How can a “proof” be wrong? How can you dutifully march through its reasoning, convince yourself that what you’ve said is right, only to have some other mathematician show it is not? What, in other words, can go wrong with a mathematical proof seemingly laid out in obeisance to a relentlessly “logical” sequence of clear-cut steps?
In a word, everything.
Perhaps the most familiar, if trivial, example of such a failure is the “proof” that purports to show that 2 = 1. For starters, let a = b. Now, multiply both sides of the equation by a, leaving
a2 = ab
Add a2 − 2ab to both sides of the equation:
a2 + (a2 − 2ab) = ab + (a2 − 2ab)
Which reduces to:
2(a2 − ab) = a2 − ab
Now divide both sides by a2 − ab, leaving
2 = 1
Voilà. Treating both sides of the equation with scrupulous equality we reach a result defying common sense.
What’s gone wrong? Nothing but that scourge of many an elementary proof: we have divided by zero, a mathematically impermissible operation that gives an “answer” devoid of meaning.
But nowhere, a voice interrupts, does the proof say we should divide by zero.
On the contrary: we started off with a = b, then toward the end divided by a2 − ab, which is zero.
Even at his most innocent, Ramanujan would never commit such a gaffe. And yet it suggests how, busy manipulating symbols and without appreciating every nuance of what he was doing, he could go wrong.
As a mathematician, you can slide into trouble in numerous ways. You can differentiate a function without realizing the function cannot be differentiated. Or you can write off later terms of a series on the assumption that they are of a lower “order” than earlier terms, when, in fact, they contribute substantially to the series sum. Or you can assume that an operation correct for a finite number of terms is correct for an infinite number. Or you can integrate a function between two points, yet fail to note where the integral may be undefined, and so carry through your proof such meaningless quantities as “infinity minus infinity.”
In his work with primes, Hardy wrote, Ramanujan’s proofs
depended upon a wholesale use of divergent series. He disregarded entirely all the difficulties which are involved in the interchange of double limit operation; he did not distinguish, for example, between the sum of a series Σ an and the value of the Abelian limit
or that of any other limit which might be used for similar purposes by a modern analyst.
These were all quite technical, of course—like legal loopholes of which the police lieutenant is unaware. Ramanujan’s intuition steered him clear of many obstacles of which his truncated education had failed to warn him. But not all. The problem was not only that he was sometimes wrong; it was that he lacked mathematical knowledge enough to tell when he was right and when he was wrong. He stated correct and incorrect theorems with the same aplomb, the same sweet, naive confidence. And when he did offer proofs, they scarcely warranted the name.
For Ramanujan, Littlewood wrote later, “the clear-cut idea of what is meant by a proof, nowadays so familiar as to be taken for granted, he perhaps did not possess at all; if a significant piece of reasoning occurred somewhere, and the total mixture of evidence and intuition gave him certainty, he looked no further.”
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.
And now, as he would over the succeeding months and years, Hardy set to work on him, trying to overcome the deficit that was the price Ramanujan had paid for his intellectual isolation. At twenty-six, Ramanujan had long-established ways of doing things. Nonetheless, he responded. As Hardy wrote later,
His mind had hardened to some extent, and he never became at all an “orthodox” mathematician, but he could still learn to do things, and do them extremely well. It was impossible to teach him systematically, but he gradually absorbed new points of view. In particular he learnt what was meant by proof, and his later papers, while in some ways as odd and individual as ever, read like the works of a well-informed mathematician.
By early 1915, about the time Mahalanobis read him the Louvain street problem in Strand magazine, Ramanujan had already begun to shift gears, to redirect his work in ways Hardy was urging. “I have changed my plan of publishing my results,” he wrote Krishna Rao in November 1914. “I am not going to publish any of the old results in my note book till the war is over. After coming here I have learned some of their methods. I am trying to get new results by their methods.” By early the following year, he had literally set aside much of his old work, was drinking in the new: “My notebook is sleeping in a corner for these four or five months,” he wrote his childhood friend S. M. Subramanian, with whom he had briefly roomed in Summer House two years before. “I am publishing only my present researches as I have not yet proved the results in my notebook rigorously.”
Proof and rigor: he was absorbing the Gospel according to Hardy.
• • •
But while taking in the new, as Hardy would write, Ramanujan’s “flow of original ideas showed no symptom of abatement.” In his letter to Subramanian, for example, Ramanujan at one point abruptly broke into a paragraph with: “I shall now tell you [about] a very curious function,” then wrote this pattern of fractions across the page:
1, 1/2, 2/1, 1/3, 3/1, 2/3, 3/2, 1/4, 4/1, 2/4, 4/4, 3/4, 4/3
Guided by it, he ingeniously constructed a function which, bizarrely, was mathematically undefined for the fractions in his series, and existed at all only for “irrational” values, those not representable as a fraction. “A very curious function,” he had called it. Then, again: “There is another curiosity here …” And “Just imagine [how] this function [behaves] …” This was vintage Ramanujan, his delight in the function’s peculiar behavior fairly spilling across the pages of the letter.
In what, in some ways, was his greatest achievement, then, Hardy brought Ramanujan mathematically up to speed without muzzling his creativity or damping the fires of his enthusiasm. It would have been easy to sniff at his shortcomings and dutifully correct them, like a bad editor who crudely blue-pencils his way through a delicate manuscript. But he knew that Ramanujan’s mathematical insight was rarer by far than even the most formidable technical mastery. It was fine to know all the mathematical tools needed to prove a theorem—but you had to have a theorem to prove in the first place.
That was easy to forget as you flipped through the Proceedings of the London Mathematical Society. There, as in any mathematics journal, the proof was made to seem the culmination of a hundred closely reasoned steps ranging over a dozen pages. There, mathematics could seem no more than a neat lockstep march to certainty, B following directly from A, C from B, … Z from Y. But no mathematician actually worked that way; logic like that reflected the demands of formal proof but hinted little at the insights leading to Z. Rather, as Hardy himself would write, “a mathematician usually discovers a theorem by an effort of intuition; the conclusion strikes him as plausible, and he sets to work to manufacture a proof.”
The theorem itself was apt to emerge just as other creative products do—in a flash of insight, or through a succession of small insights, preceded by countless hours of slogging through the problem. You might, early on, try a few special cases to informally “prove” the result to your own satisfaction. Then later, you might go back and, with a full arsenal of mathematical weapons, supply the kind of fine-textured proof Hardy championed. But all that came later—after you had something t
o prove. Besides, it was mostly technical, like the laws of evidence; you could learn it. Rigor, Littlewood would observe, “is not of first-rate importance in analysis beyond the undergraduate state, and can be supplied, given a real idea, by any competent professional.”
Given a real idea—that was the rare commodity.
“Mathematics has been advanced most by those who are distinguished more for intuition than for rigorous methods of proof,” the German mathematician Felix Klein once noted. (Added Louis J. Mordell, an American mathematician who would ultimately succeed Hardy in his chair at Cambridge: “To very few other mathematicians are Klein’s remarks … so appropriate as to Ramanujan.”) A “real idea” wasn’t dished up, like a Tripos problem, by some anonymous mathematical Intelligence. It had to come from somewhere, had to be seen before it could be proved. 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.
Plenty of mathematical technicians, Hardy knew, could follow a step-by-step discursus unflaggingly—yet counted for nothing beside Ramanujan. Years later, he would contrive an informal scale of natural mathematical ability on which he assigned himself a 25 and Littlewood a 30. To David Hilbert, the most eminent mathematician of his day, he assigned an 80.
To Ramanujan he gave 100.
“It was impossible to ask such a man to submit to systematic instruction, to try to learn mathematics from the beginning once more,” Hardy would write. “I was afraid too that, if I insisted unduly on matters which Ramanujan found irksome, I might destroy his confidence or break the spell of his inspiration.” And so he sought an elusive middle ground where, without crimping Ramanujan’s creativity, he could teach him, as he wrote, the “things of which it was impossible that he should remain in ignorance… . It was impossible to allow him to go through life supposing that all the zeroes of the zeta function were real. So I had to try to teach him, and in a measure I succeeded, though obviously I learnt from him much more than he learnt from me.” Teaching Ramanujan, mathematician Laurence Young has written, “was like writing on a blackboard covered with excerpts from a more interesting lecture.”
Hardy held in his hands a rare and delicate flower. And the responsibility he bore for nurturing it was only redoubled by the war.
5. S. RAMANUJAN, B.A.
“At Cambridge we are in darkness,” wrote Trinity vice-master Jackson in January 1915. “No gas in the streets or courts; few electric lights and those shaded; candles on the high tables. The roads into Cambridge are blocked to prevent the approach of motors such as those which guided the East Coast Zeppelins. Rumor says that an attack on Windsor Castle was expected last week.”
In April, about a year after Ramanujan’s arrival, Jackson wrote: “In France and Flanders we make no progress. In the Dardanelles we are at a standstill. The army does not grow as it ought. We have not got ammunition for the existing army. The Germans have been preparing villainies for years.”
Wounded soldiers flooded Cambridge, almost twelve thousand being admitted to the First Eastern General Hospital by July 1915. They’d arrive on trains late in the evening at the station, where they were met by white-hooded nurses and motorized ambulances. “When they succeeded each other at frequent intervals for some time,” a Cambridge schoolgirl recalled later, “I knew that there must have been heavy fighting in Flanders or France.”
The maimed, hurt, and sick streamed into Cambridge, the healthy and strong out. Of undergraduates, Cambridge was largely deserted; normally some thirty-five hundred, they now numbered five or six hundred. College fellows served in the Foreign Office, in the War Office, in the Treasury, as well as in the army itself. One Trinity man wrote, in Jackson’s words, that “the front was like a first-rate club, as you met all your friends there.” The university’s medical laboratories, meanwhile, were put at the disposal of the First Eastern General Hospital. The chemical laboratory did research in gas warfare. The engineering laboratories began making shells and gauges for the Ministry of Munitions.
Among the many Trinity fellows to leave was Littlewood, now a second lieutenant in the Royal Garrison Artillery. He had little use for the war, but adapted, just as he had to the Tripos system, affecting a “cheerful indifference.” By late 1915, he’d been put to work on a fresh mathematical approach to fashioning antiaircraft range tables. (“Even Littlewood could not make ballistics respectable,” Hardy would write, in a gibe at applied mathematics, “and if he could not, who can?”) Almost alone among Trinity fellows he was never promoted, and ultimately was relieved of routine chores and allowed to live with friends in London. “Ballistics,” one biographer would gently put it, “did not fill all of Littlewood’s working hours during the war years”; between 1915 and 1919, he managed to collaborate with Hardy on ten joint papers, in areas of mathematics as distant as could be imagined from the war.
Still, he was away from Cambridge—and away from Ramanujan. And he, and his formidable mathematical powers, had been a prime reason to bring Ramanujan to Cambridge in the first place. So that almost before he had caught his breath in England, after barely four months, Ramanujan had been thrown, more dependently than ever, into the arms of Hardy.
Hardy would later register for military service under the “Derby Scheme,” Lord Derby’s politically shrewd move to forestall unpopular conscription through a voluntary, but socially pressured, “attestation” of readiness to serve. But Hardy, thirty-eight when the Derby Scheme was launched in October 1915, was, according to Littlewood, deemed “unfit” to serve, and spent most of the war in Cambridge.
In the First World War, unlike the Second, antiwar feeling ran high. Hardy’s activities would lead at least one obituary to assert he’d been a conscientious objector. He was not; indeed, late in the war he would write, “I don’t like conscientious objectors as a class.” Still, as Littlewood tells it, Hardy “wrote passionately about the notorious ill-treatment [to which] objectors were subject.” He belonged to an earnest, high-minded group called the Union of Democratic Control, which focused on the peace to follow what was still assumed would be a brief, if bloody conflict. Hardy was secretary of its Cambridge chapter, his old friend from the Apostles, G. Lowes Dickinson, its president. Its first public meeting was held on March 4, 1915. Later, in November, when it was announced that a meeting would take place in the Trinity rooms of Littlewood—who was a U.D.C. member in absentia and had supplied written permission—the authorities moved to block it.
Just outside the gates of Ramanujan’s castle the war raged, coming closer day by day. Comely Cambridge had been transformed into a training camp and hospital. In May 1915, the Lusitania had been sunk, hardening sentiment against Germany. By June, food prices had risen 32 percent over the previous year’s. Intellectual commerce with German mathematicians was cut off. Littlewood and other mathematicians were gone. Hardy was distracted by extra-mathematical concerns.
And yet, so far, the war had not yet reached Ramanujan. Whatever his private revulsion toward it, he yet basked in a kind of warm intellectual spring.
Since his arrival in England, he’d been writing home regularly—at first, three or four times a month, and even now, during 1915, twice a month or so, regularly assuring his family that he maintained his vegetarianism and his religious practices. His letters to friends back in India scarcely mentioned the war, but rather told of his work and his progress, inquired after family members, even occasionally gave advice. To his two brothers back in India, he sent a parcel full of books of English literature.
The persistent bother of finding and preparing food for himself undermined his sense of wellbeing somewhat. So did the English chill, and the clothes that cinched at his fat waist. And the peculiar absence of letters from Janaki. But mostly, he still rode the crest, kept working, happily and
hard, at mathematics.
In his first letter to Hardy, he had sought help in publishing his results. And now, during 1914 and 1915, letters home showed how preoccupied he was with seeing his work in print. “I … have written two articles till now,” he wrote in June 1914. “Mr. Hardy is going to London today to read a paper on one of my results before the London Math. Society.”
“I have written three papers till now. The proof sheets have come. I am writing three more papers. All will be published at the end of the vacation, i.e. in October,” he wrote in August.
“I am very slowly publishing my results owing to the present war,” he wrote in November 1914.
“It will take some months for me to write that paper systematically and publish it,” he wrote Narayana Iyer in November 1915.
After years as a mathematician known only to himself, then only to Madras, Ramanujan plainly relished the prospect of appearing in prestigious English mathematics journals. To appear in print was the only tangible sign of recognition you could hold up to family and friends, the only way the world would know what you’d done. At one point, when he learned that Ananda Rao, a mathematics student at King’s who was the son of a Madras judge and relative of R. Ramachandra Rao, was preparing an essay for a Smith’s Prize, he went straight to Hardy to ask whether he, too, might try for it. Doing mathematics satisfied deep emotional and intellectual needs in Ramanujan. But credit, kudos, appreciation satisfied quite other, just as insistent, needs. Intellectually, Ramanujan was not, and never could be, “just like everyone else.” But in this social, public realm, he was: he wanted recognition, needed it.
The Man Who Knew Infinity Page 29