The Man Who Knew Infinity

by Robert Kanigel

  As usual, it was Euler who made the first real dent in the problem. Working from an area of mathematics which later became known as elliptic modular functions, Euler detoured around the problem to construct what is known as a generating function. In theory, a generating function supplies not just a particular answer to a particular problem but all the answers to all related problems; it generates answers as fast as the numbers can be plugged in. With it, you step off into new mathematical terrain where the object of attention seems no longer to be p(n), but some new function, f(x). Except that with f(x) in hand you can go back and get what you really want, p(n). In this case, Euler’s generating function gave rise to a “power series,” which is just a series of terms each of successively higher powers and each multiplied by something, some coefficient:

  f(x) = 1 + ∑ p(n)xn

  which means:

  f(x) = 1 + p(1) x1 + p(2) x2 + p(3) x3 + …

  It was those coefficients, p(1) and so on, that potentially supplied the answers. Because if you could actually evaluate the power series, the coefficients would turn out to be not any old numbers, but just the p(n) desired. In Euler’s scheme, if you wanted, say, p(50)—which is 204,226—you’d just head out to the fiftieth term of the series, where you’d note the coefficient of x50. So if you had done it right, that part of the series would read:

  … + 204,226 x50 + …

  Now, this was not just black magic. It was natural that the generating function for partitions be a power series, because powers combine by addition. Thus 4 × 8 = 32 can be written as:

  22 × 23 = 25

  Their exponents add up. And adding things up is just what you do in partitions. (“A moment’s consideration shows that every partition of n contributes just 1 to the coefficient of xn,” Hardy would observe in tracing the algebraic links between power series and partitions.) Squint, then, and even without following the logic in detail, you can roughly discern the route between the two seemingly unconnected realms.

  This, at any rate, was how Euler set up the problem. But Euler didn’t solve it, didn’t say how to use his generating function to actually churn out the desired power series. He offered a strategy for arriving at p(n), but little more. As Hardy and Ramanujan would write in their big paper in the Proceedings of the London Mathematical Society, “we have been unable to discover in the literature of the subject any allusion whatever to the question of the order of magnitude of p(n).” No one, in other words, had a clue.

  The seeds for the attack Ramanujan and Hardy were to take came in Ramanujan’s first letter to Hardy in January 1913. There, on page 7, he gave a particular theta series—a power series of a certain kind, containing only terms with squared powers, like x1, x4, and x9—and claimed that to determine its coefficients one had but to evaluate the mathematical expression he furnished and select the integer closest to it.

  This was not quite right. “The function,” Hardy would write, “is a genuine approximation to the coefficient, though not at all so close as Ramanujan imagined.” And yet, he went on, “Ramanujan’s false statement was one of the most fruitful he ever made, since it ended by leading us to all our joint work on partitions”; indeed, Ramanujan’s function from 1913 bore a strong family resemblance to the problem’s ultimate solution, which came only in 1937.

  In the course of their work, Ramanujan and Hardy were led to what would come to be known as the circle method. The circle method made use of Cauchy’s theorem, which might seem at first as inappropriate for the job as anything you could imagine. Cauchy’s theorem lay in the domain of “analysis,” the broad area of mathematics that includes calculus and deals with “continuous” rather than “discrete” quantities; how long you’ve been pregnant is a continuous quantity, how many children you have a discrete one. Partitions were a discrete quantity; you couldn’t have 6.719 partitions just as you couldn’t have 6.719 children. It had to be 6, exactly, or 7.

  But Cauchy’s theorem had a history of being applied to such problems by now, and Hardy could be thanked for it; he was foremost among twentieth-century mathematicians in pioneering the “analytic theory of numbers,” which took the powerful tools developed over more than two centuries for continuous quantities and, through feats of mathematical legerdemain, applied them to the integers of number theory. In an account of his work with Ramanujan, Hardy noted that “the idea [of using Cauchy’s theorem had become] an extremely obvious one: it is the idea which has dominated nine-tenths of modern research in the analytic theory of numbers; and it may seem strange that it should never have been applied to this particular problem before.” It hadn’t, he went on, in part because of “the extreme complexity of the behavior of the generating function f(x) near a point of the unit circle.”

  Hardy referred to the fact that the integral at the heart of Cauchy’s theorem couldn’t be evaluated as it stood, because the “contour” over which it was to be integrated held impermissible points, where its value wasn’t defined. So they would have to find an alternative path close to the forbidden “unit circle,” systematically dissect it, make approximations as they went along. This was the new strategy.

  Basing a result on a series of approximations seemed to guarantee that it would itself be an approximation. But should they expect anything better? Hadn’t prime numbers resisted all efforts at exact calculation, forcing mathematicians to be content with no more than a rough estimation that grew relatively more accurate as n increased? Hadn’t all Ramanujan’s attempts to make the prime number theorem exact run afoul of mathematical pitfalls? Surely partitions would prove similarly intractable, making them happy with almost any decent approximation.

  To see how far off they were—to check the worth of their approximations by how close they came to the precise result—they put Major MacMahon to work.

  The son of a brigadier general, Percy Alexander MacMahon was a sixty-one-year-old mathematician who had served in the Royal Artillery—including a stint in Madras back in the 1870s—before seriously taking up mathematics. “With his moustache, his ‘British Empah’ demeanor and worst of all his military background,” it was once written of him, “MacMahon was hardly the type to be chosen by Central Casting for the role of the Great Mathematician.” But after leaving the army, he had gone on to become a professor at Woolwich, the army school, and since 1904 had been associated with Cambridge’s St. John’s College. His expertise lay in combinatorics, a sort of glorified dice-throwing, and in it he had made contributions original enough to be named a Fellow of the Royal Society in 1890. MacMahon was a whirlwind of a calculator. Sometimes, in fact, he would take on Ramanujan in friendly bouts of mental calculation—and regularly thrash him.

  But now, MacMahon was putting his calculating skills to good use and, using a simple formula that led directly from Euler’s earliest work in partitions, had arduously hand-calculated, in impossible, endless streams of numbers, the values for the first two hundred p(n). By the roughest of analogies, he was adding up 29 + 29 + 29 + … 1,000,001 times to get 29,000,029, while Hardy and Ramanujan sought a way to simply multiply 29 by 1,000,001. But they didn’t have it yet, and MacMahon’s tedious efforts supplied a benchmark against which to test it when they did.

  When they tried it early in their work, with their approximation strategy still relatively primitive, they were already encouragingly close; for p(50) and p(80), the error stood at just 5 percent. Further refinements, with correspondingly smaller errors, reinforced their conviction that they were on the right track.

  Then, in December 1916, the two men, perhaps having gone their separate ways until the start of Lent term in January, came a big step forward. Ramanujan wrote Hardy a postcard, squeezing into the tight space one more contribution to their continuing mathematical dialogue:

  It therefore appears that in order that p(n) may be the nearest integer to the approximate sum, S need not be taken beyond β log n and cannot be taken below α log n I hope you can easily prove these. Then the problem is completely solved. Major MacMahon was ki
nd enough to send me a typewritten copy of the 200 numbers. The approximation gives the exact number. I think you knew this already from him.

  Ramanujan alluded to a way of making the number of terms of the series they used to approximate p(n) itself depend on n. This, as Littlewood wrote later, was “a very great step, and involved new and deep function-theory methods that Ramanujan obviously could not have discovered by himself.” What it seemed to mean, as Hardy and Ramanujan would write, was that “we may reasonably hope, at any rate, to find a formula in which the error is of order less than that of any exponential of the type ean; of the order of a power of n, for example, or even bounded.”

  A bounded error, an error they could fix within set limits: they would have been happy with that. But their results were better still: When

  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.”

  This was the shocker. Nothing in the history of this problem, nothing in the work with primes, had prepared them for anything like it. What they had done with partitions was just what Ramanujan had thought he’d done with primes but, Hardy showed him, hadn’t. What was more, the uncanny accuracy of their results attested to the power of the approximating technique they had used to get them. The circle method it would be called, from how it let you draw oh-so-near, but never actually touch, the forbidden circular path. So subtle and inspired were the approximations it permitted that it went beyond approximation to promise exactitude. Two decades later, sure enough, Hans Rademacher came up with the missing piece of the puzzle and made the formula exact.

  Did Ramanujan suspect there was an exact solution all along? That’s what the eminent Norwegian number theorist Atle Selberg has suggested. Selberg, in fact, argues that Hardy’s insistence on certain methods of classical analysis actually impeded their efforts; and that lacking faith in Ramanujan’s intuition he discouraged a search for the kind of exact solution Rademacher produced twenty years later.

  In any case, their partitions solution was big news, the circle method they’d used to come up with it a stunning success. In late 1916, Hardy dashed off an early account, under his own name but offered “as the joint work of the distinguished Indian mathematician, Mr. S. Ramanujan, and myself,” to the Quatrième Congrès des Mathématiciens Scandinaves in Stockholm. Early the following year, a brief joint paper appeared in Comptes Rendus, as “Une Formule Asymptotique Pour le Nombre Des Partitions De n.” And a one-paragraph reference to the French journal article appeared in Proceedings of the London Mathematical Society in March. The forty-page paper setting out their work in full detail didn’t appear until 1918.

  • • •

  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.”

  For Ramanujan, it was all deliciously addictive. A decade before, his discovery of Carr had left him so singlemindedly devoted to mathematics that he no longer could function as an ordinary college student. Now, something like that was at work again, only worse. For now it was not only his own delight in mathematics that spurred him on, but the encouragement he got from Hardy. Hardy, this embodiment of all that was highest and best in the Western mathematical tradition, with his immense technical prowess and rich knowledge of the whole mathematical world of England and the Continent, was all Ramanujan could want in a colleague and mentor. And that he saw such breathtaking originality in him could do nothing to restrain Ramanujan’s eagerness to get on with their work.

  “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. Another time, he called him “freer from the emotion [of envy] than any man I have ever known.” Indeed, though Hardy deemed Ramanujan’s natural mathematical ability superior to his own, no hint survives of so much as a wisp of envy tainting his relationship with him. Despite any private twinges, he seems always to have been Ramanujan’s unalloyed friend and supporter. All his life he championed him, hailed his gifts. Recognizing Ramanujan’s genius, he wanted only to push it toward its limits.

  And that, if anything, was just the problem.

  Hardy was in many ways the best and truest friend Ramanujan ever had. He was considerate, loyal, and kind to him. And yet in at least one way, unintentionally, he probably did Ramanujan harm. For in the ardent hopes he had for him, in his unbridled wish to see that he lived up to his potential—in keeping him to the mark, driving him—Hardy only fed Ramanujan’s addiction.

  He couldn’t keep Ramanujan from deepening the hole he was digging for himself. He may even have helped him dig it deeper.

  4. DEEPENING THE HOLE

  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. “His sense of excellence was absolute; anything less was not worthwhile,” remembered an Oxford economist, Lionel Charles Robbins, who knew Hardy later. “What Dr. Johnson said of Burke applied exactly to Hardy: his presence called forth all one’s powers.”

  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 mathematician, were visiting a zoo. At one point, Polya recalled, a caged bear “sniffed at the lock, hit it with his paw, then he growled a little, turned around and walked away. ‘He is like Polya,’ remarked Hardy. ‘He has excellent ideas, but does not carry them out.’ ”

  You didn’t turn to G. H. Hardy for an unjudging sho
ulder to cry on. Once, when the mathematician Louis J. Mordell wrote him, complaining that his papers were getting picked apart by the editors of a mathematical journal because of minor stylistic faults, Hardy refused to indulge him. “I know I have spent over three hours over the journal proofs of a note of yours,” he wrote back, “and have made over thirty corrections on a page. All ‘trivialities’—so trivial that you have never noticed them, or at any rate commented on them: but a morning’s work gone west.” The remainder of Hardy’s eleven-page letter showed similar irritation. Don’t be so easy on yourself, it said.

  Hardy, in short, was a stern taskmaster. His was a personality of expectations, of high performance. From him, Ramanujan could get encouragement and, in those ways in which Hardy could express it, friendship—but little in the way of pure, uncritical nurturing.

  Hardy’s reply to Ramanujan’s first letter from India showed that same holding-to-account—Prove it—and he never let up once Ramanujan was in England. 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.