Comment by measurablefunc
1 day ago
If it doesn't correspond to the original proof then you don't know what it is actually formalizing. It could be a buggy proof of ⊥.
1 day ago
If it doesn't correspond to the original proof then you don't know what it is actually formalizing. It could be a buggy proof of ⊥.
The thing is that Navier-Stokes has a definition split off separate from the formalization, and that is what has been completed. People have looked at the definition of the final statement. This paper only mentions the proof and intermediate statement, not the final statement. The most likely case to me is that intermediate statements do not match, but the end result still holds.
Seems kinda odd then that it didn't occur to OpenAI to iterate until they reached a fixedpoint for both the informal & formal development b/c it's obvious that correspondence should have been part of their training pipeline.
Their aim was to create a correct-looking proof using as little resources as possible, not to create a correct proof.
Jesus Christ, so many people here who have no clue what they are talking about.
A proof of a theorem is different from the statement of the theorem. OpenAI has a Lean proof of the statement. That is all they need. There may be many different proofs of this statement, including NL proofs. It does not matter that these NL proofs may or may not be different from the Lean proof, at least for the correctness of the Lean proof. But of course the NL proof may be wrong. But who cares?
The statement written in Lean, is not actually a correct description of the Navier-Stokes problem. It's some other easier statement. That's why the Lean code may not be a proof.
Here's a quote from the paper.
"3.1. When the NL paper declares stronger statements than what Lean proves
We commence with an example that complements those in §2. In the following example the AI autoformalisation results in a different Lean proof of a weaker statement."
"Tracing the proof of (3.3) we find that the series arises from applying the Lean theorem coefficient_seminorm_bound, just as Figure 3 mentions. Consequently, the Lean results discussed in this section are weaker than (3.1) in the NL proof."
So the question is, was the Navier-Stokes problem framed properly in the lean code, or is some easier problem represented in the Lean code?
Here is their remark.
"Remark 3.4 (Further potential mistranslations of the Navier-Stokes proof). The above examples require careful manual checking of both the NL proof as well as the Lean proof, which is delicate and highly time consuming. Moreover, the fact that we display only two examples does not mean that these are the only cases of mistranslations."
The "statements" here are statements of lemmas, not the statement of the main theorem. The main theorem statement was written by humans (prior to the OpenAI work), so there shouldn't be any concern that an AI mistranslated it.
> the NL proof may be wrong. But who cares?
The people trying to understand the proof are probably following the natural language version. So they care.
I wouldn’t be surprised at all if that’s how this paper (which I did not read) arose.
Why would they do that, knowing that the one known to be correct is the Lean one? Just to claim that the (correct) Lean proof did not translate well to English? That would be weak, and a colossal waste of energy and time.
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It reminds me of how provably secure software was all the rage for awhile. Until people found that the idealized system/lemmas were so far from reality that the proved security was worse than meaningless because it gave a false sense of security.
In order to prove security, you must first simulate the universe.
> Who cares?
Everyone. I don't think many people working in fluid dynamics were surprised you can find a blow-up in Navier-Stokes. What would advance human knowledge is understanding the situations in which a blow-up might occur. In that context, the lean proof is necessary, but the non-lean proof is more important.
> Jesus Christ, so many people here who have no clue what they are talking about.
Indeed. If only some of those people would see the irony.
What matters most of all, as any first year student of mathematics would know, is whether the formal problem statement corresponds to the NL statement. TFA specifically states that at least some of the allegedly proven formal statements DO NOT.
No. What the paper says is that in principle, translating NL statements to Lean statements is hard. Nobody doubts that, translating informal to formal text cannot be formally proven correct, so...
Does the paper give a single example of one of the OpenAI solved theorems with a Lean certificate where the Lean statement does not correspond to the actual statement from the mathematical literature? I don't think so, but in case I am wrong, feel free to provide that example.
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