Mathematics has changed. Let's act like it

> The goal is to find solutions

> and to advance the field

Nobody seems to be disputing LLMs help with the former, but some are concerned that the latter is jeopardized by having aliens do our homework.

no, its just a language. physics constrains 1+1. "one plus one equals two" is difficult to transform

He phrases it like these criteria are requirements, but they really aren't for every problem. They make sense for those theoretical problems whose solutions' main values lie in their understandability and further development, but not every problem is like that. Some problems' solutions are just revolutionary on their own even if you contribute or understand absolutely nothing else. AI technology itself is an example of this!

To give a trivial hypothetical counterexample: if you came up with a solution for the traveling salesman problem that is both empirically sufficiently faster than the state of the art, that is an immense contribution and you absolutely deserve the credit for it regardless of whether you or anyone else understands how/why it works. It could literally even produce wrong results 50% of the time and still revolutionize humanity if it's efficient enough when it's correct. Denying them credit because they didn't understand the algorithm would be ludicrous - the results stand & speak for themselves.

Statisticians, traditional-AI researchers, etc. had to learn this the hard way when (basically black-box) neural nets became so good that their otherwise-elegant & explainable classical algorithms just stopped being able to compete in more and more applications, and got left behind. Are mathematicians looking to repeat that history?

Which is absurd on it's face for most day to day programmers, but a critical distinction for theoretical computer scientists.

The time saved using AI comes from reducing understanding.

The disaster for a problem-solution-oriented mathematician isn't that once problem-solution is easy they'll just switch to theory-explanation and machinery-building. It is that they will have to compete (and won't be able to compete) with students and post-docs who've been doing theory-explanation and machinery-building for their entire career - and there are lots of these 'cause academic pipeline produces more people who can fill a job than jobs in every imaginable category and subcategory.

"Results" in mathematics has always been the proof. (Preferably a beautiful and succinct one.)

Math is supposed to expose the workings of the epistemological foundations of the universe. The result itself (number or formula or whatever) is absolutely irrelevant to math.

(It may be important to physics, chemistry or engineering, but it isn't to math.)

I’m no mathematician, though I do have an undergrad degree in math, and unless I had a graduate degree or made substantial contributions at that level, I would never consider myself anything but a “student of mathematics,” and that too not just because I have a degree in the subject, but because I still have a great interest in certain branches of math and I keep learning in my spare time from books or from adjacent applied fields I continue to formally study. As a kid, I grew up hating math but I became interested in it later in life because of coding, computer science and formal logic, pushing myself a lot through my college years to get some foundational understanding of the subject at a deeper level than the general population, just to get my foot in the door, so to speak.

Even if I’m an outsider from the circle of mathematicians, this whole AI takeover of mathematics leaves such a bad taste in my mouth because I’ve long felt that math was one of those things that everyone could agree is a useful and beneficial thing for humanity in general, and it was one of the least divisive fields, where people from different parts of world, from different stations in life could participate, and very few points, especially today, caused any controversy (unlike just about every other subject). It feels like a great loss for humanity to lose a brunt of our engagement with it to the machines, like a whole wing of a gigantic library starting to be closed off from us indefinitely because some people decided that we don’t _need_ to access it anymore, except on an even grander scale possibly than my imaginary library metaphor.

Less capable (but good enough) mathematicians with more capital will have a huge advantage over great mathematicians who are broke.

I'm not a mathematician, but that situation rubs everyone (except those with the most capital) the wrong way.

This statement is a monstrosity, and the reason why real software engineers know vibe coding will fail. It is fundamental misunderstanding of thinking that that the intellectual work consists of knowing what you want, and coding is merely the mechanical labor of making it happen.

The coding is the intellectual work!

Its where the actual problems are discovered, where the difficult decisions are made, where vague ideas become precise, and where solutions are invented. There is not such thing as a fully formed solution waiting to be translated into code. :-)

Again just the modern variation of the idea guy who goes around saying I have a genius idea for a new startup...I just need some Developers to implement it ....

People have been dreaming running PS5 games on PC for a long time. Just when AI starting to become capable of doing that, Anthropic shut it down.

Yeah you can dream, but only what we allow.

There might be a “proof motive” in math which one could argue will drive the full automation of theorem proving. But we must recognize that profit is more universally valuable (i.e. it has value outside the industry in which it was created), whereas proofs are only valuable inside the math community (at least the pure math proofs the labs have been publishing thus far).

I guess you can dream big enough only if you have access to the really interesting models. The 20$ subscription won't get you there, and the 100$ one is already out of reach for so many people.

There’s an immediate resistance towards change rather than embracing it and finding opportunities to improve and work with it.

Maybe we should listen to people that care about negative externalities more often, and actually mitigate them.

You can drive 1000 miles and arrive at the destination on the highway quickly.

Or you can take the slow road and walk. You discover unrelated things, make new relationships, see the scenery, listen and smell what is around you and have time to contemplate things which may or not be related.

This particular tool economy robs us of all these things in the rush to the destination.

I'd sacrifice "progress" a little for some of the better bits of humanity.

Singularity? Probably not.

Tech bro dictatorship? Unfortunately a bit more likely.

None of the above? Then we have a small shot at a great future.

Having a machine that solves hard problems must be a good thing.

Like we can even know what the future of math is this early

It’s not like you can just declare that up front, it’s something that happens naturally when people spend time using the tools

In fact, they completely disregard all standards of mathematical dissemination. They just post a bunch of 100+ page papers online and call it the day.

A real mathematician polishes and polishes for months until it is understandable by the community. Then they communicate the results to a journal where there are editors and peer reviewers who come with detailed feedback which is then implemented. And just then it is ready to publish.

I wish mathematicians would just focus on this. This really is the problem, at the end of the day.

Having models get better at writing elegant proofs and explaining their results will not solve this problem - it will only make it worse, if anything.

We are all sympathetic to this. I, personally, am much less sympathetic to the absurd notion that it is "bad" that this new technology can solve important open mathematical problems at all, that we should stop trying to solve them, that things like the Mathathon are dangerous and needs to be shut down, that ugly proofs are the real problem here, so much so that you wish the open question weren't resolved at all, and so on. So silly.

But being pissed off that people are going to be out of work is something I agree 100% on. Hopefully, much like the internet, there's a way to make the whole thing a net benefit for society.

What you could argue though is that he is muddying the waters with all these guest posts on a domain that is as personal as it gets. I appreciate he's trying to be the 'Carlos Santana of mathematics' by sharing the spotlight but it might be good to highlight the distance a bit more, for instance by adding his own views.

Define "better".

Otherwise we're just creating a black box, easily manipulated by their owners, who would be ruling over human lives. That's not what people want

When AI writes proof no human can understand, it doesn't advance mathematics this way. Maybe AIs can understand their own proofs - meaning not just verify them but extract a machinery/broader-perspective (but I wouldn't count on it). But even machinery an AI isn't necessarily useful for humanity/society.

If that was true it would of course be significant.

If you had no morals, 2 trillion dollars to spend, ability to commit intellectual property theft without consequences, and most of the smartest people in the world actively working to advance your cause... ...you really think you couldn't compete with the clankers?

I would agree with you if these models were available to mathematicians, and OpenAI wasn't effectively competing with them by spending millions of dollars in compute to rush out solutions.

here is the direct link of the video which starts at 1800 seconds.

Offtopic: Fun fact but aside from archive.org there is also a way to have a link which links to any direct video at any time.

Someone made a good HN article about it that I vaguely remembered[0]

[0]: https://www.autodidacts.io/how-to-deeplink-to-specific-part-...

Some of these problems have had mathematicians dedicating their careers towards getting closer to a solution. Your comment reads like wishful thinking.

I won't even start on "AI is unconscious"...

[1] https://youtu.be/ThFq87Rp21s?is=xvgU-cKCpK0KwG_M

If we have a theorem proving machine, then we can use it to prove theorems. If we have a machine to solve the Langlands program in one day, then we'll think of more things to prove after that. There's a million unsolved questions in number theory unrelated to Langlands, and if we knew Langlands (and why it was true) we could make new conjectures that our theorem proving machine could solve, rather than having to speculate for 50 or 100 years like now.

The possibility he's not considering is that the machine might be better even at conjectures, and then there really is no role for humans once the machine already knows things our primitive meat brains are too small to understand. But what he's saying isn't "pants on head stupid".

It took us a while to make the planets line up neatly in a heliocentric system. Just last month I read that a whale mother "might" grief while carrying her stillborn child.

The limiting factor is human imagination and comprehension. No offense, but see your comment.

I think AI has already invented new math, but the guys haven't looked for it because they didn't think it possible. And even under their noses, they wouldn't understand what they see.

> It's not what you look at that matters, it's what you see. —Thoreau