Imagination in Mathematics

The eminent mathematician Felix Klein wrote, in his intimate history of nineteenth-century mathematics, that “mathematics is not merely a matter of understanding but quite essentially a matter of imagination” (cf. Klein [1979], p. 207). Klein was responding to the turbulent trajectory of imagination’s role inmathematics during the nineteenth century, which beganwith Gaspard Monge teaching new ways of representing three-dimensional figures in the plane to engineers at the École Polytechnique in Paris, and ended with Moritz Pasch fulminating against visualization in geometry. In that light Klein’s remark concerns imagination as representation of the visual. But there is another sense of imagination that mathematicians frequently employ: imagination as the ability to think of novel solutions to problems; in other words, ingenuity. This twofold usage of the term “imagination” was characterized by Voltaire, in his entry on imagination in the Encyclopédie, as distinguishing between what he called the “passive” and “active” faculties of imagination. Both are faculties of every “sensible being” by which one is able “to represent to one’s mind sensible things” (cf. Diderot and d’Alembert [1765], p. 560). The passive imagination consists in the ability “to retain a simple impression of objects”, while the active imagination consists in the ability “to arrange these received objects, and combine them in a thousand ways”. The active imagination is thus the faculty of invention, and is linked with genius, in particular in mathematics: “there is an astonishing imagination in mathematical practice, and Archimedes had at least as much imagination as Homer” (Ibid., p. 561). This distinction between imagination as faculty of representation and as faculty of reconstruction, however dubious it may be as cognitive science (on this, cf. Currie and Ravenscroft [2002]), organizes well the two primary ways that mathematicians and philosophers have understood the nature and role of imagination in mathematics. This essay will focus on just the first of these ways, since the second would make for a study of discovery in mathematics, demanding consideration of a rather di erent set of issues than those called for by the first (cf. Hadamard [1945] for such a study for mathematics, as well as Stokes, “Imagination and Creativity”, this volume). It will consider imagination in mathematics in the first sense from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.