The Man Who Knew Infinity

by Robert Kanigel

  Other laudable traits the British discerned in the Indian personality verged on damning with faint praise. Of the South’s Dravidian stock, one Briton wrote, they were “hardworking, docile and enduring. They are more sober, self-denying and less brutish in their habits than Europeans. They show greater respect for animal life, they have more natural courtesy of manner, and, as servants, attach themselves to those who treat them well with far greater affection than English servants.” However seemingly admirable, these were hardly traits that left Indians the equals of their British masters.

  It was with some such blend of due regard for his intellect, coupled with a lingering dubiety about his character and temperament, all seen across a vast social divide, that the British would tend to view Ramanujan.

  7. THE LETTER

  Through Narayana Iyer, Ramachandra Rao, Presidency College mathematics professor Middlemast, and others testifying to his mathematical gifts, it had become clear around Port Trust offices by late 1912 that Ramanujan was something special. The question facing Sir Francis and other British officials was how special? And in just what way special? And what, in any case, were they to do with him? Were his gifts trivial, or profound? Were they the gifts of the genius or, intellectually speaking, the shaman? Was Ramanujan a minor oddity who could be safely dismissed, or a prodigy demanding nurture and guidance?

  No one would go out on a limb. The harsh explanation is that, despite firm opinions, they were afraid to, on the chance that events might prove them wrong and history judge them harshly. The kinder and simpler explanation—and the more likely—is that they just didn’t know, and knew they didn’t know.

  Narayana Iyer, of course, thought he did know; he worked with Ramanujan every day and saw his abilities up close. If Ramanujan was to achieve his promise, it was clear, he needed the British solidly in his corner. Narayana Iyer lobbied on his behalf with Sir Francis.

  Drawing on his connections, Ramachandra Rao also tried to gain Spring’s ear. “Dear Sir Francis,” wrote C. L. T. Griffith, a forty-year-old civil engineering professor at Madras Engineering College, on November 12, 1912, apparently at Ramachandra Rao’s behest. “You have in your office an Accountant, on Rs 25, a young man named S. Ramanujan, who is a most remarkable mathematician. He may be a very poor accountant, but I hope you will see that he is left happily employed until something can be done to make use of his extraordinary gifts.” Since few could follow, much less meaningfully critique, Ramanujan’s work, he went on, he was writing another mathematician (M. J. M. Hill, in London) for advice, and sending him some of Ramanujan’s papers. “If there is any real genius in him,” wrote Griffith, “he will have to be provided with money for books and with leisure, but until I hear from home,” he added, hedging his bets, “I don’t feel sure that it is worthwhile spending much time or money on him.”

  Among those to whom Spring turned for advice was A. G. Bourne, Madras’s director of public instruction, who advised that Ramanujan be sent to see one or both of two Madras mathematicians he indicated by name. Then, he added: “If his genius is so elusive or mysterious that good mathematicians, possessed besides of much common sense, cannot recognize and appreciate it even if it carries them beyond their scope, I should doubt its existence.”

  Two weeks later Ramanujan went to see W. Graham, Madras’s accountant general, one of those Bourne had suggested. “Whether [Ramanujan] has the stuff of great mathematicians or not I do not know,” Graham wrote after seeing him. “He gives me the impression of having brains.” Gives me the impression … With such care did he word his assessment, no one could possibly fault him should he prove wrong. Confusing matters more, he suggested that “it is possible his brains are akin to those of the calculating boy.”

  Graham was referring to those freaks of nature, some of them today described as idiot savants, who though lacking real understanding of higher mathematics, possess a peculiar ability to perform extremely rapid calculations—to unerringly multiply and divide long strings of ten-digit numbers, or give the day of the week on which a thousand-year-old battle occurred, or perform similarly trivial computing tasks.

  In fact, just as some artists of surpassing brilliance are no good at drawing straight lines or representing the human figure, so does mere facility in arithmetic—whether extracting square roots, or balancing books, or working out tricky word problems—have nothing to do with real mathematics. A mathematician may be adept at such skills, just as the artist may be adept at routine draftsmanship or figure drawing. But possession of such skills does not predict mathematical talent.

  Ramanujan was more than ordinarily good in arithmetic calculation; on the other hand, his wasn’t a skill developed to freakish proportions. And certainly in no other way did he resemble the “calculating boy” model. Nonetheless, it was one more among various possibilities as, during late 1912, British officialdom in Madras groped with the question of what to do with Ramanujan.

  Griffith, to whom Graham had also written, wrote Spring the next day: “I think I was right in writing to Prof. Hill,” said he, “and we must wait his opinion.”

  Micaiah John Muller Hill was Griffith’s professor from twenty years before, at University College in London, and a teacher known more for the patience and care he lavished on his students than for his mathematical researches. Around mid-December Griffith heard from him at last. He could not look through all Griffith had sent him just now, Hill apologized, but a glance was sufficient to show that Ramanujan had fallen into some pitfalls; some of his results were simply absurd. Should he want to overcome his evident lacks, Bromwich’s Theory of Infinite Series was the text to consult. If still interested in publication, he ought to write the secretary of the London Mathematical Society. But, Hill warned, “He should be very careful with his [manuscripts. They] should be very clearly written, and should be free from errors; and he should not use symbols which he does not explain”—as he had in the published paper on Bernoulli numbers Griffith had sent him.

  But Hill’s letter didn’t answer the question: Had Ramanujan something extraordinary to offer the world? What was the nature and extent of his genius, if genius it was? “What you say about him personally is very interesting”—presumably a reference to his unusual intellectual history—“and I hope something may come of his work,” was about all Hill would add.

  A few days later, Hill wrote his former student again. It was a curious letter, still not definitive, but this time more encouraging. On the one hand, Ramanujan’s paper on Bernoulli numbers, he said, was riddled with holes. “He has in fact observed certain properties of the earlier Bernoulli numbers and assumed them to be true of them all without proof. For [this and other] reasons, I feel sure that the London Mathematical Society would not have accepted the paper for their Proceedings.” On the other hand, he said, “Mr. Ramanujan is evidently a man with a taste for mathematics, and with some ability.” His educational deficit was hurting him. He needed to get that Bromwich book, he said again, this time citing the specific chapter that would clear up Ramanujan’s misunderstandings.

  And then, in a personal aside, Hill said perhaps the most revealing thing of all. “When I was a student in Cambridge, 1876–9, these things were not properly understood,” he wrote, referring to the subtle but crucial points undermining Ramanujan’s work, “and the modern theory has only recently been established on a firm basis. Many illustrious mathematicians of earlier days stumbled over these difficulties, so it is not surprising that Mr. Ramanujan, working by himself, has obtained erroneous results. I hope he will not be discouraged.”

  1876–9. Hill’s Cambridge years, as it happened, coincided exactly with those of George Shoobridge Carr, author of the book so important to Ramanujan. Here, then, was the first hint of the price Ramanujan had paid in finding no more recent inspiration: he had missed out on all that had been learned in the mathematical capitals of Europe over the past forty years. Ramanujan’s mathematics, in effect, was trapped in a time warp. No wonder he had gone astray.
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  Hill, who scarcely remembered his old student Griffith, had been sufficiently intrigued by Ramanujan’s work to write two long letters in response to it. Of course, it came to nothing more than that. He was not offering to take him on as a student. Why, if anything, he had judged Ramanujan’s first paper unfit for publication. Still, though he did not fully understand all Ramanujan had done, his reply probably contained more serious, reasoned, professional advice than Ramanujan had gotten all his life. And it was encouraging enough to quell most lingering suspicion among the British in Madras that maybe Ramanujan was more crank than genius.

  For some time now, many had advised Ramanujan that no one in India properly understood him, that he’d not be able to find there the expertise and encouragement he needed, that he should instead write to Cambridge, or elsewhere in the West, for help. One who did was Singaravelu Mudaliar, his old professor at Pachaiyappa’s College, to whom he had drawn close during his brief time there. Another was Bhavaniswami Rao, one of Ramanujan’s professors at Kumbakonam College. A third was his friend Narasimha, with whom he had lived in Park Town a couple of years before. More recently, Narayana Iyer probably gave him similar advice.

  If Ramanujan needed convincing to look toward the West, now, in the wake of Hill’s letter from England, he needed no more. Events had conspired to tell him that he was, in effect, too big for Indian mathematics, and that he was apt to get a more sympathetic hearing from European mathematicians.

  India was a quarter of the way around the globe from Europe, but the mail was cheap, reliable, and—long before airmail shrank the world—surprisingly fast; people grumbled if letters to England took as long as two weeks. And so, in late 1912 and early 1913, it was to the international mails that Ramanujan turned. In letters drafted with the help of Narayana Iyer, Sir Francis Spring, and perhaps P. V. Seshu Iyer, he began to write leading mathematicians at Cambridge University, including with his letters samples of his work.

  He wrote to H. F. Baker, who held a long string of high honors as a mathematician, including a fellowship of the Royal Society, and had been president of the London Mathematical Society until two years before. Could Baker offer him help or advice?

  Either through the kind of formulaic letter of polite discouragement that important men learn to write, or by returning his unsolicited material without comment, or by ignoring his letter altogether, Baker said no.

  Ramanujan wrote to E. W. Hobson, an equally distinguished mathematician, also a Fellow of the Royal Society, and holder of Cambridge’s Sadleirian chair in pure mathematics.

  Hobson, too, said no.

  On January 16, 1913, Ramanujan wrote to still another Cambridge mathematician, G. H. Hardy, who at thirty-five, a generation younger than the other men, was already setting the mathematical world of England on its ear. Could Hardy help him?

  And Hardy said yes.

  CHAPTER FOUR

  Hardy

  [G. H. Hardy to 1913]

  1. FOREVER YOUNG

  He was a study in perpetual youth.

  One day in the spring of 1901, Hardy took his friend Lytton Strachey to the private green behind Trinity College, to which as a fellow of the college he enjoyed access, for a game of bowls. “He is the mathematical genius,” Strachey wrote his mother, “and looks a babe of three.” Even into his thirties, Hardy was sometimes refused beer and at least once, while at lunch with other Trinity dons, he was mistaken for an undergraduate.

  He had ice-clear eyes, a finely chiseled face, and in 1913, straight, close-trimmed hair. He was beautiful. He didn’t think so, of course, and could scarcely bear to look at himself. His college rooms had no mirrors, and in a hotel room he would cover any with towels, shaving by touch. But he alone was deceived. Even when past fifty, his looks were arresting. His skin, wrote a friend from those years, the novelist C. P. Snow, was tanned to “a kind of Red Indian bronze. His face was beautiful—with high cheek bones, thin nose, spiritual and austere… . [Cambridge] was full of unusual and distinguished faces, but even there Hardy’s could not help but stand out.” He was not, by every yardstick, handsome, at least not “ruggedly” so; his features were too delicate for that. And pursed, ungenerously thin lips, turned down a little at the corners, hinted at a judgmental streak in him.

  Hardy was forever judging, weighing, comparing. He rated mathematicians, the work they did, the books and papers they wrote. He held firm opinions on everything, and expressed them. When a Cambridge club to which he’d belonged moved to change its official colors, Hardy took six pages to attack the plan. He faulted a sacrosanct academic tradition of almost two centuries’ standing, and condemned it, unrelentingly, for more than twenty years. All his enthusiasms, peeves, and idiosyncrasies were like that—sharp, unwavering, vehement. He hated war, politicians as a class, and the English climate. He loved the sun. He loved cats, hated dogs. He hated watches and fountain pens, loved The Times of London crossword puzzles.

  In The Case of the Philosophers’ Ring, a Sherlock Holmes mystery written half a century after the death of Arthur Conan Doyle, the characters include Ramanujan and Hardy. In it, author Randall Collins pictures Hardy as a sort of White Rabbit hopping around the Fellows Garden at Trinity in white flannels and cap, cricket bat in hand, frantically searching for his cricket gloves, crying, “There’s a match due to begin, and I can’t find them. I’m late, I’m late!” In a prefatory note, Collins abjures all claim to historical accuracy. But in Hardy, he’s close to the mark.

  Hardy was a cricket aficionado of almost pathological proportion. He played it, watched it, studied it, lived it. He analyzed its tactics, rated its champions. He included cricket metaphors in his math papers. “The problem is most easily grasped in the language of cricket,” he would write in a Swedish mathematical journal; foreigners failed to grasp it at all. His highest accolade was to rate a mathematical proof, say, as being “in the Hobbs class”—leaving the benighted to imagine the philosopher Thomas Hobbes, not the legendary Surrey cricketer Jack Hobbs. Hardy would play the game into his sixties. His sister would be reading to him about cricket when he died.

  Hardy judged God, and found Him wanting. He was not just an atheist; he was a devout one. As an undergraduate, he was told that to be excused from chapel he had to inform his devout parents; he agonized over what to do—but ultimately wrote them with the crushing news. God, it would be said of him, was his personal enemy. Yet his friends included clerics, and some of his infidel posturing was just that—another of the harmless games he never tired of playing. “It’s rather unfortunate,” he once grumbled to a friend as a church’s six o’clock chimes sounded the end of a sunny day of cricket at Fenner’s cricket ground in Cambridge, “that some of the happiest hours of my life should have been spent within sound of a Roman Catholic Church.”

  Shy and self-conscious, he disliked small talk; cricket, of course, was not small talk. He abhorred formal introductions, would not shake hands, would walk, face down, along the street, ignoring those who might expect him to exchange how-do-you-dos. He was “one of the most strange and charming of men,” wrote Leonard Woolf, who knew him at Cambridge long before Woolf married Virginia Stephen and, with Strachey and others, launched the Bloomsbury literary movement. His eccentricities would ossify with age, become caricatures of themselves, the stuff of story. But his personality, temperament, and values were already largely formed when he heard from Ramanujan.

  Ramanujan knew nothing of this side of Hardy, of course. He knew him only as a mathematician. And in 1913, at the age of thirty-five, Hardy was already a famous one. He had appeared in the mathematical literature for fifteen years, counted more than a hundred papers, and three books, to his credit. He was a Fellow of Trinity College, the mecca of Cambridge mathematics, and hence English mathematics. He had been named to the Royal Society, Britain’s most elite body of scientists, in 1910. Indeed, more than sniping at God, or delighting in cricket, or fashioning sly conversational gambits over dinner, Hardy cared about discovering mathematical t
ruth. A brilliant mathematician, he was also a major influence on other mathematicians. A whole school had begun to form around him. He had served on the Council of the London Mathematical Society for three years, would later occupy numerous other posts within the mathematical community. “My devotion to mathematics is indeed of the most extravagant and fanatical kind,” he would write. “I believe in it, and love it, and should be utterly miserable without it.” His mathematical research, he would say, was “the one great permanent happiness of my life.”

  Hardy spoke beautifully. He batted out sparkling bons mots the way he did cricket balls from the popping crease—provoking, challenging, asserting. He was scrupulously honest, fastidious about giving others their due, once even admitted that the pro-God position in a debate had been better argued. He was endlessly amusing—but it was all like the gauzy silken shimmer of a woman’s dress, meant to distract and disguise more than reveal. Conversation, one of his research students would say, was to him “one of the games which he loved to play, and it was not always easy to make out what his real opinions were.”

  C. P. Snow once reported that the longer you spent in Einstein’s company, the more extraordinary he seemed; whereas Snow found that the longer you spent with Hardy, the more familiar a figure he seemed to become—more like most people, only “more delicate, less padded, finer-nerved”; that his formidable wall of charm and wit shielded an immensely fragile ego; that within lay someone simple, caring, and kind.

  There is a picture of Hardy from middle age that shows him slouched in an upholstered wicker chair, one leg crossed over the other, right hand cocked at the wrist, lightly gripping a cigarette, left arm suspended at an unlikely angle across the back of the chair. A wisp of hair slips down over his forehead. He does not look relaxed; in no photograph of Hardy does he ever look relaxed. Always there’s that haunted look in his eyes, like “a slightly startled fawn,” as Leonard Woolf once said of him. There he sits, brows knitted, lips pursed, peering out over the tops of his reading glasses, imperious and forbidding. Someone spying this picture once said, “To sit that way you have to have been educated in a public school” (or what to Americans is a private boarding school).