14
MATHEMATICS

Brian Rotman

Literature and mathematics are ancient arenas of human imagination. Viewed as systems of written signs, they are almost alien to each other. The signs of literature are those of speech, their medium (in the West) an alphabet and a dozen punctuation marks. Mathematical signs are those of invented thought, their medium an unlimited menagerie of written symbols and diagrams. Literature concerns thoughts, passions, and actions of persons describable in language; mathematics concerns symbolically notated virtual objects, actions, and relations detached from persons thinking them. Of course, mathematics, like anything else in the world, can be spoken about and enter literature as content, as a topic in a novel, a play, a poem, or a philosophical treatise. Mathematics can also, less obviously and more interestingly, enter and impinge on literature through its form. This is because mathematics, though classified as a science, is equally an art. In this it differs radically from the other sciences, all of which are defined by their relation to the physical world through experiment and prediction. True, many mathematical ideas are ideal versions of physical phenomena – hence the utility of the subject for the sciences. Nevertheless, nothing empirical can enter into a mathematical argument. No physical fact can ever prove or refute the truth of a mathematical theorem.

This freedom from empirical reality, its virtually unconstrained powers of imagination, is what enables mathematics to be an art, and what in turn allows aspects of itself – its structures, styles of thought, ambience, preoccupations, methods, aesthetics – to impact other art forms, such as music, architecture, and literature. “We all believe mathematics is an art,” declares Emil Artin (Sinclair et al. 2006: 21), voicing a credo shared by a majority of his fellow mathematicians. Certainly, mathematicians invent fictive universes, employ metaphor, metonymy, and similitude and create narratives in imaginary worlds every bit as multilayered and complex as those of literature. And, as with any art, aesthetic considerations are central. “Beauty,” the mathematician G.H. Hardy insists, “is the first rule. There is no permanent place in the world for ugly mathematics” (Hardy 1940: 85). Equations, formulas, theorems, proofs, and even definitions have affect. Proofs, for example, can be beautiful for their elegance (economy of means), ingenuity (unusual or surprising twist), insight (revelation of why something is true), as well as for their generality, sheer heuristic power, fecundity, and conceptual depth.

The elevated status of mathematics in the West has its roots in Greek philosophical thought. For Plato, who taught that “Geometry will draw the soul toward truth and create the spirit of philosophy” (Plato 2000: VII, 52), mathematical truths were the nearest things to the pure, unchanging ideal forms that lay behind the world of everyday reality. Before him, the Pythagoreans had proclaimed that the entire universe was constructed out of numbers, an understanding echoed at the dawn of modern science (and frequently since) by Galileo, for whom “the grand book” of the universe “was written in the language of mathematics” (Galileo 1953: 183).

From the beginning, then, mathematics has been understood as both the language of the physical world and the model of pure thought – a duality perfectly brought together by Euclid in his Elements. Starting from self-evident truths – axioms and postulates – about circles, lines, and points, and using only strict logical reasoning, Euclid showed how one could deduce all the known truths of geometry. For two millennia Euclid’s Elements was the paradigm of rigorous, systematic thought that philosophical literature might aspire to. In his magnum opus, Ethics of 1677, starting from axioms and using strict deductive reasoning, Baruch Spinoza sought to do for the sphere of human ethics what Euclid accomplished for the geometry of the plane (Spinoza 1985). Three centuries later Ludwig Wittgenstein, in his celebrated Tractatus (Wittgenstein 1922), strove, through numerically ranked logical propositions, to arrive at what can be said (ultimately not said) about the underlying logical form of a proposition.

For philosophical tracts, Euclid’s method offers a narrative and rhetorical structure. But in the case of literature, even a single number can act as a structuring principle. Such is the case with the number three and Dante Alighieri’s Divine Comedy. Observe first the deep-lying association between three and Christianity, a religion whose defining mystery springs from a triad formed by interposing a third mediating figure of the man-God Jesus Christ between the Judaic poles of earthly Man and heavenly God. This, in turn, begets a trisection of the one God into the holy trinity of Father, Son, and Holy Ghost. A further, much later triad, incorporated into Christianity in the century of Dante’s birth, introduces Purgatory – a dwelling place for souls not destined for Hell but yet unready for Heaven. It is this last that organizes Dante’s poem into its three sections narrating the journey through the regions of the Inferno, Purgatory, and Paradise. Each of these thirds is divided into 33 cantos (with an introductory canto making 100 in all). More fundamentally, each canto consists of three-line tercets written in the terza rima form introduced by Dante; an interlocking rhyme scheme, aba, bcb, cdc, ded, etc., of line endings which injects a pulse of three – two steps forward and one back – across each tercet of Dante’s poem, a scheme that propels the Comedy forward through a series of linked meditative recoils.

A very different focus on numbering and a single number is at the heart of William Shakespeare’s King Lear (Rotman 1991: 78–86). If the Christian tropism to three organizes the narrative and poetic structure of the Divine Comedy, the arithmetic of capitalism and the number zero constitute a symbolic armature of Shakespeare’s play. The play’s action is initiated when Lear’s daughter Cordelia utters “nothing” when asked to measure her love for him as her elder sisters had done. “Nothing will come of nothing, speak again,” Lear demands. Out of this “nothing” much that is anything but nothing – murder, treachery, torture, and the destruction of a kingdom – will unfold. And its vehicle will be unnatural measurement, quantification, a relentless calculating, counting, and enumerating that will reduce human relations to acquisition. All will be captured within a buying and selling of loyalty, status, inheritance and, above all, the natural bonds of love, which reflects the disruptive entry of contemporary mercantile capitalism and its system of numbers based on zero into Elizabethan society. Lear’s entourage – bargained down by his daughters from 100, to 50, 25, 10, to 1, at which point he is asked, “What need one?”–is reduced to zero. As indeed is Lear himself when his Fool chastises him, “Thou art an O without a figure. I am better than thou art now; I am a fool, thou art nothing.”

Shakespeare’s play was written at a time when mathematics was entering intellectual, aesthetic, and literary discourse in England and Europe in new, increasingly prominent ways. Not only was calculation with zero-based numerals displacing Roman notation, but the even more abstruse art of algebra, fundamental to the seventeenth-century explosion of mathematics, was beginning to surface. But arithmetic and algebra aside, it was geometry that served as the face of mathematics and arena of inspiration. One strange literary example is a recently unearthed play, Blame Not Our Author (Anonymous 1613), an eccentric drama whose learned protagonists are figures of Euclidean geometry: Square, Rectangle, Compass, Line, Circle, Triangle, Semicircle, Rhombus, and Ruler. The theatrical action involves much physical cavorting and capering centering on Square’s repeatedly foiled attempts to become a circle (Mazzio 2004). On a wider canvas, geometry can be seen to be deeply entangled with the literary and philosophical writing of the period, not only as a narrative model, as with Spinoza, but also internally within the forms of prose. For example, Arielle Saiber (2005) shows how the employment of figures and tropes in Giordano Bruno’s writings, whose publication caused him to be burned at the stake, are frequently literary equivalents and parallels of geometrical ideas.

Geometry (and Euclid) has persisted as an icon of abstraction and mathematics. Poets in more recent times still treat it so. In Vachel Lindsay’s “Euclid” the great geometer and his colleagues gathered on a beach to make figures in the sand are merely a foil for a child’s innocent eye:

A silent child stood by them
From Morning until noon
Because they drew such charming
Round pictures of the moon.

(Lindsay 1925)

In a sonnet of Edna St. Vincent Millay, the beauty of Euclid’s mathematics is beyond all else:

Euclid alone has looked on Beauty bare.
Let all who prate of Beauty hold their peace,
And lay them prone upon the earth.

(Millay 1956)

Lautréamont, in his Songs of Maldoror,inflates this vision into a numinous halo surrounding mathematics by:

O austere mathematics! … you set an extreme coldness, a consummate prudence and an implacable logic. … Arithmetic! Algebra! Geometry! Imposing trinity! Luminous triangle! He who has never known you is without sense! He merits the ordeal of the most cruel tortures for in his ignorant carelessness there is a blind contempt. … But you, O concise mathematics, by the rigorous fetters of your tenacious propositions and the constancy of your iron-bound laws you dazzle the eyes with a powerful reflection of that supreme truth whose imprint is manifest in the order of the universe.

(Cited in Badiou 2006: 19–20)

For others, mathematics’ naked, cold beauty and implacable concision were anathema. For the speaker of John Keats’s narrative poem Lamia, the “rule and line” of geometry (for which read mathematics) was indeed a “cold philosophy,” but one that disenchanted whatever it touched and emptied the world of awe and wonder:

There was an awful rainbow once in heaven:
We know her woof, her texture; she is given
In the dull catalogue of common things.
Philosophy will clip an Angel’s wings,
Conquer all mysteries by rule and line,
Empty the haunted air, and gnomed mine –
Unweave a rainbow.

(Keats 1998: Part 2)

As indicated, mathematics can relate to literary texts in different ways. It can inflect the form of a work, but be unimportant or even unmentioned in its content. Dante in his Comedy is undoubtedly consumed by threes, but the number itself is scarcely mentioned; Spinoza shapes his Ethics after Euclid, but geometry is not what his text is about. On the other hand, the works by St. Vincent Millay, Lindsay, Lautréamont, Keats, and Shakespeare all variously mention geometry or arithmetic or number, as divinely beautiful or threateningly reductive, but in no case is their structure, rhetoric, or narrative shaped by anything mathematical.

A feature of mathematics, evident to even a casual observer, is its preoccupation with problems. The subject is organized around agendas set by unresolved difficulties, open questions, conjectures, and hypotheses. Some, like Fermat’s last theorem, or Goldbach’s conjecture – which asks whether every even number is the sum of two prime numbers – are grand questions that remain open for centuries, and attempts to solve them engender entire fields of new mathematics. Some, like Cantor’s Continuum Hypotheses, cannot, even in principle, be resolved. But, less grandly, problems are the bread and butter of mathematics, manifest as everyday responses to questions and sundry logical challenges. Grappling with them is an essential part of learning and practicing, and indeed loving, the art.

Nobody has better and more famously recreated this aspect of mathematics in literature – though a playful mix of logical provocation, word play, riddles, conundrums, strange paradoxical inversions, and inspired nonsense – than the mathematician Lewis Carroll (Charles Dodgson). His poem Jabberwocky starts:

’Twas brillig, and the slithy toves
Did gyre and gimble in the wabe;
All mimsy were the borogoves,
And the mome raths outgrabe.

(Carroll 2007: 99–100)

The poem continues in the same vein, dancing in and out of nonsense. Alice is puzzled by it –“Somehow it seems to fill my head with ideas – only I don’t exactly know what they are!”–a response that could capture for many, lost in the fog of mathematical abstraction, their encounter with algebra. It is also not too remote from Bertrand Russell’s dictum: “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true” (Russell 1917: ch. 4). Carroll has much fun with problems involving numbers as well as acknowledging the less attractive feeling they induce: “The different branches of Arithmetic – Ambition, Distraction, Uglification, and Derision” (Carroll 2007: 68). Mostly, however, it’s logic rather than numbers that interests Carroll, and Alice is no match for the Red Queen’s mad version of it:

“It’s very good jam,” said the Queen.
“Well, I don’t want any to-day, at any rate.”
“You couldn’t have it if you did want it,” the Queen said. “The rule is jam tomorrow and jam yesterday but never jam to-day.”
“It must come sometimes to ‘jam to-day,’” Alice objected.
“No it can’t,” said the Queen. “It’s jam every other day; to-day isn’t any other day, you know.”

(Carroll 2007: 129)

Equally defeating is the more imperiously obscure logical boast by the Red Queen:

Alice laughed: “There’s no use trying,” she said; “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When
I was younger, I always did it for half an hour a day. Why, sometimes
I’ve believed as many as six impossible things before breakfast.”

(Carroll 2007: 131–32)

The boast here, though perhaps an extravagant feat on an empty stomach, makes perfect logical sense in light of the reductio ad absurdum reasoning that is fundamental to mathematics. According to this method of proof, one demonstrates the truth of a proposition by assuming (believing) it to be false (impossible) and deducing from that assumption a flat contradiction (an absurdity).

A less playful, more purposeful literary deployment of mathematics occurs in a work of Carroll’s contemporary, Edwin A. Abbott, in his satirical novella Flatland: a romance in many dimensions (Abbott 1884). In this strange tale, reminiscent (but composed independently) of the play Blame Not Our Author mentioned earlier, male characters are depicted as two-dimensional geometrical shapes, triangles, squares, pentagons, and so on. They form a caste system, status being determined by number of sides: those of the artisan class are triangles, the professionals are squares and higher, with many-sided polygons, who approximate to perfect circles, forming the priesthood. Female characters are mere one-dimensional lines, who are required to sway as they approach to avoid being mistaken for points. The narrator, a square named A. Square, encounters a three-dimensional sphere, a denizen of Spaceland, mysteriously able to evade the confines of his world, whose aim is to enable Flatlanders to overcome their illusory denial of a third dimension. A. Square also dreams of visiting the one-dimensional world of Lineland whose monarch, a single point, is unable to grasp the idea of a being outside his own linear world. Eventually, the narrator is imprisoned for publicly maintaining the existence of Spaceland. Notwithstanding its satire of Victorian society, Flatland is obsessed with “space”–its “dimensions,” the nature of boundaries, entrances, exits, pathways, and the insides and outsides of regions. As such, it is surely a product of the interest in the mysterious “fourth dimension,” which the mathematical invention of non-Euclidean geometries had recently introduced into the very idea of space and which became an object of consuming fascination, particularly by artists, beginning in the second half of the nineteenth century (Henderson 1983).

A fertile appropriation of mathematics for literary purposes is as a constraint, a set rule or pattern for a work to follow. The use of constraints is widespread within literature, quite independently of mathematics. The very notion of a poetic “form” implies conformity to a constraint as, very differently, in the sonnet, terza rima, haiku, iambic pentameter, epigram, limerick, and so on – not to mention in sub-literary devices such as puns, anagrams, palindromes, and acrostics. For instance, the Danish poet Inger Christiansen employed a wide range of mathematical constraints and patterns, making use of Fibonacci numbers, of algebraic forms, and of the self-similarity property of fractal geometry, where aspects of a whole poem are mirrored in portions of it. Her book-length work of social criticism in verse, it (Christensen 2006), contains series of ever smaller sections, like Russian dolls, which induce a philosophical layering and deepening of meanings. Other works incorporate cyclic permutations: the volume of her poems Butterfly Valley: a requiem (Christensen 2004) includes a chain of 14 sonnets, each beginning with the last line of the previous one, with a fifteenth sonnet consisting of the first lines of the preceding 14 poems.

Christensen’s fascination with the poetic possibilities of mathematical patterns echoes a wider contemporary movement interested in creation through formal devices. In 1960, a diverse group of poets, novelists, mathematicians, and philosophers announced in Paris the foundation of a literary workshop, later to be called Oulipo (ouvroir de littérature potentielle), dedicated to exploring the use of mathematical rules, procedures, and constraints to produce literary texts. Over the years the group has experimented with a variety of procedures: poems written to be read from the surface of a möbius strip; “lipograms”–stories excluding a particular letter of the alphabet; the “larding” of a source text by the repeated insertion of new sentences; narratives organized according to pathways determined by mathematical graphs; and sundry kinds of rules of permutation and substitution of letters, phrases, and so on. One of the group’s earliest productions was 100,000,000 000,000 Poems by one of its founding members, Raymond Queneau. The work offers 10 alternatives for each of its 14 lines. Selecting a first line, then a second, and so on, produces a single sonnet, one of the hundred thousand billion (10 to the power 14) poems of the title. For example (parentheses enclose other possible selections):

The wild horse champs the Parthenon’s top frieze
Since Elgin left his nostrils in the stone
The Turks said just take anything you please
(Upon his old oak chest he cuts his cheese)
And loudly sang off-key without a tone
O Parthenon you hold the charger’s strings
The North Wind bites into his architraves
Th’outrageous Thames a troubled arrow slings
To break a rule Britannia’s might might wave
(Victorious worms grind all into the grave)
Platonic Greece was not so talentless
A piercing wit would sprightliest horses flog
Socrates watched his hemlock effervesce
Their sculptors did our best our hulks they clog
(Lobsters for sale must be our apologue)
With Marble souvenirs then fill a slum
For Europe’s glory while Fate’s harpies strum

(Mathews and Brotchie 1998: 15–33)

The writer Georges Perec produced two of Oulipo’s most celebrated works, La disparition and Life: a user’s manual. The first, in purely formal terms the simpler of the two, is a novel-length lipogram, written entirely with words not containing the letter e. The work is full of strange juxtapositions engendered by its constraint and is obsessed, as well as being haunted, by absences. It has been translated – rewritten – into e-less English as A Void by Gilbert Adair (Adair 2005). Life: a user’s manual is a complex, labyrinthine text, perhaps the most mathematically schematized novel ever written. Featuring and inspired by a jigsaw puzzle, it tells of the interconnecting lives, histories, fates, and passions of the inhabitants of a Parisian apartment building, connected (more or less) by the doomed attempt of a painter, Bartlebooth, to give meaning to his life. The building has 10 floors, each with 10 units (rooms and stairs) facing the street, imagined as simultaneously visible as if the front façade had been removed. Perec employed two mathematical schemes to construct his novel. Firstly, the contents and characters occupying each unit are constrained by a 10x10 mathematical array known as a Graeco-Latin square, subsequently explained by Perec, in a simplified 3x3 case, as follows:

Imagine a story 3 chapters long involving 3 characters Jones, Smith, Wolkowski. Supply the 3 individuals with 2 sets of attributes: first, headgear – a cap C, a bowler hat H, a beret B; second, something handheld – a dog D, a suitcase S, and a bouquet of roses R. Assume the problem to be that of telling a story in which these 6 items will be ascribed to the 3 characters in turn without their having the same 2. The following formula:

which is nothing more than a very simple Graeco-Latin square – provides the solution. In the first chapter, Jones has a cap and a suitcase, Smith has a beret and a bouquet of roses, Wolkowski has bowler hat and a Dog.

(Mathews and Brotchie 1998: 172)

And so on for the other chapters. The story would have to “invent situations to justify these successive transformations.” In the novel, “instead of 2 series of 3 items, 21 times 2 series of 10 items are permutated in this fashion” obliging Perec to include 42 themes in every chapter (Mathews and Brotchie 1998: 172). To determine the order of narration through the apartments, Perec uses a second mathematical scheme, a Knight’s Tour – the path that a knight takes on an 8x8 chessboard visiting each and every square only once – modified to work for his 10x10 array of apartments.

Another prominent Oulipian, the mathematician Jacques Roubaud, who has written several mathematically inflected novels, enunciates two principles at work in some of the group’sefforts. One, when “a text written in accordance with a restrictive procedure refers to the procedure,” for example, Perec’s La disparition, which, among other things, recounts the disappearance of the letter that constitutes it; two, when “a text written according to a mathematically formulable procedure includes the consequences of the mathematical theory that it illustrates” (Mathews and Brotchie 1998: 218). Examples of this second principle are harder to find. Roubaud cites his own novel Princess Happy, concerning four individuals, which takes account of the algebra of the four-element mathematical group.

We might ask about the opposite to constraints as sources of literary creation, about the presence in a work of the mathematically unconstrained. Could that which is unbounded – such as the infinite progression of whole numbers – make its mark in a literary text? We might, for example, find – or at least impute – the presence of infinity in Laurence Sterne’s satirical novel Tristram Shandy (1759), whose eponymous hero devotes himself to telling the story of his life. But (for various hilarious reasons) he writes at a snail’s pace, taking two years to narrate two days of his life, and laments he’ll never finish the task. However long he lives, the number of years and days can’t possibly match: logically, the whole of his life must exceed the part. Sterne wrote a century before Georg Cantor showed that for infinite collections wholes and parts obey a different logic. The even numbers, for example, despite being a part, only half, as it were, of the numbers, can indeed be tallied off with the whole, with all the numbers. Both collections have the same magnitude, which Cantor designated as aleph-zero – the smallest infinite number. Bertrand Russell, writing in the wake of Cantor’s theory of infinity, maintained that if Tristram Shandy “had lived for ever, and not wearied of his task, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten” (Russell 1903: 358). Thus, year 1 could be devoted to writing the events of day 1, year 2 to day 2, and so on, where every day would eventually be accounted for. Russell does not of course say what it would mean to live without end.

Jorge Luis Borges, who admired Sterne’s novel and knew of Russell’s solution to Tristram Shandy’s predicament, engages with infinity on several occasions. His perhaps most celebrated fiction, “The Library of Babel” (1964), is saturated with mathematical ideas (Bloch 2008), most notably infinity. It depicts “The universe (which others call the library)” as a vast, topologically strange construction of hexagonal cells, “a sphere whose exact center is any hexagon and whose circumference is unattainable” (Borges 1964: 79), a formula which plays on “God is a sphere whose centre is everywhere and whose circumference is nowhere,” the hermetic dictum repeated by the medieval theologian-mathematician Nicholas de Cusa. The library comprises books of precious wisdom, incomparable nonsense, and unimaginable, arbitrary content. Its ontology follows that of mathematics – x exists if the definition of x is not contradictory: “It suffices that a book is possible for it to exist” (84). But everything is written in a fixed alphabet of letters, so that (since each book in the library is unique) the number of books is finite. And yet, is not the universe, the library, the world, infinite? To which the narrator responds that “the library is unlimited and cyclical”–a traveler setting off into the universe would return eventually from the opposite direction. The story ends with the suggestion that perhaps the whole library could be replaced by a single book “containing an infinite number of infinitely thin leaves” (86).

A later work, “The Book of Sand” (Borges 1979: 87–91), features just such a book. It begins: “The line is made up of an infinite number of points; the plane of an infinite number of lines; the volume of an infinite number of planes; the hypervolume of an infinite number of volumes” (87), at which point it breaks off and plunges into its tale of a nondescript, nighttime visitor offering to sell the narrator what he calls a “Bible.” The narrator is baffled by its unknown script and incomprehensible pagination. He’s told it was acquired for a few rupees from an Untouchable and is called the Book of Sand “because neither the book nor the sand has any beginning or end” (89). Disbelieving, the narrator searches for a first and then a last page, but in vain. “The number of pages in this book,” the stranger tells him, “is no more or less than infinite.” The narrator agrees to accept it in exchange for a precious Wyclif Bible. But the infinite book invades his dreams and comes to seem monstrous, “a nightmarish object, an obscene thing that affronted and tainted reality itself” (91), and he buries it in a dusty corner of the Argentine National Library.

The material I’ve chosen to illustrate the nexus of literature and mathematics here is of course selective, with many omissions. Most notably, I’ve included nothing of the great mass of contemporary science-fiction novels where things mathematical are most at home. In this genre of speculative fiction, mathematics – fractal geographies, topologies of time and space, algebraic encryption, chaotic attractors, algorithmic life-forms, computational intelligences, and much else – is an ever-present source of imagination. This profusion of mathematically inflected novels of possible, impossible, and virtual worlds (too numerous and various to summarize here) reflects a larger phenomenon, namely the ongoing upheaval of contemporary thought and consciousness engendered by computation-based digital media and technologies of the virtual (Rotman 2008). Once again, as in the seventeenth century, although on a wider canvas, an explosion of mathematical language and ideas is impinging on the written text in new and interesting ways.

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