A paper by Keith Weber
Reidar Mosvold’s math-ed blog recently provided a link to an article by Keith Weber (Reference ) about a very good university math student he referred to as a “syntactic reasoner”. He interviewed the student in depth as the student worked on some proofs suitable to his level. The student would “write the proofs out in quantifiers” and reason based on previous steps of the proof in a syntactic way rather than than depending on an intuitive understanding of the problem, as many of us do (the author calls us semantic reasoners). The student didn’t think about specific examples – he always tried to make them as abstract as possible while letting them remain examples (or counterexamples).
I recommend this paper if you are at all interested in math education at the university math major level — it is fascinating. It made all sorts of connections for me with other ideas about how we think about math that I have thought about for years and which appear in the Understanding Math part of abstractmath.org. It also raises lots of new (to me) questions.
Weber’s paper talks mostly about how the student comes up with a proof. I suspect that the distinction between syntactic reasoners and semantic reasoners can be seen in other aspects of mathematical behavior, too, in trying to understand and explain math concepts. Some thoughts:
Other behaviors of syntactic reasoners (maybe)
1) Many mathematicians (and good math students) explain math using conceptual and geometric images and metaphors, as described in Images and metaphors in abstractmath.org. Some people I think of as syntactic reasoners seem to avoid such things. Some of them even deny thinking in images and metaphors, as I discussed in the post Thinking without words. It used to be that even semantic reasoners were embarassed to used images and metaphors when lecturing (see the post How “math is logic” ruined math for a generation).
2) In my experience, syntactic reasoners like to use first order symbolic notation, for example
and will often translate a complicated sentence in ordinary mathematical English into this notation so they can understand it better. (Weber describes the student he interviewed as doing this.) Furthermore they seem to think that putting a formula such as the one above on the board says it all, so they don’t need to draw pictures, wave their hands [Note 1], and so on. When you come up with a picture of a concept or theorem that you claim explains it their first impulse is to say it out in words that generally can be translated very easily into first order symbolism, and say that is what is going on. It is a matter of what is primary.
The semantic reasoners of students and (I think) many mathematicians find the symbolic notation difficult to parse and would rather have it written out in English. I am pretty good at reading such symbolic notation [Note 2] but I still prefer ordinary English.
And maybe not
Now it may very well be that syntactic thinkers do not all do all those things I mentioned in (1)-(3). Perhaps the group is not cohesive in all those ways. Probably really good mathematicians use both techniques, although Weyl didn’t think so (quoted in Weber’s paper). I think of myself as an image and metaphor person but I do use syntax, and sometimes even find that a certain syntactic explanation feels like a genuinely useful insight, as in the example I discussed under conceptual in the Handbook.
Distinctions among semantic thinkers
Semantic thinkers differ among themselves. One demarcation line is between those who use a lot of visual thinking and those who use conceptual thinking which is not necessarily visual. I have known grad students who couldn’t understand how I could do group theory (that was in a Former Life, before category theory) because how could you “see” what was happening? But the way I think about groups is certainly conceptual, not syntactic. When I think of a group acting on a space I think of it as stirring the space around. But the stirring is something I feel more than I see. On the other hand, when I am thinking about the relationships between certain abstract objects, I “see” the different objects in different parts of an interior visual space. For example, group is on the right, stirring the space-acted-upon on the left, or the group is in one place, a subgroup is in another place while simultaneously being inside the group, and the cosets are grouped (sorry) together in a third place, being (guess what) stirred around by the group acting by conjugation (Note ).
This distinction between conceptual and visual, perhaps I should say visual-conceptual and non-visual-conceptual, both opposed to linguistic or syntactic reasoning, may or may not be as fundamental as syntactic vs semantic. But it feels fundamental to me.
Weber’s paper mentions an intriguing sounding book (Reference ) by Burton which describes a three-way distinction called conceptual, visual and symbolic, that sounds like it might be the distinction I am discussing here. I have asked for it on ILL.
- Handwaving is now called kinesthetic communication. Just to keep you au courant.
- I took Joe Shoenfield’s course in logic when his book Mathematical Logic  was still purple.
- Clockwise for left action, counterclockwise for right action. Not.
- Leone L. Burton, Mathematicians as Enquirers: Learning about Learning Mathematics. Springer, 2004.
- Keith Weber, How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Proof copy available from Science Direct.
- Joseph Shoenfield, Mathematical logic, Addison-Wesley 1967, reprinted 2001 by the Association for Symbolic Logic.