No, no, I think you're missing the point. Mathematics is such that all mathematicians, in particular Brouwer and Hilbert, will read a proof and say "yes, this f

No, no, I think you're missing the point. Mathematics is such that all mathematicians, in particular Brouwer and Hilbert, will read a proof and say "yes, this follows" or "no, it doesn't follow". The agreement is on the argumentation, i.e. what can be said given a set of assumptions. This doesn't mean there are no disagreements in the Philosophy of Mathematics. The place Brouwer and Hilbert disagrees is about what can be assumed at all times. The disagreement is on the foundation that mathematics lies on. This is also not the only disagreement, there are various models of what mathematics really is. You can ask N philosophers of mathematics (who are also trained in math) about a proof and they'll all agree that the mechanism of the proof follows. What they may disagree on is whether the assumptions are sound, and what "mathematics" really is. Some will focus on the social aspect and point out that "proof" ultimately means "convinces all mathematicians". Some will focus on the psychological aspect. Some will push mathematics outside of material realm etc etc.