" I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well.
Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools?
We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions."
I was at the workshop that resulted in the Leiden Declaration in Fall 2025. The majority vibe was that this was inevitable, but hard to predict whether it would be in one year or 30 years.
I guess they showed this to the advisory group they created. I guess the group tried reading the work for a day and they could only think of telling them to release the results to the community. I now understand why the group had this suggestion.
There was a Wired Magazine article from either the late 90s or early 2000s that made a prediction that this sort of thing would eventually be possible, likely within my lifetime. I believe the context was "distributed computing" models of the time, like SETI.
I've never been able to find that article as an adult, but I would love to know who wrote it.
Gina Kolata allegedly in the New York Times in 1996 on the Robbins conjecture (noting that computers had started to contribute to math research in some sense), and a longer piece in Math Horizons by her the following year ("Computer Math Proof Shows Reasoning Power"). I didn't immediately find the NYT article, so I don't know if it might be a hallucination.
I was involved in some distributed computing stuff in the late 1990s and early 2000s and I don't really remember people in that community talking about proofs but there may have been a "if we had a mechanical proof-checker, could we do distributed searches for valid proofs that it would accept?" conversation somewhere at some point. There were definitely volunteer distributed computing projects working on pure math; I remember the Optimal Golomb Ruler search (https://en.wikipedia.org/wiki/Golomb_ruler). So, that could possibly have shaded over into "can we find proofs this way too?". At the time it probably would have been based on brute force searches through proof space rather than clever optimization, though.
The idea that you can lexicographically list all proofs in some formalism and then mechanically determine if any is valid is quite clear from Gödel's construction of the function Bew in "On Formally Undecidable Propositions", but he points out that you don't know where to stop because you don't know how long a valid proof would potentially have to be (so "is this a valid proof of this claim?" can be decided mechanically in a limited time, while "is there any valid proof of this claim?" can't be! maybe the shortest valid proof is 49 steps long but you eventually stopped checking after looking at all 7-step proofs, or something).
It's really impossible to predict which discoveries will "matter", have a direct impact on other fields, or an impact in making other mathematics or physics discoveries.
Only after a world's worth of experts look at these results and then mull over if and how their own fields are impacted by this new info will we be able to answer this question.
It can take decades to really know the full significance. You know, the whole "We stand on the shoulders of Giants", well the Giants just grew a few inches all at once.
Lots. To give an example Terrance Tao was lambasted skeptics on this site for stating it in 2024.
https://unlocked.microsoft.com/ai-anthology/terence-tao/
" I expect, say, 2026-level AI, when used properly, will be a trustworthy co-author in mathematical research, and in many other fields as well.
Then what? That depends not just on the technology, but on how existing human institutions and practices adapt. How will research journals change their publishing and referencing practices when entry-level math papers for AI-guided graduate students can now be generated in less than a day—and with the far better accuracy of future AI tools? How will our approach to graduate education change? Will we actively encourage and train our students to use these tools?
We are largely unprepared to address these questions. There will be shocking demonstrations of AI-assisted achievement and courageous experiments to incorporate them into our professional structures. But there will also be embarrassing mistakes, controversies, painful disruptions, heated debates, and hasty decisions."
He's pretty damn smart that guy.
Terrance, Reinmann and Hebert walks into a bar...
> He's pretty damn smart that guy. This is probably the understatement of the year. I am literally ROFLing.
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I was at the workshop that resulted in the Leiden Declaration in Fall 2025. The majority vibe was that this was inevitable, but hard to predict whether it would be in one year or 30 years.
I guess they showed this to the advisory group they created. I guess the group tried reading the work for a day and they could only think of telling them to release the results to the community. I now understand why the group had this suggestion.
I predicted, over 2 years ago, that theorem proving would fall way before other problems that people believe are harder.
https://news.ycombinator.com/item?id=41072330
There was a Wired Magazine article from either the late 90s or early 2000s that made a prediction that this sort of thing would eventually be possible, likely within my lifetime. I believe the context was "distributed computing" models of the time, like SETI.
I've never been able to find that article as an adult, but I would love to know who wrote it.
Some candidates suggested to me by an AI:
Gina Kolata allegedly in the New York Times in 1996 on the Robbins conjecture (noting that computers had started to contribute to math research in some sense), and a longer piece in Math Horizons by her the following year ("Computer Math Proof Shows Reasoning Power"). I didn't immediately find the NYT article, so I don't know if it might be a hallucination.
John Horgan in Scientific American in 1993 (https://www.scientificamerican.com/article/the-death-of-proo...). There's also a retrospective on the topic by the same author in Scientific American in 2022 (https://www.scientificamerican.com/article/should-machines-r...).
Natalie Wolchover in Quanta (but reprinted in Wired) in 2013 (https://wired.com/2013/03/computers-and-math).
I was involved in some distributed computing stuff in the late 1990s and early 2000s and I don't really remember people in that community talking about proofs but there may have been a "if we had a mechanical proof-checker, could we do distributed searches for valid proofs that it would accept?" conversation somewhere at some point. There were definitely volunteer distributed computing projects working on pure math; I remember the Optimal Golomb Ruler search (https://en.wikipedia.org/wiki/Golomb_ruler). So, that could possibly have shaded over into "can we find proofs this way too?". At the time it probably would have been based on brute force searches through proof space rather than clever optimization, though.
The idea that you can lexicographically list all proofs in some formalism and then mechanically determine if any is valid is quite clear from Gödel's construction of the function Bew in "On Formally Undecidable Propositions", but he points out that you don't know where to stop because you don't know how long a valid proof would potentially have to be (so "is this a valid proof of this claim?" can be decided mechanically in a limited time, while "is there any valid proof of this claim?" can't be! maybe the shortest valid proof is 49 steps long but you eventually stopped checking after looking at all 7-step proofs, or something).
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Ted Kaczynski,the Unabomber, made the same prediction 30 years ago.
Fucking even called LEAN the “hottest shit under the sun”—which it is. You, legend you!
And do any of them actually matter? Will the fact that Noodleheinz's Third Postulate now has a proof affect anyone?
It's really impossible to predict which discoveries will "matter", have a direct impact on other fields, or an impact in making other mathematics or physics discoveries.
Only after a world's worth of experts look at these results and then mull over if and how their own fields are impacted by this new info will we be able to answer this question.
I'm reminded of a great TV Show, James Burke's Connections. Where discoveries in one area of science would revolutionize or fundamentally change a completely different area. https://www.youtube.com/watch?v=XetplHcM7aQ&list=PL5HjoPOFFC...
It can take decades to really know the full significance. You know, the whole "We stand on the shoulders of Giants", well the Giants just grew a few inches all at once.