Here is the Amazon description:
Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Lévi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field, remains an important tool for exploring the increasingly complex problem of mental life.
The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought.
Hadamard emphasizes a four step process of invention, consisting of
Preparation: conscious effort attacking the problem, including analysis of various methods and approaches; the outcome often appears fruitless
Incubation: a period of subconscious effort while the conscious mind is occupied with other matters
Illumination: the solution forms in the conscious mind
Verification: the solution is verified through conscious effort
This description agrees with my own experience. It seems that certain activities like walking are good for Incubation, and the shower is a good time for Illumination. Over the past weeks I've been deliberately trying to follow this method in my own research, with modest success :^) I noticed that it is also useful for overcoming writer's block -- put aside the paragraph you are struggling with and walk around for five or ten minutes before resuming. The walk often allows a small conceptual reordering, enough to proceed.
Incidentally, I often found my father and other professors and scientists I grew up around to be absent minded. I was never that way, and therefore wondered if I was fundamentally not of the scientific type (a smart kid, but not nerdy enough!). However, what I've found is that a lifetime of working on abstract problems has now given me the ability to call forth a problem of my choice and work with it in my head, leading to the tendency to occasionally detach from my surroundings. So, I appear (e.g., to my wife and kids!) to be the scientist type, but as a consequence of developing a particular functional ability rather than due to a specific predilection or tendency.
This paper contains a nice discussion of Hadamard's essay, and describes a modern study using Hadamard's original questionnaire. I liked these comments on the role of talking about problems:
... it is also clear from the mathematicians' responses that, while working in the initiation phase, they have a much higher regard for transmission of mathematical knowledge through talking than through reading. ...
Jerry: I assimilate the work of others best through personal contact and being able to question them directly. [..] In this question and answer mode, I often get good ideas too. In this sense, the two modes are almost indistinguishable.
George: I get most of my real mathematical input live, from (good) lectures or one-on-one discussions. I think most mathematicians do. I look at papers only after I have had some overall idea of a problem and then I do not look at details.