Much of math (or science) research has the strange quality of being mostly curiosity-driven, but having giant benefits that occasionally spin out to the public. Some questions are more urgent and practical. My feeling is that the more directly practical a question is, the more likely the research community is to support AI usage in that question. The annoying thing about recent AI advances is that they target questions on the wrong end of the spectrum: Erdos problems are exactly the sort of "useless" questions that people might answer purely for the love of the game. The sort of questions that a young person might cut their teeth on and gain confidence. Solving questions like these automatically, I think, is not good for the long-term health of research. 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.

To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect. Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable. Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.

Do you not think that solutions to erdos problems might end up stepping stones to other important problems? Either by introducing new tools, or by proving things that were previously unproven that end up helping in unexpected ways? That's often how math goes, isn't it?

This is, indeed, how math often goes.

I think nuance gets lost in these conversations. Your distinction between the practical and the theoretical is important. Practicality is important - everything we do is a matter of practicality of means or method , even how we pursue theoretical ends - but two points. First, there is more to life than the practical. Some truths are known for their own sake, even if they also tell us about still more profound truths (also known for their own sake) or may have incidental practical relevance and consequences in some other context. Second, while the theoretical terminus is the truth for its own sake, the practical terminus is always something other than itself. Well, what is that "something else"? You can't have an infinite regress of practicality. The meaning of a proximate, practical end is always other than itself. The practical requires an end beyond itself to justify it. I agree that most people don't seem to inquire much about such ultimate ends. Their thoughts are confined to the proximate. Of course, how have they determined what the proximate should be? Something for people to contemplate. Where science is concerned, it depends. On the one hand, there are fields that are certainly more theoretically oriented. It's not "the game" that motivates theory - that would make it mere recreation, with the truth taking a backseat - but the truth. (For this reason, I hesitate to call Erdos theoretically motivated. AFAICT, he was motivated by the challenge of problem solving and not the truth, insight, and understanding to be gained which would have been merely incidental and instrumental for him.) However, I would also say a good chunk of science is motivated by a background motivation of technology production and the mastery of nature. Think Francis Bacon who viewed science as an instrument of power and showed a preference for the "how" over the "what" (τόδε τι) or the "why" (τὸ διότι). This set the tone for a great deal of modern science. A great deal does less explaining and more predictive modeling, because predictive modeling can be sufficient for control. Indeed, a truly theoretical causal account and understanding of a thing's nature can be less useful as a practical instrument than a merely predictive model. Now, AI is a practical tool. I think they can be enormously useful as research aids, even in theoretical contexts, provided that one 1. understands their nature; 2. understands the purpose of the theoretical activity undertaken. What is their nature? Well, they're statistical models that can unearth interesting and useful correlations and patterns. But they are not reasoning and knowing things. Their results are generated mechanically and mindlessly. Knowing this means taking their results with a healthy skepticism and a critical eye. What about the purpose of theory? By analogy, think of a student in school who uses AI to complete all his assignments. Has he satisfied the purpose of those assignments? No, because the purpose of the assignments isn't to produce the effect - the solutions - per se, but to learn something. Theoretical work is like that; it's purpose is to understand and to grasp some truth. An AI can be used to assist this process, just as a calculator or a search engine can, but if you use it in a manner that circumvents that purpose instead of supporting it, then you're not achieve that purpose and wasting your time. What's the point?

It worked for humans. It took a lot of us, but eventually we accepted zero as a number. Then negative numbers. Then "imaginary numbers" as a useful trick, and then as meaningful. In our case, hundreds of millions, but we got there.

unify general relativity with quantum mechanics. The continuum hypothesis. The traveling salesman problem in polynomial time.

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

> 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).

Culturally, mathematics is a jobs program for nerds. The field very explicitly takes pride in working on problems that have no obvious applications, and most practitioners are funded publicly or supported by private endowments, with zero pressure to deliver specific results. Of course, this produces useful results every now and then, but it's not like we pursued ruthless efficiency / maximum rate of knowledge advancement before. We just let them do their thing, essentially treating them as artists and letting them pursue the craft for its own sake. If we weren't interested in maximum throughput before, why is that an objective now?

Hardy would agree with the viewpoint that you espouse but it would be pushed back against by Arnol'd, Poincare, Gauss, Von Neumann, and even Grothendiek: Arnol'd and Poincare were vituperatively against the division between "pure" and "applied" mathematics; they considered mathematics and physics interchangeable, and Arnol'd lamented that the field had lost a large amount of funding/prestige/relevance due to groups like the Bourbaki that took a purely aesthetic view; Gauss had a critical view of problems like Fermat's last theorem (he felt that you could construct infinitely many such problems, and felt that attempting to prove it was a generally useless endeavor), along with outright calling pure mathematics worthless; but while Von Neumann and Grothendiek were more moderate, both were critical of the field losing motivation/quality as it strayed away from empirical science into—quoting Von Neumann—"abstract inbreeding". Arnold's polemics are perhaps the most infamous and easily found online (see "On Teaching Mathematics"), but the written opinions of Poincare et seq. are also easy to find. Even today the vast majority of research funding for mathematics, at least in the United States, is dolled out for highly applied fields like partial differential equations. The field does not even close to unanimously (contemporarily or historically) "explicitly take pride" in working on problems that have no obvious application, or being a "jobs program for nerds": the notion of such "pure" or "nonapplied" mathematics is at the very least a highly fractious and controversial subject, with a number of big names taking opposing viewpoints (often vehemently). I think your picture of the field is over-represented on the internet, much like the fixation on certain niche fields: Category Theory, Homotopy Type Theory or, worst of all, outright dubious fields like Geometric Algebra; fields with a large number of online promoters, but with much less funding and relevance in the actual academic space. Of course there are reputable people with PHDs that feel this way,—but I can only imagine that there's a legion of tyros, pop math consumers, and undergraduate students who disproportionately promote this viewpoint.

What’s wrong with geometric algebra? The same could’ve been said about linear algebra 100 years ago.

> It's not like we treat math as a charity project for eccentrics who like blackboards. Love it! XD I agree, and I think, as with physics, mathematical research produces building blocks whose utility won't be realised until later.

Except, it was true before and it is still true today that the best "artists" whether graphic or mathematic, are the ones that do somehow manage to cross the chasm of pure research and providing a tangible benefit to their benefactors. That aspect of understanding your customer is not changed by the presence of AI.

Pure mathematics != applied mathematics

Probably one of the funniest things to read on a site like this, when you consider that eg. Boolean algebra was entirely abstract and had little practical purpose for almost 100 years until Shannon picked it up for use in circuits

Boole was trying to improve logic for humans, "The Laws of Thought". So it has a connection to human problems, and eventually to practical matters. He could instead have been working on something much more abstract and much less useful. By which I'm trying to make an abstract point about the inevitability of staying somewhat down to earth. I mean "pure" curiosity is great, except it isn't ever really pure, and abstract mathematics isn't ever totally abstract, it's just sort of meta in relation to practical things that humans care about.

> So it has a connection to human problems, and eventually to practical matters. But in relation to parent post I was replying to, it did not provide an answer or solution to anything. It has much closer relation to philosophy than anything. Focusing on only ‘solutions’ in any field is shortsighted because you can’t know how the dots will connect. Someone’s seemingly pointless curiosity or experiment can unlock something unexpected, just like Boole

Fermat's last theorem probably doesn't. While attempts to solve it have led to many mathematical discoveries, the theorem itself feels little more than a piece of trivia. I'm less familiar with the implications of the Goldbach conjecture. In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.

Maths pretty much is a jobs program for nerds though. It occasionally produces results that are practically useful for society but there's absolutely no way the vast majority of today's maths research falls into that category. There are definitely exceptions, like crypto. I still think it would be pretty silly to stop maths research anyway. And anyway part of the job of maths researchers is to teach maths to undergrads and that's obviously enormously useful to society. But on the scale of "how useful is this research to society" it's dead last after engineering, chemistry, biology, and physics. Well maybe computer science would be last actually!

> “The tech industry proceeds in accordance with commercial logic, which is antithetical to the values of mathematics,” declaration co-author Michael Harris of Columbia University As a former physicist and current data scientist/engineer, I know for a fact that commercial utility drives math research and researchers. Math is a tool to solve problems. Some mathematicians might only love the process of using the tool, but commercial logic absolutely drives mathematician attention to develop commercially useful tools.

Yea, math is a crazy thing. just "a tool to solve problems" is a wild take. On one extreme, there's an edgy but logical / plausible hypothesis that we live in a universe of mathematical objects, and at the other, math also discovers a lot of questions, the exact opposite of solving problems.

Is there a useful abstraction that doesn't help solve a problem someone has?

Math for non mathematicians is a tool. Math for mathemeticians is an art in the same way an artisan takes pride in his work. That's why there's a disconnect when you go from math for engineers to the stuff above it. It feels less useful and very different

Absolutely, but at least in the pure / less applied fields, access to computation hasn't really been that critical. The more towards the pure and theoretical, less so. But now you have people like Gowers and Tao, pure mathematicians, hyping up what the SOTA models can do - and I figure they both are getting access and tokens us mortals can't afford. So I guess the question is - will everything be as expensive as applied fields?

I'm curious about whether we will start discovering new maths in the next few years that provide insight into unsolved CS or Physics problems!

Well I rather like to be paid more than a mathematician so left academia rather quickly. In my case corporate modelling mostly involves making prediction models based on shitty data and metrics to make poorly contrived business decisions that lose millions of dollars.

lol.... but they are still data driven decisions, everyone loves those, especially when you lose millions of dollars and need to justify it.

> “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.