before gödel’s theorems can be formally stated, you have to make a lot of assumptions about axioms, and you have to pick which kinds of logical rules are “valid”, etc. and that all feels way more dicey to me than the actual content of gödels theorems.
i definitely agree that gödels theorems can help to undercut the idea that math is this all knowing, objective thing and there’s one right way to do everything. but to me personally, i feel like the stuff that’s very close to the foundations is super sketchy. there are no theorems at that level, it’s just “we’re going to say these things are true because we think they are probably true”.
Indeed, and the reason why I think the incompleteness theorems are the nail in the coffin, is that otherwise you could at least prove that certain sets of axioms worked together, then you’d have some basis for the math you use, even if it was self-referential
But it turns out that is impossible, and what we choose as our foundation for the math we use is pretty much arbitrary. Although of course we have reasons to use what we use
how does that follow from Gödel’s incompleteness theorems?
Godel’s second theory of incompleteness states that a formal system cannot prove its own consistency
I think that’s as close as you can get to “math is not objective”
before gödel’s theorems can be formally stated, you have to make a lot of assumptions about axioms, and you have to pick which kinds of logical rules are “valid”, etc. and that all feels way more dicey to me than the actual content of gödels theorems.
i definitely agree that gödels theorems can help to undercut the idea that math is this all knowing, objective thing and there’s one right way to do everything. but to me personally, i feel like the stuff that’s very close to the foundations is super sketchy. there are no theorems at that level, it’s just “we’re going to say these things are true because we think they are probably true”.
Indeed, and the reason why I think the incompleteness theorems are the nail in the coffin, is that otherwise you could at least prove that certain sets of axioms worked together, then you’d have some basis for the math you use, even if it was self-referential
But it turns out that is impossible, and what we choose as our foundation for the math we use is pretty much arbitrary. Although of course we have reasons to use what we use