arXiv:2606.28572v1 Announce Type: new Abstract: The axiom of choice has divided the foundations of mathematics for over a century, but the distinction between classical and constructive proofs has remained a philosophical and methodological one. We use Lean 4's kernel-level tracking of axiom dependence to show that the axiom of choice has a measurable geometric correlate in proof space that obeys a one-parameter mixture law and has operational consequences for neural theorem provers. To do this, we partition $471{,}260$ declarations of Mathlib by transitive dependence on the axiom of choice an
The proliferation of neural theorem provers and advanced formal verification tools like Lean 4 has enabled new experimental approaches to foundational mathematics.
This research provides a quantifiable method to understand the impact of fundamental mathematical axioms on AI systems, potentially leading to more robust and explainable AI in formal reasoning.
The understanding of how axiomatic choices in mathematics manifest in neural networks is advancing from philosophical debate to measurable, geometric properties.
- · AI researchers in formal methods
- · Developers of neural theorem provers
- · Formal verification tooling providers
- · Mathematics education
- · AI systems lacking axiomatic awareness
- · Purely philosophical approaches to mathematical foundations
AI models trained for mathematical reasoning will be better equipped to handle axiomatic dependencies.
This understanding could lead to the development of 'axiomatic debugging' tools for AI, improving their reliability and transparency in logical tasks.
Future AI systems might dynamically select axiomatic frameworks based on problem requirements, blurring the lines between classical and constructive mathematics in computation.
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Read at arXiv cs.LG