The ultimate barrier to their relationship, of course, was, as Hardy would write, that “Ramanujan was an Indian, and I suppose that it is always a little difficult for an Englishman and an Indian to understand one another properly.” It was Kipling’s verse all over again: “East is East and West is West and never the twain shall meet.” For the English, India was impenetrable, scarcely possible to understand. But ensuring that the cultural gap blocked a closer friendship between them was that Hardy scarcely tried to bridge it.
Hardy, C. P. Snow once said of him, “would have been the first to disclaim that he possessed deep insight into any particular human being.” The untidy contours of human personality were not his home turf. Yes, he might ask Ramanujan about his knowledge of art or philosophy—the European kinds of things that one might brilliantly discuss with advantage at High Table. And he knew something of Ramanujan’s tastes in literature and politics. But he talked to him scarcely at all about his family, or South India, or the caste system, or the Hindu gods. He didn’t probe, he didn’t pry. “I rely, for the facts of Ramanujan’s life, on Seshu Iyer and Ramachandra Rao,” Hardy began an account of him later—not facts he’d gleaned from Ramanujan himself.
Even so safely neutral a matter as Ramanujan’s mathematical influences in India never profited from Hardy’s questioning. “Here I must admit that I am to blame,” Hardy would write, “since there is a good deal which we should like to know now and which I could have discovered quite easily. I saw Ramanujan almost every day, and could have cleared up most of the obscurity by a little cross-examination.” But he never did, never stepped past the mathematics of the moment, “hardly asked him a single question of this kind… .
I am sorry about this now, but it does not really matter very much, and it was entirely natural. [Ramanujan] was a mathematician anxious to get on with the job. And after all I too was a mathematician, and a mathematician meeting Ramanujan had more interesting things to think about than historical research. It seemed ridiculous to worry about how he had found this or that known theorem, when he was showing me half a dozen new ones almost every day.
Mathematics, then, was the common ground of their relationship—perhaps the only one other than their mutual pleasure in having found one another. Like many an Englishman, Hardy hid behind his reserve, disdaining any too-presumptuous an intrusion into Ramanujan’s private life. He was not ideally suited to draw out a lonely Indian, to ease his adjustment to an alien culture, to shelter him from the English chill.
9. RAMANUJAN, MATHEMATICS, AND GOD
Exemplifying the distance between the two men was Hardy’s refusal to view Ramanujan, in matters of religious belief, as any different from the usual run of atheists, agnostics, and skeptics he knew among Cambridge intellectuals; or indeed, to see his mind as flavored by the East at all.
• • •
In the 1930s, E. T. Bell would remark that Ramanujan had broken the rules by which mathematicians evaluate their own. “When a truly great [algorist, or formalist] like the Hindu Ramanujan arrives unexpectedly out of nowhere, even expert analysts hail him as a gift from Heaven,” he wrote, crediting him with “all but supernatural insight” into hidden connections between seemingly unrelated formulas.
Supernatural insight.
A gift from Heaven.
It is uncanny how often otherwise dogged rationalists have, over the years, turned to the language of the shaman and the priest to convey something of Ramanujan’s gifts. Hardy was the first Western mathematician to thoroughly examine Ramanujan’s notebooks, but over the next seventy-five years many others would, too. And repeatedly they have been reduced to inchoate expressions of wonder and awe in the face of his powers, have stumbled about, groping for words, in trying to convey the mystery of Ramanujan.
“We have no idea how he did the marvelous things he did, what led him to them, or anything else,” says mathematician Richard Askey, a Ramanujan scholar at the University of Wisconsin in Madison. Says Bruce Berndt, after years of working through Ramanujan’s notebooks: “I still don’t understand it all. I may be able to prove it, but I don’t know where it comes from and where it fits into the rest of mathematics.” He adds at another point, “The enigma of Ramanujan’s creative process is still covered by a curtain that has barely been drawn.”
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.
Ramanujan, in the language of the Polish emigré mathematician Mark Kac, was a “magician,” rather than an “ordinary genius.”
An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark.
Mystery, magic, and dark, hidden workings inaccessible to ordinary thought; it is these that Ramanujan’s work invariably conjures up, a sense of reason butting hard up against its limits.
But at reason’s limits does something else take over? Do we here flirt with spiritual or supernatural forces outside our understanding?
It’s an unlikely, anachronistic, even heretical notion today, with science and Western rationalism everywhere triumphant. But there is scant reason to doubt that Ramanujan himself thought so. South Indians of otherwise presumably rationalist bent—accountants, lawyers, mathematicians, professors—recall him as wholly at ease in the spiritual world to which his mother and grandmother introduced him as a child, and ready to see in it a source of mathematical inspiration.
T. K. Rajagopolan, a former accountant general of Madras, would tell of Ramanujan’s insistence that after seeing in dreams the drops of blood that, according to tradition, heralded the presence of the god Narasimha, the male consort of the goddess Namagiri, “scrolls containing the most complicated mathematics used to unfold before his eyes.”
R. Radhakrishna Iyer, a classmate at Pachaiyappa’s College, recalled one day asking Ramanujan about his research only to have him reply, in Radhakrishna’s words, “that Lord Narasimha had appeared to him in a dream and told him that the time had not come for making public the fruits of his research.”
And it was on a day about a year before he left for England, in 1912 or early 1913, that Ramanujan, showing his work to mathematics professor R. Srinivasan, made the statement in which he pictured equations as products of the mind of God.
Ramanujan’s friends invariably noted his interest in astrology and his penchant for interpreting dreams. His boyhood friend Anantharaman would tell how once, when his brother had a dream, Ramanujan read it as foretelling a death in the street behind their house. (There was.)
K. Gopalachary, a friend of Ramanujan from Madras days, said that Ramanujan even attributed his early interest in mathematics to a dream—a dream about, of all things, a street peddler hawking pills.
Ramanujan’s belief in hidden forces and the powers of the supernatural was never, at least back in India, something about which he felt the need to apologize or keep quiet. It was no mere ma
tter of private conviction to him, consigned to the periphery of his life, or some pet theory about which he merely liked to speculate. Time and again, he acted on it.
Thus, in the hectic year before he left for England, he found time to prepare astrological projections, fixing auspicious times for religious functions for relatives and friends. He was convinced, from studying the lines on his palm, that he would die before he was thirty-five, and told his friends as much. Anantharaman would record that he attributed to a temple near Trichinopoly the power to cure mental ailments and advised sufferers to go there. And while still at Pachaiyappa’s College, Ramanujan dreamed of a family whose child was near death, went to its parents, and in obedience to the dream, asked them to move the child to another house. “The death of a person can occur only in a certain space-time-junction point,” he said.
One evening before he left for England, Ramanujan returned to his house in Triplicane in an electric streetcar (which had been introduced to Madras in 1895, years before anywhere else in India). The driver, enjoying himself, was alternating sudden stops with sharp accelerations, jerking his passengers around unmercifully. Said Ramanujan, sitting with a friend on the long bench behind the driver: “That man imagines he has the power to go slow or fast at his pleasure. He forgets that he gets the power through the current that flows in the overhead wires… . That,” he said, invoking the Hindu term for the illusion, or vanity, thought to deflect humans from God, “is the way maya works in this world.”
Ramanujan, then, was steeped in the belief system of his culture. So many stories, from so many quarters, taking so many forms, over so many years, add up to nothing like the skeptical rationalist Hardy imagined, nothing like a man mechanically adhering to the quaint orthodoxies of his family and his caste.
Yet that is indeed what Hardy insisted. “I do not believe in the immemorial wisdom of the East,” he would one day declare in a lecture devoted to Ramanujan,
and the picture which I want to present to you is that of a man who had his peculiarities like other distinguished men, but a man in whose society one could take pleasure, with whom one could drink tea and discuss politics or mathematics; the picture in short, not of a wonder from the East, or an inspired idiot, or a psychological freak, but of a rational human being who happened to be a great mathematician.
Ramanujan’s religion, Hardy insisted, “was a matter of observance and not of intellectual conviction, and I remember well him telling me (much to my surprise) that all religions seemed to him more or less equally true.”
He was sure, Hardy wrote on another occasion, that “Ramanujan was no mystic and that religion, except in a strictly material sense, played no important part in his life.”
For years, C. P. Snow would keep a large photo of Hardy in his otherwise almost bare study and otherwise respected him in every way. But from Hardy’s insistence that Ramanujan “did not believe much in theological doctrine, except for a vague pantheistic benevolence, any more than Hardy did himself,” Snow took care to distance himself. “In this respect,” he wrote, “I should not trust his insight far.”
Still, taking their cue from Hardy, most Western observers, and some Indians, have wholly detached Ramanujan’s mystical streak from his mysterious ability to forge new mathematical linkages. Hailing the one, they’ve dismissed the other, and written off his credulousness—his weakness for astrology, his arrant superstitions, his devotion to Namagiri—as an unfortunate eccentricity peripheral to his mathematical inventiveness but which has somehow to be stomached for the sake of it.
For Ramanujan, though, the split was not so sharp. The man whom Hardy met in England in April 1914 was a man still of South India, who had grown up on the Indian gods and the relaxed fluidity of Hindu belief. In him, the natural and the supernatural, Jacobi and Namagiri, Number and God, found a common home, stood in something like an easy intimacy.
Did Ramanujan’s religious belief bestow on him his mathematical gift? Certainly not, since otherwise all those with kindred beliefs would share it. Nor did he necessarily gain mathematical insights through anything like the means he thought he did and to which he assigned credit. Nor, to state the obvious, does the fact that Ramanujan believed what he believed mean that what he believed is so. On the other hand, in how the mystical streak in him sat side by side, apparently at perfect ease, with raw mathematical ability may testify to a peculiar flexibility of mind, a special receptivity to loose conceptual linkages and tenuous associations.
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.
Ramanujan’s belief in the Hindu gods, it stands repeating, did not explain his mathematical genius. But his openness to supernatural influences hinted at a mind endowed with slippery, flexible, and elastic notions of cause and effect that left him receptive to what those equipped with more purely logical gifts could not see; that found union in what others saw as unrelated; that embraced before prematurely dismissing. His was a mind, perhaps, whose critical faculty was weak compared to its creative and synthetical.
It is the critical faculty, of course, that keeps most people safe—keeps them from rashly embracing foolishness and falsehood. In Ramanujan, it had never developed quite as fully as the creative—thus giving him the credulousness, the appealing innocence, upon which all who knew him unfailingly remarked. Without that protective screen, as it were, he risked falling prey to the silly and the false—as many, over the years, would view his belief in palmistry, astrology, and all the rest of the esoterica to which he subscribed.
And yet, without that screen did he thus remain more open to the mathematical Light?
• • •
Hardy was not, to say the least, at home in this mental universe.
“I have always thought of a mathematician as in the first instance an observer” he said in a Cambridge lecture in 1928, “a man who gazes at a distant range of mountains and notes down his observations.”
His object is simply to distinguish clearly and notify to others as many different peaks as he can. There are some peaks which he can distinguish easily, while others are less clear. He sees A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B. B is now fixed in his vision, and from this point he can proceed to further discoveries. In other cases perhaps he can distinguish a ridge which vanishes in the distances, and conjectures that it leads to a peak in the clouds or below the horizon.
But about the veiled process by which one might come to discern those peaks in the first place, Hardy remained largely silent. Indeed, rarely in a long life of doing mathematics and writing about it did he choose to discuss the creative process behind it, not even in his Mathematician’s Apology, otherwise so rich with his insights into the mathematician’s world. Always it was the product of that process, the theorem itself, that interested him. He might want to establish, through proof, whether or not it was true. Or, perhaps, to evaluate its place in the mathematical firmament; in the Apology, for example, he speaks of theorems almost as an art critic might the works in a gallery show, evaluating them by this or that yardstick of mathematical beauty.
Almost the only time he did write about mathematical creativity came many years later, near the end of his life, in a review of a book by Jacques Hadamard, The Psychology of Invention in the Mathematical Field. What philosophers or poets could say about the creative process in mathematics, Hardy felt, was next to nothing. But Hadamard w
as a mathematician, and a great one. He had run up the highest score ever recorded on the entrance examination to the Ecole Polytechnique, France’s premier school of science. He had, with the Belgian Charles J. de la Vallée-Poussin, proved the prime number theorem. What he had to say about “invention in the mathematical field” was worth listening to.
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.
But beyond this, Hardy seemed to say in every word of his review, he thought it best not to venture. Too soon are we thrust upon the Unknowable; it was better to meekly sidestep the issue than mire our explanations in foolishness. Indeed, he lauded Hadamard for being “wisely diffident and tentative in his conclusions.”
How unconscious activities “are related to those of a more normal [sic] kind, to fully conscious work or reflection on the fringe of consciousness, how they function and what is the proper language in which to describe them, are terribly difficult questions,” Hardy wrote. When Hadamard did offer a tentative explanation for what Hardy called “the most puzzling question” of all—how you seize one from among the welter of ideas your unconscious serves up—Hardy made explicit how he felt: “It may be so,” he wrote, “though I cannot say that I find it very convincing; but I am no psychologist, and my distaste for all forms of mysticism may be prejudicing me unduly.”
The Man Who Knew Infinity Page 36
