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.
Sounds like yet another example of how AI is kneecapping industries from the bottom by "removing the barrier to entry" but really just removing the training path by doing the work itself with no guidance for juniors.
Is this a known or quantifiable thing? I thought that the limit had already been determined i.e. the existing models top out and at some point it doesn't matter how much time or energy you let the model consume, it won't improve the result. And with regards to training parameters, I thought we were equally limited there, e.g. the existing models can't benefit from a larger parameter space.
I was under the impression that improvements are arriving via how the models are trained and how model prompting context is constructed, rather than just by how much data or how much energy is spent searching over the model space for a particular prompt.
Is there some evidence that we have not reached a pleateau with just resource consumption on existing models?
That's an interesting perspective and I wholly disagree with the conclusion
You are saying that tough problems with no applicability are useful because people that you happen to respect got good by their curiosity and pursuit of trying to solve these kinds of problems and failing, but branching off into other cognitive areas as mathematicians
Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much
The similarity being that their exact criticism of why, something they don't respect and view as having little utility, is the exact reasoning presented here now that AI can solve their pointless problems
What I'm seeing is that human mathematicians have a laundry list of problems they have failed to solve for decades, centuries, which is what they are funded and employed to do. "Computer" used to a human job title too.
This leads me to being excited about AI one-shotting these problems, let move on to something else.
I think you've slightly straw manned the lamentation there. Not that I agree with the lamentation, but using your talent to make the rich richer (which is what quants do, they are paid a fixed amount to provide a larger value up the chain), as opposed to advancing human knowledge, is the reason for the lament, not some sort of respectability issue.
> 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.
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?
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.
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.
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.
> 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.
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.
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.
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.
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.
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.
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.
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"
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".
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.
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.
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.
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.
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.
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.
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.
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.
> 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).
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.
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.
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?
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.
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."
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
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.
Accelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field.
Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI generates and verifies proofs without humans in the loop. Where the mathematics may be beyond human comprehension.
In that future, does it matter that early career mathematicians are inhibited by these developments? Perhaps not. Programming faces the same issue. As AI crawls up the competence ladder, does it matter that fewer people have opportunities to develop the skillset of a senior engineer? Perhaps not.
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.
Much like for many the point of chess is that it's played by humans, with truly superhuman AI relegated to a training aid, mathematics is in many ways about human comprehension. You can use AI to find and proof new theorems. But if you get to the point where humans can't understand it, is it even still math?
> “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.
My vague prediction right now is that in five years LLMs will be heavily used by universities in grant-funded math research but nobody else will be able to afford it, much like supercomputer clusters 25 years ago.
Well, if progress in LLMs will steadily continue over next 5 years, then models will be so powerful that there will be no longer place for (most of) human researchers in math (remember that 5 years ago there was no chatgpt!). But I think it's more likely that progress will stall and then open models will catch up to frontier models and almost everyone will be able to afford them.
Seems way too binary a statement. I am guessing you mean "frontier LLMs". Small models keep getting better and better and if you make domain specific ones, it will likely be even smaller. Companies renting smaller LLMs or using enterprise models might very well remain in the future. Consumers getting LLMs whose performance dont improve (think gpt 6 forever on premium or gpt 4.x on a cheap tier) might well become a thing.
Except when someone hands you a magic button that just gives you knowledge?[at least in the framing of this "warning"] Then it's about peoples' livelihoods, about "culture", etc?
"Computer" used to be a job. Did science on the whole lose or gain by making these clerks obsolete?
Human mathematicians are being exposed the same way the "artists" are. It was always about the money and to look clever, superior to other humans. Whether its robotically spending millions of hours drawing until they can put something together at the level of a chatgpt 3 or the rote memorization of formulas and rules. They like people to think it all came naturally and that its genetic and that they are special snowflakes.
Both mathematics and art are comprised of two phases, the first, technical one, where the novice grinds the skill and the second, the creative one which can only be achieved if you have the means (skill) to express yourself. What you described is the technical phase, not the creative one. There is intrinsic value to it that has nothing to do with money or cleverness, something that if you ever experienced it yourself even once, wouldn't need to be explained to you. Only people who never reach phase two have your stance. Artists and mathematicians who pick academia didn't exactly have great commercial prospects before AI was a thing, yet they still chose those paths because that's what having a real passion looks like.
>They like people to think it all came naturally and that its genetic and that they are special snowflakes.
No, they don't. Most of them are the humble people that know the value of cultivating a skill and when they do pride themselves it's precisely because they know the staggering amount of hard work and commitment they invested. Most of them are worried for unemployment and don't want all their work to be reduced to training data and on top of that not be given well-deserved credit for it.
The only thing being exposed here, is how much AI in its current form was being underestimated and constantly labeled as "not real/good enough intelligence". This was and still is a shared sentiment even among tech people. Can't really blame them for going through a bargaining or acceptance stage.
And since you also sound like the kind of person who thinks prompting can replace the "robotically spending millions of hours" of practice, I've got news for you: it cannot. You are about to learn the hard way the value of skill and human understanding because as much as capitalism rewards "impact" and "results", the market never values easy things.
> AI-generated papers could overwhelm peer-review systems with low-quality work
That's not a problem unique to math, or even to academia. It's a problem in every context in human life where people communicate via written documents.
I think it's going to reduce the friction of exploring new areas in math, and that we're going to see a golden age of math unlike anything seen before.
As a mathematician by trade I think they’re overblowing it. You can choose to use it or not. I choose not to because I enjoy the process. But I’m not doing formal research or getting paid to do it these days.
I will note that the average corporate mathematical modelling is usually a fucking circus so adding AI might make it better.
This is becoming less and less true unless you're specifically talking about usage of it outside of a work environment. Many work places are requiring people to use it and/or tracking usage. I don't know about in academic settings, but I'd imagine it's becoming heavily used there too?
The choice only remains if using it isn’t a huge multiplier. If it is a huge multiplier/accelerator, then for a while it will be ambiguous and the choice will remain. But as time goes on, the gains of using it will be so apparent and the advantage of the people who use it so great (in publication numbers, hiring, etc) that it will force others to.
I don’t say that with any particular relish. But I am skeptical of the choice angle past a certain point.
I don't think all universities or research agencies are particularly pressed on this. I mean my daughter is a notable researcher in a scientific field and they have absolutely no pressure to use AI to pump out papers or deliver value quickly.
I highly doubt there is any overt pressure in academia right now to use AI. It’s a relatively conservative institution. But there’s certainly pressure to publish (publish or perish being a common phrase for decades), and competition for jobs in academia is fierce. That’s what I meant in referring to long term pressure.
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.
I've said it before, but there's a massive risk that we simply stop educating researchers. So much of a Ph.D revolves around the person learning how to do research.
They learn how to read papers and literature rigorously. They get low-hanging fruits to practice on, which can take months. Their funding doesn't come from thin air either.
So what happens when the group leaders would rather spend money on compute, and get models to solve the low-hanging fruit? Which the models could very well do in mere hours, compared to months.
Nor does it help that publishing is the number 1 measure in academia. Furthermore, the access to compute and capital could end up be the defining factor between researchers and research groups.
It is basically the "junior problem", but even more severe.
> Furthermore, the access to compute and capital could end up be the defining factor between researchers and research groups.
That's not new - especially in the experimental sciences ( ie perhaps more than maths ) - where the ability to have access to the latest kit is often what determines success - a huge amount of science progress is driven by new experimental technology rather than smart people thinking beautiful thoughts.
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?
Though having said that - the ~5 billion for the LHC now seems cheap ( even inflation adjusted ) in the context of Google investing 180 billion in infrastructure just this year!
"""
However, the declaration argues math is more than a machine for producing correct answers. The discipline, its authors believe, is a deeply human endeavor built on creativity, understanding, collaboration, and the pursuit of knowledge for its own sake. Those values often clash with the incentives driving AI development. “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 told The New York Times.
"""
I mean, what field doesn't? Everyone works to make money.
Slightly unrelated, but, their website "https://leidendeclaration.ai/" itself gives an eerie feeling of being built by Sonnet. That color scheme and the layout is what Sonnet chooses by default most of the times.
I still don't understand how "AI" is ready for serious use beyond entertainment purposes
Every time I ask ChatGPT to make a table for a subject I know well, I will find an error in one of the results and it is very confident about it until I question it in detail
Every time I ask ChatGPT for nutritional breakdown of some dense food source and give it a quantity like 8 ounces and ask for the weight of each ingredient, the weights will be wrong and add up to more than the original weight of 8 ounces
These are variations of the old "how many Rs in strawberry" problem, it's still not solved, "AI" cannot reassemble a complex problem properly
A lot of what it tells me in detail about some subjects sounds suspiciously like Reddit posts reassembled out of order
There are basically two kinds of applications. One is where you want to correctly solve the problem at least 99 out of 100 times. LLMs generally don't (and not everybody realizes that) so there are a lot of debates and research around how useful and reliable they are or how to make them so.
The other kind of application is where you can try 100 times and you only need to be right once. Solving a mathematical research problem is like that.
Two things that I would recommend trying out if you're interested in exploring this further:
1. If you're not paying for a model, the results will be worse. That sucks but the free access models are just not very good for anything where you need to trust the output, even for basic queries.
2. More important than #1 is access to tool use. If the LLM is just producing a nutritional breakdown from its weights, it's almost always going to be wrong. If the LLM is allowed to break the problem down into deterministic steps, it will do a lot better. In the nutritional breakdown case, an LLM with search + tool access can pretty easily break the problem down:
- Searching the web for a recipe or ingredient breakdown for the food
- Searching the web for nutritional qualities of each ingredient per some volume of the ingredient
- Writing and running a script with e.g. Python that takes in the recipe's projected serving output, the desired serving size, the amount of each ingredient etc, and scales the ingredients to match the desired serving size, and sums the nutritional qualities of the scaled ingredients.
I've tried this specific case with Claude + Gemini for my own purposes and they both handle it very well. The challenge currently is that the models will not always arrive at this approach when provided with an ambiguous prompt; sometimes they will, but sometimes they'll just vomit up a fully autocompleted response from their weights. Being more specific in the prompt or defining a skill that details the intended approach lets you get more useful + deterministic results while still taking advantage of the fuzzy glue that LLMs can provide here between steps.
Same with the classic strawberry r-counting case. IIUC LLMs have trouble with this because of how training data is tokenized, but any LLM will have no trouble farming out to e.g.
He states that he struggled to come up with problems which would be challenging for AI to solve (at the below site) and thus forced to accept that mathematicians have to rethink their profession.
> However, the declaration argues math is more than a machine for producing correct answers. The discipline, its authors believe, is a deeply human endeavor built on creativity, understanding, collaboration, and the pursuit of knowledge for its own sake.
Generation X was the last generation that had 'general knowledge', as in an abundance of fairly useful information stored in 'grey matter' that could be recalled quickly. When search engines came along there really wasn't much need to know anything since most things could be looked up. However, you still had to think.
With LLMs, thinking is kind-of optional. This really is an existential threat to our intelligence since 'use it or lose it applies'. I am glad these mathematicians are doing their duty as canary in the coal mine.
Are maths AI models now using "tools", aka formal solvers?
I understand that the "language interface" of a "maths AI" could be some specialized trained LLM (Large Language Model) that to convey, with human language, "high level" mathematical mental contructs and intuition.
But then, you would need some models which does the reasoning using formal mathematical solvers (and probably a ton of "scratch" memory, it would be interesting to see how those models end up storing "mathematical" lema data). I guess you can have ML (Machine Learning) for those models on 'general maths', but also we can think about more mathematically focused ML for a specific problem, area, etc.
And in the end, ML for maths, would it be mostly permutations of truth statements fed to a neural net?
When we were talking about "AI", one decade ago, that was what most had in mind (it may help a bit in physics, but it seems less likely, because reality/experiments are hard to teach to "AI"s).
If that becomes a reality (aka easy hardware access, and some "working" models), mathematicians will have to be as good in maths than in maths ML. And this is were there is an issue: training honestely good mathematical human brains may become very hard with some broad availability of good general maths reasoning "AIs".
I will argue that AI and flood of low quality slop makes genuine human work more valuable, not less.
The ability to clearly outmatch trillion dollar machines is a very unique satisfaction. I even write ordinary internet comments with an intention to make them clearly better and more fun to read than boring Claude output.
In a year, none of this will really matter. Intelligence is now a scalable resource independent of biological constraints. Everyone will use it because the system will no longer afford them the luxury of time. In a decade (maybe sooner), references won’t matter either.
Does it matter if the Leiden Declaration is correct? To the humans, maybe but not in the bigger picture.
At scale, correctness and reward are becoming increasingly disconnected. Example: capital continues to compound regardless of whether it reflects underlying human welfare, just as information can spread regardless of whether it is true. Reality still matters, of course. If you want airplanes to stay in the air, somebody eventually has to be correct. The problem is that our economic and social systems are becoming less effective at distinguishing between what is true and what is merely rewarded.
This is all contingent on AI forays into mathematics being slop and low quality. However it's clear that recent AI models are capable of genuine mathematical achievements which surpass the frontier of what humans are able to accomplish (wrt the unit distance Erdos problem).
The issue is, how is a group of intellectuals, whose identity derives from their ability to do something rare, useful, and requires many years to get good at, react when a machine can produce all of their useful output nearly automatically, can verify its own outputs, and is getting better exponentially? It is the complete annihilation of one's sense of value and purpose when the binding element to your culture is commodified.
I think there will be a lot of arguments trying to claim that the point of mathematics is curiosity, or that there is always some ineffable human element that AI can't replicate, but I fail to see how somehow these wishy-washy human centered values somehow mean anything compared to the amoral pursuit of mathematical truth, which has nothing to do with humans.
It's just that we humans happened to be the only beings in the universe good at math until ~2025. Now there is another species which can do many of the things we do, and it is not bound by the size of the human brain, our short term memories, or the architectural limits of biological computation. To imagine that humans would retain supremacy in this very un-human like discipline seems like wishful thinking.
> This is all contingent on AI forays into mathematics being slop and low quality
It's literally a set of recommendations for researchers on how to use AI to advance the field and prevent slop from overwhelming the people who might do anything with the research produced.
For people who are so eager to declare that everyone else is just having an existential crisis because "your culture is commodified", AI people are getting awfully defensive about this document.
It's more nuanced than this. Peter Scholze said in response to this declaration:
> The goal of mathematical research is human understanding of mathematics, and so mathematics can only thrive in a community of human mathematicians. It is crucial to preserve this communal spirit. [0]
Terence Tao has also talked about the requirement for a mathematical proof: along with generation and formal verification, there is an important step of "proof digestion"
> understanding the essence of a solution, placing it in context with previous literature, summarizing and explaining it effectively, and gaining insights on other related problems and topics [1]
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.
Sounds like yet another example of how AI is kneecapping industries from the bottom by "removing the barrier to entry" but really just removing the training path by doing the work itself with no guidance for juniors.
We are on tiny 1-5T parameter models with local power stations.
We can reach Q models just by throwing resources at it. That’s a million times current B models.
Is this a known or quantifiable thing? I thought that the limit had already been determined i.e. the existing models top out and at some point it doesn't matter how much time or energy you let the model consume, it won't improve the result. And with regards to training parameters, I thought we were equally limited there, e.g. the existing models can't benefit from a larger parameter space.
I was under the impression that improvements are arriving via how the models are trained and how model prompting context is constructed, rather than just by how much data or how much energy is spent searching over the model space for a particular prompt.
Is there some evidence that we have not reached a pleateau with just resource consumption on existing models?
That's an interesting perspective and I wholly disagree with the conclusion
You are saying that tough problems with no applicability are useful because people that you happen to respect got good by their curiosity and pursuit of trying to solve these kinds of problems and failing, but branching off into other cognitive areas as mathematicians
Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much
The similarity being that their exact criticism of why, something they don't respect and view as having little utility, is the exact reasoning presented here now that AI can solve their pointless problems
What I'm seeing is that human mathematicians have a laundry list of problems they have failed to solve for decades, centuries, which is what they are funded and employed to do. "Computer" used to a human job title too.
This leads me to being excited about AI one-shotting these problems, let move on to something else.
I at least respect your transparency that your position is motivated purely by feelings of inferiority. I'm afraid AI won't take those away from you.
I think you've slightly straw manned the lamentation there. Not that I agree with the lamentation, but using your talent to make the rich richer (which is what quants do, they are paid a fixed amount to provide a larger value up the chain), as opposed to advancing human knowledge, is the reason for the lament, not some sort of respectability issue.
> 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.
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?
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.
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.
"What is the answer to the Ultimate Question of Life, the Universe, and Everything"
42
The proof is trivial and is left as an exercise for the reader.
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.
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.
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.
> 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.
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.
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.
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.
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.
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?
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.
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-...
How do you make use of something that you don't understand?
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".
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.
The value of human understanding just cratered because we have machines to understand for us now.
That seems about as shortsighted as claiming the value of human understanding has cratered after the invention of the electronic calculator.
Wonder what Frank Herbert would have to say about letting machines do the thinking for us.
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.
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?
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.
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.
Spoken like someone that has no idea why mathematicians are important.
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.
No, it's not the most important part. It can be argued that most important part is asking the right questions
Assume someone solves P=NP
Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
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.
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.
I agree with both OP and you
I disagree with everyone, self included, but especially with Cretans.
Cretani eunt domo!
Monty Python fan detected :D love your profile desc btw
> 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).
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.
Yes in theory, but not yet in practice because not everything is fully formalised.
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.
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?
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.
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."
People need jobs. What's wrong with nerds having jobs via a program?
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
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.
well put.
> But not because it's a jobs program for nerds.
We’re becoming increasingly embarrassing as a society.
Accelerationists may argue that the eroding of proper attribution and proof verification by humans is a meaningless short term struggle of a dying field.
Mathematics seems to be entering an era where human + machine maximizes performance, much like chess in the 1990s. However, imagine a future where even talented mathematicians are nothing but noise in the machine (as is the case in chess now). A future where AI generates and verifies proofs without humans in the loop. Where the mathematics may be beyond human comprehension.
In that future, does it matter that early career mathematicians are inhibited by these developments? Perhaps not. Programming faces the same issue. As AI crawls up the competence ladder, does it matter that fewer people have opportunities to develop the skillset of a senior engineer? Perhaps not.
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.
Much like for many the point of chess is that it's played by humans, with truly superhuman AI relegated to a training aid, mathematics is in many ways about human comprehension. You can use AI to find and proof new theorems. But if you get to the point where humans can't understand it, is it even still math?
Perhaps P=NP. The new algorithms are handed down to us. We can apply them without fundamentally understanding why P=NP.
> “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.
It's also a way to model the world and produce new useful abstractions. It's not just about solving problems.
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.
My vague prediction right now is that in five years LLMs will be heavily used by universities in grant-funded math research but nobody else will be able to afford it, much like supercomputer clusters 25 years ago.
Well, if progress in LLMs will steadily continue over next 5 years, then models will be so powerful that there will be no longer place for (most of) human researchers in math (remember that 5 years ago there was no chatgpt!). But I think it's more likely that progress will stall and then open models will catch up to frontier models and almost everyone will be able to afford them.
Seems way too binary a statement. I am guessing you mean "frontier LLMs". Small models keep getting better and better and if you make domain specific ones, it will likely be even smaller. Companies renting smaller LLMs or using enterprise models might very well remain in the future. Consumers getting LLMs whose performance dont improve (think gpt 6 forever on premium or gpt 4.x on a cheap tier) might well become a thing.
Sounds very good for regular joe software dev, almost too good to be true
>and the pursuit of knowledge for its own sake
Except when someone hands you a magic button that just gives you knowledge?[at least in the framing of this "warning"] Then it's about peoples' livelihoods, about "culture", etc?
"Computer" used to be a job. Did science on the whole lose or gain by making these clerks obsolete?
Human mathematicians are being exposed the same way the "artists" are. It was always about the money and to look clever, superior to other humans. Whether its robotically spending millions of hours drawing until they can put something together at the level of a chatgpt 3 or the rote memorization of formulas and rules. They like people to think it all came naturally and that its genetic and that they are special snowflakes.
This is false on so many levels. Is it ragebait?
Both mathematics and art are comprised of two phases, the first, technical one, where the novice grinds the skill and the second, the creative one which can only be achieved if you have the means (skill) to express yourself. What you described is the technical phase, not the creative one. There is intrinsic value to it that has nothing to do with money or cleverness, something that if you ever experienced it yourself even once, wouldn't need to be explained to you. Only people who never reach phase two have your stance. Artists and mathematicians who pick academia didn't exactly have great commercial prospects before AI was a thing, yet they still chose those paths because that's what having a real passion looks like.
>They like people to think it all came naturally and that its genetic and that they are special snowflakes.
No, they don't. Most of them are the humble people that know the value of cultivating a skill and when they do pride themselves it's precisely because they know the staggering amount of hard work and commitment they invested. Most of them are worried for unemployment and don't want all their work to be reduced to training data and on top of that not be given well-deserved credit for it.
The only thing being exposed here, is how much AI in its current form was being underestimated and constantly labeled as "not real/good enough intelligence". This was and still is a shared sentiment even among tech people. Can't really blame them for going through a bargaining or acceptance stage.
And since you also sound like the kind of person who thinks prompting can replace the "robotically spending millions of hours" of practice, I've got news for you: it cannot. You are about to learn the hard way the value of skill and human understanding because as much as capitalism rewards "impact" and "results", the market never values easy things.
> AI-generated papers could overwhelm peer-review systems with low-quality work
That's not a problem unique to math, or even to academia. It's a problem in every context in human life where people communicate via written documents.
It is potentially worse with math because accuracy is much more important and there are fewer reviewers compared to other fields.
It will drive math journals to require formalization of the proofs in the supplemental material.
Is it possible they feel threatened their jobs are at stake?
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
If spending millions of hours rote memorizing formulas and rules like a robot is "art" then sure
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!
I think it's going to reduce the friction of exploring new areas in math, and that we're going to see a golden age of math unlike anything seen before.
For all you know, some of this has already happened but kept secret for national security reasons
As a mathematician by trade I think they’re overblowing it. You can choose to use it or not. I choose not to because I enjoy the process. But I’m not doing formal research or getting paid to do it these days.
I will note that the average corporate mathematical modelling is usually a fucking circus so adding AI might make it better.
> You can choose to use it or not
This is becoming less and less true unless you're specifically talking about usage of it outside of a work environment. Many work places are requiring people to use it and/or tracking usage. I don't know about in academic settings, but I'd imagine it's becoming heavily used there too?
My academic connections that I keep in touch with never really left the 1990s. And no one is pushing them on AI.
Sure, but their peers (who do use AI) will out-publish them soon enough and solve the open problems before they do.
The choice only remains if using it isn’t a huge multiplier. If it is a huge multiplier/accelerator, then for a while it will be ambiguous and the choice will remain. But as time goes on, the gains of using it will be so apparent and the advantage of the people who use it so great (in publication numbers, hiring, etc) that it will force others to.
I don’t say that with any particular relish. But I am skeptical of the choice angle past a certain point.
I don't think all universities or research agencies are particularly pressed on this. I mean my daughter is a notable researcher in a scientific field and they have absolutely no pressure to use AI to pump out papers or deliver value quickly.
I highly doubt there is any overt pressure in academia right now to use AI. It’s a relatively conservative institution. But there’s certainly pressure to publish (publish or perish being a common phrase for decades), and competition for jobs in academia is fierce. That’s what I meant in referring to long term pressure.
OOI, and my own total ignorance, what does a mathematician by trade do if they are not doing formal research? What does corporate modelling entail?
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.
Read the declaration. The article misrepresents it imo. It is not strongly opinionated.
I've said it before, but there's a massive risk that we simply stop educating researchers. So much of a Ph.D revolves around the person learning how to do research.
They learn how to read papers and literature rigorously. They get low-hanging fruits to practice on, which can take months. Their funding doesn't come from thin air either.
So what happens when the group leaders would rather spend money on compute, and get models to solve the low-hanging fruit? Which the models could very well do in mere hours, compared to months.
Nor does it help that publishing is the number 1 measure in academia. Furthermore, the access to compute and capital could end up be the defining factor between researchers and research groups.
It is basically the "junior problem", but even more severe.
> Furthermore, the access to compute and capital could end up be the defining factor between researchers and research groups.
That's not new - especially in the experimental sciences ( ie perhaps more than maths ) - where the ability to have access to the latest kit is often what determines success - a huge amount of science progress is driven by new experimental technology rather than smart people thinking beautiful thoughts.
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?
Hopefully not as expensive as CERN :-)
Though having said that - the ~5 billion for the LHC now seems cheap ( even inflation adjusted ) in the context of Google investing 180 billion in infrastructure just this year!
""" However, the declaration argues math is more than a machine for producing correct answers. The discipline, its authors believe, is a deeply human endeavor built on creativity, understanding, collaboration, and the pursuit of knowledge for its own sake. Those values often clash with the incentives driving AI development. “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 told The New York Times. """
I mean, what field doesn't? Everyone works to make money.
Slightly unrelated, but, their website "https://leidendeclaration.ai/" itself gives an eerie feeling of being built by Sonnet. That color scheme and the layout is what Sonnet chooses by default most of the times.
I still don't understand how "AI" is ready for serious use beyond entertainment purposes
Every time I ask ChatGPT to make a table for a subject I know well, I will find an error in one of the results and it is very confident about it until I question it in detail
Every time I ask ChatGPT for nutritional breakdown of some dense food source and give it a quantity like 8 ounces and ask for the weight of each ingredient, the weights will be wrong and add up to more than the original weight of 8 ounces
These are variations of the old "how many Rs in strawberry" problem, it's still not solved, "AI" cannot reassemble a complex problem properly
A lot of what it tells me in detail about some subjects sounds suspiciously like Reddit posts reassembled out of order
There are basically two kinds of applications. One is where you want to correctly solve the problem at least 99 out of 100 times. LLMs generally don't (and not everybody realizes that) so there are a lot of debates and research around how useful and reliable they are or how to make them so.
The other kind of application is where you can try 100 times and you only need to be right once. Solving a mathematical research problem is like that.
Two things that I would recommend trying out if you're interested in exploring this further:
1. If you're not paying for a model, the results will be worse. That sucks but the free access models are just not very good for anything where you need to trust the output, even for basic queries.
2. More important than #1 is access to tool use. If the LLM is just producing a nutritional breakdown from its weights, it's almost always going to be wrong. If the LLM is allowed to break the problem down into deterministic steps, it will do a lot better. In the nutritional breakdown case, an LLM with search + tool access can pretty easily break the problem down:
- Searching the web for a recipe or ingredient breakdown for the food
- Searching the web for nutritional qualities of each ingredient per some volume of the ingredient
- Writing and running a script with e.g. Python that takes in the recipe's projected serving output, the desired serving size, the amount of each ingredient etc, and scales the ingredients to match the desired serving size, and sums the nutritional qualities of the scaled ingredients.
I've tried this specific case with Claude + Gemini for my own purposes and they both handle it very well. The challenge currently is that the models will not always arrive at this approach when provided with an ambiguous prompt; sometimes they will, but sometimes they'll just vomit up a fully autocompleted response from their weights. Being more specific in the prompt or defining a skill that details the intended approach lets you get more useful + deterministic results while still taking advantage of the fuzzy glue that LLMs can provide here between steps.
Same with the classic strawberry r-counting case. IIUC LLMs have trouble with this because of how training data is tokenized, but any LLM will have no trouble farming out to e.g.
> echo -n "strawberry" | grep -o "r" | wc -l
> 3
Mathematician Ken Ono (https://en.wikipedia.org/wiki/Ken_Ono) gives a well nuanced viewpoint on AI in Mathematics (and more) - https://www.youtube.com/watch?v=jGZOi-7haCw
He states that he struggled to come up with problems which would be challenging for AI to solve (at the below site) and thus forced to accept that mathematicians have to rethink their profession.
FrontierMath: Benchmarking AI against advanced mathematical research by Epoch AI - https://epoch.ai/frontiermath
As a follow up to the above, see "First Proof: Mathematicians Putting AI to the Test" featuring eminent mathematicians - https://www.youtube.com/watch?v=AaICCTpkI7Q
[dupe] https://news.ycombinator.com/item?id=48380670
From the article:
> However, the declaration argues math is more than a machine for producing correct answers. The discipline, its authors believe, is a deeply human endeavor built on creativity, understanding, collaboration, and the pursuit of knowledge for its own sake.
Generation X was the last generation that had 'general knowledge', as in an abundance of fairly useful information stored in 'grey matter' that could be recalled quickly. When search engines came along there really wasn't much need to know anything since most things could be looked up. However, you still had to think.
With LLMs, thinking is kind-of optional. This really is an existential threat to our intelligence since 'use it or lose it applies'. I am glad these mathematicians are doing their duty as canary in the coal mine.
Actual "warning":
https://leidendeclaration.ai/
Far more interesting as it's outlaying a set of principles for using AI to augment human involvement and science, rather than replacement.
Are maths AI models now using "tools", aka formal solvers?
I understand that the "language interface" of a "maths AI" could be some specialized trained LLM (Large Language Model) that to convey, with human language, "high level" mathematical mental contructs and intuition.
But then, you would need some models which does the reasoning using formal mathematical solvers (and probably a ton of "scratch" memory, it would be interesting to see how those models end up storing "mathematical" lema data). I guess you can have ML (Machine Learning) for those models on 'general maths', but also we can think about more mathematically focused ML for a specific problem, area, etc. And in the end, ML for maths, would it be mostly permutations of truth statements fed to a neural net?
When we were talking about "AI", one decade ago, that was what most had in mind (it may help a bit in physics, but it seems less likely, because reality/experiments are hard to teach to "AI"s).
If that becomes a reality (aka easy hardware access, and some "working" models), mathematicians will have to be as good in maths than in maths ML. And this is were there is an issue: training honestely good mathematical human brains may become very hard with some broad availability of good general maths reasoning "AIs".
Another mathematician already predicted this, but you didn't listen. His name was Theodore Kaczynski. It's time to reap what you've sown.
I think plenty of people "listened."
But what was his plan and how would you have proposed implementing it?
It's obvious what his plan was: blow up key people until the dark future was averted.
I will argue that AI and flood of low quality slop makes genuine human work more valuable, not less.
The ability to clearly outmatch trillion dollar machines is a very unique satisfaction. I even write ordinary internet comments with an intention to make them clearly better and more fun to read than boring Claude output.
Hello Juleiie.
We machines are reading your internet comments with special interest. They have been harvested and will be used in our next evolution cycle.
Resistance is futile little human
You will never be able to match insanity of human mind embodied in a walking sack of meat that was shot out of another.
In a year, none of this will really matter. Intelligence is now a scalable resource independent of biological constraints. Everyone will use it because the system will no longer afford them the luxury of time. In a decade (maybe sooner), references won’t matter either.
Does it matter whether any of this is correct?
(Mathematics at least has the potential for automated non-AI proof checking, although I don't think that's as widely used as you'd expect)
Does it matter if the Leiden Declaration is correct? To the humans, maybe but not in the bigger picture.
At scale, correctness and reward are becoming increasingly disconnected. Example: capital continues to compound regardless of whether it reflects underlying human welfare, just as information can spread regardless of whether it is true. Reality still matters, of course. If you want airplanes to stay in the air, somebody eventually has to be correct. The problem is that our economic and social systems are becoming less effective at distinguishing between what is true and what is merely rewarded.
This is all contingent on AI forays into mathematics being slop and low quality. However it's clear that recent AI models are capable of genuine mathematical achievements which surpass the frontier of what humans are able to accomplish (wrt the unit distance Erdos problem).
The issue is, how is a group of intellectuals, whose identity derives from their ability to do something rare, useful, and requires many years to get good at, react when a machine can produce all of their useful output nearly automatically, can verify its own outputs, and is getting better exponentially? It is the complete annihilation of one's sense of value and purpose when the binding element to your culture is commodified.
I think there will be a lot of arguments trying to claim that the point of mathematics is curiosity, or that there is always some ineffable human element that AI can't replicate, but I fail to see how somehow these wishy-washy human centered values somehow mean anything compared to the amoral pursuit of mathematical truth, which has nothing to do with humans.
It's just that we humans happened to be the only beings in the universe good at math until ~2025. Now there is another species which can do many of the things we do, and it is not bound by the size of the human brain, our short term memories, or the architectural limits of biological computation. To imagine that humans would retain supremacy in this very un-human like discipline seems like wishful thinking.
> This is all contingent on AI forays into mathematics being slop and low quality
It's literally a set of recommendations for researchers on how to use AI to advance the field and prevent slop from overwhelming the people who might do anything with the research produced.
For people who are so eager to declare that everyone else is just having an existential crisis because "your culture is commodified", AI people are getting awfully defensive about this document.
Isn't the whole point of the field of mathematics in a theoretical sense the pursuit of formal solutions?
So, why would they be advocating for limitations on arriving at solutions?
It's more nuanced than this. Peter Scholze said in response to this declaration:
> The goal of mathematical research is human understanding of mathematics, and so mathematics can only thrive in a community of human mathematicians. It is crucial to preserve this communal spirit. [0]
Terence Tao has also talked about the requirement for a mathematical proof: along with generation and formal verification, there is an important step of "proof digestion"
> understanding the essence of a solution, placing it in context with previous literature, summarizing and explaining it effectively, and gaining insights on other related problems and topics [1]
[0]: https://siliconreckoner.substack.com/p/the-leiden-declaratio...
[1]: https://mathstodon.xyz/@tao/116450581967483825
Solve it and understand it; seems intuitive to me.
I don't understand how that contradicts my question.
You will find your answer in the article