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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undergrad and grad level texts do not show these specific problems
be the change you want to see in the world and show them!
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the problem is they usually require a lot of background to explain. but i got good stuff out of reading dieudonne's history of algebraic and differential topology, eg
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Really want to read this but holy moly that hardcover price batman
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it’s on libgen lol dw