[note: I dabble, at best. This is likely wrong in some ways, so I look forward to corrections. ]
I find myself appealing to basic logical principles like the law of non-contradiction. Even if we can’t currently prove certain axioms, doesn’t this just reflect our epistemological limitations
It’s REALLY hard to distinguish between “unprovable” and “unknown truth value”. In fact, this is recursively hard—there are lots of things that are not proven, but it’s not known if they’re provable. And so on.
Mathematical truth is very much about provability from axioms.
rather than implying all axioms are equally “true”?
“true” is hard to apply to axioms. There’s the common-sense version of “can’t find a counterexample, and have REALLY tried”, which is unsatisfying but pretty effective for practical use. The formal version is just not to use “true”, but “chosen” for axioms. Some are more USEFUL than others. Some are more easily justified than others. It’s not clear how to know which (if any) are true, but that doesn’t make them equally true.
Not a correction (because this is all philosophy) but the problem with this “hard formalism” stance:
Mathematical truth is very much about provability from axioms.
is that statements of the form “statement x follows from axiom set S” are themselves arithmetical statements that may or may not even be provable from a given standard axiom system. I would guess that you’re implicitly taking for granted that Σ1 and Π1 statements in the arithmetic hierarchy have inherent truth in order to at least establish a truth value for such statements. Most people do this implicitly; it’s equivalent to saying that every Turing machine either halts or it doesn’t (and the behavior has nothing to do with someone’s axiom system).
[note: I dabble, at best. This is likely wrong in some ways, so I look forward to corrections. ]
It’s REALLY hard to distinguish between “unprovable” and “unknown truth value”. In fact, this is recursively hard—there are lots of things that are not proven, but it’s not known if they’re provable. And so on.
Mathematical truth is very much about provability from axioms.
“true” is hard to apply to axioms. There’s the common-sense version of “can’t find a counterexample, and have REALLY tried”, which is unsatisfying but pretty effective for practical use. The formal version is just not to use “true”, but “chosen” for axioms. Some are more USEFUL than others. Some are more easily justified than others. It’s not clear how to know which (if any) are true, but that doesn’t make them equally true.
Not a correction (because this is all philosophy) but the problem with this “hard formalism” stance:
is that statements of the form “statement x follows from axiom set S” are themselves arithmetical statements that may or may not even be provable from a given standard axiom system. I would guess that you’re implicitly taking for granted that Σ1 and Π1 statements in the arithmetic hierarchy have inherent truth in order to at least establish a truth value for such statements. Most people do this implicitly; it’s equivalent to saying that every Turing machine either halts or it doesn’t (and the behavior has nothing to do with someone’s axiom system).