Harriss & Stange: ‘Mathematical Illustration: Proof, Intuition and AI’ (arXiv 2610.02655)
Edmund Harriss and Katherine E. Stangearxiv.org
Why it matters
Two mathematicians argue that illustration (drawing, models, computer graphics) is a form of mathematical knowledge that needs its own standard of rigour, as an answer to the question the Leiden Declaration raises about AI-generated proofs; accepted in the Notices of the AMS per the authors’ guest post on Tao’s blog.
Summary
arXiv preprint (math.HO), v1 submitted 2 Oct 2026; the PDF itself is dated October 5, 2026. 12 pages, 6 figures; the arXiv comment says the work was developed at the IHP trimester “Illustration as a Mathematical Research Technique” (January 5 to April 3, 2026). Authors: Edmund Harriss (Department of Mathematical Sciences and School of Art, University of Arkansas) and Katherine E. Stange (Department of Mathematics, University of Colorado Boulder). Starting from Gaspard Monge’s drawing-based curriculum (“Drawing was knowing”, quoting historian Frédéric Brechenmacher), they trace how twentieth-century formalism set illustration aside and how it persisted (Hadamard, Coxeter, Conway, Thurston, the Geometry Center). Using the philosophy of mathematical practice (Manders, De Toffoli, Giardino) and examples such as early Mandelbrot set images and shader renderings of “algebraic starscapes”, they argue for developing “rigorous illustration”. They use the Leiden Declaration “as a symbol and primary source” without arguing for or against it. On AI they note that vibecoding has made computer visualization much faster and more accessible. The paper has an “AI Declaration” (quoted below). Read from the PDF text (two-column layout; we skimmed the middle sections) on 2026-10-13. The authors’ guest post on Tao’s blog of 11 Oct says the article is “now accepted in the Notices”: see 2026-10-11-stange-harriss-illustration-beyond-proof-tao-blog.
Archived text
- “If an LLM provides a rigorously verifiable but incomprehensible or unenlightening proof of a famous conjecture, what have we gained?”
- “It is not our goal here to argue for or against the declaration.”
- “We recognize that aspects of this article could be out of date in a matter of months!”
- AI Declaration: “The authors made use of large language models (specifically Claude) during the preparation of this manuscript for bibliographic assistance, structuring and proofreading. All text and arguments are the authors’ own.”
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Terence Tao, Professor of mathematics, UCLA