Why AI Labs Prefer Erdős to Riemann скачать в хорошем качестве

Why AI Labs Prefer Erdős to Riemann

Dear QF Community, This is a special post that serves as a follow-up to the many emails we receive from community members asking for our perspective on recent claims about AI progress in mathematics. These messages often arrive after viral headlines circulate on social media, usually amplified by influencers who, in many cases, do not have a background in mathematics. Important note: This post should not be interpreted as a harsh rebuke of the many well-intentioned mathematicians, AI researchers, and engineers who are genuinely working to make AI useful for mathematical research. Their efforts are valuable and, in many cases, technically impressive. At the same time, it is important to remain open and honest about the level of hype that often surrounds these developments and the risk of overstating their significance. A podcast narration version of this post will also be uploaded to Spotify (here (https://open.spotify.com/show/0vVuYQA...) ) and YouTube (here (   • All-in-Maths  ) ). Why AI Labs Keep Returning to Erdős-Style Problems It is easy to assume that if an AI lab is serious about mathematics, it should be aiming straight for the most famous peaks in the field. Why spend time on extremal combinatorics, olympiad-style constructions, or finite search problems when mathematics still contains giants such as the Riemann Hypothesis (https://en.wikipedia.org/wiki/Riemann...) , the Hodge Conjecture (https://en.wikipedia.org/wiki/Hodge_c...) , or the unresolved questions associated with the famous Hilbert’s problems (https://en.wikipedia.org/wiki/Hilbert...) ? The answer is not that these labs regard Erdős-style problems as more important than the great central questions of modern mathematics. The real issue is that current AI systems have a very uneven mathematical profile, and their strongest abilities happen to align far more naturally with some parts of mathematics than with others. At present, AI systems perform best in settings where success can be checked quickly, where the objects involved are discrete rather than highly continuous, and where one can tell almost immediately whether a proposed move has improved the situation. This is one reason combinatorics, graph theory, colouring problems, and extremal constructions appear so often. Many problems in that world can be framed as search tasks with a clear objective. One can test a candidate construction, score it, discard it, refine it, and repeat. From an engineering point of view, that is a far more manageable setting than one in which progress is difficult even to measure. An open problem such as the Riemann Hypothesis belongs to an entirely different mathematical setting. Its difficulty is not simply a matter of scale. Any serious attempt to engage with it draws on advanced areas of modern mathematics, including complex analysis, number theory, spectral theory, and a long tradition of conceptual development. It is not the kind of problem for which a system can generate thousands of candidates, receive a clear signal, and improve steadily through repeated trial. The issue, then, is not mathematical prestige, but how well the problem matches the limits of current AI systems. That distinction matters more than many people realise. It is often lost in headlines, which can leave readers with the impression that AI is on the verge of replacing mathematicians or transforming mathematical research in some sudden and universal way. The reality is much narrower! Progress so far has been strongest in areas where problems can be formalised clearly, checked quickly, and explored through repeated search. That is impressive in its own right, but it is very different from saying that machines are close to taking over the deeper conceptual work on which much of mathematics still depends. As such, a lab building AI for mathematics is not currently asking which problem carries the greatest historical status. It is asking where a model has a realistic chance of making contact with the underlying structure of the task. If a model proposes a graph construction, a colouring, or a finite configuration, the proposal can often be checked automatically. If it proposes a proof in a formal environment, a proof assistant can determine whether the reasoning is valid. That matters enormously, because it means the system is not being rewarded for producing something persuasive or elegant in tone. It is being judged against an external standard that does not care how convincing the output sounds. This is one of the main reasons Erdős-style domains have become such attractive testing grounds. Representation also plays an important role. Finite combinatorial objects are relatively easy to encode computationally. Graphs, sets, hyper...