Math proofs need to produce the correct output correctly, which is not quite the same thing.

This looks like an AI IPO PR powerplay, because at this point the proofs haven't been checked and it may not be possible for a human to check them - because proofs should be clear, not horribly written and noisy.

The noise is suspicious because it's the difference between brute forcing and cognition. A human proof won't just be logically correct, it will be cognitively distilled and coherent. It may still take years to understand it, but the logical flow will be straightforward, not obfuscated.

You want the path through the maze to be as short as possible and the map to be as clear as possible.

This sounds like the opposite. There may be a genuine path through the maze, but if it's too convoluted and takes too long it will be impossible to confirm.

I think the next step is to demand that proofs either be human-scale or they prove that a human-scale proof is impossible and the machine proof is as good as it gets.

I suspect that's possible without tripping over the halting problem. (But I can't prove it.)

Isn't that just the default experience with AI these days? In small enough scale, AI models can express their ideas clearly. But the larger and more complex the ideas are, the less suitable the outputs are for human consumption. I guess AI models think too different from humans, and nobody has trained them to communicate complex ideas in the way human experts in that particular topic expect.

This proof dump reminds me of the glut of low-quality drive-by PRs overwhelming open-source repos.

It also aligns with the fear that these proofs present a risk to the ecosystem by out-competing attempts at more human-readable proofs. Perhaps though we end up with more math influencers who edit and annotate these proofs to bring them back to us.

Serious question:

Why would anyone believe this (also) is not simply example N+1 of this is the worst it will ever be, as opposed to recognizing this as what will almost certainly prove to be an awkward moment, soon to be replaced by another order of magnitude of cleaner, clearer, more intelligible, etc.?

Ximm's Law: every critique of AI assumes to some degree that contemporary implementations will not, or cannot, be improved upon.

But up until now, the mathematics community has valued the "prove" part much more highly than the "deliver an insight" part. Mostly because with a little work they went hand in hand.

And going from zero proofs to one proof (even a sloppy one) is a big deal regardless of whether it was written by AI or a human.

No, haven’t you heard? Since the AI bubble began we’ve collectively decided that outcomes are all that matters. /s

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Ted Chiang is incredible, my favorite writer.

I also recommend "Exhalation", though that has nothing to do with AI.

It's a mixed bag I think.

For example, the statement of e.g. Fermat's last theorem in Lean should be understandable to anyone who played The Natural Number Game [0] and knows a bit of mathematics and programming. For the proof, you trust the compiler.

The statement of other theorems can be much more delicate, and the Lean formalization may require an extensive introductory section which will need to be carefully checked.

Then there are the cases where no Lean formalization is currently available, and all we have right now is an often impenetrable pdf in the OpenAI repo. I would not at all be surprised if some of those claims contained logical gaps.

Time will surely tell, but there are certainly doubts and lots people are very busy checking these results.

[0] https://adam.math.hhu.de/#/g/leanprover-community/nng4

There's an entire paper claiming that many of these AI-generated Lean proofs are formulated incorrectly / mistranslated: https://arxiv.org/abs/2610.08144

Also a ton of comments in this thread: breathless frothing hype declaring mathematics is over and assuming these proofs are exactly what they claim they are at face value, giving the company with a vested interest in everyone unquestioningly believing this is all real every conceivable benefit of the doubt

If you haven't made progress in Quantum Computing in the last 10 years, lecturing others can become a popular pastime.

Having stuff explained to you in patronizing tones? How horrible Scott!

more like 3/4 of the comments but yea

Yes, their parents are going around saying oof, the kids definitely didn’t get it from Roblox, or YouTube, or their peers.

I'll just leave this here https://owl.purdue.edu/owl/general_writing/academic_writing/...

The issue is ownership, we have no means of distributing the knowledge from the AI or rewarding those who could help.

We are quickly moving to a world where all symbolic and numeric reasoning for economic purposes is performed by AI.

I'm a research mathematician. From what I can tell, the answer is roughly comparable to: if you posed 8,000 challenging open problems to the human math community, you might expect to see 372 of them solved within five years.

Probably some combination of: some of the 372 problems were easier than the rest; the AI got lucky on these 372; there were existing papers out there in the literature which proved especially helpful for these 372; and other similar factors.

My guess would be that these particular problems were vulnerable to an attack which built on recent advances and potentially tied in something unexpected from a distant area of mathematics. "Harder" is becoming harder to define. Harder for humans is probably not harder for LLMs.

If it took 3h for one of them, perhaps there was a time/compute budget cutoff along with a sorting based on some relevance.

There must be an element of luck, if they ran the remaining problems again with the same time constraints presumably a bunch would be solved

Often times the constant (2 in this example) is a conjectured minimum, so anything below that is a noteworthy result. Think of it as breaking through some theoretical limit.

Many people thought it could never be less than 2. They proved that it can. What is the true value? Nobody knows, now.

for say FFT/integer multiplication or 3SUM, we have natural algorithms that have existed a long time with a given complexity (O(n \log n) and O(n^2), respectively). Given how long these natural algorithms have been the best algorithms we have, it is natural to conjecture they are optimal. Showing an O(n(\log n)^{.99999}) algorithm exists shows that these optimality conjectures are false.

Now, there are some critiques you can have of this. Namely, it is possible that these novel algorithms have significant trade-offs that make them almost never worthwhile in practice. "Fast" matrix multiplication algorithms are typically of this form. So perhaps this all points towards a deficiency in big O notation, which can be deceptive. But, for people who care about optimizing asymptotic complexity, it is still interesting.

to get some intuition about why this is such a big deal, look up the history of strassen's algorithm, which solved matrix multiplication in less than O(n^3). this was a truly stunning result because it seemed intuitively obvious that the output matrix had n^2 cells each of which was calculated via an independent O(n) loop over a row/column of the input matrices, so how could you do better than n^3. but once strassen proved that you could do some clever tricks and reduce the overall time to something less than O(n^3) it started an entire cottage industry of people getting better and better algorithmic bounds. the initial breakthrough was a qualitative one, independent of how much it improved things in numerical terms.

https://hideoushumpbackfreak.com/algorithms/algorithms-stras...

You just run it again and again and again

I've asked my students whether they still want to learn maths even if there will be a machine that will answer any question instantly and they will be homeless. They said yes.

(To my credit, I have warned them since more than a year ago that we will reach this point.)

To some extent this was discussed in the article, and in a way I think their goal is actually the same as it was: become the first human to understand something.

It's just that we lost one of the important ways to demonstrate understanding.

Pro AI blogging sponsored by ClosedAI, XTX markets and the Simons Foundation.

> what problems do these people reorient towards after this?

The same problem almost every person on earth is going to have to reorient to in the next decade, which is: how do we eat and stay housed when we have no real economic value?

They will continue to prove theorems and make discoveries, except now they will have AI to help them so hopefully progress will be faster. At the same time, new challenges will open up, for example how do you verify what the AI is doing and how do you explain it.

Math isn't about collecting random theorems, progress in math is about gaining understanding of new systems, and the theorems are guideposts to aid in that understanding.

You can prove 1000 theorems and not really increase any understanding about a subject, but gain knowledge of 1000 random facts. For example, I can write down some complicated equation and ask you "does this have a solution in the integers"? And if you do a maze of very complex and tedious algebra to show that there is a solution, you would have proved a theorem, but you would not have done much to move math forward at all.

On the other hand, if you introduce some completely new technique, say you take my equation and turn that into an algebraic surface, and then you count some special curves that live on this surface using geometric ideas, and then you show that if the number of such curves is odd, there must be a solution in the integers, and in this specific case, it is odd, so there is a solution -- well, then you have really pushed math forward and people will celebrate your proof, even though no one really cares if the equation I wrote down has a solution in the integers.

For example, there is a long history of failed attempts to prove Fermat's last theorem driving algebra and number theory forward by introducing the concept of ideals, for example, and this concept ended up much more important than whether Fermat's theorem is true or false, which is not too much more than a piece of trivia.

Or for example, the recent proof of the Poincare conjecture relies on the machinery of the Ricci flow introduced by Richard Hamilton, who then applied it to solve a number of open problems, but Perelman was able to take it even more forward to solve Poincare. So Ricci flow was massively important machinery.

For this reason, we celebrate people like Gromov, who didn't really prove that many theorems but introduced amazing machinery -- for example, the h-principle, or Gromov Compactness -- these were ideas and math is about the ideas. The ideas are then applied, using laws of logic, to form theorems.

So mathematicians will need to mine these proofs to see if there are any new techniques - new machinery - being introduced, or if the AI just used the existing machinery more efficiently. Here too, we are just looking at AI as a form of search, which it is really good at, since there are so many thousands of papers and so many ideas, that there might be a connection between two areas that lead to a solution and the human mathematician, not knowing all known results, can't make that connection. In the future, we may wonder how anyone did math without AI, much like we would wonder how anyone can be a writer without access to a dictionary or reference work. Is the AI just searching through a catalogue of known ideas and connecting them or is the AI coming up with genuinely new stuff like Ricci flow or the h-principle?

What is interesting is seeing whether we can get AI to actually discover new machinery for us. That would be huge.

And then we need to find efficient ways to detect these ideas and describe them.

Really this is very exciting and opens up whole new workstreams for mathematicians.

priesthood

Food and shelter /s

You can certainly do that - but it's quite the break from tradition to release a paper in the state described. Why they did is a really interesting question! It may be that AI math requires approaches that humans don't find intuitive and what you are describing is actually counter productive (because, in summarizing the work in a way humans understand, you're removing the context an AI would use to further the work an AI did). It also might be that OpenAI could have done that and chose not to - or maybe they tried and this was the best they could do. No matter what I don't think anything about how to react to a paper being released in this state is obvious.

Isn't AI well-suited to tasks #1 and #2, though?

#3 at this point might need more human intuition; but that might be a 2026 problem.

I’m still wondering why UT Austin is letting him teach a course (CS395T AI Alignment Theory) so completely outside his field of expertise (Quantum Computing).

The reaction/fallout would have been substantially improved even if OpenAI had just made a commitment to not scoop external researchers using an internal model until X months after the model had been made available to the public.

That would have given grad students who've been grinding towards a PhD for years a fighting chance to see if they could leverage the model to push their work forward, rather than watching years of work potentially turn to dust via a tool they don't even have access to.

It wouldn't delay the progress of mathematics by any meaningful amount in the long run (an X month delay is nothing) for OpenAI to take this approach, and would help somewhat to preserve the health of mathematics as a field. Without it, the motivation for any young mathematician to devote years to a new problem must be sapped knowing there's an uneven playing field... an OpenAI team with access to colossal tools months before they'll ever be able to get access, willing to scoop anyone as soon as they can, perhaps without even taking the time to completely understand the proof.

I don't see any long-term benefit to OpenAI with their current strategy. This is an internal model; it's not available for sale at the moment. They've said they're not even going to bother claiming the Millenium Prize money for Navier-Stokes. It feels like kicking over hundreds of other people's chessboards just because they can.

Also, the independent authors might have spent some time to actually understanding the results and producing a readable manuscript. The AI papers are pretty badly written.

Evolution did statistically guided brute forcing. Doesn't mean that biology has no value

That is not a correct description of what the AI did.