Well, if mathematics is to say anything about the world, then the foundations must be sound (and drawn from experience which brings us back to empiricism). Some

Well, if mathematics is to say anything about the world, then the foundations must be sound (and drawn from experience which brings us back to empiricism). Sometimes the problem isn't necessarily with the concepts in our heads but the formalizations we choose to proceed from. Set theory is a great example. Take Russell's paradox, for instance, and the different responses to it (e.g., ZFC, Lesniewski's mereology). I'm reminded of Duhem's rather humorous "German Science" where he characterizes German mathematics as being disproportionately oriented toward rigorously deducing nonsense from arbitrary axioms as opposed to the French tendency to rely on intuition to arrive at sensible axioms but neglecting systematic rigor afterwards.