dailynewsnblog
A general purpose AI model from OpenAI has produced a (dis)proof of an important conjecture. Tim Gowers writes:
AI has now solved a major open problem — one of the best known Erdos problems called the unit distance problem, one of Erdos’s favourite questions and one that many mathematicians had tried.
A number of prominent mathematicians comment. I enjoyed Thomas Bloom’s comments:
This was one of Erdős’ favourite problems – he first asked it in 1946 [14] and returned to it many times. (The site www.erdosproblems.com, on which it is Problem #90, currently lists 14 separate references, and there are no doubt more.) The influential collection of ‘Research Problems in Discrete Geometry’ by Brass, Moser, and Pach [8] describes it as ‘possibly the best known (and simplest to explain) problem in combinatorial geometry’. For an AI to produce a solution to a problem of this calibre is both surprising and impressive.
…On examining the construction, it becomes more clear how people had missed this before – it requires the confluence of several different unlikely events: that a good mathematician is
(1) spending significant time in thinking about the unit distance conjecture in the first place;
(2) seriously trying to disprove it, despite the oft-repeated belief of Erdős that it is true;
(3) believes that there is mileage in generalising the original construction to other number fields,
and so is willing to expend significant time in exploring such constructions; and
(4) sufficiently familiar with the relevant parts of class field theory to recognise that the appropriately phrased question about infinite towers of number fields with appropriate parameters can be solved using existing theory.The AI met all of these criteria, and its success here echoes previous achievements: it often produces the most surprising results by persevering down paths that a human may have dismissed as not worth their time to explore, combining superhuman levels of patience with familiarity with a vast array of technical machinery.
…perhaps some in the area will be a little disappointed with how little this tells us: it does not introduce any powerful new geometric tools, or hitherto unsuspected structural results, that a proof of the unit distance conjecture would likely have called for. Still, while perhaps not the proof of a conjecture that we had hoped for, no doubt this construction and the ideas involved will have a major impact in discrete geometry.
One aspect of this proof should not be overlooked: while the original proof produced by AI was completely valid, it was significantly improved by the human researchers at OpenAI and the many other mathematicians involved in the present paper. The human still plays a vital role in discussing, digesting, and improving this proof, and exploring its consequences.
The frontiers of knowledge are very spiky, and no doubt the coming months and years will see similar successes in many other areas of mathematics, where long-standing open problems are resolved by an AI revealing unexpected connections and pushing the existing technical machinery to its limit. AI is helping us to more fully explore the cathedral of mathematics we have build over the centuries; what other unseen wonders are waiting in the wings?
One way of putting this is that the mathematicians are now acknowledging that the AI’s are “one of us”. Gooble Gobble! Read the AIs chain of thought to understand why. I asked Claude how many people the world could understand the proof:
A rough tiered estimate, treating “understand” as “could read the 42-page note and follow the argument without needing to learn new machinery from scratch”:
Tier 1 — could referee it cold (real working knowledge of class field towers + the Ellenberg–Venkatesh circle): roughly 150–400 people worldwide. This is essentially the active algebraic number theory community working near arithmetic statistics, plus a handful of arithmetic-geometry-adjacent combinatorialists. The author list itself is a decent proxy for the upper crust of this group.
Tier 2 — could understand it with a week or two of focused effort and some Wikipedia/textbook chasing (strong number theorists or combinatorialists outside the immediate subfield, plus sharp grad students past quals at top programs): roughly 2,000–5,000. Think most tenure-track number theorists, the top tier of extremal combinatorics, and arithmetic geometers generally.
Tier 3 — could grasp the structure of the argument from a Quanta-style exposition without verifying the steps: 50,000–200,000+, i.e., most working mathematicians and a chunk of physicists/CS theorists. This is not what you asked, but it’s where most of the public “understanding” will sit.
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