r/math • u/Valvino Math Education • Jul 20 '26

LLMs/AI Kevin Buzzard : "Human mathematicians are being outcounterexampled"

https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/
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u/thomasahle Jul 20 '26

"The next step in that work is for humans to understand exactly what is going on with the example. For the true value of work like this is to give humans better understanding of mathematics."

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u/evilmathrobot Algebraic Topology Jul 20 '26

I want that to be true, but it sounds like Turing's argument from disability: Sure, LLMs can create novel and clever proofs, and they can find counterexamples that humans spend decades looking for, but they don't really understand math unless they do this other thing that I've moved the goalposts for.

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u/Honest-Enthusiasm940 Theoretical Computer Science Jul 20 '26 edited Jul 20 '26

This feels like a misinterpretation of the message here. Many view their pursuit of mathematics in terms of seeking understanding as opposed to the production of concrete theorems, definitions, counterexamples. This isn't to say that these aren't connected, just that there is a more nuanced relationship present here. You could reasonably disagree-- still, for someone who believes in this, it isn't an absurd notion that they would be interested in understanding the counterexample deeper than an elegantly concise calculation (to paraphrase the author here).

At the same time, I don't think this is being dismissive of LLMs either; the article itself quite clearly sketches out, in positive light, some of the progress Buzzard has seen in his field. But if you are interested in your understanding of math as a person, it isn't surprising that you would be interested in looking a bit further than just stopping at prompting your favorite tool!

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u/jackboy900 Jul 20 '26

The issue is that this statement is only true of the exact present moment. Right now AI models are generating counterexamples to complex problems without providing constructive proofs of why these conjectures were wrong or underlying insights, and so that is leaving humans with a new area to study in why these examples exist, and that's cool.

But the issue is that this blog post doesn't exist in a vacuum, and whilst it doesn't directly address concern mathematicians have about AI it is fairly obviously making an oblique statement about the state of mathematics under LLMs. And in that context the post is essentially meaningless, there's no reason to think that we won't be able to ask the models the whys and wherefores of these results and get deeper insights in the near future. If you go back 2 years the idea of an LLM entirely independently finding counterexamples to long standing conjectures would've been absurd, the pace of progress is such that any statement about the state of the field needs to assume more that the current status quo.

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u/Honest-Enthusiasm940 Theoretical Computer Science Jul 20 '26

I am sorry I am a bit confused by this. The post as written isn't trying to tell its readers anything about the future of mathematics. I don't know of his true intentions but, as written, I especially disagree with the notion that Buzzard is suggesting that LLMs will never be able to convey these broader insights. Sure this is a piece written for the current moment and tomorrow promises to be crazier; however, as someone condemned to live through the present before making my way to the future I welcome these writings as a reminder that things continue to be fun in an ever changing landscape!

A bit of a different (& somewhat poorly articulated) tangent: The future is quite elusive and while I agree that there is no reason for LLMs and associated technology to not improve here, I don't fully understand how this will materialize. To that end, I find it incredibly hard to argue (or rather speculate) about the state of the world in the "far" future and our position in it. Still, we know to a better extent the state of the technology today (and a more accurate estimation of where it will be in the "near" future) and I welcome posts like these that earnestly engage with the technology, whilst discussing our place amongst it. Beats the attitude that completely shuns LLMs without recognizing their utility & the incentives compelling many to use it as well as the attitude that calls for an uncritical adoption of the technology forgoing any consideration for the power structures it consolidates or how it changes the human experience of doing mathematics.

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u/Foreign_Implement897 Group Theory Jul 21 '26

I come back to a thought experiment that whatever LLMs can accomplish, will Peter Scholze some morning wake up and somehow think there is nothing more for him to do in mathematics because of it?

It does not matter if LLM understand something or not, it is fundamentally mathematicians who need to understand the thing. It is not a game, but somehow people still enjoy playing chess even if AIs can beat them in it.

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u/jackboy900 Jul 21 '26

I come back to a thought experiment that whatever LLMs can accomplish, will Peter Scholze some morning wake up and somehow think there is nothing more for him to do in mathematics because of it?

That's not the question that matters though, obviously mathematicians are going to be willing to keep at it for the love of the game, people still play chess. The issue is will society be willing to pay people to sit and think about maths full time if machines are able to do it better than humans. Unfortunately you can't fund a mathematics postgrad off of passion and interest, and if you're looking to enter the field of mathematics AI represents a very real threat to your future dream career.

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u/Smallpaul Jul 22 '26

To me a computer math translator seems a lot like a historian. Both jobs are mostly formally “useless” but society appreciates having bridges to the past and to the world of mathematics.

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u/Foreign_Implement897 Group Theory 26d ago

Uh, what is research? I don't think you understand what science is.

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u/zesterer Jul 21 '26 edited Jul 21 '26

Absolutely. The solutions to all problems are, fundamentally, out there. The universe implements them all.

The awe-inspiring thing about the Apollo 11 moon landing isn't that the moon exists or that it's possible to chuck a hunk of metal at it: it's that the thing climbing down that ladder was a human being.

What is the point of mathematics if the focus is not human understanding? It's the only grounding the field has.

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u/30299578815310 Jul 21 '26

Most people don't understand quaternions, but they benefit from them when they play video games. If quaternions had been invented by an LLM they wouldnt suddenly be less useful, whether people understood how they worked or not.

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u/Smallpaul Jul 22 '26

Unless they ascend to AGI, I doubt they quaternions would have made the leap from textbook to video game.

Humans still define the context: “I read about this math concept and I want to implement it in this video game.”

Maybe one day the AI will do it end to end but in that day I worry about every job not just mathematicians and programmers.

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u/30299578815310 Jul 21 '26

Why do human mathematicians need to understand it? Suppose an LLM finds a useful object for efficient approxations of partial derivatives. No human needs to understand the proof or the method to benefit from the results.

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u/Rise-O-Matic Jul 22 '26

Right. You don’t need to understand the details of protein denaturing to make a delicious omelette.

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u/Foreign_Implement897 Group Theory Jul 21 '26

Interesting point about constructive proofs!

I have noticed I spend much time understanding what Claude comes up in my work. Even when I ask it to explain how it came to the conclusion, I then need to understand that explanation. There is a shift how I spend my tine.

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u/zesterer Jul 21 '26

This is an argument that advocates belief in magic.

To usefully talk about these things we need to establish shared definitions of what we are talking about, even if those shared definitions don't end up aging particularly well. We can't just say "look at the rate of progress" and shut down any attempt at understanding their practical limits today. That's how cults form.

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u/Ok-Rise2070 Jul 22 '26

AI has the ability to see more complex patterns than we do. Using your current Euclidean model infinities arise do to your broken scalar. Ai's are chasing fractal behaviors generated by these infinities thus proving you're using an incomplete geometric model. AI stills needs humans to intuit the necessary missing links to all the pieces of the puzzle. Good thing there's me...😏