still don't think i understand the pythagorean theorem, all things considered
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it would take awhile to elaborate more fully on what i mean but here's a chunk of it: all the visual / geometric proofs take for granted that whatever "area" means it's something that's preserved under translations and rotations. it's unclear exactly what is needed for this
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there's a book called "the mathematical mechanic" that provides "physical proofs" of the pythagorean theorem. they all seem to be saying the same thing about translation- and rotation-invariant physics but i don't quite get what that thing is
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here's the first proof, that derives the pythagorean theorem from the impossibility of perpetual motion (and, implicitly, that the concept of torque is meaningful and behaves nicely wrt rotations). i cannot claim to really understand what is going on here
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that does seem to be the obvious guess ๐ค i haven't quite integrated my understanding of noether's theorem into this situation yet tho
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i mean, yes, i know how to define areas using *shudder* lebesgue measure and prove that they're invariant, but it's conceptually unsatisfying to do things that way. why did we think that was going to work? because we have some *pretheoretic* understanding of area. what's *that*?
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