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Interesting generally means having some sort of sufficiently broad applicability to be useful and sufficiently narrow applicability to have enough explanatory p
by TimPC 4y ago
Interesting generally means having some sort of sufficiently broad applicability to be useful and sufficiently narrow applicability to have enough explanatory power. If you are too narrow you don’t get applied in many situations and if you are too broad you’re generally too watered down to prove anything significant about the situations in which you apply. Some exceptions which makes everything harder.
1+1=2 is a theorem but a very uninteresting one because it applies exactly once.
Every number n has a double successor S(S(n)) is also uninteresting because while it applies more broadly it’s too watered down.
There are no numbers n with n greater than two such that x^n + y^n = z^n is extremely interesting.
- pharmakom 4y agoYou are defining useful relative to human understanding. My point is this may not matter in an AI future.
- TimPC 4y agoThe underlying concept I’m getting at is that there is more than just human understanding here. Our understanding of mathematics is highly subjective but it seems to me theorems do have some sort of true value with some of them being more useful for doing mathematics and others being things you prove once and never use again. A robot could prove infinite theorems about every number having a double successor, triple successor and so on. If it does so it’s not just human subjectivity saying it hasn’t done any meaningful mathematics.