I'm reading Logicomix, which is fantastic. At some point, it portrays Hilbert saying something like: axioms are not necessarily true, they are just the starting point of the logical process for a proof.
I would go further to say that the logical process itself is just a pile of assumptions. The axioms of logic are assumptions. The rules for logical inference are assumptions. We even assume we've applied the rules correctly when we carry out a proof.
Logic is not so much true as it is useful. The real "grounding" of logic is just that it seems to work often, and where it works often, we trust it more. But it never seems good to say we trust it completely for all problems, and just because it fails sometimes on paradoxes and such doesn't mean that it isn't useful for any problems.
For example, I think of something like modus ponens -- if a is true and a implies b, then b is true -- and wonder, how do we know that? And I don't think we do know it. It's just that, if I build a mental model of a physical system using this rule, then the predictions made by my mental model about real world events turn out to be true, based on my experience. And of course, building a mental model "using" the rule implies some sort of interpretation of the rule, by my brain, which may or may not be the same as the interpretation that other brains give to this rule when they use it to build mental models.
So, I guess I feel less and less sure that math and logic have a life -- an intrinsic truth -- beyond just being problem solving tools, subject to interpretation by brains and computers.