Jacob's Ladder

8 The Garden of Magnitudes

(1958–1974)

When they were still in the eighth grade, Grisha Lieber and Vitya Chebotarev went to the Department of Mathematics and Mechanical Engineering at the university to enroll in study clubs. About twenty boys and two (chance) girls began to live there in a unique, highly rarefied atmosphere. But even in this hothouse of young talent, Vitya stood apart. That year, he took first place among all the Moscow school-age children, and, what was most surprising, he also came in first among the ninth-graders. One year later, he won a prize at the first Math Olympiad of schoolchildren in Bucharest—true, only second place. This was more of a surprise than a disappointment to him. By that time, he was already used to being the best and the brightest in his age group. But he was not conceited, because he was a natural-born scientist, and there was no greater reward for him than solving a task or problem.

In the autumn of the ninth grade, Grisha took a book over to Vitya, who was home sick with tonsillitis. The book was Hausdorff’s Set Theory, a prewar edition, somewhat battered and dog-eared, which had passed through many hands and minds before coming to Vitya and changing his life in the most profound manner.

In the evening, after he had taken his prescribed pills and gargled before going to sleep, Vitya sprawled out on the divan, picking up the little book Grisha had brought over with instructions to guard it carefully—it was valuable. He had never seen anything like it! His sleep, and his tonsillitis, and his very sense of reality deserted him. He was hooked. With every page he read, he felt he changed, even physically. For several years, he had been trying to solve the most disparate and intricate problems, thinking he was doing mathematics; but it wasn’t until this night that he felt he had set foot in the realm of true mathematics. It was a whole new world of wondrous and varied sets. In the morning, he looked out the window and noticed that the world had not changed a bit, and it was incomprehensible to him how buildings could even remain standing, and not collapse in a heap, when there were such wonders in the world as he had discovered in this little book.

Vitya had never read the well-known lines of Mandelstam, but he experienced the same emotion the poet describes in his obscure, redolent, pulsing words:

And I step out of the space of the world

and into the garden of magnitudes

and I rend the illusory permanence

and self-evidence of reasons.

And your very own primer, eternity,

I read all alone, in solitude—

Wild, leafless book of medicinal lore—

problem set of enormous roots.

In short, he had ended up in that garden. It was impossible to imagine anything more wonderful.

By the tenth grade, Vitya had become a real mathematician. His massive, somewhat convex forehead—like that of a child with a mild case of hydrocephalus—contained the brain in which the expanding universe moved, breathed, bubbled, and frothed. All other natural functions of the organism—eating, drinking, sleeping, etc.—were mere obstacles to the constant working of the rest of his highly functioning mind. Very little interested him apart from mathematics, and even his friendship with Grisha waned a bit. Grisha no longer satisfied him as an interlocutor. To be more precise, the pleasure he got from the music of mathematics so exceeded all other joys, including that of spending time with other people, that he began avoiding everything “extraneous.” Even physiological growth into manhood was an inconvenience or a malady to him, something like tonsillitis, which prevented him from studying. During that period in adolescence when young people are in the throes of hormonal revolution, Vitya found a simple method for releasing the tension that gripped him: overtaxing his mind.

Nora, who inhabited the outskirts of the world of Vitya’s interests, took the initiative at this timely moment to change her status from tutor in literature to friend and companion in sexual activity, and readily accepted his newfound manly maturity. She was a premature, illegitimate offspring of the sexual revolution, of which she knew nothing—if you didn’t take into account Marusya’s bold but old-fashioned pronouncements about the full emancipation of women in the socialist world, spoken in a whisper for fear the neighbors would overhear her.

Vitya was grateful to Nora for liberating him from the yoke of his hormones, a relief he experienced immediately after each one of their short, stormy meetings. Business meetings … The marriage prank they staged for the graduation ceremony had no effect on their relationship. Sometimes Vitya went to see Nora, with a friendly but purposeful aim; sometimes Nora would call him. They would come together, then part ways, without discussing when they would see each other again. Sometime or other … Vitya devoted all his energy to another romance—mathematics. Nora did her artwork with supreme pleasure, attended lectures on the history of theater, and read books.

Vitya was admitted to the Department of Mathematics and Mechanical Engineering, and from the first year of his studies dived headlong into set theory, a field of mathematics that had arisen relatively recently, in the mid-nineteenth century—a field that seemed to appeal greatly to madmen and suicides. It also beckoned to Vitya. Human fates, characters, and biographies did not yet stand behind the names of theories. It wasn’t until a few years later, when the Russian translation appeared of a multivolume work on mathematics and its history written by a group of mathematicians who had adopted the pseudonym Nicolas Bourbaki, that Vitya found out about the founder of the entire field. Georg Cantor, born in St. Petersburg, was the originator of the notion of actual infinity. A philosopher, musician, and Shakespeare scholar, he lost his way in the world of his own creation and died in a psychiatric clinic in Halle. He left behind (in addition to all of the above) what is known as Cantor’s Problem or the Continuum Hypothesis, which, as the next generation of mathematicians would claim, was possible neither to prove nor disprove. Vitya learned about the death of Felix Hausdorff, who took his own life in 1942 before he could be sent to a concentration camp, about Hausdorff spaces and the Hausdorff Paradox, which were his legacy to his descendants, and much else besides, concerning not so much mathematics as mathematicians.

Vitya spent the entire fourth year writing about computational functions, which thrilled his department chair, who was also quite an eccentric fellow.

The university administration, forced to consider the outstanding achievements of the department head, a world-renowned scholar, forgave his eccentricity; but Vitya, his student, did not get off so lightly. In those days, the Communist Party representative called the shots, and the dean’s office was beholden to him. The students were kept on a tight leash—mandatory Komsomol meetings, political briefings, “volunteer” social work. From time to time, Vitya was taken to task for disregarding the laws of existence. Once, he was barred from taking an exam for skipping PE classes. Another time, he was nearly expelled from the university for what became known as the “potato-carrot incident.”

Every September, all the students were sent “potato-picking” on communal farms. Those who were best suited to the conditions of Soviet life managed to procure medical exemptions beforehand. As secretary of the Housing Management Committee, Varvara Vasilievna had connections all over the district, and getting hold of the desired certificate would have been a breeze for her; but Vitya hadn’t even thought to ask for one, and now he had to fulfill his Komsomol obligations.

The students worked with great enthusiasm this time, since Dennikov, the Komsomol organizer in their class, promised that they would be released as soon as they dug up all the potatoes from the immense field. Inspired by this promise, the young people worked from sunrise to sunset, took in the entire harvest in two weeks, and were glad that they had earned fifteen extra days of free time. Toward the end of the harvest, however, Dennikov disappeared. He had been recalled on important Komsomol business, and the Party comrade who replaced him announced that now they would pick carrots. The rains started the same day.

The students howled and went into the carrot fields. Not all of them, however—some of them refused on principle, and left for home. Vitya also left—not out of principle, but because of illness. He had a terrible cold and a fever, so he took to his bed and gave himself over to mathematical dreams. He experienced what in later years he would call “intuitive visualization.” He even attempted to describe this experience of the world of sets as forests or lacework of profoundly beautiful ties and nodes, moving in space, and having no connection whatsoever with crude reality, where the teakettle boiled, and sometimes all the water boiled off; where the indestructible cockroaches persecuted by Varvara Vasilievna roamed the kitchen; where the exhaust fumes from Nikitsky Boulevard came in through the window and filled up his ground-floor basement room. The attempt failed.

Misty visions the mind couldn’t comprehend alternated with a semiconscious state in which the shadow of Nora was present. She offered him amazing objects on a large flat plate made of bright metal, and these objects were algorithms, and they were alive—they stirred slightly and interacted with one another. Vitya felt that there was some exquisite idea he had to write down, but something was missing, something was always missing. A tall man was walking down a long corridor with a shining hole at the end and carrying the same dish that Vitya had seen in Nora’s hands, and on the dish were the same creatures—which were the theory of functions and functional analysis. The man’s name was Andrei Nikolayevich, and Vitya desperately needed to have this man notice him, but because of some unspoken law he didn’t dare call out to him, so Vitya had to wait until the man noticed him first. Then the scenery changed, and the tall man went away; the dish with the algorithms ended up in Vitya’s hands, only they were all dead and didn’t stir any longer, and horror engulfed him.

He was sick for a long time, and there were complications. Just at the time he returned to the university, there was a meeting at which the students who had abandoned the potatoes, or, rather, the carrots, were expelled from the Komsomol. Their fate was predetermined: after expulsion from the Komsomol, expulsion from the university followed automatically. Vitya Chebotarev’s case was considered separately. He had a medical certificate attesting to his illness, but it was dated two days after the exodus from the carrot fields.

Logically, he was guilty and did not deserve pardon; but from a humane perspective, he really had been ill. Moreover, there were other factors of a purely medical nature that had bearing on the matter: the two days prior to the issuing of the certificate could have been the incubation period of the illness, when the symptoms had not yet manifested but the infection was already hard at work trying to undermine the host organism.

In short, taking into account the above-mentioned circumstances, they gave Vitya a break in the form of a reprimand, though all the other offenders were kicked out of the Komsomol.

While he was sitting in the Komsomol meeting, he made an effort to remember why he had ever joined. This detail had completely vanished from his memory. Then he remembered—his mother had insisted. Yes, Varvara Vasilievna considered it necessary. She herself was a Party member, and knew there were certain matters in which you had to be like everyone else, and even a bit better—so as not to violate the laws of life. Vitya, who had never objected to his mother’s trivial demands, had submitted his application for acceptance to the Komsomol in the eighth grade with the same insouciance with which he applied for a marriage certificate two years later.

In matters that did not interest him, Vitya never made a show of principle. This time, however, he suddenly sensed the injustice of it all: they had been deceived. They were promised that if they dug up all the potatoes, they would be allowed to leave, but they were prevented from leaving. So in what way were they to blame? For believing what they were told? This was fraud!

“What do you think you’re doing, you idiot? Stuff it!” Slava Berezhnoi, a friend and fellow student, whispered. “You won’t help us, you’ll only make things worse for yourself.”

And that’s just what happened: Vitya was expelled, too. Shocked and shaken to the core, he went home and lay down on the divan. And refused to speak. Varvara Vasilievna wasn’t able to ferret out the facts from him, but she put two and two together and decided that Nora, her mythical daughter-in-law, was the culprit in Vitya’s depression. By this time, she and Nora had already met, and Varvara Vasilievna managed to get hold of her telephone number, which wasn’t all that difficult for someone who worked in the Housing Management Committee. She called Nora, but wasn’t able to get a straight answer from her. She believed Nora was beating around the bush.

A week later, Slava Berezhnoi came to visit him and explained everything to Varvara Vasilievna. But Vitya refused to discuss anything with Slava, either, and was taciturn the whole evening. Nevertheless, Varvara Vasilievna, now that she understood everything, went to the university and straight to the Party committee, where she spoke with the department representative heart to heart, communist to communist. He understood the situation on a human level: it’s hard for a single woman, a soldier’s widow, to raise a son by herself … It must be said that Varvara Vasilievna embellished the truth somewhat, elevating the rather unseemly circumstances. She wasn’t exactly a soldier’s widow, and not really a widow at all, but she did say things that were perfectly true: Vitya had fallen into a depression, and Varvara Vasilievna had managed to pull him out of it with good medication, an effort that took her three whole months. Vitya was reinstated in the Komsomol, and they didn’t expel him from the university. The head of his department also weighed in on the matter: though frightened, the old crank didn’t want to lose an outstanding student. “The future of Soviet mathematics”—those were his very words.

Though Vitya remained enrolled at the university, he was granted a leave of absence; he had been traumatized by the whole affair. He had discovered that life consisted of more than a roll with salami for breakfast, mathematics, and the episodic Nora. These heretofore unknown difficulties were something he had wanted neither to notice nor to know about. He had developed no immunity to hardships like this, a fact that would cause him no end of suffering later in life.

Unlike her son, Varvara Vasilievna was adept at handling the everyday contingencies of life, and it was not for nothing that she worked for the Housing Management Committee. She acquired a valuable certificate from the Neuropsychiatry Clinic attesting that Vitya Chebotarev was subject to fits of depressive psychosis, though he was otherwise absolutely healthy. And, as later life would prove, she had not done this in vain.

Everything fell into place. Vitya successfully defended his honors thesis and was admitted into the graduate program. Three years later, he was ready to defend his doctoral dissertation on an absolutely new subject: “Computable Operations on Sets.” It is impossible for a nonmathematical mind to grasp, and even for some mathematical minds, but at the preliminary defense, Professor N, a brilliant proponent of the newest branch of “constructive mathematics”—not yet widely accepted, but very highly regarded in the Department of Mathematical Logic—advanced sharply critical views, reproaching the defendant for failing to follow the principles of this very constructive mathematics. Vitya did not accept his premises, and calmly held forth, insisting that the most constructive objects, including his beloved algorithms, could be viewed within the framework of classical logic and mathematics, a framework that was accepted in all the other departments. This unleashed a dispute for which Vitya’s dissertation was only a pretext, because underlying the scholarly issues was a discord in personal relationships that Vitya was not privy to. Vitya listened to the noisy quarrel and couldn’t make heads or tails of what either his defenders or his opponents were arguing for or against. When he tried to interject something, he couldn’t get a word in edgewise—so he quietly stood up and left the auditorium.

The argument dragged on for a long time after he left; the preliminary defense was aborted. Vitya followed the well-worn path to the divan, where he lay for another three months.

Varvara Vasilievna also followed the beaten path—to the Neuropsychiatry Clinic, where medication was prescribed for her son, after which he gradually recovered.

Meanwhile, the year 1968 had come and gone. Vitya was not aware of a single event rocking the socialist world. When his buddy Slava Berezhnoi, who dropped in from time to time to talk about important things, discovered the absolute political infantilism of his friend, he said, “You are simply another Luzin.”

Vitya’s feathers were somewhat ruffled, since he held Luzin in very high esteem as a mathematician.

“What do you mean by that, Slava? What does Luzin have to do with anything?”

Slava related an anecdote that Professor Melnikov had told during a lecture. After the war, the eminent Luzin took part in a seminar, at which he said, “In 1917, the greatest event of my life took place. I began to study trigonometric series.”

“And? What did he say next?” Vitya said, his curiosity piqued. (He also had a great deal of respect for Melnikov.)

Slava was astonished by his naïveté. “Nothing. Everyone associated the year 1917 with another event!”

“Which one?” said Vitya.

Slava waved his hand, exasperated. “Vitya, 1917 was the year of the October Revolution!”

“Oh … Oh, I see.”

Vitya’s dissertation adviser, who was also the head of the department, paid him a visit at home two weeks after the preliminary defense fell through. By this time, Vitya was already coping with his new trauma, and had started thinking about the future again. The two specific criticisms made by his opponent that had ultimately prevented the defense from passing, which concerned Lemma 2.2 and Theorem 6.4, contained a kernel of thought that Vitya began mulling over deeply. He himself had already discerned a few defects, or, rather, inconsistencies, in his dissertation. This disturbed him, and he plunged into the debris of variable and branching sets, which reached far beyond the boundaries of the poor three-dimensional world.

The head of the department spent two hours in the semi-basement apartment by the Nikitsky Gates, and left, saddened that his student had abandoned the actual (according to his notions) field of mathematics and leapt into a realm where those damaged by the burden of highly developed intellectual capacities grazed. These were the professional risks of being a mathematician, and the professor had already witnessed two such breakdowns in his life. Regrettable. A very capable young man, perhaps even with a dash of genius, Vitya had completed all his coursework but now refused to defend his dissertation. No job, of course. No means of subsistence. What could the professor do for him? No, there was no way he could help him.

But in this case the professor was at least partly mistaken. For six months, Vitya gnawed at the lock and bars of the theorem, and managed to wriggle out of the seemingly hopeless situation in an unexpected, almost miraculous way. He sat down and wrote a paper. Then he called Nora. She received him, somewhat bewildered but quite glad. He spent three days with her, and a glimmer of tenderness even appeared in their relations.

As he was leaving, he asked Nora, “Maybe we ought to get married for real? Things are going so well.”

“But where are they going? We’re already married,” Nora said, laughing. “Live together, you mean? At your place?”

“Well, no,” Vitya said, imagining life with Nora and Varvara under the same roof. “Perhaps if we live at your place?”

“Here? I’m sorry, that’s not going to work.”

Nora was surrounded by people of all stripes and colors—a most varied assortment of artists, actors, people with one foot in the theater or none at all, gifted, fascinating people who adored being on display. Not one of them was truly unique, or free of at least a touch of vulgarity and superficiality. They all aspired to be geniuses; but geniuses they were not. Vitya came closest of all of them to being a genius. Nora had already guessed this when they were still in school, and required no evidence or proof of the fact. But she couldn’t very well keep him in the house!

Vitya had a few mathematician friends who knew his worth—his trusty friend Grisha Lieber, and Slava Berezhnoi. Who needs lots of friends, anyway? Vitya was not fine-tuned emotionally, and was incapable of conversation on subjects of general interest, so he was condemned to a life of strictly mathematical friendship.

It was Slava Berezhnoi—who had been expelled from the university over the “carrot affair,” afterward graduating from evening courses at the Moscow Higher Technical University and taking up computer programming in the early days of the field—who found a position for Vitya in the computing center. The work suited Vitya down to the ground. From the theory of algorithms to programming was just a short leap. Mathematics had never promised to be of any practical use to Vitya; it was a captivating mental game. On the other hand, algorithms written in an artificial, simple, and logical language facilitated the solution to the most various problems, problems that were actually not connected to mathematics at all.

The directors of the center valued him, and Slava took more pride in Vitya’s achievements than in his own. For the first time in his life, Vitya received a salary, which he spent on books about mathematics and on expensive sweets. He had more than just a sweet tooth—he was addicted to glucose. Couldn’t live without it.

His work left him some spare time. He moved away from the strictly defined tasks of programming, solved several problems that he, in part, created, and even wrote two papers for a scholarly journal. One of them, however, the one Vitya considered to be the more successful, was returned to him with a negative review, very rude in tone. This offended him, and he took back both articles. Faced with this unfair criticism, Vitya, after a moment’s consideration, sent the papers off to an American mathematical journal. He only found out they had published them a year later.

At the same time, thanks to Vitya’s blunt honesty, he became embroiled in a conflict with the director of the center, Bogdanov. By the standards of the day, Bogdanov was a decent man, but a careerist. Not long before, he had received a secret award from the government—part of the work of the computing center was classified, involving sensitive military matters—for a new program that was intended to put the West at a disadvantage, with the aim of not merely catching up to it but outstripping it altogether.

Bogdanov was the nominal head of the project, but he took no part in developing or implementing it. He could not have done so even if he wished to, because he understood almost nothing about computer programming. A Party man, not a scientist, he compensated for his inadequacies by insisting on adding his name to the collection of authors.

A group of five people were involved in the project, including Vitya as the senior member, and, as the junior member, Amayak Sargsyan, who was a student intern from the Moscow Institute of Physics and Technology. He had a fine mind, it must be said.

There was a great deal Vitya did not know about the administrative organization of the computing center. The computer itself took up an entire building. It was filled with punch cards and young women, who were in charge of delivering them from point to point, so computations, among other things, involved their rushing energetically between floors to the rhythmic clip of their high heels. Vitya did not even suspect the existence of another, invisible level, that of relations between people. In short, at a certain moment, when the program was ready to be sent to the higher-ups for final approval, Vitya noticed that the name of Bogdanov, who had made no contribution to developing the program, headed the list of authors, while the name of the highly competent student, who had helped Vitya a great deal, especially in debugging the program, was missing altogether.

Vitya went to see Bogdanov during his office hours. Perhaps if he had begun the conversation more diplomatically the matter would have had a different outcome. But Vitya began by saying that it was unjust for Bogdanov to put his own name first on the list of authors, since he didn’t have a clue as to the merits and faults of the program, whereas the name of Sargsyan, who had taken part in the project and made a genuine contribution, for some unknown reason was absent. Bogdanov answered dryly that he would look into it.

After this meeting, Vitya could no longer gain admittance to Bogdanov’s office. He showed up at the weekly office hours time after time, until the secretary whispered to him that he should just give up—it was pointless. Vitya then forced his way into the office and caused a bona-fide scandal. He even shouted something about the interests of the state, which the director wasn’t taking into account. Poor Amayak Sargsyan was immediately fired from the computing center. He was not permitted to defend his honors thesis, and, being an exceptionally thorough and conscientious person, he wasn’t able to write a new one in time. Vitya’s hunger for justice resulted in a great deal of misfortune for Amayak, but it strengthened his faith in humanity.

A month and a half later, Vitya himself lost his job. He felt both indignant and dejected—not so much because his name was also removed from the list of authors as because he did not understand the first thing about this whole predatory and cruel operation.

Vitya lay down on the divan again without a word. He did not intend to search for a new job, and he refused to answer his mother’s questions. Varvara Vasilievna, who still continued to hope that her son was a genius, began to doubt the wisdom of that elderly psychiatry professor who, shortly before his death, had predicted for Vitya a unique and outstanding role in life. Where was it, then? Where was it?

Vitya himself was completely oblivious to the idea of any unique mental endowments or gifts he might have had. After he was dismissed from the computing center, he continued, through inertia, to think up new programs. When he had been lying in bed for a long time, he realized that his initial program could be improved. He began elaborating something that he could no longer have presented to anyone. Such was the program that his own mind was running: his brain could not live without intellectual activity, just as an ordinary person cannot live without food. He would have been glad to do something else, to live in some other way, but he knew not how. He sank deeper and deeper into a sleepless depression, until Varvara Vasilievna realized it was time to call in the doctors. This was the same trap that had snared him before his defense of the ill-starred dissertation.

It was cold and rainy, an autumnal spring. Tengiz had gone away forever. Again. Nora intended to start a new life. She called Vitya and invited him over. He came. While he was eating frankfurters he told Nora about what a bastard his boss had turned out to be. He explained to her how a good computer program differed from a bad one. Nora listened to him with half an ear, then made a beeline for the bedroom.

Vitya fulfilled the task entrusted to him seriously and honestly. And a new life began: for Nora, a pregnancy; and for Vitya, ever deeper depression.

Yurik was born early in the year 1975.