The Man Who Knew Infinity

by Robert Kanigel

  First Gauss, Abel, and Cauchy had risen above the looser, intuitive nostrums of the past; later in the century, Weierstrass and Dedekind went further yet. None of them were English. And the English professed not to care. Why, before the turn of the century, Cauchy—the Cauchy, Augustin Louis Cauchy, the Cauchy who had launched the French school of analysis, the Cauchy of the Cauchy integral formula—was commonly referred to around Cambridge as “Corky.”

  Since Newton’s time, British mathematics had diverged off on a decidedly applied road. Mathematical physics had become the British specialty, dominated by such names as Kelvin, Maxwell, Rayleigh, and J. J. Thomson. Pure math, though, had stultified, with the whole nineteenth century leaving England with few figures of note. “Rigor in argument,” J. E. Littlewood would recall, “was generally regarded—there were rare exceptions—with what it is no exaggeration to call contempt; niggling over trifles instead of getting on with the real job.” Newton had said it all; why resurrect these arcane fine points? Calculus, and the whole architecture of mathematical physics that emanated from it, worked.

  And so, England slept in the dead calm of its Tripos system, where Newton was enshrined as God, his Principia Mathematica the Bible. “In my own Tripos in 1881, we were expected to know any lemma [a theorem needed to prove another theorem] in that great work by its number alone,” wrote one prominent mathematician later, “as if it were one of the commandments or the 100th Psalm… . Cambridge became a school that was self-satisfied, self-supporting, self-content, almost marooned in its limitations.” Replied a distinguished European mathematician when asked whether he had seen recent work by an Englishman: “Oh, we never read anything the English mathematicians do.”

  The first winds of change came in the person of Andrew Russell Forsyth, whose Theory of Functions had begun, in 1893, to introduce some of the new thinking—though by this time it wasn’t so new anymore—from Paris, Göttingen, and Berlin. Written in a magisterial style, it burst on Cambridge, as E. H. Neville once wrote, “with the splendour of a revelation”; some would argue it had as great an influence on British mathematics as any work since Newton’s Principia. By the standards of the Continent, however, it was hopelessly sloppy and was soundly condemned there. “Forsyth was not very good at delta and epsilon,” Littlewood once said of him, referring to the Greek letters normally used for dealing with infinitesimally small quantities. Still, it helped redirect the gaze of English mathematicians toward the Continent. It charted a course to the future, but did not actually follow it.

  That was left to Hardy.

  • • •

  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.” Norbert Wiener, the American mathematical prodigy who would later create the field known as “cybernetics,” attended Hardy’s lectures. “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 regret: so great was Hardy’s personal magnetism and enthusiasm, it was said, that he sometimes diverted to mathematics those without the necessary ability and temperament.

  But as lucid as were his lectures, it was his writing that probably had more impact. Later, speculating about what career he might have chosen other than mathematics, Hardy noted that “Journalism is the only profession, outside academic life, in which I should have felt really confident of my chances.” Indeed, no field demanding literary craftsmanship could fail to have profited from his attention. “He wrote, in his own clear and unadorned fashion, some of the most perfect English of his time,” C. P. Snow once said of him. That Hardy’s impressions of Ramanujan would be so relentlessly quoted, and would go so far toward fixing Ramanujan’s place in history, owes not alone to his close relationship with Ramanujan but to the sheer grace with which he wrote about him.

  Hardy didn’t much like his early style, he decided later, terming it “vulgar.” Of course, he didn’t much like his glorious good looks, either. “Everything Hardy did,” Snow once wrote of him, “was light with grace, order, a sense of style.” And his writing exemplified that. He wrote, for the Cambridge Review, about the philosophy of Bertrand Russell. His obituaries of famous mathematicians were rounded, gracious, and wise. He could write about geometry and number theory for lay audiences. And his Mathematician’s Apology, which became a classic, is almost mesmerizing in its language’s hold on the reader.

  He applied his gifts even to the most densely mathematical of his work. In collaborations, it was almost always he who wrote up the joint paper and shepherded it through publication. “He supplied the gas,” recalled Littlewood, who was content if what he had to say was simply correct. Hardy wanted more; the “gas,” as he once defined it, was the “rhetorical flourishes,” the equivalent of “pictures on the board in the lecture, devices to stimulate the imagination of pupils.” A reviewer would say of one of his mathematical texts that Hardy had “shown in this book and elsewhere a power of being interesting, which is to my mind unequalled.”

  Thought, Hardy used to say, was for him impossible without words. The very act of writing out his lecture notes and mathematical papers gave him pleasure, merged his aesthetic and purely intellectual sides. Why, if you didn’t know math was supposed to be dry and cold, and had only a page from one of his manuscripts to go on, you might think you’d stumbled on a specimen of some new art form beholden to Chinese calligraphy. Here were inequality symbols that slashed across the page, sweeping integral signs an inch and a quarter high, sigmas that resonated like the key signatures on a musical staff. There was a spaciousness about how he wrote out mathematics, a lightness, as if rejecting the cramped, ungenerous formalities of the printed notation. He was like a French impressionist, intimating worlds with a few splashes of color, not a maker of austere English miniatures.

  All through the first decade of the twentieth century Hardy used his pen to seduce a generation of young English mathematical students into taking seriously the new Continental rigor. When a review of Bertrand Russell’s Principles of Mathematics ran in the Times Literary Supplement in 1903, it was Hardy who wrote it. While English mathematics often turned a deaf ear to events across the Channel, Hardy used the pages of the Mathematical Gazette to comment on foreign books. In 1903, he reviewed Einleitung in die Funkionentheorie, by Stolz and Gmeiner; in 1905 Leçons sur les fonctions des variables réeles, by Borel.

  Meanwhile, he held English and American texts to strict account for their lapses. In one Mathematical Gazette review in 1907, for example, he wrote of an American calculus text by W. Woolsey Johnson. Oh, it wasn’t bad of its type, he allowed. But its type was a breed of English book that, while forgivable thirty years before, was no longer. Perhaps it was all right to pass over theoretical difficulties in laying the foundations of calculus, he wrote.

  But there are different ways of passing over difficulties. We may simply and absolutely ignore them: that is a course for which there is often much to be said. We may point them out and avowedly pass them by; or we may expand a little about them and endeavour to make our conclusions plausible without professing to make our reasoning exact.

  But, Hardy went on:

  There is only one course for which no good defence can ever be found. This course is to give what profess to
be proofs and are not proofs, reasoning which is ostensibly exact, but which really misses all the essential difficulties of the problem. This was Todhunter’s [the author of a kindred text] method, and it is one which Prof. Johnson too often adopts.

  Whereupon Hardy launched into a mathematical example to show how the author employed arguments “entirely destitute of validity.”

  Hardy felt he could do better, and in September 1908 completed A Course of Pure Mathematics, the first rigorous exposition in English of mathematical concepts other texts sloughed over in their rush to get to practical applications or cover broad expanses of mathematical ground. Such rigor was sorely needed, said Hardy in his preface. “I have [very rarely] encountered a pupil who could face the simplest problem involving the ideas of infinity, limit, or continuity with a vestige of the confidence with which he could deal with questions of a different character and of far greater intrinsic difficulty.”

  Like everything else Hardy ever wrote, his textbook was readable. This was not simply page after gray page of formula. His were real explanations of difficult ideas presented in clear, cogent English prose. What in other hands would be buried in a sea of abstractions, in Hardy’s fairly jumped out at you, sometimes as the culmination of a passage actually verging on suspenseful.

  Early on, for example, he addressed “rational” numbers, numbers like 6, 2/3, 112/3890, or 19 that can be expressed as ordinary fractions or integers. Between any two numbers representing points on a line segment, he showed that more rational points can always be squeezed in. Between 1/2 and 2/3, you can fit a 3/5. Between 3/5 and 2/3, you can fit 5/8. And so on, forever, resulting in an infinity of such points. Then he goes on:

  From these considerations the reader might be tempted to infer that these rational points account for all the points of the line, i.e. that every point on the line is a rational point. And it is certainly the case that if we imagine the line as being made up solely of the rational points, all other points (if any such there be) being imagined to be eliminated, the figure which remained would possess most of the properties which common sense attributes to the straight line and would, to put the matter roughly, look and behave very much like a line.

  Something is coming, the reader rightly suspects, without knowing just what.

  Hardy then showed that within the same line segment there was, roughly speaking, another infinity of points that could be crowded into the interstices between these rational numbers—the “irrational” numbers, which cannot be expressed as fractions, and whose properties he then proceeded to explore.

  This loving attention to fundamentals was just what English mathematics needed. As one review of the book commented, “When Mr. Hardy sets out to prove something, then, unlike the writers of too many widely read textbooks, he really does prove it… . If the book is widely read, I for one shall hope to avoid in the future the many weary hours that have usually to be spent in convincing University students that ‘proofs’ which they have laboriously learned at school are little better than nonsense.”

  Hardy’s book was widely read. For the next three-quarters of a century, and through ten editions and numerous reprints, it became the single greatest influence on the teaching of English mathematics at the university level. Through it—and through his lectures at Cambridge, through his papers and reviews, through his relationships with other mathematicians—Hardy made rigor no longer the preserve of a few Teutonic zealots but something that bordered on the mathematically fashionable.

  • • •

  At the root of Britain’s mathematical backwardness, Hardy was sure, lay the Tripos system. Originally the means to a modest end—determining the fitness of candidates for degrees—the Tripos had become an end in itself. As Hardy saw it, English mathematics was being sapped by the very system designed to select its future leaders.

  Around 1907, he became secretary of a panel established to reform it. But in fact, he championed its reform only as a first step toward doing away with it altogether, and only because he saw no hope, just then, for more radical change. As he later told a meeting of the Mathematical Association, “I adhere to the view … that the system is vicious in principle, and that the vice is too radical for what is usually called reform. I do not want to reform the Tripos but to destroy it.”

  Hardy did not oppose examinations in general; he saw a place for them, a sharply limited one—as a floor, a minimum standard necessary to earn a degree. “An examination,” said he, “can do little harm, so long as its standard is low.” But the Tripos laid no such meager claims; it meant to appraise, to sift, to grade. Undergraduates, as Hardy pictured them, exhausted “themselves and their tutors in the struggle to turn a comfortable second [class] into a marginal first.” That, in his view, was the problem: the Tripos distorted teaching and learning alike, and English mathematics was the loser for it.

  With others among the younger dons, Hardy succeeded in forcing changes through a reluctant senate, the university’s governing arm. Chief among them was abolishment of the Order of Merit; a degree candidate still took the Tripos but, beginning in 1910, was ranked only by broad category—as Wrangler, Senior Optime, or Junior Optime. There would no longer be a Senior Wrangler to which to aspire, no longer the merciless pressure it created, no longer the ambition-driven need for coaches. Overnight, the most notorious abuses of the Tripos system were eliminated.

  But Hardy’s more ambitious goal was futile; the Tripos, in modified form, exists still—in part because while many pointed out its failings, few did so with Hardy’s ferocity. There was a mild-manneredness in the English personality that Hardy, when it came to the Tripos, trespassed. “It is useless to propose anything revolutionary to Englishmen,” it would be pointed out to a mathematical audience some few years later. “Existing institutions always have merits, which are as deep-seated as their defects are patent… . Our English way is to alter the defective institution a little bit at a time, so that it comes a little nearer to what we desire.” Hardy’s friend Littlewood, while no fan of the Tripos, was also less heated about it. “I do not claim to have suffered high-souled frustration,” he wrote of his experience with it. “I took things as they came; the game we were playing came easily to me, and I even felt a satisfaction of a sort in successful craftsmanship.”

  Hardy’s enmity, then, was something different, almost beyond reason. Plainly, his own experience influenced him. Back in 1896, the prospect of two years of Tripos tedium had nearly deflected him from mathematics altogether. But in the end, he had meekly surrendered; he had acquiesced to a coach, climbed on board the System, put “real” mathematics aside. When he did at last pit himself against the Tripos, he could almost be said to have “failed” it; for someone as competitive as he, that’s what being Fourth Wrangler meant. The Tripos, in a sense, had beaten him.

  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.

  In Hardy coexisted a stern, demanding streak with an indulgent liberal-mindedness, a formidable and forbidding exterior with a soft and fragile core. He would later claim to have scant interest in his less able students. But this, by all accounts, was nine-tenths bluster; he never failed any of them. “He simply couldn’t think that way,” Mary Cartwright, a former research student, told a friend, “because he was so kind to the weak ones.”

  Hardy disdained social niceties, ever kept his distance, arrogantly dismissed God. Yet he could be kind and endlessly obliging. Even the obituaries he wrote showed a largeness of spirit that, as someone once put it, “must have made every mathematician wish that he could have seen his own career described in the same generous terms.”


  So he was demanding, distant, emotionally astringent—and large-hearted, caring, and kind. It takes no straining of the facts to lay this split to the respective influences of his mother and father. But whatever their source, these two contrasting strands wound through his personality always. And both would emerge in his relationship with Ramanujan over the next seven fateful years.

  • • •

  Winter 1913. Europe stirred, armies marshaled. The world was restless with change. Picasso’s first cubist drawings had appeared barely a year before. In Paris, Diaghilev, whose Russian Ballet had given its first London performance two years before, prepared for the premiere of Stravinsky’s tempestuous Le Sacre du Printemps, in which a maiden dances herself to death. In England, George V was King, Edward having died suddenly in 1910. In 1911, the Parliament Bill had stripped the House of Lords of its veto power on acts of Commons. All through Britain, workers struck and militant suffragettes smashed windows. Ireland seethed.

  But in Cambridge, things were as they always were. Hardy neared his thirty-sixth birthday with his face bearing scarcely a mark of it. He’d visit Bertrand Russell in Nevile’s Court and discuss Bergson and the philosophy of religion; once, Norbert Wiener and his father met him there and took him to be an undergraduate. In 1912, Hardy published nine more papers, including his first collaborative one with Littlewood, “Some Problems of Diophantine Approximation.” His first key paper on Fourier series was coming out later in 1913, the revised edition of his popular textbook the following year. Hardy’s friend from the Apostles, Leonard Woolf, recently back from Ceylon, found Cambridge much as he’d left it; on the train back to London, he wrote, “I felt the warmth of a kind of reassurance. I had enjoyed my weekend. There was Cambridge and Lytton and Bertie Russell and Goldie, the Society and the Great Court of Trinity, and Hardy and bowls—all the eternal truths and values of my youth—going on just as I had left them seven years ago.”