A week-ago post on the role of rigor versus intuition in mathematics, initially focusing on Hilbert versus Poincare:
"In some sense, both Hilbert and Poincaré were right. Mathematics is a matter of intuitions, but rigor is essential to get the intuitions right. Only then, when the intuitions are right, can we do complex reasonings on difficult problems….The author also links to this related Terry Tao post that I've linked to before (where he discusses "pre-rigorous," "rigorous," and "post-rigorous" math education, interweaving with intuition):
"I believe the role of mathematics in schools to be precisely to use rigor to get intuitions right. Or, at least, less wrong."