>> At least for the foreseeable future you still would like people to become interested and develop skills in these fields. These developments, and especially how they are presented, directly discourage that. This assumption may well turn out to be correct, but it is not self-evident. Nearly everyone who has ever got interested in mathematics got discouraged at some point and they left the field. Mathematics is very hard. Those very few that remained certainly have talent, but they also have characteristics that are necessary for success in a competitive field, which are perhaps less valuable per se. Such characteristics as may be over-represented in males for instance. This is not a point about gender differences, but about the intrinsic merit of different success factors. It seems equally possible that the above assumption will turn out to be diametrically incorrect. People that would have been discouraged before LLMs will now retain their curiosity longer. Democratisation is surely a possible outcome. Arguably, chess has never been as popular and accessible. And that discipline fell to AI three decades ago.

Also worth noting these are reactions to what is essentially equity in access to skills and knowledge. I personally do wonder (worry) about where all of this pans out and what society looks like post generative llms. But at the same time there is a particular flavor of amusement that I can't help feeling watching folks simultaneously balance, "llms produce nothing of value" and "llms are so harmful and dangerous to our culture that we need to start policing use within our community" Where that harm essentially stems from devaluing hard earned skills within the community. And while I do not take joy in the displacement of labor, never in my wildest dreams could I have anticipated how harsh and irrational of a reaction to the equity of these skills could be. Which, I would like to point out, though hard earned were earned under the tremendous privilege to pursue these goals in the first place. Llms are an amplifier of an individuals intuition and taste. That these supposed pillars of the community are not bravely exploring how to push and wrangle these bounds, and instead are retracting into conservative stances under the guise of human centric morality is (IMHO) demonstrative of lack of confidence and creativity within these fields more generally. I believe that this lack of creativity and imagination is how we find ourselves in the personal fable you're noting: the experts are so myopic that they can't even imagine how they're field can be disrupted until it's disrupted outside of their control, and feel the need to control rather than explore.

An issue I see is who controls the information. The next generation may not recieve the knowledge, it may be gatekept by industry who *will* own the gate. The future may not have access unless we fight to ensure they do. This is how I read the article.

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.

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.

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.

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.

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.

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-...

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".

The proof is trivial and is left as an exercise for the reader.

This reminded me of my 11 yr old who, when I give her math problems to solve, is too focused on “getting the right answer”. I’ve told her plainly, I don’t care if you get the right answer right now, I want to see your reasoning. She has yet to understand this.

The struggle itself is virtuous.

Agreed. As someone who was always curious but had difficulties learning math the way it's taught at the university, AI teaching me the way no professor ever could is a blessing. I fail to see the point of the memo besides: we got here first and we decide what math is because we can. I'm really optimistic about AI and the value it brings in education. Gatekeepers will complain, but ultimately, will either adapt or be left behind.

>AI makes the math world more accessible than before. If you have a question about a proof in the lecture, you can just ask it. I think that is great, really! but does anyone remember asking a TA or teacher or prof or parent and getting told you can work it out for yourself, or maybe just given a hint? What if that is an essential part of learning, having to work through things you don't understand, but that you have the tools, the foundation, to figure out. A calculator can't teach you math. A forklift can't build your strength. This is really a double edged sword, as far as education or accessibility goes. You have to constantly ask... what do I lose by not figuring it out myself?

Yeah, among other factors, that "figure it out" mentality put me off in the end. Especially because often you need to show the same mentality unless you want to overkill proofs and spend more time on them than assigned to you. I sometimes miscalibrated and pointed out some details that didn't need pointing out in my proofs while in other proofs, I skipped over too many details for the TA. Of course I agree that if the student just asks LLM to do their homework, they have not learned anything. But it's sad if one can't ask questions about a proof or such. Having the LLM around to review the homework submission is also useful, to make sure that the arguments are solid.

You will have to learn to voluntarily figure things out for yourself without being pushed towards that. In a sense it's analogous to the presence of cheap calorie dense foods. In order to not be overweight you have to be mindful of and regulate your food intake in various ways. Alternatively, perhaps universities will provide access to fine tuned models that are mindful of such things.

You can but sadly most people ask for awnsers.

If spending millions of hours rote memorizing formulas and rules like a robot is "art" then sure

> “The tech industry proceeds in accordance with commercial logic, which is antithetical to the values of mathematics,” I briefly studied at a pure math department. We were learning linear algebra and I found the symbol heavy, proof oriented approach very difficult and unintuitive. But when I squinted at the diagrams I realized, oh wait, this actually has dozens of practical applications! Across dozens of different fields! How fantastic! And the textbook, for some reason, chose to mention precisely none of them. Which I found quite disappointing, because it made the whole thing seem quite abstract (which it actually wasn't), and made it harder to understand. I mentioned this to my colleagues, who became extremely upset, and informed me that I was in the wrong department.