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June 17 - June 26, 2016
This is a story, too, about what you do with genius once you find it. Ramanujan was brought to Cambridge by an English mathematician of aristocratic mien and peerless academic credentials, G. H. Hardy, to whom he had written for help. Hardy saw that Ramanujan was a rare flower, one not apt to tolerate being stuffed methodically full of all the mathematical knowledge he’d never acquired in India. “I was afraid,” he wrote, “that if I insisted unduly on matters which Ramanujan found irksome, I might destroy his confidence and break the spell of his inspiration.”
This is also a book, then, about an uncommon and individual mind, and what its quirks may suggest about creativity, intuition, and intelligence.
Letters from an Indian Clerk,
Hardy, it turned out, was the third English mathematician to whom Ramanujan had appealed; the other two declined to help. And Hardy did not just recognize Ramanujan’s gifts; he went to great lengths to bring him to England, school him in the mathematics he had missed, and bring him to the attention of the world. Why Hardy?
Hardy, I learned, was a bizarre and fascinating character—a cricket aficionado, a masterful prose stylist, a man blessed with gorgeous good looks who to his own eyes was so repulsively ugly he couldn’t look at himself in the mirror. And this enfant terrible of English mathematics was, at the time he heard from Ramanujan, working a revolution on his field that would be felt for generations to come.
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.
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.
There were four castes, these accounts recorded: Brahmins, at the top of the heap; Kshatriyas, or warriors; Vaisyas, or merchants and traders; and Sudras, or menials. A fifth group, the untouchables, lay properly outside the caste system.
A Synopsis of Elementary Results in Pure and Applied Mathematics.
Was he respected as a mathematician? Was he deemed a dutiful son, a good Brahmin? Did he hold an important scholarship? Had he won a prize? The answers, as outward markers of acceptance or success, counted—and certainly never more so than now, as a teenager, at an age of exquisite sensitivity to the opinions of others.
Ramanujan would insist that its height was, of course, only relative: who could say how high it seemed to an ant or a buffalo? One time he asked how the world would look when first created, before there was anyone to view it. He took delight, too, in posing sly little problems: If you take a belt, he asked Viswanatha and his father, and cinch it tight around the earth’s twenty-five-thousand-mile-long equator, then let it out just 2π feet—about two yards—how far off the earth’s surface would it stand? Some tiny fraction of an inch? Nope, one foot.
Ramanujan had lost all his scholarships. He had failed in school. Even as a tutor of the subject he loved most, he’d been found wanting. He had nothing. And yet, viewed a little differently, he had everything. For now there was nothing to distract him from his notebooks—notebooks, crammed with theorems, that each day, each week, bulged wider.
Ramanujan needed no vision of monkeys chomping on guavas to spur his interest. For him, it wasn’t what his equation stood for that mattered, but the equation itself, as pattern and form. And his pleasure lay not in finding in it a numerical answer, but from turning it upside down and inside out, seeing in it new possibilities, playing with it as the poet does words and images, the artist color and line, the philosopher ideas.
And what about numbers, like the square root of − 1, which seem impossible or absurd? A negative number times a negative number, after all, by mathematical convention is positive; so how can any number multiplied by itself give you a negative number? No ordinary number, of course, can; those so defined are called “imaginary,” and assigned the label i; . Such numbers, it turns out, can be manipulated like any other and find wide use in such fields as aerodynamics and electronics.
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 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.
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,
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E. T. Bell put it in Men of Mathematics,
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.
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.”
he used to bemoan his wretched condition in life. When I encouraged him by saying that being endowed with a valuable gift he need not be sorry but only had to wait for recognition, he would reply that many a great man like Galileo died in inquisition and his lot would be to die in poverty. But I continued to encourage him that God, who is great, would surely help him and he ought not to give way to sorrow.
“He was so friendly and gregarious,” one who knew him later in Madras would say of him. He was “always so full of fun, ever punning on Tamil and English words, telling jokes, sometimes long stories, and going into fits of laughter when relating them. His tuft would come undone and he would try to knot it back as he continued to tell the story.” Sometimes he’d start laughing before reaching the punch line, garble its telling, and have to repeat it. “He was so full of life and his eyes were mischievous and sparkling. . . . He could talk on any subject. It was hard not to like him.”
People didn’t take to Ramanujan because he was sensitive to them, or because he was especially considerate. They might not understand his mathematics. They might even flirt with the idea that he was a crank. And they might, in the end, be unable to help him. But, somehow, they couldn’t help but like him.
Want the sine of 30 degrees? Just plug into the equation its radian equivalent (π/6, or about .5236), and add up as many terms as you want to get a value as accurate as you want. Here, even three terms are enough to get you to .500002—quite close to the correct .500000; the series converges rapidly.
Thus, this alternating infinite series—adding a bit here, subtracting something a little smaller there, and so on through an infinite number of terms—inexplicably equals just what you get from dividing one leg of a triangle by another.
“that the ordinary [mathematical] reader, unaccustomed to such intellectual gymnastics, could hardly follow him.”
“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.
“that some of the happiest hours of my life should have been spent within sound of a Roman Catholic Church.”
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
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.
Indeed, Hardy would later be offered as an “example of how far the English educational system can bring out the personal powers and capabilities of a man.”
It was not just mathematics in which Hardy was gifted, but all to which he took his hand. Yet a fragility, a diffidence, sometimes undermined it. He was painfully shy and self-conscious and could scarcely bear going up before the whole school to accept a prize. Sometimes he’d give wrong answers to ensure he wouldn’t have to do so. “Over-delicate,” his friend Snow described him later. “He seems to have been born with three skins too few.”
“Cranleigh boys may wander where they please, and this freedom is characteristic of the establishment throughout.”
he grew up with an emphasis on intellect and learning that, in ages past, only the aristocracy enjoyed.
Distinguish himself he did. Cranleigh had nourished him but now offered him, at age twelve, nothing more. His parents, who had the highest aspirations for their prodigiously intelligent son, kept tabs on scholarships for which he might be eligible. Around this time, about the scarcest and most coveted scholarship of all was to Winchester, one of England’s most hallowed public schools and a traditional proving ground for the mathematically gifted. Its scholarship exam was among the toughest of its kind; one year in the 1860s, 137 boys competed for seven scholarships. To prepare for it, you
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“gentlemanly rebels and intellectual reformers.”
reserved, patrician, conservative social and political leaders. But those who did rebel often became distinguished rebels.
read Tyndall and Huxley,
Second Wrangler. So had J. J. Thomson, discoverer of the electron. Then there was thermodynamicist Lord Kelvin, then still William Thomson, surely the best mathematician of his year. Everyone, including himself, thought he was a shoo-in for Senior Wrangler. “Oh, just run down to the Senate House, will you, and see who is Second Wrangler,” he asked his servant. The servant returned and said, “You, sir.” Someone else, his name today forgotten, had proved better able to master Tripos mathematics.
granted professional success—a fellowship at a good college, say—to those doing well on the exam, not those demonstrating a bent for research, or boldness in pursuing it.
Bertrand Russell, who ranked as Seventh Wrangler when he took the Tripos in 1893, and who would make numerous contributions to mathematical philosophy in the years ahead, wrote later how preparing for it “led me to think of mathematics as consisting of artful dodges and ingenious devices and as altogether too much like a crossword puzzle.” The Tripos over, he swore he’d never look at mathematics again, and at one point sold all his math books.
But the young Hardy, new to Cambridge and deeply disenchanted, began to think he just couldn’t go through with it. It was all too stupid. Maybe he would quit mathematics altogether.
When Jordan died almost three decades later, it would be Hardy who wrote his obituary in Proceedings of the Royal Society and took the opportunity to comment on the book that changed his life: “To have read it and mastered it is a mathematical education in itself,” Hardy wrote, “and it is hardly possible to overstate the influence which it has had on those who, coming to it as I did from the elaborate futilities of ‘Tripos’ mathematics, have found themselves at last in [the] presence of the real thing.”
“Surely?” “Just as surely?” For English mathematicians untouched by the precision of the Continent, such notions were obvious, scarcely worthy of another thought. But Jordan actually stated these seemingly self-evident truths as theorems and set about trying to prove them rigorously. In fact, he couldn’t do it, or at least not completely; his proofs were laced with flaws, and his successors had later to correct them. But they invoked just the kind of close, sophisticated reasoning that Hardy, coming upon Jordan now, at the age of barely twenty, found beguiling.
“He was enough of a natural competitor,” wrote Snow, “to feel that, though the race was ridiculous, he ought to have won it.”
truth “with the tenacity of a bulldog and the integrity of a saint.”
“There were to be no taboos, no limitations, nothing considered shocking, no barriers to absolute freedom of speculation,” wrote Bertrand Russell, who was inducted six years before Hardy.
A younger mathematician who knew Hardy much later, when during the 1920s he was at Oxford, says that there was indeed a “rumor of a young man” then. Later, when Hardy visited America in the 1930s, he would impress the mathematician Alan Turing, himself homosexual, as, in the words of his biographer, Andrew Hodges, “just another English intellectual homosexual atheist.”
“I have never done anything ‘useful,’ ” is how he would put it years later. “No discovery of mine has made or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” He would never say so, perhaps he did not even see it, but he had taken Moore’s sensibilities and applied them to mathematics. “Hardyism,” someone would later dignify this doctrine, so hostile to practical applications; and Hardy’s Mathematician’s Apology, written almost half a century later, would embody it on every page.
