Post by @aceroe@mathstodon.xyz

2026-09-09 02:00 UTC

@tao@mathstodon.xyz (1/4) I don’t really agree with the idea that “good problems are being consumed by AI like a non-renewable resource.” Problems are not the whole of mathematics. An open problem is better understood as an entrance, not as mathematics itself. What truly matters is what structures we come to understand through that entrance, what mechanisms we discover, what new connections we uncover, and what new questions are generated from them. If a problem is valuable only while it remains unsolved, and its research value rapidly disappears once an answer is found, then perhaps what is actually scarce is not mathematics, but difficulty itself. AI is certainly flattening the difficulty landscape very quickly. But: Flattening the difficulty landscape is not the same as flattening the mathematical landscape. For a long time, we have often used questions like “where are people stuck?” and “how many years have mathematicians spent on this?” as proxies for importance. If AI makes that way of navigating mathematics increasingly unreliable, that only means we need better criteria for mathematical value. It does not mean mathematics itself is being depleted.

Replies (1)

@aceroe@mathstodon.xyz 2026-09-09 02:00

@tao@mathstodon.xyz (2/4) There is also an important tension here. In the fourth part of this argument, Tao explicitly says that what matters is not merely obtaining a raw solution, but extracting insight from the solution process: understanding why the problem was difficult, what new ideas appeared, and what the solution teaches us about nearby problems. I agree with that. But if an AI-generated solution already contains structures, mechanisms, and constructions worth investigating further, then why should we describe the event primarily as the consumption of a problem, rather than the opening of a new research entrance? Whether a solution contains insight should be decided by analyzing the mathematics in the solution itself—not by looking first at who or what produced it. A human proof is not insightful merely because a human spent years finding it, and an AI proof is not devoid of insight merely because it was found quickly. The relevant question is: What is actually inside the solution? When a problem is solved unusually quickly, the more important questions should begin afterward: What mathematics appeared in the solution that we had not seen before? Which parts reflect genuine structure, and which are merely proof technology? Can those structures be generalized? What do they tell us about neighboring problems?

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