Perfect Rigour

by Masha Gessen

  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 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:11 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,12 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 dissertations13 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 West14 in developing its atomic capability. The effort to assemble an army of physicists and mathematicians15 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 effort16 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,17 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.18 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 ostracized19 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 coinventors20 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 realized that they would never be admitted to the Academy and that the most they could hope for was being able to defend their doctoral dissertation at some institute in Minsk, if they could secure connections there,” said Sergei Gelfand, the American Mathematics Society publisher, who happens to be the son of one of Russia’s top twentieth-century mathematicians, Israel Gelfand, a student of Kolmogorov’s. “These people attended seminars at the university and were officially on the staff of some research institute, say, of the timber industry. They did very good math, and at a certain point they even started having contacts abroad and could even get published occasionally in the West—it was hard, and they had to prove that they were not divulging state secrets, but it was possible. Some mathematicians came from the West, some even came for an extended stay because they realized there were a lot of talented people. This was unofficial mathematics.”

  One of the people who came for an extended stay was Dusa McDuff,21 then a British algebraist (and now a professor emeritus at the State University of New York at Stony Brook). She studied with the older Gelfand for six months and credits this experience with opening her eyes to both the way mathematics ought to be practiced—in part through continuous conversation with other mathematicians—and to what mathematics really is. “It was a wonderful education,22 in which reading Pushkin’s Mozart and Salieri played as important a role as learning about Lie groups or reading Cartan and Eilenberg. Gelfand amazed me by talking of mathematics as though it were poetry. He once said about a long paper bristling with formulas that it contained the vague beginnings of an idea which he could only hint at and which he had never managed to bring out more clearly. I had always thought of mathematics as being much more straightforward: a formula is a formula, and an algebra is an algebra, but Gelfand found hedgehogs lurking in the rows of his spectral sequences!”

  On paper, the jobs that members of the mathematical counterculture held were generally undemanding and unrewarding, in keeping with the best-known formula of Soviet labor: “We pretend to work, and they pretend to pay us.” The math
ematicians received modest salaries that grew little over a lifetime but that were enough to cover basic needs and allow them to spend their time on real research. “There was no such thing as thinking that you had to focus your work in some one narrow area because you have to write faster because you had to get tenure,” said Gelfand. “Mathematics was almost a hobby. So you could spend your time doing things that would not be useful to anyone for the nearest decade.” Mathematicians called it “math for math’s sake,”23 intentionally drawing a parallel between themselves and artists who toiled for art’s sake. There was no material reward in this—no tenure, no money, no apartments, no foreign travel; all they stood to gain by doing brilliant work was the respect of their peers. Conversely, if they competed unfairly, they stood to lose the respect of their colleagues while gaining nothing. In other words, the alternative mathematics establishment in the Soviet Union was very much unlike anything else anywhere in the real world: it was a pure meritocracy where intellectual achievement was its own reward.

  In after-hours lectures and seminars, the mathematical conversation in the Soviet Union was reborn, and the appeal of mathematics to a mind in search of challenge, logic, and consistency once again became evident. “In the post-Stalin Soviet Union it was one of the most natural ways for a freethinking intellectual to seek self-realization,” said Grigory Shabat, a well-known Moscow mathematician. “If I had been free to choose any profession,24 I would have become a literary critic. But I wanted to work, not spend my life fighting the censors.” Mathematics held out the promise that one could not only do intellectual work without State interference (if also without its support) but also find something not available anywhere else in late-Soviet society: a knowable singular truth. “Mathematicians are people possessed of a special intellectual honesty,” Shabat continued. “If two mathematicians are making contradictory claims, then one of them is right and the other one is wrong. And they will definitely figure it out, and the one who was wrong will definitely admit that he was mistaken.” The search for that truth could take long years—but in the late Soviet Union, time stood still, which meant that the inhabitants of the alternative mathematics universe had all the time they needed.