100 years ago everyone was worrying about whether the foundations of mathematics were consistent. We figured out some details that we had been taking for granted, like the axiom of choice, how to prevent Russell's paradox, and the inevitable difference between "true" and "provable" (ie Godel's incompleteness). Now that part is pretty much solved, we are fairly confident that the foundations of mathematics are consistent.
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. It seems like it should be possible, in theory, but it takes too much work. So most mathematicians prefer communicating their proofs in PDFs rather than in a more rigorous format.
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. It seems like it should be possible, in theory, but it takes too much work. So most mathematicians prefer communicating their proofs in PDFs rather than in a more rigorous format.