Do you have any sources on that? I can not find anything about those interacting or Perelman saying that he does not care about the axioms of his work.
Recent Podcast by with Wolfram, maybe the quote is apocryphal or it was by another mathematician, anyway the point was that there is this "toxic" culture of always focusing on formal correct systems while not appreciating the creativity required todo these things. I remembers at some point in the proof of the Poincare conjecture Perelman uses big O notation to argue that one concept is related to another, using a concept from analysis that everyone know in a novel way to proof something new.