Mathematical Explanations: An Analysis Via Formal Proofs and Conceptual Complexity

This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion.

Automated summary

This paper examines internal mathematical explanations, which refer to the proofs of mathematical theorems that seem to provide an explanation for the theorem itself. The paper aims to rigorously analyze these explanations in two steps: first, by demonstrating how to transform informal proofs into a formal presentation using proof trees and element decomposition; second, by showing that math proofs with explanatory power exhibit an increase in conceptual complexity from assumptions to conclusions.