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> I'm curious what probability you think applies to propositions like "2 + 2 = 4" and "All triangles have 3 sides" and "All triangles have less than 11 sides".
by paulsutter 10y ago
> I'm curious what probability you think applies to propositions like "2 + 2 = 4" and "All triangles have 3 sides" and "All triangles have less than 11 sides".
Those are great examples, thanks. All true, and there's nothing interesting about them.
- kazga 10y ago"All triangles have 3 sides" might be an uninteresting triviality, but "The sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse in a right-angled triangle" is neither trivial nor uninteresting and yet it has a probability of 1.
- jholman 10y agoYes, exactly. And "this code has a mathematical error in it" is often interesting, often non-trivial, and often has probability 1 (and often probability 0). And these things are exactly the sort of thing that "differ from [trivialities about triangles] only in degree of complexity". Note that "all triangles have 3 sides" is probably an axiom, but "all triangles have less than 11 sides" is a trivial theorem.
- paulsutter 10y agoI love this example the most. It's the exception that proves the rule. If you need to dig this hard to find something interesting with a probability of 1, that's pretty good evidence that the vast majority of interesting statements are not of the true/false variety. Although I don't find it interesting, I am open-minded enough to ... embrace.. the .. uh.. diversity of the world, that allows some people, to find that interesting.
- kazga 10y agoPythagorean Theorem is "digging hard" and "not interesting"? Mind explaining? The language that contains all Turing Machines that halt on all inputs is not decidable. Or e^(iy) = sin(y) + i * cos(y) Are those uninteresting trivialities to you?