There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds.
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?
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.
I think we generally did that because that seemed to be the best known process for maximizing the quantity of useful mathematics that they occasionally stumble upon.
It's not like we treat math as a charity project for eccentrics who like blackboards. What we want is new mathematical discoveries that have a huge positive impact on other areas of the world. It's just that math and/or human brains are such that seemingly the best way to find those discoveries was to let mathematicians wander around randomly in mindspace.
If a more guided structured process produced more results, we'd probably do that. But it doesn't seem to, so we don't. I don't think anyone knows yet what the best process for producing useful mathematics with humans + AIs looks like.
Love it! XD
I agree, and I think, as with physics, mathematical research produces building blocks whose utility won't be realised until later.
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Do you see the problem with your reasoning?
A proof being latent in an LLM is no more significant than a proof being latent in a book, a theorem prover, or the axioms themselves. Einstein's papers were latent in the genetic code of his parents and the environment of his time. That doesn't mean general relativity was "already done" before Einstein was born.
By your logic, no computation has ever accomplished anything because the output was always implicit in the inputs.
The entire purpose of computation is extracting information from representations where it's difficult to see into representations where it's easy to see.
So no, this isn't a problem with the original reasoning. It's a problem with yours.
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.
Is that what is offending you so much?
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.
> 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.
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.
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).
(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.
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.
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.
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.
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-...
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.
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".
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.
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.
> …
> 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.
You have machines that can aid and expand your (human) understanding greatly, that wasn't possible without machines.
Machines don't think. They aren't human. They have no soul, agency/free will, self-reflection/awareness, moral imperatives or ethics.
You've been watching too much Terminator, son.
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.
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
Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
But it is a moot point anyway. Cook and Levin are very well known already in TCS, and credit is not directly enumerable like money, so "more than a lot of credit" doesn't make too much sense.
For this problem in particular, asking the right kind of question was really important for the field and led to a lot of discoveries even before it will be answered.
In contrast, class NP and NP-completeness quickly became central concepts in theoretical computer science.
It seems like a key problem here is that peer-review is expected but not explicitly funded/rewarded while it is probably one of the aspects where humans still add a lot of value. Academia’s incentives are hugely misaligned (… as usual unfortunately).
> There might be more to maths than that, but that is definitely the most important part. I love science funding. But not because it's a jobs program for nerds.
I can produce an infinite number of verifiably correct papers, if that's all that matters.
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
1 + 5 = 6
A machine that takes longer and longer to prove propositions in ever more inscrutable ways is hardly useful at all.
The machine too needs to produce more generalizable and comprehensible systems, for it to scale up its own conceptualization. Needing to load all the new mathematics in the context window won't be great either.
Is an 80 year old unsolved problem maybe unsolved because it was never prioritized? Some problems stay unsolved because few people consider them worth working on.
Who is going to validate the results? Or do we skip that, with the risk of flooding the literature and collective understanding with unverified proofs?
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!
There are many theorems that aren't directly interesting, but whose proof requires techniques that are of substantial further interest, that lead to new domains, and/or new practical applications. Simply being handed a proof for those theorems isn't enough--we require the ability to apply those techniques in the real world, or discover further areas of mathematical research that build on that proof or its techniques.
It may be that AI can build on its own work for the long-term, but so far, AI does best at exploration in areas that have precisely specified and measurable goals. Actually creating understanding, and making use of mathemtical results outside of pure mathematics is more challenging than simply creating proofs.
I think the field will figure out how to make use of AI, and it will be better off for it. But that is not the same as just saying "answers good, grog want more answers."
We’re becoming increasingly embarrassing as a society.
Of course, there may be some valid arguments that everyone should have a jobs program in the form of ubi or something similar. But I feel thats very different to arguing for mathematicians specifically
1. (generally) incredibly cheap to fund, and
2. (occasionally) has extremely out-sized commercial impacts.
This is to say that jobs programs for math (and more generally fundamental research) have lead to extremely positive ROI for society, which is the typical justification given for funding them.
Which makes it not a "jobs program" as the term is generally used.
To go back to the analogy of jobs programs for alcoholics, it is somewhat similar if there was a small chance every time an alcoholic defecated in public gold came out. This fact might be used to support a jobs program for alcoholics, on the basis of it being positive ROI to society. At the same time, the "job" any individual alcoholic is doing in this setup is not particularly useful to society, so one might still call it a jobs program.
Finally, a job AI will never beat me at.