This is a really great piece! In past years I read the Goldstein and Nagel & Newman books, and this piece increased and clarified my understanding. Before, I didn't quite see how people drew broad "epistemic " or universal conclusions from Godel, and now I do, or at least have made major progress.

The finite versus infinite framing works very well for me in the above regards. Like you, I'm more on Wittgenstein's side and against the hubris of Platonism, of "universalism."

The concepts of foundations and axioms look problematic, because they look like mental exercises rather than observations. Would it be fair to say the foundation of mathematics is counting, not logic or any other construct? Could part of the answer to the question you end with be that any solution must be without axioms? That axioms cripple solutions before they get off the ground?