Jean Piaget, "Mathematical Epistemology and Psychology"
Evert W. Beth and Jean Piaget, Mathematical Epistemology and Psychology, Translated from the French by W. Mays, Synthese Library, Springer, 1966
Part Two, Chapter IX, Section 50, “Invention and Discovery”, pp. 200-201:
For that matter every division into “conscious” and “unconscious” in the process of mathematical invention remains relative to the defects of our introspection. The topologist Leray who knows by experience what an invention is, has even maintained (in a discussion at “The Institute for Advanced Study” at Princeton) that when examined closely, the original idea whose novelty is characteristic of a discovery, only seems to arise from the unconscious at the moment of illumination because we have forgotten that we have seen it beforehand. According to Leray creative work consists first of all of a series of trials in many different directions, trials to which we ourselves do not attribute an equal importance, some seeming more reliable (being orientated in classical directions but in fact incapable of leading to the solution of the new problem) and the others more speculative (precisely because directed towards the new). Amongst the latter may be found, amongst others, the right idea to which we attribute no value at first, so much does it appear contrary to our thought up to then. As the work thus proceeds consciousness becomes more and more crowded with it, like a blackboard on which we write our formulae in order to retain them, a board each corner of which ends by being filled with less and less legible writing. Then the work of the preparatory phase ends and we enter the second phase, characterised by the cessation of enquiry and by the underlying work which Poincaré attributed to unconscious automatism. Now, according to Leray, the unconscious plays at this point only a negative role: it erases from the blackboard all the useless developments and retains only the important ones. On coming back later to its conscious efforts, we thus see that we only have at our disposal a few lines of enquiry and the one which we had neglected then appears more important than it seemed before: in this case we rapidly arrive at the solution looked for, and if it may seem quite new it is simply because we had forgotten having glimpsed it before in passing.
Even if we have never made any discoveries in mathematics, we cannot help recognising the frequent occurrence of the process thus described by Leray. For example, I have often had the impression of having found a new idea, whereupon, trying to exploit it, I put my hand on some forgotten notes where it was already present, inadequately separated from an unimportant context. In the child himself it sometimes happens in the course of the free questioning by means of which we study the solution of problems, that the subject gives the correct solution long before he believes in it and only returns to it after having considered other less sound hypotheses (and without being in the least conscious that he is then rediscovering a possibility previously envisaged).