How Will AI Change the Field of Mathematics? (theverge.com) 46
"AI went from being very terrible to seemingly genuinely quite good at a professional level in a very short space of time..." says the Verge's AI reporter.
But AI systems "are still truly, truly terrible at some areas of math... even the days of the week."
If you look at academic math papers, a lot of the time you won't see numbers... So they're still terrible, but they're now also very good at this other part. As to why, at some point you reach a critical mass of what these systems can do. We saw it with writing, we've seen it with programming. They're very good at forging connections between different areas, applying old methods in new ways, those kinds of things. It appears that the newer models they're training have apparently reached that level where it clicks, and now it can do math...
There are areas now where it seems to be producing work that is on par with good mathematicians, alongside other parts where yeah, it can't count. All caught up in that is whether it's going to rewrite employment structures or funding structures... One would hope [math researchers] would branch out into all of these new exciting areas or pose new questions. But AI won't do that, and that's the concern. And then that would leave the field quite sterile, and it will have all of these things that have been done, and maybe nothing left to pursue... [A] lot are scared that it's closing off the field. So by definition, those breakthroughs that lead to something surprising and new that you can say, "Oh, this works here," may not be happening anymore...
There were some bleak responses from graduate students I saw in essays posted online. Where's their place in this as future researchers? Do they have a place in this? Is it as glorified AI proof checkers? That will be quite an unsatisfying career, I imagine. Or maybe not, I don't know. We will see. I think anything used properly will be a net boon.
They also had an interesting response when asked if AI democratizes access to high-level mathematical proofs: A lot of the mathematicians I spoke to were almost quite weary of this, actually. They love the idea, in theory, of democratizing access. They're also quite fed up with AI-generated or -assisted papers that are flooding every publication imaginable, as well as the pre-print servers that they use in these fields. Some of those I spoke to said things like, "Oh, I got three emails this week alone with people being like, 'Hey, is this legit?'" Because they thought they'd solved something with ChatGPT or with Claude, and they also don't have the mathematical skills to check whether they've actually solved something.
On the flip side, there are parts where they said, "Well, we've got a talented undergrad who's done something that a talented undergrad would probably have never managed, and here they are doing grad-level work and they've produced a paper that is legit." And in the bigger scheme of things, a few I spoke to said, "Well yeah, a lot of these are in the ivory tower. Having access to this kind of thing globally could really boost access to the kind of things here." On the flip side, the cost. These things cost a lot to run.
There are areas now where it seems to be producing work that is on par with good mathematicians, alongside other parts where yeah, it can't count. All caught up in that is whether it's going to rewrite employment structures or funding structures... One would hope [math researchers] would branch out into all of these new exciting areas or pose new questions. But AI won't do that, and that's the concern. And then that would leave the field quite sterile, and it will have all of these things that have been done, and maybe nothing left to pursue... [A] lot are scared that it's closing off the field. So by definition, those breakthroughs that lead to something surprising and new that you can say, "Oh, this works here," may not be happening anymore...
There were some bleak responses from graduate students I saw in essays posted online. Where's their place in this as future researchers? Do they have a place in this? Is it as glorified AI proof checkers? That will be quite an unsatisfying career, I imagine. Or maybe not, I don't know. We will see. I think anything used properly will be a net boon.
They also had an interesting response when asked if AI democratizes access to high-level mathematical proofs: A lot of the mathematicians I spoke to were almost quite weary of this, actually. They love the idea, in theory, of democratizing access. They're also quite fed up with AI-generated or -assisted papers that are flooding every publication imaginable, as well as the pre-print servers that they use in these fields. Some of those I spoke to said things like, "Oh, I got three emails this week alone with people being like, 'Hey, is this legit?'" Because they thought they'd solved something with ChatGPT or with Claude, and they also don't have the mathematical skills to check whether they've actually solved something.
On the flip side, there are parts where they said, "Well, we've got a talented undergrad who's done something that a talented undergrad would probably have never managed, and here they are doing grad-level work and they've produced a paper that is legit." And in the bigger scheme of things, a few I spoke to said, "Well yeah, a lot of these are in the ivory tower. Having access to this kind of thing globally could really boost access to the kind of things here." On the flip side, the cost. These things cost a lot to run.
Millennium Prize (Score:3)
I predict an AI will be instrumental or outright solve a Millennium Prize.
Re:Millennium Prize (Score:5, Interesting)
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Or maybe a LLM will hallucinate that such a paper exists, and then someone will Porcupine Tree it into reality.
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Hasn't this been happening forever, before LLMs?
https://xkcd.com/978/ [xkcd.com]
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Porcupine Tree is a band that started out as one guy pretending to be a long-forgotten psychedelic rock band from the 1960s. He ultimately achieved enough success that it is an actual band in its own right and has been for decades. By pretending the band existed, he set up the conditions for a real band to exist instead. Kinda like Tom Scholz and Boston, except taken to the next level.
Re: Millennium Prize (Score:2)
It will certainly gather context over every flipping obscure paper, in every language, and use it to achieve its goals.
Why not use AI to answer his spam? (Score:2)
"Oh, I got three emails this week alone with people being like, 'Hey, is this legit?'" Because they thought they'd solved something with ChatGPT or with Claude, and they also don't have the mathematical skills to check whether they've actually solved something.
So would it be better for those inquirers to remain wallowing in their ignorance rather than have AI try to help them?
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"To avoid three emails a week becoming thirty a day, perhaps it would be best to remove the monetary chase from these mathematical challenges."
I doubt it would help. There are legions of people who want to claim some sort of mathematics fame to the tune of: I beat all the best mathematicians to this.
These people regularly contact famous mathematicians with their proofs of fuck-all. I think the noise level will just continue to go up: now I have a proof and I used QUANTUM AI to solve this.
Just an example, on
Can we please get that crash soon? (Score:1, Troll)
All these morons believing in magic are getting on my nerves.
Re: Can we please get that crash soon? (Score:2)
It's not magic. It's probablility statistics, just like the rest of us.
If AI ever achieves a 4 digit IQ, we'll see whose nerves are being trampled.
Re: Can we please get that crash soon? (Score:1)
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You are just a moron that does not understand Science at all. The actual state-of-the-art is that we do not know how a human mind works, despite your hallucinations.
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No, we do not know AT ALL how a human mind works.
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No, we do not know AT ALL how a human mind works.
We just don't know all the details, but your characterisation is incorrect, unless you going to start invoking dualism.
Re: Can we please get that crash soon? (Score:2)
I feel like I'm talking to a Luddite Language Model.
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Yes, it must be magic.
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I do not think you know what "doing mathematics" means. All LLMs can really do for mathematics is better searching. That will find a few overlooked things, because the mathematical literature is very dispersed and hard to search. But that is it. Incidentally, even a fields medal does not usually qualify you do understand LLMs. But many experts in one field have no understanding of how limited their insight is in another. So, yes, I am absolutely not above to call a fields medal winner that makes a stupid st
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I do not think you know what "doing mathematics" means.
I'm reasonably confident I have a pretty good understanding of what doing math is like given that I'm a published mathematician. Heck, I'll be a little egotistical and just to my own recent paper directly https://cs.uwaterloo.ca/journals/JIS/VOL29/Zelinsky/zel14.html [uwaterloo.ca]. But this also isn't terribly relevant.
All LLMs can really do for mathematics is better searching.
LLMs have been very good at doing searches. But they've done a lot of things that are pretty obviously not mere searches. For example, in the case of Erdos 1196, the LLM solution constructed a Markov chain
AI will engage in nomenclature confusions. (Score:2)
and spawns severe nomenclature problems. AI can make any mistake it thinks
fits the 'mold', and probably will. Soon.
The problem of teaching and communicating mathematics won't be solved by AI. At its best,
it'll identify contradictions. At its worst, it'll produce contradictions.
The real answer... (Score:3)
We don't know.
It might help a bit.
It might revolutionize math.
It might help invent a radically different kind of math that answers very difficult questions in physics and engineering.
It might hit a wall.
The wrong way to look at it is like a labor union boss who criticizes it because it might affect employment of mathematicians.
Seems obvious! (Score:2)
Mathematicians invented imaginary numbers because they refused to admit they were wrong which logically means that AI will invent hallucinated numbers to explain why they gave a non-sense answer to a basic math question.
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Mathematicians invented imaginary numbers because they refused to admit they were wrong
Shhh! Don't let them find out that x^2 + 1 = 0 has no solutions at all. It's our best kept secret. You have to learn, like, at least three secret handshakes and get an A in the entire calculus series before you can learn about that.
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Just in case anyone reads the parent and thinks it might be true, it's bullshit.
The cost on human minds is high (Score:3)
It is really stunning to go back to people like Euler and see how brilliant they were before they had computers. So much of math was invented from avoiding tedious calculation.
But the hard truth is that mathematician may be another fleeting job of history. Think of the genius that went into violin making, glass work, all these fields before industrial processes replaced them. How valuable was a top 10,000 piano virtuoso in 1730, before we could just listen to beat one recorded? So many unique and valuable geniuses just erased. But at the end of the day, humans just need food and shelter. Love of math, like love of piano, will not die but social need will.
The strange thing is how many people are still choking in mines wondering when the roof will cave in, still wiping butts in nursing homes, still risking life and limb fighting fires. The social perception of valuable math genius robbed some of their heaven and left others in hell.
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Nice post and an interesting comparison, but I think there's one big difference: live music remains hugely popular and a big business despite recording technology, but you don't see a lot of people showing off their arithmetic skills for profit. Today's mathematicians are more akin to composers.
There's also an interesting historical side to the analogy, as the separation between composers and performing musicians is relatively modern (and only really applies to classical art music genres anyway). And as
Counter examples (Score:2)
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In 2025 Claude had pass rate of 2% in FrontierMath. In 2026 Claude had 88%. Perhaps we should just wait a few years and see if the pass rates stop increasing?
Source:
https://epoch.ai/frontiermath/... [epoch.ai]
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I always find your comments well-reasoned and carefully considered. May I be allowed to speculate, that, absent publishing the source code and defining its attendant processes (perhaps human-guided), we have no idea what the actual f*ck is going on here ? Given that OpenAI borrows its marketing a bit too much from the writings of Mr. Eric BlAIr. From what I've read, the LLM was "unaware" that it had discovered the counter-example, and continued yakking away. But that would be very human too...
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AI is just a stochastic parrot/tinman! (Score:2)
A short list of major math AI successes (Score:3)
Erdos 1196, completely solved by the AI. https://www.erdosproblems.com/1196 [erdosproblems.com]. This was the first major unambiguous success by an AI where the solution was clearly not the AI just finding something from a really obscure part of its training data. The method involved a clever construction of a Markov chain on the natural numbers with the chain weighted via the von Mangoldt function.
The Unit Distance Conjecture. This is a really cool and easy to understand problem so I'm going to take a moment to talk about what it is. Suppose you want to put n points in the plane and you want to maximize the number of points which are exactly distance 1 away from each other. Note this is the same as just picking some distance k and making lots of points all distance k from each other and then rescaling. Now, the most naive thing to do is put one point in the center of a circle and then put n-1 points on the edge of that circle. That gets you n-1. But if you instead put your n points in a grid, and make it as close to a square as you can get, then you can get around a constant times n^2 points all the same distance. The conjecture was that this was close the best possible. In particular, that for any epsilon>0, you could not do better than O(n^(2+epsilon)). The AI constructed a clever tower of fields to disprove the conjecture. This was previously discussed also on Slashdot here https://science.slashdot.org/story/26/05/21/0351218/openai-claims-it-solved-an-80-year-old-math-problem [slashdot.org]
The Cycle Double Cover Conjecture https://arxiv.org/abs/2607.15399 [arxiv.org] was a long-standing problem in graph theory. Curiously, the problem's solution is something that a skilled undergrad could have plausibly come up with but apparently no one did. The proof uses some standard graph theory tools and then connects it to some ideas from linear algebra, but linear algebra is already pretty often used in graph theory.
The Jacobian conjecture for n=3 https://en.wikipedia.org/wiki/Jacobian_conjecture [wikipedia.org] This is a conjecture that was major enough that I've mentioned it when I've taught multivariable calculus. n=2 is still open. This is one where there may have been some amount of human input but we don't know.
OpenAI released a set of 10 major problems here https://openai.com/index/ten-advances-in-mathematics/ [openai.com]. I'm not going to go into all 10 of these in detail, but I will note that both the Ramsey problem and the sphere packing problem are not too complicated to understand. I had also given some thought to both of those problems and got nowhere. The sphere packing problem was close to things I thought about in grad school, and the Ramsey problem is close to some things I've been thinking about it. Of the other 8, 6 we're ones I was familiar with simply because they were well known. For the other 2 of those, I've discussed them now with people in those subfields and confirmed that OpenAI's description of these as major problems is not unreasonable. We don't have direct confirmation that there was minimal human input here.
But by far the biggest breakthrough was recently when Anthropic released a proof which increased the percentage of non-trivial zeros of the Riemann zeta function [on the critical line from around 41% to 67.2% . https://www.anthropic.com/research/riemann-zeta [anthropic.com]. A clarification: You don't get to prove the Riemann hypothesis when you get 100% on the line. The Riemann hypothesis is that all non-trivial zeros live on the critical line. But that's stricter than 10
AI change mathematics ? (Score:1)
WTF.. garbage statement. AI will NOT change math at ALL. It will hopefully extend math comprehension by a large margin, but I seriously doubt it "changes" any of the underlying supports. Maybe AI will suddenly discover new numbers ??
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1. You're being a silly pedant. They mean changing how advanced math is done. Mathematicians using AI to generate and check proofs, for example.
2. You might be surprised at how your pedantic interpretation might turn out to be true as well. It happened around the beginning of the 20th century when mathematicians engaged in a program to formalize the foundations of mathematics by adopting axiomatic set theory. It's happening now with a new formalization program advocating machine verifiable proofs. And there
AI is state space search (Score:2)
More work completed will lead to more work to do (Score:1)
Just as human accountants still have a job after the "paper spreadsheet" was replaced the with the "computer spreadsheet" in the 70s-80s, as more work is completed, then more work will be given.
I figure with AI will be the same.
Humans do a good job at thinking of more work to do.
As the more math problems that are solved, may lead to future questions to work on.
This is somewhat related to Jevons paradox.
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Just as human accountants still have a job after the "paper spreadsheet" was replaced the with the "computer spreadsheet" in the 70s-80s,
I do not believe this is true.
It's true that the job of Accountant still exists, but it is not true that the number of humans employed to maintain and transact the the ongoing financial operation of a business in the 70s-80s is the same as the number today. In the 70s-80s we had entire rooms filled with "bookkeepers", accounts payable, accounts receivable, payroll, bursars, finance, etc. And we needed to hire multiple duplicates of them to run the books for every local branch, because so much of the work wa
Not can we, but should we? (Score:2)
It's entirely possible that AI will soon be able independently to prove all of known mathematics, and then go on to resolve all the outstanding open problems. And then maybe it will start proposing new, interesting conjectures.
Do we really want this to happen? There are so few genuine pleasures in human life. Do we really need to build machines to take away yet another one, the esoteric pleasure of discovering a new mathematical proof?
Look what happened to the human race when machines came on the scene t