[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.

[fc417fc802]

Micromanagement wasn't the message I took from that. Rather the level of human involvement required which (it seems like) the two of you more or less agree on.

The meaning I took was how far it's possible to travel from the shore - ie the scope of the state space. The mathematics we're exposed to is all quite shallow compared to what will (presumably) be possible between digital formalization and massive ML models. But the latter probably can't ever be understood by regular biological humans.

[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).

[datsci_est_2015]

For me it was a “Modern Algebra” course required for my mathematics major, where I managed to squeak by with a B, but it was definitely a filter course for research-level mathematics. It was very clear in the class of a few dozen students who the top 5 or so were based on their questions during lectures and office hours, as well as when they blessed us mere mortals with their presence at our study groups.

(Aside, this was one of the only undergrad courses where I felt I needed to attend study groups in order to not fail.)

The first exam was easy to pass based on intuition alone, as the topics were isomorphic to concepts I was familiar with like geometry or algebra. The midterm was a wake up call when it was made clear that just understanding the homework wasn’t sufficient, you were going to be asked to prove things that were much more difficult than what I’d ever encountered, and under time pressure (I had been doing math proofs since age 13 in geometry, and I was 22 at that point).

Maybe if you did discrete math, combinatorics, or linear algebra I would say it was 5x to 10x more abstract and difficult. Probably 2x more difficult and abstract than Theory of Calculus, if you had taken that or a similar course.

Edit: I also do endurance running and play soccer into my 30s. Seeing people run literally twice as fast as me (world record pace), and playing against former college athletes is equally as humbling. The time has passed for me to have anything near their ability haha.

[grayclhn]

Algebra is the class where I learned I shouldn’t try to figure out how to prove theorems named after people during tests.

And I think you’re underestimating the jump from discrete math and linear algebra to abstract algebra… I think I attended each of those classes and opened their textbooks a total of 3 times each and did fine - once for each exam. But fml abstract algebra and measure theory were rough.

[dehsge]

For myself it was learning what a limit is in calculus, then learning about vector spaces, then learning about metric spaces and then learning about different topological spaces.

Then I had to relearn how a limit worked.

From a proof with epsilon delta inequalities. To a proof with showing for some n dimensional metric spaces that has all the properties needed to converge does in-fact converge. Finally to a proof that for any space that is metric there is an isometric function into that metric space that also converges.

And that does touch measure theory, functional analysis or set theory. So there’s still so so much more for me to learn.

[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.