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

https://www.nature.com/articles/d41586-021-01627-2

by Wxc2jjJmST9XWWL • 5 years ago

344 136 5 years ago

nature.com

qntmfred• 5 years ago

A quick intro from quanta magazine to Kevin Buzzard's work on computer-assisted proof systems

https://www.youtube.com/watch?v=HL7DEkXV_60&t=295s

lvh• 5 years ago

I recommend the following post, by the author of the proof, for deeper context. Especially near the end, they talk about some of the things they're trying to accomplish with it in plain(ish) English.

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

deadbeef5714• 5 years ago

See also [1] for a follow-up blogpost that is less technical.

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

ajarmst2• 5 years ago

The article is interesting, but that lede is incoherent. Many mathematicians accept computer proofs the way chess grand masters accept computer players. Computer “assistants” that generate proofs that humans cannot follow or understand will always be controversial, and the proofs they generate, even though accepted as valid, will always be decorated with an asterisk.

363849473754• 5 years ago

This is a proof assistant, not an automated theorem prover. The user has to supply* the mathematics and the proof checker formally verifies whether or not the steps are correct. It doesn’t have any creativity (that’s up to the mathematician).

*I should have clarified there is some proof generation, see the comment below by opnitro, but I meant the meat and potatoes of novel non-trivial proofs currently has to be supplied by the user.

ajarmst2• 5 years ago

Thanks for the clarification.

opnitro• 5 years ago

That's not quite true. Many of these proof assistants support some level of automation and proof search. I haven't used Lean specifically, but it's quite common in Coq for projects to write proof search techniques specific to the problem domain and utilize them in their proofs.

LudwigNagasena• 5 years ago

I don’t understand the chess analogy. How is it in any way similar?

kevinbuzzard7• 5 years ago

Lean and the other theorem provers turn mathematical proofs into levels of a computer puzzle game, much like a chess puzzle.

6gvONxR4sf7o2• 5 years ago

If not an asterisk, they’ll just have less impact. A proof generates a fact. The best facts and proofs are useful in that they help other things. You work becomes useful for my work, which may become useful for others.

CuriousCosmic• 5 years ago

It's still just as useful. The fact that's proven is what helps other proofs.

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.

kevinbuzzard7• 5 years ago

Presumably the asterix denotes "actually true"? ;-)

fjfaase• 5 years ago

For more details about the actual proof, see: 'Blueprint for the Liquid Tensor Experiment'

https://leanprover-community.github.io/liquid/index.html

xvilka5• 5 years ago

Interesting choice of the proof assistant though - some specific parts of the Lean's core are not completely decidable, moreover the upcoming Lean 4 version is incompatible with many libraries and proofs written for Lean 3. See also the discussion[1] if the Coq is suitable for number theory as quotients are ubiquitous here.

[1] https://github.com/coq/coq/issues/10871

ulber• 5 years ago

I wasn't aware of Lean not being sound and a quick search didn't come up with anything related to that. Could you link a source?

martincmartin• 5 years ago

See other comment. It's sound but not decidable.

deadbeef5714• 5 years ago

The claim that Lean's core is not completely sound is FUD. Completely bogus.

You might be confused by the fact that Lean's definitional equality is not decidable, but that doesn't mean it isn't sound. Nobody has ever found a proof of `false` so far in Lean 3.

The choice for Lean is actually quite natural: (i) it has a large and coherent library of mathematics to build such a project upon. And (ii), it has a substantial user base of mathematicians somewhat familiar with the subject at hand (i.e., condensed mathematics).

The initial challenge by Peter Scholze was directed at all theorem proving communities. As far as I know, only the Lean community took up the challenge.

(Concerning Lean 4: yes, mathlib will have to be ported. And yes, people are working hard on this. I think that none of the theorem provers so far are ready for wide-scale use in mathematics. And that's why it is important to iterate fast. The Lean community is not afraid of redesigns and large refactors when discovering better approaches.)

xvilka5• 5 years ago

You might be confused by the fact that Lean's definitional equality is not decidable, but that doesn't mean it isn't sound. Nobody has ever found a proof of `false` so far in Lean 3.

You are right, my bad. Taking my words back on that.

A bit more details from the Pierre-Marie Pédrot:

Lean does not escape this limitation, as it defines the equality type inductively as in Coq. Rather, in a mastermind PR campaign, they successfully convinced non-experts of type theory that they could give them quotient types without breaking everything around.
The first thing they do is to axiomatize quotient types, which is somewhat fine and can be done the same in Coq. As any use of axioms, this breaks canonicity, meaning your computations may get stuck on some opaque term. (While slightly cumbersome, axiomatizing quotients already solves 99,9% of the problems of the working mathematician).
Now, were they do evil things that would make decent type theorists shake in terror, they add a definitional reduction rule over the quotient induction principle. We could do the same in the Coq kernel, but there is a very good reason not to do this. Namely, this definitional rule breaks subject reduction, which is a property deemed more important than loss of canonicity.
In Lean, you can write a term t : A where t reduces to u, but u doesn't have type A (or equivalently may not typecheck at all). This is BAD for a flurry of reasons, mark my words. Reinstating it while keeping quotients requires an undecidable type checking algorithm, which is another can of worms you don't want to wander into. The same kind of trouble arises from similar Lean extensions like choice.

kzrdude• 5 years ago

Shouldn't it be easy to automatically port theorems? If they are well defined there can't be ambiguity

generationP• 5 years ago

As to decidability: I suspect you mean that Lean's foundations are non-constructive. I'd rather they weren't, but with what Peter Scholze is doing, I'd be surprised if the arguments could be made constructive within just a few years from their conception. This usually takes much longer (e.g., constructive proofs of the Quillen-Suslin theorem appeared in print just a few years ago).

About incompatibility: Last time I checked (some 3 or so years ago), Coq was not backwards compatible either, and libraries had to be ported manually with each update. Sadly, as this is the one greatest anti-feature that is currently putting me off proof assistants, but probably it needs some time, and the quickest way to get there is to maximize usage. From what I know about Coq and Lean, I suspect Lean will stabilize faster than Coq does, due to it being more "declarative" (Coq is based too much on complex tactics, which are hard to make stable by design).

Barrin92• 5 years ago

The abiltiy to automate proofs creates some interesting questions about the nature of mathematics. The article remined me of Erdos saying that you "don't need to believe in God, but you do need to believe in the book", 'the book' here being an imagined collection of mathematical proofs that are so simple, clear and beautiful that they immedieately stand out to any mathematician.

I don't mind proof assistants as a way to gain new insights into mathematics, but I worry that maths is drifting into a direction where it turns more into hermeneutics than actual mathematics. The automation of proofs isn't the only thing, I also was very scpetical about the whole process of Shinichi Mochizuki's proof of the abc conjecture.

deadbeef5714• 5 years ago

As explained in another comment, there is only very mild proof automation going on in this Lean project. Every non-trivial idea has to be supplied to the computer by a human being.

The whole circus around Mochizuki's proof of the abc conjecture was dealt with quite well by the social structure of the mathematical community. Many people looked at the proof. Many people got stuck. Several experts got stuck at exactly the same point. And Mochizuki refuses to acknowledge that there is a problem at that point of his "proof". But a consensus was reached in the mathematical community (maybe minus RIMS).

kungito• 5 years ago

What's the name of this point? It's really hard to find any progress on this topic regarding Mochizuki other than some popular articles without any content.

citrusynapse• 5 years ago

Made an account just to share this incredible book that follows Erdos's life & exploits:

https://en.wikipedia.org/wiki/The_Man_Who_Loved_Only_Numbers

civilized2• 5 years ago

Automated proofs aren't remotely comparable to Mochizuki's abc stuff. Automated proofs just handle a lot of complicated case by case checking that humans could in principle do, it's just too much work, like counting to a trillion. Automated proof systems are incapable of the brilliant but unreliable intuitive leaps that mathematicians can make.

Mochizuki's stuff is simply a hypercomplicated pile of nonsense unintelligible to the mathematical community.

davnn• 5 years ago

Automated proof systems are incapable of the brilliant but unreliable intuitive leaps that mathematicians can make.

I would add current automated proof systems. IMHO we are just at the beginning that mathematicians realize the usefulness of technology, which was basically neglected ever since.

Ericson23143• 5 years ago

On the contrary proof assistants help one systematize and organize existing concepts. They are a great way to revise the basic curriculum and revolutize pedagogy. They make the creative process of defining new mathematical objects more accessible, and easier to motivate via unlocking more code reuse. They are a great way to revise the basic curriculum and revolutize pedagogy.

zant2• 5 years ago

I strongly agree on this one. As a programmer and someone who's always been interested in mathematics, having a way to check proofs without relying on trust of my own logic or a third agent, but solely on a prover like Lean or Coq is a really big difference.

visarga• 5 years ago

but I worry that maths is drifting into a direction where it turns more into hermeneutics than actual mathematics

Related to the unexplainability of AI. We're just grasping for more than we can hold in our little working memory.

gigatexal• 5 years ago

For anyone frustrated that the article doesn’t say what specific part of math has the most to gain it’s here:

“ The crucial point of condensed mathematics, according to Scholze and Clausen, is to redefine the concept of topology, one of the cornerstones of modern maths. A lot of the objects that mathematicians study have a topology — a type of structure that determines which of the object’s parts are close together and which aren’t. Topology provides a notion of shape, but one that is more malleable than those of familiar, school-level geometry: in topology, any transformation that does not tear an object apart is admissible. For example, any triangle is topologically equivalent to any other triangle — or even to a circle — but not to a straight line.

Topology plays a crucial part not only in geometry, but also in functional analysis, the study of functions. Functions typically ‘live’ in spaces with an infinite number of dimensions (such as wavefunctions, which are foundational to quantum mechanics). It is also important for number systems called p-adic numbers, which have an exotic, ‘fractal’ topology.”

dr_kiszonka• 5 years ago

I know nothing about topology. If you have time, could you please explain this sentence?

"For example, any triangle is topologically equivalent to any other triangle — or even to a circle — but not to a straight line."

Is it because triangles and circles are "2D" and lines aren't?

yongjik• 5 years ago

Crudely speaking, topologists consider spaces as if they're made of rubber - a mathematically perfect rubber that can be made infinitely thin or stretch to infinity. So, a circle can be made with an infinitely thin circular rubber ring, and you just pinch three points and stretch, and you get your triangle, in any shape.

But you can't get a straight line - to do that you need scissors to cut one point of a circle (to be precise, remove a single point) - and then you can stretch the remainder to infinity and now you have your line.

77pt77• 5 years ago

No. Both are 1-d (lines).

It's because triangles are loops (closed) and straight lines aren't.

hopfenspergerj• 5 years ago

If you remove a point from a line, it breaks into two pieces.

QuesnayJr• 5 years ago

You can give a function that maps the points of the triangle to the points of the circle in such a way that this function is continuous. (Continuous basically means "no jumps".)

You can give this function explicitly. Let's assume that the triangle is drawn on a piece of paper with an x and y axis, such that the intersection of the axes (the origin) is inside the triangle. For each point on the triangle, draw a line from the origin to the point. Take the angle between that line and the x-axis. Use that angle to map it onto a circle.

sabujp• 5 years ago

    "Proof assistants can’t read a maths textbook, they need continuous input from humans, and they can’t decide whether a mathematical statement is interesting or profound — only whether it is correct, Buzzard says. Still, computers might soon be able to point out consequences of the known facts that mathematicians had failed to notice, he adds."

we're closer to this than people realize

GPerson2• 5 years ago

Why do you say this?

kevinbuzzard7• 5 years ago

I'm not quite sure what you're asking about. I'm saying that we can't yet take the Wiles and Taylor-Wiles proof of Fermat's Last Theorem, feed it into a machine, and get a Lean proof of Fermat's Last Theorem.

kevinbuzzard7• 5 years ago

I agree, but I think my statement is accurate today in 2021. I would love to see funds directed towards this sort of question. The big problem is that at high level so so much is skipped over, and you still sometimes have to struggle to put undergraduate-level mathematics into Lean -- this is why UG maths is such a good test case.

mherrmann• 5 years ago

Very nice to see you here Kevin. We never interacted but I do still remember a lecture you gave at Imperial in '06 where you filled in for Prof. Liebeck and started with Lemma 1: "I am not Professor Liebeck" ;-) Thank you for the nice memory and your important work on / with Lean.

zozbot2345• 5 years ago

The big problem is that at high level so so much is skipped over

This is an issue, but there's an established practice of writing formal sketches where the gaps in the proof are explicitly marked, and future tooling might bring ways to address these gaps once a full formal context is provided.

One issue is that Lean has little or no support for declarative proof, which is by far the most natural setting for these "proof sketches", and also brings other advantages wrt. complex proofs. (Coq has the same issue; some code was written to support declarative proofs, but it was too buggy and bitrotted, so it got removed.)

gwern• 5 years ago

It's worth noting that GPT-f already gets a big performance boost from pretraining on Arxiv etc (https://arxiv.org/pdf/2009.03393.pdf#page=7) despite those sources containing next to no Metamath or anything that looks like a raw Metamath proof, just regular natural language & LaTeX discussing math...

astrange• 5 years ago

How well does text extraction from a PDF work? I almost never try it but thought there were random spaces in the output and such things.

Haga• 5 years ago

Can't usefulness be approximated like Google search results of old, by connectedness to other theories.

amelius• 5 years ago

we're closer to this than people realize

At least give a proper reference to what you're alluding to, please.

Also, closeness in AI has shown to be a misleading concept.

gerdesj• 5 years ago

A reference you might like to note is in a response - that kevinblizzard bloke probably has a fair old handle on this stuff. Note how he is quoted throughout the article.

This is about some pretty creative uses of computing in maths and bugger all to do with AI (whatever that is.)

If you put enough blood, sweat and tears into codifying mathematical concepts into Lean, you can feed it a mathematical thingie and it can tell you if that thingie is correct within its domain of knowledge. If you get an "out of cheese error", you need to feed it more knowledge or give up and take up tiddlywinks.

This explains Lean in terms I can understand: https://www.quantamagazine.org/building-the-mathematical-lib...

AtlasBarfed2• 5 years ago

Simply having a linked graph of related concepts might show "impact diffs" in theorems.

I recall that the Fermat's proof linked several normally disparate areas to get to the meat of the issue.

Simply tagging those relations to identified sub-fields of study will probably help give guidance to impacts of theories, maybe farm them out to advanced students for quick review.

SneakyTornado29• 5 years ago

Last time someone talked about automating proofs, a whole new field was invented (computer science)

denial• 5 years ago

Does anyone know how condensed mathematics would fit into the modern theory of PDEs (which is heavily based on functional analysis)? Perhaps it's a relic of the sort of math Scholze works on, but it looks far too abstract to provide an impetus for people in those fields to embrace it. Topology, on the other hand, is relatively easy to define and work with (though there are some quirks with dual spaces of continuous linear functionals I've seen aesthetic objections to). Or does it "contain" topology in some sense, allowing people to continue working with notions of convergence obtained from norms?

deadbeef5714• 5 years ago

I think that right now it is not clear why condensed/liquid mathematics would be useful for PDEs. On the other hand, your question

Or does it "contain" topology in some sense, allowing people to continue working with notions of convergence obtained from norms?

has a positive answer. You can, if you want, swap out topological spaces, and use condensed sets instead, and just continue with life as usual.

At the same time, all of this is in fast paced development, so hopefully we will see some killer apps in the near future. But I expect them more in the direction of Hodge theory and complex analytic geometry.

wolverine8763• 5 years ago

Thanks for sharing your expertise. Would you be open to sharing your background? Obviously it's not required, but it would help contextualize what you're saying for the interested non-mathematician; otherwise we're kinda stuck with 'some guy on the Internet said ...' syndrome. :)

est2• 5 years ago

is there a simpler version of LEAN suitable for high school student level math? Sympy?

deadbeef5714• 5 years ago

You might enjoy http://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game...

It's a game implemented in lean, where you work your way through the basic facts about natural numbers.

Tainnor5• 5 years ago

sympy is not a proof assistant, but a symbolic computer algebra system

est2• 5 years ago

Well I'd say it's enough to automatically solve many high-school level problems, automatically. Like a poor man's Mathematica or WolframAlpha.

Tainnor5• 5 years ago

This is going to be an exciting area.

I spent some time previously playing with Coq. It's very powerful, but even proving the simplest undergraduate maths statements (say, about group theory) can prove very challenging. I believe that part of this is that Coq uses different mathematical foundations than traditional mathematics, which mostly uses set theory (ZFC, although most people don't care about the specifics). So it can be hard or unnatural to express something like "subgroup". I don't know if Lean fares better in that respect. Coq documentation is also IMHO almost impossible to understand unless you're already very deeply knowledgeable about the system.

We will probably still need some more iterations, to get more user friendly assistants with better documentation and to get adequate teaching resources etc.

raphlinus• 5 years ago

Lean's foundations are similar to Coq. I think the ergonomics are a bit better.

Most activity in proof systems is based in type theory these days, but set theoretical systems do exist, of which Metamath is the most mature. That said, Metamath is seriously lacking in automation, so there is an element of tedium involved. That's not because of any fundamental limitations, but I think mostly because people working in the space are more motivated to do things aligned with programming language theory. There was also a talk by John Harrison a few years ago proposing a fusion of HOL and set theory, but I'm not sure there's been much motion since then.

I believe a fully featured proof assistant based in set theory would be a great contribution.

[1] http://aitp-conference.org/2018/slides/JH.pdf

robinzfc• 5 years ago

It depends on what you mean by "fully featured" but Isabelle supports ZF logic [1] and also you can do ZFC in Isabelle/HOL [2]. And, of course there is Mizar [3].

[1] https://isabelle.in.tum.de/dist/library/ZF/ZF/index.html

[2] https://www.isa-afp.org/entries/ZFC_in_HOL.html

[3] http://mizar.org/

Ericson23143• 5 years ago

You don't need subgroups, you just need injective homomorphisms.

Jweb_Guru7• 5 years ago

That's a nice idea but it tends to fall flat when you can't actually construct an explicit function exhibiting the homomorphism in question (which does happen from time to time). A lot of equivalences in areas like group theory only hold classically.

zozbot2345• 5 years ago

ZFC was only ever developed as a proof of concept, not as a practical foundation for formal math. Structural set theories, type theories or category-based foundations are actually a lot easier to work with, and otherwise quite equivalent in power.

Tainnor5• 5 years ago

This is missing the point. Modern mathematics textbooks, especially undergratuate ones, are written with set theoretic foundations. It requires a lot of effort to reformulate all of mathematics into equivalent formulations. That makes it harder to get buy-in from many working mathematicians, and it also makes the subject less approachable for, say, undergraduates.

deadbeef5714• 5 years ago

In my experience, the difficulty is very rarely the transition from set theory to type theory. I find this almost transparent in practice.

The issue is rather that you need to deal with edge cases that are usually swept under the rug, or that you need to spend a full page working out the details of a proof that everyone recognizes as obvious. It would be great if computers could give even more assistance with these tedious parts of formalization, and I'm very glad that people are working hard on realizing this.

ganzuul• 5 years ago

Kurt Gödel with a bat in dark alleyway of obviousness haunts my dreams.

chriswarbo3• 5 years ago

One area I've been playing with is "theory exploration", which takes a set of definitions and produces lemmas that are 'interesting' (i.e. irreducible, via some scheme like Knuth-Bendix rewriting).

(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.

ixaxaar• 5 years ago

Hey type hacking is hard. Agda is kinda easier for someone who knows haskell, but hard nonetheless.

kmill3• 5 years ago

Here's the definition of a subgroup in Lean's mathlib: https://leanprover-community.github.io/mathlib_docs/group_th...

Given a group G, there is a type `subgroup G` of subgroups of G, and a subgroup is a structure consisting of a set of elements of G along with some proofs that it has the identity, it's closed under multiplication, and it's closed under taking inverses. Lean has a coercion facility using typeclasses, and there is a typeclass instance that will use the subgroup's set coerced to a type when the subgroup is used in a place that expects a type. This coerced type has a group typeclass instance, so a subgroup "is" a group.

The big complexity in all of this doesn't seem to be ZFC vs type theory, but rather how do you implement a mathematician's use of synecdoche and implicit coercions. Synecdoche is the figure of speech where you refer to a thing by its parts or vice versa -- for example, "let G be a group" and then using G as if it were a set, even though a group consists of a set, a binary operation on the set, a choice of identity element, and a few axioms that must hold. Mathlib uses typeclasses to implement the synecdoche -- a type is a group if there is a typeclass instance that provides the rest of the structure. As I understand it, Coq's Mathematical Components library uses structures and unification hints instead (though I've failed to understand how it works exactly), but I have heard that they can be very fiddly to get right.

I think you'd have to find solutions to these same problems no matter the foundations. At least with types, there's enough information lying around that, for example, group structures can be automatically synthesized by the typeclass resolution system in many common situations.

mjreacher• 5 years ago

Is there any opportunity for interested undergrads to learn about this more (since I doubt we could contribute)?

foooobar7• 5 years ago

If you're interested in interactive theorem proving with Lean (and not condensed mathematics), the Lean community landing page is a good place to start. https://leanprover-community.github.io/

Especially the "Natural Number Game" under "Learning resources" has been successful in teaching folks the very basics for writing proofs. Once finished, a textbook like "Theorem Proving in Lean" can teach the full basics. Feel free to join the Lean Zulip at any point and ask questions at https://leanprover.zulipchat.com/ in the #new members stream.

Mathlib has plenty of contributions from interested undergrads :)

FabHK• 5 years ago

TIL: Fields medallists ask questions on MathOverflow... [1]

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

hansor• 5 years ago

They did the same with Prolog in 70" and... failed. Nothing new.

contrarian_5• 5 years ago

i love the time that we live in. people talk about there being a glut of podcasts, but i can type "Peter Scholze" into youtube and there is a full hour-long interview with him hosted by another mathematician. you can see her channel is very new and almost certainly a part of this latest wave of pandemic podcasters. its so great to take an interest in a random person from a nature article and be able to immediately get a visceral idea of who he is and what hes about.

https://www.youtube.com/watch?v=HYZ3reRcVi8

bpcpdx• 5 years ago

This kinda gives me chills because I can't help but think of Warhammer 40k.

Essentially our civilization progressed to the point where computers started to take more of a role in new discoveries much like in the real world. As computers got more powerfull and AI developed naturally scientific advancement sped up. But there came a point when the science and math became too advanced for humans to even comprehend, so computers did it. Then there came a point when things were so advanced that even scientists couldn't even ask the right questions so an AI intermediate would have to be used.

Then things get weird and you have the 40k universe.

Anyways I know I probably butchered it a little but that's the gist of it and I can totally see things progressing in the real world up to the point where things get weird in 40k.

runeks• 5 years ago

But systems known as proof assistants go deeper. The user enters statements into the system to teach it the definition of a mathematical concept — an object — based on simpler objects that the machine already knows about. A statement can also just refer to known objects, and the proof assistant will answer whether the fact is ‘obviously’ true or false based on its current knowledge.

As far as I can see this is just programming. How is this different from writing in Java

  int i = “hello”

and seeing that the Java compiler rejects this “thesis”?

Of course, we need more complex types than “int” and “String”, but in principle it’s the same.

fspeech• 5 years ago

Yes and the isomorphism is known as Curry-Howard Correspondence https://en.wikipedia.org/wiki/Curry-Howard_correspondence

zant2• 5 years ago

Yeah you can say that. You can also say that Machine Learning is just programming. Or in a similar way you can also say that First Order Logic is just programming.

However, the cool thing about programming is that it lets us represent a lot of different things. In this case you're representing the construction and interaction of mathematical objects, with a language that targets a specific proof management system to verify this constructions.

But yes, it is "just programming", and some functional languages even support proofs to some extent like Scala or Agda.

AtlasBarfed2• 5 years ago

The Lean version of the theorem was 10,000s lines of code...

With verifiers like this as a useful tool, I'm guessing in the coming years this LoC will be dwarfed.

I'm a bit surprised a theorem of major complexity reduced to that small a LoC count.

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

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