Perfect Rigour

by Masha Gessen

  Whatever the reason, his not being a part of the military effort left Kolmogorov free to devote his considerable energies to creating the world for mathematicians that he had envisioned since he was a young man. Kolmogorov and Alexandrov both hailed from Luzitania, Luzin’s magic land of mathematics, and they sought to re-create it at their dacha outside of Moscow, where they would invite their students for days of walking, cross-country skiing, listening to music, and discussing their mathematical projects.

  “The way our graduate group interacted with Kolmogorov was almost classically Greek,” said one of the countless memoirs published by his students; virtually everyone who had contact with Kolmogorov seemed to have been moved to write about him. “Through the woods or along the shore of the Klyazma River9 the muscular mathematician would be moving briskly, on foot or on skis, surrounded by young people. The shy students would be rushing behind him. He talked almost without stopping—although, unlike perhaps the ancient Greeks, he talked less of mathematics and more of other things.” Kolmogorov believed that a mathematician who aspired to greatness had to be well versed in music, the visual arts, and poetry, and—no less important—he had to be sound of body. Another of Kolmogorov’s students wrote in his memoir10 that he was singled out by the teacher for wrestling well.

  The mix of influences that shaped Kolmogorov’s idea of a good mathematical education would have been an odd combination anywhere, but in the Soviet Union in the middle of the twentieth century, it was extraordinary almost beyond belief. Kolmogorov hailed from a wealthy Russian family that founded a school of its own in Yaroslavl, a town about a hundred and fifty miles north of Moscow. There they published a children’s newspaper to which Kolmogorov, along with other family members, contributed. Here is a math problem he authored at the age of five:11 How many different patterns can you create with thread while sewing on a four-hole button? Don’t try solving this one until you have some time; I know two professional mathematicians,12 both students of Kolmogorov’s, who each came up with a different response.

  In 1922, Kolmogorov13—nineteen, a student at Moscow University, and already an emerging mathematician in his own right—started teaching mathematics at an experimental school in Moscow. Incredibly, the school was modeled after the Dalton School, the famous New York City institution immortalized by, among others, Woody Allen in the film Manhattan. The Dalton Plan,14 which lay at the foundation of both the Dalton School and the Potylikha Exemplary Experimental School where Kolmogorov taught, called for an individual instruction plan for every student. Each child would map out his own path for the month and proceed to work independently. “So every student spent most of his school time at his desk,15 or going to the small school libraries to get a book, or writing something,” Kolmogorov recalled in his final interview. “The instructor would be sitting in the corner, reading, and the students would approach him in turn to show what they had done.” This might have been the first sighting of the figure of the instructor reading quietly behind his desk; decades later, the math-club coach would take up this position.

  It was always a boys’ club. Kolmogorov himself referred to his students affectionately as “my boys,” reporting to Alexandrov, in a letter from a trip taken with his students in 1965, “In just three hours at an elevation of 2400 meters16 all my boys got so badly sunburned (parading around in their swimming trunks or without them) that they could barely sleep for two nights following.” The casual happy homoeroticism of Kolmogorov’s view of his students seemed to come from an entirely different time and place. Before the Iron Curtain sealed off the Soviet Union from the rest of the world, Kolmogorov and Alexandrov had done some traveling. Alexandrov, who was seven years older, had traveled extensively before the two met, but the pair spent the 1930–1931 academic year abroad,17 some of it together. They started out in Berlin, where all culture, and gay culture in particular,18 was flourishing. They imported all they could: books, music, ideas. “Interesting that this idea19 of a truly beloved friend seems to be purely Aryan: The Greeks and the Germans seem always to have had it,” Alexandrov wrote to Kolmogorov in 1931, a few years before the reference to Aryans would have had a different connotation. “The theory of a lone friend is a difficult one to fulfill in the contemporary world,” Kolmogorov lamented in response. “The wife will always have pretensions to that role,20 but it would be too sad to consent to this. In Aristotle’s times, these two sides of the issue never came into contact: The wife was one thing, and the friend quite another.” Kolmogorov brought back from Germany collections of verse by Goethe, who would always be his favorite poet. In all their letters to each other, Kolmogorov and Alexandrov included detailed reports of concerts attended and music heard, and when vinyl records became available, they started collecting them. Alexandrov hosted weekly classical-music evenings at the university; he would play records and lecture on the music and the composers; after Alexandrov’s death, Kolmogorov21—already nearing eighty and crippled by Parkinson’s disease—took over as host.

  Classical music and male bonding, mathematics and sports, poetry and ideas added up to Kolmogorov’s vision of the ideal man and the ideal school. At the age of forty, Kolmogorov wrote up a plan22 “of how to become a great man should I have sufficient desire and diligence.” The plan called for completing his research work by the age of sixty and devoting the rest of his life to teaching secondary school. He followed the plan: in the 1950s he enjoyed a second creative flowering, publishing as prolifically as he had in his thirties—very unusual for a mathematician—and then he stopped and turned his full attention to teaching children.

  In 1935, Kolmogorov and Alexandrov organized23 the first Moscow mathematical competition for children, helping to lay the foundation for what would eventually become the International Mathematical Olympiad. A quarter century later, Kolmogorov teamed up with Isaak Kikoin,24 an unofficial kingpin of Soviet nuclear physics who had run similar competitions in physics. Since the only value the State seemed to assign their sciences was military, the two conspired to make Soviet leaders believe that elite, specialized math-and-physics high schools could supply the country with the brains it needed to win the arms race. The project was championed by a young Central Committee member named Leonid Brezhnev—then five years away from becoming the Soviet leader. The Soviet of Ministers issued a decree25 creating the school in August 1963, and it opened in December of that year. Half a dozen similar schools soon opened in Moscow, Leningrad, and Novosibirsk. Kolmogorov’s students ran most of them, and he personally oversaw the shaping of the curriculum.

  That August, Kolmogorov organized26 a summer mathematics school in a town outside of Moscow. Forty-six high-school seniors who had done well in the All-Russian Mathematical Olympiad attended. Kolmogorov and his graduate students taught workshops, lectured the boys in mathematics, and took them hiking in the surrounding woods. In the end, nineteen boys were chosen27 to attend the new mathematics-and-physics boarding school in Moscow.

  They landed in a strange new world. Kolmogorov, who had been dreaming up the school for forty years, had developed not only a method of individual instruction based on the Dalton Plan but an entirely new curriculum. Lectures in mathematics28—a number of them presented by Kolmogorov himself—aimed to introduce ideas from the world of real research while taking into account the students’ varied backgrounds, for Kolmogorov emphasized choosing students who exhibited the presence of what he called “a spark from God”29 rather than a thorough knowledge of high-school mathematics. In addition, the boarding school was probably the only one in the Soviet Union that offered a high-school course in the history of antiquity.30 The curriculum also included more hours of physical education instruction31 than regular Soviet schools did. Finally, Kolmogorov himself lectured the students in music,32 the visual arts, and ancient Russian architecture. He also took the boys on boating, hiking, and skiing trips.33 “We liked the trips and the poems,” one of the students wrote in a memoir. “And few of us unders
tood the music:34 that required at least some background. Fortunately, [Kolmogorov] kept quiet on the importance of the social sciences.” In other words, Kolmogorov not only rushed to impress his students with his version of Renaissance values but also shielded them from the Marxist indoctrination to which they had been subjected in secondary school and which they would be forced to endure once again at the university.

  Kolmogorov’s goal was not just to create a handful of elite institutions for talented mathematicians but also to teach real mathematics to as many children as could learn it. He developed a curriculum that took schoolchildren out of the business of adding and subtracting and getting confused, and into the business of thinking about mathematics in clear and interesting ways. He oversaw a curriculum-reform effort35 that introduced the use of simple algebraic equations with variables and of computers as early as possible. In addition, Kolmogorov sought to revamp the secondary-school understanding of geometry,36 opening the way to comprehending non-Euclidean ideas. In the mid-1970s I attended one of the schools chosen to try out the new textbooks (this was not a specialized math school but an “experimental” school open to a much broader range of children). It must have been in third grade that I shocked my father, a computer scientist, with my understanding of the concept of congruence. It made perfect sense to me: two triangles, for example, were considered congruent if they were exactly the same in every way. The word equal, which older textbooks had used, was clearly less precise.

  Bizarrely, it was the subject of introducing congruence to schoolchildren that forced Kolmogorov’s first serious confrontation with the Soviet system—something he had avoided for decades, through luck and care. In December 1978 the seventy-five-year-old Kolmogorov was dressed down at a general meeting of the mathematics section of the Soviet Academy of Sciences. One after another, Kolmogorov’s colleagues rose to criticize him for the term congruence, for a difficult new definition of vectors used in the textbooks he oversaw, and for the introduction of set theory as the cornerstone of the math curriculum. These, the speakers claimed, were examples of a larger failing: the reform—and its authors—were evidently anti-Soviet. “These things can provoke nothing but disgust,”37 opined Lev Pontryagin, one of the leading Soviet mathematicians. “This is a disaster. This is a political phenomenon.” Newspaper denunciations followed: authors of the curriculum reform were exposed38 as having “fallen under a foreign influence of bourgeois ideology” of set theory. They had a point. Education reform just then under way in the United States and, indeed, throughout the Western hemisphere mirrored Kolmogorov’s efforts. The New Math movement brought actual mathematicians39 into active involvement in secondary schools; set theory was introduced in early grades and formed a basis for teaching all of mathematics. The Harvard psychologist Jerome Bruner observed at the time that it had “the effect of freshening [the student’s] eye40 to the possibility of discovery.” At the third-grade level, mathematics finally became accessible enough to be dragged through the pages of the Soviet newspapers—and Kolmogorov was exposed as what he most certainly was: an agent of Western cultural influence in the Soviet Union.

  The aging Kolmogorov never recovered from the scandal. His health deteriorated catastrophically; he developed Parkinson’s and lost his sight and, eventually, speech. Some of his students believed the illnesses were set off41 by the public disgrace and by a head trauma that resulted from what may have been an attack: walking through a university building, Kolmogorov was struck by a heavy door that he thought might have been swung deliberately by someone he then saw rushing away. As long as Kolmogorov was able, and perhaps a little longer, he continued to lecture at the boarding school. He died at eighty-four, speechless, blind, and motionless, but surrounded by his students, who for the preceding couple of years had taken turns42 providing round-the-clock care at his house.

  The ideological conflict that made Kolmogorov’s proposed reforms impossible was real. His plan called for dividing high-school students into groups depending on their interest and abilities in mathematics, allowing the most talented and motivated to get farther faster. The entire Soviet system of secondary education was based on the concept of uniformity: everyone was to be taught the same thing at the same time,43 using the same textbooks. But the Soviet Union still craved international prestige—in fact, that need became more and more pronounced as the technological rivalries of the second half of the century heated up. Just as the world of adult mathematics had to cultivate a certain number of geniuses to showcase at international conferences, so a small world of talented children had to be allowed to exist in a sort of greenhouse setting, if only for the country to field competitors at international math and physics olympiads. And just as it was in the world of adult mathematicians, in the world of student mathematicians, the space for comfortable existence was too small to accommodate all whose talents warranted inclusion; to get in, a Jewish child had to be twice as good as a non-Jewish child and four times as good as the child of an apparatchik.

  Possibly because there were so few schools, they were all fairly similar, shaped in the Kolmogorov mold of increased emphasis not only on math and physics but also on music, poetry, and hiking—in no small part because Kolmogorov’s students influenced most of these schools directly. They were all subject to heightened scrutiny: Kolmogorov’s boarding school was visited frequently by ideology inspectors, who became especially vigilant following the denunciation of his curriculum reform. School supporters were often called upon to defend it before the authorities, who claimed that “elite education is not allowable in our society”44; Moscow’s School 2 was apparently the object of many denunciations45—written by concerned parents and outraged Soviet-issue teachers—that eventually had its cofounders fired; and School 239 lost some of its most popular teachers to KGB pressure46 while its principal was frequently reprimanded for admitting too many Jewish children47 (according to historical lore, two out of four Leningrad math schools were shut down48 in the 1970s for having too many Jewish students). And the feature that united all the math schools was the sheer concentration of student brainpower, teacher talent, and intellectual urgency: the children had only two or three years to spend at the school, which was always on the verge of being discovered and shut down by the authorities.

  The selection of teachers assembled at these schools matched that of the best Soviet universities. In fact, for the most part, they were the same people. Kolmogorov brought his students to teach at his school, and those students, in turn, drafted their own students. Some teachers came because their children attended the school; some were strong-armed for the same reason. School 2 graduates recalled that when members of Moscow’s intellectual elite flocked to the school, the director set the price of admission: those parents who were college instructors49 had to offer electives at the school. As a result, the school’s bulletin boards overflowed with announcements50 of elective courses offered by some of the top names in various fields—more than thirty courses at one time. Clearly, if there had been more schools like these, the concentration of outstanding instructors at them would not have been so high. By trying to keep the number of schools low, the Soviet authorities had in fact created hotbeds of freethinking.

  “What made the school different51 was that the students’ talents and intellectual achievements made them more popular and significant,” remembered a Boston computer scientist who graduated from a Leningrad math school in 1972. In the world outside the school, peers respected one another for athletic achievements while the establishment rewarded proletarian provenance or Komsomol (the Communist youth organization) eagerness. Inside the school, the ideological demands of the outside world were flouted: some schools allowed students not to wear uniforms52 (though they were still required to put on a jacket and tie and keep their hair cropped); some teachers read forbidden works of literature aloud in class53 (though they avoided naming the author or the work). “What can be more beneficial at sixteen or seventeen54 than not having to li
e?” author Mikhail Berg wrote in a memoir of his years at a Leningrad math school. “You had an interview, you were admitted, and you became a member of a community in which the percentage of anything Soviet was many times lower than outside of it. You had to pay for the opportunity to breathe in this microclimate: Every day, spine bent, you had to deliver gifts to the altar of the idols—the two sisters Mathematics and Physics and their mother, Logic. Mathematics and rigid logic simply left no space for ideology: It could mix with logic no more than water can mix with kerosene.” Granted, these were still Soviet schools, complete with Komsomol organizations, denunciations, and “primary military training” classes, but compared with the rest of the country, the boundaries placed on speech and thought had been broadened so much as to seem almost nonexistent. The schools managed to create a bubble that resisted the pressure of the Soviet state. It protected both the students who paid mathematical dues in order to gain a measure of intellectual freedom, like Berg, and those who paid intellectual dues—studying antiquity, for example—in order to gain the freedom to study mathematics, like Perelman.