Formalizing all indexed mathematics as a benchmark for general reasoning, with the example of implementing dilatations of categories
2026-06-02 • Databases
DatabasesHuman-Computer Interaction
AI summaryⓘ
The authors discuss how mathematics is uniquely precise because every statement is logically proven from basic rules. They look at tools called interactive theorem provers that help write and check these proofs in a way a computer can understand. The paper explores the big challenge of turning all known math into a fully verifiable and constantly updated collection stored in a database-like system. To show how this works in practice, the authors share their ongoing work on a specific math topic called categorical algebra, demonstrating how complex math can be formalized step-by-step.
formal rigorinteractive theorem proverformalizationmathematical proofcategory theorycategorical algebralocalizationdilatationformal librariesmathematical knowledge database
Authors
A. Mayeux
Abstract
Formal rigor distinguishes mathematics from other disciplines, in the sense that mathematical statements are derived from explicit axioms by logically verifiable steps. Interactive theorem provers support this by expressing definitions, theorems, and proofs in a fully formal language and verifying them mechanically. We consider the benchmark problem of formalizing all published mathematics as a machine verifiable and continuously updated corpus of mathematical knowledge. This viewpoint treats mathematics as a structured database of interdependent results and raises questions about scalability and organization of large formal libraries. As a case study, we present an ongoing formalization in categorical algebra, namely dilatations of categories, extending classical localizations and illustrating what such an implementation looks like in practice.