pianist; I saw posters of her concerts but never heard her play. Leon’s brother, Zygmunt, was a schoolteacher and well-known poet.

  The thoughtful forty-seven-year-old man facing the camera second from the left is Father (1883–1952), second of Grandfather’s four sons, a deeply principled and fiercely independent man and a major figure in my life. Two older siblings married young and joined their spouses’ families far from Warsaw—but not Father. Selfless as a son, brother, husband, and father, he held on until very late in life before he finally complained in front of me that outside of his family he had done nothing he enjoyed, while his youngest brother, Szolem, had never done anything but.

  Szolem is the thirty-one-year-old man at the far left of the photograph. Grandmother was fifty at the time of her sixteenth pregnancy; she never quite recovered and died before World War I. Intellectually and financially, her last child was largely raised by Father.

  At that momentous dinner, Szolem was both a host
and an honored guest, and probably the main interpreter. As a child, he had lived at that very same apartment, and he was the first in Father’s family to attend an academic—rather than a religious or trade—high school and a university other than medical. He moved to Paris, where he had been rapidly and thoroughly accepted, and was visiting his birthplace in triumph. He was one of four professors on the way to represent France at a big event to be held in June 1930 in Kharkov, in eastern Ukraine, namely the First Congress of Mathematics of the Union of Soviet Socialist Republics. No one would influence my scientific life as much as Szolem.

  At the place of honor to Grandfather’s right sits the most senior guest, Jacques Hadamard (1865–1963). As far back as I can remember, I heard him described as a great scientist, arguably the greatest mathematician in France at that time and one widely believed to deserve much broader recognition than he now has. In different ways, he also turned out to have an enormous influence on my life in very specific and important areas. I think of him as my grandfather of the mind.

  To Grandfather’s left sits mathematician Paul Montel (1876–1975), who only a few years before had supervised Szolem’s doctoral dissertation. I interacted occasionally with Montel, after a time when he had become rather remote. Szolem praised the work of two mathematicians directly inspired by Montel, Gaston Julia (1893–1978) and Pierre Fatou (1878–1929). Around 1980, I had the privilege of joining Montel’s scientific progeny by discovering the Mandelbrot set, the work that made our family name known far and wide. It revived sober words and formulas of Montel’s school in the 1910s and transformed them into both active scientific research and continuing infatuation among the young. It even created a popular fad.

  The mathematician Arnaud Denjoy (1884–1974), who sits in the center, only affected one corner of my work, not an important one.

  Finally, next to Aunt Helena stands her husband, Loterman, handsome, sweet, and cultured. As my private tutor, he was to help mold a very peculiar “formal” education. In Warsaw at that stage of the Depression, regular employment was a privilege that neither Loterman, nor Mother’s younger brother, nor many other relatives ever seemed to enjoy. But strong family solidarity somehow saved them all from menial jobs. The Lotermans vanished in the Holocaust.

  Father

  One guest at that momentous dinner demands a closer look. Father was sixteen when Szolem was born, attending a trade high school, where he learned to be a bookkeeper. He was a lifelong self-improver and an extremely widely read, clear-minded, and scholarly person. He was fascinated with how machines worked and very handy with tools—in wartime, he taught me many tricks.

  (Illustration Credit 1.2)

  A man is known by his heroes. Next to the lens grinder and philosopher Baruch Spinoza, Father held special admiration for Charles Proteus Steinmetz, a cripple who became a prolific inventor and an advocate for competent and honest government in Schenectady, New York, where he worked. Father’s creativity in mathematics was never tested, but he had a phenomenal practical gift for numbers: when adding two-foot-long columns, his sharp pencil just flew down and up and he never made a mistake.

  An episode during the German occupation of France illustrates his intelligence, independence, and daring. At one point, when he was imprisoned at a collection point on the way to the death camps, a Resistance group broke in and overcame the guards. They shouted that they could only open the gates but not defend the camp, told everyone to run, and disappeared. Father found himself in a long line of prisoners walking toward the nearest city. He sensed trouble, switched to a side road, and watched in horror as a Stuka plane of the Waffen SS—alerted by the guards—strafed the prisoners. Walking home, he kept to back roads, sleeping in abandoned shacks along the way. Other war survivors describe being in a herd on the way to the death camps, noticing a way out, and taking it instantly. That is the kind of man Father was.

  He was a reluctant businessman forced by fate into professions related to clothing—the “needle” or “rags” trade. This activity brought him no fulfillment and played a small role in my life. He did not train me in “business wiles.” Never an “organization man,” he always managed to be on his own, or at worst with one partner.

  Among my earliest recollections are visits to his wholesale business in ladies’ hosiery: stockings, tricots, and gloves. His shop was located at Ulica Nalewki 18, a major shopping street in the Jewish quarter, on the ground floor in the back of what seemed to me a big courtyard. World War II turned Warsaw to rubble, and Nalewki was rebuilt as a short paved street along a park, where no one lives—a mere shadow of its former teeming self. Recently, a well-wisher mailed me a copy of an old trade directory showing that Nalewki 18 housed a high proportion of similar businesses—proof that Father had chosen the best possible location in Warsaw. His business was one of the few to be “registered” (whatever that meant), to have a telephone, and to be listed in boldface letters. He had done well.

  The door from the street to the courtyard was always “guarded” by beggars. Father’s regular suppliers and buyers often had to stay overnight with us because, although Warsaw had palaces and flophouses, it had no affordable business hotels. Father’s business moved on confidence and credit, and both collapsed a year after that momentous dinner. I recall vividly a visitor who wondered what happened. Mother brought in and opened a big suitcase filled with copies of invoices: “None have been paid because everybody is bankrupt; that is what happened.”

  Unbent, Father went to Paris in 1931 to seek a better life and brought his family to join him in 1936. Having escaped Poland, he also attempted to escape the needle trade for activities closer to his personality and ambitions, including trying to be a freelance inventor. One of his gadgets, which he called Géographie amusante “Terra,” even received a patent. However, Paris too was affected by the Depression—though less horribly than Poland and the United States—and he could not make a living that way. Soon he was forced to be realistic and became the junior partner at a tiny manufacturer of cheap children’s clothing.

  After the war, he found a job as an accountant for the U.S. Army. Mother argued that, having passed sixty, he should play it safe, give up independence, and take a quiet salaried job, at least “until our situation settles down.” As she aged, this became a favorite phrase of hers—though I never tired of reminding her that their “situation” had remained unsettled since 1914.

  Instead, Father started yet another new business, an amazing feat that circumstances made even more difficult. It was accomplished, almost single-handedly and on a very limited budget, in a tenement far from the garment district. He ordered his cloth from old-fashioned cloth makers in distant mill towns, then cut it himself, sometimes with my help. His margin would disappear if too much cloth fell on the floor as worthless rags, and at that time stylishness was of low priority. The actual tailoring was outsourced to homebound housewives in some outlying suburb. The son of one of the seamstresses, a truck driver, moonlighted as Father’s dispatcher.

  Then Father became his own salesman. He traveled alone to little towns and sold his goods directly to mom-and-pop merchants who wandered from one country fair to another—people of an altogether different culture. He had once visited those merchants, and he reminded them after the war that he was available, punctual, and inexpensive. Practically all his customers and suppliers returned.

  Father’s boldness worked. He did well enough to move up close to the proper garment district, buying an apartment/workshop in the declining hat district (a neighborhood currently favored by Kurds from Turkey!).

  As Father was fighting his final illness, increasing general prosperity destroyed his niche profession. When Mother was not watching, I sold cheap the remaining rolls of low-quality wool and packages of unfashionable clothing, then told a charitable lie by boasting of having made a killing. This was a lesson in real economics, in how elusive and how quickly changing the notion of monetary value can be.

  Mother

  When that 1930
picture was taken, my immediate family was avoiding the oppressive heat of Warsaw by summering in Świder, a bathing spot on the Vistula River. Most days Father was in town for business, but Mother stayed with us, which is why she did not take her rightful place in that momentous dinner party.

  (Illustration Credit 1.3)

  Here she is at three stages of her life: a young woman in a formal studio photograph in 1935, a lioness in a 1942 identity-card photo during the war, and a relaxed grandmother in 1962.

  Mother was born in a sizable town she spoke of as “Shavlee,” which Poles write as “Szawli.” Today it is Šiauliai in Lithuania, northwest of the capital, Vilnius. As a child, she lived in St. Petersburg, the capital of the Russian Empire, which included the Grand Duchy of Lithuania. Her family went on to Warsaw, in part because her mother was sickened by the wet, freezing winters farther north.

  Born not long after Napoléon’s disastrous campaign to conquer Russia, Mother’s paternal grandfather had a streak of wildness, left home as a teenager, and walked to St. Petersburg. Eventually, he came back home to start a family. He held remarkably advanced ideas. As confirmed by several cousins of Mother who scattered around the world, he pioneered by insisting that all his granddaughters become doctors. At age ninety-four, he fell off the horse he was riding and died. On that horse he looked like any other denizen of the ghetto, but imperial Russia ruled its Jews in many different ways. He had met and impressed an extremely rich man named Sergei Yulyevich Witte and helped him run his estates. They went on writing to each other, even in 1905, when his former employer had become a count of imperial Russia and the czar’s prime minister. That good man did not last, and his successor helped move that empire toward the abyss.