Illustration: A Mathematician’s Apology by G.H Hardy
In January of 1913, a 25-year-old autodidact by the name of Srinivasa Ramanujan wrote a letter to the British mathematician G.H. Hardy. He had already written letters to a number of prominent academics, but Hardy was the only one to respond. A little more than a year later Ramanujan boarded a ship in Madras bound for London. Soon, he was in Cambridge, where he would collaborate with Hardy and live out the duration of the First World War. In England, however, Ramanujan’s health began to deteriorate — in part because of the weather, in part because of the food. More than anything else, it was this poor health that led Ramanujan to board a ship at the conclusion of the war and return to India, having left a definitive mark on modern mathematics.
Hardy had recognized Ramanujan’s brilliance almost immediately. His approach was highly intuitive, which is notable in a field dominated by logical demands. But his thinking also had its blindspots. As Hardy once said of him, “The limitations of his knowledge were as startling as its profundity.” Ramanujan, to be sure, was an anomaly, and Hardy wondered what Ramanujan would have been capable of if he had had the kinds of privileges Hardy enjoyed. But part of the beauty of Ramanujan’s story is that he was not limited by the kinds of knowledge out of which men like Hardy made careers. Having a relatively poor mind for math myself, my interest in Ramanujan has less to do with his research on Bernoulli numbers than with the fact that, at the conclusion of the war and in terrible health, he returned to India, where he died at the age of 33.
But you will forgive me, Comrades. It seems I have overshot my mark. I have not begun at the beginning. Let me retrace my steps and begin again.
First, let it be assumed that everywhere we touch the world meaning proliferates, and that this fecundity is what makes the dual activities of reading and writing so pleasurable. Let it also be assumed that this pleasure can produce, as an unintended consequence, a kind of paranoia, by which I mean both the delirium the word more commonly implies and something else entirely. The word paranoia comes from the Greek root nous, meaning mind, and the prefix para– (as in words such as paradox, paraphrase, and paradise) from something like “analogous or parallel to” but also “beyond.” I am interested, here, in appraising a mind that is like mind, separate from or just beyond it. A mind untethered to mind. What language would such a mind speak? I want to describe a position from which thinking can address the world, but buoyantly, as though thought could suspend us above the melee of daily life.
I also want to examine where writing leads us, and how it takes us there. Literature, although it may not be an entirely useless endeavor, comes pretty damn close sometimes. The fact that everything can be found to have some use, however provisional, does not mean that those phenomena nearest to uselessness on an imaginary spectrum are, for all intents and purposes, useful. In a house, one wants every square foot to be accounted for, to have some purpose. In my own house, we have even found a use for the space beneath the stairs. But while architecture may abhor impracticality, in literature it may be a cause for celebration.
Useful or not, let us assume that any product of a culture is inseparable from the conditions that produce it. What to make of the fact, then, that one of the most radical of American art forms — abstract expressionist painting — was secretly supported by the CIA, and that many of the most lauded names in 20th century American art were unwitting weapons in an artistic cold war conducted alongside the actual one? Such an extreme case may be beside the point, but to say that my writing exists in a world separate from and untouched by Lady Gaga’s meat dress, on one hand, and, on the other, the ongoing horrors in the eastern Congo, is to claim for art a privilege that seems to me at once self-congratulatory and self-deceptive. But I have almost no interest in a programmatic or didactic art. I only bring up this business about the Congo to suggest that the buoyant position I have just described is not insouciance, is never exempt from the realities it addresses or fails to address.
In our public library, the one I go to with my family, there are versions of the story of the pied piper available in the children’s section. I find the story horrifying, and leafing through some pages this past week I was struck by the fact that the faces of the children, as they are being led out of town, are bright and cheerful. In fact, like the piper, the children are dancing down the street, across the bridge, and into the countryside beyond. The children’s happiness would be obscene if they knew any better, but under the spell of the piper’s flute they don’t.
The smiles aren’t gallows humor, precisely, but a strange detachment. I think of a famous photograph taken on 9/11 that shows a group of people apparently chatting it up along the Brooklyn waterfront, completely unbothered by the smoking buildings in the background. The image, as was subsequently revealed, is more complex than that, but there is also the unrelated photograph of beachgoers outside Naples, who continue soaking up the sun and chatting on their cellphones while the bodies of two drowned teenage girls — whom nobody bothered to save — lie in open sight some ten feet away. Indifference, yes, and one could certainly write an essay dealing with the social, psychological, and artistic dimensions of that indifference. My interest, though, is in a kind of lightness that acknowledges the bodies on the beach or the burning buildings by barely acknowledging them at all.
As elsewhere, one of my guides here is W. G. Sebald, who performed this task with a kind of relentlessness that is as stunning as it is deeply sad. The unnamed subject of each of Sebald’s books is, by his own admission, the concentration camps, and yet, with a few exceptions, he touches on them so lightly that you could be lulled, by his long, languorous sentences, into thinking the books were about something else: herring, say, or the rise of the Dowager Empress. That they are not is a function of a very Sebaldian principle: atrocity needs no exaggeration. If you look closely enough you see how it saturates all that surrounds it, drawing the energy of the world into its deep and abhorrent abyss. But lightness, in Sebald and elsewhere, provides more than a cover. Lightness is a strategy, much as I distrust that word. It is a method for dealing with and channeling other energies.
A single memo at the end of Georges Perec’s Species of Spaces, an incomparably light text, illustrates what I mean. The memo is a description of a landscaping project to be completed outside of the crematorium ovens at Auschwitz; the correspondents are two of the camp’s administrators. This is likely the very place where Perec’s own mother was murdered. So why include the letter? What practical use could it serve? It is typical of the method I have in mind that the lightness here is heavy-light, much as Coleridge argued for the virtues of the clear-obscure. In the case of Species of Spaces, what the letter represents is so heavy that it could sink any text that included it, which is also why Perec says nothing whatsoever about it. But the opposite is equally important: without some counterbalance, the purely light text drifts off like a loose balloon. Montaigne writes: “Let the mind awaken and quicken the heaviness of the body: let the body arrest the lightness of the mind and fix it fast.” Meanwhile the balloon floats higher and higher. I can see it now, there above the plains. It’s pretty, yes, but I can’t hold it. It’s pretty, yes, but it won’t last.
In 1940, G.H. Hardy published a famous book-length essay, A Mathematician’s Apology. Hardy begins his work with the melancholy admission that “Exposition, criticism, appreciation is work for second-rate minds” — melancholy because this is precisely what Hardy is undertaking in his essay. The book is premised on the painful realization that his creative period as a mathematician is over and now all that’s left to him is to reflect on what it has meant.
In the Apology, Hardy opposes what he calls real with trivial mathematics. He argues that while the latter is, on the whole, useful, “real [or pure] mathematics does not, ‘do good’ in a certain sense.” Real mathematics, he says, has no effect on war. Trivial mathematics, on the other hand, can be applied to a wide variety of combat scenarios. Mathematics can be used to plot the trajectory of a missile, but the study of pure mathematics generally has no comparable application. While it’s curious to note that Hardy’s particular field of study has since been applied to cryptology, this does not negate his central premise, and just this past Saturday I spoke with a “pure” mathematician who assured me that 99% of what he does is as useless as number theory was in Hardy’s time.
Hardy ends his Apology by saying that he has never done anything useful, and that nothing he has done “is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world.” It should probably go without saying that, as writers, we may find ourselves in a similar situation. I’m reminded of W.H. Auden’s axiom that “Poetry makes nothing happen.” Like it or not, the phrase I often heard repeated at Naropa — that poetry propels the century forward a few inches — is only a slight revision of Auden’s phrase, since, given the severity of our situation as a species — overpopulation, climate change, global poverty — the difference between nothing and a few inches may be piddling.
The poet and critic Maggie Nelson has written her own revision of Auden, but with a slight twist. Writing, she says in her marvelously troubling book Bluets, changes nothing. Like fucking, she says, it leaves everything pretty much the way it is. A biologically determined activity, fucking happens because on some level we have to fuck. We’re hardwired that way. Is writing any less biologically determined than fucking: do we write because we have to write? (And among American writers, to paraphrase Frank O’Hara, which of them are better than fucking?) How conscious of a decision, for a writer, is the one to write? Like fucking, it must be slightly more conscious than breathing, but somewhat less conscious than choosing to order a cheeseburger for lunch.
Writing is not unlike what producing mathematical proofs was for Hardy, or better yet Ramanujan. Did I mention that Ramanujan’s stay in England was exactly contemporaneous with the First World War? Did I mention that Hardy’s Apology was undertaken in 1939, in the deep crease of the Second? That its composition came only after the heart attack that same year that left him as incapable of playing real tennis as conducting real math?
According to Italo Calvino, lightness is one of the supreme virtues of literature, and by lightness he means something that loosens objects from their inevitable heaviness, a “weightless gravity” in which the world is suspended. Privations produce a desire for levitation, he writes, and it is precisely this need to hover above or beyond the surface of our reality that literature addresses.
There are a couple of ways to read this, the first of which has a distinctly escapist edge. In short, to banish actual privation from one’s writing and to adopt, instead, the fantasies literature allows: ordered worlds, clear justice, happy endings. There are plenty of literary examples of this in both realist novels and fantasy fiction: the world of the sitcom, even when cloaked in distinctly darker tones. The second possibility is something like Hamlet’s melancholy, which Calvino describes not as a “dense, opaque” sadness, “but a veil of minute particles of humours and sensations, a fine dust of atoms.” Veils must be remarkably light, although having never worn one I can’t say with any certainty. In any case, veil is the operative word in Calvino’s description, and functionally speaking a veil does not reinvent the world but filters and distorts it. A veil destabilizes the wearer’s relationship to objects, and vice versa. Through one, reality isn’t absent: it’s just fuzzy.
Calvino’s lecture on lightness also touches on the hero Perseus, who kills the gorgon Medusa by viewing her reflection in the polished surface of his shield. You may never find a lighter story about a beheading than this one, but the act is still grisly. Sure, it’s a fantastic horror, but it’s a horror nonetheless. And within it are the seeds of the sadism and misogyny that have dominated much of human history. Calvino’s point is that Perseus only succeeds in overcoming the terrible beauty of the gorgon by approaching it indirectly. In Perseus’s hands lightness is not the opposite of weight but something like its casing or shell. Or, to reverse the formulation, weight is not the opposite of lightness but its lining.
Calvino goes to great lengths to make just this point: lightness is not the abolition of reality. Perseus, he says, does not refuse “the reality in which he is fated to live; he carries the reality with him and accepts it as his particular burden.” His challenge, like the writer’s, is not to get bogged down by “the weight, the inertia, the opacity of the world,” or rather, to address that weight without succumbing to it. There’s an equal but opposite risk, however: what is it exactly that has us running for Harry Potter and vampire fiction, while on our iPods some mewling adolescent whines about love and miscommunication? Such culture is pure veil with nothing on the other side of it, a shield in which no gorgon lurks and where the image is instead, and narcissistically, one’s own. That’s what so offensive about the lightness present in much that our culture produces — offensive not to any sense of propriety, but to the world itself.
Nor is it elitist to say so. Our culture retains a deep distrust of intellect and nurtures a nearly paralyzing fear of what it does not understand. That many readers still find Gertrude Stein opaque, for example, has less to do with Stein’s approach to language and reality than with our failure, albeit over several generations, to incorporate her challenge to language into everyday life. Difficulty is not to be avoided. As Chögyam Trungpa often said, we need to lean into the sharp points. Not impale ourselves, but not run away, either.
A few years ago, I went to see a gifted body-worker. She spent a generous amount of time working on me, probably two hours, but in all that time only one thing stands out in my memory: the single, extraordinarily light touch at the site of an injury I had sustained months before. It’s possible she didn’t even touch the site, but much of my body’s energy was clearly pooling itself there. All it required was the slightest gesture to be released into the air. The experience was electric. I shuddered. It was deeply moving, even though it only lasted a second or two. After that, she moved on. More remarkable yet, while she was aware of my injury and its general location, I don’t remember saying with any exactitude where it was. I suppose I didn’t need to. Perhaps all of us wear our wounds far more openly than we suppose. For those who know how to look, they mark us all too clearly.
A skilled body-worker herself, Bhanu Kapil performs something similar in her recent book, Schizophrene. The work contains its share of violence, including an assault, a race riot, and a scene from post-partition India, in which the writer’s young mother catches a glimpse of a row of women tied to trees along the new border, their stomachs cut out. Kapil doesn’t dwell on this violence, and gives us little context. As a result the touch, despite the heaviness of the situations she describes, feels light and even — although I say it with great care — electric. Perhaps the best example in the book is a single page that reads, in quotation marks, “He dragged her down the stairs by her hair to the room where we were eating.” That’s it. Nothing more. We don’t learn who the pronouns refer to, we don’t know who the speaker is, and the context of both the act and the statement are purposefully withheld. All that remains is the horror, alone on the page, and then we move on. We have touched the site of the injury and that is perhaps sufficient. In witnessing, we have validated.
But to what end? The intention here can hardly be to stop domestic violence, as noble a goal as that is and even if, in one’s daily life (as opposed to one’s writing life), one is fundamentally committed to it. Am I saying what is already obvious to everyone? That the writer’s personal and social commitments are inseparable from her work, and at the same time they are also incommensurable with it? So why write a scene of such violence into a work unlikely to have much of an effect on the continuation of that violence? For me, and I suspect for Bhanu as well — although you would have to ask her — the answer always comes back to the body and to the necessity of writing and living from the curious mixture — I might even call it the hybridity — at one’s core. And if indeed writing is a biological imperative, if literature is less of a social construct than a physical one, how can the body of the text avoid reflecting the body of the person who produced it? And if this body, any body, is present when “He drag[s] her down the stairs by her hair to the room where we [a]re eating,” to deny expression to that memory, felt perhaps in the scalp, would be as painful and as pointless as exaggerating it.
Hardy argues that his work — real or pure mathematics — is indefensible from certain moral positions. Literature, he suggests, if it’s of any permanent value, is equally indefensible. How can you defend your decision to study creative writing when about half of the world’s population lives on around $2 a day? Maybe you have never seen what $2 a day looks like. If so, you’ll have to take my word for it when I tell you that the scale on which life is lived in this way is a sin so great that I know of no word in any language to name it. Moreover, literature is unlikely to ameliorate that poverty. So where does the value of writing reside? What are its uses? Is the most that can be said for it that it is harmless? And can’t this very harmlessness be construed as a kind of ethical lapse?
Perhaps a case can be made for the text as a kind of devotional object — the rite within write — and while this is really outside the scope of my consideration here, it bears mentioning that Ramanujan — have we forgotten about him? — was apparently a deeply religious man. Hardy was of course an atheist — how could he not have been? — but his writings on the aesthetics of pure mathematics strike a distinctly mystical tone. So as the battle of the Somme raged in Europe, claiming more than a million lives in just five months, these two men were writing papers on the qualities of integers. Like high priests in some hidden temple, albeit a secular one named Cambridge University, they reverently performed their duties while the world around them went to the dogs.
Hardy writes in his Apology that “When the world is mad, a mathematician may find in mathematics an incomparable anodyne.” Not a cure, mind you, but something like a painkiller. Maybe such an anodyne can also take the form of a light touch, maybe at or near the spot where the nose meets the eye socket. Maybe it can even take the form of a book — Bhanu’s Schizophrene, say. The question her book begs, after all, is whether writing also can serve as a balm. Can it heal? Or does it, at most, speak to the body’s need for healing? Does it suggest the balm it cannot itself provide?
During the war it was difficult for Ramanujan to leave England. Before his departure he was even hospitalized for a time. Hardy would often come to visit, and one day he found himself thinking about the number of the taxi he was riding in, 1729. Once at Ramanujan’s bedside, Hardy said the number struck him as quite uninteresting, mathematically speaking. Ramanujan protested: it was the lowest number that could be expressed as the sum of two cubes in two different ways. I don’t know about you, but the mind that could grasp this complexity instantaneously is so foreign to me as to induce a kind of naïve awe. Literature may have recourse to its own such inducements, but I can’t imagine any as dramatic. Maybe it’s just that purely mathematical realities are so far removed from daily life, whereas literature is linked to daily life in a kind of perpetual antagonism. Whatever the reasons, Ramanujan’s apprehension of this curious fact about a number strikes me as both profound and profoundly useless, and it may be this very conjunction that makes the anecdote so moving.
Hardy concludes that it’s wrong to talk about the usefulness of pure mathematics or art, since there is practically no use for either. The best he can come up with is that he pushed the century forward a few inches and added a small portion to human knowledge. Say what you will, Hardy tells us, I collaborated with the best minds of my generation as something like a peer. Still, one can’t help feeling, reading his book, that Hardy sees his life as a failure, especially as he grasps so clearly that his best powers as a mathematician have left him. Maybe it was this that led Hardy, at the end of the Second World War, to attempt suicide. Unfortunately, as he told his friend C.P. Snow, he found he had no aptitude for it and took far more pills than were necessary. Or maybe it was his inability to justify a life spent playing, essentially — albeit with numbers — while a massive orgy of violence claimed, all told, tens of millions of lives. Inevitably, the question with which he begins his Apology, that of the ultimate value of his work, remains unanswered and, at least by Hardy, unanswerable. (One can reasonably assume that if Hardy knew his life’s work was being vulgarized in every credit card transaction that takes place, he would roll in his grave.)
In his memoir, Fugitive Days, former Weather Underground member Bill Ayers recounts his first contacts with leading figures in the anti-war movement and in Students for a Democratic Society (SDS). He remembers the early rallying cry: try to live in a way that does not make a mockery of your values. This exhortation also haunts Hardy’s Apology, and is the real source of the book’s melancholy. As tempted as I am to revise that earlier catchphrase into something like write in a way that does not make a mockery of your values, I don’t know whether our work can ever align precisely with our values, though I’m also not saying we shouldn’t try.
A light touch does not negate reality, nor, I might add, are all silences complicit. Anne Carson, in her moving elegy for her brother, talks about a muteness or opacity “which likes to show the truth by allowing it to be seen hiding.” And near the very end of Schizophrene, Bhanu Kapil gives us a page that has been totally blacked out. It’s an opaque square of ink and it defies you to see through it, or to place words on the page. There is some truth here, or perhaps some horror, that is inaccessible to us — something we are not allowed to see but are allowed to see hiding. The page has a famous precedent: in Laurence Sterne’s Tristram Shandy, a black page appears after the death of one of the characters. And as in Sterne’s famous book, the effect of the page is not, paradoxically, a sense of heaviness. Or at least, not a sense of heaviness alone: some weight has been lifted and this one page, of all the pages in the book, is allowed to levitate, to unhinge the book from its subject. It accomplishes this through a lightness so dark it’s opaque, through ink so dense it’s mute.
One could say it’s the most useless page in the book.
Someone who makes a living writing ad copy or political speeches or satire might say look what I’ve done with words. All would tell you — as would certain champions of the Virtues of Literature — that language has a profound capacity to shape who we are, how we think, what we buy. I agree. Wholeheartedly. But literature? Strangely enough, those black pages in Schizophrene and Tristram Shandy might be the purest form of literature there is, even though I’ll grant you it’s an impossible and undesirable ideal.
Language as tool, language as literature: even if the two occasionally overlap, it’s unlikely any work of literature will change your life in the way that an instruction manual will teach you how to run your new microwave. Whatever use there is to literature has nothing to do with this kind of efficacy. Instrumental language reduces possibility — push this button — but when literary language touches the world, meaning proliferates.
Like Hardy before me, I may be leaving you with a melancholy picture. I wanted to say something about writing, about usefulness not correlating at all with beauty. But I find myself thinking, at the end of these thoughts, not about writing or even mathematics. My thoughts are far from Ramanujan and Hardy, from Calvino and Kapil. I’ve even forgotten the words to the Radiohead song about the pied piper that inspired me to begin this piece, a song I haven’t listened to in years but which I remember moving me through its muteness. Instead, I am thinking of the island of Takuu.
Takuu is part of Papua New Guinea, a brittle ring of sand the highest point of which is no more than one meter above sea level at high tide. Five hundred people call the place home, but the island, says the writer and cartographer Judith Schalansky, is sinking. The beach narrows with every storm. Pieces of land often disappear overnight. The older inhabitants refuse to admit what is happening. They build dykes. They pray. The young people, though more realistic about their prospects, are no more useful than the rocks and brushwood bundled in nets and cast on the shore. All day long, they “drink the juice of the coconut palm, fermented in the hot sunlight,” and though it isn’t precisely the way I spend my days, here at my end of the world, I wonder whether, given a tiny taste of that coconut juice, it could be.