At least two books I have read, claim that proofs are not supposed to be intuitive and are meant to be terse and mechanical. I find this confusing as mathematicians are humans too. Differential geometry once upon a time was seen as simply a theoretical mathematical concept which had no real world applications. Einstein proved this wrong by using differential geometry in his theory which actually implies that Euclidean geometry is an approximation and differential geometry is more accurate for our world. Doesn't this say that Einstein could just 'feel' the way the world was non - Euclidean?
Differential geometry may seem hostile at times but if one is to translate those mysterious proofs and mechanisms to real world situations, surely Einstein could just speak mathematics as if it were German to him? When we are proving theorems on rings, fields and number systems, the proofs all materialise mysteriously and we are at times satisfied with the way they are non-intuitive. Satisfied simply because we can recall the entire proof and have a logical flow as well. Logical flow will do when we are studying the proof. But didn't someone actually come up with the proof? That person too must have had his 'Eureka' moment when he suddenly notices something that immediately gives him an idea how to prove something.