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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How is area not conserved with rototranslation in Euclidean geometry?
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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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a simpler version of this to think about is length. why do we live in a universe where long, thin objects have a notion of "length" that doesn't change as we move them around? (note that this wouldn't be true if we moved at relativistic speeds so this isn't tautological)
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Wait I thought we were talking about Euclidean geometry not the real world.
(Btw I just discovered that De Morgan proved the parallel postulate. Took only 2200 years)
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