Table of contents

Kanigel, Robert. The Man Who Knew Infinity. Washington Square Press, 2018. Kindle edition.

Comment on 2018-08-20 The deities that led to Ramanujan’s infinity and is embodied in this biography pacified the mind.

Prologue

P176 Ramanujan was a man who grew up praying to stone deities; who for most of his life took counsel from a family goddess, declaring it was she to whom his mathematical insights were owed; whose theorems would, at intellectually backbreaking cost, be proved true—yet leave mathematicians baffled that anyone could divine them in the first place.

P211 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.

P237 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.

P241 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.

P245 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.”

Chapter One: In the Temple's Coolness [1887 to 1903]

1. Dakshin Gange

P256 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.

P277 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

P387 In many parts of South India, the land was, for much of the year, a bleak brown. But here, midst the rice fields of the Cauvery, the landscape suddenly thickened with lush greenery in a rich palette of shades and textures. Farmers nursed delicate infant rice seedlings in small, specially watered plots whose rich velvety green stood out against neighboring fields. When, after thirty or forty days, the plants were healthy and strong, laborers individually scooped them out with their root pods and transplanted them to large, flooded fields; these made for a softer green. There the plants grew until a yellower hue signaled they were ready for harvest.

P401 Kumbakonam was more cosmopolitan than its surroundings, was a center for the work of eye, hand, and brain, which needs a degree of leisure to pursue.

笔记 - P402 闲暇是思想之始。

5. The Goddess of Namakkal

P616 It would take a few minutes for his eyes to adjust to the shadows. There, in the Sarangapani temple’s outer hall, it seemed gloomy after the bright sun outside. What light there was swept in from the side, softly modeling the intricate sculpted shapes, the lions and geometrically cut stone, of the hall’s closely spaced columns. Away further from the light, nestled among the columns, were areas favored by bats for nesting. Sometimes Ramanujan could hear the quick, nervous swatting of their wings. Or even see them hanging from the ceiling, chirping away, then abruptly fluttering into flight. Unlike Western churches which, architecturally, drew you higher and higher, here the devout were pulled, as it were, inner and inner.

P625 Always the temple stirred with little bright devotional fires, the chanting of mantras, the smell of incense in small shrines and dark niches devoted to secondary deities. The closer one approached to the central shrine itself, the darker it grew—more mysterious, more intimately scaled, progressively smaller, tighter, closer. What from the noisy street beyond the temple walls might have seemed a fit site for great public spectacles, here, inside, within stone grottos blackened by centuries of ritual fire presided over by bare-chested Brahmin priests, was a place for one man and his gods.

P691 South India was a world apart. All across India’s northern plains, the centuries had brought invasion, war, turmoil, and change. Around 1500 B.C. light-skinned Aryans swept in through mountain passes from the north. For eight centuries, Buddhism competed with traditional Brahminism, before at last being overpowered by it. Beginning in the tenth century, it was the Muslims who invaded, ultimately establishing their own Moghul Empire. One empire gave way to another, the races mingled, religions competed, men fought.

P708 In South India an undiluted spirituality had had a chance to blossom. If the North was like Europe during the Enlightenment, the South was, religiously, still rooted in the Middle Ages. If Bombay was known for commerce, and Calcutta for politics, Madras was the most single-mindedly religious. It was a place where there was less, as it were, to distract you—just rice fields, temples, and hidden gods. Here, in this setting, with the secular world held at bay, within a traditional culture always willing to see mystical and magical forces at work, Ramanujan’s belief in the unseen workings of gods and goddesses, his supreme comfort with a mental universe tied together by invisible threads, came as naturally as breath itself.

P736 The three chief deities in the Hindu pantheon, Brahma, Siva, and Vishnu, were traditionally represented as, respectively, the universe’s creative, destructive, and preserving forces. In practice, however, Brahma, once having fashioned the world, was seen as cold and aloof, and tended to be ignored. So the two great branches of Brahminic Hinduism became Shaivism and Vaishnavism. Shaivism had a kind of demonic streak, a fierceness, a malignity, a raw sexual energy embodied in the stylized phallic symbol known as a lingam that was the centerpiece of every Shaivite temple. Think of sweeping change, of cataclysmic destruction, and you invoked Lord Siva. Vaishnavism, befitting its identification with the conserving god Vishnu, had more placid connotations. One contemporary English account likened it to the Spirit of Man—a distinctly gentler idea. Figuring largely in Vaishnavism were Rama and Krishna, heroes of Indian legend, and two of the incarnations, or avatars, in which Vishnu appears.

P768 Ramanujan’s maternal grandmother, Rangammal, was a devotee of Namagiri and was said to enter a trance to speak to her. One time, a vision of Namagiri warned her of a bizarre murder plot involving teachers at the local school. Another time, many years earlier, before Ramanujan’s birth, Namagiri revealed to her that the goddess would one day speak through her daughter’s son. Ramanujan grew up hearing this story. And he, too, would utter Namagiri’s name all his life, invoke her blessings, seek her counsel. It was goddess Namagiri, he would tell friends, to whom he owed his mathematical gifts. Namagiri would write the equations on his tongue. Namagiri would bestow mathematical insights in his dreams.

Chapter Two: Ranging with Delight [1903 to 1908]

1. The Book of Carr

P885 the typical mathematics text methodically worked through a subject, setting out a theorem, then going through the steps of its proof. The student was expected to dutifully follow along behind the author, tracking his logic, perhaps filling in small gaps in his reasoning.

P904 “I have, in many cases,” he explained, merely indicated the salient points of a demonstration, or merely referred to the theorems by which the proposition is proved. . . . The difference in the effect upon the mind between reading a mathematical demonstration, and originating one wholly or partly, is very great. It may be compared to the difference between the pleasure experienced, and interest aroused, when in the one case a traveller is passively conducted through the roads of a novel and unexplored country, and in the other case he discovers the roads for himself with the assistance of a map.

2. The Cambridge of South India

P943 He got his hands on the few foreign-language math texts in the library and made his way through at least some of them; mathematical symbols, of course, are similar in all languages.

笔记 - P944 Mathematics is a universal language.

P961 Still, he managed to hang on for a few months, showing up for class enough to earn a certificate in July 1905 attesting to his attendance. The effort must have taxed him. He’d lost the scholarship, and everybody knew it. His parents were under a heavy financial burden; he knew that, too. He felt pressure to do well in his other subjects, yet he didn’t want to lay mathematics aside for their sake. He was torn and miserable. He endured the situation until he could endure it no longer. In early August 1905, Ramanujan, seventeen years old, ran away from home.

笔记 - P963 对数学难以割舍的感觉，令其只忍受了一个月。

3. Flight

P966 As the hot breeze poured through the open windows of the railway car, Ramanujan watched the South Indian countryside slip by at twenty-five miles an hour. Villages of thatched roofs weathered to a dull barn-gray; intense pink flowers poking out from bushes and trees; palm trees, like exclamation points, punctuating the rice field flatness. From a distance, the men in the fields beside the tracks were little more than brown sticks, their dhotis and turbans white cotton puffs. The women were bright splashes of color, their orange and red saris set off against the startling green of the rice fields. A snapshot might have recorded the scene as a charming bucolic tableau, but Ramanujan saw people everywhere engaged in purposeful activity. Men tending cattle. Women, stooped over in the fields, nursing the crops. Sometimes they worked alone, sometimes together in groups of a dozen or more, baskets perched atop their heads, fetching water from streams. Occasionally, a child with its mother would glance up from the surrounding fields and wave at the train bearing Ramanujan north to Vizagapatnam.

P1035 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. Tales of Ramanujan’s youth reveal a boy content to camp out on the pial of his house and work at mathematics, outwardly oblivious to the raucous play of his friends out on the street. Often, wrapped up in mathematics, he was oblivious. At other times, though, he must have wanted to be part of it. His thirst for public acknowledgment of his gifts, his pain when denied it, and his sensitivity to social slight, show how deeply, at another level, he really cared.

笔记 - P1041 A sensitive heart.

4. Another Try

P1131 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.

笔记 - P1133 摆脱了世俗成就的困扰和对其的孜孜以求, 才能够获得思想的真正自由, 打开真理殿堂的大门。

5. The Notebooks

P1138 After Ramanujan’s death, his brother prepared a succession of handwritten accounts of the raw facts, data, and dates of his life. And preserved in their original form as they are, they remind us of a world before computers and word processors made revision easy and routine: we see rude scrawls growing neater, more digested and refined, as they are copied and recopied through successive versions.

笔记 - P1140 如切如磋, 如琢如磨。要想臻于完美的境界不知要经过多少次的反复提炼。

P1145 Broken into discrete chapters devoted to particular topics, its theorems numbered consecutively, it suggests Ramanujan looking back on what he has done and prettying it up for formal presentation, perhaps to help him find a job. It is, in other words, edited. It contains few outright errors; mostly, Ramanujan caught them earlier. And most of its contents, arrayed across fifteen or twenty lines per page, are entirely legible; one needn’t squint to make out what they say. No, this is no impromptu record, no pile of sketches or snapshots; rather, it is like a museum retrospective, the viewer being guided through well-marked galleries lined with the artist’s work.

P1169 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.

P1190 That happens often in mathematics; a notion at first glance arbitrary, or trivial, or paradoxical turns out to be mathematically profound, or even of practical value.

P1196 Ramanujan’s notebooks ranged over vast terrain. But this terrain was virtually all “pure” mathematics. Whatever use to which it might one day be put, Ramanujan gave no thought to its practical applications. He might have laughed out loud over the monkey and the guava problem, but he thought not at all, it is safe to say, about raising the yield of South Indian rice. Or improving the water system. Or even making an impact on theoretical physics; that, too, was “applied.” Rather, he did it just to do it. Ramanujan was an artist. And numbers—and the mathematical language expressing their relationships—were his medium.

P1201 Ramanujan’s notebooks formed a distinctly idiosyncratic record. In them even widely standardized terms sometimes acquired new meaning. Thus, an “example”—normally, as in everyday usage, an illustration of a general principle—was for Ramanujan often a wholly new theorem.

P1258 Mathematics is full of similarly simple ideas lurking behind alien terminology.

P1276 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.

6. A Thought of God

P1290 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.

P1301 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.

P1331 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.”

P1333 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

P1346 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.

P1356 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.

P1362 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?

Chapter Three: The Search for Patrons [1908 to 1913]

1. Janaki

P1378 It was in many ways an apt match, between two persons of equally meager social standing.

P1396 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.

2. Door-to-Door

P1477 Ramanujan’s notebooks were no longer for him merely a private record of his mathematical thought. As the preceding incident suggests, they were his legacy. And they were a selling document, his ticket to a job—“evidence,” as his English friend Neville would later put it, “that he was not the incorrigible idler his failures seemed to imply.” Propelled by necessity, he had begun calling on influential men who, he thought, could give him a job. And slung under his arm as he called were—just as photographers have their portfolios, or salesmen their display cases—his notebooks. Ramanujan had become, in the year and a half since his marriage, a door-to-door salesman. His product was himself.

P1510 His humor ran toward the obvious. His puns were crude. His idea of entertainment was puppet shows, or bommalattam; or else simple street dramas, terrukutu, that ran all night during village festivals and to which Ramanujan would go with friends, cracking jokes and telling stories along the way. Ramanujan wore his spirits on his sleeve. There was something so direct, so unassuming, so transparent about him that it melted distrust, made you want to like him, made you want to help him.

P1579 But many of Ramanujan’s formulas, he’d written in the margins of the sheet of paper Ramanujan had sent him, seemed intriguing indeed. It was just that he could hardly throw the weight of his reputation behind someone working in areas so unfamiliar to him.

3. “Leisure” in Madras

P1593 The word leisure has undergone a shift since the time Ramachandra Rao used it in this context. Today, in phrases like leisure activity or leisure suit, it implies recreation or play. But the word actually goes back to the Middle English leisour, meaning freedom or opportunity. And as the Oxford English Dictionary makes clear, it’s freedom not from but “to do something specified or implied” [emphasis added]. Thus, E. T. Bell writes of a famous seventeenth-century French mathematician, Pierre de Fermat, that he found in the King’s service “plenty of leisure”—leisure, that is, for mathematics.

P1599 In his Report on Canara, Malabar and Ceded Districts, Thackeray spoke of the “leisure, independence and high ideals” that had propelled Britain to its cultural heights. The European “gentleman of leisure,” free from the need to earn a livelihood, presumably channeled his time and energy into higher moral and intellectual realms. Ramanujan did not belong to such an aristocracy of birth, but he claimed membership in an aristocracy of the intellect. In seeking “leisure,” he sought nothing more than what thousands born to elite status around the world took as their due.

P1609 As a Brahmin, Ramanujan may also have felt freer to seek the sort of constructive idleness he thought he needed—and perhaps even, in some measure, conceived as his due. Traditionally, Brahmins were recipients of alms and temple sacrifices; earning a livelihood was for them never quite the high and urgent calling it was for others. Uncharitably, it might be said that Ramanujan exhibited a prima donna-like self-importance that left him unwilling to study what he had no wish to study, or to work for any reason but to support his mathematics. Less harshly—and, on balance, with greater justice—he was a secular sanyasi.

4. Jacob Bernoulli and His Numbers

P1694 reverence for the past was no substitute for present achievement—surely hoped it did.

5. The Port Trust

P1824 Early the following year, K. S. Srinivasan, a student at Madras Christian College who’d known Ramanujan back in Kumbakonam, dropped by to see him at Summer House. “Ramanju,” he said, “they call you a genius.” Hardly a genius, replied Ramanujan, “Look at my elbow. That will tell you the story.” It was rough, dirty, and black. Working from his large slate, he found the quick flip between writing hand and erasing elbow a lot faster, when he was caught up in the throes of his work, than reaching for a rag. “My elbow is making a genius of me,” he said. Why, Srinivasan asked, didn’t he just use paper? Can’t afford it, replied Ramanujan. He was getting money from Ramachandra Rao. But that only went so far. Paper? He’d need four reams of it a month. Another friend from the Summer House days, N. Ramaswami Iyer [no relationship to the “Professor”] also recalled Ramanujan’s “huge appetite” for paper. Ramaswami pictured him lying on a mat, his shirt torn, “his long hair carelessly bound up with a piece of thin string,” working feverishly, notebooks and loose sheets of plain white paper piled up around him. A friend from Pachaiyappa’s who met him in Madras a little later, T. Srinivasacharya, recalled that, for want of paper, Ramanujan would sometimes write in red ink on paper already written upon.

P1903 Cows and bullocks, chickens and goats, roamed freely. In one street, metalworkers would squat in front of their tiny stalls, hammering out shapes, or tossing scraps of tin into little buckets to be melted. The next street would be clogged with bullocks, shouldering huge sacks of grain. Then streets of jewelry stalls, of textile shops, oilmongers, basketweavers, fruit and vegetable wholesalers . . . And everywhere, driving it all, was muscle power, black-haired men, shoeless and shirtless, clad only in their dhotis, ribs and muscles pushing out against glistening dark brown skin, straining as they pulled carts, or bent low under heavy loads upon their backs, or whipping their animals through the dusty streets.

P1916 “I used to see him many times running to his office via the Beach Road,” recalled a friend from Summer House days, referring to the road that ran right up beside the Port Trust offices. “With his coat, tail and all, flying in the breeze, and his long hair coming undone, a bright namam [his trident-shaped caste mark] adorning his forehead, the young genius had no time to waste; he was always in a hurry.”

P1919 Janaki would later recall how before going to work in the morning he worked on mathematics; and how when he came home he worked on mathematics. Sometimes, he’d stay up till six the next morning, then sleep for two or three hours before heading in to work. At the office, his job probably included verifying accounts and establishing cash balances.

P1925 a friend found him around the docks during working hours, prowling for packing paper on which to work calculations.

P1929 Narayana Iyer, a member of the Mathematical Society and long its treasurer, was not just Ramanujan’s immediate superior, but his colleague. In the evenings, they would retire to the elder man’s house on Pycroft’s Road in Triplicane. There, they’d sit out on the porch upstairs overlooking the street, slates propped on their knees, sometimes until midnight, the interminable scraping of their slate-pencils often keeping others up. Sometimes, after they had gone to sleep, Ramanujan would wake and, in the feeble light of a hurricane lamp, record something that had come to him, he’d explain, in a dream.

6. The British Raj

P1982 (Indeed, years later, asked as part of a survey what most struck them about England, students from former Asian and African colonies invariably mentioned the sight of white men doing manual labor.)

P1984 That there was an ineradicable split between Englishman and Indian the British themselves were eager to acknowledge. “East is East and West is West and never the twain shall meet,” wrote Kipling. Indians might work, even live with you in your bungalow, noted one old India hand, Herbert Compton; but in the end, “there is no assimilation between black and white. They are, and always must remain, races foreign to one another in sentiment, sympathies, feelings, and habits. Between you and a native friend there is a great gulf which no intimacy can bridge—the gulf of caste and custom. Amalgamation is utterly impossible in any but the most superficial sense, and affinity out of the question.”

P2002 Herbert Compton, who had run a plantation for years, observed in a book published in 1904 that “whilst you can polish the Hindu intellect to a very high pitch, you cannot temper the Hindu character with those moral and manly qualities that are essential for the positions he seeks to fill.” A retired member of the Indian Civil Service, Sir Bampfylde Fuller, marveled at how Hindu boys could flock to classrooms and libraries, and pursue Western literature and science, yet unaccountably cling to . . . well, Indian ways. Indians, he said, were unduly sentimental, wildly inconsistent. “An Englishman is constantly disconcerted by the extraordinary contradictions which he observes between the words and the actions of an educated Indian, who seems untouched by inconsistencies which to him appear scandalous. . . . They give eager intellectual assent to [European] ideals, yet live their lives unchanged.”

7. The Letter

P2042 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.

笔记 - P2046 A mathematician is not a calculating machine.

P2050 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.

笔记 - P2057 The critiques of mediocrity over prodigy

P2057 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.

笔记 - P2060 We have heard too many times of “interesting” from others’ mouth, which have led to nothing.

P2070 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.

笔记 - P2072 脱离学术圈的代价。

Chapter Four: Hardy [G. H. Hardy to 1913]

1. Forever Young

P2121 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.

笔记 - P2126 将兴趣当成专业，是专业人士的特质（idiosyncrasy）之一。

3. Flint and Stone

P2329 At Winchester, Hardy found much against which to rebel. From outside the ancient complex, he confronted the original college wall, all flint and stone, pierced by tiny slotted windows; these formidable architectural details stemmed from the Peasants’ Revolt, and from bloody town-gown battles at Oxford, both still of recent memory in the fourteenth century when the school was founded. Scholarship students like Hardy lived in a sort of intellectual ghetto within the college, a fortresslike complex of medieval, gray stone buildings, worlds apart from the sunny openness of Cranleigh. As a new student, Hardy was grilled on “notions”—a vast lexicon of jargon and slang peculiar to Winchester. Some of them went back to the Latin, some were submerged in the mists of the school’s medieval past. Collectively, they defined good form and bad, as Winchester, across the span of centuries, had come to see them. “Tugs” was stale news. To “brock” was to bully. A “remedy,” derived from the Latin remedium, meant “a holiday.” A “tunding” was a flogging at the hands of a prefect, or senior student officer. Learning your notions made for no idle study. There were thousands of words, with whole published glossaries given over to them, some graced with exquisite drawings and illuminated capitals. All had to be memorized.

P2408 He was a Fellow of Trinity! He sat at the high table now, and the grave portraits of the founders and the illustrious dead looked down upon him approvingly. The ardours, the sorrows, the struggles of the race, were all over; only the brilliant achievement remained. The great cloud of witnesses that looked down from those old rafters overhead upon those who feasted there had never approved a more nobly earned success in the rich intellectual history of the past of Trinity.

4. A Fellow of Trinity

P2459 Cambridge men were deemed wholly unable to cook, clean, or otherwise care for themselves. They were served by bedmakers and “gyps,” who brought coal up to feed the fireplace, took in the mail, fetched lunch, set out bed linen, towels, and tea cloths, served tea. Students were invariably addressed as Mister and treated with no little respect.

P2558 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.

P2582 A thirty-three-year-old man with a huge, bushy mustache, mutton-chop sideburns, and a vast bald oval of a head, Love had been named, a few years before, a Fellow of the Royal Society, Britain’s most distinguished scientific body. In 1893, he’d finished his two-volume Treatise on the Mathematical Theory of Elasticity, summarizing what was then known of how materials deform under impact, twisting, and heavy loads.

P2597 Say you draw a circle on a piece of paper; obviously, the circle divides the paper into two regions—within the circle, and outside it. Now, say you’ve got two points on the paper, both of which lie outside the circle: surely you can connect them (not necessarily with a straight line) without cutting the circle, right? And just as surely, if one point lies outside the circle and the other inside, then any continuous line linking them has to cut the circle . . . “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. “I shall never forget,” Hardy later wrote of Jordan’s book, whose second, much-improved edition had just appeared in 1896, “the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.”

笔记 - P2607 This is also my impression when I took the functional analysis course given by Prof Bu Shangquan, which I still often reminisce fondly up to now.

5. “The Magic Air”

P2676 Homosexuality was elevated almost to the status of an art form or aesthetic doctrine. “The Higher Sodomy,” Apostles termed it—the assertion that the love between man and man could be higher and finer than that of man for woman, thus raising homosexual relationships to an almost spiritual plane.

笔记 - P2678 Women could also join the elite games. Unfortunately, they were absent.

P2740 That Hardy’s life was spent almost exclusively in the company of other men, that he scarcely ever saw a woman, was, in those days, not uncommon. After all, among Havelock Ellis’s thousand or so British “geniuses,” 26 percent never married. In the academic and intellectual circles of which Hardy was a part, such a monastic sort of life actually represented one pole of common practice.

6. The Hardy School

P2795 In 1903, he was named an M.A., which at English universities was normally the highest academic degree. (Cambridge didn’t offer the doctorate, a German innovation, until after World War I, hoping to lure Americans otherwise drawn to Germany.)

P2798 he was never actually a college tutor. He was there to do research.

笔记 - P2799 The conflict between tutoring and research.

P2803 In much of this work, he was refining and enhancing ideas suggested by Camille Jordan in the book that had so inspired Hardy as an undergraduate.

笔记 - P2804 A bud formed in the early years.

P2830 Havelock Ellis once wrote that “by inborn temperament, I was, and have remained, an English amateur; I have never been able to pursue any aim that no passionate instinct has drawn me towards.” There was, as in Ellis, a streak of disdain in the English character for mere necessity; the amateur, bless his heart, did what he did for love, for the sake of beauty or truth, not because necessity compelled it. This streak had gained more reasoned form in the philosophy of G. E. Moore, Hardy’s Apostolic “father.” His Principia Ethica represented, in the words of Gertrude Himmelfarb, a “manifesto of liberation” stressing love, beauty, and truth. “And even love, beauty, and truth were carefully delineated as to remove any taint of utility or morality. Useless knowledge was deemed preferable to useful, corporeal beauty to mental qualities, present and immediately realizable goods to remote or indirect ones.” G. H. Hardy’s mathematics would emerge as the consummate manifestation, within his own field, of Moore’s credo. “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. That mathematics might aid the design of bridges or enhance the material comfort of millions, he wrote, was scarcely to say anything in its defense. For such mathematics, he bore only contempt. It is undeniable that a good deal of elementary mathematics . . . has considerable practical utility. [But] these parts of mathematics are, on the whole, rather dull; they are just the parts which have least aesthetic value. The “real” mathematics of the “real” mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly “useless.”

P2867 Newton was the premier genius of his age, the most fertile mind, with the possible exception of Shakespeare’s, ever to issue from English soil. And yet he would later be called “the greatest disaster that ever befell not merely Cambridge mathematics in particular but British mathematical science as a whole.” For to defend his intellectual honor, as it were, generations of English mathematicians boycotted Europe—steadfastly clung to Newton’s awkward notational system, ignored mathematical trails blazed abroad, professed disregard for the Continent’s achievements. “The Great Sulk,” one chronicler of these events would call it.

P2873 Continental mathematics laid stress on what mathematicians call “rigor,” the kind to which Hardy had first been exposed through Jordan’s Cours d’analyse and which insisted on refining mathematical concepts intuitively “obvious” but often littered with hidden intellectual pitfalls. Perhaps reinforced by a strain in their national character that sniffed at Germanic theorizing and hairsplitting, the English had largely spurned this new rigor. Looking back on his Cambridge preparation, Bertrand Russell, who ranked as Seventh Wrangler in the Tripos of 1893, noted that “those who taught me the infinitesimal Calculus did not know the valid proofs of its fundamental theorems and tried to persuade me to accept the official sophistries as an act of faith. I realized that the Calculus works in practice but I was at a loss to understand why it should do so.” So, it is safe to say, were most other Cambridge undergraduates.

P2882 it is possible to blithely sail on past these intellectual perils, concentrate on the many practical applications that fairly erupt out of calculus, and never look back.

P2939 Thought, Hardy used to say, was for him impossible without words. The very act of writing out his lecture notes and mathematical papers gave him pleasure, merged his aesthetic and purely intellectual sides.

P2957 There is only one course for which no good defence can ever be found. This course is to give what profess to be proofs and are not proofs, reasoning which is ostensibly exact, but which really misses all the essential difficulties of the problem.

P2992 At the root of Britain’s mathematical backwardness, Hardy was sure, lay the Tripos system. Originally the means to a modest end—determining the fitness of candidates for degrees—the Tripos had become an end in itself. As Hardy saw it, English mathematics was being sapped by the very system designed to select its future leaders. Around 1907, he became secretary of a panel established to reform it. But in fact, he championed its reform only as a first step toward doing away with it altogether, and only because he saw no hope, just then, for more radical change.

笔记 - P2996 Chinese education still faces similar headache without any hope. Time and patience are needed to solve this problem.

P3019 Hardy’s vehemence suggests a peculiar rift within his personality. Here was a man—a friend would one day liken him to “an acrobat perpetually testing himself for his next feat”—who set up rating scales at the least provocation, loved competitive games, grilled new acquaintances on what they knew, held up mathematical work to the highest standards—yet swore eternal enmity to the Tripos system which, in a sense, was the ultimate rating scale, the ultimate test.

笔记 - P3022 This is similar to my attitude toward Chinese Gaokao.

Chapter Five: “I Beg to Introduce Myself ...” [1913 to 1914]

1. The Letter

P3092 The whole letter, all ten pages of it, was written out in large, legible, rounded schoolboy script distinguished only by crossed t’s that didn’t cross. His handwriting had always been neat; but here, if possible, it was neater still, as if he realized the gulf of skepticism that divided him from Hardy and dared not let an illegible scrawl widen it. It was a wise precaution; the gulf was indeed great. For Hardy, Ramanujan’s pages of theorems were like an alien forest whose trees were familiar enough to call trees, yet so strange they seemed to have come from another planet; it was the strangeness of Ramanujan’s theorems that struck him first, not their brilliance. The Indian, he supposed, was just another crank.

P3107 The Indian manuscript scraped and tugged at his composure with, as Snow wrote, its “wild theorems. Theorems such as he had never seen before, nor imagined.” Were they wild and unimaginable because they were silly, or trivial, or just plain wrong, with nothing to support them? Or because they were the work of some rare flower of exotic genius? Or maybe they were merely well-known theorems the Indian had found in some book and cleverly disguised by expressing in slightly different form—making it just a matter of time before Hardy found them out? Or perhaps it was all a practical joke? Hoaxes, after all, were much in vogue just then. Many Englishmen holding high positions in the Indian Civil Service had endured the mathematical Tripos or were otherwise versed in mathematics—well versed enough, perhaps, to pull off such a stunt. And how best to dupe your old Cambridge friend Hardy? Why, you’d garb familiar “theorems” in unfamiliar attire, purposely twist them into weird shapes. But who in India was adept enough to do it? Maybe the hoax had originated in Europe. But would the perpetrator have gone to the trouble of securing a genuine Madras postmark . . . ?

笔记 - P3116 真正的突破性、创新性成果虽然不一定但是相对容易获得青睐。

P3142 For three years, he left to assume a lectureship at the University of Manchester. There, in “exile,” he endured an oppressive load of lecturing, conferences, and paperwork—the normal lot of faculty members at provincial universities.

笔记 - P3144 Similar to present situation of Chinese universities.

P3160 “Nowadays,” somebody said later, “there are only three really great English mathematicians: Hardy, Littlewood, and Hardy-Littlewood.” Because Littlewood disdained bright, sparkling company and stayed away from mathematics conferences, some—at least in jest—doubted he existed at all.

P3163 Littlewood had recently moved into rooms on D Staircase of Nevile’s Court. Pausing in the arched doorway at the staircase’s base, he could sight through the portico to the arches across the court framed within it. It was an arresting view—perhaps one reason he remained there for sixty-five years, until his death in 1977. From Hardy’s rooms, it was just nineteen steps down the winding stone staircase, then forty paces through the gate into Nevile’s Court and around to D Staircase. Yet normally the two men communicated by mail or college messenger and did not, in any case, routinely run off to confer with one another in person. And so, that winter evening in 1913, to let Littlewood know he wished to meet with him after Hall, Hardy sent word by messenger.

笔记 - P3169 保持距离的君子之交。

P3227 in a classic Hardy flourish, he added: “They must be true because, if they were not true, no one would have the imagination to invent them.”

P3241 Indeed, as the mathematician Louis J. Mordell would later insist, “It is really an easy matter for anyone who has done brilliant mathematical work to bring himself to the attention of the mathematical world, no matter how obscure or unknown he is or how insignificant a position he occupies. All he need do is to send an account of his results to a leading authority,” as Jacobi had in writing Legendre on elliptic functions, or as Hermite had in writing Jacobi on number theory.

P3271 many who claim the mantle of “new and original” are indeed new, and original—but not better.

P3288 Snow wrote that Hardy preferred the downtrodden of all types “to the people whom he called the large bottomed: the description was more psychological than physiological. . . . [They] were the confident, booming, imperialist bourgeois English. The designation included most bishops, headmasters, judges, and . . . politicians.” When Hardy attended a cricket match and knew none of the competitors, he would tap his own favorites on the spot. These “had to be the under-privileged, young men from obscure schools, Indians, the unlucky and diffident. He wished for their success and, alternatively, for the downfall of their opposites.”

笔记 - P3293 ⟪论语·宪问篇第十四 14.10⟫ 子曰：”贫而无怨难, 富而无骄易。”

P3298 Each time Hardy had opened himself, he had come away enriched. Now, something wildly new and alien had presented itself to him in the form of a long, mathematics-dense letter from India. Once again, he opened his heart and mind to it. Once again, he would be the better for it.

2. “I Have Found in You a Friend ...”

P3305 It was not until a windy Saturday, February eighth, the day following his birthday, that Hardy sat down to deliver to Ramanujan the verdict on his gifts that Cambridge already knew. “Trinity College, Cambridge,” he wrote at the top, and the date, then began: “Dear Sir, I was exceedingly interested by your letter and by the theorems . . .” With an opening like that, Ramanujan would, at least, have to read on. But in the very next sentence, Hardy threw out his first caveat: “You will however understand that before I can judge properly of the value of what you have done, it is essential that I should see proofs of some of your assertions.”

P3310 Proof. It wasn’t the first time the word had come up in Ramanujan’s mathematical life. But it had never before borne such weight and eminence. Carr’s Synopsis, Ramanujan’s model for presenting mathematical results, had set out no proofs, at least none more involved than a word or two in outline. That had been enough for Carr, and enough for Ramanujan. Now, Hardy was saying, it was not enough. The mere assertion of a result, however true it might seem to be, did not suffice.

P3368 In fact, it was not just Walker, Spring, and others in the Madras mathematical community who had been fortified by Hardy’s letter. It was Ramanujan himself. For all his confidence in his mathematical prowess, Ramanujan needed outside approval, affirmation. Now he had it. Hardy’s letter took him seriously. And the pronouncement delivered by this unseen F.R.S., this man reputed to be the finest pure mathematician in England? It was no vague, empty one filled with glowing accolades that Ramanujan might, in an anxious moment, dismiss, but rather nine pages of specific, richly detailed comment: a statement Ramanujan had written on his sixth page of theorems about a series expressible in terms of pi and the Eulerian constant could be deduced from a theorem in Bromwich’s Infinite Series (the book M. J. M. Hill, in his letter of two months before, had advised that he consult). A theorem on the same page involving hyperbolic cosines Hardy himself had proved in Quarterly Journal of Mathematics. Hardy knew.

P3376 In assessing Ramanujan’s work, Hardy had informally broken it down into three broad categories—those results already known or easily derived from known theorems; those curious, and perhaps even difficult, but not terribly important; and those promising to be important indeed, if they could be proved. In Hardy’s mind, plainly, it was those in this third category that weighed most heavily.

3. “Does Ramanujan Know Polish?”

P3515 Bruce Berndt, an American Ramanujan scholar, would see in that curious split a message for mathematicians today: “We might allow our thoughts to occasionally escape from the chains of rigor,” he advised, “and, in their freedom, to discover new pathways through the forest.”

参阅《数学学习中的严谨与漏洞》。

4. A Dream at Namakkal

P3550 For an orthodox Hindu—and Ramanujan came from a very orthodox Hindu family—traveling to Europe or America represented a form of pollution. It was in the same category as publicly discarding the sacred thread, eating beef, or marrying a widow. And, traditionally, it had the same outcome—exclusion from caste. That meant your friends and relatives would not have you to their homes. You could find no bride or bridegroom for your child. Your married daughter couldn’t visit you without herself risking excommunication. Sometimes, you couldn’t go into temples. You couldn’t even get the help of a fellow casteman for the funeral of a family member. Here was the grim, day-to-day meaning of the word outcaste.

P3558 However much he may have yearned for England, Ramanujan found in Madras little reprieve from tradition’s hold. Madras was no oil-fed Houston, or railroad-driven Chicago, where ambitions surged and dreams were meant to be lived. Its population was static, barely changing in the past decade. It was flat, built low, spread out over the countryside; there were no Himalayan peaks to draw the imagination upward, no great towers to serve as symbols of human aspiration. More than the other great cities of India, Madras clung in spirit to the villages and towns of the surrounding countryside.

5. At the Dock

P3747 So was Narayana Iyer who had worked so closely with Ramanujan, the incessant clicking and scraping on their slates keeping people in his house up all night. “My father made a strange request to him,” his son N. Subbanarayanan would record many years later. “As a memento my father wanted to exchange his slate with Ramanujan’s slate, [a request that] was granted. Perhaps my father thought that he may get an inspiration from the slate during [Ramanujan’s] absence.”

Chapter Six: Ramanujan's Spring [1914 to 1916]

2. Together

P3884 A few of Ramanujan’s results were, Hardy could see, wrong. Some were not as profound as Ramanujan liked to think. Some were independent rediscoveries of what Western mathematicians had found fifty years before, or a hundred, or two. But many—perhaps a third, Hardy would reckon, perhaps two-thirds, later mathematicians would estimate—were breathtakingly new. Ramanujan’s fat, mathematics-rich letters, Hardy now saw, represented but the thinnest sampling, the barest tip of the iceberg, of what had accumulated over the past decade in his notebooks. There were thousands of theorems, corollaries, and examples. Maybe three thousand, maybe four. For page after page, they stretched on, rarely watered down by proof or explanation, almost aphoristic in their compression, all their mathematical truths boiled down to a line or two.

P3895 Around that time, Hardy was visited by the Hungarian mathematician George Polya, who borrowed from him his copy of Ramanujan’s notebooks, not yet then published. A few days later, Polya, in something like a panic, fairly threw them back at Hardy. No, he didn’t want them. Because, he said, once caught in the web of Ramanujan’s bewitching theorems, he would spend the rest of his life trying to prove them and never discover anything of his own.

P3910 After the second letter, Littlewood had written Hardy, “I can believe that he’s at least a Jacobi.” Hardy was to weigh in with a tribute more lavish yet. “It was his insight into algebraical formulae, transformation of infinite series, and so forth, that was most amazing,” he would write. In these areas, “I have never met his equal, and I can compare him only with Euler or Jacobi.”

P3920 Euler and Jacobi were not just generic “great mathematicians”; it was not capriciously that Hardy and Littlewood had compared Ramanujan to them. Rather, these two men represented a particular mathematical tradition of which Ramanujan, too, was part—that of “formalism.” Formal, here, carries no suggestion of “stiff” or “stodgy.” Euler, Jacobi, and Ramanujan had (along with deep insight) a knack for manipulating formulas, a delight in mathematical form for its own sake. A “formal result” suggests one fairly bubbling up from the formulas themselves, almost irrespective of what those formulas mean. Computers today manipulate three-dimensional contours regardless of whether they represent economic forecasts or car bumpers. Some painters care as much for form, line, or texture as they do subject matter. The mind of the mathematical formalist works along similar lines. All mathematicians, of course, manipulate formulas. But formalists were almost magicians at it, uncannily selecting just the tricks and techniques needed to obtain intriguing new results. They would replace one variable in an equation by another, thus reducing it to simpler form. They would know when to integrate a function, when to differentiate it, when to construct a new function, when to worry about rigor, when to ignore it.

笔记 - P3929 “when to worry about rigor, when to ignore it.” 这一点尤为重要！所以，对于自己的数学学习来说，在透彻理解的前提下可以考虑省掉一些非常基础的逻辑推演。这些对于缺乏训练的初学者是必须的，自己目前则没有必要过于拘泥。

P3932 Useful formulas tended to be found early in the development of a new mathematical field, pointing the way to future work. But as a field matured and these early formulas were applied and extended, they often grew too complicated to be useful. By Ramanujan’s time, something like this had happened in branch after branch of mathematics.

P3946 For Hardy’s part, confronting the mystery of Ramanujan’s mind would constitute, as his friend Snow had it, “the most singular experience of his life: what did modern mathematics look like to someone who had the deepest insight, but who had literally never heard of most of it?” Ramanujan “combined a power of generalization, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without a rival in his day,” Hardy would conclude. As for his ultimate influence, Hardy couldn’t, at the time he wrote, say; in a sense, its very peculiarity undercut it. “It would be greater,” he suggested, “if it were less strange.” But, he added, “One gift it has which no one can deny—profound and invincible originality.”

P3953 Having gone so out on a limb to bring Ramanujan to Cambridge, Hardy, after familiarizing himself with the notebooks, probably felt a little relieved, too. And proud. “Ramanujan was,” he would write, “my discovery. I did not invent him—like other great men, he invented himself—but I was the first really competent person who had the chance to see some of his work, and I can still remember with satisfaction that I could recognize at once what a treasure I had found.”

P3975 (Two Canadian mathematicians, the brothers Jonathan M. Borwein and Peter B. Borwein, have noted that thirty-nine decimal places will fix the circumference of a circle around the known universe to within the radius of a hydrogen atom.)

P3984 finding new ways to express it can reveal hidden links between seemingly disparate mathematical realms.

P3988 Archimedes took another geometric approach and came up with a value of pi equal to between 310/70 and 310/71.

P4008 Some of his results, it turned out, had been anticipated by European mathematicians, like Kronecker, Hermite, and Weber. Still, Hardy would write later, Ramanujan’s paper was “of the greatest interest and contains a large number of new results.” If nothing else, it was astounding how rapidly some of his series converged to pi. Leibniz’s series, on page 209, is lovely—but almost worthless for getting pi; three-decimal-point accuracy demands no fewer than five hundred terms. Some of Ramanujan’s series, on the other hand, converged with astonishing rapidity. In one, the very first term gave pi to eight decimal places. Many years later, Ramanujan’s work would provide the basis for the fastest-known algorithm, or step-by-step method, for determining pi by computer.

3. The Flames of Louvain

P4073 “The depravation of Germany—its gospel of iniquity and selfishness—is appalling,” wrote Jackson. “For though I never thought the Prussians gentlemen, I had a profound respect for their industry and efficiency, and I attributed to them domestic virtues. As it is, their good qualities subserve what is evil.”

P4077 Feeling against Germany swelled. Even Ramanujan was caught up in it, writing his mother about the German advance across Belgium. “Germans set fire to many a city, slaughter and throw away all the people, the children, the women and the old.”

P4084 In late August, pursuing an explicit policy of brutalization against civilian populations, German troops began burning the medieval Belgian city of Louvain, on the road between Liege and Brussels. House by house and street by street they set Louvain to the torch, destroying its great library, with its quarter million books and medieval manuscripts, and killing many civilians. The burning of Louvain horrified the world, galvanized public opinion against Germany, and united France, Russia, and England more irrevocably yet. “The March of the Hun,” English newspapers declared. “Treason to Civilization.” It was an early turning point of the war, doing much to set its tone. Louvain came to symbolize the breakdown of civilization.

P4108 Ramanujan’s continued fraction comprised within a single expression all the correct answers. Mahalonobis was astounded. How, he asked Ramanujan, had he done it? “Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, Which continued fraction? And the answer came to my mind.”

4. The Zeroes of the Zeta Function

P4112 The answer came to my mind. That was the glory of Ramanujan—that so much came to him so readily, whether through the divine offices of the goddess Namagiri, as he sometimes said, or through what Westerners might ascribe, with equal imprecision, to “intuition.” And yet, it was the very power of his intuition that, in one sense, undermined his mathematical development. For it blinded him to intuition’s limits, gave him less reason to learn modern mathematical tools, shielded him from his own ignorance.

P4121 His ideas as to what constituted a mathematical proof were of the most shadowy description. All his results, new or old, right or wrong, had been arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account.

P4153 The logarithmic response is ubiquitous in nature, as, for example, in the realm of the human senses. Double the amount of light in a room and you scarcely notice; the eye can respond to both the glare of the noonday sun and to the flicker of a match a mile away because its response is logarithmic. If light intensity rises by a factor of a thousand, which can be written as 103, the response, in principle, may be only about three times as much instead of a thousand. Well, it was just such a slow logarithmic growth in this braking effect that mathematicians long thought they saw with prime numbers.

P4193 The renowned German mathematician David Hilbert once said that, were he awakened after having slept for a thousand years, his first question would be, Has the Riemann hypothesis been proved?

P4197 “There are regions of mathematics in which the precepts of modern rigour may be disregarded with comparative safety,” Hardy would write, “but the Analytic Theory of Numbers is not one of them.”

P4207 Still, Ramanujan was wrong, which now, in England, under Hardy’s tutelage, he came to understand. “His instincts misled him,” wrote Hardy. And that was the point. Ramanujan’s “instincts,” sure as they were, in some ways better than those of any other mathematician of his day, were not good enough.

P4210 A car mechanic reliant on mechanical instinct may “know” how an engine works yet be unable to set down the physical and chemical principles governing it. For a writer, it may be enough to “know” that one scene should precede another and not follow it, without being able to explain why. But mathematicians are not normally content to guess, or assume, or assert that something is true; they must prove it, or feel they have—or as Hardy would put it, “exhibit the conclusion as the climax of a conventional pattern of propositions, a sequence of propositions whose truth is admitted and which are arranged in accordance with rules.”

这便是数学与工程学科的不同之处。

P4219 Many other examples like this, where seemingly “obvious” patterns prove not to be patterns at all, appear all through number theory and elsewhere in mathematics.

P4225 A year before he heard from Ramanujan, Littlewood had proved that, if you went far enough, the prime number theorem was destined to sometimes predict less, not more, than the actual number of primes. Later, someone found the number below which this reversal was guaranteed to take place. And it was a number so big you had to laugh—a number more than the number of particles in the universe, more than the possible games of chess. It was, Hardy would say, “the largest number which has ever served any definite purpose in mathematics.” And it made for the ultimate illustration of how intuition could serve you badly, and so must always be subject to proof.

P4257 As a mathematician, you can slide into trouble in numerous ways. You can differentiate a function without realizing the function cannot be differentiated. Or you can write off later terms of a series on the assumption that they are of a lower “order” than earlier terms, when, in fact, they contribute substantially to the series sum. Or you can assume that an operation correct for a finite number of terms is correct for an infinite number. Or you can integrate a function between two points, yet fail to note where the integral may be undefined, and so carry through your proof such meaningless quantities as “infinity minus infinity.”

在自由奔放与谨小慎微之间拿捏分寸、保持平衡也是数学（至少在实践的意义上）能够作为一门艺术的特征之一。

P4267 Ramanujan’s intuition steered him clear of many obstacles of which his truncated education had failed to warn him. But not all. The problem was not only that he was sometimes wrong; it was that he lacked mathematical knowledge enough to tell when he was right and when he was wrong. He stated correct and incorrect theorems with the same aplomb, the same sweet, naive confidence. And when he did offer proofs, they scarcely warranted the name.

P4273 It was just Ramanujan’s luck, then, to be thrown in with Hardy, whose insistence on rigor had sent him off almost single-handedly to reform English mathematics and to write his classic text on pure mathematics; who had told Bertrand Russell two years before that he would be happy to prove, really prove, anything: “If I could prove by logic that you would die in five minutes, I should be sorry you were going to die, but my sorrow would be very much mitigated by pleasure in the proof.” Ramanujan, Intuition Incarnate, had run smack into Hardy, the Apostle of Proof.

P4298 In what, in some ways, was his greatest achievement, then, Hardy brought Ramanujan mathematically up to speed without muzzling his creativity or damping the fires of his enthusiasm. It would have been easy to sniff at his shortcomings and dutifully correct them, like a bad editor who crudely blue-pencils his way through a delicate manuscript. But he knew that Ramanujan’s mathematical insight was rarer by far than even the most formidable technical mastery. It was fine to know all the mathematical tools needed to prove a theorem—but you had to have a theorem to prove in the first place. That was easy to forget as you flipped through the Proceedings of the London Mathematical Society. There, as in any mathematics journal, the proof was made to seem the culmination of a hundred closely reasoned steps ranging over a dozen pages. There, mathematics could seem no more than a neat lockstep march to certainty, B following directly from A, C from B, . . . Z from Y. But no mathematician actually worked that way; logic like that reflected the demands of formal proof but hinted little at the insights leading to Z. Rather, as Hardy himself would write, “a mathematician usually discovers a theorem by an effort of intuition; the conclusion strikes him as plausible, and he sets to work to manufacture a proof.”

Ramanujan的天赋令人称奇赞叹，而Hardy对Ramanujan的发现与引导则是伟大和感人至深了。

P4308 The theorem itself was apt to emerge just as other creative products do—in a flash of insight, or through a succession of small insights, preceded by countless hours of slogging through the problem. You might, early on, try a few special cases to informally “prove” the result to your own satisfaction. Then later, you might go back and, with a full arsenal of mathematical weapons, supply the kind of fine-textured proof Hardy championed. But all that came later—after you had something to prove. Besides, it was mostly technical, like the laws of evidence; you could learn it. Rigor, Littlewood would observe, “is not of first-rate importance in analysis beyond the undergraduate state, and can be supplied, given a real idea, by any competent professional.” Given a real idea—that was the rare commodity.

P4314 “Mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof,” the German mathematician Felix Klein once noted. (Added Louis J. Mordell, an American mathematician who would ultimately succeed Hardy in his chair at Cambridge: “To very few other mathematicians are Klein’s remarks . . . so appropriate as to Ramanujan.”) A “real idea” wasn’t dished up, like a Tripos problem, by some anonymous mathematical Intelligence. It had to come from somewhere, had to be seen before it could be proved. But where did it come from? That was the mystery, the source of all the circular, empty, ultimately unsatisfying explanations that have always beset students of the creative process. Here, “talent” came in, and “genius,” and “art.” Certainly it couldn’t be taught. And certainly, when in hand, it had to be nurtured and protected.

5. S. Ramanujan, B.A.

P4394 “I think it may be necessary,” he wrote, “to stay here a few years more as there is no help nor references in Madras for my work.”

即便是有如此天赋的顶级数学家也不能闭门造车。

Chapter Seven: The English Chill [1916 to 1918]

1. High Table

P4537 The college fellows in their black academic robes solemnly trooped into the great candlelit Hall. A webbed understructure of wooden arches graced the high ceiling. Dryden and Tennyson, Newton, Thackeray, Francis Bacon, and other Trinity notables glowered down from their portraits on the walls. Silver adorned bare wood tables; the tablecloths of peacetime were gone now for fear the light they reflected might beckon German zeppelins. Once all the fellows were seated, the senior among them, sitting at one end of the long table, recited a Latin grace. And the meal began. This was High Table, so called because the tables at which the fellows ate were set on a platform built up about four inches from the floor. At Trinity, as elsewhere in Cambridge, it was the focal point of the college’s social life, and long-standing notions of conversational good form governed it. For one thing, you never got too serious. “If one has done a hard day’s thinking one does not want to work at conversation,” Littlewood would say. “Dinner conversation is in fact easy and relaxed. No subject is definitely barred, but we do not talk shop in mixed company, and, Heaven be praised, we abstain from the important and boring subject of politics.” It was a place not of great profundity but of wit, wine, and release—release from the high tension of translating ancient Greek, or writing about the fall of Constantinople, or proving a new theorem in the theory of numbers. Here, the Important receded into distant memory. Here, trifles had their day.

2. An Indian in England

P4607 One later visitor to England, the Indian Nirad Chaudhuri, wrote how “One evening, when dining at a club, I tried in my innocence to open a conversation across the table, and I admired the skill with which the intrusion was fended off without the slightest suggestion of discourtesy.” Fended off, though, it was.

P4617 Laurence Young, the son of two mathematical contemporaries of Hardy and Littlewood at Cambridge, told how when Littlewood, as an old man, visited him in Wisconsin he would row him out on to the lake to view the sunset. Littlewood never said a word, and Young, in time, surmised that he was bored. But when one day Young suggested that the water might be too rough for their excursion, “his face dropped . . . and I quickly looked again at the lake and pronounced it smooth. This is typical of Cambridge—what you admire, you merely do not speak of.” Littlewood himself would remark that “When you make your speech at a Trinity Fellowship Election, do not expect them to break into irrepressible applause; no one will blink an eyelid.” It was a wall erected around one’s feelings, a great silence of the emotions. In Cambridge, the emphasis was on ideas, events, things, work, games—anything, it seemed, but the deeply personal.

P4631 Even if he didn’t always let his mind wander back home, there were times when the small, familiar things of South Indian life insinuated their way into his awareness. The smells of his mother’s cooking on Sarangapani Sannidhi Street, or of burning cow dung in the streets of Madras. The bright colors of religious festivals parading down the streets of Kumbakonam, accompanied by the strumming and jingling of the musicians. The reds and oranges of sari-clad women along the banks of the Cauvery, white dhotis setting off the dark brown skin of laborers in the fields. The vibrant greens of the vegetables, the coconuts and bananas and mangos sold down by the market near the river. And always, the bright blue sky and high overhead sun.

3. “A Singularly Happy Collaboration”

P4677 Mathematician Norbert Wiener would one day note how, in one sense, number theory blurs the border between pure and applied mathematics. In search of concrete applications of pure math, one normally turns to physics, say, or thermodynamics, or chemistry. But the number theorist has a multitude of real-life problems before him always—in the number system itself, a bottomless reservoir of raw data. It is in number theory, wrote Wiener, where “concrete cases arise with the greatest frequency and where very precise problems which are easy to formulate may demand the mathematician’s greatest power and skill to resolve.”

P4774 we proceeded to test this hypothesis by means of the numerical data most kindly provided for us by Major MacMahon, we found a correspondence between the real and the approximate values of such astonishing accuracy as to lead us to hope for even more. Taking n = 100, we found that the first six terms of our formula gave 190568944.783 + 348.872 − 2.598 + .685 − .318 + .064 190569291.996 while p(100) = 190569292; so that the error after six terms is only .004. Similar precision applied to p(200). Their method was supplying an answer whose error was not just “bounded”—which could, after all, mean bounded but large—but small enough to round off to the nearest integer. “These results,” wrote Hardy and Ramanujan, “suggest very forcibly that it is possible to obtain a formula for p(n), which not only exhibits its order of magnitude and structure, but may be used to calculate its exact value for any value of n.”

P4803 Two decades later, sure enough, Hans Rademacher came up with the missing piece of the puzzle and made the formula exact.

P4813 Ramanujan and Hardy: as a mathematical team, they would remind Pennsylvania State University mathematician George Andrews of the story of the two men, one blind and the other lacking legs, who together could do what no normal man could. They were a formidable pair. On the strength of their work on partitions alone, which by itself justified Ramanujan’s trip to England, their names would be linked forever in the history of mathematics. Cambridge mathematician Béla Bollobás has observed that while Hardy furnished the technical skills needed to attack the problem, I believe Hardy was not the only mathematician who could have done it. Probably Mordell could have done it. Polya could have done it. I’m sure there are quite a few people who could have played Hardy’s role. But Ramanujan’s role in that particular partnership I don’t think could have been played at the time by anybody else. Whatever the proper assignment of credit, “We owe the theorem,” Littlewood would write, “to a singularly happy collaboration of two men, of quite unlike gifts, in which each contributed the best, most characteristic, and most fortunate work that was in him. Ramanujan’s genius did have this one opportunity worthy of it.”

4. Deepening the Hole

P4857 At one point, writing to Ramanujan, who was then in the hospital, about their current work, his eagerness for the mathematical fray plainly warred with his concern for his friend’s health: “If I get out any more I will write to you again. I wish you were better and back here—there would be some splendid problems to work at. I don’t know if you feel well enough to think about such difficult things yet.” Then, a postscript: “At present you must do what the doctors say. However you might be able to think about these things a little: they are very exciting.” As maddeningly equivocal as the letter was, Ramanujan would have been obtuse indeed to miss its message: The work awaits you. And so, if any part of Ramanujan wanted to relax, pursue his pet philosophical notions, investigate the psychic theories of the English physicist Oliver Lodge that so intrigued him, take the train into London to visit the zoo, as he did once or twice, or otherwise stray from mathematics, it got scant support from Hardy. Hardy’s urgings, to be sure, found fertile soil in Ramanujan’s own obsessive bent.

P4873 By early 1917, he was a man on a mission, propelled toward his destiny, oblivious to all but mathematics. After three years in Cambridge, his life was Hardy, the four walls of his room, and work. For thirty hours at a stretch he’d sometimes work, then sleep for twenty. Regularity, balance, and rest disappeared from his life. He was not the first man to sacrifice his health on the altar of mathematics. Jacobi gave this retort to a friend who worried that excessive devotion to his work might make him sick: “Certainly I have sometimes endangered my health by overwork, but what of it? Only cabbages have no nerves, no worries. And what do they get out of their perfect wellbeing?” And before Jacobi, there was Newton himself. “Never careful of his bodily health,” E. T. Bell wrote of him, “Newton seems to have forgotten that he had a body which required food and sleep when he gave himself up to the composition of his masterpiece. Meals were ignored or forgotten.”

P4897 Mens sana in corpore sano,—a sound mind in a sound body,—is

P4911 Whatever the scientific validity to such a view, it was true that national identities were less blurred in those days, making for more to adapt to; India was more inviolably India, England more England. There were not, as there are today, Indian restaurants and food stores littering the streets of London and Cambridge. As for the climate, there was no air-conditioning or central heating to moderate its extremes. You either endured it or got out. That was why well-heeled Englishmen journeyed to Italy or Spain during the winter; or why, in India during the worst of the summer heat, they visited hill stations like Simla in the North, or Ootacamund in the South. Those who could not escape, meanwhile, faced the full force of the heat, eased only by feeble fans or, midst the damp and chill of English winters, were left to the mercies of ineffectual coal fires.

P4924 Ramanujan did indeed keep to the diet of his “cult,” and from even his first days in England, before the war, doing so had posed problems for him. In a letter to a friend soon after his arrival, he complained about “the difficulty of getting proper food. Had it not been for the good milk and fruits here I would have suffered more. Now I have determined to cook one or two things myself and have written to my native place to send some necessary things for it.” In other letters, he asked that certain provisions be sent him or no longer sent him, gave accounts of shipments gone astray or arriving in poor condition, expressed thanks for those that did arrive intact. Monthly, Narayana Iyer sent him powdered rice in tin-lined boxes. Others sent him spices or pickled fruits and vegetables. Back in India, Janaki later recalled, Ramanujan would sometimes abruptly stop eating, or else hurry through it, to pursue a mathematical thought; meals were something to be dispensed with. But now, with no one to cook for him, they had become an awful bother. Preparing from scratch South Indian meals—assembling the ingredients, soaking and grinding lentils, cutting vegetables, boiling rice, and so on, all the way through the final cooking—took time. And time, while working on what to him were the most seductively challenging problems in the world, he resented having to give up. So, as Neville’s wife, Alice, for one, would report, Ramanujan sometimes cooked only once a day, or sometimes only once every other day, and then at weird hours in the early morning. The simplest “solution,” of course, was a species of asceticism. Back in Madras, he had enjoyed mango, or banana, or jackfruit with his rice and yogurt. And there was brinjal, prepared that special way his mother did it, and . . . But the great days of good South Indian meals were gone. He had written Subramanian in 1915, “I am not in need of anything as I have gained a perfect control over my taste and can live on mere rice with a little salt and lemon juice for an indefinite time.”

7. Trouble Back Home

P5215 Seven decades later, Janaki, eighty-eight years old, was a stooped old woman, living in Triplicane, Madras, with her forty-five-year-old adopted son, Narayanan, his wife and three children. Their modest house stood behind a low wall with an iron fence, a few feet back from a busy street accented by coconut palms heavy with fruit. Inside, wrapped in a burgundy sari, Janaki sat on a bare wood bench against a wall, where she had only to look up to see a bronze bust of her late husband, garlanded with flowers, the gift of his admirers from around the world. Her skin was glossy, stretched over bones barren of fat. Hunched and frail, she got around the house only by painfully pushing a wooden chair, which functioned as a walker. Nearly deaf, she could hear Narayanan only when he shouted through a rolled-up magazine into her ear. As she replied, in loud staccato bursts, to questions asked her, her face would sometimes grow contorted with the effort of simply listening and speaking. At other times, it would break out into a broad, captivating smile. According to some who knew her, Janaki was more confident and assertive now than in years past. And yet, such was a daughter-in-law’s place as she had been brought up to accept it that even now, nearing ninety and known to be bitter about her treatment at Komalatammal’s hands, she took pains to show respect for her long-dead mother-in-law. Through Narayanan, she expressed gratitude to her for the opportunity to marry Ramanujan. And she asked that certain difficulties in their relationship be couched in properly respectful circumlocutions; that it be said, for example, that they were simply “not able to see eye to eye”; and that she fled the household at one point merely because she “wanted to change the atmosphere.”

8. The Nelson Monument

P5308 If you stand in the middle of London’s Trafalgar Square, which commemorates Admiral Nelson’s defeat of the French in 1805, at the base of the Nelson Column you’ll see, up close, sculpted depictions of his various naval campaigns. But as you lift your gaze to the top of the great 167-foot-tall fluted column, Nelson himself is just a cloaked figure in a three-cornered admiral’s cap, too high to make out even the bare outline of his features. One day, offered the hypothetical choice between being commemorated by such a statue, glorious but distant, and a lower, more approachable one, Hardy would choose the former. He preferred the safety of barriers, privacy, distance.

9. Ramanujan, Mathematics, and God

P5348 Something of this same enigmatic flavor makes its way into Littlewood’s account of Ramanujan’s work with partitions. Attempting to trace the progress of Ramanujan’s thinking, he ultimately throws up his hands, frustrated and perplexed: There is, indeed, a touch of real mystery [here]. If only we knew[the result in advance], we might be forced, by slow stages, to the correct form of Ψq. But why was Ramanujan so certain there was one? Theoretical insight, to be the explanation, had to be of an order hardly to be credited. Yet it is hard to see what numerical instances could have been available to suggest so strong a result. And unless the form of Ψq was known already, no numerical evidence could suggest anything of the kind—there seems no escape, at least, from the conclusion that the discovery of the correct form was a single stroke of insight.

P5418 Despite his emphasis on rigor, G. H. Hardy was not blind to the virtues of vague, intuitive mental processes in mathematics. Bromwich, he would write, for example, “would have had a happier life, and been a greater mathematician, if his mind had worked with less precision. As it was, even the best of his work is a little wanting in imagination. For mastery of technique in a wide variety of subjects, it would be difficult to find his superior, but he lacked the power of ‘thinking vaguely.’ ” And some such extraordinarily developed ability to “think vaguely” may have been among Ramanujan’s special gifts.

P5450 Hardy agreed with Hadamard that unconscious activity often plays a decisive part in discovery; that periods of ineffective effort are often followed, after intervals of rest or distraction, by moments of sudden illumination; that these flashes of inspiration are explicable only as the result of activities of which the agent has been unaware—the evidence for all this seems overwhelming.

P5495 Hardy, though, did not admit to such ambivalence. For him, the whole spiritual realm was just so much bunkum. He knew—this was his faith—that wherever Ramanujan’s genius came from, there was something straightforward to explain it. He would write: I have often been asked whether Ramanujan had any special secret; whether his methods differed in kind from those of other mathematicians; whether there was anything really abnormal in his mode of thought. I cannot answer these questions with any confidence or conviction; but I do not believe it. My belief is that all mathematicians think, at bottom, in the same kind of way, and that Ramanujan was no exception. Ramanujan’s mathematics, he was saying, was the product of the reasoned working of a reasoning mind, and nothing more needed to be said.

笔记 - P5502 这才是客观理性的评价。

10. Singularities at X = 1

P5554 Undeterred by the Trinity rebuff and hoping to boost Ramanujan’s morale, Hardy set about trying to get his friend the recognition he felt he deserved. On December 6, 1917, Ramanujan was elected to the London Mathematical Society. Then, two weeks later, on December 18, Hardy and eleven other mathematicians—Hobson and Baker were among them, as were Bromwich, Littlewood, Forsyth, and Alfred North Whitehead, Bertrand Russell’s collaborator on Principia Mathematica—together put him up for an honor more esteemed by far than any fellowship of a Cambridge college: they signed the Certificate of a Candidate for Election that nominated him to become a Fellow of the Royal Society. The Royal Society was Britain’s preeminent scientific body, going back to 1660 when Christopher Wren and Robert Boyles helped found it. There were, at about the time Hardy put up Ramanujan, 39 foreign members, including the Russian Ivan Pavlov, the American Albert Michelson (of Michelson-Morley experiment fame), and 6 other Nobel Prize winners. The Royal Society counted in all 464 members in physics, chemistry, biology, mathematics, and every branch of science. Being an F.R.S. meant that forevermore those three little letters would be appended to your name, appear on your own scientific papers, and on letters addressed to you. It was the ultimate mark of scientific distinction. Younger scientists lusted after it, older scientists lamented their lack of it.

P5580 There was no question in Hardy’s mind, or Littlewood’s, or anyone else’s, that Ramanujan merited the honor. Still, few candidates made it the first time out, and by normal practice his nomination was premature. Hardy had been thirty-three years old when elected in 1910. Littlewood himself had made it only the previous February, also at age thirty-three—more than a decade, and dozens of notable mathematical papers, beyond his early glory as a Senior Wrangler. Ramanujan, still twenty-nine at the time of his nomination, had contributed to European mathematics for just a few years and still had a modest publication record, at least in number. But Hardy’s concern for Ramanujan’s health moved him to press his claim with unusual urgency. J. J. Thomson, discoverer of the electron, winner of the Nobel Prize in 1906, and then president of the Royal Society, had asked him to outline the circumstances surrounding Ramanujan’s candidature. “If he had not been ill I would have deferred putting him up a year or so,” Hardy admitted: “not that there is any question of the strength of his claim, but merely to let things take their ordinary course. As it is, I felt no time must be lost.”

笔记 - P5589 A friend indeed.

P5624 Just what triggered Ramanujan’s desperate bound onto the tracks doesn’t come down to us. Certainly, though, if the mere refusal of his dinner guests to accept a third helping could foment such a storm of shame in him that he had to get up and leave, deeper humiliation might spark action more precipitous still. And in 1917 he had certainly experienced his share of it. Rejected by Trinity. Seemingly abandoned by his wife. Left sick and dependent in the sanatorium, helpless even to command the food he wanted. Unable to produce the work he felt his friends expected of him. Confronted by the knowledge that much of his past work had been rediscovery and, viewed blackly enough, a waste of time.

11. Slipped from Memory

P5833 Rogers was, in spirit, a gifted amateur who, despite his abilities, never pursued his mathematical career with the single-minded devotion so necessary, then as now, to establish a big name: “He did things, and did them well, because he liked doing them, but he had nothing of the professional outlook, and his knowledge of other people’s work in mathematics was vague. He had very little ambition or desire for recognition.”

Chapter Eight: “In Somewhat Indifferent Health” [From 1918 on]

1. “All the World Seemed Young Again”

P5874 “He has apparently been approached (with a view to return) directly by several friends,” Hardy wrote. “It is possible, I think, that the suggestion has not been made in the most tactful way possible; at any rate, it seems to have turned him rather against the idea of going”; something they said had pushed one of Ramanujan’s numerous buttons. Mindful of Ramanujan’s sensitivities, and of no mind to trespass on them, Hardy advised Dewsbury that “the suggestion would best be made more or less officially and by letter simultaneously to him and to me.” Offer Ramanujan a university position that left him free to do research and occasionally visit England and he, Hardy, would favor his return—and Ramanujan, he felt sure, would, too.

笔记 - P5879 Tactics is necessary.

2. Return to the Cauvery

P6036 For three months Ramanujan stayed at the bungalow on Luz Church Road. And here, he and Janaki began to forge something like a real relationship. Janaki had been just thirteen when he left. Now she was eighteen. As they never really had before, they began to talk, perhaps now each coming to discover how Komalatammal had intercepted their letters, perhaps finding time for physical intimacy as well.

P6104 “I have a friend who loves me more than all of you, who does not want to leave me at all,” Ramanujan told Sarangapani later. “It’s this tuberculosis fever.”

3. The Final Problem

P6137 As with so many other subjects that interested Ramanujan, theta functions could be represented as infinite series. Strict rules governed the formation of these series, and so long as you observed them, they always had particularly intriguing properties. For example, they were “quasi doubly periodic” and they were “entire functions.” Over the years since Jacobi had first studied them, theta functions had come to exert a profound impact on fields ranging from mathematical physics to number theory. Books devoted to them typically wound up bridging vast terrain, yielding connections to fields to which at first they might seem wholly unrelated.

P6235 During his last months, Ramanujan drew closer to Janaki, with whom, when he wasn’t exploding at her in a fit, he now had a warmer, more relaxed relationship. “He was uniformly kind to me,” Janaki recalled. “In his conversation he was full of wit and humor,” was forever cracking jokes. As if trying to cheer her up, he plied her with tales of England—of his visits to the British Museum, and the animals he’d seen there, of the time in Cambridge when an English guest eating a South Indian dish he had prepared chomped down on a piece of hot pepper . . . His life came down a little from the heights of mathematics to small things, human things. He would summon Janaki with a little bell, or would tap with a stick. He told C. S. Rama Rao Sahib, Ramachandra Rao’s son-in-law, how much he craved the rasam he had enjoyed almost a decade before in those destitute days of 1910 at Victoria Hostel—and loved it when somebody brought him some. Still, it was not an easy death. Whatever cheerfulness or equanimity he could muster—and some of his friends preferred later to remember him that way—papered over a grim despair. He was sullen and angry much of the time. His mood was volatile, resting on a hair trigger. Janaki felt later—so had Hardy in England, and so did many of his friends in India—that his illness had affected his mind. He was forever raving at one thing or another. There was a tray kept nearby with lentils, spices, and rice, and once, in a fit of anger and pain, Ramanujan pounded it all together with a stick. Narasimha Iyengar, hurt at finding Ramanujan so sullen and cold when he greeted him at the Central Station, visited him now again on Harrington Road. “I found to my great grief that though physically living, he was mentally dead to the world, even to his once dear friends.” Toward the end, “he was only skin and bones,” Janaki remembered later. He complained terribly of the pain. It was in his stomach, in his leg. When it got bad, Janaki would heat water in brass vessels and apply hot wet towels to his legs and chest; “fomentation,” it was called, standard therapy at the time. But through all the pain and fever, through the endless household squabbles, through his own disturbed equanimity, Ramanujan, lying in bed, his head propped up on pillows, kept working. When he requested it, Janaki would give him his slate; later, she’d gather up the accumulated sheets of mathematics-covered paper to which he had transferred his results and place them in the big leather box which he had brought from England. “He wouldn’t talk to anyone who came to the house,” said Janaki later. “It was always maths. . . . Four days before he died he was scribbling.” Early on April 26, 1920, he lapsed into unconsciousness. For two hours, Janaki sat with him, feeding him sips of dilute milk. Around midmorning or perhaps a little earlier, he died. With him were his wife, his parents, his two brothers, and a few friends. He was thirty-two years old. At the funeral later that day, most of his orthodox Brahmin relatives stayed away; Ramanujan had crossed the waters and, too sick on his return to make the trip to Rameswaram for the purification ceremonies his mother had planned, was still tainted in their eyes. Ramachandra Rao arranged the cremation, through his son-in-law and Ramanujan’s boyhood friend, Rajagopalachari. At about one in the afternoon, his emaciated body was put to the flames on the cremation ground near Chetput. The next day, assigning it Registration No. 228, a government clerk officially recorded his death.

4. A Son of India

P6301 I have no uncles or cousins to protect me. We have no property, as you might have known very well. I have a great desire to study and I wish “to neglect worldly ends, all dedicated to closeness and the bettering of my mind” (Tempest). I have no taste, sorry to say, for Mathematics. I like to read Shakespeare, Wordsworth, Tennyson, and wish to travel in the fairy land “half flying half on foot.” I do not know how to feed them. Therefore, I humbly request you to write to the Madras University to give a monthly allowance to us. I have been told that my brother is entitled to get a sum from the Cambridge University. In short, I am a young man. One day passes with great difficulty. I do not know how to protect them. Unless any arrangement is made to support us, we have to go a-begging from door to door. I entrust the whole case into your hands. It is your bounden duty to protect us.

P6320 she asked Hardy to intercede on her sons’ behalf with the government and with the India Office in London. She wanted for them high positions in the post office department—the better-educated Tirunarayanan as probationary superintendent of post offices, and Lakshmi Narasimhan as inspector of post offices in Madras.

P6334 She had learned to embroider and work a sewing machine in Bombay, and so now she scratched out a living making clothes and teaching tailoring to girls. In 1937, S. Chandrasekhar, the astrophysicist, asked by Hardy to try to find a good photograph of Ramanujan next time he was in India, tracked her down in Triplicane. She was, Chandrasekhar reported back, “having a rather difficult life, some of her unscrupulous relatives having swindled her out of such financial resources as R[amanujan] had left her,” and unable even to get hold of a copy of her husband’s Collected Papers, published a few years before.

P6475 For some years, his work went into eclipse, as new areas of mathematics, wholly distant from those Ramanujan had pursued, became fashionable. The Collected Papers, while it had its disciples, was no best-seller, even by scholarly standards. Just 42 books were sold the first year, 209 the second, and an officer of Cambridge University Press predicted in a letter to Hardy late in 1929 that it would be another ten years before the first printing, of 750, was sold out. “When I came to the United States [after World War II],” Freeman Dyson would recall, “I was all by myself as a devotee.” To the mathematical avant-garde of the day, Ramanujan’s work was no more than a “backwater,” a vestige of the nineteenth century. But that, in time, would change.

笔记 - P6480 Recognition and approbation of contemporaries are not mandatory.

6. Better Blast Furnaces?

P6601 A pure mathematician must leave to happier colleagues the great task of alleviating the sufferings of humanity.

P6643 Computers, scarcely the dream of which existed in 1920, have also drawn from Ramanujan’s work. “The rise of computer algebra makes it interesting to study somebody who seems like he had a computer algebra package in his head,” George Andrews once told an interviewer, referring to software that permits ready algebraic manipulation. Sometimes in studying Ramanujan’s work, he said at another time, “I have wondered how much Ramanujan could have done if he had had MACSYMA or SCRATCHPAD or some other symbolic algebra package. More often I get the feeling that he was such a brilliant, clever and intuitive computer himself that he really didn’t need them.” Then, too, a modular equation in Ramanujan’s notebooks led to computer algorithms for evaluating pi that are the fastest in use today.

P6677 “What we do may be small, but it has a certain character of permanence,” wrote Hardy of the work of pure mathematicians, “and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.”

Epilogue

P6992 Later that year, the Royal Society notified him that he was to receive its highest honor, the Copley Medal. “Now I know that I must be pretty near the end,” he told Snow. “When people hurry up to give you honorific things there is exactly one conclusion to be drawn.”

P6999 The 1939 heart attack began the long physical and emotional slide that led to his suicide attempt. And it was in its wake, about a month after France fell to the Nazis, that he put the finishing touches to A Mathematician’s Apology, his paean to mathematics. Snow saw the Apology “as a book of haunting sadness,” the work of a man long past his creative prime—and knowing it. “It is a melancholy experience for a professional mathematician to find himself writing about mathematics,” wrote Hardy. Painters despised art critics? Well, the same went for any creative worker, a mathematician included. But writing about mathematics, rather than doing it, was all that was left him.

P7006 I still say to myself when I am depressed, and find myself forced to listen to pompous and tiresome people, “Well, I have done one thing you could never have done, and that is to have collaborated with both Littlewood and Ramanujan on something like equal terms.” Ramanujan. All these twenty years later, Ramanujan remained part of him, a bright beacon, luminous in his memory. “Hardy,” said Mary Cartwright, his student during the 1920s and whom Hardy would describe as the best woman mathematician in England, “practically never spoke of things about which he felt strongly.” Yet at one remove from his listener, on the printed page, he became a little freer. And there he revealed Ramanujan’s hold on him: “I owe more to him,” he wrote, “than to any one else in the world with one exception [Littlewood?] and my association with him is the one romantic incident in my life.”

P7035 From then on, over the next thirty-five years, Hardy did all he could to champion Ramanujan and advance his mathematical legacy. He encouraged Ramanujan. He acknowledged his genius. He brought him to England. He trained him in modern analysis. And, during Ramanujan’s life and afterward, he placed his formidable literary skills at his service. “Hardy wrote exquisite English,” the Manchester Guardian would say of him, citing especially his obituary notice of Ramanujan as “among the most remarkable in the literature about mathematics.” To mathematician W. N. Bailey, it was “one of the most fascinating obituary notices that I have ever read.” And it was Hardy’s book on Ramanujan, more than anything he knew about him otherwise, that convinced Ashis Nandy to make Ramanujan a prime subject of his own book. Hardy’s pen fired the imagination, shaping Ramanujan’s reception by the mathematical world.

P7048 Someone once said of Hardy that “conceivably he could have been an advertising genius or a public relations officer.” Here was the evidence for it.