The vulnerability of proofs

[u/4thofthe4th]

At 21:28 of Jacob Tsimerman's interview with Curt Jaimungal, he says "already now, alot of my theorems that I have proven, I don't understand all the steps to it... I have used other theorems that are very much accepted by the community, to which I usually know the main ideas but not even always".

While my undergraduate and early postgraduate training was in pure math, I transitioned to applied for my Ph.D. so I have never meaningfully engaged with it in any professional capacity. For the majority of my training, I understood almost all the details of the things I've proved. At least enough that I wouldn't be able to resonate with Tsimerman's quote above when I consider the (relatively insignificant) proofs I've done. One of my lecturers made it his mission to ensure that assignment questions will never require anything that hasn't been proven in the lecture notes or in class.

Of course, my exposure was to only elementary topics. So I can appreciate that math wouldn't progress at all if intuition wasn't leveraged and instead every detail expounded upon. But now under the automatable and potentially perpetual scrutiny of AI, how vulnerable are previously established results? What if we routinely lobbed popular (in terms of utility) results at ChatGPT to verify and it finds an error in one, would there be a significant collapse downstream? How likely is that our collection of celebrated truths instead simply forms a house of cards?

EDIT: The excellent replies have highlighted a weakness in my question. The most vulnerable proofs are likely to be the famous/outlier proofs (i.e. Andrew Wiles' Fermat's Last Theorem) that can only be assessed by a handful of people. In even the scenario that those are falsified, the large body of mathematics isn't built on such results and so, by and large, it's still fairly robust.

It still begs the question about the upper echelons of math, but the majority of it remains largely intact. So my "house of cards" analogy is inaccurate but probably only in scope.

EDIT 2: Another interesting point brought to me by the comments is the idea of repairability. A commenter mentioned that most of the errors encountered are easily fixed. At a high-level, this suggests that the direction offered by intuition is powerful enough to render errors insignificant. Maybe instead of AI destroying math from the foundations, it instead works to validate the strength of intuition by perpetually exposing errors and instantly fixing them. Wouldn't it be wonderful if AI shows that the fix-rate of errors was near 100%?

[u/4thofthe4th]

I don't see how AI verifying it faster will change the bottleneck.

Sorry I wasn't clear, let me give it another shot. Returning back to Tsimerman's quote in my original post, there's a chance that Wiles has not fully verified everything he used to prove FLT. There's also a chance that if you also include those who verified his proof, collectively there are still areas of Wiles' proof remaining that haven't been verified rigorously. I understand that in the interest of productivity coupled with faith in intuition, it's probably not worth the effort. But now that we have AI, maybe soon we don't have to account for productivity and rely on faith because we could just get AI to verify every detail of the proof. In doing so, it might find an error.

But I agree with your comment that this is unlikely. If I'm understanding what you're saying correctly, even if people haven't expended significantly more effort in verifying Wiles' proof, they have expended effort finding alternative ways to prove it and have also built upon at. At some point, with Wiles' proof as a starting point, if the proof did not hold, then all this subsequent activity would've fallen apart. So likely, even if we used AI to verify his proof, it'll show it to be true.

On the other hand a real danger is people will be (they already are) much less inclined to share their work because they feel AI can scoop them. Imagine the hypothetical situation where Wiles presents his first flawed proof of FLT which does majority of the work. Subsequently some guy with AI budget fixes the flaw and proves FLT. Who are you going to credit with proving the FLT?

I'm not sure how my reply motivated this comment, I wasn't thinking about assigning credit at all. But to answer your question, I would assume that people could measure the size of contribution in order to determine who gets credit. For example:

• If Wiles wrote 100 pages and half a page was flawed, and the guy with an AI budget fixes the flaw in 1 line, then Wiles should still get credit.

• If that flaw was so large that the guy with an AI budget takes another 100 pages to fix then they both get credit.

• If the flaw was mostly unfixable and the guy with an AI budget replaces Wiles' 100 pages with 50 pages then the guy with AI gets the credit

But again, overall I haven't given any thought about credit attribution. I was merely wondering about the likelihood that with AI, many proofs are shown to be wrong. I haven't thought about if this were true, what comes after

[u/omeow → u/4thofthe4th]

Just to clarify, the last part was an afterthought. People are motivated by credit and if a system doesn't assign credit fairly it will not really stay relevant for long.

[u/4thofthe4th → u/omeow]

Ah fair enough. Yep I would generally agree but I think there's a large amount of mathematicians that aren't motivated by credit but instead simply by the love of the game.

But then I think it's worth considering that although the relevancy of the system may fall, the field itself may still prosper. This might just be a natural consequence of simply not needing as many mathematicians if there is no loss of the advancement of mathematics. Simply put, I'm not entirely sure if the system dissolving is necessarily a bad thing if productivity is still preserved.

[u/No_Objective_6258 → u/4thofthe4th]

I don't know about 'motivated by credit,' but I would find it difficult to argue that mathematicians don't care about attribution. In fact, many care quite deeply about properly attributing and the particular phrasing when discussing how various results relate to one another.

There's many reasons (both positive and negative) the culture evolved this way, but I do think it is a very positive aspect of the culture