And then, the coffee finished, we set to work. In those early days I was still trying to hammer home to him the importance of writing proofs—a futile effort, I see now. Such values must be imparted early to a mathematician; in Ramanujan's case, I realize, it was already too late. Still, I tried.
I have very particular ideas about proof. I believe that proofs should be beautiful and, to the extent that this is possible, concise. A beautiful proof should be as slender as one of Shelley's odes, and, like an ode, it should imply vastnesses. I tried to impress this upon Ramanujan. “A good proof,” I told him, “must combine unexpectedness with inevitability and economy.” There is no better example than Euclid's proof that there are infinite prime numbers—a proof I am going to walk you through now, just as I walked him through it so many years ago, not because you don't know it (I should hardly wish to insult you by implying such ignorance) but because I want to call attention to aspects of the proof of which your professors, in teaching it, may not have taken note.
This is, of course, a reductio ad absurdum proof, and so we begin by assuming the opposite of what we want to prove: we assume that there is only a finite number of primes, and we call the last prime, the largest prime, P. We must also remember that, by definition, any non-prime number can be broken down into primes. To choose a random example, 190 breaks down as 19 × 5 × 2.
Assuming, then, that P is the largest prime number, we can write out the primes in sequence, from smallest to largest, and the sequence will look like this:
2, 3, 5, 7, 11, 13, 17, 19, 23 … P
Then we can propose a number, Q, that is 1 greater than all the primes multiplied together. That is to say,
Q = (2×3×5×7×11×13…×P) + 1
Either Q is prime or it is not. If Q is prime, this contradicts the assumptions that P is the largest prime number. But if Q is not prime, it must be divisible by some prime, and that cannot be any of the primes up to and including P. So the prime divisor of Q must be a prime bigger than P, which again contradicts our original assumption. Therefore there is no greatest prime. There is an infinity of primes.
I cannot tell you what pleasure I continue to take, even today, in the beauty of this proof; in the brief yet extraordinary journey it represents, from a seemingly reasonable proposition (that there is a greatest prime) to the inevitable yet utterly unexpected conclusion that the proposition is false. Nor would I be telling you the truth if I said that Ramanujan was oblivious to the beauty of the proof. He understood that beauty; he appreciated that beauty. And yet his appreciation was rather akin to mine of the novels of Mr. Henry James. That is to say, I admire them yet I cannot love them. In the same way I never had the sense that Ramanujan had much love for the proof. What he loved were numbers themselves. Their infinite flexibility and yet their rigid order. The degree to which natural laws, many of which we barely understand, check our ability to manipulate them. Littlewood thought him an anachronism. According to Littlewood, he belonged to the age of formulae, which ended a hundred years ago. If he had been German, if he had been born in 1800, he would have changed the history of the world. But he was born too late, and on the wrong side of the ocean, and even if he never admitted this, I feel sure that he knew it.
These were, I believe, days of great happiness for Ramanujan, no matter what Mrs. Neville might say to the contrary. Nor was she in any sense out of the picture. One weekend, for instance, I recall her hauling Ramanujan off to London, to meet Gertrude for a visit to the British Museum. He might have made friends. Sometimes I would see him in the company of other Indians. Above all, he worked, and before the summer was out, he published his paper on modular equations and routes to π.
Occasionally I would visit him in his rooms, which were on the ground floor of Whewell's Court. They were extraordinarily tidy and contained almost nothing in the way of possessions, aside from the requisite bed and dresser and, for some reason, a pianola that did not work. He lived ascetically, like one of those Hindu mystics about whom one reads from time to time. From the little kitchen an odor of curry and that clarified butter so beloved by the Indians—ghee, it is called—always emanated. If a shadow of trouble passed over our conversations during those days, it was due to his wife's failure to write him any letters. Forget that the poor thing was barely literate: he longed for some communication from her, in addition to which, in India, there were apparently scribes and such to whom you could go when you needed a letter written for you. From his mother, letters arrived regularly, pages densely filled with a script as mysterious to me as the language of theorems must be to any non-mathematician. His wife, though, wrote nothing, even though he wrote to her, unfailingly, once a week.
One wonders what would have happened had the war not broken out. Many wonder this, for all sorts of reasons. There is of course no answer.
PART FOUR
The Qualities of the Isle
1
GERMANY INVADES BELGIUM, and at first Hardy feels as he does about a beautiful proof: the onset of war seems at once inevitable and unexpected.
Almost everyone he speaks to now claims to have seen it coming, yet as he looks back over the past month, he can remember only Russell saying that he saw it coming. Instead domestic crises—strikes, unrest in Ulster—dominated the conversation at high table. The assassinations in Sarajevo, of course, provoked a bit of comment. Yet Servia was so far away! A small, primitive country. Nothing that happened there could touch Cambridge.
Russell, by contrast, was in a panic. Most of July he spent shuttling between London and Cambridge, running about announcing that he knew no one who was in favor of war, that everyone he had spoken to considered the prospect of war folly. As if public opinion ever influenced the decisions of government. As if saying that something could never happen would stop it from happening.
The day after the news broke, he chased Hardy down in the middle of Great Court. “So it's come,” he said, not with any sort of “I told you so” glee, but in a tone at once stunned and drawn. “Everything we believe in is over.” And now declarations of war are being presented like visiting cards.
It's all very bewildering to Hardy. War with Germany, after all, means war with Göttingen, beloved Göttingen, land of Gauss and Riemann. Yet Germany has now invaded Belgium on its way to France. To protect Belgium, England must forge an alliance with Russia—savage, autocratic Russia—and all in order to defeat Germany, land of Göttingen, land of Gauss and Riemann … How à propos that Russell alone predicted the worst! Hardy's imagination spins in an infinite regress, the barber who shaves only those men in his town who do not shave themselves. And the town (where else?) is Göttingen.
As soon as war is declared, the tone among his acquaintances changes from one of dismissal to one of denial. Instead of reassuring one another that Britain will remain neutral, they start reassuring one another that the war, should it actually commence, will be swift. Over by Christmas. Lord Grey, for instance, has just admitted to secret talks with France. Might these lead to a quick armistice? Heartening words resonate through New Court and Nevile's Court, but behind them Hardy can hear the thin, ceaseless babble of despair.
“It is the end,” Russell says. He is just back from yet another trip to London. The day before war broke out, his lover, Ottoline Morrell, summoned him, as her husband was to give a speech to Parliament, urging the British government not to enter into the fray. Unable to gain admission to the gallery, Russell paced up and down Trafalgar Square and was appalled to hear the men and women sitting under the lions voicing enthusiasm, even delight at the prospect of war. “Today is not yesterday,” Russell says, speaking of the reactions of the “average” person. Yet even here at Cambridge, where supposedly no one is average, subterranean rumblings of patriotism sound. Even among the brethren. Rupert Brooke, for instance, has said he's ready to volunteer—“no doubt the influence of that odious little Eddie Marsh,” Russell says—while Butler has offered up all the facilities of Trinity College to the war effort. “It is the
end,” Russell repeats, then goes back to London because he cannot bear to be too far from the center of things. “Horrible as it is,” he says, “I have to get the news as soon as it arrives.”
The irony, of course, is that often the news arrives at Trinity sooner than it does at the office of the Times. The brethren have enviable connections—Keynes to the treasury, Marsh, through Churchill, to 10 Downing Street. Norton writes to Hardy that he saw Marsh at a party in London, “parading about in evening dress, immaculate, enjoying his importance.” Brooke was with him. “He is living in Marsh's flat. He has spurned Bloomsbury and boys in favor of manliness and uniforms. Yet isn't it funny that he should have chosen Edwina, of all people, as his mentoress?”
And in the meantime, it does not stop being summer. That is the heartbreak. Cambridge has more or less emptied out for the long Vacation. Littlewood stays in Treen, returning, presumably, only when Dr. Chase takes up residence. Hardy divides his weeks between Trinity and his mother's house in Cranleigh. When he's at Trinity, whole days pass during which he sees no one but Ramanujan, with whom he takes walks along the river, and sometimes sits on the banks. Heliotropic by nature, he raises his face to the sun whenever it passes between the clouds. Truth be told, he appreciates the quiet. It seems inconceivable that the world could end in such a season.
He tries, as much as he can, to see Ramanujan. Standing in shadowed profile before the river, his arms folded behind his back and his stomach protruding slightly, he might be the silhouette of a Victorian gentleman, cut from black paper and pasted against a white ground. Restraint and discipline, a certain aloofness, or perhaps even elusiveness: these are his most distinguishing traits. Except when they're talking mathematics, he rarely speaks except when spoken to, and when he is questioned, almost always answers by dipping into what Hardy envisages as a reserve of stock replies, no doubt purchased on the same shopping trip in Madras during which he was supplied with trousers, socks, and underwear. Replies such as: “Yes, it is very lovely.” “Thank you, my mother and wife are well.” “The political situation is indeed very complex.” Here he is, after all, in English clothes and on English land, and still Hardy can't begin to penetrate his carapace of cultivated inscrutability. Only occasionally does Ramanujan let something slip, a whiff of panic or passion slips through (Hobson! Baker!), and then Hardy feels the man's soul as a mystery, a fast-moving prickle beneath his skin.
Mostly, those afternoons, they talk mathematics. Definite integrals, elliptic functions, Diophantine approximation. And, of course, primes, their diabolical tendency to confound, of which Hardy wants to make sure Ramanujan never loses sight. For example, Littlewood has of late made another important discovery. It has to do with a refinement that Riemann made of Gauss's formula for counting primes. Up until recently, most mathematicians took it for granted that Riemann's version would always give a more accurate estimate than Gauss's. But now Littlewood has shown that, though Riemann's version may be more accurate for the first million primes, after that Gauss's version is sometimes more accurate. But only sometimes. This discovery is of vast importance to about twenty people. Unfortunately, half of those people are in Germany.
As they walk, he asks Ramanujan if he knows the story of Rie-mann's terrible housekeeper, and when Ramanujan waggles his head, he tells it. “Of course,” he concludes, “the story's probably false.”
“How old was he when he died?”
“Thirty-nine. He died on Lake Maggiore, of tuberculosis. So why would the housekeeper feel compelled to burn his papers? It all seems suspiciously convenient, a way of saying, ‘Yes, there's a proof out there, you've just got to find it.’”
Ramanujan is silent for a moment. Then he asks Hardy about Göttingen, and Hardy tells him what little he knows of the place; he describes the ratbaus on the front of which is emblazoned the motto “Away from here there is no life,” and the cobbled streets down which, in his imagination, Gauss and Riemann—freed, now, from the constraints of time—stroll together as they discuss the hypothesis. Every few paces, when Riemann comes to a crucial juncture in his lost proof, the pair stops, diverting the passersby as a rock diverts a stream. Likewise, when they are talking mathematics, Hardy and Ramanujan sometimes stop; only this time of year, there are few passersby to impede.
He asks Ramanujan about his childhood. Did he ever play chess? Again, Ramanujan waggles his head. He only learned chess once he arrived in Madras, he says. However, when he was very small, he and his mother played a game with eighteen pieces, fifteen representing sheep and three representing wolves. “When the wolves surrounded a sheep, they would eat it. But when the sheep surrounded a wolf, they would immobilize it.”
“I would imagine,” Hardy says, “that it would be rather difficult for the sheep to win.”
“Yes. Very quickly, however, I was able to calculate the probabilities of the game, and from then on, whether playing wolf or sheep, I always won over my mother.”
“Did she mind?”
“No, not at all.”
“How old were you?”
“Six. Five, perhaps.”
Hardy is not surprised. At five he was beating his own mother at chess.
“Both my parents were, as they say, mathematically minded,” he says. “My mother especially. Not that she ever had the chance to cultivate her talents. She was a schoolteacher.”
Ramanujan says nothing.
“And your parents?”
“They are poor people. They did not have education. My father is a gumasta, a simple accounts clerk.”
They stop to look at the river. No punts glide by. Hardy hears birdsong, the faint whoosh of branches in the breeze. Ramanujan turns to face him, as he rarely does, and his eyes, so black and deep, astonish Hardy. Such eyes, he thinks, would drive even the most rigorous mind to bad poetry: Liquid pools of molten ore, / These portals to a world beyond … At night, sometimes, in his head, he works on the poem, which he never writes down.
“Hardy,” he says, “is it true that in Belgium the Germans are setting fire to whole villages?”
“That's what the newspapers tell us.”
“And that they are killing the children and throwing away the old people?”
“So I am told.”
Ramanujan frowns. “I am worried about two young men from Madras who are coming here to study. Ananda Rao and Sankara Rao. They are carrying much food for me, including tamarind.”
“There's no need to worry,” Hardy says. “No one's going to attack a British passenger ship.”
“But they are not traveling on a British ship. They are traveling on an Austrian ship. Their intention was to get to Austria and come here by train. What will happen to them now?”
“Oh, an Austrian ship.” A robin flies by. “Well, once they arrive in Trieste, they'll just … I don't see why anyone should give them trouble. After all, they're students.”
Again, Ramanujan frowns. “Last night I dreamed that they were trapped in a burning village,” he says. “I saw them burn.”
“Oh, I shouldn't think that would happen. They're not going anywhere near Belgium.”
Silence now. They continue their walk. Ramanujan has his eyes fixed on the ground in front of him. And for a moment Hardy, turning to glance at him, asks himself a terrible question, a question he berates himself for even entertaining. Is what really worries Ramanujan the fate of the young men or of his tamarind?
2
THE FRIDAY AFTER war is declared, Hardy takes part in an expedition to Leintwardine Manor, on the Welsh border, to watch an open air performance of The Tempest. Alice Neville organized the trip at the beginning of July, before anyone guessed that war was imminent. She has people near Leintwardine, and the performance is to benefit some charity to which they have connections. On Thursday, Hardy sent her a note asking if, under the circumstances, she might not prefer to cancel the outing, and she replied that she could see no reason why they should. “There is no fighting in Hertfordshire, or so I'm told,” she wrote. This bo
th irritated and disappointed Hardy, who had been hoping, at the very least, that the war would give him a reasonable excuse not to do things he did not want to do.
And so that Friday morning he finds himself gathered at the Nevilles' house, along with Ramanujan, Littlewood, Neville's brother Eddie, and Eddie Neville's friend Mr. Allenby. Of all the invitees, only Gertrude has begged off, pleading a fictitious cold. Ramanujan and the Nevilles will ride with Eddie in his Jowett, Hardy and Littlewood with Mr. Allenby in his Vauxhall.
Once they set off, Cambridge thins out quickly, giving way to open countryside. The weather is fine. Still, Hardy's dismay at the roar and stink of the Vauxhall inhibits his ability to take any pleasure in the view. He sits in the back, alone. Littlewood is up front with Allenby, who has red cheeks and heavy jowls. Like the elder Neville, he lives north of London, in High Barnet. Both are members of a motoring club; “mad for motoring,” he tells Littlewood, who nods and smiles in that irritating, inevitable way of his—Littlewood, with his confounding ability to make himself at home anywhere, no matter how dire the circumstances. Hardy, on the other hand, finds that the older he gets, the more ill at ease he feels venturing beyond the walls of Trinity. He's never liked cars, and Allenby is not what you would call a conservative driver. He takes the turns with a ferocity that brings Hardy's heart to his throat, all the while laughing and chattering with Littlewood above the engine's clamor, while Hardy hallucinates rifle tips pointing up from the roadside hedgerows. Hours turn into weeks, then years, the rifle tips seem with every mile to poke him more urgently in the bowels, until at long last they pull up outside Leintwardine Manor. Hardy is let out of the backseat. He has been sitting for so long, his legs feel as if they're going to collapse under him. He needs a loo. He stumbles over to Ramanujan, who looks complacent enough, if a bit dusty.
“Enjoy the ride?” he asks.
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