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*?
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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Maybe your new quest is to find a clearer, simpler, less shuddery way compared to slicing an area like prosciutto?
But seriously is integration the only tool we have at hand? Before calculus we had no other formalism to calculate areas?
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before calculus we had stuff like cavalieri's principle but it's unclear how rigorous it was compared to the kind of stuff we could do after analysis was put on a reasonably secure foundation