[kian]

>>> The Pythagorean theorem becomes intuitively true not when you have some deep insight about Euclidean space, but when, on seeing a right triangle, three proofs of it spring instantly to mind.

To be honest, this sounds like orienting one's self in the 'space of mathematics'. Is it not possible that, just like one can navigate by landmarks (proofs) or by the space itself (deep understanding), that there are in fact two roads to intuition in mathematics, of which ones is practice and fluency, and the other is deep insight and understanding?