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On the latter point, that's a pretty major dispute in philosophy and mathematics. The axiomatic Hilbert-style approach probably has the upper hand in mathematics, but a Russell-style logicist approach that sees foundations themselves as subject to rational analysis is still a major position, and among philosophers, the idea that axioms are arbitrary and rationality only applies past that is probably the minority view (though existentialists hold a view somewhat like that). Not that that's necessarily practical advice.

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