[svisser]

The opening sentence of a linked article sums it up quite well: "If nobody understands a mathematical proof, does it count?"

The abc conjecture may be solved today but only when a sufficient number of people understand and accept the proof as a proof.

[pacala]

Formalize it with Isabelle / Coq / Lean. Then it counts.

[tobhahn]

Math is more about figuring out why things are true than what is true.

Mathematicians want to understand Mochizuki. They couldn't care less if hn thinks it counts.

[SomeStupidPoint]

I don't know why this got flagged.

At the point that we no longer understand the relationship between the formal proof witnesses (and really, the class of possible witnesses) and the axioms we choose, we can no longer do mathematics, because we can no longer meaningfully explore axioms -- our ability to make guided changes is destroyed by our inability to understand their effect.

It's important for the community to understand something of why a thing, not just that it's true, because that's why drives the development of mathematics forward. (And indeed, particularly so in the ABC conjecture, which sits at a node between the nature of multiplication and addition, which don't usually have much to do with each other.)

I actually wonder if US (and perhaps other) math education is harmful here: the focus on rote learning and just knowing that a thing is true (to mechanistically apply it) has conditioned people to not understand why the hesitance over proofs that humans don't understand -- for most of those people, they never understood the proofs anyway.

[pacala]

That is an excellent point. The future is mechanical proofs. Which will bring tools for proof search, proof refactoring, proof minimization, proof navigation, theorem generation. We'll be able to tackle _much_ harder problems, while still being able to get the gist of it.

Sometimes I'm saddened that this future may come slower than we'd like due to imperfect funding structures. But I've grown a lot of patience over the years :)

[SomeStupidPoint]

I'm actually very pro-formalization and mechanical verification -- both for mathematics and computer science. $HOBBYPROJECT involves automated theorem proving, while I'm trying to convince $DAYJOB to adopt some formal methods.

I was just pointing out that the person got flagged for commenting that "witness and dump" isn't actually very useful for mathematics as a field, except as a signal that we should investigate a topic further. But in the case of the ABC conjecture, we already have plenty of incentive to investigate.

I think mathematics and science have a lot of learn from computer science in terms of managing large models, proofs, etc -- and that we'll get a lot of automatic tools. That will all be really great.

But there are proofs that are basically just brute-forcing a solution for which we have no higher-level understanding, and those don't really add much by way of knowledge to mathematics. At the point that those are all we can generate for "big" problems, we may be in trouble.

[pacala]

Another excellent point. Right now it's a "winner takes all" competition. It matters to prove a result, and much less to provide an "elegant" proof. I can only hope for a future where we measure the algorithmic entropy of a proof [log proof length][0], and results like "ABC theorem proof using half the bits as best known proof" become notable.

[0] https://en.wikipedia.org/wiki/Kolmogorov_complexity

[haskellandchill]

I'm interested in your $HOBBYPROJECT. I'm hoping to develop some software similar to edukera.com, ie using proof assistants as educational tools for mathematics. Would love to swap some cool links and references. Thanks!

[fourthark]

Fully agree. But person probably got flagged for commenting that what is important to hn may not be what's important to mathematicians.

[Koshkin]

> The future is mechanical proofs.

That almost sounds like "the future is mechanically composed novels" (or music). Understanding why something is true is just as important as knowing that it is. Mechanical proofs will be impossible for humans to understand, so the value of such proofs will be rather limited (in that people will still continue searching for a "real" proof).

[pacala]

Mechanical proofs and understanding are not mutually exclusive, that's the whole point! Just like mechanical processes [aka software] are still understandable. A stronger conjecture is that mechanical proofs will enable _better_ understanding, as tools and metrics are developed for tackling "understanding".

[dsacco]

> The future is mechanical proofs.

For some value of "future" - sure, maybe. But the vast majority of theorem solutions do not lend themselves to mechanical proofing, and it takes great effort to do it at all.

I don't really agree with your thesis here. I don't see how we're going to get to a future where we're tackling much harder proofs, because the hardness is already what prevents them from being so easily mechanized and checked in an automated manner. We have no problem coming up with new theorems, either - half the job of a pure math researcher is coming up with interesting questions that are too hard to solve immediately but not quite so hard that they're inaccessible.

The only way to make the proofs "harder" is to make theorems that are even further removed from our current mathematical capabilities. Otherwise you're stepping into the undecidable territory. Simply put - solving open problems is a major activity in mathematics because it develops new mathematics, not because the actual end result is useful in of itself. If I prove to you that Pi is normal, the fundamental contribution is (hopefully) a method that is partially or fully generalizable to other domains. No one really cares if Pi itself is normal, and most already expect it to be. To mechanize that process (if it's even possible at all) requires that the mathematics for solving it already exists, which means that the problem is most likely either 1) uninteresting, 2) overlooked.

[chuckdries]

People have a lot of very valid complaints about common core, but at its core (hehe) it does seem to attempt to start to remedy your complaint about US math education at least at the high school level.

Hopefully that can propagate upward.

[skybrian]

I don't see these approaches as necessarily opposed. It seems like in principle, formalizing a proof and verifying that it doesn't have any errors could be a preliminary step towards understanding it? (Though in practice, perhaps this is overly roundabout.)

[SolarNet]

Also, once enough people figure it out, one (or some) of them can make better explanations.

[sanxiyn]

Indeed. We now actually have some math results of interest for which we have only formal proofs and no human proofs. https://arxiv.org/abs/1509.05468 is an example.

[mikebenfield]

We are very far from the point where that is feasible for many new results in math.

[MikkoFinell]

>"If nobody understands a mathematical proof, does it count?"

Just wait until general AI really kicks off, then all of new math will be like that. It's not that humans are bad at logical thinking, our weakness is memory. That won't be the case for an artificial agent with instantaneous perfect recall of everything it has ever seen.

[BucketSort]

lol. I think the resurgence of interest in constructionist mathematics via Homotopy Type Theory will lead to better proofs ( since all proofs in HoTT are like programs and can be computationally verified in a straightforward way ).

[LeifCarrotson]

Even an AI will have various levels of cache. Some memories will be register-level instant, some will be thousands of miles and quite a few servers away, and others in between.

[MikkoFinell]

That's a good point. Lets throw "instantaneous" out, having the ability to store and perfectly recall data is a critical advantage for any thinking entity.

[posterboy]

>instantaneous

as the memory will exceed ram and then disk-space, that will be less true.

[KGIII]

Mathematics is a language. You don't really need to memorize it, you can read it. Unfortunately, most people aren't exposed to anything higher than arithmetic.

[posterboy]

yeah, well, have fun reading e.g. the stacks project[1] of round about 4000 pages without remember the necessary steps leading up to a corollary.

[1] https://stacks.math.columbia.edu/browse - abstract algebra as far as I can tell

[KGIII]

You're sure as hell not going to memorize it.

[Koshkin]

I remember having to memorize the multiplication table. And that was just the beginning...