[stlee42]

According to Gödel, it's probable that there are true mathematical formulas that cannot be proven.

So from our point of view, we cannot know given a statement whether it can be proven or not.

So all true statements in pure mathematics that we know are a posteriori true, since before we had the experience of proving it, we could not know if it's a statement that could be proven or disproven.

[wk_end]

The Incompleteness Theorem is a technical result about formal systems. It says nothing about the provability of mathematical formulae, since math isn't a formal system.

If you accept the a priori/posteriori distinction, you're abusing words to claim that pure mathematics - the classic example of a priori knowledge! - is a posteriori. A priori knowledge isn't knowledge that we come to know is true through an experience (like proving something), it's knowledge that is true regardless of any particular experience.

[amw-zero]

Of course math is a formal system. If you take Zermelo-Frankel Set Theory as the foundation of math, it has a very clear, formal set of axioms.

And we do not know for certain that math is objective truth. If we did, there would be no philosophy of mathematics.

Reasoning within mathematics is objective, because math is a formal system. But to think that we know anything about anything is frankly pure arrogance. We don’t know why we are here or what our universe even is at a fundamental level. Math is a human-imposed construct that we use to try and make sense of it all.

[jbay808]

> It says nothing about the provability of mathematical formulae, since math isn't a formal system.

Ok, I might accept that math isn't a formal system, but what do you think a proof is?

[dwohnitmok]

Godel's incompleteness theorems are really syntactic results rather than semantic results. The main consequence they have for modern mathematics is that there is no "one axiom system to rule them all," since you can always extend many systems with a new axiom, which doesn't change much, since mathematics since time immemorial has been engaged in the practice of tweaking various axioms and seeing what consequences emerge.

Indeed Godel's completeness (not incompleteness) theorem (which applies to most mathematical settings such as anything that uses ZFC) is a strong indication that everything mathematicians would care to prove is in fact provable.

[n4r9]

It's been a long time since I studied this stuff properly, but is it true to say that you can tell that a statement is provable (or otherwise) if it can be couched in the language of first order logic?

[Supermancho]

> So from our point of view, we cannot know given a statement whether it can be proven or not.

Like most proofs, there are statements which we know are true within constraints to a domain (axioms).