I wonder how many imperfect, inelegant, sprawling, large _systems_ I will have to learn well in my lifetime.
(Parts of human biology would be one if I were a medical student, for instance, but I'm not.)
Will that include any programming languages that fit that mould (say C++)?
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Honestly, the only thing that I've seen to at least approach perfection, elegance, and brevity is mathematics. As a rule of thumb, human-made things are generally bad.
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Even mathematics (as a human construct) is full of legacy cruft and hard to change conceptual edifices. That said, it's hard to prevent the aching beauty of the foundations the universe from shining through it's flawed construction. Just need to take the time to look.
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But perhaps as we delve deeper we'll discover we're actually living in the abandoned legacy simulation project, with a terrible train-wreck of logical foundations that we can never reconcile to arrive at any coherent model of anything… 😱🤖
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The amount of "hackiness" is different per-field; does analysis have more than algebra?
But mathematical hackery, when it permeates disciplines, tends to have a quality closer to people taking care to write cache-aware code than "that function throws errors on string inputs"
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You can black-box a lot of things in a way that leaves the exterior ripe for being fitted into a hole in a large, sensical structure. Maths feels like it leaks less implementation details than most things.
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Naturals in set theory LOL. Look ma, no abstraction boundary!
I don’t know if that changes the conclusion, but it’s worth exploiting the availability bias: pointing to that example and asking again if math feels clean.
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See the _tends to_ above, and the hedging: of course there is going to be some selection bias due to the kinds of maths I know more about. If I were a differential equations person, I think I'd have other things to say. :)
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I’m not pushing a thesis, just sincerely curious. And mathematicians do have abstraction boundaries in practice, even when they’re not part of their formal language.
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It just seems very hard to refactor mathematics. So much relies on stuff in people's head - seems very hard to change things lower down the stack – do people ever bother? (Univalent foundations?) But then, we are fighting similar battles in software… and without any proofs… 🤪
Wait what do you think logicians and type theorists and some category theorists have spent their time doing if not refactoring foundations? HoTT is just one of the most visible attempts (not sure how much by merit or by marketing, but it's closer to critical mass).
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And it is possible: compare Weierstrass et al.'s refactoring of calculus or Bourbaki's influence (which pedagogically was a net negative, mind you). It just seems less necessary nowadays; when most mathematicians point out set theory serves its current purpose, they're right.
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