Mathematics and the Imagination: A Brief Introduction

"[I]f mathematics is the study of purely imaginary states of things, poets must be great mathematicians."

—Charles Sanders Peirce¹

I prove a theorem and the house expands:the windows jerk free to hover near the ceiling,the ceiling floats away with a sigh.

—Rita Dove²

A few years ago, mathematician, writer, and regular contributor to NPR Keith Devlin wrote that mathematics is about rendering the invisible visible and about inventing symbolic worlds into which the mind can enter. "Is there a link between doing mathematics and reading a novel?" Devlin asks. "Very possibly," he answers.³ Imagining a conversation between two invented characters or the intricate imagery of a poem arguably requires a similar kind of mental process as imagining "the square root of minus fifteen," as mathematician Barry Mazur has demonstrated.⁴ "Of all escapes from reality," wrote mathematician Giancarlo Rota, "mathematics is the most successful ever."⁵

Not that literature is about escaping from reality, of course. It and all the visual and performing arts, as well as every discipline in the humanities and sciences for that matter, often share with mathematics a common goal: that of describing and/or addressing the "really real." Questions of reality, truth, and certainty are at the core of the philosophy of mathematics: Does mathematics afford us entry into reality and truth? Does it provide us with certainty? Contrary to Platonist beliefs about the ability of mathematics to give us these things is the stance that mathematics is actually about multiple realities, relative truths, complexities, and ambiguities. In essence, doing pure mathematics (not merely doing computations) is an exercise in imagination—and imagination, an exercise in abstraction.

One need only recall a few key moments in the history of mathematics—the discovery of irrational numbers by the ancient Greeks; the development of non-Euclidean geometry; Kurt Gödel's findings concerning undecidable propositions and the "incompleteness of mathematics"; chaos theory—as well as the debates and controversies surrounding these topics, to see how closely mathematics dances with uncertainties. Perhaps what mathematics shows us is that the "really real" is, in fact, a whirl of ambiguity, and that, as mathematician William Byers has written, mathematics requires thinking in terms of contradiction and paradox.⁶ Or perhaps, as Edwin Hutchins and others have argued, we exteriorize thinking through our physical environment, through marks, instruments, and the physical configurations of objects in our built environment.⁷ A related claim, but one that is vigorously contested, is the notion that George Lakoff and Rafael Núñez have proposed: that mathematics exists only because the human brain does—it is a product of it, just as metaphors and anything we make are.⁸ And so, the age-old question remains: Are mathematical objects and concepts transcendental entities, Platonic ideas hovering in the out-there to be discovered, or are they invented by the human mind?

Thinking about mathematics inspires many kinds of questions: metaphysical questions on the status (existence) of mathematical objects and concepts; epistemological questions on how we "know" mathematics to be true and verify it as such; semiotic questions about the nature of mathematical language; as well as questions regarding how the mind visualizes, categorizes, systematizes, abstracts, and articulates entities that are imagined, or, in many cases, are barely imaginable at all. Among the early landmark thinkers in the philosophy of mathematics there have been those, like Plato and Kant, who have seen pure mathematics as part of the world of forms and only accessible via reason; Aristotle, who saw mathematics not as something separate from the world of sensation but as related to the way the mind performs its thought; Giordano Bruno, who believed mathematics (geometric thinking in particular) to be the link between the human and celestial worlds; Galileo and Kepler, who proposed that the universe was written in mathematics; Descartes, who built a philosophy around the certainty he believed mathematics offered; and Gottfried Wilhelm Leibniz, who saw mathematical propositions as not true of particular eternal objects or of idealized objects resulting from abstraction but as true because their...