bonus chapter 5

An Automorphic Reading of Thomas Pynchon’s Against the Day (Interrupted by Elliptical Reflections on Mason & Dixon)

All mathematics leads, doesn’t it, sooner or later, to some kind of human suffering.

—Against the Day, p. 541

Thomas Pynchon, postmodern author, is commonly said to have a nonlinear narrative style. Inger H. Dalsgaard suggests that “a novel like Against the Day may be read in non-linear fashion, in keeping with the operations of a time machine.”1 No critic, however—not even the “seventeen of the foremost heavyweights from over forty years of Pynchon criticism”² who contributed to the Cambridge Companion to Thomas Pynchon—seems to have taken seriously the possibility, to be explored in this chapter, that his narrative style might in fact be quadratic.

Google gives no matches whatsoever for “quadratic narrative style,”³ and this hypothesis—more precisely, that Pynchon’s major novels are structured by conic sections, at a rate of roughly one per book—sheds no light on the deeper import of his writing. Just as with an “obvious” proof in mathematics, however, once you’ve seen the thesis, it’s impossible to stop seeing it.

This “insight” came to me instead of a solution to another puzzle that had been troubling me since I started reading Against the Day in the spring of 2008. The prominence of mathematics in this book is exceptional even for Pynchon. Two of the main characters are at least part-time mathematicians; Hilbert, Minkowski, and Gibbs make cameo appearances; several chapters are set in the Göttingen mathematics department; and among Pynchon’s signature silly ditties there is this romantic number:

Her idea of banter

Likely isn’t Cantor

Nor is she apt to murmur low

Axioms of Zermelo,

She’s been kissed by geniuses,

Amateur Frobeniuses

One by one in swank array,

Bright as any Poincaré …

and so on in that vein.

It was when I came upon the word automorphic on page 409,

Earth making its automorphic way round the sun again and yet again …,

that I began to wonder what was going on—and then on pages 452 and 453, there it was again:

periodic functions, and their generalized form, automorphic functions

as a prelude to a scholarly discussion of time travel:

Time no longer ‘passes,’ with a linear velocity, but ‘returns,’ with an angular one. All is ruled by the Automorphic Dispensation. We are returned to ourselves eternally, or, if you like, timelessly.

Between the two mentions of automorphic is a scene reminiscent of Odysseus’ voyage to the river Styx, in which one of the Chums of Chance crosses a recognizably non-Euclidean landscape:

… the more “respectable” parts of town … at each step were receding, it strangely seemed, disproportionately farther as the young men went on.

The puzzle was: did all this technical and, for the most part, legitimate mathematics serve as mere atmospheric accompaniment to the turn of the century’s hesitant exploration of the relations between time, space, and light? So that reviewers could write, with Luc Sante in the New York Review of Books:

My own eyelids drooped when the subject was mathematics, for example, but that is something I am profoundly ignorant about …

and still feel qualified to shower the book with unstinting praise? Or, for that matter, to write, like Louis Menand in The New Yorker:

I can’t do the math, but I think that the idea behind “Against the Day” is something like this … [continuing with an elaboration of what I just wrote above],

but concluding in disappointment that “Pynchon must have set out to make his readers dizzy and, in the process, become a little dizzy himself.”4

Or was all this mathematics there for a reason integral to the structure of the book, inaccessible, perhaps by design, to “profoundly ignorant” reviewers who “can’t do the math”? Perhaps I was missing a cryptic message intended expressly for people on familiar terms with the word “automorphic,” for participants in the Langlands program—for people exactly like me, in other words….

I’d better stop here to reassure the reader that I have not lost my marbles. Paranoia is one of Pynchon’s favorite topoi, and although this is very much a self-referential paranoia—as when. in The Crying of Lot 49, the secret network of communication Oedipa Maas believes may have been created for her benefit was indeed created by the author himself—Pynchon’s reclusiveness combined with his choice of theme must make the interpretation of his books a magnet for all sorts of cranks as well as genuine paranoids. So I repeat that what impels me to write is not the belief that I have somehow penetrated “the bright, flowerlike heart of a perfect hyper-hyperboloid,” to quote a passage from the last page of Against the Day, to which I will soon return but that here can stand for the book itself or for the entirety of Pynchon’s work. Whatever I’ve seen is just something I can’t help noticing, meaningless or not.

Returning to my narrative, it was not reassuring to discover that Christophe Claro, who had recently translated Against the Day into French to considerable acclaim in the literate press, chose to render automorphic as “automorphique,” as in “fonctions automorphiques,” instead of the correct term. “automorphe,” although he was supposedly indirectly in communication with Pynchon himself.⁵ If Pynchon uses mathematics as background music, an imprecise translation makes no difference; but if it serves a structural purpose, the choice of word may be very important.

I’m afraid I can’t solve the puzzle. But I can say that when you start to look, you find an awful lot of hyperbolas in Against the Day. For example: the hyperbolic geometry to which I alluded in connection with automorphic functions; the “Automorphic Dispensation,” which seems to be a “function … by which, almost as a by-product, ordinary Euclidean space is transformed to Lobachevskian” (p. 453); and that “perfect hyper-hyperboloid” that “only Miles” Blundell, the one character to have apprehended the meaning of space-time, “can see in its entirety.” There are (hyperbolic) wave equations (and a whole family of Vibes) and the “noted Quaternionist V. Ganesh Rao of Calcutta University” who, by rotating himself in an imaginary direction, performs something “like reincarnation on a budget, without the element of karma to worry about” (pp. 130, 539).

I remembered having understood Pynchon’s V. as the convergence, V-like, of two narrative lines.6 Gravity’s Rainbow, obviously dominated by the image of the parabola and full of explicit references to the shape, such as:

He had noted this parabola shape around on Autobahn overpasses, sports stadiums u.s.w. and thought it was the most contemporary thing he’d ever seen. Imagine his astonishment on finding that the parabola was also the shape of the path intended for the rocket through space (p. 298).

has also been compared to a parabola in its narrative structure, not least by Salman Rushdie.⁷ The obvious guess was that Mason & Dixon, which I hadn’t read, would turn out to have been written under the sign of the ellipse. And sure enough, here’s what I found on page 555 of M&D:

In a slowly rotating Loop, or if you like, Vortex, of eleven days, tangent to the Linear Path of what we imagine as Ordinary Time, but repeating itself,—without end.

M&D’s main characters are Astronomers; there are orreries and orbits “as elegant as Kepler’s,” and then this vision, near the end of the book, when the explorers are at the point of being turned back:

“In the Forest … ev’ryone comes ‘round in a Circle sooner or later. One day, your foot comes down in your own shit. There, as the Indians say, is the first Step upon the Trail to Wisdom.”

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

The infinite boredom of conic sections … their calm and tantalizing respectability …

—F. Scott Fitzgerald, This Side of Paradise

But does M&D have an elliptical or even merely circular structure? You can find plenty of ellipses, not to mention circles, in all Pynchon’s novels, as well as the ellipses that look like this…. For that matter, you can find circles in practically every story ever recorded, starting with Gilgamesh. Besides, what better literary representation of a double line (figure β.8) than the border between Pennsylvania and Maryland, mapped by Mason AND Dixon?⁸

Of course I decided I had no choice but to read M&D, bitterly regretting I had not done so ten years sooner and, of course, unlike the book’s eponymous heroes, finding exactly what I was seeking, specifically ellipses of all shapes and sizes: the word Ellipse to start with, or rather to end with, since it occurs twice and rather superfluously in close succession in chapters 75 and 76.

M&D comes closest of all Pynchon’s novels to the traditional sense of closure. The account of the Mason-Dixon expedition is neatly sandwiched between introductory and concluding sections, as it is enclosed comfortably within the story told by the Rev^d Cherrycoke to his family circle, itself reflected in the thematically overdetermined “Mirror in an inscrib’d Frame” that appears on the novel’s second page. The main narrative is, in turn, studded like a Fruit-Cake with digressions: side trips to New York and Virginia, a tale of a Tub, and a chapter devoted to the mechanickal Duck’s love story and another one on the fairy tale of the Court Astronomers Hsi and Ho…. The Ghastly Fop episode that interrupts the novel without warning in chapters 53 and 54 looks like an exception: far from being self-contained, its unexpected fusion with Cherrycoke’s narrative leads directly to this dialogue between Captain Zhang and Dixon:

“We happen to be the principal Personae here, not you two! Nor has your

Line any Primacy in this, being rather a Stage-Setting … ”

“And Mason and I,—”

“Bystanders. Background. Stage-Managers of that perilous Flux,—little more.”

“Eeh.” Dixon thinks about it. “Well it’s no worse than Copernicus, is it …?”

This blurring of multiple levels of fiction may be a pinnacle of novelistic self-referentiality, but it may also be a sly reminder of the simple geometric fact that an ellipse has not one but two foci….9 And perhaps we should understand the double Mason-Dixon line as what remains of an ellipse when its two foci, in this case the two main characters, are sent to infinity in opposite directions.

In my initial sketch for this chapter, written in 2008, I set out to write down unquiet thoughts about automorphy in Against the Day that came to me only some time after I had put the book aside. Having turned my attention to Mason & Dixon with a Thesis to defend or discard, it’s not surprising that my reading turned up a great deal more in the way of material. I just mention in passing

—the “Geometry more permissive than Euclid,” apparently spherical rather than hyperbolic, in chapter 33,

—the Möbius smoke ring in chapter 34,

—and the ruminations on Time, the “Space that may not be seen,” the “true River than runs ‘round Hell” in chapters 32–34.

Time is undoubtedly of thematic import in M&D, and this particular sequence of attempts to circumscribe Time is initiated by the swallowing, by a character known only as R.C., of a Watch that operates by Perpetual-Motion. Having completed his reflections on the consequences of swallowing Time, Pynchon opens chapter 35 with the Cherrycoke’s family’s most heated, and most quoted, argument, on the relation between History, Truth, and ‘Novel.’ Philadelphia lawyer Ives LeSpark puts it this way:

Time on Earth is too precious. No one has time, for more than one Version of the Truth.

For the author, swallowing Time amounts to enclosing it in a novelistic structure. And Time, in the form of the watch inside R. C. who is buried in a tomb in the middle of a story-within-Cherrycoke’s-story-within-Pynchon’s-story at the heart (or should I say focus?) of the book, is the cosmic principle around which all the plot’s planets revolve.

Most suggestive of the Mason & Dixon’s cyclical structure is the prediction in its very last line:

We’ll fish there. And you too.

Addressed to Mason by his two older sons, after (or as?) they “ensign their Father into his Death,” the last three words lose none of their poignancy if they are taken literally as a prediction, which becomes true only on the condition that the narrative is meant to recommence at that point from the beginning. As it is a convenient way to reconcile Mason’s abrupt decision near the end of his life to return with his family to Pennsylvania with this summary of his life, made shortly before Dixon’s death:

To leave home, to dare the global waters strange and deep, consort with the highest Men of Science, and at the end return to exactly the same place, us’d,—broken….

Gluing the novel’s front and back together in this way is also the only way to validate what the talking “British Dog” promises Mason and Dixon at the end of the penultimate chapter:

The next time you are together, so shall I be, with you.

The words Mason speaks “the next time,” back at the front of the book in chapter 3, could well serve as my motto for this chapter:

Isn’t it worth looking ridiculous, at least to investigate this English Dog, for its obvious bearing upon Metempsychosis….

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Determining how, if at all, Against the Day’s narrative structure is hyperbolic is more challenging, but here are some thoughts. As a hyperbola has two connected components—bilocation?—one would expect Against the Day to have two nonoverlapping narrative arcs. So it is significant that the Chums of Chance and the main characters of the Traverse family narrative never meet. The Chums open the novel with a landing of the airship Inconvenience and close it with the same airship returning to the sky, to “fly toward grace.” The Traverses naturally spend much of their time underground in mines or tunnels or underwater in a submarine or, in one case, a torpedo. The two arcs do come very close in three successive chapters set in the Low Countries—in Oostende, to be precise, exactly in the middle of the book.¹⁰

Legitimate Pynchon scholars have also noted the presence of two narrative arcs. Nina Engelhardt, one of the rare professional readers to take Pynchon’s mathematics seriously, has made the ingenious suggestion that the two narratives echo Against the Day’s frequent references to complex numbers and quaternions and their real and imaginary parts. The Chums, recurring heroes of a series of adventure books, live—like the square root of −1—on an imaginary axis, while the Traverse family and their companions traverse the all-too-real landscape of war and class struggle. Webb Traverse is “murdered by men whose allegiance […] was to that real axis and nothing beyond it” (p. 759), while his son Reef encounters the Chums in his imagination, reading one of their books; and his future companion Yashmeen meets them in her dreams. This is appealing and also makes sense—not least because it’s so easy to extend the name (0, 0) of the meeting point of the real and imaginary coordinate axes to spell out Oostende. The hyperbolic and real/imaginary readings are no less mutually exclusive than two strands in one of Finnegan’s Wake’s multilingual puns—and remember that Joyce boasted that he had “put in [Ulysses] so many enigmas and puzzles that it will keep the professors busy for centuries arguing over what I meant….”11

Against the Day is filled out by a host of secondary characters who bounce or vibrate from one narrative strand to the other—mostly more or less mad scientists (Heino Vanderjuice, Merle Rideout, V. Ganesh Rao, Roswell Bounce) obsessed with time travel and quaternions, but also including the detective Lew Basnight and (at least tangentially) members of the Vibe clan. The plot is beginning to look like a hyperbola whose two arcs are joined by a sinusoidal curve whose graph one might hope to find in a mathematical analysis of double refraction in calcite (Iceland spar, the title of the second part of the book). But this is too simplistic. Calcite is merely uniaxial birefringent; for hyperbolic interference patterns you have to look at biaxial birefringent minerals, as in figure 5.1.

I leave these speculations to a later version of this text that no one may ever need to write. But let me insist again that the hypothesis would be no less frivolous if it turned out to be in some sense correct. Pynchon is such a relentlessly playful writer—I would have said relentlessly silly, except that Freud made the point that play (unlike silliness, perhaps) is not opposed to seriousness—that frivolousness must often be part of his design. Choosing to structure his successive novels by conic sections, if that is in fact what Pynchon did, may have been a private joke, just like the recurring trope of entropy in his work may be an extended allusion to C. P. Snow’s comparison of ignorance of the second law of thermodynamics to not having read Shakespeare. Thanks largely to Pynchon,¹² it’s hard to find a critic ignorant of the second law of thermodynamics, though if Sante and Menand are typical, “don’t know much trigonometry” is still a popular refrain among the literary Elect. Or, entropy may have been an arbitrary (dis)organizing principle, a disciplinary constraint to guarantee the nonlinearity of his narratives—or maybe a little of both.

Figure 5.1. http://www.uwgb.edu/dutchs/Petrology/intfig1 htm.

Even if you’re willing to grant this interpretation, you may well ask, so what? Is Pynchon deliberately broadcasting conspicuously futile signals across the Two-Culture divide? Is his goal to delimit a bounded zone of reference from which the habitual insiders are excluded but which, in any case, can be mined for depthless interpretation, at best? Does it matter, for example, that the initials R.C. in the story of the perpetual motion watch clearly allude to the “Ticking Stone” of George Alfred Townsend’s Tales of the Chesapeake, which features Mason and Dixon as minor characters—also known as the Ticking Tomb, a tourist attraction in Landenberg, Pennsylvania? Are Pynchon’s conic sections like the Figure in the Carpet in Henry James’s novella?

[T]he thing without the effort to achieve which [the writer] wouldn’t write at all, the very passion of his passion, the part of the business in which, for him, the flame of art burns most intensely? … It stretches, this little trick of mine, from book to book, and everything else, comparatively, plays over the surface of it…. I live almost to see if [the secret] will ever be detected.

To answer questions like these you would have to consult a licensed Pynchonian—someone like Inger Dalsgaard, for example, who wrote:

[L]iterary scholars naturally tend to look less at what natural sciences described in fiction say themselves about the real world or reality than at what fictional texts or literary strategies which include or mirror scientific fields, theories or methods say about the world.13

Or like Michael Harris, professor of English at Central College, Iowa, whose article “Pynchon’s Postcoloniality” sees M&D as a “meditation on lines, boundaries, borders, margins [but not ellipses, MH¹⁴], and their consequences,” quoting Captain Zhang to the effect that drawing a straight “Line upon the Earth is to inflict … a long, perfect scar … hateful Assault.” Though his essay aimed to “reinscribe Pynchon as, at least in part, a political writer”—the mapping of the Mason-Dixon line is a clear allegory for the colonial conquest of America and prefigures the subsequent drawing of the closed curves of reservations—he adds that, with Pynchon, “almost nothing can be taken literally.”¹⁵ What does quadratic narrative form say about the world? Does a postcolonial border run between the Two Cultures?

I don’t know how to answer those questions. Nor have I figured out how or whether Pynchon’s other novels fit in this tableau of conic sections. By a process of elimination, it would make sense to associate Vineland with the circle, which is what you get when the two foci of an ellipse coincide. But apart from the name of the main character (Zoyd Wheeler) and a throwaway allusion to the Arctic Circle Drive-In, I’ve found nothing more conclusively circular in Vineland than in any of the other novels. It’s true that the circle is the most visibly symmetric of the conic sections; and group theory, the branch of mathematics concerned with symmetry, figures prominently in Vineland, through the saga of Weed Atman, “a certain mathematics professor, neither charismatic nor even personable” at College of the Surf in Trasero County, California. The magic of marijuana transformed this absent-minded group theorist to the center of a “moving circle of focused attention” of a cluster of groupies and the leader by consensus of “The People’s Republic of Rock and Roll,” a short-lived utopian community—a self-governing “relaxed field”—smashed by the violence of the state. Gunned down by a comrade as a result of the machinations of the supremely sinister FBI agent Brock Vond, Atman becomes a symbol of nostalgia for an unfinished revolution—just like Galois, popularly considered the inventor of group theory.

Shortly after Vineland appeared, Joseph W. Slade published an analysis in Critique in which he lists parallels—too many to be coincidental—between the details of Atman’s story and popular accounts of Galois’ life. “What probably attracts Pynchon to group theory,” he speculates, “is that it can support contradictory schemes of reality.”16 Slade has produced another mathematical reading of a Pynchon novel, one that again elicits the reply so what?—from Slade himself, who warns that “almost any work of fiction is subject to analysis by group theory.” One more thing, though: in Pynchon’s major novels, much attention is paid to the movement of objects through space. Newton’s law of universal gravitation, one of the oldest of all differential equations, tells us that the motion of an object once launched from the earth is determined forever—predestined, one might say, in keeping with the Calvinistic spirit of Gravity’s Rainbow—by its initial conditions, in this case the starting point and its initial velocity. If a projectile is launched at a speed below escape velocity, it will fall to the earth, like a V-2 rocket, its path tracing a parabolic arc. If launched precisely at escape velocity, earth’s gravity will restrain its motion to an elliptic orbit. But if it attains and exceeds escape velocity, it will recede endlessly into space along a hyperbolic trajectory. At the end of the Chums of Chance’s journey in Against the Day,

it is no longer a matter of gravity [for the Inconvenience]—it is an acceptance of sky.

There aren’t any more conic sections, except for the most degenerate of all, the single point, a possible plan for a novel in which absolutely nothing happens. On the other hand, there’s literally no limit to what Pynchon might do with cubic narrative structures….