Perfect Rigour

A Genius and the Mathematical Breakthrough of a Lifetime

Masha Gessen

• Publisher: Icon Books
• Year: 2011
• ISBN:
  9781848313095
• Topic: Technology & Engineering
• Subtopic: Science & Technology Biographies
• Index: Technology & Engineering

In 2006, an eccentric Russian mathematician named Grigori Perelman solved one of the world's greatest intellectual puzzles. The Poincare conjecture is an extremely complex topological problem that had eluded the best minds for over a century. In 2000, the Clay Institute in Boston named it one of seven great unsolved mathematical problems, and promised a million dollars to anyone who could find a solution. Perelman was awarded the prize this year - and declined the money. Journalist Masha Gessen was determined to find out why. Drawing on interviews with Perelman's teachers, classmates, coaches, teammates, and colleagues in Russia and the US - and informed by her own background as a math whiz raised in Russia - she set out to uncover the nature of Perelman's astonishing abilities. In telling his story, Masha Gessen has constructed a gripping and tragic tale that sheds rare light on the unique burden of genius.

PERFECT RIGOR

ESCAPE INTO THE IMAGINATION

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,”1 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 teaching2 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,3 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,4 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 promoted5 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,6 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 case7 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,8 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,9 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 1960s10 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 itsel...

Table of contents

• Cover
• Title page
• Copyright
• By same author
• Contents
• Prologue
• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11
• Epilogue
• Acknowledgments
• Endnotes