[datsci_est_2015]

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for humans to identify what is meaningful and what is not in the slop.

[canjobear]

Your model of what AI is good at is wrong. Generative AI is not good at wandering off into novel esoteric abstract corners while maintaining correctness, it is good at things that are close to its training data. I suspect that humans will long outperform AI in the domain of "novel esoteric abstract useless math" whereas AI will outperform humans in the domains of (1) making connections between already-well-understood concepts, things that seem obvious in retrospect but which no human figured out just because of the accidents of what people happened to focus on, and (2) proving things that require long, tedious, intellectually unsatisfying calculations, which would cause a human mathematician to give up for boredom.

[datsci_est_2015]

My understanding is that we’re talking about “tool-assisted” proof generation, which provides some guard rails but would still allow significant creativity. Tools like Lean, Coq, etc.

[sunshowers]

Elliptic curves over reals and the complex numbers had some physical/scientific meaning, but elliptic curves over finite fields had none before cryptography.

[yaris]

Works of Shinichi Mochizuki immediately come to mind. He is not AI but provides very good examples of math that is useless because it is incomprehensible by (other) humans.

[seanmcdirmid]

Do AIs produce answers whose work is incomprehensible to humans? It seems like you could just have the AI elaborate multiple times until you were satisfied with the explanation and documentation of what went into figuring out the answer. It’s not like the AI is one shotting the answer in a single opaque query anyways.

[datsci_est_2015]

Like other commenters, I think you’re also underestimating the complexity of esoteric higher level math.

Consider the “Magnus Carlsen” of mathematics, who is more capable of understanding mathematics than any other human. But then also realize that that individual has probably devoted their entire career into a specific subdomain of mathematics. Within other deep recesses of mathematics, this Magnus equivalent will be less capable than their peers without years of rewiring their brain to understand the esoteric concepts and properties within that other subdomain.

LLMs will be able to dig deeper and broader than any human mathematician, and find results that are completely useless to humans because it would take more than an entire lifetime to “speak the language” of the concepts the LLMs have produced. The only way those results can become useful to humans is if then the LLM itself finds a way for it to be practical to humans once again.

So, no, I don’t think this represents the “democratization” of mathematics where mathematicians are no longer necessary because anyone can just prompt the LLM to explain it. The bar for entry level mathematics is lower, for sure, but research level mathematics will continue to be unapproachable for anyone who hasn’t devoted their career to it.

[storus]

> “speak the language” of the concepts the LLMs have produced

LLMs don't produce concepts, they just predict next tokens; they can't invent new concepts, only synthesize what is in their probability distribution already. They can mix/fuse vast areas of math together that are inaccessible to individual mathematicians, but can't create new concepts not present in their probability distribution.

[seanmcdirmid]

I don't get it. LLMs don't have ego, they don't have the ability to say "no, this should be obvious, I'm not going to explain further", they are just token predictors, and given context, they can generate more tokens. If you don't understand how the answer was derived? You just ask more questions and it isn't going to get bored or annoyed, it will just try to answer the questions.

Is that what is offending you so much?

[datsci_est_2015]

No, it doesn’t sound like you get it. It has nothing to do with the properties of LLMs and everything to do with the complexity of mathematics.

Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly.

Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand.

My point is the human is a critical piece to the puzzle, but not just any human, a career mathematician.

[bawolff]

> Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly.

> Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand.

This feels like a little bit of a jump to me. AIs arent actually alive so of course someone is going to have to pose the question. They arent going to just do stuff on their own. And of course mathmaticians are going to need to interpret the results if we are to glean anything beyong if the conjecture is true or false.

But you seem to be suggesting that mathematicians will have to micromanage every step. That seems like a bit of a jump which i dont see much evidence for.

[duchef]

> Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly.

I'm really interested in this anecdote. I have never experienced this but have a reasonable academic background (BSc, MSc, MD) - and I am certainly not the person you're describing. Could you elaborate? Is this something more exclusive to pure mathematics (my bsc/msc are CS).

[seanmcdirmid]

> Have you ever been exposed to concepts that are so complex that you feel like you could devote your entire lifetime to trying to understand it and still fall short? It’s a very humbling experience, especially if you have classmates who pick it up effortlessly.

I do have a PhD so I kind of know how that feels. I watched my entire field (PL) get eaten up by AI though, the problems that I thought were huge 10 years ago are just silly footnotes now.

> Without a human holding the reins, consider an LLM a rudderless superboat speeding erratically towards the horizon, finding and proving meaningless theorems that not even your most talented classmate could ever begin to understand.

I don't disagree with that. LLMs are a tool, a super fast pattern matcher, research, token predictor. I don't expect it to go out and define its own esoteric (or useful) problems to pursue without human interaction. That's for the humans to do.

I don't understand what that has to do with my original comment though. I wasn't addressing what problems the LLMs were answering, just how to review and dissect the answers that they would come up with.

[BalinKing]

Excluding supergeniuses, pure mathematics—even at a very basic, undergraduate level—simply can't be understood passively. Even with an infinitely patient AI teacher who could answer any question on-demand, it'd still require a massive amount of work to actually understand anything in research-level mathematics. Basically every single word in a mathematical definition is a term of art, and (IME) if one doesn't grok each of those words at a fairly deep level, the new definition never really makes too much sense. And this applies recursively: each of the words has some thoroughly inscrutable definition of their own.

Of course it'd be super helpful to have, say, a teacher who could tailor explanations to anyone's precise background (e.g. where possible, using examples that come from the student's field of study when explaining some abstract concept). Or, if some definition comes with some precondition that has no obvious purpose, perhaps an omniscient teacher could explain why it's there with concrete counterexamples.[0] But even granting all this, I think that mathematical intuition is necessarily based on a lot of hard work actually exploring definitions on one's own, with pencil-and-paper and a lot of thought. That is to say, even though the process could probably be sped up a lot with a nigh-omniscient teacher[1], I doubt that a student wouldn't still need years of training to even have a clue what's going on.

(I'm saying all this, by the way, as someone who is terrible at all this and has very little mathematical maturity[2]—I'm speaking from my own frustrating experience....)

[0] c.f. Lakatos' excellent book Proofs and Refutations

[1] without the "curse of knowledge," or else we're back to square one of "answers that are correct but useless"

[2] e.g. the "post-rigorous stage" described in https://terrytao.wordpress.com/career-advice/theres-more-to-...

[ironman1478]

How do you make use of something that you don't understand?

[seanmcdirmid]

The whole history of technology has been based on knowledge encapsulation and specialization, you don’t need to be a mechanic to drive a car.

[epgui]

It’s easy to imagine this being a problem both in quality and in volume. Verifiable work is less valuable than verified work. And noise is always costly.

[pfdietz]

It's not that it's incomprehensible, it's that it appears to be wrong.

[jmorenoamor]

Sorry but I couldn't agree less.

Deep esoteric research and trivial looking boring research can be as useful as state of the art trending areas.

"Jobs for nerds" as has been stated, has given surprising and unexpected advances, or leveraged incredible advancements.

An standard and boring bacteria in a specific Spanish biome, gave us CRISPR-Cas. There ar hundreds of examples.

True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society.

[datsci_est_2015]

> Sorry but I couldn't agree less.

> …

> True knowledge is, and will be, a human endeavor, deiven by human curiosity. Promoting curiosity is the sign of a developed society.

Unless I misunderstand, it sounds like you do agree? My point is that without human mathematicians LLM output is meaningless, and without human mathematicians holding the reins, LLMs would probably quickly devolve into “proving” things that are not only completely unintelligible by humans, but have no utility.

Your examples of esoteric mathematical concepts are anecdata. The vast majority of esoteric mathematics does not have utility. Mathematics is an incredibly large space of concepts. Consider the number of provable theorems in number theory alone, perhaps even related to specific subsets and sequences of numbers. The vast majority of the findings in that domain will not be isomorphic to some real world problem, they will be trivia.

We will need mathematicians to separate the signal from the noise.

[psychoslave]

Esoterism is mostly a social tool to keep those not initiated excluded from the private club. Most of the time mathematics becomes tricky less due to unfathomable intrinsic complexity, and more due to the way it’s communicated.

LLMs don’t give a shit about social side effects, leave alone on unconscious level, because they are void of any intention. At most they are tuned on their thin edge layer to lean toward this or that kind of output, but that’s it.

Now the landscape shift as it’s sold (I guess) is that anyone can take a postdoc gibberish infused with the hard gained academic winks and subtle references and turn it into a ELI5 "does it have any applicability for my concrete issue at stake, prove it through Lean, good let’s deploy".

[datsci_est_2015]

When I use the word “esoteric”, I mean it at an absolutely hyperbolic level. Like exploring new-but-basically-useless axiom spaces, and creating concepts for which there exists no clean metaphor in time-space - like quantum mechanics on steroids. And then creating multiplicatively more complex concepts by combining those concepts together.

There’s no way to “ELI5” this type of complexity. I’m talking about concepts exponentially more esoteric than quantum mechanics, and even within quantum mechanics there is nothing to ELI5 for a concept like “spin”. The best you can do is say that it’s a property of a particle. But imagine the words “property” and “particle” are also completely meaningless to you because they’re built on even more layers of conceptual mathematical abstraction.