🤖 AI Summary
This paper addresses the challenges of cross-logical comparability, visualization, and automated reasoning for mathematical theorems. Methodologically, it introduces the novel concept of *proof vectors*: theorems are encoded as high-dimensional vectors whose axes correspond to foundational axiom systems (e.g., Hilbert’s geometry, Peano arithmetic, ZFC); axiom embeddings, cosine similarity, and clustering enable quantitative assessment of logical similarity and heatmap-based visualization; and the Atlas-GPT prototype implements semantic mapping from natural-language theorems to proof vectors. Key contributions include: (1) the first interpretable, axiom-system-agnostic theorem vectorization paradigm; (2) cross-disciplinary automatic clustering of theorems by underlying logical foundations; and (3) the first interactive “Axiomatic Atlas” prototype system, supporting mathematics education, formal verification, and AI-assisted mathematical discovery.
📝 Abstract
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.