Cities without landmarks
Niagara Falls, U.S./Canada
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 intrinsiclikeness of math and narrative.
An equally dubious, yet more interesting, angle of approach pursued by Circles Disturbedis 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 did take 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.
This debate over the status of mathematical entities maps rather neatly onto a debate in literary theory over the status of fictional characters, as Uri Margolin points out in his essay “A Narratological Perspective.” Neither Emma Bovary nor a perfect circle exist in the real world, yet we are reluctant to relegate them to a pure creation of mind, and so we argue: is Emma the collective creation of Flaubert and his readers, or is she the words on the page that make up Madame Bovary? Is a circle (as the Platonists would have it) an idealized, intelligible entity that exists outside our mind, or is it (as Aristotle believed) the abstraction of a ring you might put round your finger? How you feel about language, and fiction especially, might just be how you feel about math. Both, as Margolin writes, require the “activities of human minds and human expressive means (words, symbols) to become part of the world in which we ourselves exist.”
These stories from history also point to another commonality between fiction and mathematics. In the histories related by Alexander, Peter Galison, Federica La Nave, and Colin McLarty, a younger mathematician solves a longstanding problem in a way that outraged the mathematical establishment and led, in the case of Paul Gordon responding to David Hilbert, to a cry of “This is not Mathematics, it is Theology!” Yet when the innovation proved essential, the conceptual framework expanded to make room for the “theology.” In “Visions, Dreams, and Mathematics,” Barry Mazur writes, “Things become particularly interesting not when these templates fit perfectly but when they don’t, and yet, despite this, their explanatory force, their unifying force, is so intense we are impelled to reorganize the very constellation they are supposed to explain.” Mazur is interested in things that don’t fit, the mistakes and impossibilities, the interruptions in the intellectual landscape that open up new fields. His comments suggest that even if math doesn’t seem like a very good model for fiction, math history might be. The field does not progress in an orderly, logical extension of reason, but rather from jolts, from the surprises that might attract a storyteller’s attention.
The collection gains momentum when the writers turn their attention to the formal similarities between a narrative and a proof: a proof, like a narrative, is a series of events arranged in a particular sequence. But is a proof merely a device for understanding mathematical truth, or the truth itself? Platonist belief in mathematical objects as logical, idealized entities outside our minds helped spur acceptance of various mathematical innovations, such as imaginary and irrational numbers. However, to argue that a proof is both a narrative and a mathematical object, one needs the contrarian Aristotelian perspective. Otherwise, math is relegated to Plato’s timeless, idealized realm, and the sequence of steps in a proof that deliver these timeless truths become merely its humble vehicle. G. E. R. Lloyd lays the intellectual framework for the Aristotelian view by close-reading a passage from Metaphysics in which Aristotle describes the potential for meaning released by the action of a proof. According to Lloyd, the drawn figures “store” the potential to actualize, or demonstrate, these truths, but these truths are not “released” until the sequence of steps is performed. Aristotle believed that an actuality (the drawn triangle) is prior to the truth that it reveals, that the truth springs from the concrete reality. Thus, the unfolding of the proof, which moves us from a concrete, drawn figure to abstract truth, is both a narrative and the central movement, the heart, of mathematics.
By now I felt I had become more nuanced in my understanding of what “the relation between math and literature” could mean. There is the way the two are produced and understood, and the cognitive requirements of each. There is the way certain standard elements of each genre (the proof, the story) are structured. Both literature and math have a history, and both are improved by vivid detail. Some of these similarities struck me as promising, but others seemed inane. It was interesting that Wallace had written a book on math, and interesting that he claimed to have structured his masterpiece on a fractal. These biographical facts begged for interpretation. But it seemed to me, more and more, that a dozen top scholars had failed (though failed in a most interesting fashion) to help me with this task.
Enter Apostolos Doxiadis, novelist, mathematician, and author of the longest essay in the book. His severely heroic author photo, his lack of institutional affiliation, and his grand title ‘A Streetcar Named (Among Other Things) Proof’ all promised not the timidity of the small, well-fortified claim, but the thrill of bold arguments and definite answers.
Doxiadis delivers. Literature and math, he argues, share a history and a structure; they come from the same source, at least in the Western tradition — or, at the very least, literature has lent some of its most striking elements to math. To prove this, Doxiadis traces the persistence of several formal elements in Homeric storytelling to rhetoric and geometric proofs. He claims that these formal elements, first prized for their beauty, were adopted by logical and mathematical proofs as mnemonic and persuasive devices, and eventually as a structure for logical deductions.
The two formal elements to which Doxiadis devotes most of his time are the closely related chiasmus (a usually short phrase structured as a A-B-B-A, such as “truth is beauty and beauty, truth”) and ring composition (identical to the chiasmus except for the addition of a central, non-repeating element, for example, A-B-C-B-A). The earliest examples of ring composition occur in the Homeric epics, where they probably aided memorization, as well as giving aesthetic pleasure. Doxiadis shows how this structure was copied by the earliest rhetoricians in their speeches of public praise, and then copied again by rhetoricians trying to sway public opinion. The repetitive, singsong structure of a ring composition appears logiclike, important in a politician or lawyer’s speech: the listener is walked through the deductions to the central argument and then backed away again, the deductions repeated in reverse order. Doxiadis diagrams and close-reads several panegyrics and murder trial transcripts from Ancient Greece, demonstrating that the speeches are indeed structured as ring compositions, and argues that their use persisted across genres as their efficacy in persuasion, as well as inherent beauty, was discovered and exploited.
As the Ancient Greeks slowly adapted the legal proof for the mathematical one, instances of chiasmus and ring composition continued to appear. In a Greek trial, the two opposing sides traditionally gave a highly structured speech with a prologue, narration, proof proper, and epilogue. This format corresponds exactly with the later structure of a mathematical proof, which included a statement of aims, manipulation of form (the “narration,” as Lloyd argued), proof proper, and restatement of aims. This structure, roughly chiasmuslike, is replicated on a micro or sentence level in both the legal and mathematical proof. The poetry of the Iliad and the Odyssey, their hypnotic rhythm — Doxiadis offers, among many examples: “Let us eat / For even Niobe ate / This was her story / She ate / So let us also eat” — was borrowed and transmuted into the structure of logic and proof.
The persistence of chiasmus and ring composition across genres could be viewed simply as an interesting example of “exaptation,” as Doxiadis initially characterizes it, an instance of one field coopting another’s style for its own purposes. Doxiadis, in his conclusion, doesn’t claim much more than this, saying only that he has hoped to provide an account of the “cultural transformation from narrative roots to their logical progeny,” in hopes of illuminating their “structural similarities.” But might these similarities point to something more profound? Perhaps the aims of art and math are not as far apart as they appear. Both hope to persuade their audience of their truth. Both strive to illuminate and provoke what Bernard Tessier calls a “resonance […] between our conscious thought and the structure of the world.” Perhaps, then, David Foster Wallace’s use of fractals was a kind of reverse streetcar, in which literature borrows a mathematical and natural structure for its own aims. Doxiadis’ conclusions imply that it is not so strange for literature to borrow from math. Apparently, math has been borrowing from literature for a long time.
Jan Christoph Meister, one of the narratologists, concludes his essay with the words of Ludwig Wittgenstein, who believed that “mathematics is a purely self-referential language game” and “‘mathematical truth’ is essentially nonreferential and purely syntactical in nature.” As anyone familiar with The Broom of the System knows, Wittgenstein was an important influence on Wallace, who viewed the philosopher’s work with fascination, but trepidation. The view that language does not exist beyond one’s head could so easily elide into the conviction that life was meaningless.
It is significant, then, that Wallace sided with the Platonists, with those who believe that math and language are not mere solipsism. In the concluding pages of Everything and More, Wallace writes, “Gödel and Cantor both died in confinement, bequeathing a world with no finite circumference. One that spins, now, in a new kind of all-formal Void. Mathematics continues to get out of bed.” This Beckettian formation of the “all-formal Void” begetting the need to “get out of bed” is an almost paradoxical call for persistence in the face of misunderstood nothingness. Wallace was drawn to paradoxes, to infinite recursions and eternal loops. He was drawn to fractals — shapes infinitely self-similar, “objects and concepts at the very farthest reaches of abstractions, things we literally cannot imagine.” But it mattered to him that these impossibilities are real, as Platonists believe, because he wanted a reason to get out of bed in the morning and figure them out. Structuring his masterpiece as a fractal was a hopeful act, a prayer, that fiction might also live outside our minds.
And, yet, after reading an early draft, Michael Pietsch, Wallace’s editor at Little, Brown, convinced the author to abandon his design. The universe of Infinite Jest, one imagines, had surpassed the intricacy of its structuring fractals; the rhythm of story and character had started to generate its own logic. This universe, it turns out, was proof enough.
Niagara Falls, U.S./Canada
Sydney Opera House, Sydney, Australia
Mount Rushmore, South Dakota, U.S.
Eiffel Tower, Paris, France
Colosseum, Rome, Italy
Taj Mahal, Agra, India
Siena Cathedral, Siena, Italy
Christ the Redeemer, Rio de Janeiro, Brazil
Arc de Triomphe, Paris, France
Lost City of Petra, Jordan