I have a question about proofs. While reading proofs I would sometimes encounter a step that I just don't get. I can verify that it follows from the previous steps and is correct, but not why it was taken. It just seems arbitrary. The question is: do mathematicians usually have a reason for these steps that they do not include in a proof, because it has no place in the formalism, or are these steps just the result of sudden revelations and random ideas and even the proof's author does not know why the proof is correct, only that it is? Do mathematicians "think in proofs" or is there a difference between what is written down as a proof and how it was conceived?