That's just plainly false. Mathematics is all about intuition.
That's how we all know that Riemann hypothesis is true, regardless of whether there's a proof of it. Or that Fermat's Last Theorem holds.
That's how we still do math, even though we never really had a solid system of axioms for it -- and Godel showed that, in some sense, it's not even possible.
That's how even famous theorems were proven after many attempts, with people accepting proofs with errors in them.
That's how Calculus was invented before the notions of the limit, derivative, and integral -- the most fundamental ones! -- were solidly written down. Newton and Leibniz built up the math upon heresy, and everyone took it, because it felt right.
What you are writing about is surprise that mathematics still gives to practicing mathematicians.
And your first quote is a trick on how to extend one's intuition, not abandon it!