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June 17 - June 26, 2016
If he wants to be useful, he must work in a humdrum way, and he cannot give full play to his fancy even when he wishes to rise to the heights. “Imaginary” universes are so much more beautiful than this stupidly constructed “real” one; and most of the finest products of an applied mathematician’s fancy must be rejected, as soon as they have been created, for the brutal but sufficient reason that they do not fit the facts.
Samuel Hynes has written, in The Edwardian Turn of Mind,
“the greatest disaster that ever befell not merely Cambridge mathematics in particular but British mathematical science as a whole.”
As a spokesman for the new rigor, Hardy exerted his impact not alone by what he had to say, but through the force, grace, and elegance with which he said it, both in print and in person. In lectures, his enthusiasm and delight in the subject fairly spilled over. “One felt,” wrote one of his later students, E. C. Titchmarsh, “that nothing else in the world but the proof of these theorems really mattered.”
“In all my years of listening to lectures in mathematics,” he would write, “I have never heard the equal of Hardy for clarity, for interest, or for intellectual power.” Around this time, a pupil of E. W. Barnes, director of mathematical studies at Trinity, sought Barnes’s advice about what lectures to attend. Go to Hardy’s, he recommended. The pupil hesitated. “Well,” replied Barnes, “you need not go to Hardy’s lectures if you don’t want, but you will regret it—as indeed,” recalled the pupil many years later, “I have.” Others who missed his lectures may not, in retrospect, have felt such
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“pictures on the board in the lecture, devices to stimulate the imagination of pupils.”
“shown in this book and elsewhere a power of being interesting, which is to my mind unequalled.”
When a review of Bertrand Russell’s Principles of Mathematics ran in the Times Literary Supplement in 1903, it was Hardy who wrote it.
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.
“pretended to prove the prophetic wisdom of the Great Pyramid, the revelations of the Elders of Zion,
“wild theorems. Theorems such as he had never seen before, nor imagined.”
A Mathematician’s Miscellany, Littlewood
In the words of two of his biographers, he was “a rough-hewn earthy person with a charm of his own.” He was strong, virile, vigorous. He had been a crack gymnast at school, had played cricket, would become an accomplished rock climber and skier and, even into his eighties, could be seen hiking through the East Anglian countryside around Cambridge.
“the man most likely to storm and smash a really deep and formidable problem: there is no one else who can command such a combination of insight, technique, and power.”
Imagine cutting a hot dog into disclike slices. You could wind up with ten sections half an inch thick or a thousand paper-thin slices. But however thin you sliced it, you could, presumably, reassemble the pieces back into a hot dog. Integral calculus, as this branch of mathematics is called, adopts the strategy of taking an infinite number of infinitesimally thin slices and generating mathematical expressions for putting them back together again—for making them whole, or “integral.” This powerful additive process can be used to determine the drag force buffeting a wing as it slices through
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And then, 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.”
“a mathematician of the highest quality, a man of altogether exceptional originality and power.”
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.”
Hardy, on the other hand, was a generation younger and had a penchant for the unorthodox and the unexpected.
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.
“I have found a friend in you who views my labours sympathetically.”
Got by treading on the groove! Ramanujan was flying high. Four years of hawking his mathematical wares had left him neither shy about going after what he needed nor above stooping to self-pity: “I am already a half-starving man,” he wrote Hardy now. “To preserve my brains I want food and this is now my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship either from the University or from Government.”
Hardy, we may be sure, was not put off by any of this. “Good work,” he once wrote, “is not done by ‘humble’ men.”
As in his letters to Hardy earlier that year, Ramanujan was attacking definite integrals that resisted every effort to reduce them to simpler, more useful forms, defeated the whole arsenal of mathematical tools brought to bear on them. Ramanujan was fashioning new tools.
Like a screwdriver, saw, or lathe, a mathematical “tool” is supposed to do something; those used to evaluate a definite integral perform mathematical operations on it that, one hopes, get it ready for the next tool—the next theorem or technique—and ultimately lead to a solution. But just as a screwdriver tightens screws but can’t saw wood, a mathematical tool may work for evaluating one integral but not others. If you don’t know in advance, you try it. If it doesn’t work, you try something else.
“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.”
“I should expect [his work] to be good, but he is not a man like Littlewood of whom one would say it must be good.”
Was he, Berry inquired, following the lecture? Ramanujan nodded. Did he care to add anything? At that, Ramanujan stood, went to the blackboard, took the chalk, and wrote down results Berry had not yet proved and which, Berry concluded later, he could not have known before.
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.
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.
In 1977, the American mathematician Bruce Berndt took up where Watson and Wilson left off. After thirteen years of work, having published three volumes devoted solely to the notebooks, he is still at it today, the task unfinished.
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.”
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.
Ramanujan’s mathematics, if it fit any category, fit this one. And yet, Hardy could see that if Ramanujan possessed conjurer’s tricks, they were ones of almost Mephistophelean potency. Ramanujan was a formalist who undermined the stereotypes. “It is possible that the great days of formulae are finished and that Ramanujan ought to have been born 100 years ago,” Hardy would write. But, he acknowledged, “He was by far the greatest formalist of his time,” one whose mathematical sleight of hand no one could match, and whose theorems, however he got them, later generations of mathematicians would
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But, he added, “One gift it has which no one can deny—profound and invincible originality.”
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.”
which in a single, strange, beautiful statement of mathematical truth ties trigonometry and geometry to natural logarithms and thence to the whole world of “imaginary” numbers. Pi even shows up in the mathematics of probability. Drop a needle onto a table finely scored by parallel lines each separated by the length of the needle and the chance of its intersecting a line is 2/pi. Again and again pi pops up. So finding new ways to express it can reveal hidden links between seemingly disparate mathematical realms.
“The limitations of his knowledge were as startling as its profundity,” Hardy would write.
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.
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.
“a man who will eat a hot fruit pie is unfit for decent society.”
Stubborn, self-driven, self-willed, he was every bit the product of his country and its customs. Unlike some Madrasi intellectuals, he’d never lived a Westernized life in India;
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
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“A character so remarkably free from the petty meannesses of human life . . . the most generous of men.” That’s what C. P. Snow once said of Hardy.
Hardy was an aristocrat of the intellect, raised to value high achievement and dismiss anything less. As a mathematician, he was all Ramanujan could want. But he was also a formidable and distant figure who demanded always, and only, the best from him.
J. C. Burkill, a Cambridge undergraduate beginning in 1919 who later himself became a prominent mathematician, remembers feeling always intimidated by Hardy—always “below him,” as he puts it. Whereas Littlewood, say, came across as thoroughly human and accessible, chatting away amiably in Hall, Hardy was busy being brilliant. “When he was conversing,” says Burkill, “he felt he had to be on a high plane.” The great Hungarian mathematician George Polya told how Hardy once expressed disapproval of Polya’s failure to pursue a promising mathematical idea. The two of them, with a third
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Mens sana in corpore sano,—a sound mind in a sound body,—is a maxim of universal and practical application.
If there is one lesson which our students in India must learn from English students, it is this—to pay as great an attention to their bodily as to their mental development.
But so, too, almost certainly, did what today might be called “lifestyle” factors. While consensus on their influence eludes the research community, the evidence gives strong credence to Dr. Wingfield’s impression that “overwork, overplay, overworry, undernourishment [and] lack of necessary sunshine and fresh air” help transform otherwise failed bacterial attacks into successful ones.
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.
