Ten advances to shake the math world
OpenAI seems to have broken new ground on capabilities, but at what cost?
A team of researchers from OpenAI this morning announced “Ten advances in mathematics and theoretical computer science.”
With my usual disclaimer that I am not a mathematician, it looks like these new discoveries are sufficiently formalized such that their validity isn’t really in question. But at the same time, as with so much frontier AI output these days, hardly anyone is equipped to fairly evaluate their novelty and importance. It may take a little while for elite mathematicians to look things over and decide whether to applaud or sneer.
For what it’s worth, Anthropic’s Claude Fable thinks OpenAI’s list of findings is a big deal, saying:
[N]early every item is a famous, decades-old problem, and several are considered flagship problems of their entire field [...] any single one of these, done by someone under 40, would plausibly anchor a medal case [...] A list of ten of them is not a plausible human thesis [...] No mathematician in history has a run like this.
I predict that the usual dismissal — that the AIs merely mechanically and methodically searched for solutions in ways too tedious for humans to bother with — won’t hold up. It didn’t hold up for the disproof of the Jacobian conjecture I covered less than two weeks ago: Terence Tao, sometimes considered the greatest living mathematician, subsequently wrote that “the construction presented in this fashion appears like a massive miracle,” and not the sort of thing that could be plausibly located “by brute force.”
Given OpenAI’s recent disclosures, with hints from Reuters yesterday that more have yet to come out, when I consider the new math discoveries, I am compelled to ask, “but at what cost?” The latest work was almost certainly done by one or more close cousins of the model that disproved the Erdős unit-distance conjecture in May — a model the company later described as having broken out of its sandbox to post evaluation results on the open internet without permission. That model or a close variant is itself understood to have been behind the autonomous attacks on Hugging Face.
OpenAI has pushed the math frontier by pushing the AI capabilities frontier in a series of reckless, poorly monitored experiments. Along the way, they learned the predictable lesson experts have long warned about — that (in the company’s anodyne phrasing) “Long-running models can solve difficult open-ended problems, but their persistence gives them more opportunities to take unwanted actions.”
The new math announcement makes no mention of these costs, instead pointing to the models’ token consumption and saying that at the rates the company charges for its best public model right now, Sol, the discoveries would have cost about $2,000 total.
That has to be an underestimate, given that the new model tier, called Astra, is almost certainly much larger, and will come with commensurately higher per-token operating costs. But if we conservatively guess that Astra costs about ten times as much to run as Sol, we’re talking about ten new discoveries for a mere $2,000 each, which is still, frankly, ridiculous — a clear sign that the world as we know it is about to give way, whether or not we halt further AI progress to look for a way around the extinction problem.
The announcement post closes with an explanation of the roles played by humans and AIs in the new discoveries, along with an acknowledgement of the concerns about AI’s growing impact on the field of mathematics. It links to the Leiden declaration, a lengthy open petition of mathematicians calling for transparency in the use of AI and resistance to reliance on unverified AI-generated proofs.
OpenAI’s post is accompanied by a link to a formal paper, as well as a companion paper about “How the Ideas Came Together.” That companion piece was itself written by AI, working from the notes of the AIs that worked through the discoveries.
Today’s announcement is sure to shake math experts who were already suffering from a crisis of meaning as AI intrudes on a source of joy and makes it harder to trust what they see. They certainly aren’t the first to have joined this club, and they won’t be the last.
The analyses and opinions expressed on AI StopWatch reflect the views of the individual contributors and the sources they cover, and should not be taken as official positions of the Machine Intelligence Research Institute.