I think there are probably a few mathematicians that have deal with this problem better than the ones that have been simultaneously active in a completely different area, and from the ones that I know, I would suggest you to read Gian-Carlo Rota.
In particular, I would suggest you the book Indiscrete Thoughts, and chapter 9 ( The reductionist concept of the mind) from his essay The Pernicious Influence of Mathematics upon Philosophy.
In this chapter he asks: What does a mathematician do when trying to work on a mathematical problem? He starts citing Polya and saying that the most important step is look at other attemps, then he tells about mathematicians looking for the story of problems in order to solve them...A real story or an ideal reconstructed story a gifted mathematician may create.
He explains that the real nature of a mathematical problem is something that mathematicians are daily discovering (not something fixed or completely known) by a ''hand-in-hand'' passing of solutions of mathematical problems, and he associated this to an historic process.
He compares the way the mathematical thinking differ from the philosophers: hard thinking and blank mind is not the way how a mathematician thinks.
He concludes saying that ''the process of the working of the mind, which may be of interest to physicians but is of no interest to mathematicians, is confused with the progress of thought that is required in the solution of any problem''.
I would also suggest you to try to extract your own conclusions from the bibliographies, quotes and behaviour of your favourite mathematicians. You may start with the list of Alex Bellos: Pythagoras, Hypatia, Cardano, Euler, Gauss, Cantor, Erdös, Conway, Perelman and Tao.