Illustration Beyond Proof

A group of mathematicians, historians, artists and educators examined what it would mean for a mathematical illustration to be rigorous, arguing that illustration is an essential mode of mathematical knowing distinct from, but complementary to, proof. The central claim is that theorem production has become an overvalued proxy for understanding, a problem intensified by AI’s ability to generate proofs; understanding also requires stories, pictures, insights and community digestion. Philosophical and historical perspectives are marshaled to show that drawing and making have long been epistemic practices (Monge’s École Polytechnique, debates in the 1990s), and contemporary voices insist mathematics should credit nonformal outputs - visualisation, pedagogy and exposition - alongside formal results.

Concrete examples and practical proposals illustrate the case. Visual experiments such as algebraic “starscapes” revealed novel structures when shader-induced distortions became mathematically meaningful; the discovery of a three-dimensional aperiodic monotile relied on shape illustrations that guided later rigorous distillation. The authors propose treating an illustration’s affordances as properties that proofs can certify, adopt art’s crit method and “thinking-through-making” pedagogy, and expand infrastructures like Geometry Labs to validate and credit visual and material modes of research. The overall call is to recognise, support and integrate illustration, education and practice into the mathematical ecosystem.