The Man Who Knew Infinity

by Robert Kanigel

  You could see it in the little book kept over the years that recorded suggestions and criticisms of meals served at High Table. Hardy had been among several to declare in 1909 that Berlin pudding and Hirsch-horn fritters ought never appear again on the menu. Later, he and Littlewood defended plum pudding with wine and camperdown sauce against those who would see it blacklisted; over the days and weeks, alliances were built around the issue, compromises forged. At another point, Hardy was embroiled in a battle over whether fruit pies should be served hot, subscribing to the view that, as another fellow put it, “a man who will eat a hot fruit pie is unfit for decent society.”

  It was good fun, evenings of camaraderie that eased days of hard solitary work. Littlewood got his share of it over the years, before he left for the army. So did Neville. So, of course, did Hardy, for whom High Table, with its light, frothy repartee, was his natural terrain. During the war, officers attached to the hospital unit or quartered at the college showed up. So may have a few Indians. But not Ramanujan. For him, dinner in Hall was a slice of Trinity life from which he was excluded.

  “Mutton, sir? Beef, sir?” the waiters would bustle around the noisy hall. That was the problem.

  Ramanujan was a vegetarian unusually strict in his orthodoxy—if not for South Indians generally then at least for those in England. In crossing the seas he had defied Brahminical strictures. He had forsaken his tuft. He mostly wore shoes and Western clothes. But as he had promised his mother, he clung fiercely to the proscriptions most central to Brahminic life, on food.

  The story is told of a hungry Brahmin who requests food from a man he meets along the road. He knows nothing of the man’s caste, or character, but he is too hungry to ask. Belly full, he sets out again and, after a few minutes, reaches the house of a Brahmin, who puts him up for the night. That evening, he spies a gold statue in the house, aches to have it, and under cover of night, spirits it away. All the next day, his guilt mounts. Finally, shaken by remorse, he returns to the house to give it back. “Oh, I knew you would steal something,” says the host. “You see, I saw you take food from that man yesterday, and I knew he was a thief.”

  It was this spirit that was built into Brahminic food prohibitions—that from whom you take your food matters. In accepting food from just anyone, who knows what sins he has committed in this or a former life?

  Brahmins varied, naturally, in the details of their observance. Some forswore onions and garlic on the theory that these foods raised sexual appetites, while others extended the prohibition even to cabbage and potatoes. Remembered details differ on what Ramanujan would and would not eat. One friend has him eating eggs, another not. Some said he refused onions, or even tomatoes. But all recall the absolute rigidity with which he clung to his observance.

  Early in his stay, he had once or twice ordered fried potatoes from the college kitchen. Fried in lard, joked another Indian, also a Tamil Brahmin, trying to get his goat. That was all it took. Whether true or not, Ramanujan never again ordered anything from the college kitchen.

  Instead, he cooked in the tiny, small-windowed alcove, equipped with electricity and a little gas stove, just off his sitting room. When he could get the ingredients, he ate what he’d eaten back in India—rice, yogurt, fruits, rasam, and sambhar, a thick, spicy, potato-laced vegetable stew. His friend Mahalanobis recalled him standing over that little stove, stirring vegetables over the fire. But it was something he did much more often alone.

  From his window, Ramanujan could look out over the roof of the adjacent building to the spired steeple atop the college Hall. There, at High Table during the long winter evenings of 1916, the candles flickered, the conversation hummed. But of all that, Ramanujan never shared.

  2. AN INDIAN IN ENGLAND

  Ramanujan was not the first Indian to come to England for an education and feel isolated from the alien world around him. Mohandas K. Gandhi, the future apostle of nonviolence and leader of the Indian independence movement, arrived in England in 1887, the year of Ramanujan’s birth. He wrote later:

  I would continually think of my home and country. My mother’s love always haunted me. At night the tears would stream down my cheeks, and home memories of all sorts made sleep out of the question. It was impossible to share my misery with anyone. And even if I could have done so where was the use? I knew of nothing that would soothe me. Everything was strange—the people, their ways, and even their dwellings. I was a complete novice in the matter of English etiquette and continually had to be on my guard.

  Between 1892 and 1906, Cambridge University admitted about twenty Indians per year, during Ramanujan’s time a little more. At any one moment, something like a thousand Indians were scattered among English colleges. In any case, their numbers and the magnitude of their problems in adjusting to English life were enough to spur government studies of them in 1907, in 1922, and again in later years. And the recurring theme of these studies, which is ubiquitous, too, in stories told by Indians over more than a century, was the maddening reserve, the unfathomable distance, of the ordinary Englishman.

  Sometimes, to be sure, Indians experienced downright racial prejudice; it was common enough, for example, that ugly rumors of it reached Ramanujan’s mother in India. But more often, it was the peculiar shyness of the English that left Indians feeling so adrift. “We think it must be admitted,” concluded a report by the Lytton Committee on Indian Students in 1922, “that the British and the Indian student each has his racial characteristics which imposes an initial barrier to intimacy.” Whereas the Indian, for his part, tended to be oversensitive to any hint of patronage, “the British has a reserve which causes him to be slow in making friends even with his own countrymen, and he is apt to regard with suspicion the first attempts of any stranger to cultivate his acquaintance.”

  Observed a student in a much later study: “The initial difficulty is to break the extraordinary reserve of the English people, their correct but cold behavior, formal, unemotional, courteous and decent to a degree, but detached, both to take sides or involve themselves.”

  A still later student complained of having to stock a ready supply of pleases and thank-yous. To him, it was just crude barter: you give me something, I give back a thank-you. Far worse, though, was English indifference. Some Indians learned to accept, even embrace, it as “respect for privacy,” but most simply saw it as unfeeling and cold. The conversational task of the Englishman, it could seem, was to be scrupulously correct yet remain untouched and unmoved, taking care always to convey the clear impression that, at bottom, he just didn’t give a damn. One Indian student would dream of dying, his body going undiscovered for weeks, reeking and rotting all the while. Neville would read the fact “that none of us, as far as I know, ever pressed [Ramanujan] for the true reason for his initial refusal to come to England” in 1913 as typical of “the reticence of his English friends.”

  Nowhere were these traits more prevalent, of course, than in the patrician class. In the House of Lords just before World War I, one of their number perceived a polite “detachment almost amounting to indifference.” Another complained of the “profound weariness and boredom” its members affected. One later visitor to England, the Indian Nirad Chaudhuri, wrote how “One evening, when dining at a club, I tried in my innocence to open a conversation across the table, and I admired the skill with which the intrusion was fended off without the slightest suggestion of discourtesy.” Fended off, though, it was.

  Back home, Indians recalled, people would come up to you, sit down, start talking, and in five minutes know all about you—whether you were married, had children, where you were from, what kind of work you did. Whereas in England, returning Indians advised their countrymen, formal introductions were de rigueur. One story, set in India, told of a swimmer whose cries for help sent everyone rushing to his aid. Everyone, that is, save the lone Englishman, who sat where he was, apparently unmoved. “Oh,” he replied when asked later why he’d not helped, “were we introduced?”

  C
ambridge boasted its own brand of aloofness. A book aimed at Indian students in England told how even college porters went about their duties “without the least concern about our new comer and with an air of indifference.” And a student of Ramanujan’s tutor, E. W. Barnes, once set Barnes apart from most other Cambridge dons who, he said, gave “the impression that they were not greatly interested.”

  Laurence Young, the son of two mathematical contemporaries of Hardy and Littlewood at Cambridge, told how when Littlewood, as an old man, visited him in Wisconsin he would row him out on to the lake to view the sunset. Littlewood never said a word, and Young, in time, surmised that he was bored. But when one day Young suggested that the water might be too rough for their excursion, “his face dropped … and I quickly looked again at the lake and pronounced it smooth. This is typical of Cambridge—what you admire, you merely do not speak of.” Littlewood himself would remark that “When you make your speech at a Trinity Fellowship Election, do not expect them to break into irrepressible applause; no one will blink an eyelid.”

  It was a wall erected around one’s feelings, a great silence of the emotions. In Cambridge, the emphasis was on ideas, events, things, work, games—anything, it seemed, but the deeply personal.

  At first Ramanujan may have marveled at these peculiarities of the English, much as Western tourists do the sight of untended cattle on the streets of Delhi or Madras. But he was not one to eagerly embrace foreign ways; rather, he was apt to stand apart from them. Stubborn, self-driven, self-willed, he was every bit the product of his country and its customs. Unlike some Madrasi intellectuals, he’d never lived a Westernized life in India; he was too much, and too recently, a son of Kumbakonam. He maintained an Indian’s deep-felt deference to the wishes of his parents. He’d cried when his kutumi was cut off. In England, he remained steadfastly vegetarian, kept a poster of an Indian deity in his room, and each morning performed the Brahmin’s fastidious morning ritual. He changed into a new, ritually pure dhoti, applied the namam caste mark on his forehead, performed his devotions, then wiped it off. Only then, if he were going out, would he don Western clothes.

  Even if he didn’t always let his mind wander back home, there were times when the small, familiar things of South Indian life insinuated their way into his awareness. The smells of his mother’s cooking on Sarangapani Sannidhi Street, or of burning cow dung in the streets of Madras. The bright colors of religious festivals parading down the streets of Kumbakonam, accompanied by the strumming and jingling of the musicians. The reds and oranges of sari-clad women along the banks of the Cauvery, white dhotis setting off the dark brown skin of laborers in the fields. The vibrant greens of the vegetables, the coconuts and bananas and mangos sold down by the market near the river. And always, the bright blue sky and high overhead sun.

  Among the English, Ramanujan could not long forget his foreignness. His musical accent was alien to their ears. His skin was darker by many shades than theirs—which, the winter chill bringing color to their cheeks, was more a rosy pink than anything you could call “white.” Everywhere were churches and chapels, Christian crosses, and Jesus Christ. Here, in this strange land, families scattered, children paying parents nothing like the respect that was a law of social life in India. Meanwhile, all Cambridge resounded to the sound of tramping feet bound for an alien war.

  Even the prevalence of body odors among the English mystified him—until, the story goes, one day he was enlightened about it at a tea party. A woman was complaining that the problem with the working classes was that they failed to bathe enough, sometimes not even once a week. Seeing disgust writ large on Ramanujan’s face, she moved to reassure him that the Englishmen he met were sure to bathe daily. “You mean,” he asked, “you bathe only once a day?”

  Ramanujan was not the kind of chameleonlike figure who does well at the tough job of reshaping himself to fit a foreign culture; he was not flexible enough, could not sink down effortlessly into his new English life. Nor could he long be immune to that succession of subtle, slight rebuffs the ordinary Englishmen dispensed, with scarcely a thought, every day. He would have needed to experience few incidents of aloofness and reserve to damp his sunny openness and send him running back to the cozy den of his mathematical research.

  And that, it seems, is what he did. He withdrew. “I remember him during those years, though I never spoke to him,” recalled B. M. Wilson later. “He was in fact very rarely seen.” For long stretches, he scarcely left his room. Back in India, he liked to work in the cool of the night, the better to escape the midday heat. Now, even without the heat, he worked at night, alone.

  Ramanujan was not the first foreigner to retreat into his shell in a new country; indeed, his was the typical response, not the exceptional one. One later study of Asian and African students in Britain observed that a sense of exclusion “from the life of the community … constituted one of the most serious problems with which they were confronted … [and had] a serious psychological effect” upon them. Another study, this time of Indian students in particular, reported that while 83 percent of them saw friends more or less every day back in India, just 17 percent did while in England.

  During the Easter Term of 1916, Hardy was a member of the Trinity Sunday Essay Society. So was E. H. Neville. So was Bertrand Russell. Ramanujan was not. While Hardy played tennis or whatever cricket could still be had in wartime England, Ramanujan remained steadfastly sedentary. When some of his friends joined the Majlis, the Indian students’ debating society (where, it has been said, “succeeding generations of scions of Indian aristocracy picked up their nationalism and radicalism”), Ramanujan did not. When Ramanujan’s paper on a type of Diophantine equation was read to the Cambridge Philosophical Society on October 30, 1916, it was Hardy who read it, not Ramanujan. When Hardy read a joint paper at a meeting of the London Mathematical Society on January 18, 1917, Littlewood was there to hear it, and so was Neville, and so was Bromwich—but not Ramanujan.

  There, in Bishop’s Hostel, Ramanujan was a shut-in.

  From the small west window of his sleeping alcove, the Gothic windows and stone walls of New Court loomed just a few feet away. The east window got whatever morning sunlight wasn’t blocked by the building across the way. But there usually was not much sun to block. The English skies were notoriously overcast. And around Christmas, Cambridge—which is as far north as Labrador—would be mostly dark by four in the afternoon. Even the long summer days were, by Madras standards, sunless; a typical July brought rain a dozen days a month.

  The war only made things worse. Cambridge and surrounding East Anglia, which projected out into the North Sea toward the Continent, were particularly exposed to zeppelin raids. In part to protect King’s College Chapel, streets were kept dark at night. Cambridge, someone would remember, “was wrapped in a medieval gloom for some three years and men had to grope their way about the streets as best they could.” If the England streetscape was normally gray and drear, now it was black, fairly pressing you back inside.

  In South India, the boundaries between inside and outside were not so fixed and immutable as they were in England, where you were forever trying to escape the chill. Walls and windows were more permeable. Insects, smells, and sounds brought outside inside. Chipmunks and lizards scampered through window shutters. Whereas in Cambridge, amidst the stony, solid permanence of its five-hundred-year-old walls, there was an ever-present sense of demarcation and division.

  So that in the winter, especially, Ramanujan’s apartment could feel like a prison. A plush-lined prison, perhaps, but a prison nonetheless. And while driven back into it by the English reserve, the winter chill, and the dark streets and wartime gloom, he was lured back into it by the delight he got from his work with Hardy.

  3. “A SINGULARLY HAPPY COLLABORATION”

  Mathematician Norbert Wiener would one day note how, in one sense, number theory blurs the border between pure and applied mathematics. In search of concrete applications of pure math, one normally turns
to physics, say, or thermodynamics, or chemistry. But the number theorist has a multitude of real-life problems before him always—in the number system itself, a bottomless reservoir of raw data. It is in number theory, wrote Wiener, where “concrete cases arise with the greatest frequency and where very precise problems which are easy to formulate may demand the mathematician’s greatest power and skill to resolve.”

  In 1916, one such problem lay in the area of number theory known as “partitions.”

  On the surface, the problem was so simple it went back to almost the first days of grammar school: 2 + 2 = 4. But that’s just one way to add up numbers to get 4. There are others. Like 1 + 3. Or 1 + 1 + 2. Or 1 + 1 + 1 + 1. And, lastly, to be scrupulously complete about it, just plain 4 itself. These (aside from mere rearrangements of the same numbers) are the only ways of adding up integers to get 4. Count them up, and you get five different ways, or “partitions.” Mathematicians say that the number of partitions of 4 is 5. Or,

  p(4) = 5

  More generally p(n), read “p of n,” represents the number of partitions of any number, n, and is known as the partition function. What, the mathematician wonders, can be said about it? How could we evaluate it for any n?

  In principle, it’s not difficult to go through all the possibilities for a given number and add them up. In principle. The problem is that the number of partitions rises very fast. The number of partitions of 3, p(3), is just 3 (3, 1 + 1 + 1, 1 + 2). But by the time you get up to 10, p(10) = 42. And p(50) = 204,226. Two hundred four thousand, two hundred twenty-six different ways of adding up the integers. Just listing them all, one every five seconds, would take two weeks. And that’s just for n = 50. So the question is, can you find a formula for p(n) that sidesteps the awful arithmetic and spits out the number of partitions for any number you please?