Geometric Measurements of the Axiom of Choice in Neural Proof Embeddings

What Happened

This paper introduces a novel methodology for quantifying the "axiomatic distance" between mathematical proofs using neural embeddings. Specifically, the researchers leverage Lean 4's kernel-level axiom tracking to create geometric representations of proofs, then measure how far a proof is from relying on the axiom of choice. The work bridges formal verification systems with representation learning, treating the axiom of choice not as a binary toggle but as a continuous spectrum within proof space.

The core innovation lies in using the theorem prover's built-in dependency tracking to generate labeled training data for neural embeddings. By recording which axioms each proof step invokes, the system learns to map proofs into a vector space where Euclidean distance correlates with axiomatic divergence. This allows for quantitative comparisons between classical proofs (which use the axiom of choice) and constructive proofs (which avoid it).

Why It Matters

For over a century, the axiom of choice has been a philosophical fault line in mathematics. Classical mathematicians accept it for its power; constructivists reject it for its non-constructive nature. This work transforms that debate from a binary philosophical stance into an empirical, measurable phenomenon. It demonstrates that formal proof assistants like Lean 4 can serve as experimental platforms for foundational questions.

More broadly, this represents a convergence of three trends: the maturation of interactive theorem provers, the rise of neural embeddings for structured data, and growing interest in "proof mining" — extracting insights from the corpus of formalized mathematics. The ability to geometrically measure axiomatic reliance could help mathematicians identify proofs that minimize controversial assumptions, or automatically classify proofs by their foundational commitments.

Implications for AI Practitioners

For those working on AI-assisted theorem proving, this paper offers a concrete technique for embedding proofs in a way that captures not just syntactic structure but foundational dependencies. This could improve retrieval systems for proof libraries, where a search might prioritize proofs that avoid certain axioms. It also suggests a path toward more interpretable AI reasoning: if an AI system's proof can be geometrically located relative to known axioms, users gain a transparency tool for understanding the system's logical commitments.

For the broader AI community, the approach of using a formal system's internal tracking as supervision for embeddings is transferable. Any domain with explicit dependency tracking — code compilers, hardware verification, legal reasoning — could adopt similar methods. The key insight is that formal systems already record the "axiomatic cost" of each step; neural embeddings can make that cost geometrically navigable.

Key Takeaways

• Researchers used Lean 4's axiom tracking to create neural embeddings that measure a proof's reliance on the axiom of choice as a continuous geometric property
• This transforms a century-old philosophical debate into an empirical, quantifiable framework using formal proof assistants
• The methodology provides AI theorem provers with a new tool for retrieving, comparing, and explaining proofs based on their foundational assumptions
• The technique generalizes beyond mathematics to any domain with formal dependency tracking, offering a blueprint for embedding structured logical dependencies