发表机构
University of Arkansas; University of Colorado Boulder（阿肯色大学; 科罗拉多大学博尔德分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文追溯数学图解传统，结合哲学分析，主张发展严格图解以辅助证明生成与洞见确证，并呼吁数学界重视其在研究中的作用。
AI 中文摘要
在加斯帕尔·蒙日（Gaspard Monge）的法国教学传统中，几何是通过绘制和操作模型来学习的：绘图即认知。二十世纪初的形式主义将图解置于直觉一侧，或许对发现有用，但与证明相分离。然而，图解在数学中持续活跃，并为数学知识提供了一种模型，因为证明可由自动化系统生成，而《莱顿宣言》（Leiden Declaration）再次追问数学知识为何物。我们追溯了一条贯穿阿达马（Hadamard）、考克斯特（Coxeter）、康威（Conway）、瑟斯顿（Thurston）以及几何中心（Geometry Center）的持久图解传统，并借鉴数学实践哲学，包括曼德斯（Manders）关于精确主张与共精确主张的区分、德·托福利（De Toffoli）关于具有证成力之图示的论述，以及贾尔迪诺（Giardino）的表征可供性。通过施瓦茨（Schwartz）极小平坦环面、曼德布罗特集（Mandelbrot set）的早期图像以及代数星空景观的着色器渲染等例子，我们主张发展严格的图解，以激发并确证其所提供的洞见。最后，我们呼吁数学界认识到图解在促进数学研究与理解发展中所发挥的作用。
英文摘要
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.
Comments12 pages, 6 figures, work developed at the IHP trimester "Illustration as a Mathematical Research Technique" from January 5 to April 3, 2026
