A New Style of Proof for Mathematics Organized as a Network of Axiomatic Theories

TL;DR

This paper introduces a novel proof style for mathematics that leverages the structure of theory graphs, combining traditional and formal proof strengths to better organize and utilize mathematical knowledge.

Contribution

It proposes a new proof style that exploits the structure of theory graphs, enhancing the organization and understanding of mathematical proofs.

Findings

The new proof style effectively integrates traditional and formal proof advantages.

It improves the organization of mathematical knowledge within theory graphs.

The approach enhances the utility of proofs in complex mathematical structures.

Abstract

A theory graph is a network of axiomatic theories connected with meaning-preserving mappings called theory morphisms. Theory graphs are well suited for organizing large bodies of mathematical knowledge. Traditional and formal proofs do not adequately fulfill all the purposes that mathematical proofs have, and they do not exploit the structure inherent in a theory graph. We propose a new style of proof that fulfills the principal purposes of a mathematical proof as well as capitalizes on the connections provided by the theory morphisms in a theory graph. This new style of proof combines the strengths of traditional proofs with the strengths of formal proofs.