TITLE: Is it usual to have no intuition to certain proofs and simply do them mechanically?
QUESTION [15 upvotes]: 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.

REPLY [3 votes]: To quote Spivak from his preface to 'Calculus':

...precision and rigour are neither deterrents to intuition nor ends in themselves, but the natural medium in which to formulate and think of mathematical questions.

But the question may arise why take one author's words and disregard another's. Quite simply because saying things like proofs are 'supposed' to be non-intuitive is not true in any way. Not only is it untrue but it also confuses and discourages amateur mathematicians. 
To an uninformed person, Einstein's field equations are nothing more than a bunch of Greek letters put together. If that person asks what those symbols actually mean, he will get explanations about how the world really is non-Euclidean and about the 4th dimension and whatnot. It may seem terse, unapproachable and even false. Even a trained physicist may accept the equation but he or she may not agree that it is intuitive. 
But when one hears of the way Einstein himself got his idea. The way he was staring at everyday objects and thinking of them in different situations; thought experiments as they were called. The way he just let his imagination loose like a child, but a mathematically inclined one. The little ideas were so simple that when one hears of them he will immediately connect with them. (Although the mathematics behind them will seem intuitive only to a handful)
Every proof you see, I guarantee you, has intuition behind it. If it seems like a bunch of symbols strewn here and there it is one's incompetency to understand the language clearly. 
Whether what is intuitive to one is intuitive to another or not is a different story. But one thing is certain; the more you explore, the more you think, the more thought experiments you conduct, the more mistakes you make...you have a higher chance to connect with the most impenetrable proofs