∑ Fascinating indeed! This is how MOST research works, but I didn’t appreciate quite how true it is in math (or at least current number theory). Lots of tasty morsels in this thread; h/t
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Sounds more or less right, also makes math sound much healthier than other subjects, particularly wrt the error rate. Note the implication that the results are mostly correct and almost always meaningful!
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Just about to launch a long rant about this! :)
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I was going to do a second whole rant about how the natural numbers are bad actually, contrary to “God made,” because you can multiply them, and you get division that way, and if you have division you get primes, and if you get primes you get number theory, and that is AWFUL,
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whereas the reals are wonderful, people are scared because they encapsulate actual Lovecraftian infinities with tentacles, but you get continuity with them, so you can do calculus, and calculus is great, calculus is actually the foundation of THE WHOLE MODERN WORLD,
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Excellent explanation of real numbers here:
twitter.com/QiaochuYuan/st
Take-home point: "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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glad you asked! so there's a certain kind of person who will respond to this by describing one of the standard constructions of the real numbers. this is completely beside the point, and you can tell because there's more than one standard construction twitter.com/Jakob__B/statu…
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Yes, that’s nice… although I guess I’m agnostic about that conclusion.
I had a major spiritual crisis when I was 19 and no one would tell me what a real number was. I read about the Cantor & Dedekind constructions and they satisfied me at the time.
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No, I was 18. Anyway, since then I’ve rethought the question several times and mostly my take is “no one else understands this either, but it doesn’t seem to be a problem, so don’t worry about it”
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Which is exactly what my math profs would have said when I was 18, except they were scared to admit the “yeah I don’t understand it either” part.
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it is kind of funny how it doesn't seem to a problem eh
Well when I did real analysis in 1979 the prof was (in retrospect, I didn’t understand this at the time) still nervous about tentacles emerging from the Dungeon Dimension, but I guess it’s been well over a century since the crisis and they haven’t escaped yet so we can all relax
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Is it a problem for computers trying to prove things?
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maybe! my (very limited and poorly informed) understanding is that at least some of the cool new proof assistants get around this issue using type-theoretic foundations (which aren't without their own problems)
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