# David Foster Wallace, mathematician

## He's long been celebrated for his fiction's grotesque hyperrealism, but few acknowledge its bold use of fractals

Topics: David Foster Wallace, literature, LA Review of Books, math, Infinite Jest, Fractals, Entertainment News

God has particular languages, and one of them is music and one of them is mathematics.

*— David Foster Wallace, *The Boston Globe*, 2003*

TO THE EXTENT THAT HE WAS AT HOME anywhere, David Foster Wallace was at home in the world of math. As an undergraduate, he studied modal logic; *Everything and More*, his book on infinity, explained Georg Cantor’s work on set theory to a general audience, and *Infinite Jest* includes a two-page footnote that uses the Mean Value Theorem to determine the distribution of megatonnage among players in a nuclear fallout game.

But Wallace didn’t just talk *about *math. He structured his work with it. In a 1996 *Bookworm* interview with Michael Silverblatt, Wallace explained that he modeled *Infinite Jest *after a Sierpinski Gasket, a type of fractal in which a triangle is infinitely subdivided into smaller triangles using the midpoint of its borders. Pressed by Silverblatt on why he chose such a formation, Wallace elaborated: “Its chaos is more on the surface; its bones are its beauty.”

Now, many people agree that *Infinite Jest *is a singular novel, *sui generis*, akin perhaps only to *Moby-Dick *in its originality, but the qualities that earn the book that praise — its grotesque hyperrealism, exuberant asides, and melding of academese and slang, its spikes and spurts of kindness and abjection — seem to have nothing to do with Wallace’s experimental use of fractals. Wallace’s genius lies in his guts, his encyclopedic imagination, his eyes and ears, but not, it appears, in his tricks with advanced math. And yet perhaps the fact that the casual reader remains oblivious to the Sierpinski Gasket is proof of its success. Traditional narrative structures — the Fichtean curve, Aristotle’s rising action — are designed to keep us engaged and organized, yet remain invisible; a well placed climax pops and hooks, even if we don’t notice its strategic placement. And as an organizing principle, the fractal has an intuitive logic: the best novels already have a fractionalized quality — each chapter, and indeed every paragraph and sentence, reproduce in miniature its central conflict and arc. Wallace’s comment to Silverblatt made me wonder if fractals, or some other mathematical pattern, might generate order from everyday experience without the ordinary contrivances of plot.

The more I thought about it, the more it seemed that a novel structured like a fractal would make cellular sense to a reader, even if she never consciously thought *Ah, a fractal*. Fractals occur so often in nature, in such a wide array of phenomenon, that it’s difficult to avoid the conclusion that the universe is, on some deep level, intricately patterned. Study snowflakes, mountains, coastlines, cauliflower, rivers, blood vessels, DNA, and ferns for long enough, and it is easy to believe that fractals are indeed one of God’s languages. To make a novel in the shape of a fractal, then, is to claim literature as another example of this unifying language.

Yet, is a fractal beautiful or true in the same way that a story can be beautiful or true? A story, a good one, surprises us. The interruption of expectation, the breaking of routine, is the engine of storytelling, whereas fractals, and math more generally, are concerned with patterns and the rules underlying these patterns. How, then, could math structure a tale of breaking?

Well. To solve a complex problem, it can help to start with a simple one.* Circles Disturbed: The Interplay of Mathematics and Narrative*, edited by Harvard mathematician Barry Mazur and the novelist Apostolos Doxiadis, is five hundred pages long and far from simple. But the anthology’s 15 essays — written by philosophers, historians, English professors, literary theorists, and, of course, mathematicians — conjecture so widely and freely about any and all sorts of links between literature and math that I soon became convinced that an answer to my question, or a clue to it, must fall within their confines. So I began to read and quickly realized: Wallace would have been *delighted *by this book.

Absurdities abound. Despite the subtitle *The Interplay of Mathematics and Narrative*, three of the contributors found it fit to also call their essays “Mathematics and Narrative,” and another three couldn’t resist such iterations as “Vividness in Mathematics and Narrative,” “Non-Euclidean Mathematics and Narrative,” and “Deductive Narrative and […] Mathematics.” In one essay, which casually tosses around the term “non-Euclidean culture” as if it were commonplace, there is a pagelong footnote detailing the difference between modernity (“The Enlightenment [is] sometimes given a central role”) and modernism (“primarily an aesthetic category”). Elsewhere, an android tries to talk to a ghost, and a proof is read as a Fryean quest.

The essays written by the humanities professors initially seemed to hold the most hope for answering the question posed by Wallace’s use of fractals. Rather than search for the narrative in math, they search for the math in narrative, which is exactly where Wallace’s elliptical comments on *Infinite Jest *seem to lead. But I soon found I was in the strange world of narratology, an earnest if bewildering discipline, somewhat comical in its inelegant lurchings, which often deploys the tools of mathematics — the diagram, mostly — to understand literature. Two of the three contributors come from this burgeoning field, and parsing their belabored schematic explanations of perspective and character, I saw clearly the way in which math and literature are profoundly different enterprises.

For instance, Jan Christoph Meister, in his essay “Tales of Contingency, Contingencies of Telling,” imagines a Story Generator Algorithm, or SGA, capable of passing a literature-specific Turing test. Imagining such a machine, he claims, allows literary theorists to be more precise about what, exactly, a literary character is. To that end, his essay is replete with diagrams of the SGA, each accompanied by vigorous arrows leading from “Ontology” to “Goal-Setting Interface” and on down to “Recruiter,” “Verbalizer” and other carefully labeled boxes. His definition of subjectivity is timid and hamfisted; he writes: “Filtering and constraining the flow of information by a mediating instance necessarily results in a certain normative and cognitive bias, which, to repeat, we generally interpret as a sign of subjectivity,” which makes Jane Eyre sound less like a person and more like a water pump. David Herman, in “Formal Models in Narrative Analysis,” similarly struggles to explain subjectivity. On Joyce’s *The Dead*, he has this to say:

[T]he scene outside the party becomes proximate to Gabriel’s mind’s eye through the same process of transposition that allows readers to relocate, or deictically shift, to the spatial and temporal coordinates occupied by Gabriel in the world of the narrative.

Herman concludes his essay with some suggestions about how to conduct “research” on “focalization theory” and other areas of “narrative analysis.”

This would all be funny, but tolerable, if it led to genuine insight. But, I reckon, even a half-conscious preteen in thrall to *Twilight *(never mind a first-year English PhD student) would be able to offer a better definition of perspective than Meister and Herman. Most fiction readers and writers have an intuitive sense of how narrative subjectivity operates*, *such that they can say that “The Dead” starts broadly and gradually becomes more intimate, or that *Crime and Punishment *feels more close and interior than *War and Peace. *Yes, this loose sense is unscientific, and yes, people disagree, and yes, ultimately there are no right answers. It’s mysterious. But my objection is not, or not only, a sentimental objection to the intrusion of science’s cold pinching fingers. Rather, it’s that this mathematical approach is so clearly worse than the humanistic one — more complex, but less precise. It reaches for Herculean absurdities to explain the most common thing. It flips eggs with snow shovels.

There is another problem. Applying mathematical modeling to a novel does not prove its intrinsic “interplay” with math. You can graph, model, and chart anything, but it does not follow that anything is related to math — for if that’s the case, the relation is meaningless. This triad of humanities essays, by focusing on the commentary and analysis of literature, rather than literature itself, fails to make a claim about the intrinsic*likeness *of math and narrative.

An equally dubious, yet more interesting, angle of approach pursued by *Circles Disturbed*is that of math history, with its thousands of years of stories. From there, it is a short leap to claiming, as Amir Alexander does in the book’s opening essay, that the supposedly timeless problems of algebra and geometry have always been solved in culturally specific ways. For instance, under the sway of Enlightenment discourse, 18th century mathematicians strove to bring mathematical truths into harmony with the natural world. This distinctly Aristotelian approach made mathematics subservient to physical reality; solutions were judged by their efficacy. Discoveries about infinitesimals, differentials, and other elements of calculus made during this time had almost extraordinary power to explain the natural and scientific world, yet lacked rigorous proof; as Wallace comments in *Everything and More, *“Without anyone explicitly saying so, math began to operate inductively.”

With the surge of Romanticism in the 19th century, Alexander argues, mathematicians longed to transcend the physical world, with all its limitations and imperfections. Internal logic and consistency were now valued above utility. Mathematicians such as Évariste Galois and Niels Henrik Abels manipulated triangles with less than 180 degrees and other geometric objects with no real world equivalent. This non-Euclidean geometry solved problems that had dogged mathematicians for centuries, but more importantly, it grew from deductions. Just like their poet counterparts, Romantic mathematicians had liberated themselves from the mundane and imperfect world.

Variations on the divide that separated the 19th century mathematicians from their Enlightenment predecessors resurfaces again and again in the anthology. This debate has its roots in an argument between Plato, who held that mathematical entities existed outside both our minds and the real world, in his plane of Ideal Forms, and Aristotle, who believed that while “physical hoops or rings come to be and pass away, [perfect] circularity does not […] exist as a separate intelligible entity.” In other words, Platonists believe that mathematical entities exist apart from the world they describe, but an Artistotelian holds that this abstraction is simply a crutch for our minds. Though at first this distinction seems piddling — after all, Plato and Aristotle still agree on a circle’s properties — it will turn out to matter quite a lot to math history.

For instance, as Wallace relays in *Everything and More*, an argument from Aristotle about the “actuality” of infinity delayed the invention of calculus by 1700 years:

On the one hand, Aristotle’s argument lent credence to the Greeks’ rejections of ¥ and of the “reality” of infinite series, and was a major reason why they didn’t develop what we now know as calculus. On the other hand, granting infinite quantities at least an abstract or theoritical existence allowed some Greek mathematicians to use them in techniques that were extraordinarily close to being differential and integral calculus […] But, back to the first hand, a big reason it

didtake 1700 years was the metaphysical shadowland Aristotle’s potentiality concept had banished ¥ to, which served to legitimate math’s allergy to a concept it couldn’t really ever handle, anyway.

Elsewhere in *Circles Disturbed *we learn about the 16th century Italian mathematician Rafael Bombelli, who saw that in order to solve cubic equations of irreducible case he would need to perform calculations with the square root of (-1). As a mathematical Platonist, Bombelli was undisturbed that the square roots of negative numbers do not correspond to anything in our world. He reconceptualized numbers as abstract calculating devices, rather than as quantities, and argued that since these imaginary numbers were essential for solving equations and made internal sense, they should be accepted. Irrational numbers, which upset the Greeks so terribly that Arkady Plotnitsky refers to their discovery as “the disaster of the diagonal,” required a similar faith that a number with infinite digits could indeed be legitimate.

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