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Big advance on simple-sounding math problem was a century in the making

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129 points isaacfrond | 1 comments | 15 Oct 24 13:10 UTC | HN request time: 0.212s | source
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nomilk ◴[15 Oct 24 16:30 UTC] No.41850289[source]▶
How do mathematicians come to focus on seemingly arbitrary quesrions:

> another asks whether there are infinitely many pairs of primes that differ by only 2, such as 11 and 13

Is it that many questions were successfully dis/proved and so were left with some that seem arbitrary? Or is there something special about the particular questions mathematicians focus on that a layperson has no hope of appreciating?

My best guess is questions like the one above may not have any immediate utility, but could at any time (for hundreds of years) become vital to solving some important problem that generates huge value through its applications.

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1. Xcelerate ◴[16 Oct 24 14:36 UTC] No.41859530[source]▶
The problems mathematicians tend to be interested in are those that have a short description in a formal language yet the proof is likely not very compressible via the language. I.e., you can easily get the index of any statement in the formal language of PA but finding the index of either the statement or its negation in an enumeration of the theorems of PA is significantly more difficult. It’s equivalent to the halting problem in a general sense because the statement and its negation may be independent of PA in which case the enumeration program never halts.

What does this mean? These problems are like the Busy Beaver analogue for formal mathematical systems, which implies that if you solve one, in theory you solve entire classes of lower complexity problems simultaneously (similar to how BB(5) solves the halting problem for all programs below a certain complexity).