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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Wait, literally zero?
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You're NEVER going to pick exactly 4?
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this is why i needed "precise technical sense" - when you start doing probability with uncountable sets "probability zero" is not the same thing as "impossible" 😅 for some people this is a good enough reason to reject doing probability with uncountable sets
The width of "exactly 4" on the number line is zero, so it has zero probability-mass/measure, but it's OK because intervals of the form (4−ε, 4+ε) will have some probability, no matter how small ε is
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