The Axiom-Based Atlas: A Structural Mapping of Theorems via Foundational Proof Vectors
This work addresses the challenge of visualizing and comparing mathematical knowledge for researchers and AI systems, though it appears incremental as it builds on existing ideas in structural mathematics.
The paper tackles the problem of representing mathematical theorems structurally by mapping them as proof vectors over foundational axiom systems, enabling quantitative similarity metrics and grouping by logical structure, with a prototype assistant introduced for applications in automated reasoning and education.
The Axiom-Based Atlas is a novel framework that structurally represents mathematical theorems as proof vectors over foundational axiom systems. By mapping the logical dependencies of theorems onto vectors indexed by axioms - such as those from Hilbert geometry, Peano arithmetic, or ZFC - we offer a new way to visualize, compare, and analyze mathematical knowledge. This vector-based formalism not only captures the logical foundation of theorems but also enables quantitative similarity metrics - such as cosine distance - between mathematical results, offering a new analytic layer for structural comparison. Using heatmaps, vector clustering, and AI-assisted modeling, this atlas enables the grouping of theorems by logical structure, not just by mathematical domain. We also introduce a prototype assistant (Atlas-GPT) that interprets natural language theorems and suggests likely proof vectors, supporting future applications in automated reasoning, mathematical education, and formal verification. This direction is partially inspired by Terence Tao's recent reflections on the convergence of symbolic and structural mathematics. The Axiom-Based Atlas aims to provide a scalable, interpretable model of mathematical reasoning that is both human-readable and AI-compatible, contributing to the future landscape of formal mathematical systems.