All of it adds to their conviction that they are dealing with genius on a scale neither has ever imagined, much less encountered. What Ramanujan has not been taught he has, using his own peculiar language, reinvented. Better yet, building on this foundation, he has constructed an edifice of astonishing complexity, originality, and strangeness. Little of which Hardy indicates in his reply, which he tries to keep as understated as he can manage. So far as the first group of results goes, he forgoes confessing his amazement and offers only consolation, telling Ramanujan: “I need not say that if what you say about your lack of training is to be taken literally, the fact that you should have rediscovered such interesting results is all to your credit. But you should be prepared for a certain amount of disappointment of this kind.” To the second and third groups, he devotes more attention. “It is of course possible that some of the results I have classed under (2) are really important, as examples of general methods. You always state your results in such particular forms that it is difficult to be sure about this.”
They worked hard, he and Littlewood, on that last sentence. First Hardy wrote “peculiar” before “forms,” but worried that the word might put Ramanujan off. Then he crossed out “peculiar” and wrote “odd,” which was worse. It was Littlewood who came up with “particular,” the slightly arch connotation of which (Hardy imagined Lytton Strachey saying it) he doubted that Ramanujan would pick up on.
“It is essential that I should see proofs of some of your assertions,” he writes next. “Everything depends on rigorous exactitude of proof.”
His conclusion mixes encouragement with caution. “It seems to me quite likely that you have done a good deal of work worth publication; and, if you can produce satisfactory demonstrations, I should be very glad to do what I can to secure it.” Then he signs his name, puts the letter in an envelope, addresses it, has it stamped, and on the morning of February 9th—the day after his thirty-sixth birthday—slips it into the mouth of the postbox outside the Trinity College gates. For a time in his childhood he believed that all the postboxes in the world were connected by a system of underground tubes; that when you mailed a letter, it actually sprouted legs and ran to its destination. Now he imagines his letter to Ramanujan scuttling along the passageways underlying England, crossing the Mediterranean and the Suez Canal, tirelessly trudging forward until it reaches an address he can hardly visualize: Accounts Department, Port Trust, Madras, India. And now he only has to wait.
6
New Lecture Hall, Harvard University
ON THE LAST DAY of August, 1936, Hardy wrote on the blackboard behind him:
“I am sure that Ramanujan was no mystic,” he said as he wrote, “and that religion, except in a strictly material sense, played no important part in his life. He was an orthodox high-caste Hindu, and always adhered (indeed with a severity most unusual in Indian residents of England) to all the observances of his caste.”
Even as he spoke, though, he was doubting himself. He was giving, he knew, the script: the authorized version of his own opinion, already at odds with other versions of Ramanujan's story, in particular those circulating in India, where the youth's piety and devotion to the goddess Namagiri were situated at the dramatic heart of his mathematical discoveries.
Hardy did not—could not—believe this. His atheism was not merely part of his official identity; it was part of his being, and had been since his childhood. Still, even as he uttered them, he had to admit that his words simplified considerably not just the real situation but his own feelings about it.
He would have liked to put down his chalk at that moment, turn to his audience, and say something else. Something along the lines of:
I don't know. I used to think I did. But as I get older it seems that I know less and less rather than more and more.
I used to believe that I could explain anything. Once, at Gertrude's request, I attempted to explain the Riemann hypothesis to some girls at St. Catherine's School. This was during the early spring of 1913, when we were still awaiting Ramanujan's reply to the first letter. I really thought that leading these girls through the steps of the Riemann hypothesis would be easy, that I would awaken in them a fascination that would last them the rest of their lives. And so, while Gertrude stood attendance with Miss Trotter, the maths mistress, a pale-faced young woman whose hair, though she could not have been more than thirty, was already white, I lectured those girls in their starched pinafores. They gazed up at me with eyes that were either love-struck or vacant or contemptuous. One chewed her hair. The Riemann hypothesis might be the most important unsolved problem in mathematics, but that did not make it a subject of interest to twelve-year-old girls.
“Imagine,” I said, “a graph, like any ordinary graph, with an x axis and a y axis. Let us say that the x axis is the ordinary number line, with all the ordinary numbers lined up in succession, and that the y axis is the imaginary number line, with all the multiples of i lined up in succession: 2i, 3.47i, 4,678,939i, and so on. On such a graph, as on any graph, you can draw a point, and then connect the point, with lines, to points on the two axes. In this case, the numbers that correspond to these points on the plane are called complex numbers because each one has a real part and an imaginary part. You write them like this: 2i+1. Or 4.6i + 1736.34289 or 3i + 0. The part with the i is the imaginary part, and corresponds to a point on the imaginary axis, while the other part is the real part and corresponds to a point on the real axis.
“So this is the Riemann hypothesis: you take the zeta function and you feed it with complex numbers. And then you look at your answers, and you see at which points the function takes the value of zero. According to the hypothesis, at every point where the function takes the value of zero, the real part will have a value of ½; or, to put it another way, all the points at which the function takes the value of zero will line up along the line of ½ on the x axis, which is called the critical line.
“To prove the hypothesis, you must prove that not a single zeta zero will ever be off the critical line. But if you can find just one zeta zero off the line—just one zeta zero where the real part of the imaginary number isn't ½ —then you'll have disproved the Riemann hypothesis.
So half the job is looking for a proof—something airtight, theoretical—but half the job is hunting for zeros. Counting zeros. Seeing if there are any that are off the critical line. And counting zeros involves some pretty knotty math.
“And what will you have done, if you find the proof? You'll have eliminated the error term in Gauss's formula. You'll have revealed the secret order of the primes.”
That was it, more or less. Of course there was a lot that I left out: the so-called trivial zeros of the zeta function; and the need, when exploring the zeta function landscape, to think in terms of four dimensions; and most crucially, the complex series of steps that leads from the zeta function to the primes and their calculation. Here, if I had tried to explain, I would have foundered. For there is a language that mathematicians can speak only among themselves.
After the lecture, the students applauded politely. Not for very long, but politely. I inspected their faces. They wore expressions of boredom and relief. Already, it was obvious, they were thinking ahead: to hockey practice, or Gertrude's art class, or a secret rendezvous with a boy. “Are there any questions?” Miss Trotter asked, her voice as colorless and icy as her hair, and when no girl spoke, she filled in the void with her own words. “Do you believe, Mr. Hardy, in your heart of hearts, that the Riemann hypothesis is true?”
I thought about it. Then I said, “Sometimes I do, sometimes I don't. There are days when I wake convinced that it's just a question of counting zeros. Somewhere there must be a zero off the line. Then there are days when a stab of insight slices right through me and I think I've made a step toward a proof.”
“Could you give us an example?”
“Well, a few weeks ago, when I was taking my morning walk—I take a walk every morning—it suddenly occurred to me how I might pro
ve that there is an infinite number of zeros along the critical line. I hurried straight home and wrote down my ideas, and now I'm very close to completing the proof.”
“But that means you've proven the Riemann hypothesis,” Miss Trotter said.
“Far from it,” I said. “All I'll have proven is that there is an infinite number of zeros along the critical line. But that does not mean that there is not an infinite number of zeros not along the critical line.”
I watched her as she tried to untangle the triple negative. Then I looked at Gertrude. It was clear that she had got the point before I'd even made it.
Afterward, when we were walking home together, I said to my sister, “This is why the Indian's letter interests me so much. If, as he claims, he's significantly reduced the error term, then he may be on to Riemann.”
“Yes,” Gertrude said. “He may even be the man to prove Riemann. How would you feel if he did?”
“I'd be delighted,” I said. She smirked. Of course she doubted— and was right to—my pose of selflessness. We stepped into the house, where what seemed an infinity of maids was in the throes of an orgy of cleaning, Mother supervising their activities. One swabbed the floor, another scrubbed the windows, a third was beating pillows. Suddenly I saw the maids as zeros of the zeta function. I imagined them lining up, drawn as if by magnets along the critical line. There is a secret history through which a monstrous housekeeper strides, destroying all she touches. According to O. B., a famous musician went deaf after taking his housekeeper's advice that he treat an earache by stuffing cotton dipped in ether in his ears. And of course there was Riemann's legendary housekeeper, who, upon learning of his death (if the story is to be believed), threw all of his papers—including a reputed proof of the hypothesis—into the fire. How clearly I can envision that scene! The summer of 1866, warm weather, and this vigorous woman—in many ways the most important figure in the history of mathematics—methodically feeding the pages into the stove's stinking maw. Feeding and feeding, scrawled sheet after scrawled sheet, until, as legend has it, Riemann's Göttingen colleagues arrive in a mob. Cry out for her to stop. Patiently they sort through what they have salvaged, praying that the proof will have survived her reign of terror, while in the background … what does she do? Does she weep? Probably not. I see her as plump, methodical. Energy without imagination. No doubt she goes about her business. Scrubbing down floors. Washing pots.
The irony, of course, is that Riemann wasn't even there. He didn't witness, even in death, the conflagration. He had gone to Italy, hoping the balmier weather would improve his health. He was thirty-nine when he died. Consumption.
And do you think his housekeeper imagined that somehow the papers themselves might be tainted?
People understood so little about contagion in those days.
I can't stop thinking about this woman. What I find most monstrous about her is her efficiency. It has a bloodthirsty edge. In my mind I try to place myself at the scene in Göttingen. I try to explain to her, after the fact, the importance of the documents she has destroyed. In response she simply gazes at me, as if I'm a perfectly benign idiot. Her belief in her own rectitude is impregnable. This is the side of the German character that I preferred, before the war, not to contemplate, because I could not reconcile it with my dream image of the German university town down the cobbled streets of which Gauss and Hilbert strolled arm in arm, in defiance of fact, in defiance even of time. Ideas and ideals have a homey smell, rather like coffee. And yet in the background there always lurks this housekeeper with her ammonia and her matches.
7
SATURDAY EVENING, he goes to the weekly meeting of the Apostles, which is held on this occasion at King's, in the rooms of Jack Sheppard, classicist. He goes mostly out of boredom, because he is impatient to receive Ramanujan's reply, and hopes that the meeting will distract him from trying to guess at its content. In his coat pocket he carries the first page of Ramanujan's original letter, as, when he took the tripos, he carried the first volume of Jordan's Cours d'analyse.
It is his habit to arrive exactly twenty minutes late to the meetings, thereby avoiding both the awkwardness of being the first to arrive and the ostentation of being the last. Fifteen or so men between the ages of nineteen and fifty stand gathered on Sheppard's Oriental carpet, trying to look as if they're smart enough to deserve to belong to such an elite society. Although some are active undergraduate members, most are angels. (The Society's stock is rather low at the moment.) But what angels! Bertrand Russell, John Maynard Keynes, G. E. Moore: himself excepted, Hardy thinks, these are the men who will determine England's future. And why should he be excepted? Because he is merely a mathematician. Russell has political aspirations, Keynes wants to rebuild the British economy from the ground up, Moore has published Principia Ethica, a work that many of the younger Apostles regard as a kind of Bible. Hardy's ambition, on the other hand, is merely to prove or disprove a hypothesis that perhaps a hundred people in the world even understand. It's a distinction in which he takes some pride.
He counts the other angels in the room. Jack McTaggart is here, pressed up against the wall, as always, like a fly. So is suave little Eddie Marsh, who, in addition to serving as Winston Churchill's private secretary, has recently established a reputation as a connoisseur of poetry. Indeed, he has just published an anthology entitled Georgian Poetry, to which one of the major contributors is Rupert Brooke (no. 247), whom everyone thinks very handsome and whom, at the moment, Marsh is chatting up. Of the more significant angels, only Moore and Strachey are missing, and Strachey, Sheppard tells Hardy, is due any minute on the train from London. For this is no ordinary meeting. Tonight two new Apostles are to be born into the society. One of the “twins,” Francis Kennard Bliss, has good looks and a talent for playing the clarinet to recommend him. The other, Ludwig Wittgenstein, is a new arrival from Austria via Manchester, where he went to learn to fly an aeroplane. Russell says he's a metaphysical genius.
To distract and amuse himself, Hardy plays a game. He pretends that he has not, in fact, come alone to the meeting, but that he has brought along a friend. No matter that such a thing would never be done, or that the “friend,” though he answers to the name Ramanu-jan, bears an uncanny physical likeness to Chatterjee, the cricketer: so far as the game goes, the young man standing next to him is the author of the letter in his pocket, fresh off the boat from India and eager to learn Cambridge ways. He wears flannel trousers that ripple when he walks, like water touched by a breeze. A shadow of beard darkens his already dark cheeks. Yes, Hardy has studied Chatterjee with care.
He takes his friend on a tour of the rooms. Until recently, they belonged to O. B., who kept them filled with visiting royals, Louis XIV furniture, Voi che sapete, and handsome representatives of the Royal Naval Service. But then O. B., much to his dismay, was forced into superannuation and Italian retirement, and Sheppard took the rooms over. His sordid motley of domestic possessions looks forlorn and miserly in a space so accustomed to grandiose gestures. A portrait of his mother, stout and contemptuous, gazes across the Hamlet chair at a pianola that doesn't work. On the wall are some photographs of Greek statues, all of them nude, several missing limbs, none, Hardy notes, missing the dainty penis-and-balls set that the Greeks considered so elegant, especially when compared to those huger, crasser appendages that figured so prominently in O. B.'s badinage, and continue to figure prominently in Keynes's. And what does his friend from India think of Keynes? At the moment, the rising star of British economics is lecturing a rapt audience of undergraduates on the comparative size of “cock-stands” in Brazil and Bavaria. Wittgenstein is standing alone in a corner, staring at one of the photographs. Russell is directing at Sheppard a foul-odored expatiation on the liar's paradox from which poor Sheppard has intermittently to turn away, if only to catch his breath.
“Imagine a barber who each day shaves every man in his town who doesn't shave himself. Does the barber shave himself?”
“I should think
so.”
“All right, then the barber's one of the men who doesn't shave himself.”
“Fine.”
“But you just said he did shave himself.”
“I did?”
“Yes. I said the barber shaved every man who didn't shave himself. If he does shave himself, then he doesn't shave himself.”
“All right, then he doesn't shave himself.”
“But you just said he did!”
“Hardy, come save me,” Sheppard says. “Russell is tying me to a spit and is about to roast me.”
“Ah, Hardy,” Russell says. “Yesterday Littlewood was telling me about your Indian, and I must say, it sounds quite exciting. On the brink of proving Riemann! Tell me, when will you be bringing him over?”
Hardy is rather taken aback to learn that Littlewood has been talking Ramanujan up. “I'm not sure we will,” he says.
“Oh yes, I've heard about the fellow,” Sheppard says. “Living in a mud hut and scribbling equations on the walls with a stick, isn't that right?”
“Not exactly.”
“But Hardy, couldn't this be someone's idea of a joke? Mightn't your Indian be, I don't know, some bored Cambridge man trapped in an observatory in the wilds of Tamil Nadu, trying to while away the hours by having you on?”
“If so, the man's a genius,” Hardy says.
“Or you're a fool,” Russell says.
“Isn't there a point, though, where it comes to the same thing? Because if you're clever enough to construct something this brilliant as a joke … well, you've defeated your own intention, haven't you? You've proven yourself a genius in spite of yourself.”
Sheppard laughs—a wheezy, girlish laugh. “A puzzle worthy of
Bertie!” he says. “And speaking of puzzles, Wittgenstein ought to be a client of your maddening barber, Bertie. Look at the cuts on his chin.”
The Indian Clerk Page 5
