This is a continuation of my post Syntactic and semantic thinkers, in which I mentioned Leone Burton’s book  but hadn’t read it yet. Well, now it is due back at the library so I’d better post about it!
I recommend this book for anyone interested in knowing more about how mathematicians think about and learn math. The book is based on in-depth interviews with seventy mathematicians. (One in-depth interview is worth a thousand statistical studies.) On page 53, she writes
At the outset of this study, I had two conjectures with respect to thinking style. The first was that I would find the two different thinking styles,the visual and the analytic, well recorded in the literature… The second was that research mathematicians would move flexibly between the two. Neither of these conjectures were confirmed.
What she discovered was three styles of mathematical thinking:
Style A: Visual (or thinking in pictures, often dynamic)
Style B: Analytic (or thinking symbolically, formalistically)
Style C: Conceptual (thinking in ideas, classifying)
Style B corresponds more or less with what was called “syntactic” in  (based on ). Styles A and C are rather like the distinctions I made in  that I called “conceptual” and “visual”, although I really want Style A to communicate not only “visual” but “geometric”.
I recommend jumping through the book reading the quotes from the interviews. You get a good picture of the three styles that way.
Visual vs. conceptual
I had thought about this distinction before and have had a hard time explaining what “conceptual” means, particularly since for me it has a visual component. I mentioned this in . I think about various structures and their relationship by imagining them as each in a different part of a visual field, with the connections as near as I can tell felt rather than seen. I do not usually think in terms of the structures’ names (see ). It is the position that helps me know what I am thinking about.
When it comes time to write up the work I am doing, I have to come up with names for things and find words to describe the relationships that I was feeling. (See remark (5) below). Sometimes I have also written things down and come up with names, and if this happened very much I invariable get a clash of notation that didn’t bother me when I was thinking about the concepts because the notations referred to things in different places.
I would be curious if others do math this way. Especially people better than I am. (Clue to a reasonable research career: Hang around people smarter than you.)
1) I have written a lot about images and metaphors , . They show up in the way I think about things sometimes. For example, when I am chasing a diagram I am thinking of each successive arrow as doing something. But I don’t have any sense that I depend a lot on metaphors. What I depend on is my experience with thinking about the concept!
2) Some of the questions on Math Overflow are of the “how do I think about…” type (or “what is the motivation for…”). Some of the answers have been Absolutely Entrancing.
3) Some of the respondents in  mentioned intuition, most of them saying that they thought of it as an important part of doing math. I don’t think the book mentioned any correlation between these feelings and the Styles A, B, C, but then I didn’t read the book carefully. I never read any book carefully. (My experience with Style B of the subtype Logic Rules diss intuition. But not analysts of the sort who estimate errors and so on.)
4) Concerning A, B, C: I use Style C (conceptual) thinking mostly, but a good bit of Style (B) (analytic) as well. I think geometrically when I do geometry problems, but my research has never tended in that direction. Often the analytic part comes after most of the work has been done, when I have to turn the work into a genuine dry-bones proof.
5) As an example of how I have sometimes worked, I remember doing a paper about lifting group automorphisms (see ), in which I had a conceptual picture with a conceptual understanding of the calculations of doing one transformation after another which produced an exact sequence in cohomology. When I wrote it up I thought it would be short. But all the verifications made the paper much longer. The paper was conceptually BigChunk BigChunk BigChunk BigChunk … but each BigChunk required a lot of Analytic work. Even so, I missed a conceptual point (one of the groups involved was a stabilizer but I didn’t notice that.)
 Leone Burton, Mathematicians as Enquirers: Learning about Learning Mathematics. Kluwer, 2004.
 Keith Weber, Keith Weber, How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Proof copy available from Science Direct.
 Post on this blog: Syntactic and semantic thinkers.
 Post: Thinking without words.
 Post: Proofs without dry bones.
 Abstractmath.org article on Images and Metaphors.