The Man Who Knew Infinity

by Robert Kanigel

  Not much had changed by the time Hardy reached Cambridge a generation later. Among the twenty or so colleges, two—Girton and Newnham—had been established for women in the previous two decades. But though women, with the lecturer’s consent and chaperoned by a woman don, could attend university lectures, by 1913 they still kept mostly to themselves and played little part in undergraduate life. Until 1882, college fellows couldn’t marry, but even after that most fellows remained bachelors. In 1887, a proposal was made to offer degrees to women; it was soundly defeated. Ten years later, on a May day in 1897, a straw-hatted mob thronged outside the Senate House, where the matter was again being taken up, demonstrating against the measure. A woman was hanged in effigy. A large banner advised (after Act II, Scene I of Much Ado About Nothing) “Get you to Girton, Beatrice. Get you to Newnham. Here’s no place for you maids.”

  It was an almost laughably artificial environment, with dons left woefully ignorant of domestic life. One time at St. John’s College, the story goes, an elderly bachelor at High Table congratulated someone on the birth of his son. “How old is the little man?” he asked.

  “Six weeks,” came the reply.

  “Ah,” said the bachelor don, “just beginning to string little sentences together, I suppose.”

  About the only time Hardy and other fellows encountered women was among the bedmakers who tidied up college rooms—and they were said to be selected for their plainness, age, and safely married status, presumably so as to minimize the distraction they represented to students and fellows of the colleges.

  Within so exclusively male a setting, steeped in a sterner sense of public morality, and free of today’s pervasive sexual drumbeat, even passionate and devoted friendships took physical form less frequently. David Newsome, writing of late Victorian Cambridge, alludes to

  romantic friendship within exclusively male communities, a phenomenon so normal and respected throughout the period … that the greatest care must be taken to avoid slick and dismissive judgements… . In the nineteenth century the normality of both men and women forming highly emotional relationships with those of their own sex, of the same age or sometimes older or younger … was neither questioned as necessarily unwholesome, nor felt to inhibit the same relationship with the opposite sex leading to perfectly happy marriage.

  One physician in the late 1890s wrote Ellis and Symonds of several such cases of passionate, yet presumably nonphysical relationships:

  In all these, I imagine, the physical impulse of sex is less imperative than in the average man. The emotional impulse, on the other hand, is very strong. It has given birth to friendships of which I find no adequate description anywhere but in the dialogues of Plato; and beyond a certain feeling of strangeness at the gradual discovery of a temperament apparently different to that of most men, it has provoked no kind of self-reproach or shame. On the contrary, the feeling has been rather one of elation in the consciousness of a capacity of affection which appears to be finer and more spiritual than that which commonly subsists between persons of different sexes… . In all these cases, a physical sexual attraction is recognized as the basis of the relation, but as a matter of feeling, and partly also of theory, the ascetic ideal is adopted.

  It was just such kinds of relationships, remote to American life today, that C. P. Snow, who knew Hardy as well as anyone, imputes to him. Hardy, he wrote, did not normally form close, demonstrative bonds among even those he called his friends.

  But he had, scattered through his life, two or three other relationships, different in kind. These were intense affections, absorbing, nonphysical but exalted. The one I knew about was for a young man whose nature was as spiritually delicate as his own. I believe, though I only picked this up from chance remarks, that the same was true of the others. To many people of my generation, such relationships would seem either unsatisfactory or impossible. They were neither the one nor the other; and unless one takes them for granted, one doesn’t begin to understand the temperament of men like Hardy … nor the Cambridge society of his time.

  Despite suggestive evidence, then, one cannot conclude that Hardy was a practicing homosexual. And yet, in one sense, it doesn’t matter. Either he led an almost wholly asexual life, scarcely knowing what he was as a sexual being, and submerging any sexual desires behind a screen of Victorian propriety, or he led a secret sexual life so elaborately and successfully hidden that even friends knew nothing of it and those who did kept quiet. In either case, he would have required a vast architecture of personal defenseworks to pull it off, heroic acts of will performed day in and day out over the years. And though made somewhat easier and more ordinary by the times in which he lived, it would, in the end, have had to exact its toll.

  And it did. There was a hauntedness to Hardy that you could see in his eyes. “I suspect,” remembered an Oxford economist, Lionel Charles Robbins, who knew him later, that “Hardy found many forms of contact with life very painful and that, from a very early stage, he had taken extensive measures to guard himself against them. Certainly in his friendlier moments—and he could be very friendly indeed—one was conscious of immense reserves.” Always, he kept the world at bay. The obsession with cricket, the bright conversation, the studied eccentricity, the fierce devotion to mathematics—all of these made for a beguiling public persona; but none encouraged real closeness. He was a friend of many in Cambridge, an intimate of few.

  In the years after 1913, Hardy would befriend a poor Indian clerk. Their friendship, too, would never ripen into intimacy.

  6. THE HARDY SCHOOL

  In 1900, Hardy became a Fellow of Trinity College. In 1901, he won one of two Smith’s Prizes, named after a former master of Trinity College, and since 1769 the blue ribbon of Cambridge mathematics.

  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.)

  In 1906, be became a Trinity lecturer. He gave six hours a week of lectures, usually in two courses, elementary analysis and the theory of functions. He occasionally gave informal classes during this period, but he was never actually a college tutor. He was there to do research.

  Hardy would later say he blossomed slowly. In a sense, that was true; most of his more important mathematical contributions lay in the future. But already in the first decade of the twentieth century he was batting out papers at a prodigious clip, ten or a dozen a year, most of them on integrals and series. Like “Research in the Theory of Divergent Series and Divergent Integrals,” which appeared in Quarterly Journal of Mathematics in 1904; and “On the Zeros of Certain Classes of Integral Taylor Series,” in the Proceedings of the London Mathematical Society in 1905. 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.

  In later years, Hardy himself would set little stock in the work he did during this period. “I wrote a great deal during the next ten years, but very little of importance,” he would say of the period before he met Littlewood and Ramanujan; “there are not more than four or five papers which I can still remember with some satisfaction.” Still, by 1907, they added up to a corpus substantial enough that, on October 31 of that year, he was put up for membership in the Royal Society.

  Many of the top names in the Cambridge mathematical establishment went to bat for him. A. E. H. Love, who had introduced Hardy to Camille Jordan when he was still an undergraduate, did. So did E. W. Hobson, one of those who would, in a few years, hear from Ramanujan. So did T.J. I’A. Bromwich, whose book on infinite series Ramanujan had been urged to consult just before he wrote Hardy. Like most who would place the coveted F.R.S. after their names, Hardy didn’t get it first time out. But in 1910, he was elected, at the age of thirty-three. A London photographer took his picture for the occasion and did a little retouching around the eyes and mouth. But it was probably a r
eflex action; Hardy still looked boyish.

  Many years later, Hardy would insist that none of the mathematics he had done during his career was ever in the least “useful.” But around now came one exception. During the previous century an Austrian monk named Gregor Mendel had done experiments in crossing tall pea plants with dwarfs, found that each generation of the progeny bore fixed, predictable proportions of dwarf and tall, and so laid the basis for the science of genetics. At the time, of course, nobody cared, and Mendel’s experiments lay forgotten. But the publication of his work in 1900, sixteen years after his death, sparked a flurry of interest in his insights, and the next few years saw them the subject of much active debate.

  One controversy surrounded the fate of recessive and dominant traits in succeeding generations. A recessive trait was normally “silent”; it needed to be represented in both parents to show up in the children. A dominant trait, on the other hand, needed only one copy of the gene. An article in the Proceedings of the Royal Society of Medicine argued that Mendelian genetics predicted that a dominant trait, like the stunted finger growth known as brachydactylism, would tend to proliferate in the population. That this ran flatly counter to the evidence, the author asserted, undermined Mendel.

  But no, showed Hardy, in a brief letter to the American scholarly journal Science in 1908, a dominant trait would not proliferate in the population. Assign symbols to the probabilities of a particular gene type, work through the simple algebra, and you’d find that the proportion of each gene would tend to stay fixed generation after generation. In other words, if matings took place randomly—that is, not skewed one way or the other by Darwinian natural selection—dominant traits would not take over and recessive traits would not die out. A German physician, Wilhelm Weinberg, showed something similar the same year, and the principle became known as the Hardy-Weinberg Law. It exerted a marked impact on population genetics. It was applied to the study of the genetic transmission of blood groups and rare diseases. It appears today in any scientific dictionary or genetics textbook.

  Hardy himself, of course, set no great store by it. As for the mathematics, it was trivial. Besides, it was not, well, useless. And in Hardy’s mind that made it verge on the execrable.

  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.”

  Hardy went on to pity the mathematical physicist who might use mathematical tools to understand the workings of the universe: was not his lot in life a little pathetic?

  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.

  Not so in real mathematics, Hardy continued, which “must be justified as art if it can be justified at all.” It was in this light that the Hardy-Weinberg Law, hopelessly useful as it was, and barren of anything in the least mathematically beautiful, violated his principles.

  While these aesthetic principles, as it were, may not yet have been so fully formed or so well articulated at the time Hardy heard from Ramanujan in early 1913, they had certainly by then jelled—and had begun to find an outlet in the school of English mathematics forming around him. It was a school with scant interest in anything so dull and pragmatic as, say, genetics; or even, for that matter, with the mathematical physics that had for years been the special English strength. It was a school, rather, that embraced the Continental purity of Camille Jordan.

  • • •

  Alone in their island kingdom, cut off from the Continent, the British were an insular people, suspicious and intolerant of all things foreign. At the end of the nineteenth century and beginning of the twentieth, however, they were probably even more so, basking in the glow of empire, fat and happy. “It must have seemed like a long garden party on a golden afternoon—to those who were inside the garden,” Samuel Hynes has written, in The Edwardian Turn of Mind, of British attitudes during this period. “But a great deal that was important was going on outside the garden: it was there that the twentieth-century world was being made”—in mathematics, he might have added, as elsewhere.

  Since the seventeenth century, Britain had stood, mathematically, with its back toward Europe, scarcely deigning to glance over its shoulder at it. Back then, Isaac Newton and the German mathematician Gottfried Wilhelm von Leibniz had each, more or less independently, discovered calculus. Controversy over who deserved the credit erupted even while both men lived, then mushroomed after their deaths, with mathematicians in England and on the Continent each championing their compatriots. 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.

  In calculus as in mathematics generally, the effects were felt all through the eighteenth and nineteenth centuries and on into the twentieth. 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 infinit
esimal 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.

  Calculus rests on a strategy of dividing quantities into smaller and smaller pieces that are said to “approach,” yet never quite reach, zero. Taking a “limit,” the process is called, and it’s fundamental to an understanding of calculus—but also, typically, alien and slippery territory to students raised on the firm ground of algebra and geometry. And yet, 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.

  In textbooks even today you can see vestiges of the split—which neatly parallels that between Britain and the Continent in the nineteenth century: the author briefly introduces the limit, assumes a hazy intuitive understanding, then spends six chapters charging ahead with standard differentiation techniques, maxima-minima problems, and all the other mainstays of Calc 101 … until finally, come chapter 7 or so, he steps back and reintroduces the elusive concept, this time covering mine-strewn terrain previously sidestepped, tackling conceptual difficulties—and stretching the student’s mind beyond anything he’s used to.

  Well, the first six chapters of this generic calculus text, it could be said, were English mathematics without the Continental influence. Chapter 7 was the new rigor supplied by French, German, and Swiss mathematicians. “Analysis” was the generic name for this precise, fine-grained approach. It was a world of Greek letters, of epsilons and deltas representing infinitesimally small quantities that nonetheless the mathematicians found a way to work with. It was a world in which mathematics, logic, and Talmudic hairsplitting merged.