- From what I’ve noticed there are two kinds of mathematical thinking: manipulating abstract syntax, versus direct experience/perception of concrete mathematics. These two are intertwined in various ways, but in my experience people generally excel in one of these two styles of thinking and not the other. I think that many famous collaborations between two mathematicians are divided along these lines.
Susan Awbrey and I have worked a lot and written a little on a variety of “two-culture” and “cognitive style” questions from a broadly pragmatic perspective informed by the work of C.S. Peirce, John Dewey, and like-minded thinkers. The three dimensional spaces of Peirce’s triadic sign relations afford a perspective on the ways diverse thinkers can specialize their thought to different planes or facets of a sign relation’s full volume. Various issues along these lines are discussed in the following paper.