Masahiko Fujiwara is a mathematician and essayist. Born in Manchuria and raised in Tokyo, Japan, he obtained his PhD in Mathematics at Tokyo University. After stints at the University of Colorado and at Cambridge University, he became a full professor at Ochanomizu University, Tokyo. In 2010, he became Professor Emeritus, post-retirement. Fujiwara won the Nihon Essayist Club Prize for his 1977 debut Wakaki Sugakusha no America (Days of a Young Mathematician in America) and has since followed up with many publications, including the 2005 best-seller Kokka no Hinkaku (The Dignity of the Nation), of which more than 2 million copies have been sold in Japan. His latest writings in Kanken Mogo (Within my scope) now appear serially in the weekly magazine Shukan Shincho. His parents are Naoki Prize-winning novelist Jiro Nitta and best-selling author Tei Fujiwara.
Sayuri Okamoto is a contributing editor at Asymptote. She holds degrees in art history and Japanese literature and works as a translator, writer, and curator. Born and raised in Shizuoka, Japan, she is currently living and working in London (UK) and Padua (Italy). In 2014, she received a PEN/Heim Translation Fund grant for her translation of Gozo Yoshimasu's "post-3.11" poetry series Dear Monster.
In 1998, the New York Times wrote about a performance by the Fort Worth Ballet, singling out a male dancer for “his surprisingly fluid strength” while lamenting his lack of “soaring leaps” and “lively pirouettes” in a challenging routine that included Balanchine’s “Firebird.” If that seems like tepid praise, consider that the dancer was Herschel Walker, then a running back for the Dallas Cowboys. Walker had studied ballet at the University of Georgia, and while what he learned in the dance studio cannot alone account for his eight thousand two hundred and twenty-five career rushing yards, it surely fooled a linebacker or two. Nor is Walker the only football player to have seriously studied ballet: the Hall of Famer Lynn Swann is the subject of an NFL Films featurette titled “Baryshnikov in Cleats.”
What ballet is to football players, mathematics is to writers, a discipline so beguiling and foreign, so close to a taboo, that it actually attracts a few intrepid souls by virtue of its impregnability. The few writers who have ventured headlong into high-level mathematics—Lewis Carroll, Thomas Pynchon, David Foster Wallace—have been among our most inventive in both the sentences they construct and the stories they create.
As anyone who has taken a standardized test in the last half-century knows, math and “language arts” run on parallel tracks for much of one’s school career. Both begin with an emphasis on rote memorization of the basics: sentence diagrams, multiplication tables. Later, though, both disciplines become more heady: English class discards grammar in favor of the ideas lurking beneath textual surfaces, while math leaves off earthbound algebra, soaring along the ranges of calculus.
By the time you’re old enough to drive, you’ve likely decided which region of the brain you plan to use in your adult life, and which you want nothing to do with beyond the minimum requirements imposed by modern society. Long gone are the days of the catholic scholar who could quote both Pindar and Newton with ease. As the Cambridge mathematician G. H. Hardy noted in 1940’s “A Mathematician’s Apology,” perhaps the most eloquent defense of the subject on its aesthetic merits, “most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity.”
Poets have been more conversant with mathematics than fiction writers, probably because they have to pay attention to the numerical qualities of words when working in meter, forced to consider the form and even physical shape of what they write, not just its meaning. Wordsworth praised “poetry and geometric truth” for “their high privilege of lasting life,” while Edna St. Vincent Millay remarked that “Euclid alone has looked on beauty bare.”
Fiction writers have rarely expressed such earnest appreciation for mathematical aesthetics. That’s a shame, because mathematical precision and imagination can be a salve to a literature that is drowning in vagueness of language and theme. “The laws of prose writing are as immutable as those of flight, of mathematics, of physics,” Ernest Hemingway wrote to Maxwell Perkins, in 1945. Even if Papa never had much formal training in mathematics, he understood it as a discipline in which problems are solved through a sort of plodding ingenuity. The very best passages of Hemingway have the mathematical complexity of a fractal: a seemingly simple formula that, in its recurrence, causes slight but crucial changes over time. Take, for example, the famous retreat from Caporetto in “A Farewell to Arms”:
When daylight came the storm was still blowing but the snow had stopped. It had melted as it fell on the wet ground and now it was raining again. There was another attack just after daylight but it was unsuccessful. We expected an attack all day but it did not come until the sun was going down. The bombardment started to the south below the long wooded ridge where the Austrian guns were concentrated. We expected a bombardment but it did not come. Guns were firing from the field behind the village and the shells, going away, had a comfortable sound.
The procession here has an algebraic deliberateness, but that simplicity gives way to a complexity of meaning. Hemingway starts with the material (snow, wet, daylight, sun) only to end with the unexpected and intimate “comfortable sound” of the receding Austrian guns—a revelatory bit of naiveté on Frederic Henry’s part. Everything in this passage is intentional, from the plain imagery to the heightening of narrative urgency that comes with the repetition of “we expected.”
Hardy hints upon this, too: “A mathematician, like a painter or a poet, is a maker of patterns…. The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test.” But fiction that does nothing but follow rules is cold arithmetic, no matter how beautiful it is. And there are, indeed, many “craftsmen” today who can write what reviewers call lapidary prose, but who don’t come even close to wielding the axes that shatter the frozen seas inside us. That Hemingway paragraph would be inert artifice were it not for the “comfortable sound” that captures the impossible yearning of soldiers about to be sent either to slaughter or the gray twilight of retreat. The objective observation that begins the paragraph flowers into an ironic condemnation of war.
As the mathematician Terence Tao has written, math study has three stages: the “pre-rigorous,” in which basic rules are learned, the theoretical “rigorous” stage, and, last and most intriguing, “the post-rigorous,” in which intuition suddenly starts to play a part. As Tao notes, “It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark.”
In literature, that big picture means you have to extrapolate to people who are not yourself, which can be a risk as great as the potential reward—as, for example, William Styron found out when he tried to write from the voice of the rebel slave Nat Turner, quickly discovering himself branded a racist. The postmodernism of the late twentieth century, for all its excesses, at least understood that the world was now far too much with us, and fiction must tune into the frequencies of the age. David Foster Wallace came as close as anyone in the last half-century to finding that universal frequency. Wallace may have led a largely hermetic existence, and his novels aren’t exactly supermarket thrillers, but his fiction was obsessively concerned with the gulf between our small and discrete selves and the world at large.
A serious student of math in his youth, he had switched to philosophy by high school because high-level calculus did not provide the “click” of enlightenment, as D. T. Max describes in his new biography of Wallace, “Every Love Story Is a Ghost Story.” But even though Wallace may have foresworn mathematics, he never cast off the broadening spirit of the discipline. If Hemingway’s writing is algebraic in its precision, then Wallace’s is quantum calculus, a discipline that taxes the imagination by asking us to conceptualize things we cannot see, like the the way a function shows change through space and time. We must employ our private intellects to conceive forms that are, as Plato would have it, both timeless and universal. Not bad work if you can get it.
In a 2000 review of two mathematical novels in the magazine Science, Wallace wrote about “the particular blend of reason and ecstatic creativity that characterized what is best about the human mind”: “Just about anyone lucky enough ever to have studied higher math understands what a pity it is that most students never pursue the subject past its introductory levels and therefore know only the dry and brutal problem solving of Calc I or Intro Stats…. Those who’ve been privileged (or forced) to study it understand that the practice of higher mathematics is, in fact, ‘an art’ and that it depends no less than other arts on inspiration, courage, toil, etc.”
Courage is not a word I’d use to describe a lot of today’s fiction. Writing, M.F.A. students are often told, is a messy exploration of the self. The result can be a suffocating narcissism, a lack of interest in the kind of extrapolation and exploration that is necessary to both mathematics and literature. In his landmark 1921 essay “Tradition and the Individual Talent,” T. S. Eliot wrote, “What happens is a continual surrender of himself as he is at the moment to something which is more valuable. The progress of an artist is a continual self-sacrifice, a continual extinction of personality…. It is in this depersonalization that art may be said to approach the condition of science.” He went to compare the mind of an artist to a crucible in which a chemical reaction takes place.
Presently, we have become too enthralled by the notion of literature as Jackson Pollock action painting, the id flung with violence upon the canvas. The most lasting fiction has both the supremely balanced palette of Rothko and the grandeur of his themes. All this may seem like I am urging for a literature that is cold and scientific, subjecting itself to the rigors of an alien discipline. That isn’t so. I am pleading, instead, that fiction think more deeply and determinedly about how it is to be composed and what it is to say—that is the best gift mathematics could give us.
“I am interested in mathematics only as a creative art,” Hardy, the Cambridge mathematician, wrote. He meant creative in the most literal sense, contrasting serious mathematical inquiry with chess. The latter, too, requires great intelligence, but it resolves nothing of the human condition. The same distinction exists in fiction, between the diverting and the serious, the trivial and the universal. In both cases, too, formulas are but guideposts that fall away the higher you climb. In the end, you are left alone with your own variables, your own private equations.
Illustration by Istvan Banyai.