Autoformalisation is increasingly used to verify mathematical texts, including those generated by AI, as in OpenAI's announced proof of blow-up of solutions to the Navier-Stokes equations. In this process, an AI system translates the text from a natural language (NL) into a formal language such as Lean. Once this translation is done, the argument expressed in the formal language can easily be mechanically verified. The purpose of this article is to demonstrate why this process may offer no confidence in the original NL argument, owing to the various difficulties in performing the translation semantically faithfully. In particular, we highlight that the problem of resolving ambiguities in mathematical NL text, which is necessary in order to provide semantically faithful translation, is arbitrarily high up in the Solvability Complexity Index (SCI) hierarchy/arithmetical hierarchy (the SCI $= \infty$). Hence, informally, providing semantically faithful AI autoformalisation is harder than any computational problem including the Halting problem (which has SCI $= 1$). To demonstrate the effect of this result we provide several examples of AI mistranslations of NL statements and proofs into Lean in practice, resulting in mismatches between NL proofs and their Lean `verifications'. These include OpenAI's announced Navier-Stokes proof. In particular, we show that the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations.
I think this paper is worth reading, and does highlight a valuable caution about the relationship between natural language proofs and Lean formalizations, especially in the case that they were generated by agents. There really is significant room for slippage.
That said, it’s important not to draw too strong a conclusion. The most likely scenario is that there is slippage between the Lean proofs and the natural language proofs. This does not by itself imply that the natural language proof is fundamentally wrong, or that the theorems have not been proven.
As long as the Lean statement properly captures the proposition in question, the Lean proof is almost certainly accurate. And if the Lean proof doesn’t capture the proposition, you’ll hear about if for the major results (“OpenAI didn’t really solve Navier-Stokes” would be enormous news).
I’ll share a comment I made on HN: https://news.ycombinator.com/item?id=49995154.
I think this paper is worth reading, and does highlight a valuable caution about the relationship between natural language proofs and Lean formalizations, especially in the case that they were generated by agents. There really is significant room for slippage.
That said, it’s important not to draw too strong a conclusion. The most likely scenario is that there is slippage between the Lean proofs and the natural language proofs. This does not by itself imply that the natural language proof is fundamentally wrong, or that the theorems have not been proven.
As long as the Lean statement properly captures the proposition in question, the Lean proof is almost certainly accurate. And if the Lean proof doesn’t capture the proposition, you’ll hear about if for the major results (“OpenAI didn’t really solve Navier-Stokes” would be enormous news).
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