rhetorical wrote:First, a definition: The way I am interpreting the question "Can we truly prove anything" involves "prove" to be proving something in the physical world around us.
Perhaps that is the origin of this disagreement, as you may be thinking of only theoretical mathematics. If not, then would you mind showing me where you disagree, and where you agree?
Do you agree that:
Mathematics is based on assumptions
If those assumptions are incorrect, so is mathematics
You seem to be very fundamentally misunderstanding what function the "assumptions" of mathematics serve. Don't think of axioms as physical laws, but as rules of a game. We can define the rules for checkers, and there is no reasonable way to say whether those rules are right or wrong, only whether or not the resulting game is interesting and worth playing. If the rules you define end up being contradictory, you just change them in such a way that you still keep the fun/interesting parts of the game. In math, that amounts to saying that even if a contradiction were found in ZFC, it wouldn't invalidate the achievements mathematics has had, particularly in the realms of science and engineering, it would just mean that we'd have to find a new set of axioms that skirted the contradiction and still gave us a system with all the properties that make math so nice. Math's history has many examples where just this sort of thing has occurred, and it never ruined mathematics or invalidated (many) results, in fact it just led to a firmer understanding of just what math is.
rhetorical wrote:Mathematics can only function correctly if the mind (be it a human mind or a computer) is purely logical, or logical enough that the probability that a certain conclusion is true is close enough to 1 so as to make accepting it as true, reasonable
If you aren't certain your mind is functioning correctly enough for math, how would you define such a probability ?
The human mind may/does have flaws that prevent it from reasoning logical or correctly all the time (i.e., it is logical or correct only some of the time.)
Again, the checkers analogy. When playing you might slip up and move a piece the wrong way, perform a move incorrectly, but these have no bearings on the rules themselves. We're human, we make mistakes, and the mistakes are corrected in the long run. Math doesn't require that we are 100% logical all the time, only that we're logical enough to eventually catch the errors when they crop up. I'm no professional mathematician, but from what I've seen I'd say that that is done reasonably well. As to some deeper meaning of truth and whether we can ever really "know" that what we've proven is objectively "true", that is not a question of mathematics.