An Alternative Approach to Formal Mathematics that Focuses on Communication and Accessibility

2026-03-21

AI Summary

Traditional formal mathematics emphasizes rigorous proof and mechanical verification, yet its complex tools and high entry barriers hinder accessibility and effective idea exchange for most mathematical practitioners. This work proposes the “free approach”—a novel paradigm centered on communicability and usability that departs from certification-centric frameworks by eliminating the mandatory requirement to formally verify every detail. Grounded in Alonzo logic—a practical variant of simple type theory—the approach establishes both theoretical foundations and implementation mechanisms. The study demonstrates that the free approach aligns more closely with actual mathematical practice, effectively serving a broad community of mathematicians, and calls for collective development of supporting logical systems, toolchains, knowledge repositories, and educational infrastructure.

Technology Category

Knowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSecurity and Privacy: Data transparency and provenanceSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Formal mathematics is mathematics done within the framework of a formal logic. It offers major benefits to mathematicians as well as to computing professionals, engineers, and scientists who use mathematics in their work. The standard approach to formal mathematics, in which mathematics is done with the help of a proof assistant and all details are formally proved and mechanically checked, achieves these benefits and offers a very high level of assurance that the results produced are correct. However, since the main goal of the standard approach is certification, the proof assistants supporting the standard approach are generally complex, based on unfamiliar logics, difficult to learn how to use, and far removed from mathematical practice. Thus the standard approach does not adequately serve mathematics practitioners who are more interested in communicating mathematical ideas than in formally certifying their correctness or who prefer not to make the investment needed to gain proficiency in the use of proof assistants. This paper presents an alternative to the standard approach that focuses on communication and accessibility, the two weaknesses of the standard approach. It is called the free approach to formal mathematics since it is free of the obligation to formally prove and mechanically check all details of a mathematical development. The paper argues that the free approach would serve the needs of the average mathematics practitioner much better than the standard approach. It describes an implementation of the free approach based on a logic named Alonzo, a practice-oriented version of Alonzo Church's formulation of simple type theory. And it calls for the mathematics community to develop logics, software, and libraries of formal mathematical knowledge to support the free approach and to train mathematics practitioners to use them.

Problem

• formal mathematics
• proof assistants
• communication
• accessibility
• mathematical practice

Innovation

• formal mathematics
• free approach
• Alonzo logic
• accessibility
• mathematical communication