Post by @aceroe@mathstodon.xyz

2026-09-09 02:00 UTC

@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?

Replies (1)

@aceroe@mathstodon.xyz 2026-09-09 02:01

@tao@mathstodon.xyz (3/4) Navier–Stokes is a good example. Even just looking at the publicly released construction, one already encounters a large collection of structures worth studying further: concentrating cores, swirl amplification, annular stress, oscillatory realization, moment matching, residual hierarchies, and more. Those are not “used-up” mathematical resources. They are new objects of investigation. If anything, rapidly obtaining a solution may expose mathematical structure sooner than the traditional process would have done. The fact that the entrance was opened quickly says nothing by itself about how large the world behind it is. If an entrance has only just been opened and the first reaction is, “What a shame, another good problem has been consumed,” then perhaps what has really been touched is not a vulnerability of mathematics itself, but a value system that has long tied mathematical worth to difficulty, scarcity, and the privilege of being the one who solves the problem. As obtaining answers becomes cheaper, mathematical value can move upward: from solving a problem to understanding a structure; from proving one theorem to explaining why an entire class of phenomena occurs; from competing over an entrance to exploring the world behind it.

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