chapter  14
MATHEMATICS
ByBRIAN ROTMAN
Pages 12

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 lan-

guage 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.