[silveraxe93]

> However, the declaration argues math is more than a machine for producing correct answers.

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.

[zerobees]

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?

[141205]

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.

[drivebyhooting]

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

[munificent]

> We just let them do their thing, essentially treating them as artists and letting them pursue the craft for its own sake.

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.

[naruhodo]

> 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.

[bdamm]

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.

[orbital-decay]

Pure mathematics != applied mathematics

[psyklic]

The most important part of math is advancing human understanding. A correct answer by itself is not as important as understanding why it is correct.

[rowanG077]

Once you now something is correct, with a proof. It is MUCH easier to understand why it is correct. Than to start from a slate that you don't even know whether something is correct or not. In that sense AI that can just solve high level math problems is immensely useful. It allows a mathematician to explore ideas at a much more rapid pace.

[terminalbraid]

Consider that since an LLM is really just an large encoding of data, the "proof" is in there already. All further work on it is effectively only rearranging words. Then all math an LLM is capable of is "done" and we have the "proof" in the LLM which by your definition is now "MUCH easier to understand" and this work is somehow sufficient.

Do you see the problem with your reasoning?

[rowanG077]

You're confusing "contains information" with "has produced a result."

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.

[datsci_est_2015]

To further this assertion, there is almost no value to deeply esoteric math that is technically correct, but completely inapplicable to any scientific reality, and completely unintelligible to humans. Consider these findings deep, dark corners in the unfathomably large hyperspace of mathematics. My guess is AI will be incredibly adept at identifying these types of findings, and it will be exceedingly difficult for humans to identify what is meaningful and what is not in the slop.

[canjobear]

Your model of what AI is good at is wrong. Generative AI is not good at wandering off into novel esoteric abstract corners while maintaining correctness, it is good at things that are close to its training data. I suspect that humans will long outperform AI in the domain of "novel esoteric abstract useless math" whereas AI will outperform humans in the domains of (1) making connections between already-well-understood concepts, things that seem obvious in retrospect but which no human figured out just because of the accidents of what people happened to focus on, and (2) proving things that require long, tedious, intellectually unsatisfying calculations, which would cause a human mathematician to give up for boredom.

[datsci_est_2015]

My understanding is that we’re talking about “tool-assisted” proof generation, which provides some guard rails but would still allow significant creativity. Tools like Lean, Coq, etc.

[sunshowers]

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

[yaris]

Works of Shinichi Mochizuki immediately come to mind. He is not AI but provides very good examples of math that is useless because it is incomprehensible by (other) humans.

[seanmcdirmid]

Do AIs produce answers whose work is incomprehensible to humans? It seems like you could just have the AI elaborate multiple times until you were satisfied with the explanation and documentation of what went into figuring out the answer. It’s not like the AI is one shotting the answer in a single opaque query anyways.

[datsci_est_2015]

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.

[storus]

> “speak the language” of the concepts the LLMs have produced

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.

[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?

[epgui]

It’s easy to imagine this being a problem both in quality and in volume. Verifiable work is less valuable than verified work. And noise is always costly.

[pfdietz]

It's not that it's incomprehensible, it's that it appears to be wrong.

[jmorenoamor]

Sorry but I couldn't agree less.

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.

[datsci_est_2015]

> Sorry but I couldn't agree less.

> …

> 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.

[psychoslave]

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".

[datsci_est_2015]

When I use the word “esoteric”, I mean it at an absolutely hyperbolic level. Like exploring new-but-basically-useless axiom spaces, and creating concepts for which there exists no clean metaphor in time-space - like quantum mechanics on steroids. And then creating multiplicatively more complex concepts by combining those concepts together.

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.

[ragebol]

"What is the answer to the Ultimate Question of Life, the Universe, and Everything"

42

[lioeters]

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

[bitwize]

The value of human understanding just cratered because we have machines to understand for us now.

[martin1975]

Not quite.

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.

[goatlover]

Wonder what Frank Herbert would have to say about letting machines do the thinking for us.

[bitwize]

"You wanna know what the best thing about humans is? You invented us! Giving you a chance to take a rest while we invented everything else!" —Wheatley, Portal 2

[epgui]

That seems about as shortsighted as claiming the value of human understanding has cratered after the invention of the electronic calculator.

[dwroberts]

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

[card_zero]

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.

[dwroberts]

> 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

[delichon]

For most engineers a mathemetician is a machine for producing correct algorithms, like a chef is a machine for producing tasty food. In both cases that overlooks the human element, but that's a critical skill for a limited mind with finite resources to grok infinite complexity. You can read that as permission to be an asshole or a neccesary compromise.

[19f191ty]

No, it's not the most important part. It can be argued that most important part is asking the right questions

[silveraxe93]

Assume someone solves P=NP

Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?

[NotOscarWilde]

That's a bit too simplistic -- if there is a small group that really pushes things forward in a big way, then maybe not, but if this result builds upon decades of prior work, then Cook and Levin might be equally or even slightly more famous than the solver group after the dust settles.

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.

[dchftcs]

If the problem resolves to P=NP, that result would probably be more celebratee than being able to formulate the problem, but being able to formulate the problem and get people interested in it is probably worth more than the average primal dual trick to prove a polylog integrality gap for some integer linear program.

[jltsiren]

Depends on the solution. If the solution is that P≠NP, the concept of NP-completeness remains the bigger contribution, unless the proof techniques are particularly interesting and lead to other major results. The same applies if P=NP but the proof does not ultimately lead to a practical algorithm. If we get a practical algorithm, the answer is more valuable than the question.

[redox99]

Do questions like goldbach conjecture, fermats last theorem, etc deserve more credit than whoever answers them?

[jltsiren]

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.

[i_am_a_peasant]

I agree with both OP and you

[psychoslave]

I disagree with everyone, self included, but especially with Cretans.

[codeduck]

Cretani eunt domo!

[i_am_a_peasant]

Monty Python fan detected :D love your profile desc btw

[conformist]

> The authors warn the consequences are already becoming visible. AI-generated papers could overwhelm peer-review systems with low-quality work …

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).

[armchairhacker]

Math is one field where you can mechanically prove a paper's findings. The only thing that would need to be judged is the (verified) statement's importance.

[conformist]

Yes in theory, but not yet in practice because not everything is fully formalised.

[smath]

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.

[solid_fuel]

>> However, the declaration argues math is more than a machine for producing correct answers.

> 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

Shall I continue? Or do you think that choosing which questions to answer might have some level of importance, in addition to getting correct answers?

[barrkel]

A statement that some proposition is true or false is usually less useful than a new framework for understanding the class of problem.

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.

[kleyd]

The wording in the declaration may be a bit romanticized. But the points are valid:

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?

[ajs1998]

Spoken like someone that has no idea why mathematicians are important.

[IshKebab]

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!

[hyperpape]

Even from the most purely instrumental perspective, what we care about is our ability to make use of correct answers, which is quite distinct from the possession of correct answers.

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."

[analognoise]

> But not because it's a jobs program for nerds.

We’re becoming increasingly embarrassing as a society.

[fragmede]

People need jobs. What's wrong with nerds having jobs via a program?

[RugnirViking]

what's wrong with artists having jobs via a program? whats wrong with struggling alcoholics having jobs via a program? athletes? politicians? there is no inherent virtue in the struggle and effort associated with great mathematical achievement. It may be satisfying and worthwhile for the solver, but not for society at large, any more than any other pleasurable activity. No, as it is, the sole reason for it is in the result itself. In increased understanding, as it flows down into the sciences, and engineering. There are other benefits, recreation and joy as experienced by others, from access to beautiful proofs, though these are never explicit goals of such programs because they are both impossible to quantify and rarely ever remotely relevant compared to the value brought by the practical value brought by maths.

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

[rspeele]

> whats wrong with struggling alcoholics having jobs via a program?

Finally, a job AI will never beat me at.

[mswphd]

for mathematicians, they do a form of fundamental research that is

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.

[orangecat]

This is to say that jobs programs for math (and more generally fundamental research) have lead to extremely positive ROI for society

Which makes it not a "jobs program" as the term is generally used.

[mswphd]

it arguably still is. The primary unit of production of the jobs of mathematicians is itself not particularly useful for society. In this sense funding them is a jobs program. It is also true that they occasionally produce things of great value, and more frequently the things they produce can be leveraged by other researchers to directly produce things of value. But neither of these are what the job of a mathematician is (either in a day-to-day sense, or even for many mathematician's careers).

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.

[fragmede]

The struggle itself is virtuous.

[psychoslave]

People need many things, there are all kind of theories ready to assess and assimilate if deemed worth it out there. A job is not part of any I’m aware of, though it can encompass some human needs in some cases, or go straight against them in some other case.

[fragmede]

Every human needs a vocation, better if it's of their choosing.

[bloqs]

well put.