This is a continuation of my post Syntactic and semantic thinkers, in which I mentioned Leone Burton’s book [1] 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 [3] (based on [2]). Styles A and C are rather like the distinctions I made in [3] 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 [3]. 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 [4]). 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.)
Remarks
1) I have written a lot about images and metaphors [5], [6]. 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 [1] 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 [7]), 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.)
References
[1] Leone Burton, Mathematicians as Enquirers: Learning about Learning Mathematics. Kluwer, 2004.
[2] Keith Weber, Keith Weber, How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Proof copy available from Science Direct.
[3] Post on this blog: Syntactic and semantic thinkers.
[4] Post: Thinking without words.
[5] Post: Proofs without dry bones.
[6] Abstractmath.org article on Images and Metaphors.
[7] Post: Automorphisms of group extensions updated.
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Here’s a paper that discusses, among other things, the barriers to integrating divergent cognitive styles:
Conceptual Barriers to Creating Integrative Universities
(Earlier conference version: Organizations of Learning or Learning Organizations)
I think I am largely a conceptual thinker, but more to the point, I think in diagrams. There is a place in TTT, C1 – C5 on the bottom of 315, top of 3.16 where, when someone asked me about it, I couldn’t even see what they said until I turned them into commutative diagrams. They should have been done as commutative diagrams in the first place and I think they are in the revision that is now posted. The point is that I could not even follow the syntactic claim, but the diagrammatic one was obvious. Does this make me visual or conceptual? Also, is my use of diagrams a result of nature or nurture? Maybe I should read the book and try to find out.
The thought of C.S. Peirce is rife with threefold classifications, some of which may be pertinent to this discussion.
Peirce took the business of representing reality, abstract of concrete, formal or material, to involve a 3-place relation among (1) the objects of the representation, (2) the elements of the representation itself, called “signs”, and (3) the elements of another representation, called “interpretant signs” or “interpretants” for short.
Any discussion of how signs denote objects and connote, suggest, or translate into other types of signs thus assumes the context of a so-called “sign relation L \subseteq O \times S \times I. Starting from a Kantian backdrop, Peirce regarded “concepts” as mental representations, very often invoked as interpretants of more public expressions.
Notions of “meaning” or “semantics” span the spectrum from relations of signs to objects (denotation or Bedeutung) to relations of sign to interpretants (one way of thinking about connotation or Sinn).
When it comes to the different ways that representations perform their role in a sign relation, Peirce recognized three basic modes of operation: Indexical, Iconic, and Symbolic.
Almost any sign we meet in wild is likely to have some vibe in all three modes, but some are naturally more one than the others. For instance, visual representations tend to have very strong iconic characters, representing their objects by virtue of sharing a property or structure with them.