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I've always been interested by the Platonist-vs-anti-Platonist debate. I generally disagree with finitist mathematicians, but I really do find their position in

I've always been interested by the Platonist-vs-anti-Platonist debate. I generally disagree with finitist mathematicians, but I really do find their position interesting, and as a computer scientist I'm not unsympathetic to their perspective. At the end of the day, though, I always go back to an old joke my undergrad philsophy professor told me: Analytic Philosopher A: "Hey, do you believe in baptism?" Analytic Philsopopher B: "Believe in it? Hell, I've been to one."

klyrs 6y ago
Me: I believe in any number that I can represent with my computer Professor: I am suddenly concerned that my PhD student does not believe in induction Me: I am suddenly concerned that my PhD supervisor does not know the difference between converse and contrapositive
ogogmad 6y ago
Constructivism is a less extreme position than finitism. It's hard to describe it, but it's got something to do with not believing that a mathematical object exists until it's been "constructed". I'm not sure if it can adequately be described without referring to formal logic. One motivation for considering this philosophy is that mathematics becomes more "computable" when you do, but there are other reasons as well. The statement that there are infinitely many integers is true in constructive mathematics: Given any integer, you can construct some integer larger than it. Likewise, the statement that the real numbers are uncountable is also true: Given any sequence of real number, it's possible to construct a real number that's not in the sequence (using Cantor's method). The claim that all real numbers are computable doesn't have any constructive content, so its truth depends on which version of constructivism you believe in. Personally, having looked into this, I prefer versions of constructivism in which the claim is false, because then you can claim things like every continuous function on R is locally uniformly continuous (which weirdly enough, implies that not all real numbers are computable, without asserting the stronger claim that an uncomputable one exists).

klyrs 6y ago

Me: I believe in any number that I can represent with my computer Professor: I am suddenly concerned that my PhD student does not believe in induction Me: I am suddenly concerned that my PhD supervisor does not know the difference between converse and contrapositive

ogogmad 6y ago

Constructivism is a less extreme position than finitism. It's hard to describe it, but it's got something to do with not believing that a mathematical object exists until it's been "constructed". I'm not sure if it can adequately be described without referring to formal logic. One motivation for considering this philosophy is that mathematics becomes more "computable" when you do, but there are other reasons as well. The statement that there are infinitely many integers is true in constructive mathematics: Given any integer, you can construct some integer larger than it. Likewise, the statement that the real numbers are uncountable is also true: Given any sequence of real number, it's possible to construct a real number that's not in the sequence (using Cantor's method). The claim that all real numbers are computable doesn't have any constructive content, so its truth depends on which version of constructivism you believe in. Personally, having looked into this, I prefer versions of constructivism in which the claim is false, because then you can claim things like every continuous function on R is locally uniformly continuous (which weirdly enough, implies that not all real numbers are computable, without asserting the stronger claim that an uncomputable one exists).