Is it usual to have no intuition to certain proofs and simply do them mechanically?
According to my knowledge and experience, sometimes proofs are more intuitive or clear and less technical, sometimes conversely.
To balance the opinions from books your read I present two others:
Nicolas Bourbaki: “the mathematician does not work like a machine, nor as the workingman on a moving belt; we can not over-emphasize the fundamental role played in his research by a special intuition, which is not the popular sense-intuition, but rather a kind of direct divination (ahead of all reasoning) of the normal behavior, which he seems to have the right to expect of mathematical beings, with whom a long acquaintance has made him as familiar as with the beings of the real world”
Henri Poincaré wrote a paper “Mathematical Creation”. You may also look at his paper “Intuition and Logic in Mathematics” (for instance, in this book).
PS. A few of my old answers to similar MSE questions:
- Question about long proofs?
- How are long proofs "planned"?
- Big list of "guided discovery" books
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