Towards a structural analysis of interesting geometric theorems: analysis of a survey

Source abstract

Abstract A longstanding challenge in automated theorem generation is not merely to derive new statements, but to identify which of them are mathematically interesting. Motivated by Wos’ Problem 31, this paper reports the results of a first survey, within a two-stage expert-survey study on theorem interestingness in geometry, and explores whether some of the dimensions elicited from human judgements admit formally computable structural counterparts. The descriptive analysis identifies six recurring, non-exclusive and non-exhaustive dimensions of interestingness: simplicity and clarity, surprise, conceptual depth, usefulness and applicability, aesthetic value, and historical, pedagogical, or emotional relevance. We then examine these dimensions in relation to a modern geometrographic analysis of area-method proofs, considering the overall complexity of a proof, the number and complexity of its steps, the occurrence and diversity of difficult lemmas, and the Geometrographic Readability Coefficient of Proofs (GRCP). The comparison suggests that simplicity, clarity, and certain aspects of conceptual depth may have plausible structural counterparts in proof features. By contrast, surprise, usefulness, aesthetic value, and historical or pedagogical relevance remain strongly dependent on mathematical and human context. The analysis is exploratory and does not establish statistical correlations or validate GRCP as a measure of theorem interestingness. Rather, it provides an empirical and formal basis for hypotheses to be tested in the second survey and in subsequent comparisons with human judgements of proof readability.