Oh, I hope I didn't come off as talking down to you! As I said in another reply here, the intention behind this comment was pretty narrow --- there's a certain perspective on this stuff that I see pretty often on HN that I think is missing some insight into what makes mathematicians tick, and I may have been letting my reaction to those other people leak into my response to you. Sorry for math-splaining :).
Anyway, yeah, if this scenario does come to pass it will be interesting to see just how impenetrable the resulting formal proofs end up looking and how hard it is to turn them into something that humans can fit in their heads. I can imagine a continuum of possibilities here, with thousands of pages of inscrutable symbol-pushing on one end to beautiful explanations on the other.
I certainly didn't mean to dispute that! Formal proofs have a lot in common with code, and of course reading code is illuminating to humans all the time.
I meant to be responding specifically to the case where some future theorem-proving LLM spits out a thousand-page argument which is totally impenetrable but which the proof-checker still agrees is valid. I think it's sometimes surprising to people coming at this from the CS side to hear that most mathematicians wouldn't be too enthusiastic to receive such a proof, and I was just trying to put some color on that reaction.
I don't know enough about the RH examples to say what the answer is in that case. I'd be very interested in a perspective from someone who knows more than me!
In general, though, the answer to this question would depend on the specifics of the argument in question. Sometimes you might be able to salvage something; maybe there's some other setting where same methods work, or where some hypothesis analogous to the false one ends up holding, or something like that. But of course from a purely logical perspective, if I prove that P implies Q and P turns out to be false, I've learned nothing about Q.
I'm a mathematician, although not doing research anymore. I can maybe offer a little bit of perspective on why we tend to be a little cooler on the formal techniques, which I think I've said on HN before.
I'm actually prepared to agree wholeheartedly with what you say here: I don't think there'd be any realistic way to produce thousand-page proofs without formalization, and certainly I wouldn't trust such a proof without some way to verify it formally. But I also don't think we really want them all that much!
The ultimate reason I think is that what really lights a fire under most mathematicians is the desire to know why a result is true; the explanation is really the product, much more so than just the yes-or-no answer. For example, I was never a number theorist, but I think most people who are informed enough to have an opinion think that the Riemann Hypothesis is probably true, and I know that they're not actually waiting around to find out. There are lots of papers that get published whose results take the form "If the Riemann Hypothesis is true then [my new theorem]."
The reason they'd still be excited by a proof is the hope, informed by experience with proofs of earlier long-standing open problems, that the proof would involve some exciting new method or perspective that would give us a deeper understanding of number theory. A proof in a formal language that Lean says is true but which no human being has any hope of getting anything from doesn't accomplish that.
For the articles on my website, I have a pretty janky workflow where I write a LaTeX document that I compile both to a PDF and (using Pandoc) to HTML, which I render with KaTeX. I've been in the market for a while for something that's less fragile but which can still produce both a PDF and visually appealing HTML output starting from a LaTeX source, and it seems like some of the ideas listed here might be what I want! Thanks for the link. (That said, if anyone has a particular recommendation, I'd love to hear it!)
Yeah, that's definitely right --- an explicit counterexample to the Riemann Hypothesis would be very surprising and interesting, and I think that would be equally true no matter whether it was found by a person or a computer! The situation that would be mostly unhelpful is a certificate that the result is true that communicates nothing about why.
Woodworking is very far from my world, so I don't really have any grounds to judge how comparable the two things actually are. I'll say two things instead.
First, right now presumably the reason a few people still become master woodworkers is that their work is actually better than the mass-produced furniture that you can get for much less money. Imagine a world where instead it was possible to cheaply and automatically produce furniture that is literally indistinguishable from, or maybe even noticeably superior to, anything a human woodworker could ever make. Do you really think the same number of people would still spend years and years developing those skills?
Second, you've talked about business logic and "math experts at the company" a few times now, which makes me wonder if we're just referring to different things with the word "mathematics". I'm talking about a specific subset, what's sometimes called "pure math," the kind of research that mostly only exists within academia and is focused on proving theorems with the goal of improving human understanding of mathematical patterns with no particular eye on solving any practical problems. It sounds like you're focused on the sort of mathematical work that gets done in industry, where you're using mathematical tools, but the goal is to solve a practical problem for a business.
These are actually quite different activities --- the same individuals who are good at one stand a decent chance of being good at the other, but that's most of what they have in common, and even there I know many people who are much more skilled at one than the other. I'm not really asking anyone who doesn't care about pure math to start caring about it, but when I'm talking about the effect of AI on the future of the field, I'm referring specifically to pure math research.
Incomprehensible proofs are indeed still useful to some extent, and I don't think you'll find many mathematicians who would reject them as an answer to the binary question of whether the result is true.
But when you talk about "getting a lot more done," I want to ask, get a lot more done to what end? Despite what mathematicians sometimes write in their grant applications, resolving most of the big open problems in the field probably won't lead to new technologies or anything. To use the Riemann Hypothesis example again, most number theorists already think it's probably true, and there are a lot of papers being published already which prove things like "if the Generalized Riemann Hypothesis is true, then [my new result]".
No one is really waiting around just for the literal, one-bit answer to the question of whether RH is true; if we got that information and nothing else, I'm sure number theorists would be happy to know, but not a whole lot about the work being done in the field would change. It's not just being "satisfying to the curious"; virtually the entire reason we want a proof is to use the new ideas it would presumably contain to do more mathematics. This is exactly what's happened with the proof of the Poincare Conjecture, the only one of the Millennium Problems that's been resolved so far.
This is what I was lamenting in my comment earlier: the thing you're describing, where we set proof-finding models to work and they spit out verifiable but totally opaque proofs of big open problems in math, very well might happen someday, but it wouldn't actually be all that useful for anything, and it would also mean the end of the only thing about the whole enterprise that the people working in it actually care about.
I was an algebraic geometer when I was still doing research in the field, and it was definitely true in that corner of the world. Authors are alphabetical, and you usually cite the paper by listing them all, no "et al"'s. I think I didn't even know there was such a thing as "first author" until I worked in ML.
This is actually a metaphor I've used myself. I do think the woodworking community is both smaller and less professionalized than it would be in a world where industrial furniture production didn't exist. (This is a bizarre counterfactual, because it's basically impossible for me to imagine a world where industrial furniture production doesn't exist but YouTube does, but like pretend with me here for a moment.) I don't know that this is necessarily a bad thing, but it's definitely different, and I can imagine that if I were a woodworker who lived through the transition from one world to the other I would find it pretty upsetting! As I said above, I'm not claiming it's not worth making the transition anyway, but it does come with a cost.
One place I think the analogy breaks down, though, is that I think you're pretty severely underestimating the time and effort it takes to be productive at math research. I think my path is pretty typical, so I'll describe it. I went to college for four years and took math classes the whole time, after which I was nowhere near prepared to do independent research. Then I went to graduate school, where I received a small stipend to teach calculus to undergrads while I learned even more math, and at the end of four and a half years of that --- including lots of one-on-one mentorship from my advisor --- I just barely able to kinda sorta produce some publishable-but-not-earthshattering research. If I wanted to produce research I was actually proud of, it probably would have taken several more years of putting in reps on less impressive stuff, but I left the field before reaching that point.
Imagine a world where any research I could have produced at the end of those eight and a half years would be inferior to something an LLM could spit out in an afternoon, and where a different LLM is a better calculus instructor than a 22-year-old nicf. (Not a high bar!) How many people are going to spend all those years learning all those skills? More importantly, why would they expect to be paid to do that while producing nothing the whole time?
Hm, good question. It depends on what you mean. If you're asking about restricting which theorems we try to prove, then we definitely are cutting ourselves off from vast swathes of math space, and we're doing it on purpose! The article we're responding to talks about mathematicians developing "taste" and "intuition", and this is what I think the author meant --- different people have different tastes, of course, but most conceivable true mathematical statements are ones that everyone would agree are completely uninteresting; they're things like "if you construct these 55 totally unmotivated mathematical objects that no one has ever cared about according to these 18 random made-up rules, then none of the following 301 arrangements are possible."
If you're talking about questions that are well-motivated but whose answers are ugly and incomprehensible, then a milder version of this actually happens fairly often --- some major conjecture gets solved by a proof that everyone agrees is right but which also doesn't shed much light on why the thing is true. In this situation, I think it's fair to describe the usual reaction as, like, I'm definitely happy to have the confirmation that the thing is true, but I would much rather have a nicer argument. Whoever proved the thing in the ugly way definitely earns themselves lots of math points, but if someone else comes along later and proves it in a clearer way then they've done something worth celebrating too.
This is an interesting question! You're giving me a chance to reflect a little more than I did when I wrote that last comment.
I can only speak for myself, but it's not that I care a lot about me personally being the first one to discover some new piece of mathematics. (If I did, I'd probably still be doing research, which I'm not.) There is something very satisfying about solving a problem for yourself rather than being handed the answer, though, even if it's not an original problem. It's the same reason some people like doing sudokus, and why those people wouldn't respond well to being told that they could save a lot of time if they just used a sudoku solver or looked up the answer in the back of the book.
But that's not really what I'm getting at in the sentence you're quoting --- people are still free to solve sudokus even though sudoku solvers exist, and the same would presumably be true of proving theorems in the world we're considering. The thing I'd be most worried about is the destruction of the community of mathematicians. If math were just a fun but useless hobby, like, I don't know, whittling or something, I think there would be way fewer people doing it. And there would be even fewer people doing it as deeply and intensely as they are now when it's their full-time job. And as someone who likes math a lot, I don't love the idea of that happening.
oersted's answer basically covers it, so I'm mostly just agreeing with them: the answer is that you use a computer. Not another AI model, but a piece of regular, old-fashioned software that has much more in common with a compiler than an LLM. It's really pretty closely analogous to the question "How do you verify that some code typechecks if you don't understand it?"
In this hypothetical Riemann Hypothesis example, the only thing the human would have to check is that (a) the proof-verification software works correctly, and that (b) the statement of the Riemann Hypothesis at the very beginning is indeed a statement of the Riemann Hypothesis. This is orders of magnitude easier than proving the Riemann Hypothesis, or even than following someone else's proof!
I would love that too. In fact, I already spend a good amount of my free time redundantly learning the mathematics that was produced by humans, and I have fun doing it. The thing that makes me sad to imagine --- and again, this is not a prediction --- is the loss of the community of human mathematicians that we have right now.
Especially not mathematicians! No one goes into math academia for the money, and people with math Ph.D.'s are often very employable at much higher salaries if they jump ship to industry. The reason mathematicians stay in the field --- and I say this as someone who didn't stay, for a variety of reasons --- is because they love math and want to spend their time researching and teaching it.
Well, it depends on exactly what future you were imagining. In a world where the model just spits out a totally impenetrable but formally verifiable Lean proof, then yes, absolutely, there's a lot for human mathematicians to do. But I don't see any particular reason things would have to stop there: why couldn't some model also spit out nice, beautiful explanations of why the result is true? We're certainly not there yet, but if we do get there, human mathematicians might not really be producing much of anything. What reason would there be to keep employing them all?
Like I said, I don't have any idea what's going to happen. The thing that makes me sad about these conversations is that the people I talk to sometimes don't seem to have any appreciation for the thing they say they want to dismantle. It might even be better for humanity on the whole to arrive in this future; I'm not arguing that one way or the other! Just that I think there's a chance it would involve losing something I really love, and that makes me sad.
The Four Color Theorem is a great example! I think this story is often misrepresented as one where mathematicians didn't believe the computer-aided proof. Thurston gets the story right: I think basically everyone in the field took it as resolving the truth of the Four Color Theorem --- although I don't think this was really in serious doubt --- but in an incredibly unsatisfying way. They wanted to know what underlying pattern in planar graphs forces them all to be 4-colorable, and "well, we reduced the question to these tens of thousands of possible counterexamples and they all turned out to be 4-colorable" leaves a lot to be desired as an answer to that question. (This is especially true because the Five Color Theorem does have a very beautiful proof. I reach at a math enrichment program for high schoolers on weekends, and the result was simple enough that we could get all the way through it in class.)
I'm a former research mathematician who worked for a little while in AI research, and this article matched up very well with my own experience with this particular cultural divide. Since I've spent a lot more time in the math world than the AI world, it's very natural for me to see this divide from the mathematicians' perspective, and I definitely agree that a lot of the people I've talked to on the other side of this divide don't seem to quite get what it is that mathematicians want from math: that the primary aim isn't really to find out whether a result is true but why it's true.
To be honest, it's hard for me not to get kind of emotional about this. Obviously I don't know what's going to happen, but I can imagine a future where some future model is better at proving theorems than any human mathematician, like the situation, say, chess has been in for some time now. In that future, I would still care a lot about learning why theorems are true --- the process of answering those questions is one of the things I find the most beautiful and fulfilling in the world --- and it makes me really sad to hear people talk about math being "solved", as though all we're doing is checking theorems off of a to-do list. I often find the conversation pretty demoralizing, especially because I think a lot of the people I have it with would probably really enjoy the thing mathematics actually is much more than the thing they seem to think it is.
I'm a private tutor who works with adults on proof-based math. I've often had a similar thought to the one you're expressing here --- I also found proofs pretty revelatory when I first exposed to them and wondered where this magical tool had been all my life --- but I wonder how well this experience would scale to the mass of students in high school math classes.
After teaching proof-writing to my students for several years now, I've seen a lot of variation in how quickly students take to the skill. Some of them have the same experience that it sounds like you and I had, where it "clicks" right away, some of them struggle for a while to figure out what the whole enterprise is even about, and everything in between. Basically everyone gets better at it over time, but for some that can mean spending a decent amount of time feeling kind of lost and frustrated.
And this is a very self-selected group of students: they're all grown-ups who decided to spend their money and spare time learning this stuff in addition to their jobs! For the kind of high school student who just doesn't really think of themselves as a "math person", who isn't already intrinsically motivated by the joy of discovering what makes integrals tick, I think it would be an even harder sell. High school math teachers have a hard job: they have to try to reach students at a pretty wide range of interest and ability levels, and sadly that often leads to a sort of lowest-common-denominator curriculum that doesn't involve a lot of risk-taking.
Anyway, yeah, if this scenario does come to pass it will be interesting to see just how impenetrable the resulting formal proofs end up looking and how hard it is to turn them into something that humans can fit in their heads. I can imagine a continuum of possibilities here, with thousands of pages of inscrutable symbol-pushing on one end to beautiful explanations on the other.