Perfect Rigour Read online

  PERFECT RIGOR

  HOW TO MAKE A MATHEMATICIAN

  2

  How to Make a Mathematician

  IN THE MID-1960S Professor Garold Natanson offered a graduate-study spot to a student of his, a woman named Lubov. One did not make this sort of offer lightly: female graduate students were notoriously unreliable, prone to pregnancy and other distracting pursuits. In addition, this particular student was Jewish, which meant that securing a spot for her would have required Professor Natanson to scheme, strategize, and call in favors: in the eyes of the system, Jews were even more unreliable than women, and convoluted discriminatory anti-Semitic practices carried the force of unwritten law. Natanson, a Jew himself, taught at the Herzen Pedagogical Institute, which ranked second to Leningrad State University and so was allowed to accept Jews as students and teachers—within reason, or what passed for it in the postwar Soviet Union. The student was older—she was nearing thirty, which placed her well beyond the usual Russian marrying-and-having-children threshold, so Natanson could be justified in assuming that she had resolved to devote her life entirely to mathematics.

  Natanson was not entirely off the mark: the woman was indeed wholly devoted to mathematics. But she turned down his generous offer. She explained that she had recently married and planned to start a family, and with that she accepted a job teaching mathematics at a trade school and disappeared from the Leningrad mathematical scene for more than ten years.

  Ten or twelve years was nothing in Soviet time. There was a bit of new housing construction in Leningrad, and some families were able to leave the crowded and crumbling city center for the new concrete towers on its outskirts. Clothing and food continued to be in short supply and of regrettable quality, but industrial production picked up a bit, so some of the new suburban dwellers could actually buy basic semiautomatic washing machines and television sets for their apartments. The televisions claimed to be black-and-white but showed mostly shades of gray, thereby providing an accurate visual reflection of reality. Other than that, little changed. Natanson continued to teach at the Herzen, which itself grew only more crowded and crumbling. His former student Lubov found him in his office. She was older and a bit heavier. She reported that she had indeed had a baby all those years ago, and now this baby was a schoolboy who exhibited a talent for mathematics. He had taken part in a district math competition in one of those newly constructed concrete suburbs where they now lived, and he had done well. In the timeless scheme of Russian mathematics, he was ready to take up where his mother had left off.

  It all must have made perfect sense to Natanson. He himself hailed from a mathematical dynasty: his father, Isidor Natanson, was the author of the definitive Russian calculus textbook and had also taught at the Herzen, until his death, in 1963. Lubov’s boy was entering fifth grade—the age at which he could begin appropriately rigorous mathematical study in a system that had been constructed over the years for the making of mathematicians. Natanson had his eye on a young mathematics coach to whom he could direct the boy and his mother.

  So began the education of Grigory Perelman.

  Competitive mathematics is more like a sport than most people imagine. It has its coaches, its clubs, its practice sessions, and, of course, its competitions. Natural ability is necessary but entirely insufficient for success: the talented child needs to have the right coach, the right team, the right kind of family support, and, most important, the will to win. At the beginning, it is nearly impossible to tell the difference between future stars and those who will be good but never great.

  Grisha Perelman arrived at the math club of the Leningrad Palace of Pioneers in the fall of 1976, an ugly duckling among ugly ducklings. He was pudgy and awkward. He played the violin; his mother, who had studied not only mathematics but also the violin when she was a child, had engaged a private teacher when Grisha was very young. When he tried to explain a solution to a math problem, words seemed to get tangled at the tip of his tongue, where too many of them collected too quickly, froze momentarily, and then tumbled out, all jumbled up. He was precocious—a year younger than the other children at his grade level—but one of the other kids at the club was even younger: Alexander Golovanov1 had packed two grades into every year of school and would be finishing high school at thirteen. Three other boys beat Grisha in competitions2 for the first few years in the club. At least one more—Boris Sudakov,3 a round, animated, curious boy whose parents happened to know Grisha’s family—showed more natural ability than Grisha. Sudakov and Golovanov both carried the marks of brilliance: they seemed always to be rushing forward and bubbling over. They naturally fought for dominance in any room, and mathematics was simply one of many things that got them excited, one of the ways to apply their excellent minds, and one of the tools to showcase their uniqueness. Next to them, Grisha was the interested but quiet partner, almost a mirror; he was a joy for them to bounce their ideas off, but he himself rarely seemed to exhibit the same need. He formed relationships with the math problems; these relationships were deep but also, it seemed, deeply private: most of his conversations appeared to be mathematical and to take place inside his head. A casual visitor to the club would not have singled him out from the other boys. Indeed, even among the people who met him many years later, not one that I encountered described him as brilliant; no one thought he sparkled or shone. People described him, rather, as very, very smart and very, very precise in his thinking.

  Just what manner of thinking this was remained something of a mystery. Crudely speaking, mathematicians fall into two categories: the algebraists, who find it easiest to reduce all problems to sets of numbers and variables, and the geometers, who understand the world through shapes. Where one group sees this:

  a2 + b2 = c2

  the other sees this:

  Golovanov, who studied and occasionally competed alongside Perelman for more than ten years, tagged him as an unambiguous geometer: Perelman had a geometry problem solved in the time it took Golovanov to grasp the question. This was because Golovanov was an algebraist. Sudakov, who spent about six years studying and occasionally competing with Perelman, claimed Perelman reduced every problem to a formula. This, it appears, was because Sudakov was a geometer: his favorite proof of the classic theorem above was an entirely graphical one, requiring no formulas and no language to demonstrate. In other words, each of them was convinced Perelman’s mind was profoundly different from his own. Neither had any hard evidence. Perelman did his thinking almost entirely inside his head, neither writing nor sketching on scrap paper. He did a lot of other things—he hummed, moaned, threw a Ping-Pong ball against the desk,4 rocked back and forth, knocked out a rhythm on the desk with his pen, rubbed his thighs until his pant legs shone, and then rubbed his hands together—a sign that the solution would now be written down, fully formed. For the rest of his career, even after he chose to work with shapes, he never dazzled colleagues with his geometric imagination, but he almost never failed to impress them5 with the single-minded precision with which he plowed through problems. His brain seemed to be a universal math compactor, capable of compressing problems to their essence. Club mates eventually dubbed whatever it was he had inside his head the “Perelman stick”—a very large imaginary instrument with which he sat quietly before striking an always-fatal blow.

  Practice sessions at mathematics clubs the world over look roughly the same. Kids come in to find a set of problems written on the blackboard or handed to them. They sit down and attempt to solve them. The coach spends most of his time sitting quietly; teaching assistants check in with the students occasionally, sometimes prodding them with questions, sometimes trying to nudge them in different directions.

  To a Soviet child, the afterschool math club was a miracle. For one thing, it was not school. Every morning Soviet children all over the country left their identical concrete apartment blocks a little after eight and walked to their identical concrete school buildin
gs to sit in their identical classrooms with the walls painted yellow and with identical portraits of bearded dead men on the walls—Dostoyevsky and Tolstoy in the literature classrooms, Mendeleev in the chemistry classroom, and Lenin everywhere. Their teachers marked attendance in identical class journals and reached for identical textbooks that they used to impart a perfectly uniform education to their charges, of whom they demanded uniformity in return. My own first-grade teacher, in a neighborhood on the outskirts of Moscow that looked just like Perelman’s neighborhood on the outskirts of Leningrad, actually made me pretend my reading skills were as poor as the other children’s, enforcing her own vision of conforming to grade level. The first time I spent an afternoon solving math problems—around the same time Perelman was doing it, four hundred miles to the north—I sat for what seemed like an eternity, holding a pencil over a drawing of some shape. I do not remember the problem, but I remember that the solution required transposing the shape. I sat, unable to touch my pencil to paper, until a teaching assistant came by and asked me a very basic question, something like “What might you do?”

  “I might transpose it, like this,” I answered.

  “So do it,” he said.

  Apparently, this was a place where I was expected to think for myself. A wave of embarrassment covered me; I hunched over my piece of paper, sketched out the solution in a couple of minutes, and felt a wave of relief so total that I think I became a math junkie on the spot. I did not drop the habit until I was in college (and was actually busted for illegally replacing a required humanities course with advanced calculus). The joy of feeling my brain rev up, rush toward a solution, reach it, and be affirmed for it felt like love, truth, hope, and justice all handed to me at once.

  The particular math club where Perelman landed was a bare-bones operation. The coach with whom old Natanson decided to place his protégé by proxy was a tall, freckle-faced, light-haired loudmouthed man named Sergei Rukshin.6 He had one very important distinguishing characteristic: he was nineteen years old. He had no experience leading a club; he had no teaching assistants. What he did have was outsize ambition and a fear of failure to match. By day, he was an undergraduate at Leningrad State University; two afternoons a week, he put on a suit and tie and impersonated an adult math-club coach at the Palace of Pioneers.

  In the quiet, dignified mathematics counterculture of Leningrad, Rukshin was an outsider. He had grown up in a town near Leningrad, a troubled kid like any troubled kid anywhere in the world. By the age of fifteen, he had racked up several minor juvenile offenses, and the only thing he liked to do was box. He was on a clear path to trade school, then the military, followed by a short life of drink and violence—like most Russian men of his generation. The prospect terrified his parents so much that they begged and pleaded and possibly bribed until a miracle happened and their son got a spot at a mathematical high school in the city. There, another miracle happened: Rukshin fell in love with mathematics and turned all his creative, aggressive, and competitive energies toward it. He tried to compete in mathematics olympiads, but he was outmatched by peers who had been training for years. Still, he believed he knew how to win; he just could not do it himself. He formed a team of schoolchildren who were just a year younger than he and trained them, and they did better than he had. He started training upperclassmen all over Leningrad. Then he became a teaching assistant at the Palace of Pioneers, and barely a year later, when the coach with whom he had been apprenticing left for a job assignment in a different city, he became a coach himself.

  Like any young teacher, he was a little scared of his students. His first group included Perelman, Golovanov, Sudakov, and several other boys, all of whom were just a few years younger than he but poised to become successful competitive mathematicians. The only way he could prove he deserved to be their teacher was by becoming the best mathematics coach the world had ever seen.

  Which is exactly what he did. In the decades since, his students have taken more than seventy International Mathematical Olympiad medals, including more than forty gold ones; in the past two decades, about half of the competitors Russia has put forward have come from Rukshin’s now-sprawling club, where they were trained by either him or one of his students, who use his unparalleled training method.

  What exactly made his method unparalleled was not entirely clear. “I still don’t understand what he did,” admitted Sudakov, now an overweight and balding computer scientist living in Jerusalem, “even though I know a thing or two about the psychology of these things. We would come in and sit down and we would get our problem sets. We would solve them. Rukshin would be sitting there at his desk. When somebody solved one of the problems, [that student] would go over to Rukshin’s desk and explain his solution and they would discuss it. There! That’s all there was to it. Eh?” Sudakov looked at me across the table of a Jerusalem café, triumphant.

  “That’s what everyone does,” I responded, as expected.

  “Exactly! That’s what I’m talking about!” Sudakov fidgeted happily as he talked.

  I observed practice sessions7 at the club Rukshin still ran a quarter century later. It was now called the Mathematics Education Center; it included a couple of hundred children eleven and older. Just like Perelman’s group, they spent two afternoons a week at the club. At the end of each session—which lasted two hours at the lower grade levels and could stretch into the night for upperclassmen—the students got a list of problems to take home. Rukshin claimed that one of his unique strategies was adapting the list of problems to the class during the course of the session: the instructor had to go in with several possible lists and choose among them depending on what he learned about the students’ progress over the next couple of hours. Three days later, the students brought in their solutions, which, one by one, they explained to teaching assistants for the first hour of the session. In the second hour, the instructor went over all correct solutions at the blackboard. As they grew older, the students gradually transitioned to explaining their own solutions at the blackboard themselves, in front of the entire group.

  I watched the younger kids struggle with the following problem: “There are six people in the classroom. Prove that among them there must be either three people who do not know one another or three people who all know one another.” Teaching assistants encouraged them to start with the following diagram:

  Two of the half dozen children working on the problem managed to doodle their way to the fact that the diagram can develop in one of three possible ways:

  The challenge, to which two children successfully rose, was to explain that this was a graphical—and therefore irrefutable—way to show that there must be at least three people who either all know or all do not know one another. Listening to the children struggle to put this into words, battling an entire short lifetime of inarticulateness, was painful.

  Mathematicians know this as the Party Problem;8 in its general form, it asks how many people must be invited to a party so that at least m will know one another or at least n will not know one another. The Party Problem refers back to Ramsey theory, a system of theorems9 devised by the British mathematician Frank Ramsey. Most Ramsey-type problems look at the number of elements required to ensure a particular condition will hold. How many children must a woman have to ensure that she has at least two of the same gender? Three. How many people must be present at a party to ensure that at least three of them all know or all do not know one another? Six. How many pigeons must there be to ensure that at least one pigeonhole houses two or more pigeons? One more than there are pigeonholes.

  The Mathematics Education Center children—some of them, at least—would learn about Ramsey theory in time. For the moment, they had to learn to express a way of looking at the world that would ultimately make them interested in Ramsey theory and in other methods of observing order in a chaotic environment. To most individuals, children in a classroom or guests at a party are just people. To ot
hers, they are the elements of an order and their relationships the parts of a pattern. These others are mathematicians. Most mathematics teachers seem to believe some children are born with the inclination to seek patterns. These children must be identified and taught to nurture this skill, the peculiar ability to see triangles and hexagons where others see only a party.

  “That’s my biggest know-how,” Rukshin told me. “I discovered this thirty years ago: every child must be heard out on every problem he thinks he has solved.” Other math clubs had children present their solutions to the class—which meant that the first correct solution ended the discussion. Rukshin’s policy was to engage every child in a separate conversation about that child’s particular successes, difficulties, and mistakes. This was perhaps the most labor-intensive instruction method ever invented; it meant that none of the children and none of the instructors could coast at any time. “In the end we teach children to talk,” said Rukshin, “and we teach the instructors to understand the students’ incoherent speech and direct them. Rather, I should say, to understand their incoherent speech and their incoherent ideas.”

  As I listened to Rukshin and watched him teach, I struggled to place the feeling his club sessions communicated. What made them different—more emotionally engaged but also more tense than any other math, chess, or sports practice session I had ever seen? It took months for my mind to locate the analogy: these sessions felt most like group therapy. The trick really was to get every child to present his or her solution to the entire group. Mathematics was the most important thing in these children’s lives; Rukshin would not have it any other way. They spent most of their free time thinking about the problems they had been given, investing all the emotion and energy they had—not unlike a conscientious twelve-stepper who stayed connected with the program between meetings by writing out the steps. Then, at the meetings, the children laid bare their minds before the people that mattered most to them by telling the stories of their solutions in front of the entire group.