A Survey of Languages for Formalizing Mathematics

TL;DR

This survey reviews various formal languages used for representing mathematical content in computer systems, highlighting their strengths and limitations, and discusses future directions for improving these languages.

Contribution

It provides a comprehensive overview of existing formal languages for mathematics, evaluating their advantages and identifying gaps for future development.

Findings

No existing language fully meets all needs for formalizing mathematics.

Languages vary in support for verification, computation, and readability.

Future improvements are necessary for more effective mathematical formalization.

Abstract

In order to work with mathematical content in computer systems, it is necessary to represent it in formal languages. Ideally, these are supported by tools that verify the correctness of the content, allow computing with it, and produce human-readable documents. These goals are challenging to combine and state-of-the-art tools typically have to make difficult compromises. In this paper we discuss languages that have been created for this purpose, including logical languages of proof assistants and other formal systems, semi-formal languages, intermediate languages for exchanging mathematical knowledge, and language frameworks that allow building customized languages. We evaluate their advantages based on our experience in designing and applying languages and tools for formalizing mathematics. We reach the conclusion that no existing language is truly good enough yet and derive ideas for possible future improvements.