> IMHO the problem now is that the foundations of mathematics are impractical. Most mathematicians don't actually do mathematics that can be tied back to the foundational axioms.
I suppose my followup to that is, why is this a problem? Does it lead to bad mathematics? (For what definition of “bad”?) Would we be able to solve unsolved problems if we tied things back to foundations more? Mathematicians have never really done much mathematics that ties back to the foundational axioms — why change? Rigor is not really much of an end in itself - surely people do mathematics because it is useful or beautiful, or both?
I suppose my followup to that is, why is this a problem? Does it lead to bad mathematics? (For what definition of “bad”?) Would we be able to solve unsolved problems if we tied things back to foundations more? Mathematicians have never really done much mathematics that ties back to the foundational axioms — why change? Rigor is not really much of an end in itself - surely people do mathematics because it is useful or beautiful, or both?