A very interesting presentation by Vladimir Voevodsky:
It is extremely difficult to accept that mathematics is in need of a completely new foundation.
Even many of the people who are directly connected with the advances in Homotopy Type
Theory are struggling with this idea.
There is a good reason it is difficult: the existing foundation of mathematics – ZFC, and its main
contender for a new foundation – category theory, have been very successful.
It was overcoming the appeal of category theory as a candidate for new foundation of mathematics
that was for me personally most difficult.