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Dispatches from the possibly last days of human relevance

As most readers have presumably heard by now, Paul Erdös’s Unit Distance Problem from 1946—one of the central open problems from the field of discrete geometry—has been solved by GPT5.5Pro. Erdös had conjectured that, given n points in the plane, at most n1+o(1) pairs of them could be unit distance apart. Using high-powered results from algebraic number theory, GPT refuted this, constructing a set with n1+ε unit-distance pairs, for ε ~ 10-38. Shortly afterward, Will Sawin, a human (!), improved GPT’s construction to get ~n1.014 pairs. Meanwhile, the best known upper bound remains n4/3, improving Erdös’s n3/2.

The entire process seems have been one-shot: my former student Lijie Chen simply gave GPT the problem, then GPT thought for a while and output a several-page argument that, on analysis by human experts, turned out to be correct. Of course there’s selection bias here; we’re not hearing as much about the hundreds of other problems GPT was given that it didn’t solve (isn’t that the case with humans too?). Clearly, too, GPT was helped by the facts that human mathematicians had wasted most of their time trying to prove Erdös right rather than looking for a counterexample, and that, even if they did look for a counterexample, they’d need to be experts in algebraic number theory to find this one, which hardly any discrete geometers are. So, maybe that suggests that AI, right now, is "merely" picking various medium-hanging fruits that human mathematicians missed for contingent reasons? With emphasis on the "right now."

In a companion paper, OpenAI helpfully included commentary from Timothy Gowers, Noga Alon, Will Sawin, Daniel Litt, and many other experts, reflecting on the breakthrough, the path that GPT took to get to it (which can actually be seen by examining its chain-of-thought), and what this might mean for the future of mathematical research.

I heard the news maybe an hour after it broke, when some UT grad students came to my office to tell me. For what it’s worth: these students were morose, musing about how everything might soon be over for young scientists and mathematicians like themselves. I don’t know whether they’re right, but I feel like I should tell the truth about what their reaction was.

Then, a few days later, a team at DeepMind, including my UT Austin colleague Swarat Chaudhuri, announced that they were able to use a system called AlphaProof Nexus to settle nine more (!) Erdös problems, many of them in additive combinatorics, along with miscellaneous other open math problems. Notably, in this case the AI also fully formalized its proofs in Lean.

And then, just today, Jelani Nelson alerted me to a new CS theory paper, which solves a longstanding open problem about electrical flows on graphs using a proof from GPT5.5Pro.

It seems to me that we’re now over the top of this particular rollercoaster, and it will keep accelerating until we reach the bottom, wherever that might be. I don’t know whether to hope or dread that solutions to P versus NP and all our other great problems will be included in the ride—that our role, as human mathematicians, will be reduced to (at most) deciding which questions we find interesting and then understanding AI models’ answers to those questions.

But maybe that won’t happen. Maybe the new AI mathematicians will soon hit a wall, because they lack the uncomputable quantum gravity microtubules of Penrose and Hameroff, or some other magic human ingredient. The fantastical thing is that, one way or the other, we’re going to find out empirically before very long.

Readers may have also seen the news that multiple prizewinning entries in a short fiction contest called the Commonwealth Prize, give overwhelming indications of having been written by AIs. As Kelsey Piper puts it:

There are, let’s say, also some noticeable similarities in the prose style between the winning stories that were flagged for AI use. AI chatbots love metaphors and similes, and they often spit out ones that sound vaguely pleasing but are logically incoherent or ascribe properties to things that don’t make sense.

“The Serpent in the Grove” gave us, “The girl smiled like sunrise over a sink.” “The Bastion’s Shadow” says, “She carried it now in her bag, heavy as a charm.” “Mehendi Nights” describes something as “swaying against plaster like a warning bell.”

The Commonwealth Foundation, whose judges chose these stories, hasn’t exactly covered itself in glory—saying, on the one hand, that it strictly forbids AI use but on the other,...