Mathematical Illustration: Proof, Intuition and AI

Abstract

In the French pedagogic tradition of Gaspard Monge, geometry was learned by drawing and handling models: drawing was knowing. The formalism of the early twentieth century placed illustration on the side of intuition, perhaps useful for discovery but kept apart from proof. Yet illustration stayed active in mathematics and provides a model for mathematical knowledge as proofs can be generated by automated systems and the Leiden Declaration asks again what mathematical knowledge is. We trace a persistent tradition of illustration through Hadamard, Coxeter, Conway, Thurston and the Geometry Center, and draw on the philosophy of mathematical practice, including Manders' distinction between exact and coexact claims, De Toffoli's account of diagrams with justificatory force, and Giardino's representational affordances. Using examples from Schwartz's minimal flat torus, early images of the Mandelbrot set, and shader renderings of algebraic starscapes, we argue for the development of rigorous illustration, to help inspire and certify the insights it affords. We close by asking the mathematical community to recognise the role illustration plays in helping develop mathematical research and understanding.