Perfect Rigor

A Genius and the Mathematical Breakthrough of the Century

Masha Gessen

• Publisher: Mariner Books
• Year: 2009
• ISBN:
  9780547427560
• Topic: Technology & Engineering
• Subtopic: Science & Technology Biographies
• Index: Technology & Engineering

A gripping and tragic tale that sheds rare light on the unique burden of genius

In 2006, an eccentric Russian mathematician named Grigori Perelman solved the Poincare Conjecture, an extremely complex topological problem that had eluded the best minds for over a century. A prize of one million dollars was offered to anyone who could unravel it, but Perelman declined the winnings, and in doing so inspired journalist Masha Gessen to tell his story. Drawing on interviews with Perelman's teachers, classmates, coaches, teammates, and colleagues in Russia and the United States—and informed by her own background as a math whiz raised in Russia—Gessen uncovered a mind of unrivaled computational power, one that enabled Perelman to pursue mathematical concepts to their logical (sometimes distant) end. But she also discovered that this very strength turned out to be Perelman's undoing and the reason for his withdrawal, first from the world of mathematics and then, increasingly, from the world in general.

1. Escape into the Imagination

AS ANYONE WHO has attended grade school knows, mathematics is unlike anything else in the universe. Virtually every human being has experienced that sense of epiphany when an abstraction suddenly makes sense. And while grade-school arithmetic is to mathematics roughly what a spelling bee is to the art of novel writing, the desire to understand patterns—and the childlike thrill of making an inscrutable or disobedient pattern conform to a set of logical rules—is the driving force of all mathematics.

Much of the thrill lies in the singular nature of the solution. There is only one right answer, which is why most mathematicians hold their field to be hard, exact, pure, and fundamental, even if it cannot precisely be called a science. The truth of science is tested by experiment. The truth of mathematics is tested by argument, which makes it more like philosophy, or, even better, the law, a discipline that also assumes the existence of a single truth. While the other hard sciences live in the laboratory or in the field, tended to by an army of technicians, mathematics lives in the mind. Its lifeblood is the thought process that keeps a mathematician turning in his sleep and waking with a jolt to an idea, and the conversation that alters, corrects, or affirms the idea.

“The mathematician needs no laboratories or supplies,” wrote the Russian number theorist Alexander Khinchin. “A piece of paper, a pencil, and creative powers form the foundation of his work. If this is supplemented with the opportunity to use a more or less decent library and a dose of scientific enthusiasm (which nearly every mathematician possesses), then no amount of destruction can stop the creative work.” The other sciences as they have been practiced since the early twentieth century are, by their very natures, collective pursuits; mathematics is a solitary process, but the mathematician is always addressing another similarly occupied mind. The tools of that conversation—the rooms where those essential arguments take place—are conferences, journals, and, in our day, the Internet.

That Russia produced some of the twentieth century’s greatest mathematicians is, plainly, a miracle. Mathematics was antithetical to the Soviet way of everything. It promoted argument; it studied patterns in a country that controlled its citizens by forcing them to inhabit a shifting, unpredictable reality; it placed a premium on logic and consistency in a culture that thrived on rhetoric and fear; it required highly specialized knowledge to understand, making the mathematical conversation a code that was indecipherable to an outsider; and worst of all, mathematics laid claim to singular and knowable truths when the regime had staked its legitimacy on its own singular truth. All of this is what made mathematics in the Soviet Union uniquely appealing to those whose minds demanded consistency and logic, unattainable in virtually any other area of study. It is also what made mathematics and mathematicians suspect. Explaining what makes mathematics as important and as beautiful as mathematicians know it to be, the Russian algebraist Mikhail Tsfasman said, “Mathematics is uniquely suited to teaching one to distinguish right from wrong, the proven from the unproven, the probable from the improbable. It also teaches us to distinguish that which is probable and probably true from that which, while apparently probable, is an obvious lie. This is a part of mathematical culture that the [Russian] society at large so sorely lacks.”

It stands to reason that the Soviet human rights movement was founded by a mathematician. Alexander Yesenin-Volpin, a logic theorist, organized the first demonstration in Moscow in December 1965. The movement’s slogans were based on Soviet law, and its founders made a single demand: they called on the Soviet authorities to obey the country’s written law. In other words, they demanded logic and consistency; this was a transgression, for which Yesenin-Volpin was incarcerated in prisons and psychiatric wards for a total of fourteen years and ultimately forced to leave the country.

Soviet scholarship, and Soviet scholars, existed to serve the Soviet state. In May 1927, less than ten years after the October Revolution, the Central Committee inserted into the bylaws of the USSR’s Academy of Sciences a clause specifying just this. A member of the Academy may be stripped of his status, the clause stated, “if his activities are apparently aimed at harming the USSR.” From that point on, every member of the Academy was presumed guilty of aiming to harm the USSR. Public hearings involving historians, literary scholars, and chemists ended with the scholars publicly disgraced, stripped of their academic regalia, and, frequently, jailed on treason charges. Entire fields of study—most notably genetics—were destroyed for apparently coming into conflict with Soviet ideology. Joseph Stalin personally ruled scholarship. He even published his own scientific papers, thereby setting the research agenda in a given field for years to come. His article on linguistics, for example, relieved comparative language study of a cloud of suspicion that had hung over it and condemned, among other things, the study of class distinctions in language as well as the whole field of semantics. Stalin personally promoted a crusading enemy of genetics, Trofim Lysenko, and apparently coauthored Lysenko’s talk that led to an outright ban of the study of genetics in the Soviet Union.

What saved Russian mathematics from destruction by decree was a combination of three almost entirely unrelated factors. First, Russian mathematics happened to be uncommonly strong right when it might have suffered the most. Second, mathematics proved too obscure for the sort of meddling the Soviet leader most liked to exercise. And third, at a critical moment it proved immensely useful to the State.

In the 1920s and ’30s, Moscow boasted a robust mathematical community; groundbreaking work was being done in topology, probability theory, number theory, functional analysis, differential equations, and other fields that formed the foundation of twentieth-century mathematics. Mathematics is cheap, and this helped: when the natural sciences perished for lack of equipment and even of heated space in which to work, the mathematicians made do with their pencils and their conversations. “A lack of contemporary literature was, to some extent, compensated by ceaseless scientific communication, which it was possible to organize and support in those years,” wrote Khinchin about that period. An entire crop of young mathematicians, many of whom had received part of their education abroad, became fast-track professors and members of the Academy in those years.

The older generation of mathematicians—those who had made their careers before the revolution—were, naturally, suspect. One of them, Dimitri Egorov, the leading light of Russian mathematics at the turn of the twentieth century, was arrested and in 1931 died in internal exile. His crimes: he was religious and made no secret of it, and he resisted attempts to ideologize mathematics—for example, trying (unsuccessfully) to sidetrack a letter of salutation sent from a mathematicians’ congress to a Party congress. Egorov’s vocal supporters were cleansed from the leadership of Moscow mathematical institutions, but by the standards of the day, this was more of a warning than a purge: no area of study was banned, and no general line was imposed by the Kremlin. Mathematicians would have been well advised to brace for a bigger blow.

In the 1930s, a mathematical show trial was all set to go forward. Egorov’s junior partner in leading the Moscow mathematical community was his first student, Nikolai Luzin, a charismatic teacher himself whose numerous students called their circle Luzitania, as though it were a magical country, or perhaps a secret brotherhood united by a common imagination. Mathematics, when taught by the right kind of visionary, does lend itself to secret societies. As most mathematicians are quick to point out, there are only a handful of people in the world who understand what the mathematicians are talking about. When these people happen to talk to one another—or, better yet, form a group that learns and lives in sync—it can be exhilarating.

“Luzin’s militant idealism,” wrote a colleague who denounced Luzin, “is amply expressed by the following quote from his report to the Academy on his trip abroad: ‘It seems the set of natural numbers is not an absolutely objective formation. It seems it is a function of the mind of the mathematician who happens to be speaking of a set of natural numbers at the given moment. It seems there are, among the problems of arithmetic, those that absolutely cannot be solved.’”

The denunciation was masterful: the addressee did not need to know anything about mathematics and would certainly know that solipsism, subjectivity, and uncertainty were utterly un-Soviet qualities. In July 1936 a public campaign against the famous mathematician was launched in the daily Pravda, where Luzin was exposed as “an enemy wearing a Soviet mask.”

The campaign against Luzin continued with newspaper articles, community meetings, and five days of hearings by an emergency committee formed by the Academy of Sciences. Newspaper articles exposed Luzin and other mathematicians as enemies because they published their work abroad. In other words, events unfolded in accordance with the standard show-trial scenario. But then the process seemed to fizzle out: Luzin publicly repented and was severely reprimanded although allowed to remain a member of the Academy. A criminal investigation into his alleged treason was quietly allowed to die.

Researchers who have studied the Luzin case believe it was Stalin himself who ultimately decided to stop the campaign. The reason, they think, is that mathematics is useless for propaganda. “The ideological analysis of the case would have devolved to a discussion of the mathematician’s understanding of a natural number set, which seemed like a far cry from sabotage, which, in the Soviet collective consciousness, was rather associated with coal mine explosions or killer doctors,” wrote Sergei Demidov and Vladimir Isakov, two mathematicians who teamed up to study the case when this became possible, in the 1990s. “Such a discussion would better be conducted using material more conducive to propaganda, such as, say, biology and Darwin’s theory of evolution, which the great leader himself was fond of discussing. That would have touched on topics that were ideologically charged and easily understood: monkeys, people, society, and life itself. That’s so much more promising than the natural number set or the function of a real variable.”

Luzin and Russian mathematics were very, very lucky.

Mathematics survived the attack but was permanently hobbled. In the end, Luzin was publicly disgraced and dressed down for practicing mathematics: publishing in international journals, maintaining contacts with colleagues abroad, taking part in the conversation that is the life of mathematics. The message of the Luzin hearings, heeded by Soviet mathematicians well into the 1960s and, to a significant extent, until the collapse of the Soviet Union, was this: Stay behind the Iron Curtain. Pretend Soviet mathematics is not just the world’s most progressive mathematics—this was its official tag line—but the world’s only mathematics. As a result, Soviet and Western mathematicians, unaware of one another’s endeavors, worked on the same problems, resulting in a number of double-named concepts such as the Chaitin-Kolmogorov complexities and the Cook-Levin theorem. (In both cases the eventual coauthors worked independently of each other.) A top Soviet mathematician, Lev Pontryagin, recalled in his memoir that during his first trip abroad, in 1958—five years after Stalin’s death—when he was fifty years old and world famous among mathematicians, he had had to keep asking colleagues if his latest result was actually new; he did not really have another way of knowing.

“It was in the 1960s that a couple of people were allowed to go to France for half a year or a year,” recalled Sergei Gelfand, a Russian mathematician who now runs the American Mathematics Society’s publishing program. “When they went and came back, it was very useful for all of Soviet mathematics, because they were able to communicate there and to realize, and make others realize, that even the most talented of people, when they keep cooking in their own pot behind the Iron Curtain, they don’t have the full picture. They have to speak with others, and they have to read the work of others, and it cut both ways: I know American mathematicians who studied Russian just to be able to read Soviet mathematics journals.” Indeed, there is a generation of American mathematicians who are more likely than not to possess a reading knowledge of mathematical Russian—a rather specialized skill even for a native Russian speaker; Jim Carlson, president of the Clay Mathematics Institute, is one of them. Gelfand himself left Russia in the early 1990s because he was drafted by the American Mathematics Society to fill the knowledge gap that had formed during the years of the Soviet reign over mathematics: he coordinated the translation and publication in the United States of Russian mathematicians’ accumulated work.

So some of what Khinchin described as the tools of a mathematician’s labor—“a more or less decent library” and “ceaseless scientific communication”—were stripped from Soviet mathematicians. They still had the main prerequisites, though—“a piece of paper, a pencil, and creative powers”—and, most important, they had one another: mathematicians as a group slipped by the first rounds of purges because mathematics was too obscure for propaganda. Over the nearly four decades of Stalin’s reign, however, it would turn out that nothing was too obscure for destruction. Mathematics’ turn would surely have come if it weren’t for the fact that at a crucial point in twentieth-century history, mathematics left the realm of abstract conversation and suddenly made itself indispensable. What ultimately saved Soviet mathematicians and Soviet mathematics was World War II and the arms race that followed it.

Nazi Germany invaded the Soviet Union on June 22, 1941. Three weeks later, the Soviet air force was gone: bombed out of existence in the airfields before most of the planes ever took off. The Russian military set about retrofitting civilian airplanes for use as bombers. The problem was, the civilian airplanes were significantly slower than the military ones, rendering moot everything the military knew about aim. A mathematician was needed to recalculate speeds and distances so the air force could hit its targets. In fact, a small army of mathematicians was needed. The greatest Russian mathematician of the twentieth century, Andrei Kolmogorov, returned to Moscow from the academics’ wartime haven in Tatarstan and led a classroom full of students armed with adding machines in recalculating the Red Army’s bombing and artillery tables. When this work was done, he set about creating a new system of statistical control and prediction for the Soviet military.

At the beginning of World War II, Kolmogorov was thirty-eight years old, already a member of the Presidium of the Soviet Academy of Sciences—making him one of a handful of the most influential academics in the empire—and world famous for his work in probability theory. He was also an unusually prolific teacher: by the end of his life he had served as an adviser on seventy-nine dissertations and had spearheaded both the math olympiads system and the Soviet mathematics-school culture. But during the war, Kolmogorov put his scientific career on hold to serve the Soviet state directly—proving in the process that mathematicians were essential to the State’s very survival.

The Soviet Union declared victory—and the end of what it called the Great Patriotic War—on May 9, 1945. In August, the United States dropped atomic bombs on the Japanese cities of Hiroshima and Nagasaki. Stalin kept his silence for months afterward. When he finally spoke publicly, following his so-called reelection in February 1946, it was to promise the people of his country that the Soviet Union would surpass the West in developing its atomic capability. The effort to assemble an army of physicists and mathematicians to match the Manhattan Project’s had by that time been under way for at least a year; young scholars had been recalled from the frontlines and even released from prisons in order to join the race for the bomb.

Following the war, the Soviet Union invested heavily in high-tech military research, building more than forty entire cities where scientists and mathematicians worked in secret. The urgency of the mobilization indeed recalled the Manhattan Project—only it was much, much bigger and lasted much longer. Estimates of the number of people engaged in the Soviet arms effort in the second half of the century are notoriously inaccurate, but they range as high as twelve million, with a couple million of them employed by military research institutions. For many years, a newly graduated young mathematician or physicist was more likely to be assigned to defense-related research than to a civilian institution. These jobs spelled nearly total scientific isolation: for defense employees, burdened by security clearances whether or not they actually had access to sensitive military information, any contact with foreigners was considered not just suspect but treasonous. In addition, some of these jobs required moving to the research towns, which provided comfortably cloistered social environments but no possibility for outside intellectual contact. The mathematician’s pencil and paper could be useless tools in the absence of an ongoing mathematical conversation. So the Soviet Union managed to hide some of its best mathematical minds away, in plain sight.

Following Stalin’s death, in 1953, the country shifted its stance on its relationship to the rest of the world: now the Soviet Union was to be not only feared but respected. So while it fell to most mathematicians to help build bombs and rockets, it fell to a select few to build prestige. Very slowly, in the late 1950s, the Iron Curtain began to open a tiny crack—not quite enough to facilitate much-needed conversation between Soviet and non-Soviet mathematicians but enough to show off some of Soviet mathematics’ proudest achievements.

By the 1970s, a Soviet mathematics establishment had taken shape. It was a totalitarian system within a totalitarian system. It provided its members with not only work and money but also apartments, food, and transportation; it determined where they lived and when, where, and how they traveled for work or pleasure. To those in the fold, it was a controlling and strict but caring mother: her children were well nourished and nurtured, an undeniably privileged group compared with the rest of the country. When basic goods were scarce, official mathematicians and other scientists could shop at specially designated stores, which tended to be better stocked and less crowded than those open to the general public. Since for most of the Soviet century there was no such thing as a private apartment, regular Soviet citizens received their dwellings from the State; members of the science establishment were assigned apartments by their institutions, and these apartments tended to be larger and better located than their compatriots’. Finally, one of the rarest privileges in the life of a Soviet citizen—foreign travel—was available to members of the mathematics establishment. It was the Academy of Sciences, with the Party and the State security organizations watching over it, that decided if a mathematician could accept, say, an invitation to address a scholarly conference, who would accompany him on the trip, how long the trip would last, and, in many instances, where he would stay. For example, in 1970, the first Soviet winner of the Fields Medal, Sergei Novikov, was not allowed to travel to Nice to accept his award. He received it a year later, when the International Mathematical Union met in Moscow.

Even for members of the mathematical establishment, though, resources were always scarce. There were always fewer good apartments than there were people who desired them, and there were always more people wanting to travel to a conference than would be allowed to go. So it was a vicious, backstabbing little world, shaped by intrigue, denunciations, and unfair competition. The barriers to entry into this club were prohibitively high: a mathematician had to be ideologically reliable and personally loyal not only to the Party but to existing members of the establishment, and Jews and women had next to no chance of getting in.

One could easily be expelled by the establishment for misbehaving. This happened with Kolmogorov’s student Eugene Dynkin, who fostered an atmosphere of unconscionable liberalism at a specialized mathematics school he ran in Moscow. Another of Kolmogorov’s students, Leonid Levin, describes being ostracized for associating with dissidents. “I became a burden for everyone to whom I was connected,” he wrote in a memoir. “I would not be hired by any serious research institution, and I felt I didn’t even have the right to attend seminars, since participants had been instructed to inform [the authorities] whenever I appeared. My Moscow existence began to seem pointless.” Both Dynkin and Levin emigrated. It must have been soon after Levin’s arrival in the United States that he learned that a problem he had been describing at Moscow mathematics seminars (building in part on Kolmogorov’s work on complexities) was the same problem U.S. computer scientist Stephen Cook had defined. Cook and Levin, who became a professor at Boston University, are considered coinventors of the NP-completeness theorem, also known as the Cook-Levin theorem; it forms the foundation of one of the seven Millennium Problems that the Clay Mathematics Institute is offering a million dollars to solve. The theorem says, in essence, that some problems are easy to formulate but require so many computations that a machine capable of solving them cannot exist.

And then there were those who almost never became members of the establishment: those who happened to be born Jewish or female, those who had had the wrong advisers at their universities, and those who could not force themselves to join the Party. “There were people who reali...

Table of contents

• Title Page
• Contents
• Copyright
• Prologue
• 1. Escape into the Imagination
• 2. How to Make a Mathematician
• 3. A Beautiful School
• 4. A Perfect Score
• 5. Rules for Adulthood
• 6. Guardian Angels
• 7. Round Trip
• 8. The Problem
• 9. The Proof Emerges
• 10. The Madness
• 11. The Million-Dollar Question
• Acknowledgments
• Notes
• Index
• About the Author
• Connect with HMH