The Man Who Knew Infinity

by Robert Kanigel

  “Rooms” at Cambridge did not mean the bare furnished room conjured up by Americans today in “room for rent.” In fact, they were often small apartments of their own, some of them well furnished and ornate, with fine wallpaper, rugs, framed paintings, sometimes even formal dining-room tables. 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. But perks at Cambridge did not extend to the modern plumbing and conveniences that ordinary middle-class Americans were enjoying by this time. The few baths were clustered together, often across a windswept court from the student’s room. Hot and cold running water were unknown. The gyp’s first job, upon showing up at seven in the morning, was to fill a tin saucer with cold water for washing.

  Trinity was among the oldest of the Cambridge colleges; it was certainly the largest, and the most famous, formed from two smaller colleges each of which went back to the fourteenth century. It had taken its present form under Henry VIII, whose plump, strutting sculpted figure faced Hardy each time he walked past the bicycles massed outside the gate to Great Court.

  Dominated by a picturesque domed fountain that went back to 1602, Great Court was the largest among Trinity’s five courts and, indeed, the largest enclosed quadrangle in Europe, more than four hundred feet diagonally across; you could drop a baseball diamond within it and its walls would make for respectably distant outfield fences.

  Hardy was assigned a room in Whewell’s Court, across cobblestoned Trinity Street from Great Court, in a complex of buildings completed just thirty years earlier. More intimately scaled than most of the rest of Trinity, and a little isolated, Whewell’s Court was its own quiet preserve of stone and lawn. From his room on the second floor of Staircase M, Hardy could look out through arched Gothic windows to the busy sidewalk, or to the stone ramparts of Sidney Sussex College across the street. Yet within the inner sanctum of the court itself, the sound of bicycles clattering down Bridge Street or of excited conversation among students in Jesus Lane did not penetrate.

  After Winchester, of course, Trinity was not primarily an aesthetic delight for Hardy. Winchester was even older than Trinity, or almost anyplace else at Cambridge for that matter, its chapel as stately, its courts as tradition bound, its cloisters as imposing. So for nineteen-year-old Hardy, as he signed the Trinity Admissions Book in 1896, it was what Cambridge and Trinity represented that mattered. And what it represented was being at the top of the intellectual heap.

  Hardy, however, had a shock coming—a disillusioning that would lead him to weigh dropping mathematics. It came in the form of a venerable Cambridge institution called the Tripos.

  • • •

  The word is pronounced try-poss, and it originally referred to a three-legged stool. On it, in olden days at Cambridge, sat a man whose job was to dispute, sometimes humorously, sometimes aggressively, the candidate for the degree in mathematics. The word subsequently came to refer generally to the examinations mathematical candidates took to earn their degrees. Still later, the word was extended to examinations in other fields—a Classical Tripos, a Natural Sciences Tripos, and so on.

  The mathematical Tripos was impossibly arduous. You sat for four days of problems, often late into the evening, took a week’s break, then came back for four more days. The first half, which stressed quickness and counted even mere arithmetical facility, covered the easy stuff; showing even modest aptitude was enough to earn a degree. The second half weighed doubly and encompassed more difficult material. Here, sometimes on problem after problem, the brightest students, destined for distinguished mathematical careers, would not even know where to begin. It was a frightful ordeal. Recollections invariably lapse into awed hyperbole. The Tripos, wrote one English-born mathematician years later, “became far and away the most difficult mathematical test that the world has ever known, one to which no university of the present day can show any parallel.”

  But the Tripos was more than an examination. It was an institution. By the time Hardy arrived in Cambridge in 1896, it comprised the academic rituals surrounding it, the esteem accorded it, the system built to support it, even the style of mathematics it encouraged, all rolled into one. The Tripos went back, in earliest form, to 1730 and had always been difficult. But over the years, as it grew more demanding and more important, it also, in inimitable English fashion, took on the luster—and deadweight—of Tradition.

  Tripos candidates were ranked on the basis of their performance, and a whole ritual surrounded the reading of the Order of Merit at the Senate House. Those ranking in the first of three classes were deemed “Wranglers,” the topmost among them being named Senior Wrangler; in the early days of the Tripos, the disputants “wrangled” over points of logic. To learn who the Senior Wrangler was, everyone flocked to the Senate House. Even women, normally banished to the fringes of Cambridge life, showed up. Mingled reverence and sex appeal, if it can be imagined, accrued to the Wranglers, who “usually expected,” a Cambridge vice-chancellor was advised back in 1751, “that all the young Ladies of their Acquaintance … should wish them Joy of their Honour.”

  If anything, the Joy and Honour were greater by Hardy’s time, the Senior Wrangler and those just behind him in the order of merit earning applause and hurrahs from friends and college mates. At graduation, the vice-chancellor sat on the dais at one end of the Senate House, and college tutors presented to him those taking their degrees. The recipient kneeled before him, and the vice-chancellor took his hand between his own and repeated in Latin the awarding of the degree.

  Standing below the Wranglers were the Senior Optimes, or second class, and the Junior Optimes, or third. When the lowest scoring of the Junior Optimes—the bottom-ranking man in the class—went to receive his degree, his friends solemnly lowered from the Senate House gallery the Wooden Spoon. In fact it was a huge malting shovel, as large as the man receiving it, inscribed in Greek, and flamboyantly decorated. Upon rising to his feet, he’d take the ungainly thing and, shouldering it triumphantly, stride from the hall with his friends.

  Of course, the Wooden Spoon was strictly a consolation prize. The honor accorded the Senior Wrangler, on the other hand, was no laughing matter, but clung to him, like an aura, for a lifetime. “If one person were consensus All-American, a Rhodes scholar, and Bachelor of the Year,” observed one account aimed at American readers, “he would not come close to commanding the lasting distinction that came to the Senior Wrangler.” To a lesser extent, Wranglers among the first ten or so shared some of the glory and were virtually guaranteed distinguished careers. Obituaries of English mathematicians half a century later invariably noted that you had once been Senior Wrangler, or Second Wrangler, or Fourth. As a history of the Cambridge Philosophical Society accurately noted, “the Senior Wrangler was not invariably a great mathematician … [but] it was virtually certain that he could, if he wished, be an influential one.”

  Around Cambridge and to an extent elsewhere in Britain, the Senior Wrangler was a celebrity. People related to him much as they do the winner of the Kentucky Derby—without knowing the first thing about horseflesh. He was a star. The Times of London invariably recorded his elevation. Picture postcards bearing his photographic likeness were sold around town. One from around the turn of the century shows the victor sitting outside, posed on an ornate chair, polished shoes gleaming, hands demurely clasped together—the Derby winner shown off to the press.

  All this was very nice in its way, very British. No one questioned that Tripos success reflected, more or less, mathematical ability. Nor that, if anyone was to become a great mathematician, it would be the Wrangler and not the Wooden Spoon. On the other hand, precise ranking in the Order of Merit, everyone knew, meant far less than it seemed. Indeed, it was an open secret that Second Wranglers seemed to distinguish
themselves more than Senior Wranglers. Maxwell, the great mathematical physicist who united electricity and magnetism, had been Second Wrangler. So had J. J. Thomson, discoverer of the electron. Then there was thermodynamicist Lord Kelvin, then still William Thomson, surely the best mathematician of his year. Everyone, including himself, thought he was a shoo-in for Senior Wrangler. “Oh, just run down to the Senate House, will you, and see who is Second Wrangler,” he asked his servant. The servant returned and said, “You, sir.” Someone else, his name today forgotten, had proved better able to master Tripos mathematics.

  And that was the problem: there was such a thing as “Tripos mathematics,” and it bore little kinship to anything of interest to serious, working mathematicians. The Tripos was tricky and challenging, and certainly separated the Wranglers from the Wooden Spoon; in 1881, for example, the Senior Wrangler got a score of 16,368 “marks,” out of a possible 33,541, the Wooden Spoon 247. But there was a datedness to the problems, a preoccupation with Euclid, and Newton, and exercises in mathematical physics—a sphere spinning on a cylinder with the candidate asked to establish the equations governing its motion, or a problem based on Carnot’s Cycle in thermodynamics, and so on. They demanded accuracy and speed in the manipulation of mathematical formulas, a shallow cleverness, perhaps, but not real insight.

  And not even stubborn persistence; a proof demanded by a Tripos question couldn’t be too long or too involved; so you learned to look for that hidden Tripos twist. During one Tripos exam, a top student—that year’s Senior Wrangler—observed a less capable candidate making short work of a problem over which he agonized. Must be a trick, he realized—and went back and found it himself. The personal qualities encouraged by the Tripos, J. J. Thomson would make so bold as to suggest, made it excellent training—for the bar.

  In homage to the Tripos, an alternative educational system had sprung up. In the nineteenth century, university lecturers at Oxford and Cambridge inhabited a world of their own with scant contact with students. Teaching was left to tutors in the individual colleges, who were hardly up to the task of preparing students for the Tripos. So a Third Force had emerged to fill the vacuum—private coaches.

  These coaches were not out to teach you mathematics for its own sake, but to school you, for handsome fees, in Tripos mathematics. You were trained, one future Senior Wrangler wrote, like a racehorse. In groups of four or five, you and your coach went over problems. Maybe three times a week, he lectured to you. He pored over old examinations, wrote little private tracts, codifying mathematical knowledge into neat bundles. Only rarely did a coach himself produce important new mathematical results. But “produce” he did—Senior Wranglers. At one point, coach E. J. Routh churned out some two dozen of them in as many years.

  For students, the work load was prodigious. One distinguished mathematician, J. E. Littlewood, wrote that “to be in the running for Senior Wrangler one had to spend two-thirds of the time practicing how to solve difficult problems against time.” Attending lectures became a luxury. Of his professors in the late 1870s, A. R. Forsyth has written that they

  did not teach us; we did not give them the chance. We did not read their work: it was asserted, and was believed, to be of no help in the Tripos. Probably, many of the students did not know the professors by sight. Such an odd situation, for mathematical students in a University famed for mathematics, was due mainly, if not entirely, to the Tripos and its surroundings which, as undefined as is the British constitution, had settled into a position beyond the pale of accessible criticism.

  That was how things stood in Forsyth’s time, and that’s how they remained during Hardy’s. The Tripos system discouraged exploration of any area of mathematics, however personally satisfying, not apt to show up on the examination. It granted professional success—a fellowship at a good college, say—to those doing well on the exam, not those demonstrating a bent for research, or boldness in pursuing it. Tripos success became, like marriage for the prototypical Southern belle, not a happy prelude to one’s life but its culmination.

  Hardy was drawn up into the system, in his first term being handed over to R. R. Webb, the day’s foremost producer of Senior Wranglers. At Cambridge, the academic year consisted of three terms of seven or eight weeks. Typically, for ten straight terms the coaches did their work, and their students did theirs. Until finally, on a cold January in the unheated Senate House, the student would sit down to begin the Tripos ordeal. This was what Hardy had to look forward to.

  He was not alone among the mathematically gifted revolted by the banality of the Tripos. 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.

  But Russell did go through it. And so did most others. Hardy’s future colleague, Littlewood, later confided that he, too, had seen Tripos mathematics as empty. But he simply gritted his teeth and, as Hardy told the story years later, “decided deliberately to postpone his mathematical education, and to devote two years to the acquisition of a complete mastery of all the Tripos technique, resuming his studies later with the Senior Wranglership to his credit and, he hoped, without serious prejudice to his career.” And that, Hardy declared “in hopeless admiration,” was precisely what he did.

  But the young Hardy, new to Cambridge and deeply disenchanted, began to think he just couldn’t go through with it. It was all too stupid. Maybe he would quit mathematics altogether.

  • • •

  “I cannot remember ever having wanted to be anything but a mathematician,” Hardy would later write. “I suppose that it was always clear that my specific abilities lay that way, and it never occurred to me to question the verdict of my elders.” Modestly—and falsely—he would disclaim all artistic ability. As a philosopher, he decided, he would have been insufficiently original. Only as a journalist might he have made a go of it. But these ruminations lay almost half a century in the future. At the time, it was none of these fields that he weighed entering, but history …

  In about 1064, two years before being killed by an arrow in the eye at the Battle of Hastings, Harold, son of Godwin, swore to Prince William of Normandy not to contest the English throne, and in return was crowned Earl of Wessex. The scene was depicted in the famous Bayeux tapestries, and as a boy, Harold Hardy had taken that masterpiece as his model for a brightly colored illumination of surpassing grace, onto which he dutifully calligraphed the caption “Ye Crowninge of Ye Earl Harold.” Later, Winston Churchill, among others, would also tell the story of Harold, in The Birth of Britain, the first of his epic four-volume History of the English-Speaking Peoples. And it was a comprehensive history something like that which Hardy, at the age of about ten, dared undertake. Unfortunately, he approached it in overmuch detail and so never got much past the good Earl Harold.

  An historical essay Hardy had written on entering Trinity impressed his examiners so much he could as well have gotten his scholarship in history as in math. Now, recalling Headmaster Fearon’s treatment of the subject as a bright spot of the drab Winchester years, he flirted with changing fields. He might well have gone ahead with it. But in the midst of his confusion, he went to his director of studies, who brought another influence to bear on his malleable young mind in the person of Augustus Edward Hough Love.

  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. But Love did not push Hardy into his own
field. Though an applied mathematician, he had a bent for fundamentals, basic principles, abstract formulations. Once, talking with a friend who was explaining something geometrically, Love shook his head, and said he didn’t follow. “You see it is all x, y, z for me, and not your pictures at all.” And so, perhaps attuned to Hardy’s natural bent, he suggested Hardy read a mathematical text alien to the libraries of most applied mathematicians, the Frenchman Camille Jordan’s Cours d’analyse de l’Ecole Polytechnique.

  When Jordan died almost three decades later, it would be Hardy who wrote his obituary in Proceedings of the Royal Society and took the opportunity to comment on the book that changed his life: “To have read it and mastered it is a mathematical education in itself,” Hardy wrote, “and it is hardly possible to overstate the influence which it has had on those who, coming to it as I did from the elaborate futilities of ‘Tripos’ mathematics, have found themselves at last in [the] presence of the real thing.”

  Jordan and other Continental mathematicians were taking seemingly obvious mathematical concepts and subjecting them to the most searching scrutiny. For example, mathematicians spoke of “continuous functions”—relationships between variables unmarked by weird and abrupt lurches. And in just such vague, intuitive ways did they tend to think of them. But what, exactly, was a function? And what did it mean to say it was continuous? Among “analysts,” as this breed of mathematician was known, just such questions were their meat.

  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 …