Navier–Stokes Lost in Translation

Automated proof verification for AI-generated mathematics does not guarantee the correctness of the original natural language reasoning.

A new paper argues that AI systems attempting to translate natural language mathematical proofs into formal languages like Lean may not guarantee the correctness of the original natural language arguments. Researchers demonstrate that resolving semantic ambiguities in mathematical text is an infinitely complex computational problem, exceeding the difficulty of even the Halting Problem. They provide examples of AI mistranslations of natural language statements and proofs into Lean, including OpenAI's announced proof for the Navier-Stokes equations. The study specifically shows that the formal Lean proof for the Navier-Stokes blow-up does not accurately correspond to the natural language proof provided.

The autoformalization process, where AI translates natural language math into formal languages like Lean for verification, is critically flawed. This research highlights that the inherent ambiguity in natural language mathematical statements makes a semantically faithful translation an extremely complex, possibly unsolvable, computational task. This complexity means that even if an AI-generated formal proof is verified, it does not imply the original natural language explanation was correct, potentially undermining confidence in AI-assisted mathematical discoveries.

This finding has significant implications for the adoption of AI in formal mathematics and scientific discovery. If the translation step is unreliable, the verification of the formal output offers little assurance about the truth of the informal argument presented to humans. The paper suggests that current approaches to AI autoformalization may produce a false sense of security, necessitating a re-evaluation of how AI-generated mathematical results are trusted and validated, especially in high-stakes fields like physics and pure mathematics.