the reason is that thinking of circles, and geometric shapes generally, in terms of sets of points described by real number coordinates mechanically produces correct answers to classical geometric questions, about the intersections between circles, or whatever else
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thinking about points in the plane in terms of coordinates lets us reduce geometric questions to algebra. this is, broadly speaking, one of the most successful mathematical strategies ever, and mathematicians have taken it very far
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this would not work if, for example, we tried to use the rational numbers instead of the real numbers. circles with only rational coordinates would be "missing" a lot of points (with irrational coordinates) and that would produce the wrong answers to various questions
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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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in the meantime this is a good opportunity for me to highlight another thing i wish people talked about more: ime it's rarely useful (at first pass) to ask "what is X?" in mathematics and usually much more useful to ask "what does X do?" or "what is X for?"
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actually another meta-level pet peeve of mine: i think a lot of people trying to learn undergrad-to-grad-level math don't really internalize that all this stuff is tools that specific people invented in order to solve specific problems, not like eternal timeless stuff
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the whole modern axiomatic method, not to mention specific axiomatizations like metric spaces or topological spaces or whatever, all that stuff is contingent. in other timelines it could've gone another way. aliens might not have it. it can be evaluated and discarded if desired
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and imo that's up to and including the modern axiomatization of set theory and of the real numbers! we brought it into this world and we can take it out!
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QC Explains Why You Should Become A Math Crank
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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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Why would you want to replace the real numbers? They're inelegant, but why anchor on the expectation that mathematics is inherently tidy?
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hopes are not the same as expectations! i simply want mathematics to be tidy where it can be and suspect this is a place it could be tidied up
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Euler fixed it, imo
Euler's identity feels to me almost more like describing how the next dimension after the one you're starting from exists, as opposed to just a coordinate system describing two dimensions you already have
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Somehow e encodes this smooth continuity when you turn towards the perpendicular