The Man Who Knew Infinity

by Robert Kanigel

  As a research scholarship holder, Ramanujan’s only obligation was to prepare reports every three months detailing his progress. He delivered three of them in late 1913 and early 1914, all dutifully on time. Like much of his work, the theorems he described had their roots in his notebooks; some went back to notes appearing around page 180 of his first notebook, some to chapters 3 and 4 of the second. Most of them dealt with evaluating definite integrals.

  “At present,” he wrote in his first report, addressed to the Board of Studies in Mathematics and dated August 5, 1913, “there are many definite integrals the values of which we know to be finite but still not possible of evaluation by the present known methods.” The theorem he offered—it would later be called “Ramanujan’s Master Theorem”—would provide means of evaluating many of them. “This paper,” his cover letter noted, “may be considered the first installment of the results I have got out of the theorem.” More, he promised, were coming soon—as indeed they were in the second and third reports.

  As in his letters to Hardy earlier that year, Ramanujan was attacking definite integrals that resisted every effort to reduce them to simpler, more useful forms, defeated the whole arsenal of mathematical tools brought to bear on them. Ramanujan was fashioning new tools.

  Like a screwdriver, saw, or lathe, a mathematical “tool” is supposed to do something; those used to evaluate a definite integral perform mathematical operations on it that, one hopes, get it ready for the next tool—the next theorem or technique—and ultimately lead to a solution. But just as a screwdriver tightens screws but can’t saw wood, a mathematical tool may work for evaluating one integral but not others. If you don’t know in advance, you try it. If it doesn’t work, you try something else.

  One tool, which Ramanujan had apparently encountered in an 1896 textbook on integral calculus, was Frullani’s integral theorem. Now, in late 1913, Ramanujan was telling of a powerful generalization of it that could defeat a wider range of formerly unyielding integrals. To apply, Frullani’s theorem demanded that two particular functions be equal; in Ramanujan’s generalization, they didn’t have to be, thus expanding its applicability to many more cases. Back in 1902, Hardy had written a paper on the Frullani integral. But he had never seen in it what Ramanujan saw now.

  While this time Ramanujan furnished proofs for many of his assertions, more analytical mathematicians would later shoot them full of holes. Yet the results themselves—the theorems Ramanujan offered as true—were true. Bruce Berndt, an American Ramanujan scholar, would see in that curious split a message for mathematicians today: “We might allow our thoughts to occasionally escape from the chains of rigor,” he advised, “and, in their freedom, to discover new pathways through the forest.”

  4. A DREAM AT NAMAKKAL

  While life was sweet to Ramanujan during this period, his long-distance correspondence with Hardy had soured. In the cover letter to his first quarterly report, in August, he cited Hardy much as the author of a book might quote favorable review comment: “The integral treated in Ex. (v) note Art. 5 in the paper, Mr. G. H. Hardy, M.S., F.R.S., of Trinity College, Cambridge, considers to be ‘new and interesting.’ ” But by then, in fact, the two of them were scarcely writing at all. Perhaps it was the tartness in their earlier exchanges. Or Hardy’s disappointment at Ramanujan’s refusal to come to England, and frustration at communicating across so formidable a physical and cultural gap. Or simply the press of other work. In any case, while Ramanujan wrote him at least once during this period, Hardy for months failed to respond.

  Finally, though, in early January, Ramanujan found a long letter from Hardy, responding to a proof he had earlier supplied, pointing out its flaws, and showing how they had led him astray. “You will see that, with all these gaps in the proof, it is no wonder that the result is wrong.” But, Hardy went on, walking on eggs, “I hope you will not be discouraged by my criticisms. I think your argument a very remarkable and ingenious one. To have proved what you claimed to have proved would have been about the most remarkable mathematical feat in the whole history of mathematics.”

  And oh yes, there was one more thing. “Try,” he added almost carelessly, “to make the acquaintance of Mr. E. H. Neville, who is now in Madras lecturing. He comes from my college and you might find his advice as to reading and study invaluable.”

  This was not exactly a lie, but it wasn’t the whole truth either. Hardy had more in mind than getting Ramanujan the right books to read. He had made Neville the instrument of his plan. He had deputized him to bring Ramanujan to England.

  Of course, that had been his intent from the beginning. As Snow put it, “Once Hardy was determined, no human agency could have stopped Ramanujan” from coming. Even before writing Ramanujan, he had contacted the India Office to that effect. But then, the answer had come back from Madras that, for religious reasons, Ramanujan had no intention of coming.

  The way Ramanujan told the story about a year later in a letter to Hardy, he had gotten a letter in February 1913 from Arthur Davies, secretary to the Advisory Committee for Indian Students in Madras. Could Ramanujan meet him in his office the following noon? Sir Francis asked Ramanujan’s “superior officer”—this, certainly, was Narayana Iyer—to accompany him and answer any questions.

  The next day, Davies dropped the big question: Was Ramanujan prepared to go to England?

  Ramanujan’s mind raced. What did the offer mean? Go to England and study math? Go to England and take examinations, and doubtless fail them? Would it mean … ?

  But before he could speak, Narayana Iyer—whom Ramanujan depicted as “a very orthodox Brahmin having scruples to go to foreign land”—replied that no, of course Ramanujan could not go to England. Then, according to Ramanujan, “the matter was dropped.”

  That, at least, was Ramanujan’s later version of these events. But almost certainly, it was a face-saving story concocted to explain his past unwillingness to go. What blocked his going was, indeed, Brahminic “scruples to go to foreign land.” But it was not Narayana Iyer’s scruples, but those of Ramanujan’s friends, and his family, and himself.

  • • •

  “His people shunned him for having lost his caste,” it was recorded of an early eighteenth-century South Indian who had traveled to Europe in defiance of his caste. “He was practically a dead man in their estimation.” While abroad, the man had been granted numerous honors, had been named Chevalier of the Order of St. Michael. But back in India, none of it counted; he was despised. When, generations later, another Indian, T. Ramakrishna, asked his mother whether he might travel to England to study, the story was invoked in warning. Ramakrishna waited for his mother to die before defying her wishes and making the trip.

  For an orthodox Hindu—and Ramanujan came from a very orthodox Hindu family—traveling to Europe or America represented a form of pollution. It was in the same category as publicly discarding the sacred thread, eating beef, or marrying a widow. And, traditionally, it had the same outcome—exclusion from caste. That meant your friends and relatives would not have you to their homes. You could find no bride or bridegroom for your child. Your married daughter couldn’t visit you without herself risking excommunication. Sometimes, you couldn’t go into temples. You couldn’t even get the help of a fellow casteman for the funeral of a family member. Here was the grim, day-to-day meaning of the word outcaste.

  A quarter century before Ramanujan, Gandhi had met similar obstacles in going to England for his education. “Will you disregard the orders of the caste?” its Sheth, or headman, had asked him. He did—and was pronounced an outcaste. By Ramanujan’s time, scruples against foreign travel had relaxed but slightly; for all but the most adventurous, it was still taboo.

  However much he may have yearned for England, Ramanujan found in Madras little reprieve from tradition’s hold. Madras was no oil-fed Houston, or railroad-driven Chicago, where ambitions surged and dreams were meant to be lived. Its population was static, barely changing in the past decade. It was flat,
built low, spread out over the countryside; there were no Himalayan peaks to draw the imagination upward, no great towers to serve as symbols of human aspiration. More than the other great cities of India, Madras clung in spirit to the villages and towns of the surrounding countryside. Calcutta and Bombay roiled over with unmarried workers from all over India, bore a sort of rude, masculine, Wild West dynamism. But Madras was more socially cohesive, with men from the villages bringing their wives and children with them. Like Ramanujan, the one in sixteen Madrasis who were Brahmins mostly inhabited the area around Triplicane’s Parthasarathy Temple. All around Ramanujan they lived—orthodox Brahmins who, like his mother, buttressed tradition and encouraged conformity. All around him the forces of social order were securely in place.

  And Ramanujan was no rebel. If he wavered in his acceptance of the bar on foreign travel, he was not about to say so—and was certainly not, on his own, going to defy it. For him to go to England, strong outside forces would have to be brought to bear. External voices he respected would have to sanction it.

  • • •

  Around New Year’s Day in 1914, Hardy’s man, Eric Harold Neville, arrived in Madras.

  An able mathematician who had never done “anything which has aroused any real enthusiasm”: that’s how Hardy assessed Neville a few years later. “I should expect [his work] to be good, but he is not a man like Littlewood of whom one would say it must be good.” That said almost as much about Hardy as it did Neville, of course, for Neville was already an important young mathematician. Just twenty-five, he had been among the last to take the old-style Tripos—sitting for it a year early in order to have a shot at becoming the last Senior Wrangler before Hardy’s reforms took hold. (He came in second.) Two years later, in 1911, Neville won the Smith’s Prize, and a year after that was named a Fellow of Trinity College. Now, in the winter of 1913, he’d come to Madras to give a series of lectures on differential geometry at the university.

  He had, of course, one additional charge—to convince Ramanujan to come to England.

  It was in the Senate House, the university’s examination hall and offices, located across the road from Marina Beach, that in early January the two men met. On the outside, the Senate House was an extravagant architectural blend of Italianate, Byzantine, and Indo-Saracenic influences, executed in brick and stone, shot through with rose windows, minarets, parapets, and chiseled stone pillars. On the inside, it was little more than a vast void—a great hall built to seat sixteen hundred people, with only a carved, fifty-four-foot-high ceiling and stained-glass windows to recall its ornate exterior. Here Neville had come to deliver the twenty-one lectures that, over the next month, would draw mathematicians from all over South India.

  After one of the first lectures, Ramanujan was introduced to him. This was not “the uncouth, unshaven, unclean figure of Ramachandra Rao’s picture,” Neville would write later, “but a man at once diffident and eager.” As for his English, once so poor it had undermined his school career, “ten years had worked wonders, for a more fluent speaker or one with a wider and better used vocabulary I have seldom met.”

  At least three times they sat down together with Ramanujan’s notebook. Neville was stunned—so much so that when, after their third meeting, Ramanujan asked him whether he might like to take the notebook away to peruse at his leisure, it struck him as “the most astounding compliment ever paid to me. The priceless volume had never before [so Neville assumed] been out of his hands: no Indian could understand it, no Englishman could be trusted with it.”

  Perhaps, Neville came to think, Ramanujan mistrusted the Englishmen he’d met in Madras. Perhaps he mistrusted Hardy, with whom he had corresponded only at a continent’s remove. But he, Neville, a fresh wind from afar, bore none of the old baggage. A pure mathematician, naturally sympathetic to Ramanujan’s mathematics, within a year or so of him in age, he had lavished three days on the notebooks that were Ramanujan’s life’s work. He had gained Ramanujan’s trust.

  And so now he struck, acting on Hardy’s charge to him: would Ramanujan come to Cambridge? Anticipating a negative response, Neville silently marshaled his arguments. Yet now, unaccountably, he didn’t need them. “To my delight and surprise,” he wrote later, he learned “that Ramanujan needed no converting and that his parents’ opposition had been withdrawn.”

  • • •

  What miracle had wrought this transformation?

  By one account it was K. Narasimha Iyengar, a family friend with whom Ramanujan had stayed in Madras early in 1911 and with whom he had remained in touch since, who helped get Ramanujan’s mother, certainly a key obstacle, to acquiesce to the trip. Seshu Iyer also exerted pressure on her, Ramaswami Iyer and Ramachandra Rao on Ramanujan. “I lent all the weight of my influence to induce him to go,” recalled Ramachandra Rao later. So did M. T. Narayana Iyengar, the Bangalore mathematician and editor of the Journal of the Indian Mathematical Society who had worked closely with Ramanujan three years before to get his first paper ready for publication; scrupulously orthodox himself, his arguments carried added weight.

  But if these influences may be said to have ultimately triumphed, they did not by themselves change Ramanujan’s mind. Something more was needed—something, at least for public consumption, beyond his mother’s merely human will, or Ramanujan’s. Neville learned what it was: “In a vivid dream his mother had seen [Ramanujan] surrounded by Europeans and heard the goddess Namagiri commanding her to stand no longer between her son and the fulfillment of his life’s purpose.” Details differ, but this and other versions of the story agree in substance—that permission for Ramanujan to go came personally through the intervention of the goddess Namagiri, residing in her shrine at Namakkal.

  • • •

  In the vicinity of Namakkal, eighty miles west of Kumbakonam, hills abruptly push up from the otherwise flat plain. Palm and banana trees can still be seen. So can rice fields. But overall, the vegetation is markedly thinner than along the Cauvery. Mud and brick huts have given way to stone, not much seen to the east, chiseled out of the surrounding hills and from rocky outcroppings jutting out from the fields.

  One of those outcroppings, more impressive by far than the rest, gave Namakkal its name. A town of seven thousand people, about twenty miles from the nearest train station (in Karur, near where Ramanujan’s mother and grandparents originally came from), Namakkal lay at the foot of a great white rock, two hundred feet straight up and half a mile around. Throughout the rock were tiny sacred grottos, jonais, formed over the eons where rock-rooted greenery and rainfall had conspired to wear away fissures. At the top, reached by narrow steps hewn into its southwest slope, stood an old fort protected by brick battlements three feet thick. To someone, the vertical gash that the great rock made in the sky suggested the vertical white streaks of the Vaishnavite caste mark, known as the namam. Hence, Namakkal.

  It was for here that in late December of 1913, Ramanujan, his mother, Narayana Iyer, and Narayana Iyer’s son set out; Janaki had asked to go, too, but Ramanujan told her she was too young. They got off the train in Salem, a city of seventy thousand set in a picturesque valley rimmed by mountains. There they stayed at the house of Ramaswami Iyer, who was deputy collector as well as founder of the Indian Mathematical Society. From there, Ramanujan wrote back home to Kumbakonam, then set out alone with Narayana Iyer, probably in a bullock cart, for Namakkal, about thirty miles due south.

  In Namakkal, they took a little road, flanked with houses built up to the edge of it, that gently climbed and curved up from the middle of town. Soon, near the base of the great rock, they reached the stone-columned facade and giant wooden doors that guarded the temple of Lord Narasimha, the lion-faced, fourth incarnation of Vishnu—and, in a separate, smaller, pillared shrine off to the left, his consort, the goddess Namagiri.

  Ramanujan’s family was not the only one upon whom the goddess exerted a powerful hold. On certain days of the week, zealous, frenzied women would wend their way to the shrine to be
exorcised of devils. “The hall in front of the goddess,” according to one account, would be “filled with their shrieks and convulsions, until a sprinkling of sacred water over their heads by the pujaris [priests] silences them.” Within the cramped passage where the sweating priests performed their devotions, the air was full of smoke, the stone walls black with incense.

  For three nights, Narayana Iyer and Ramanujan slept on the temple grounds. Sitting there on the stone slab floor, they could look up at the sheer rock face that formed the back wall of the temple, see the scalloped battlements of the old fort silhouetted against the sky. The first two nights, nothing happened. But on the third, Ramanujan rose from a dream and woke Narayana Iyer with word that, in a flash of brilliant light or some such similar revelation, he had received the adesh, or command, to bypass the injunction against foreign travel.

  Narayana Iyer’s family today believes that, with shrewd insight into Ramanujan’s psyche and mindful of his devotion to the goddess Namagiri, he conceived the trip to Namakkal; that so strong was Ramanujan’s wish to go to England and so strong his devotion to Namagiri, that something like what happened had to happen; and that when it did, Narayana Iyer calculated, he would be there to “correctly” interpret any revelation Ramanujan might have.

  Did a vision of Namagiri actually come to Ramanujan, in all earnestness, while he slept on the columned temple grounds at Namakkal? Or, as Neville heard it, did his mother have the dream, her importunings to obey its message firmly settling matters? Or did both mother and son dream of Namagiri? Or did Ramanujan now more ardently than ever wish to go to England and, having deferred to his family for most of a year, seek a socially acceptable way to do it?

  Certainly Ramanujan always attributed his decision to divine inspiration. (So did Joan of Arc attribute hers: “I locked myself up in the attic for a day and a night and God told me that I was to become a General of Heaven and lead the armies of France and expel the English.”) Known, too, is that in February 1913, so far as the India Office was concerned, Ramanujan refused to travel to England; that in December, he visited Namakkal; that a week or two later he met Neville at the Senate House and surprised him with his willingness to embark for England; that on January 22, he wrote Hardy, asking him and Littlewood to “be good enough to take the trouble of getting me there—within,” he scribbled between the lines, “a very few months.”