Post #4678525 · Elektrine

2026-09-08 20:32 UTC

I wrote recently about how the collection of good, fruitful open problems is now being mined in a non-renewable fashion, leading to the potential scenario of these problems becoming scarce. This may seem unintuitive at first, since the set of possible problems one could ask is infinite. Perhaps the following analogy can help: a country or region can suffer a critical shortage of drinking water while simultaneously being surrounded by a massive ocean. One can easily generate any number of open problems in mathematics at will, such as working out the 10^10^10th digit of pi. But the vast majority of such problems are not worth focusing attention on: they show no particular propensity to reveal any further insights or connections to other questions, or may either be too easy or too impossible relative to known techniques to learn anything from the exercise. (1/4)

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@tao@mathstodon.xyz 2026-09-08 20:32

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)

Open ##4678524