Escape to the imagination

Chapter 1 - from the book "The Perlman Riddle - a story about a genius and the mathematical breakthrough of the century" by Masha Gessen. Aliyat HaGeg Books Publishing and Yediot Books.

The Perlman puzzle

A story about a genius and the mathematical breakthrough of the century by Masha Gessen. From English: Dafna Levy, published by Aliyat HaGeg Books and Yediot Books

In 2000, the Clay Institute of Mathematics announced in Boston the seven great unsolved mathematical problems - the Millennium Problems. And the institute also announced that whoever finds a solution to one of these seven problems will win a million dollars.

Poincaré's hypothesis, one of the seven, eluded the best mathematical minds for over a century.

In 2006, an eccentric Russian mathematician, the Jew Grigory Perlman, solved it.

How did Perlman succeed where so many good ones have failed? Who is this man, who refused to accept the award, after also refusing to accept the Fields Medal - the mathematical equivalent of the Nobel Prize?

The writer Masha Gessen - in her youth a student of the Soviet program for the gifted in mathematics - interviewed a host of people who surrounded Perlman: classmates, teachers, coaches, members of the Soviet teams for the Math Olympiads, colleagues from Russia, America and China, as well as from Israel, and tried to understand the meaning of the genius and strangeness of this enigmatic man . Is this a genius intuition? In a one-of-a-kind imagination? What is clear: Grisha Perlman has a brain with tremendous, almost inhuman, computational power. Added to this is what Henri Poincare himself said is the essence of mathematics - "perfect rigor", and with it a complete disregard for everything that is not related to mathematics. All of these allowed him to get to the bottom of - and solve - mathematical problems.

But Masha Gessen discovers that these abilities come at a price. Such a special mind is unable to deal with the gray, everyday, human reality. When Perlman encounters rivalries, bigotry, narrow-mindedness - he retreats. At first from the mathematical world, then, gradually, from the world. In Perlman's puzzle, Masha Gessen illuminates the heavy burden of genius in a different light.

Masha Gessen writes for various newspapers, including Slate, Vanity Fair, The New York Times and others. She wrote six more books. Gesen lives in Russia.

The first chapter - escape to the image

As anyone who has attended elementary school knows, math is unlike anything else in the universe. Almost every person has experienced the sense of revelation we feel when an abstraction suddenly makes sense to us. And although the relationship between elementary school calculus and mathematics is about the same as that between an essay exercise and writing a novel, the desire to understand patterns - and the childlike excitement felt by those who manage to bring a closed or unruly pattern to obey a set of logical rules - is the driving force of mathematics the whole

A large part of the excitement stems from the singular nature of the solution: there is only one correct answer, and this is why most mathematicians see their field as precise, pure and fundamental. Scientific truth is tested by experiment. Mathematical truth is tested by reasoning, which makes it more like philosophy or, better, law, a method that also assumes the existence of a single truth. While the other exact sciences take place in the laboratory or in the field and are maintained by an army of technicians, mathematics takes place in the mind. Nose breathing is the thought process that causes the mathematician to turn over in his sleep and wakes him up with a jolt to an idea, and the conversations that change, correct or confirm the idea.

"The mathematician does not need laboratories or equipment," wrote Russian number theory researcher Alexander Khinchin. "A piece of paper, a pencil and creative forces are the basis of his work. If you add to that the possibility of using a more or less decent library and a dose of scientific enthusiasm (which excites almost every mathematician), no obstacle can stop the creative work." The other sciences, as the work in them has been carried out since the beginning of the twentieth century, are by their very nature a joint activity; Mathematics is a process that occurs in units, but the mathematician always turns to another mind engaged in similar things. The tools of this conversation - the rooms where these vital discussions are held - are conferences, magazines, and nowadays the Internet.

The fact that Russia produced some of the greatest mathematicians of the XNUMXth century is undoubtedly a miracle. Mathematics was the antithesis of the Soviet way in everything. She advanced an argument; She studied patterns in a country that controlled its citizens by coercion and forced them to live in a changing and unpredictable reality; She gave importance to logic and consistency in a culture fed by rhetoric and fear; To understand it requires very specialized knowledge, which made the mathematical discourse a cipher that cannot be deciphered by an outsider; And worst of all, mathematics claimed a right to singular and knowable truths when the regime based its legitimacy on its own singular truth. All these things gave mathematics in the Soviet Union a special attraction in the eyes of those people whose thought demanded consistency and logic, which could hardly be achieved in any other field of research. This is also what made mathematics and mathematicians so suspicious. The Russian algebra expert Mikhail Tsfasman, in an attempt to explain what makes mathematics important and beautiful as mathematicians know it to be, said: "Mathematics is especially suitable for teaching us to distinguish between true and false, between what has proof and what has no proof, between what is proven and what is not appear to be. It also teaches us to distinguish between what turns out to be true and what turns out to be true, but is clearly false. It is a part of mathematical culture that is sorely lacking in [Russian] society as a whole.”

It makes perfect sense that the civil rights movement in Soviet Russia was founded by a mathematician. Alexander Yesenin-Volpin, a logician, organized the first demonstration in Moscow in December 1965. The movement's slogans were based on Soviet law, and the founders had one demand: they called on the Soviet authorities to obey the written law of the country. In other words, they demanded logic and consistency; It was a crime, and because of it, Yesnin-Wolfin was imprisoned in prisons and psychiatric wards for a total of fourteen years, and in the end he was forced to leave the country.

Soviet scholarship, and Soviet scholars, were meant to serve the Soviet state. In May 1927, less than ten years after the October Revolution, the "Central Committee" added a section to the laws of the Academy of Sciences of the Soviet Union stating exactly this. The status of an Academy member may be revoked, the section says, "if it turns out that his activities are aimed at harming the Soviet Union." From that point on, every Academy member is considered guilty of intending to harm the Soviet Union. Public hearings held for historians, literary scholars, and chemists ended in the dismissal of scholars, the denial of their academic titles and rights, and often also accusations of treason, and imprisonment. Entire fields of research - and especially genetics - were destroyed because they apparently contradicted the Soviet ideology. Joseph Stalin supervised the research personally. He even published his own scientific articles, thereby setting the research agenda in a certain field for years to come. His article on linguistics, for example, removed the cloud of suspicion that hung over comparative linguistics and condemned, among other things, the study of class distinctions in language as well as the entire field of semantics. Stalin personally promoted a staunch enemy of genetics, Lysenko tropes, and probably co-wrote a Lysenko lecture that led to a total ban on the study of genetics in the Soviet Union.

What saved the Russian mathematicians from destruction by state decree was a combination of three factors that have almost nothing to do with each other. First, chance ran and Russian mathematics was unusually strong at precisely the times when it suffered perhaps more than ever. Second, it turned out that mathematics was too vague a field for the types of tinkering favored by the Soviet leader. And the third factor - at the moment of truth it became clear that it could bring enormous benefit to the country.

In the XNUMXs and XNUMXs, Moscow could boast of a strong mathematical community; Pioneering work was done in topology, probability theory, number theory, functional analysis, differential equations and other branches that laid the foundations for the mathematics of the twentieth century. Mathematics is cheap, and that helped: when the natural sciences died out due to lack of equipment and even the lack of a musket place to work, mathematicians managed to get by with their pencils and conversations. "The lack of up-to-date professional literature was compensated, to a certain extent, by non-stop scientific communication, which could be organized and maintained in those years," Khinchin wrote about that period. A whole generation of young mathematicians, many of whom acquired part of their education abroad, got on the fast track to professorship and membership in the academy in those years.

The older mathematicians – those who built their careers before the revolution – were suspicious, of course. One of them, Dmitri Egorov, the great luminary of Russian mathematics at the beginning of the twentieth century, was arrested in 1931 and died after a hunger strike in prison. His crimes: He was religious and did not hide it, and he opposed attempts to make ideological use of mathematics - for example, he tried (unsuccessfully) to prevent a congratulatory letter from being sent from a gathering of mathematicians to the party's gathering. Those who loudly supported Agorov were expelled from the leadership of the Moscow Institute of Mathematics, but by the standards of the time it was more of a warning than a purge: no field of research was banned and no general line was imposed by order of the Kremlin. The mathematicians knew very well that they had to prepare for a bigger blow.

Indeed, in the thirties of the last century, all preparations were made for the opening of a mathematical show trial. Egorov's young partner in leading the community of mathematicians in Moscow was his first student, Nikolai Luzin, a charismatic teacher in his own right, whose many students called their circle "Lusitania", as if they were in some magical land or perhaps in a secret brotherhood united around a common imagination. Mathematics, when taught by a person with the right kind of vision, really lends itself to secret societies. As most mathematicians are quick to point out, only a handful of people in the world understand what mathematicians are talking about. When these people have the opportunity to talk to each other - or better, to form a group that learns and lives in harmony - it can be exciting.

"Lozin's militant idealism," wrote a colleague who informed about Lozin, "is expressed in detail in the following quote from a report he submitted to the Academy on his trip abroad: 'It seems that a set of natural numbers is not an absolute objective configuration. It seems to be a function of a mathematician's mind talking about a set of natural numbers at a given moment. Among the problems of arithmetic there seem to be some that cannot be solved in any way.''

Whistle-blowing was a work of thought: the recipient did not have to know anything about mathematics, and surely knew that solipsism, subjectivity and uncertainty are completely un-Soviet qualities. In July 1936, a public campaign was launched against the famous mathematician in the daily Pravda, and the fact that Luzin was "an enemy wearing a Soviet mask" was revealed.

The campaign against Luzin continued with press articles, community meetings and five days of hearings held for him by an emergency committee established by the Academy of Sciences. The newspaper articles presented Luzin and other mathematicians as enemies because they had published their work abroad. In other words, the events unfolded according to the usual scenario of a show trial. But then, the process seems to have ended in nothing: Luzin expressed public remorse and was severely reprimanded, although he was allowed to remain a member of the Academy. A criminal investigation into his infidelity was apparently shelved and quietly died.

Researchers who examined the Luzin case believe that it was Stalin himself who ultimately decided to stop the campaign of condemnations. The reason, in their opinion, is the absolute uselessness of mathematics in everything that concerns the needs of propaganda. "The ideological analysis of the case would have turned into a discussion of the question of how the mathematician understands a set of natural numbers, which seems a long way from sabotage, which in the Soviet collective consciousness was more associated with explosions in a coal mine or murderous doctors," wrote Sergey Demidov and Vladimir Isakov , two mathematicians who studied the case together when this was already possible, in the nineties. "Such a discussion would have been better conducted with more useful material for propaganda purposes, such as biology and Darwin's theory of evolution, which the great leader himself liked to discuss. Such a thing had a touch on ideologically charged and easy-to-understand topics: monkeys, humans, society, and life itself - much more promising than a set of natural numbers or a function of a real variable."

Luzhin and Soviet mathematics were lucky, very lucky.

Math made it out of the attack alive, but her legs were permanently bound. In the end, Luzin was humiliated and publicly reprimanded for his work in mathematics: for publishing in international journals, for contacts with colleagues abroad, for participating in conversations that are the lifeblood of mathematics. The message that emerged from Luzin's hearing, and was well internalized by Soviet mathematicians until the mid-sixties and largely until the collapse of the Soviet Union, was this: stay behind the Iron Curtain. They pretended that Russian mathematics was not only the most advanced mathematics in the world - that was its official slogan - but the only mathematics in the world. As a result, Soviet and Western mathematicians, unaware of each other's activities, worked on the same problems, which gave rise to some double-named concepts such as Chaitin-Kolmogorov complexity and the Cook-Levin theorem (in both cases, those who were eventually partners worked separately) . A senior Soviet mathematician, Lev Pontryagin, recounts in his memoirs that on his first trip outside the borders of the Soviet Union, in 1958 - five years after Stalin's death - when he was fifty years old and world famous among mathematicians, he had to ask colleagues if his last result was Really new; He had no other way of knowing.

"In the sixties, two people were allowed to go to France for six months or a year," said Sergey Gelfand, a Russian mathematician who now manages the publications program of the "American Mathematical Society". "When they traveled and returned, it was very useful for all Soviet mathematics, because there they could come into contact with other people and prove themselves, and prove to others, that even the most talented people, when they are constantly stewing in their own juice behind the iron curtain, do not see the full picture. They have to talk to others, and they have to read other people's works, and it works both ways: I know American mathematicians who learned Russian only to read Soviet mathematics journals." Indeed, a generation of American mathematicians is apparently able to read mathematical Russian - a difficult professional skill even for a native Russian speaker; Jim Carlson (Carlson), president of the Clay Mathematics Institute, is one of them. Gelfand himself left Russia in the early XNUMXs after being recruited by the "American Mathematical Society" to fill the gaps in mathematical knowledge created during the years of the Soviet regime: he coordinated the translation of the accumulated work of Russian mathematicians and its publication in the United States.

Thus, part of what Khinchin described as the mathematician's working tools - "a more or less decent library" and "ceasing scientific communication" was denied to the Soviet mathematicians. However, they still had the main prerequisites - "a piece of paper, a pencil and creative powers" - and most importantly, they had each other: mathematicians as a group escaped the first wave of purges because mathematics is too vague for propaganda purposes. In any case, in the almost four decades of Stalin's rule, it became clear that nothing is so ambiguous that it would be impossible to destroy it. The theory of mathematics would certainly have arrived if not for the fact that at a crucial point in the history of the twentieth century, mathematics left the realm of abstract discourse and suddenly became necessary: ​​what ultimately saved Soviet mathematicians and Soviet mathematics were World War II and the arms race that followed.

Nazi Germany invaded the Soviet Union on June 22, 1941. Three weeks later, the Soviet Air Force was annihilated: it was bombed and completely wiped out at the airports even before most of the planes had time to take off. The Russian military began refurbishing and upgrading civilian planes so that they could be used as bombers. The problem was that civilian planes were really slow compared to military planes, which put everything the military knew about targets into renewed discussion. A mathematician is needed to recalculate speeds and distances so that the Air Force can hit its targets. In fact, a small army of mathematicians is required. The greatest Russian mathematician of the XNUMXth century, Andriy Kolmogorov, returned to Moscow from the academic safe haven of Tatarstan and was put in charge of a class full of students armed with calculating machines, who recalculated the bombing and artillery tables of the Red Army. After this work was done, he proceeded to establish a new system of statistical control and forecasting for the Soviet Army.

At the beginning of World War II, Kolmogorov was 38 years old, and already a member of the presidency of the Soviet Academy of Sciences - which made him one of the handful of most influential academics in the empire - and world-renowned thanks to his work in the field of probability theory. He was also an incredibly prolific teacher: by the end of his life, he managed to supervise 79 dissertations and establish the mathematics Olympiads and the culture of Soviet mathematics schools. But during the war years, Kolmogorov left his scientific career to serve the Soviet state directly - thus proving that mathematicians are essential to the very survival of the state.

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 remained silent for months afterwards. And when he finally spoke publicly, after his so-called re-election in February 1946, he assured his countrymen that the Soviet Union would catch up with the West in developing its atomic capability. The effort to assemble an army of physicists and mathematicians that would equal that of the American "Manhattan Project" had been going on for at least a year; Young scientists were called back from the front lines and even released from prisons to join the race to the bomb.

Following the war, the Soviet Union invested enormous resources in advanced technological military research, and built more than forty entire cities where scientists and mathematicians worked in secret. The urgency of the recruitment was indeed reminiscent of the "Manhattan Project" - but the Soviet project was much larger and longer. Estimates of the number of people involved in the Soviet military initiative in the second half of the century are known for their inaccuracy, but they reach up to twelve million people, and two million of them were employed in military research institutes. For many years, a young mathematician or physicist who had just graduated was more likely to be sent to research that had a security connection, rather than to a civilian institute. These jobs meant almost complete scientific isolation: employees of the defense establishment had to pass security classifications whether or not they had access to sensitive military information, and any contact they had with foreigners was not only suspect but considered downright treason. On top of that, some of these roles required moving beyond the research cities, which provided closed and comfortable social environments, but no possibility of intellectual contact with the outside world. The mathematician's pencil and paper may be useless tools in the absence of continuous mathematical discourse. This is how the Soviet Union managed to hide some of its best mathematical minds from the eyes of the world.

Following Stalin's death, in 1953, the country changed its position regarding relations with the rest of the world: now the Soviet Union had to inspire not only fear but also respect. Thus, when it fell to the lot of the majority of mathematicians to help build bombs and missiles, to the part of a select few fell the task of building prestige. Slowly, at the end of the fifties, a tiny crack began to open in the Iron Curtain - not wide enough to allow the much-needed dialogue between Soviet mathematicians and non-Soviet mathematicians, but wide enough to present some of the most impressive achievements of Soviet mathematics.

In the seventies of the last century, the shape of the Soviet mathematical establishment took shape. It was a totalitarian system within a totalitarian system. She provided her friends not only with work and money but also with apartments, food and transportation; She determined where they would live, and when and where and how they would travel to work or recreation. To those belonging to the herd, she was a domineering and strict mother, but also caring and watchful: her children were well-fed and well-groomed, without any doubt a privileged group compared to the rest of the country. When basic goods were expensive, civil servant mathematicians and scientists could shop in specialty stores that had a greater selection of products than stores that served the general public, and were also less crowded. Since during most of the years of Soviet rule it was only possible to dream of a private apartment, ordinary Soviet citizens received their residence from the state; Members of the scientific establishment were assigned apartments through their institutions, and these apartments were usually larger and better located than those of the rest of their compatriots. And finally, one of the rarest privileges in the life of a Soviet citizen - traveling abroad - was possible for members of the mathematical establishment. The Academy of Sciences and the security organizations of the party and the state supervised this, and decided whether a mathematician could receive, for example, an invitation to give a lecture at a research conference, who would accompany him on the trip, how long the trip would last and, in many cases, where he would be accommodated. For example, in 1970, the first Soviet winner of the Fields Medal, Sergey Novikov, was forbidden to travel to Nice to receive the award. He received it a year later, when the International Mathematical Union met in Moscow.

But resources, even for members of the mathematical establishment, were scarce. The number of good apartments was always less than the number of people who wanted them, and there were always more people who wanted to go to the conference than those who were allowed to do so. It was therefore a cruel little world of knives in the back, shaped by intrigue, whistle-blowing and unfair competition. The barriers to entry to this club were high and daunting: a mathematician was required to be ideologically reliable and personally loyal not only to the party but also to the current members of the establishment, and the chances of acceptance for Jews and women were close to zero.

The establishment could have easily fired anyone for inappropriate behavior. This happened to Yevgeny Dynkin, a student of Kolmogorov, who encouraged an atmosphere of unreasonable liberalism in the school of mathematics he headed in Moscow. Another of Kolmogorov's students, Leonid Levin, describes how he was ostracized because he associated with opponents of the regime: "I became a burden to everyone I was in contact with," he wrote in his memoirs. "No serious research institute was willing to hire me, and I felt that I had no right to even participate in the seminars, since the participants were instructed to report [to the authorities] every time I appeared. My existence in Moscow began to seem pointless.” Both Dinkin and Levin emigrated. Probably shortly after Levin arrived in the United States, he learned that a problem he had described in mathematical seminars in Moscow (and relied in part on Kolmogorov's work on complexity) was the same problem defined by the American computer scientist Stephen Cook. Cook and Levin, who was appointed a professor at Boston University, are considered co-inventors of the "NP-Completeness Theorem", also known as the "Cook-Levin Theorem"; It lays the foundation for one of the seven "Millennium Problems" that the Clay Institute of Mathematics is offering a million dollars for their solution. The theorem says, in essence, that some problems are easy to formulate but require so many calculations that no machine can exist that would be able to solve them.*

And there were those who almost never became members of the establishment: those who happened to be born Jews or women, those who had the wrong instructors at their universities, and those who were unable to force themselves to join the party. "There were people who found out that they would never be admitted to the academy, and could at most hope to defend their doctoral thesis at some institute in Minsk, if they could secure connections there," says Sergey Gelfand, the publisher of the "American Mathematical Society", who happens to be the son of One of the leading Russian mathematicians in the twentieth century, the late Israel Gelfand, a student of Kolmogorov. "These people attended seminars at the university and officially belonged to the faculty of some research institute, say, of the wood industry. They did very good math, and at some point they even started making connections abroad and from time to time they could even publish in the West - it was difficult, and they had to prove that they were not revealing state secrets, but it was possible. Some mathematicians came from the West, some of them even came for a long stay, because they realized that there were many talented people there. It was informal math.”

One of the next for a long stay was the British Doza McDuff, an algebra expert (and today a professor emeritus at the University of the State of New York at Stony Brook). She studied with Gelfand Sr. for six months, and says that the experience opened her eyes regarding the way one should engage in mathematics - also through continuous conversations with other mathematicians - and regarding the true nature of mathematics. "It was a wonderful education, where reading the play Mozart and Pushkin's Salieri played just as important a role as learning about 'Lie Groups' or reading Cartan and Eilenberg. Gelfand amazed me when he talked about mathematics as if it were poetry. He once said of a long, formulaic essay that it contained the vague beginnings of an idea that he himself could only hint at but could never articulate more clearly. I have always seen something much more honest and open in mathematics: a formula is a formula, and algebra is algebra, but Gelfand found hedgehogs that peeked out of the rows of his spectral series!"

Officially, the jobs of members of the mathematical counterculture were generally undemanding and unrewarding, following the well-known Soviet labor formula, "We pretend to work, and they pretend to pay us." The mathematicians received modest salaries that increased only slightly over the years, but were enough to cover their basic needs and allow them to devote time to actual research. "There was no such need, to focus on work in one narrow field because you have to write faster, because you have to get tenure," Gelfand said. "Mathematics was almost a hobby. You could spend your time doing things that won't be used by anyone in the next decade." The mathematicians called it "mathematics for the sake of mathematics" - and drew a deliberate acceptance between them and artists who worked for the sake of art. There was no material reward in it - no tenure, no money, no apartments, no trips abroad; All that could be earned for brilliant work was respect from colleagues. Conversely, those who competed unfairly risked losing this honor without winning anything. In other words, the alternative mathematical establishment in the Soviet Union was unlike anything else anywhere else in the real world: it was pure meritocracy ("the rule of the deserving"), and intellectual achievement was its only reward.

In the seminars and lectures held after school hours, the mathematical discourse was reborn in the Soviet Union, and the attractiveness of mathematics to minds that were looking for challenges, logic and consistency proved itself again. "In the post-Stalinist Soviet Union, this was one of the most natural ways of self-realization for an intellectual who did not think in the conventional way," says Grigory Shabat, a well-known Moscow mathematician. "If I had the freedom to choose any profession, I would become a literary critic. But I wanted to work, and not waste my life fighting the censors." Mathematics promised not only intellectual work without the intervention of the state (although also without support from its side) but also something that could not be obtained anywhere else at the end of the days of Soviet society: a singular, individual truth that could be known. "Mathematicians are people of special intellectual honesty," continues Shabat, "if two mathematicians make contradictory claims, then one of them is right and the other is wrong. And they will surely understand this, and the one who made a mistake will surely admit his mistake." The search for this truth may take many years - but at the end of the Soviet Union, time stood still, and this meant that the inhabitants of the alternative mathematical universe had all the time they needed.

* A detailed explanation of the problem and the Cook-Levin theorem can be read in chapter 4 of David Harel's book, The Computer is All-Powerful, Attic Books and Yediot Books, 2007.