Which is one reason why Ramanujan, and all mathematicians, use their seemingly alien language in the first place—as a standin for long-winded verbiage. When on page 86 of the first notebook, and in many other places, Ramanujan writes Σ, the Greek letter sigma, he means, simply, “the sum of …” A notational fragment like
may be read as “the sum of all the terms of the form x to the kth power divided by k, when k goes from 1 to infinity.” That means, “whenever you see a k, replace it by 1, and note it; then by a 2, and add it to the first… . Continue in this way forever.” And that is equivalent to
Mathematics is full of similarly simple ideas lurking behind alien terminology. Want to specify a series of terms that alternate between positive and negative? That’s easy: Just include the fragment (− 1)k. As k marches up through the integers one by one, the sign alternates automatically between plus and minus, because minus-times-minus is plus and minus-times-plus is minus. Or maybe you wish to specify only odd numbers, like 1, 3, 5, 7, and so on? The expression 2n + 1 churns them out. For n equals 0, 2n + 1 equals 1. For n = 1, 2n + 1 = 3. For n = 2, 2n + 1 = 5 …
Short, sweet, concise.
• • •
If you don’t know English, you can’t write a job application, and you can’t write King Lear. But just knowing English isn’t enough to write Shakespeare’s play. The same applies to Ramanujan’s notebooks: its pages of mathematical scrawl were, to professional mathematicians, what was least difficult about them. As with the English of Lear, it was what they said that took all the work.
And work it was—in expressing mathematical entities, performing operations on them, trying special cases, applying existing theorems to new realms. But some of the work, too, was numerical computation. “Every rational integer was his personal friend,” someone once said of Ramanujan; as with friends, he liked numbers, enjoyed being in their company.
Even in the published notebooks, you can see Ramanujan giving concrete numerical form to what others might have left abstract—plugging in numbers, getting a feel for how functions “behaved.” Some pages, with their dearth of Σ’s and f(x)’s, and their profusion of 61s and 3533s, look less like mathematical treatise, more like the homework assignment of a fourth-grader. Numerical elbow grease it was. And he put in plenty of it. One Ramanujan scholar, B. M. Wilson, later told how Ramanujan’s research into number theory was often “preceded by a table of numerical results, carried usually to a length from which most of us would shrink.”
From which most of us would shrink. There’s admiration there, but maybe a wisp of derision, too—as if in wonder that Ramanujan, of all people, could stoop so willingly to the realm of the merely arithmetical. And yet, Ramanujan was doing what great artists always do—diving into his material. He was building an intimacy with numbers, for the same reason that the painter lingers over the mixing of his paints, or the musician endlessly practices his scales.
And his insight profited. He was like the biological researcher who sees things others miss because he’s there in the lab every night to see them. His friends might later choose to recall how he made short work of school problems, could see instantly into those they found most difficult. But the problems Ramanujan took up were as tough slogging to him as school problems were to them. His successes did not come entirely through flashes of inspiration. It was hard work. It was full of false starts. It took time.
And that was the irony: in the wake of his failure at school, time was one thing he had plenty of.
6. A THOUGHT OF GOD
In 1807, a hundred years before Ramanujan was to fail his F.A. exam for the last time and experience India’s educational system in all its oppressive rigidity, William Thackeray, an Englishman with experience in India as translator, judge, and civil servant, concluded his Report on Canara, Malabar, and Ceded Districts. In it, he wrote:
It is very proper that in England, a good share of the produce of the earth should be appropriated to support certain families in affluence, to produce senators, sages and heroes for the service and defense of the state; or in other words, that a great part of the rent should go to opulent nobility and gentry, who are to serve their country in Parliament, in the army, in the navy, in the departments of science and liberal professions. The leisure, independence and high ideals which the enjoyment of this rent affords has enabled them to raise Britain to pinnacles of glory. Long may they enjoy it. But in India that haughty spirit, independence and deep thought which the possession of great wealth sometimes gives ought to be suppressed. They are directly averse to our power and interest. The nature of things, the past experience of all governments, renders it unnecessary to enlarge on this subject. We do not want generals, statesmen and legislators; we want industrious husbandmen. If we wanted restless and ambitious spirits there are enough of them in Malabar to supply the whole peninsula.
If Thackeray’s sentiments mirrored British educational policy in India, no better evidence for it could be found than Ramanujan, for whom college might have seemed aimed at suppressing “haughty spirit, independence, and deep thought.” Indeed, Indian higher education’s failure to nurture one of such undoubted, but idiosyncratic, gifts could serve as textbook example of how bureaucratic systems, policies, and rules really do matter. People, as individuals, appreciated and respected Ramanujan; but the System failed to find a place for him. It was designed, after all, to churn out bright, well-rounded young men who could help their British masters run the country, not the “restless and ambitious spirits” Thackeray warned against.
Viewed one way, then, for at least the five years between 1904 and 1909, Ramanujan floundered—mostly out of school, without a degree, without a job, without contact with other mathematicians.
And yet, was the cup half-empty—or half-full?
The great nineteenth-century mathematician Jacobi believed, as E. T. Bell put it in Men of Mathematics, that young mathematicians ought to be pitched “into the icy water to learn to swim or drown by themselves. Many students put off attempting anything on their own account till they have mastered everything relating to their problem that has been done by others. The result is that but few ever acquire the knack of independent work.”
Ramanujan tossed alone in the icy waters for years. The hardship and intellectual isolation would do him good? They would spur his independent thinking and hone his talents? No one in India, surely, thought anything of the kind. And yet, that was the effect. His academic failure forced him to develop unconventionally, free of the social straitjacket that might have constrained his progress to well-worn paths.
For five solid years, Ramanujan was left alone to pursue mathematics. He received no guidance, no stimulation, no money beyond the few rupees he made from tutoring. But for all the economic deadweight he represented, his family apparently discouraged him little—not enough, in any case, to stop him. India, it might be said, left room for the solitary genius in him as it would for the sage, the mystic, the sanyasi. His friends, his mother, and even his father tolerated him, made no unduly urgent demands that he find work and make something of himself. Indeed, in looking back to Ramanujan’s early years, Neville would refer to “the carefree days before 1909.” And, in a sense, they were. In some ways, they were the most productive of his life. Ramanujan had found a home in mathematics, one so thoroughly comfortable he scarcely ever wished to leave it. It satisfied him intellectually, aesthetically, emotionally.
And, the evidence suggests, spiritually, as well. Countless stories would later attest to how, in Ramanujan, the mathematical and the metaphysical lay side by side, inextricably intertwined. Once, while a student at Pachaiyappa’s College, he is said to have warned the parents of a sick child to move him away; “the death of a person,” he told them, “can occur only in a certain space-time junction point.” Another time, in a dream, he saw a hand write across a screen made red by flowing blood, tracing out elliptic integrals.
One idea Ramanujan bruited about dealt with the quantity 2n − 1. That, a frie
nd remembered him explaining, stood for “the primordial God and several divinities. When n is zero the expression denotes zero, there is nothing; when n is 1 the expression denotes unity, the Infinite God. When n is 2, the expression denotes Trinity; when n is 3, the expression denotes 7, the Saptha Rishis, and so on.”
Ramanujan was unfailingly congenial to metaphysical speculation. In Kumbakonam, there was a gymnastics teacher, Satyapriya Rao, whose fevered outpourings even tolerant South Indians dismissed. He would stand there, by the Cauvery, staring into the sun, raving; sometimes he’d have to be chained up when he got too hysterical. Most people ignored him. But not Ramanujan, who would sometimes collect food for him; some thought he must be mad to indulge him so. Yes, Ramanujan explained, he knew the man had visions, saw tiny creatures. But in an earlier birth, he was sure, Satyapriya had earned great merit. What others wrote off as the ravings of a madman was actually a highly evolved vision of the cosmos.
Later, in England, Ramanujan would build a theory of reality around Zero and Infinity, though his friends never quite figured out what he was getting at. Zero, it seemed, represented Absolute Reality. Infinity, or ∞, was the myriad manifestations of that Reality. Their mathematical product, ∞ × 0, was not one number, but all numbers, each of which corresponded to individual acts of creation. To philosophers, perhaps—and to mathematicians, certainly—the idea might have seemed silly. But Ramanujan found meaning in it. One friend, P. C. Mahalanobis—the man who discovered him shivering in his Cambridge room—later wrote how Ramanujan “spoke with such enthusiasm about the philosophical questions that sometimes I felt he would have been better pleased to have succeeded in establishing his philosophical theories than in supplying rigorous proofs of his mathematical conjectures.”
In the West, there was an old debate as to whether mathematical reality was made by mathematicians or, existing independently, was merely discovered by them. Ramanujan was squarely in the latter camp; for him, numbers and their mathematical relationships fairly threw off clues to how the universe fit together. Each new theorem was one more piece of the Infinite unfathomed. So he wasn’t being silly, or sly, or cute when later he told a friend, “An equation for me has no meaning unless it expresses a thought of God.”
7. ENOUGH IS ENOUGH
At the age of twenty, Ramanujan was, as he’d been most of his life, fat. He was short and squat, with a full nose set onto a fleshy, lightly pockmarked face, bare of mustache or beard. His shaved forehead, with its prominent red and white caste mark, and the rest of his full black hair gathered behind his head into a tuft, made him seem even rounder, fleshier, and fuller than he was.
But there was no thick, lumbering sluggishness to Ramanujan’s bulk; if anything it was more like that of a sumo wrestler, or a Buddha, with a lightness to it, even a delicacy. He walked with head erect, a sprightliness in his gait, body pitched forward onto his toes. He had long arms, hands of a surprising, velvety smoothness, and slender, tapering fingers forever in motion as he talked.
When he grew animated, the words tumbled out. Even eating, which he did with gusto, rarely staunched the flow; he’d go on with an idea or a joke even with his mouth full. And always, his dark eyes glowed; the rest of him could sometimes seem to fall away, leaving only the light in his eyes.
Occasionally he’d drop by the college that had flunked him, to borrow a book, or see a professor, or hear a lecture. Or he’d wander over to the temple. But mostly, Ramanujan would sit working on the pial of his house on Sarangapani Sannidhi Street, legs pulled into his body, a large slate spread across his lap, madly scribbling, seemingly oblivious to the squeak of the hard slate pencil upon it. For all the noisy activity of the street, the procession of cattle, of sari-garbed women, of half-naked men pulling carts, he inhabited an island of serenity. Human activity passed close by, yet left him alone, and free, unperturbed by exams he had no wish to take, or subjects he had no wish to study.
The Hindu, South India’s premier English language newspaper, had observed in an 1889 editorial that “the Indian character has seldom been wanting in examples of what may be called passive virtues. Patience, personal attachment, gentleness and such like have always been prominent. But for ages together, India has not had amongst her sons one like Gordon, Garibaldi, or Washington… . In all departments of life,” it went on, “the Hindus require a vigorous individuality, a determination to succeed and to sacrifice everything in the attempt.” And despite his seeming indifference to worldly success, Ramanujan, inwardly, was a model of all the Hindu editorial writer could have wanted.
A determination to succeed and to sacrifice everything in the attempt. That could be a prescription for an unhappy life; certainly for a life out of balance, sneering at timidity and restraint. Sometimes, as Ramanujan sat or squatted on the pial, he’d look up to watch the children playing in the street with what one neighbor remembered as “a blank and vacant look.” But inside, he was on fire.
When he thought hard, his face scrunched up, his eyes narrowed into a squint. When he figured something out, he sometimes seemed to talk to himself, smile, shake his head with pleasure. When he made a mistake, too impatient to lay down his slate pencil, he twisted his forearm toward his body in a single fluid motion and used his elbow, now aimed at the slate, as an eraser.
Ramanujan’s was no cool, steady Intelligence, solemnly applied to the problem at hand; he was all energy, animation, force.
He was also a young man who hung around the house, who had flunked out of two colleges, who had no job, who indulged in mystical disquisitions that few understood, and in mathematics that no one did. What value was his work to anyone? Maybe he was a genius, maybe a crank. But in any case, why waste one’s time and energy in activity so divorced from the common purposes of life? Didn’t his father, working as a lowly clerk in a silk shop, do the world and himself more good than he?
For a long time his parents put up with him. But in the end they too reached their limits, grew irritated and impatient. Enough is enough, his mother decided. And sometime probably late in 1908 she moved decisively to invoke what the Indian psychologist Ashis Nandy has called “that time-tested Indian psychotherapy”—an arranged marriage.
CHAPTER THREE
The Search for Patrons
[1908 to 1913]
1. JANAKI
One day late in 1908, Ramanujan’s mother was visiting friends in the village of Rajendram, about sixty miles west of Kumbakonam. There she spied a bright-eyed wisp of a girl, Janaki, daughter of a distant relative. She asked for the girl’s horoscope—the first step in virtually every arranged marriage in India—drew her son’s horoscope on the wall of the house, compared it to that of the girl, and concluded that yes, this would make a good match. Negotiations ensued for the marriage of Ramanujan and Janaki, then about nine years old.
It was in many ways an apt match, between two persons of equally meager social standing. Janaki was an unassuming, only ordinarily pretty girl from a village so tiny it appears on none but the most detailed maps. The family had once been better off; her father dealt in jewelry-making supplies and had once owned a little property. But now, fallen on harder times, they could offer only a modest dowry, perhaps a few polished copper vessels. They could not afford to be choosy about a husband, especially since Janaki was but one of five daughters (along with one son). Most of all, they sought a family for their daughter apt to treat her kindly during those early years when, still without children, she toiled under the imperious eye and unquestioned authority of her mother-in-law.
Ramanujan, meanwhile, was no great catch. Outwardly, he was a total failure, lacking degree, job, or prospects. Janaki knew nothing of the man who was to become her husband; she would not so much as glimpse his face until their wedding. He was an ordinary young man from an ordinary family. Maybe, she thought later, her parents had heard Komalatammal tout her son as a mathematical genius; if so, she’d known nothing of it.
So far as Komalatammal was concerned it was all set, which
in Ramanujan’s family meant it was. But when husband Srinivasa learned of the arrangements, he fumed. The boy can do better than that, he protested. Many families in Kumbakonam would be proud to count him as son-in-law; in fact, two years before, when Ramanujan was off at Pachaiyappa’s College, a family in Kanchipuram had come forward with an offer, and only a death in the bride’s family had gotten in the way. What really riled Srinivasa, though, was that he’d had no say in the plans. Nothing mollified him. So the following July, when it came time to travel to Rajendram for the wedding, he stayed home.
In these events, Srinivasa’s exclusion was unusual. Arranged marriages, without a say for the bride or groom, were virtually universal, the institution of child-brides almost as much so; most girls married before puberty, though they didn’t actually live with their husbands, consummating the marriage, until later. The practice was repugnant to most Europeans, but the British, ever sensitive to local custom, did nothing to change it. In 1894, the state of Mysore had passed a law barring the marriage of girls younger than eight; a similar provision in Madras had failed.
Extending over four or five days, an Indian wedding was a glory of color and tinsel, music and ceremony. The whole economy was influenced by the scale and expense of these grand affairs, on which six months’ income might be blown with scarcely a thought. Even the poorest families unblinkingly assumed every burden—saved every spare rupee, indebted themselves to local usurers—to provide their daughters’ dowries, to buy new saris, and to pay for the meals and music of the wedding itself.
Ramanujan’s was a double wedding, Janaki’s sister Vijayalakshmi being set to marry the same day. (By December, she would be dead, prey to a severe fever.) The other bridegroom showed up on schedule. But long into the day before the wedding, and then into the night, Ramanujan and his family failed to appear. Janaki’s father, Rangaswamy, had never been entirely won over by the prospect of the match. Now, Janaki heard him say, if Ramanujan didn’t show up soon, they’d marry her off, then and there, to someone else, maybe his nephew… .
The Man Who Knew Infinity Page 9
