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

William M. Farmer

2018

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.

Keywords: Mathematical proof, traditional proof style, formal proof
style, purposes of a proof, axiomatic theories, theory morphisms,
theory graphs, little theories method, cross checks,
flexiformalization, digital mathematics libraries.