I must say one thing more, though in doing so I risk alienating a few readers. To Americans, India and Cambridge are, indeed, foreign countries. And as L. P. Hartley has written, the past is alien territory, too: “They do things differently there.” Thus, the years around the turn of the century when our story begins, a time when India was still British and Victoria still queen, represent a second foreign country to explore. Now, to these two worlds remote in place and time, I must add a third—the mathematics that Ramanujan and Hardy did, alone and together, as their life’s work.
It is tempting to concentrate exclusively on the exotic and flavorful elements of Ramanujan’s life and skip the mathematics altogether. Indeed, virtually all who have taken Ramanujan as their subject have severed his work from his life. Biographies as do exist either ignore the mathematics, or banish it to the back of the book. Similarly, scholarly papers devoted to Ramanujan’s mathematics normally limit to a few paragraphs their attention to his life.
And yet, can we understand Ramanujan’s life without some appreciation for the mathematics that he lived for and loved? Which is to say, can we understand an artist without gaining a feel for his art? A philosopher without some glimpse into what he believed?
Mathematics, I am mindful, presents a special problem to the general reader (and writer). Art, at least, you can see. Philosophy and literature, too, have the advantage that, however recondite, they can at least be rendered into English. Mathematics, however, is mired in a language of symbols foreign to most of us, explores regions of the infinitesimally small and the infinitely large that elude words, much less understanding. So specialized is mathematics today, I am told, that most mathematical papers appearing in most mathematics journals are indecipherable even to most mathematicians. Pennsylvania State’s George Andrews, who rediscovered a long-forgotten Ramanujan manuscript at Trinity College, says it took someone already expert in the narrow area of mathematics with which it dealt to recognize it—that merely being a professional mathematician with a Ph.D. would not have sufficed.
What hope, then, has the general reader faced with Ramanujan’s work?
Little, certainly, if we set as the task to follow one of Ramanujan’s proofs through twenty pages of hieroglyphics in a mathematics journal—especially in the case of Ramanujan, who routinely telescoped a dozen steps into two, leaving his reader to find the connections. But to come away with some flavor of his work, the paths by which he got there, its historical roots? These pose no insuperable problem—certainly no more than following a philosophical argument, or a challenging literary exegesis.
In one sense, at least, Ramanujan’s mathematics is more accessible than some other fields; much of it comes under the heading of number theory, which seeks out properties of, and patterns among, the ordinary numbers with which we deal every day; and 8s, 19s, and 376s are surely more familiar than quarks, quasars, and phosphocreatine. While the mathematical tools Ramanujan used were subtle and powerful, the problems to which he applied them were often surprisingly easy to formulate.
• • •
In a note at the end of the book I mention by name many who have helped me along the way. But here I wish to especially thank the people of South India as a whole for making my time there so personally satisfying.
I spent five weeks in the South, traveling to places that figured in Ramanujan’s life. I rode trains and buses, toured temples, ate with my hands off banana leaves. I was butted in the behind by a cow on the streets of Kumbakonam, shared a room with a lizard in Kodumudi. I saw Erode, the town where Ramanujan was born. I toured the house in which he grew up, participated in opening exercises at his alma mater, wandered through the grounds of the temple in Namakkal to which he came at a turning point in his life, and saw the room in which he died in Madras. The people of South India took me into their homes. They bestowed upon me every kindness. Auto rickshaw and bicycle rickshaw drivers often went to extraordinary lengths to get me to out-of-the-way places, forever struggling to understand what to them was my atrocious pronunciation of South Indian place-names. They stared at the whiteness of my skin but always treated me with gentleness and goodwill.
Spend any length of time among the people of South India and it is hard not to come away with a heightened sense of spirituality, a deepened respect for hidden realms, that implicitly questions Western values and ways of life. In the West, all through the centuries, artists have sought to give expression to religious feeling, creating Bach fugues and Gothic cathedrals in thanks and tribute to their gods. In South India today, such religious feeling hangs heavy in the air, and to discern a spiritual resonance in Ramanujan’s mathematics seems more natural by far than it does in the secular West.
Ramanujan’s champion, Hardy, was a confirmed atheist. Yet when he died, one mourner spoke of his
profound conviction that the truths of mathematics described a bright and clear universe, exquisite and beautiful in its structure, in comparison with which the physical world was turbid and confused. It was this which made his friends … think that in his attitude to mathematics there was something which, being essentially spiritual, was near to religion.
The same, but more emphatically, goes for Ramanujan, who all his life believed in the Hindu gods and made the landscape of the Infinite, in realms both mathematical and spiritual, his home. “An equation for me has no meaning,” he once said, “unless it expresses a thought of God.”
Baltimore
May 1990
CHAPTER ONE
In the Temple’s Coolness
[1887 to 1903]
1. DAKSHIN GANGE
He heard it all his life—the slow, measured thwap … thwap … thwap … of wet clothes being pounded clean on rocks jutting up from the waters of the Cauvery River. Born almost within sight of the river, Ramanujan heard it even as an infant. Growing up, he heard it as he fetched water from the Cauvery, or bathed in it, or played on its sandy banks after school. Later, back in India after years abroad, fevered, sick, and close to death, he would hear that rhythmic slapping sound once more.
The Cauvery was a familiar, recurring constant of Ramanujan’s life. At some places along its length, palm trees, their trunks heavy with fruit, leaned over the river at rakish angles. At others, leafy trees formed a canopy of green over it, their gnarled, knotted roots snaking along the riverbank. During the monsoon, its waters might rise ten, fifteen, twenty feet, sometimes drowning cattle allowed to graze too long beside it. Come the dry season, the torrent became a memory, the riverbanks wide sandy beaches, and the Cauvery itself but a feeble trickle tracing the deepest channels of the riverbed.
But always it was there. Drawing its waters from the Coorg Mountains five hundred miles to the west, branching and rebranching across the peninsula, its flow channeled by dams and canals some of which went back fifteen hundred years, the Cauvery painted the surrounding countryside an intense, unforgettable green. And that single fact, more than any other, made Ramanujan’s world what it was.
Kumbakonam, his hometown, flanked by the Cauvery and one of its tributaries, lay in the heartland of staunchly traditional South India, 160 miles south of Madras, in the district then known as Tanjore. Half the district’s thirty-seven hundred square miles, an area the size of the state of Delaware, was watered directly by the river, which fell gently, three feet per mile, to the sea, spreading its rich alluvial soil across the delta.
The Cauvery conferred almost unalloyed blessing. Even back in 1853, when it flooded, covering the delta with water and causing immense damage, few lives were lost. More typically, the great river made the surrounding land immune to year-to-year variation in the monsoon, upon whose caprices most of the rest of India hung. In 1877, in the wake of two straight years of failed monsoons, South India had been visited by drought, leaving thousands dead. But Tanjore District, nourished by the unfailing Cauvery, had been scarcely touched; indeed, the rise in grain prices accompanying the famine had brought the delta unprecedented prosperity.
No
wonder that the Cauvery, like the Ganges a thousand miles north, was one of India’s sacred rivers. India’s legendary puranas told of a mortal known as Kavera-muni who adopted one of Brahma’s daughters. In filial devotion to him, she turned herself into a river whose water would purify from all sin. Even the holy Ganges, it was said, periodically joined the Cauvery through some hidden underground link, so as to purge itself of pollution borne of sinners bathing in its waters.
Dakshin Gange, the Cauvery was called—the Ganges of the South. And it made the delta the most densely populated and richest region in all of South India. The whole edifice of the region’s life, its wealth as well as the rich spiritual and intellectual lives its wealth encouraged, all depended on its waters. The Cauvery was a place for spiritual cleansing; for agricultural surfeit; for drawing water and bathing each morning; for cattle, led into its shallow waters by men in white dhotis and turbans, to drink; and always, for women, standing knee-deep in its waters, to let their snaking ribbons of cotton or silk drift out behind them into the gentle current, then gather them up into sodden clumps of cloth and slap them slowly, relentlessly, against the water-worn rocks.
2. SARANGAPANI SANNIDHI STREET
In September 1887, two months before her child was due to be born, a nineteen-year-old Kumbakonam girl named Komalatammal traveled to Erode, her parental home, 150 miles upriver, to prepare for the birth of the child she carried. That a woman returned to her native home for the birth of her first child was a tradition so widely observed that officials charged with monitoring vital statistics made a point of allowing for it.
Erode, a county seat home to about fifteen thousand people, was located at the confluence of the Cauvery and one of its tributaries, the Bhavani, about 250 miles southwest of Madras. At Erode—the word means “wet skull,” recalling a Hindu legend in which an enraged Siva tears off one of Brahma’s five heads—the Cauvery is broad, its stream bed littered with great slabs of protruding rock. Not far from the river, in “the fort,” as the town’s original trading area was known, was the little house, on Teppukulam Street, that belonged to Komalatammal’s father.
It was here that a son was born to her and her husband Srinivasa, just after sunset on the ninth day of the Indian month of Margasirsha—or Thursday, December 22, 1887. On his eleventh day of life, again in accordance with tradition, the child was formally named, and a year almost to the day after his birth, Srinivasa Ramanujan Iyengar and his mother returned to Kumbakonam, where he would spend most of the next twenty years of his life.
“Srinivasa”—its initial syllable pronounced shri—was just his father’s name, automatically bestowed and rarely used; indeed, on formal documents, and when he signed his name, it usually atrophied into an initial “S.” “Iyengar,” meanwhile, was a caste name, referring to the particular branch of South Indian Brahmins to which he and his family belonged. Thus, with one name that of his father and another that of his caste, only “Ramanujan” was his alone. As he would later explain to a Westerner, “I have no proper surname.” His mother often called him Chinnaswami, or “little lord.” But otherwise he was, simply, Ramanujan.
He got the name, by some accounts, because the Vaishnavite saint Ramanuja, who lived around A.D. 1100 and whose theological doctrines injected new spiritual vitality into a withered Hinduism, was also born on a Thursday and shared with him other astrological likenesses. “Ramanujan”—pronounced Rah-MAH-na-jun, with only light stress on the second syllable, and the last syllable sometimes closer to jum—means younger brother (anuja) of Rama, that model of Indian manhood whose story has been handed down from generation to generation through the Ramayana, India’s national epic.
• • •
Ramanujan’s mother, Komalatammal, sang bhajans, or devotional songs, at a nearby temple. Half the proceeds from her group’s performances went to the temple, the other half to the singers. With her husband earning only about twenty rupees per month, the five or ten she earned this way mattered; never would she miss a rehearsal.
Yet now, in December 1889, she was missing them, four or five in a row. So one day, the head of the singing group showed up at Komalatammal’s house to investigate.
There she found, piled near the front door, leaves of the margosa tree; someone, it was plain to her, had smallpox. Stepping inside, she saw a small, dark figure lying atop a bed of margosa leaves. His mother, chanting all the while, dipped the leaves in water laced with ground turmeric, and gently scoured two-year-old Ramanujan’s pox-ridden body—both to relieve the infernal itching and, South Indian herbalists believed, subdue the fever.
Ramanujan would bear the scars of his childhood smallpox all his life. But he recovered, and in that was fortunate. For in Tanjore District, around the time he was growing up, a bad year for smallpox meant four thousand deaths. Fewer than one person in five was vaccinated. A cholera epidemic when Ramanujan was ten killed fifteen thousand people. Three or four children in every ten died before they’d lived a year.
Ramanujan’s family was a case study in the damning statistics. When he was a year and a half, his mother bore a son, Sadagopan. Three months later, Sadagopan was dead.
When Ramanujan was almost four, in November 1891, a girl was born. By the following February, she, too, was dead.
When Ramanujan was six and a half, his mother gave birth to yet another child, Seshan—who also died before the year was out.
Much later, two brothers did survive—Lakshmi Narasimhan, born in 1898, when Ramanujan was ten, and Tirunarayanan, born when he was seventeen. But the death of his infant brothers and sister during those early years meant that he grew up with the solicitous regard and central position of an only child.
After the death of his paternal grandfather, who had suffered from leprosy, Ramanujan, seven at the time, broke out in a bad case of itching and boils. But this was not the first hint of a temperament inclined to extreme and unexpected reactions to stress. Indeed, Ramanujan was a sensitive, stubborn, and—if a word more often reserved for adults in their prime can be applied to a little boy—eccentric child. While yet an infant back in Erode, he wouldn’t eat except at the temple. Later, in Kumbakonam, he’d take all the brass and copper vessels in the house and line them up from one wall to the other. If he didn’t get what he wanted to eat, he was known to roll in the mud in frustration.
For Ramanujan’s first three years, he scarcely spoke. Perhaps, it is tempting to think, because he simply didn’t choose to; he was an enormously self-willed child. It was common in those days for a young wife to shuttle back and forth between her husband’s house and that of her parents, and Komalatammal, worried by her son’s muteness, took Ramanujan to see her father, then living in Kanchipuram, near Madras. There, at the urging of an elderly friend of her father’s, Ramanujan began the ritual practice of Akshara Abhyasam: his hand, held and guided by his grandfather, was made to trace out Tamil characters in a thick bed of rice spread across the floor, as each character was spoken aloud.
Soon fears of Ramanujan’s dumbness were dispelled and he began to learn the 12 vowels, 18 consonants, and 216 combined consonant-vowel forms of the Tamil alphabet. On October 1, 1892, the traditional opening day of school, known as Vijayathasami, he was enrolled, to the accompaniment of ancient Vedic chants, in the local pial school. A pial is the little porch in front of most South Indian houses; a pial school was just a teacher meeting there with half a dozen or so pupils.
But five-year-old Ramanujan, disliking the teacher, bristled at attending. Even as a child, he was so self-directed that, it was fair to say, unless he was ready to do something on his own, in his own time, he was scarcely capable of doing it at all; school for him often meant not keys to knowledge but shackles to throw off.
Quiet and contemplative, Ramanujan was fond of asking questions like, Who was the first man in the world? Or, How far is it between clouds? He liked to be by himself, a tendency abetted by parents who, when friends called, discouraged him from going out to play; so he’d talk to them from the
window overlooking the street. He lacked all interest in sports. And in a world where obesity was virtually unknown, where bones protruded from humans and animals alike, he was, first as a child and then for most of his life, fat. He used to say—whether as boast, joke, or lament remains unclear—that if he got into a fight with another boy he had only to fall on him to crush him to pieces.
For about two years, Ramanujan was shuffled between schools. Beginning in March 1894, while still at his mother’s parents’ house in Kanchipuram, he briefly attended a school in which the language of instruction was not his native Tamil but the related but distinct Telugu. There, sometimes punished by having to sit with his arms folded in front of him and one finger turned up to his lips in silence, he would at times stalk out of class in a huff.
In a dispute over a loan, his grandfather quit his job and left Kanchipuram. Ramanujan and his mother returned to Kumbakonam, where he enrolled in the Kangayan Primary School. But when his other grandfather died, Ramanujan was bounced back to his maternal grandparents, who by now were in Madras. There he so fiercely fought attending school that the family enlisted a local constable to scare him back to class.
By mid-1895, after an unhappy six months in Madras, Ramanujan was once more back in Kumbakonam.
• • •
Kumbakonam was flanked by the Cauvery and the Arasalar, its tributary. Most streets ran parallel to these rivers or else marched straight down to their banks, perpendicular to the first set, making for a surprisingly regular grid system. And there, near the middle of this compact grid, on Sarangapani Sannidhi Street, a dirt road about thirty feet wide with squat little buildings close packed on either side, was Ramanujan’s house.
The Man Who Knew Infinity Page 2