1). old school philosophers looked around at the natural world and said "fuck that shit, no way there's deep order to this. Geometry though, that's hot."
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2) up until Godel/Turing/Church stuff, being formal and mathematical meant "this can be perfectly understood, it's just a matter of proving it. What would it even mean for there to be inaccessible mathematical truths? If it's there, I could find it.
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3) no oracle for the halting problem. Infinite hierarchies of uncomputability. Arithmatical statements that can't be proven. Rice's theorem "almost all non trivial program behavior is undecidable"
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4) this ain't an attack on "1+1=2". That's pretty safe. Point is that math has the same upper realms of unknowability/unprovability that empirical investigation of the world has. It just took us a while to find it.
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5) fyi I haven't yet figured out how this claim matters, except that I expect it to prevent me from making some amount of confused claims about the nature of math/science/epistemology and building a castle around them
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6) so "math is lame" "you can't trust math" "you should be as sure of scientific theories as you should be of 1+1=2" are not my point.
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7) math has a complex hierachy of statements going from decidable, to semi-decidable, and upwards forever increasing in complexity. Empirical questions have a similar complexity hierarchy! "sun rises tomorrow" = verifiable. Wait and check.
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8) empirical and formal questions of the same complexity are more similar to each other than questions of the same "type" that are at different orders of complexity. Formal/Empirical is less useful than verifiable/unverifiable
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9) but demons! stray photons! illuuuuuusions!!!

nothing illusion args from applying to you thinking about math
"If I did the proof right, this is true"
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10) you become more sure of empirical facts as you measure multiple times, get independent verification, all that. Mathematical facts are highly abstract patterns in reality, abstract enough that you can "measure" them by thinking. Yes, the lab is in your head, no, it aint magic
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