Mathematical Reality Capture: from the age of discoveries to mathematical topography and surveys

Abstract

A proof answers a precise question, but mathematical understanding also grows through the relationships that its answer makes visible. A point measurement likewise differs from a photograph or a three-dimensional reconstruction of a building. In this work, we argue that, with the help of AI, mathematics should move from a discipline focused on isolated investigations of mathematical objects towards one capable of mapping them systematically. We therefore introduce the idea of "Mathematical Reality Capture", inspired by digital surveying and the reconstruction of a scene from several viewpoints. An elementary example shows how comparing algebra and geometry can illuminate the same phenomenon, provided that we distinguish diversity of explanations from validity of proof. From this analogy we derive criteria for building verifiable atlases, an experiment with which to assess their usefulness, and a reflection on the role of mathematicians in interpreting them.