Categories, axioms, constructions in SageMath: Modeling mathematics for fun and profit

These are the slides of a presentation at the Hausdorff Research Institute for Mathematics in Bonn on May 22 of 2024, at the occasion of the trimester program "Prospects of Formal Mathematics".

Abstract:

General purpose computational math systems such as SageMath systems provides thousands of mathematical objects and tens of thousands of operations to compute with them. We believe that a system of this scale requires an infrastructure for writing and structuring generic code, documentation, and tests that apply uniformly on all objects within certain realms.

In this talk, we describe the infrastructure implemented in SageMath back in the early '10. It is based on the standard object oriented features of Python, together with mechanisms to scale (dynamic classes, mixins, …) thanks to the rich available semantic (categories, axioms, constructions). We relate the approach taken with that in other systems (e.g. GAP), and discuss open problems. This is meant as a basis for discussions: how are the equivalent challenges tackled in proof systems? Is there ground for cross-fertilization?