Hadamard investigates these kinds of issues at length in his book The psychology of invention in the mathematical field. He gives several examples of famous mathematicians dreaming about solutions, including Poincare. His conclusion is that the unconscious definitely plays a decisive role in mathematics, and that sleep often has to do with it, but that it differs from person to person how to tap in to it.

It is (necessarily) a bit pseudoscientific, but has some great tidbits. For example, did you know Mobius' grandson, who was a psychologist into the then-popular phrenology, actually went around measuring mathematicians' heads, trying to locate the "bump" in the skull where mathematical ability should lie?

Hadamard investigates these kinds of issues at length in his book The psychology of invention in the mathematical field. He gives several examples of famous mathematicians dreaming about solutions, including Poincare. His conclusion is that the unconscious definitely plays a decisive role in mathematics, and that sleep often has to do with it, but that it differs from person to person how to tap in to it.

It is (necessarily) a bit pseudoscientific, but has some great tidbits. For example, did you know Mobius' grandson, who was a psychologist into the then-popular phrenology, actually went around measuring mathematicians' heads, trying to locate the "bump" in the skull where mathematical ability should lie?