Mathematicians welcome computer-assisted proof in ‘grand unification’ theory

This has me in awe about the depth of mathematics, the pace of progress, the miracle of specialisation. I have a degree in an applied-math-y adjacent field, and understand nothing. (And, btw, I was astonished how knowledgable some commenters right here were, and then realised that we have the (co-)authors of the results themselves here! gotta love HN.)

With that said, here some (non-mathematical) snippets I found interesting (apart from the great word "sheafification"):

> Why do I want a formalization?

> — I spent much of 2019 obsessed with the proof of this theorem, almost getting crazy over it. In the end, we were able to get an argument pinned down on paper, but I think nobody else has dared to look at the details of this, and so I still have some small lingering doubts.

> — while I was very happy to see many study groups on condensed mathematics throughout the world, to my knowledge all of them have stopped short of this proof. (Yes, this proof is not much fun…)

> — I have occasionally been able to be very persuasive even with wrong arguments. (Fun fact: In the selection exams for the international math olympiad, twice I got full points for a wrong solution. Later, I once had a full proof of the weight-monodromy conjecture that passed the judgment of some top mathematicians, but then it turned out to contain a fatal mistake.)

> — I think this may be my most important theorem to date. (It does not really have any applications so far, but I’m sure this will change.) Better be sure it’s correct…

> In the end, one formulates Theorem 9.5 which can be proved by induction; it is a statement of the form ∀∃∀∃∀∃ (\forall \exists \forall \exists \forall \exists), and there’s no messing around with the order of the quantifiers. It may well be the most logically involved statement I have ever proved.

> Peter Scholze, 5th December 2020 [2]

Question: What did you learn about the process of formalization?

Answer: I learnt that it can now be possible to take a research paper and just start to explain lemma after lemma to a proof assistant, until you’ve formalized it all! I think this is a landmark achievement.

Question: And about the details of it?

Answer: You know this old joke where a professor gets asked whether some step really is obvious, and then he sits down for half an hour, after which he says “Yes, it is obvious”. It turns out that computers can be like that, too! Sometimes the computer asks you to prove that A=B, and the argument is “That’s obvious — it’s true by definition of A and B.” And then the computer works for quite some time until it confirms. I found that really surprising.

Question: Was the proof in [Analytic][4] found to be correct?

Answer: Yes, up to some usual slight imprecisions.

Question: Were any of these imprecisions severe enough to get you worried about the veracity of the argument?

Answer: One day I was sweating a little bit. Basically, the proof uses a variant of “exactness of complexes” that is on the one hand more precise as it involves a quantitative control of norms of elements, and on the other hand weaker as it is only some kind of pro-exactness of a pro-complex. It was implicitly used that this variant notion behaves sufficiently well, and in particular that many well-known results about exact complexes adapt to this context. There was one subtlety related to quotient norms — that the infimum need not be a minimum (this would likely have been overlooked in an informal verification) — that was causing some unexpected headaches. But the issues were quickly resolved, and required only very minor changes to the argument. Still, this was precisely the kind of oversight I was worried about when I asked for the formal verification.

Question: Were there any other issues?

Answer: There was another issue with the third hypothesis in Lemma 9.6 (and some imprecision around Proposition 8.17); it could quickly be corrected, but again was the kind of thing I was worried about. The proof walks a fine line, so if some argument needs constants that are quite a bit different from what I claimed, it might have collapsed.

Question: Interesting! What else did you learn?

Answer: What actually makes the proof work! When I wrote the blog post half a year ago, I did not understand why the argument worked, and why we had to move from the reals to a certain ring of arithmetic Laurent series. [...]

Question: So, besides the authors of course, who understands the proof now?

Answer: I guess the computer does, as does Johan Commelin. [Note: = deadbeef57 here on HN][3]

[1] https://mathoverflow.net/questions/386796/nonconvexity-and-d...

[2] https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-e...

[3] https://xenaproject.wordpress.com/2021/06/05/half-a-year-of-...

[4] http://www.math.uni-bonn.de/people/scholze/Analytic.pdf

A computer assisted proof is just as correct or helpful, it just may be more complicated of a proof initially. Given time said proof can be simplified but having the proof in the first place allows you to move away from assumptions into proofs or alternatively even open new doors that weren't known to exist.

Keep in mind that these computer aided proofs are equivalent to pen and paper proofs but because you can rely on software to guarantee you haven't made any mistakes, you can make more complex proofs that still work.

It's the same as with programming. You can write overly complex and opaque proofs in the same way you can write bad, slow, or hard to read code. It still "works" but it's not ideal and is often a first revision in a series of steps towards the clean, fast, and easy to read final products.

(Thanks to Curry-Howard, we can also view this as generating test suites/specifications for software libraries!)

Notable tools include Hipster, IsaScheme, IsaCoSy, QuickSpec and Speculate.