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It's not a contradiction. In the constructive sense, we don't know the real numbers except the ones that can be talked about using concepts that are more intrin
by skulk 5y ago
It's not a contradiction. In the constructive sense, we don't know the real numbers except the ones that can be talked about using concepts that are more intrinsic to our reality such as rational numbers or circles.
You can express the idea that a sequence converges to a limit in constructive mathematics (if not, I'd like to know why) and you can prove that that limit is not rational, nor is it algebraic. That is to say, real analysis works in a constructive setting. At least that's my math-enthusiast understanding of it.
Traditional mathematics even asserts the existence of numbers whose concrete properties we can't even possibly know, such as Chaitin's constant: https://en.wikipedia.org/wiki/Chaitin%27s_constant https://en.wikipedia.org/wiki/Chaitin%27s_constant
My understanding is that these are the kinds of numbers you lose out on when you switch to being a constructivist. I guess you could say that means you can't fit "all of existing mathematics" in it, but to me that feels a bit vacuous.
- AnimalMuppet 5y agoWell, I guess I think the same about the constructivist position. > In the constructive sense, we don't know the real numbers except the ones that can be talked about using concepts that are more intrinsic to our reality such as rational numbers or circles. I accept that as being true. But then, to say that you can fit "all of existing mathematics" feels like you're saying that you can fit everything that fits, and everything that doesn't fit is defined to not be "real" in some sense. That is, it feels circular - like you're so far down the rabbit hole that you can't remember that there's a world above ground that also exists and has validity.
- skulk 5y agoI guess strictly, it is a contradiction because indeed "all of existing mathematics" cannot be derived purely constructively. The way I choose to see it, however, is constructivism is the activity of seeing how far you can go with the least possible axiomatic foundation. If, at some point, you need something like the Law of Excluded Middle, you just assert the specific instance that you need as an axiom (and not as a structural rule). In this sense, then all of mathematics is valid from a constructivist viewpoint, what changes is that now you care about where you assert certain axioms.
- R0b0t1 5y agoYou don't lose out on the fields you mentioned when you switch to constructivism. Your assertions become more precise. If anything is lost the mainstream thinking is it was not consistent to begin with.