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@tao@mathstodon.xyz

2026-09-08 20:32 UTC

Working out whether a question is actually worth highlighting is a lengthy, deliberate, and subjective process, often informed by historical experience on what good mathematics was generated (or not generated) while working on earlier problems of this type. In particular, being aware of the "difficulty landscape" in a field - what questions are very easy to answer with known methods, which ones can be solved but only with some effort, and which ones are impossible - is of crucial importance in making such determinations. And some problems only become interesting after an external connection is made. For instance, there could hypothetically be an unexpected connection between the Riemann zeta function and the 10^10^nth digits of pi for various n... at which point the previous question of determining the 10^10^10th digit suddenly becomes relevant again. (I should emphasize though that this specific scenario is incredibly unlikely to actually be the case; I use it only as a hypothetical iilustration.) Every new advance in mathematics, whether it comes from technique, technology, or infrastructure (such as access to libraries of past literature) reduces the difficulty of solving problems. This is generally a good thing; but it comes at the cost of flattening out the difficulty landscape of a field, to the point where one can no longer discern its geometry to the extent that promising questions can be extracted within the range of applicability of the tool. Often this effect is counteracted by the ability of such a tool to enlarge the radius of the sphere of results one can plausibly reach, creating new boundaries to fruitfully explore. (2/4)

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