The mathematician Jacques Hadamard carried out an interesting study on this subject and presented his findings in a small book called, "The Psychology of Invent

The mathematician Jacques Hadamard carried out an interesting study on this subject and presented his findings in a small book called, "The Psychology of Invention in the Mathematical Field". Here are a couple of excerpts from it that I found compelling (he's talking about a sort of 'schematic' thinking with something like relationship diagrams, and much of his motivation is the contention that thinking is only done linguistically): "Indeed, every mathematical research compels me to build such a schema, which is always and must be of a vague character, so as not to be deceptive. I shall give a less elementary example from my first researches (my thesis). I had to consider a sum of an infinite number of terms, intending to valuate its order of magnitude. In that case, there is a group of terms which chances to be predominant, all others having a negligible influence. Now, when I think of that question, I see not the formula itself, but the place it would take if written: a kind of ribbon, which is thicker or darker at the place corresponding to the possibly important terms; or (at other moments), I see something like a formula, but by no means a legible one, as I should see it (being strongly long-sighted) if I had no eye-glasses on, which letters seeming rather more apparent (though still not legible) at the place which is supposed to be the important one." "About the mathematicians born or resident in America, whom I asked, phenomena are mostly analogous to those which I have noticed in my own case. Practically all of them -- contrary to what occasional inquiries had suggested to Galton as to the man on the street -- avoid not only the use of mental words, but also, just as I do, the mental use of algebraic or any other precise signs; also as in my case, they use vague images." My experience with thinking in abstract domains matches this perfectly—and I bet it can be trained/taught. I've written more on the subject here, btw: http://westoncb.blogspot.com/2013/01/an-architecture-for-deliberate-thought.html http://westoncb.blogspot.com/2013/01/an-architecture-for-del...