so the real numbers "fill in gaps" that are "missing" in the rational numbers, and once you've filled those gaps the real numbers can implement euclidean geometry
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so, okay, that's pretty good, why does anyone have a problem with the real numbers then (and they do)? the problem is that we pay a very bizarre price for filling in the gaps: almost every specific real number is literally indescribable, because there are too many of them!
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the problem is that no matter how you choose to describe real numbers, there are only countably many possible descriptions (e.g. only countably many programs that spit out strings of digits), but uncountably many real numbers! it's wack
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this makes some people very uncomfortable (and i think that discomfort is justified). what is "real" about the vast majority of the real number line being inaccessible to any form of description whatsoever???
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(and i mean *vast* majority - in a precise technical sense the probability of a randomly chosen real number being describable is literally zero)
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for the purposes of simpler questions in euclidean geometry you can get away with working with a much smaller set of numbers, the algebraic reals, which are all describable
but you actually need all of the real numbers to do calculus. and we need calculus for a million things
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so the real numbers, as usually constructed, are (this is very much in-my-opinion) this philosophically unsatisfying technical kludge we put up with because it lets us put geometry and calculus and a million other things on a rigorous foundation
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i have hopes that someday someone will find a more philosophically satisfying replacement for the real numbers but it's very unclear to me what that would look like
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I'm curious if you have any similar reaction to the idea that complex numbers seem... very... used in real life? They embed all this stuff about reals, but do you think they're more/less natural/disconcerting?
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[I'm just a big complex number stan who never minded uncountability but I do love me some algebraic closure.]
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i think the most "practical" argument in favor of the complex numbers is actually that you appear to need to do them to do quantum mechanics. that's very compelling but also i have no idea why quantum mechanics works like that although i've seen some stuff purporting to explain