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.
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.
LLMs don’t give a shit about social side effects, leave alone on unconscious level, because they are void of any intention. At most they are tuned on their thin edge layer to lean toward this or that kind of output, but that’s it.
Now the landscape shift as it’s sold (I guess) is that anyone can take a postdoc gibberish infused with the hard gained academic winks and subtle references and turn it into a ELI5 "does it have any applicability for my concrete issue at stake, prove it through Lean, good let’s deploy".
There’s no way to “ELI5” this type of complexity. I’m talking about concepts exponentially more esoteric than quantum mechanics, and even within quantum mechanics there is nothing to ELI5 for a concept like “spin”. The best you can do is say that it’s a property of a particle. But imagine the words “property” and “particle” are also completely meaningless to you because they’re built on even more layers of conceptual mathematical abstraction.
Do you see the problem with your reasoning?
A proof being latent in an LLM is no more significant than a proof being latent in a book, a theorem prover, or the axioms themselves. Einstein's papers were latent in the genetic code of his parents and the environment of his time. That doesn't mean general relativity was "already done" before Einstein was born.
By your logic, no computation has ever accomplished anything because the output was always implicit in the inputs.
The entire purpose of computation is extracting information from representations where it's difficult to see into representations where it's easy to see.
So no, this isn't a problem with the original reasoning. It's a problem with yours.
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.
Do you think Stephen Cook and Leonid Levin deserve more credit than whoever solved it?
But it is a moot point anyway. Cook and Levin are very well known already in TCS, and credit is not directly enumerable like money, so "more than a lot of credit" doesn't make too much sense.
For this problem in particular, asking the right kind of question was really important for the field and led to a lot of discoveries even before it will be answered.
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.
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).
Is an 80 year old unsolved problem maybe unsolved because it was never prioritized? Some problems stay unsolved because few people consider them worth working on.
Who is going to validate the results? Or do we skip that, with the risk of flooding the literature and collective understanding with unverified proofs?
There are 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."
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
We’re becoming increasingly embarrassing as a society.
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.
We can reach Q models just by throwing resources at it. That’s a million times current B models.
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.
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.
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.
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?
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.
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
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?
I don’t say that with any particular relish. But I am skeptical of the choice angle past a certain point.
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.
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.
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!
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.
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
(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)
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.
Far more interesting as it's outlaying a set of principles for using AI to augment human involvement and science, rather than replacement.
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
> 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.
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".
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.
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