A general-purpose AI model from OpenAI has produced (dis)evidence for an important hypothesis. Tim Gowers writes:
AI has now solved a big open problem – one of Erdos’ most famous problems called the unit distance problem, one of Erdos’ favorite questions and one that many mathematicians had tried to solve.
A number of prominent mathematicians comment. I enjoyed Thomas Bloom’s words:
This is one of Erdős favorite problems – he first asked it in 1946 [14] and returned to it many times. (The site www.erdosproblems.com, which contains Problem #90, currently lists 14 separate references, and there are no doubt more.) The influential ‘Research Problems in Discrete Geometry’ collection by Brass, Moser, and Pach [8] describes it as ‘probably the best-known (and easiest to explain) problem in combinatorial geometry’. For AI to generate a solution to a problem of this nature is surprising and surprising.
…When examining the construction, it becomes clearer how people missed this before – it requires the confluence of several different unlikely events: that a good mathematician
(1) spending significant time in thinking about the unit of distance of the original assumption;
(2) by attempting to refute it, despite Erdős’s repeated belief that it is true;
(3) believes that there is a distance of kilometers in combining the initial construction of some number fields,
and is willing to spend significant time in inspecting such properties; again
(4) familiar enough with the relevant parts of class field theory to see that a well-posed question about infinite towers of number fields with suitable parameters can be solved using the existing theory.AI met all of these criteria, and its success here echoes previous achievements: it often produces the most amazing results by persisting in ways that a human might dismiss as unworthy of its testing time, combining superhuman levels of patience and familiarity with a host of technological devices.
…perhaps others in the field will be a little disappointed by how little this tells us: it does not scare the powerful new tools of geometry, or the hitherto unsuspected structural results, which would require proof of the unit distance hypothesis. Nevertheless, although it is perhaps not the proof of the hypothesis that we expected, undoubtedly this construction and the ideas involved will have a great impact on abstract geometry.
One aspect of this proof should not be overlooked: although the first AI-generated proofs were completely valid, they were greatly improved by the human researchers at OpenAI and many other mathematicians involved in the current paper. Man still has an important role in discussing, digesting, and developing this evidence, and evaluating its results.
The frontiers of knowledge are very spiky, and no doubt the coming months and years will see the same success in many other areas of mathematics, where long-standing open problems are solved by AI that reveals unexpected connections and pushes existing technological machines to their limits. AI helps us fully explore the cathedral of mathematics we have built over the centuries; What other unseen wonders are waiting in the wings?
Another way to put this is that mathematicians now agree that AIs are “one of us”. Gooble Gobble! Read a series of AI thoughts to understand why. I asked Claude how many people in the world could understand this testimony:
A hard graded scale, which treats “comprehension” as “can read a 42-page note and follow an argument without needing to learn new material from scratch”:
Tier 1 – can umpire cold (actual working knowledge of class field towers + Ellenberg–Venkatesh circle): probably 150-400 people worldwide. This is basically an algebraic number theory community that works around arithmetic equations, and a number of arithmetic-geometry-adjacent integrators. The author list itself is a decent proxy for the upper crust of this group.
Phase 2 – we can figure it out with a week or two of concentrated effort and Wikipedia/textbook cramming (powerful coders or integrators outside of the subfield, and sharp graduate students who have gone through graduate programs): probably 2,000–5,000. Think of many teure-track number experts, the upper class of extremal combinatorics, and mathematical geometers in general.
Phase 3 – we can capture the structure of the argument from the Quanta-style objection without verifying steps: 50,000–200,000+i.e., lots of working mathematicians and lots of physicists/CS. This is not what you asked, but this is where most of the “understanding” of society will reside.
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