In the Jungle of the Infinite

Godfrey Harold Hardy was a pure mathematician and the archetypal Cambridge don: fellow of Trinity College, unmarried, his life a mixture of research, cricket, and college society; his features so finely chiseled that C.P. Snow described his face as “beautiful.” He was eccentric in a disarmingly English way: he abhorred telephones and waged a quiet but very personal vendetta against God. Srinivasa Ramanujan Iyengar was born into a poor Brahmin family, contracted smallpox at the age of two, flunked out of college twice, went through an arranged marriage with a girl of ten, and was described by Ramachandra Rao as “a short, uncouth figure, stout, unshaved, not overclean.” He was a Hindu, as all Brahmins are, and worshiped the goddess Namagiri of Namakkal. It is hard to imagine two people more unlike each other. Yet their lives became so strongly entwined that it is difficult to mention one without, in the same breath, referring to the other. The Man Who Knew Infinity is really a biography of them both, although Ramanujan takes pride of place: perspicacious, informed, imaginative, it is to my mind the best mathematical biography I have ever read.

In January of 1913, when Europe was becoming enmeshed in what would become the First World War, Hardy received a large manila envelope, postmarked Madras. Inside was a sheaf of papers, and a covering letter:

Dear Sir,

I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education…. After leaving school I have been employing the spare time at my disposal to work at Mathematics…. I am striking out a new path for myself.

Prominent mathematicians—and Hardy was one of England’s greatest—receive packages like this all the time. Mostly they come from cranks who imagine they have squared the circle or trisected the angle, problems that mathematicians know to be insoluble. Hardy could easily have thrown the package away; but a page of strange formulas caught his eye. He recognized a few, but others were quite unusual.

The letter continued:

I would request you to go through the enclosed papers. Being poor, if you are convinced that there is anything of value I would like to have my theorems published…. Being inexperienced I would very highly value any advice you give me. Requesting to be excused for the trouble I give you.

I remain, Dear Sir, Yours truly,

S. Ramanujan.

Hardy quickly convinced himself that this was no crank, but a self-taught mathematician of the highest order. He decided that Ramanujan should be brought to England. Thus began their paradoxical friendship.

The centenary of Ramanujan’s birth took place four years ago. An hour-long television program was made about him, and innumerable articles appeared in newspapers and magazines. The Shobana Jeyasingh Indian Dance Company created and performed a new dance, Correspondences, to celebrate his life and works. His name should be a household word—but it is not, because he was a great mathematician, not a great artist, musician, or writer. Perhaps this biography will change that.

Ramanujan’s life and his mathematics were inseparable; it is impossible to understand one while ignoring the other. The Man Who Knew Infinity is not just a brilliant biography of Ramanujan, the genius born in a hovel. Nor is it a “life and works,” part one Life, part two Works. It is a sensitive and intimate portrait of Ramanujan, the human being, who lived for mathematics. Robert Kanigel puts the mathematics where it belongs, into the daily activity of Ramanujan’s life. He shows us what it is like to be a mathematician, to be driven by the secret beauty of this most mysterious of endeavors. You do not need to have any sympathy at all with mathematics to read The Man Who Knew Infinity with pleasure; but by the time you have finished it, some of Ramanujan’s love of his subject will probably have rubbed off on you, and you will have begun to appreciate the hypnotic fascination that it exerts upon those who make it their life’s work.

Kanigel gives a vivid account of Ramanujan’s childhood in the India of the British Raj. His father was a clerk in a shop that sold silk saris to prosperous farmers when they married off their daughters, his mother a strong-willed and obsessive woman who would do whatever was necessary to ensure that he learned what to do, and what not to do, to become a good Brahmin boy.

Ramanujan grew up in Kumbakonam in the province of Tamil Nadu in southern India. His mother, Komalatammal, sang devotional songs at a temple near their home on Sarangapani Sannidhi Street, a straw-roofed house with a frontage of no more than twenty feet. The five or ten rupees a month that she earned made an enormous difference, for her husband earned only twenty. Three brothers and sisters died before Ramanujan was eight. His grandfather suffered from leprosy.

The caste barriers of India were at their strongest when it came to food. Brahmins are strict vegetarians, and so was Ramanujan. They could dine only with other Brahmins, be served by Brahmins, eat food prepared by Brahmins. Every morning Ramanujan took a ritual bath in the Cauvery River. (“He heard it all his life,” Kanigel writes, “the slow, measured thwap…thwap…thwap…of wet clothes being pounded clean on rocks jutting up from the waters of the Cauvery River.”) Even a poor Brahmin took enormous pride in cleanliness—and in education. Brahmins were making an uneasy transition from sacred to secular, from guru to professional. At the age of ten Ramanujan passed his primary examinations—English, Tamil, arithmetic, and geography—with the top score in the district. By the age of twelve he was asking mathematical questions whose answers were beyond the ability of his teachers. In 1904 he won the school prize for mathematics, his teacher remarking that a score of one hundred percent understated his achievements.

At some time in 1903 he came across a copy of George Shoobridge Carr’s A Synopsis of Elementary Results in Pure and Applied Mathematics. It is a very odd book: a list of some five thousand equations, ranging over the whole of mathematics. There were no detailed explanations, but there was a sense of flow about the book—each formula was a logical extension of those before. It was a map of the mathematical territory that showed only the hilltops, but it located those accurately in relation to each other. It may sound a thoroughly boring piece of hackwork, but it changed Ramanujan’s life. With boundless inventiveness and enthusiasm, he moved quickly through the countryside between the hilltops, and then began to discover new hilltops of his own.

Unfortunately, he neglected everything else to do it. When he should have been listening to lectures on Roman history, he would sit scribbling formulas. He lost his college scholarship, ran out of money, and ran away from home—heading for the college founded by Pachaiyappa Mudaliar, and a second attempt. Everybody was struck by Ramanujan’s gift for mathematics. Unfortunately he had to study other subjects, such as physiology. (“Procure a rabbit which has been recently killed but not skinned”—not very appealing for a vegetarian.) By 1907 he had failed his exams, twice, and had to leave. Ramanujan was alone with his mathematics.

He began to compile a notebook modeled on Carr’s Synopsis, recording his results but not the thoughts that led to them. His sources were out of date, his approach old-fashioned; but his genius would not so much transcend those difficulties as cut through them as if they never existed. As Kanigel puts it: “He was like a species that has branched off from the main evolutionary line, and, like an Australian echidna or Galápagos tortoise, had come to occupy a biological niche all his own.”

By now Ramanujan was back at home: fat, unemployed, pottering around with his notebooks. No mother would be impressed, especially not Komalatammal. She found him a wife and married him off. Although Janaki would not join him until after she reached puberty, still three years away, it marked a major change for Ramanujan. He was now a grihasta, a family man with responsibilities. He went looking for a job.

In December 1910 he was introduced to Ramachandra Rao, the district collector of Nellore, a man with the right connections to do something for an impoverished genius. It took four visits before Rao was convinced. He paid Ramanujan twenty-five rupees a month from his own pocket, and looked for a suitable scholarship. For the first time in many years, Ramanujan was happy. He published a paper in the Indian Journal of Mathematics. He got a job as a Class III, Grade IV clerk in the accounts section of Madras Port Trust. One of the attractions was that he could take away used wrapping paper, on which to write yet more formulas. His wife joined him—but so did his mother. He scarcely spoke to Janaki, he never got a chance to. If his mother was away, his grandmother acted as chaperon. Until she had children of her own, a new wife’s position was little better than that of a slave.

C.L.T. Griffith, an engineering professor at Madras Engineering College, recognized Ramanujan’s ability and wrote about him to his former professor Micaiah Hill at University College in London. Hill was guardedly positive but did not wish to take Ramanujan on as a student. Ramanujan wrote to H.F. Baker, a fellow of the Royal Society and previous president of the London Mathematical Society. No success. He tried E.W. Hobson, Sadleirian Professor at Cambridge. Another failure. Discouraged, but resolved to make one final attempt, he wrote to Hardy.

Hardy was not easily convinced. He and his colleague John Edensor Little-wood pored over Ramanujan’s list of formulas. Some were very familiar, others “scarcely possible to believe,” as Hardy later wrote. They tried to prove some of the simpler-looking results, and soon decided that Ramanujan was keeping a great deal up his sleeve. Just one of Ramanujan’s formulas would subsequently keep three mathematicians busy for a decade. In an uncharacteristic burst of romanticism, Hardy declared that the results “must be true, because if they were not true, no one would have the imagination to invent them.”

Hardy tried to get Ramanujan to Cambridge. Brahminic scruples—Ramanujan’s, his friends’, his family’s—intervened. Ramanujan had no wish to become an “outcaste,” as Gandhi had become a quarter of a century earlier when he went to England for an education. Instead, Hardy persuaded the University of Madras to bend the rules and give Ramanujan a scholarship, and the two mathematicians corresponded regularly. Hardy insisted that Ramanujan should provide proofs of his remarkable results, Ramanujan offered excuses and revealed little of his methods. When Eric Neville, another Trinity fellow, visited Madras to lecture, Hardy instructed him to persuade Ramanujan to come to England.

This time Ramanujan said yes. What had changed his mind? Amazingly, it was Komalatammal. She announced that in a dream she had seen her son surrounded by Europeans, and the goddess Namagiri had commanded her not to stand between him and his life’s purpose.

One of the most memorable sections of the Indian dance production of Correspondences occurs just before Ramanujan leaves Madras for England. In vivid mime, he is seen going to a tailor to be fitted out in an English suit. Nothing could better dramatize the changes about to occur in his life. On March 17, 1913, with a second-class ticket on the steamer Nevasa, Ramanujan sailed for England.

There were two main tasks confronting Hardy, now that Ramanujan was in Cambridge. He had to tease out Ramanujan’s methods, so that others could benefit from his insights; and he had to fill gaps in Ramanujan’s education, since there were major omissions in Carr’s Synopsis. Together they began to go over Ramanujan’s notebooks and turn them into “real” mathematics that could be published. Between 1914 and 1920, either with Hardy or on his own, Ramanujan wrote some three dozen research papers. Several are classics.

Hardy now understood why Ramanujan had been unwilling to provide proofs. He didn’t have any, not in the rigorous sense that would satisfy a professional mathematician. What he had was a mixture of intuition and calculation, enough to satisfy him that he was right. Tamil culture places great value on patterns and numbers: perhaps this influenced him as a child. Ramanujan claimed that the goddess Namagiri told him formulas in dreams. Despite this, Hardy’s view was that “all mathematicians think, at bottom, in the same kind of way, and Ramanujan was no exception.” His strange patterns of thought were caused by his unusual upbringing. But Hardy added: “He combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling.”

Ramanujan was strong on algebraic calculations and the manipulation of symbols, and had a genius for formulas. He liked intriguing special results, and would often conceal any general methods that might lie behind them. This was a style employed by many of the giants of mathematics during the eighteenth and nineteenth centuries. But by the early twentieth century, the emphasis was on general methods, sweeping theories, broad structure rather than individual detail. Ramanujan was out of date.

Hardy had to deal with this as best he could, without destroying the quintessential quality of Ramanujan’s mind. He had to exploit his friend’s talents in the most effective way, but he could not try to change their nature. So remarkable were those talents, however, that even being old-fashioned scarcely blunted them. Indeed, it can now be argued that Ramanujan was not behind the times, but ahead of them. The fashion in mathematics goes in cycles, swinging from the concrete to the abstract and back again over a period of sixty to a hundred years.

Changes in mathematical style, brought about in part by the digital nature of modern electronics, have placed renewed emphasis on mathematics of the kind that Ramanujan loved. Every year, his work grows in stature. Bruce Berndt, at the University of Illinois, Urbana, recently edited the first part of a three-volume work, Ramanujan’s Notebooks, aiming to supply proofs of all of his formulas. He writes:

With a more conventional education, Ramanujan might not have depended on the original formal methods of which he was proud and rather protective…. If he had thought like a well-trained mathematician, he would not have recorded many of the formulas which he thought he had proved but which, in fact, he had not proved. Mathematics would be poorer today if history had followed such a course.

One theorem can suggest the flavor of Ramanujan’s mathematics. It should really be expressed as a formula—a curious formula, rather offbeat, even mysterious—for that is how Ramanujan conceived it.^* But it can be explained in words. It is about partitions: ways to express a given number as a sum of smaller ones. Take a multiple of five, i.e., a number that is the product of five and another number, and add four to it. Then the number of different partitions of this number is always a multiple of five. For example the number 9 (1 x 5 + 4) has precisely thirty partitions, and 30 is indeed divisible by 5. In Ramanujan’s formula, the divisibility by 5 leaps to the eye: all else is hidden in the depths. Before Ramanujan made his discovery, the arithmetical properties of partitions were a total mystery. It’s not that nobody could prove anything about them: it’s that nobody realized there was anything to prove.

One of Ramanujan’s greatest works, jointly with Hardy, is also on partitions. A central idea in the analytic theory of numbers is to find good approximations to number-theoretic quantities, such as the number of primes in a given range. In 1918 Hardy and Ramanujan found an approximate formula for stating the number of partitions. At Ramanujan’s insistence, they refined this formula until it was such a good approximation that it gave the exact answer. For example, there are precisely 190,569,292 partitions of the number 100. Their formula gives 190,569,291.996. Since the answer must be a whole number, the error of 0.004 can safely be ignored. This kind of accuracy was wholly unprecedented.

Ramanujan did not find it easy to adapt to Cambridge life. When an Indian student visited him in 1913, he asked Ramanujan whether he was warm at night. He replied that he always slept with his overcoat on, wrapped in a shawl. Wondering whether his friend had enough blankets, the student took a look. There they were, neatly piled beneath the bed. Nobody had shown Ramanujan what to do with them. As a Brahmin and a vegetarian, Ramanujan did not frequent the college dining hall. He cooked for himself. In wartime England, the ingredients he needed were not easily obtained. He was often miserable. On Sundays he would have Indian friends over for rasam, a thin peppery soup; but when his guests once refused third helpings, Ramanujan was so offended that he left in the middle of dinner, took a taxi to Oxford, and did not show up for four days.

It is still a mystery precisely what disease laid Ramanujan low in the spring of 1917. The earliest diagnosis was a gastric ulcer, then cancer was suspected, then blood poisoning. At the sanatorium in Mendip Hills he was treated for tuberculosis, and that remains the most likely suspect. The treatment required plenty of “fresh air” and Ramanujan froze. They tried to feed him scrambled eggs on toast. Back home in India, trouble was brewing between Janaki and Komalatammal. Ramanujan’s mother wouldn’t let his wife write to him, or read his letters. Ramanujan knew only what his mother told him: when a letter from Janaki finally did get through, his reply held no warmth or feeling. His morale had hit rock bottom.

There were a few bright spots. In a second sanatorium, at Matlock, he learned from Hardy that he had been elected a fellow of the Royal Society, the first Indian to be so honored. Subsequently he was made a fellow of Trinity. In A Mathematician’s Miscellany Littlewood says: “I am the only person who knows the facts, and they should be put on record, if only as illustrating the fantastic state of the College just after the 1914–1918 war.” He describes how Ramanujan’s election was opposed by several fellows at Trinity. One “went about openly saying that he wasn’t going to have a black man as a Fellow.” The fact that Ramanujan already had an FRS was seen by the opposition as a dirty trick.

By now he was out of Matlock and had gone to a small hospital “overlooking a perfectly proportioned little square in the heart of London, on which George Bernard Shaw had lived in the 1890s and Virginia Woolf for four years until 1911.” He moved again to Putney. Visiting him one day, Hardy happened to remark that the taxi he had just taken was number 1729. “Rather a dull number,” Hardy ventured. “No, Hardy,” Ramanujan protested. “It is a very interesting number. It is the smallest number expressed as the sum of two cubes in two different ways.” His morale improved, and suddenly he was doing mathematics again—more deep properties of his beloved partitions. Ramanujan seemed much improved, and he had gained fifteen pounds in weight. In April 1919 he returned to India. The long voyage may have been a mistake, for by the time he arrived in Madras his health had once more deteriorated. Janaki was not there to greet him: Komalatammal had not informed her he was coming—indeed, had no idea where she was. But Janaki read of it in the newspaper, learned that Ramanujan wanted her, and that was enough: mother-in-law notwithstanding, she came to meet him in Madras. For three months they stayed at a small bungalow on Luz Church Road, finally beginning to get to know each other. Janaki had been thirteen when she married him: now she was eighteen. Ramanujan wished to go to the river for a ritual cleansing. “Janaki wanted to go with him. Ramanujan said yes. Komalatammal said no. And Ramanujan insisted, yes.” Finally he was master in his own house.

In January of 1920 he wrote to Hardy, about a new discovery: his “mock theta functions.” The classical theta functions were invented for such purposes as calculating the perimeter of an ellipse. They are a rich source of intricate formulas. Ramanujan, purely for reasons of formal beauty, had invented his own variation on the classical theme. It was a mathematical goldmine, and he was stimulated to produce some of the best mathematics he had ever done. All that year he worked on them, producing something like 650 new formulas. George Andrews of Pennsylvania State University tackled them half a century later. Among one group of five superficially similar results: “the first one took me fifteen minutes to prove, the second an hour. The fourth one followed from the second. The third and fifth took me three months.”

There was nothing wrong with Ramanujan’s mind, only with his body. “He was only skin and bones,” Janaki recalled. Although in great pain, he still was consumed by mathematics, calculating on a slate, copying results on to paper. “It was always maths…. Four days before he died he was scribbling.” He died in April 1920. He was cremated later the same day, near Chetput.

Ramanujan has been called a magician, a sorcerer, a gift from heaven, an enigma. “It is uncanny,” Kanigel remarks, “how often otherwise dogged rationalists have, over the years, turned to the language of the shaman and the priest to convey something of Ramanujan’s gifts.” Richard Askey, at the University of Wisconsin, is quoted as saying, “We have no idea how he did the marvelous things he did, what led him to do them, or anything else.” Mathematics is a jungle, the Jungle of the Infinite. Most of us see only broad paths driven through it by pioneers, trampled flat by generations of fellow students. A few go out and hack new paths of their own. But Ramanujan was a creature of the jungle, who could move through it at will without leaving any traces of his passing. Only a creature of the jungle can understand another. He was, indeed, the man who knew infinity.