math, representations, understanding math

Three kinds of mathematical thinkers

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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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math, proof, representations, understanding math

Naive proofs

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The monk problem

A monk starts at dawn at the bottom of a mountain and goes up a path to the top, arriving there at dusk. The next morning at dawn he begins to go down the path, arriving at dusk at the place he started from on the previous day. Prove that there is a time of day at which he is at the same place on the path on both days.

Proof: Envision both events occurring on the same day, with a monk starting at the top and another starting at the bottom at the same time and doing the same thing the monk did on different days. They are on the same path, so they must meet each other. The time at which they meet is the time required.

The pons asinorum

Theorem: If a triangle has two equal angles, then it has two equal sides.

Proof: In the figure below, assume angle ABC = angle ACB. Then triangle ABC is congruent to triangle ACB since the sides BC and CB are equal and the adjoining angles are equal.

PATriangle

I considered the monk problem at length in my post Proofs Without Dry Bones.  Proofs like the one given of the pons asinorum, particularly its involvement with labeling, recently came up on the mathedu mailing list.  See also my question on Math Overflow.

Naive proofs

These proofs share a characteristic property; I propose to say they are naive, in the sense Halmos used it in his title Naive Set Theory.

The monk problem proof is naive.

For the monk problem, you can give a model of a known mathematical type (for example model the paths as  smoothly parametrized curves on a surface) and use known theorems (for example the intermediate value theorem) and facts (for example that clock time is cyclical and invariant under the appropriate mapping) to prove it.  But the proof says nothing about that.

You could imagine inventing an original set of axioms for the monk problem, giving axioms for a structure that are satisfied by the monk’s journeys and their timing and that imply the result.  In principle, these could be very different from multivariable calculus ideas and still serve the purpose. (But I have not tried to come up with such a thing.)

But the proof as given simply uses directly  known facts about clock time and traveling on paths.  These are known to most people.  I have claimed in several places that this proof is still a mathematical proof.

Every proof is incomplete in the sense that they provide a mathematical model and analyze it using facts the reader is presumed to know.  Proofs never go all the way to foundations.  A naive proof simply depends more than usual on the reader’s knowledge: the percentage of explication is lower.  Perhaps “naive” should also include the connotation that the requisite knowledge is “common knowledge”.

The pons asinorum proof is naive.

This involves some subtle issues.  When I first wrote about this proof in the Handbook I envisioned the triangle as existing independently of any embedding in the plane, as if in the Platonic world of ideals.  I applied some labels and a relabeling and used a known theorem of Euclid’s geometry.  You certainly don’t have to know where the triangle is in order to understand the proof.

That’s a clue.  The triangle in the problem does not need to be planar. It is true for triangles in the sphere or on a saddle surface, because the proof does not involve the parallel axiom. But the connection with the absence of the parallel axiom is illusory.  When you imagine the triangle in your head the proof works directly for a triangle in any suitable geometry, by imagining the triangle as existing in and of itself, and not embedded in anything.

Questions

  1. How do you give a mathematical definition of a triangle so that it is independent of embedding?  This was the origin of my question on Math Overflow, although I muddled the issue by mentioning specific ways of doing it.
  2. (This is a variant of question 1.)  Is there anything like a classifying topos or space for a generic triangle?  In other words, a category or space or something that is just big enough to include the generic triangle and from which mappings to suitable spaces or categories produce what we usually mean by triangles.
  3. Some of the people on mathedu thought a triangle obviously had to have labels and others thought it obviously didn’t.  Specifically, is triangle ABC “the same” as triangle ACB?  Of course they are congruent.  Are they the same?  This is an evil question. The proof works on the generic isosceles triangle.  That’s enough.  Isn’t it?  All three corners of the generic isosceles triangle are different points.  Aren’t they?  (I have had second, third and nth thoughts about this point.)
  4. You can define a triangle as a list of lengths of edges and connectivity data.  But the generic triangle’s sides ought to be (images of) line segments, not abstract data.  I don’t really understand how to formulate this correctly.

Note

1.  I could avoid discussion of irrelevant side issues in the monk problem by referring to specific times of day for starting and stopping, instead of dawn and dusk.  But they really are irrelevant.

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language

BC and AD

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I forget half the time that to be politically correct I should use BCE and CE instead of BC and AD.  I have a better idea.  I decree that from now on, BC means Backward Counting and AD means Advancing Denumeration.  At least in what I write.  Unless I forget.

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math

Characterizing triangles unembeddedly

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I just posted this question on mathoverflow (I recommend looking into this new forum):

The mathedu mailing list has a recent longish thread which discusses among other things whether we should teach triangles as labeled or unlabeled to high school students (this is a vast oversimplification of the thread).  I have long been concerned with how we think (informally and formally) about mathematical objects, as for example my unfinished article here about the many ways of thinking about function.  So naturally, I started to consider how we think about triangles.

Consider circles.   Most informal and formal descriptions involve an embedding into R^2, but they *can* be characterized as manifolds (even as Riemannian manifolds) of dimension 1 with specific properties, independent of any embedding. This sort of thing has turned out to be a major way to think about all sorts of spaces.  So can we describe triangles in a similar way?

Unfortunately, manifolds are far removed from my usual mathematical work (category theory).  What I *think* I understand is that there can be *piecewise* linear manifolds, even Riemannian ones.  So perhaps we can say a triangle is a piecewise linear manifold of dimension 1 with certain properties.  Now, I want to define a triangle so that it comes complete with information about the lengths of its sides and what the three angles are.  Riemannian manifolds have a way to specify length and angles, and I can believe you can make the sides have specific lengths.  But the angles?  It seems to me that the tangent spaces (like those on a circle) result in all angles being 0 or pi, except at the corners where they don’t exist.  But I may not understand the situation correctly.

So my question is:  Is there a known methodology that allows triangles to be characterized independent of embeddings in such a way that incorporates information about side lengths and angles?

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language

More about pronouncing "a" and "the"

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I eavesdropped on my grandson dictating a story for his granny to type.  He always pronounced “a” as a schwa.  I mentioned in an earlier post that Barak Obama and Hilary Clinton both normally pronounce it “ay” in speeches and wondered if this is a generational change.  The grandsonian evidence suggests not.  But:

  1. Do Obama and Clinton pronounce it that way in ordinary conversation?  I bet not.
  2. Is there a Speech Making School for Politicians that has them do this, or is this the result of their unconsciously adopting a Speech Making Register?  I’ll bet the latter.

The grandson also regularly said “thee” for “the” before vowel sounds and used the schwa before consonants.  This makes me want to go back to You Tube and eavesdrop on politicians some more.

I might add that if he stopped after “the” to let poor old granny catch up with her typing, he used the schwa even thought what came next was a vowel.  However, this may not prove anything since he may not have had the next word in mind.

Years ago, I was fascinated listening to John Kennedy, who pronounced r at the end of a word only if the next word began with a vowel, as in “The far east” but “The fah boundaries of…”.  I thought that it was remarkable that he did this even if he paused before second word.

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abstractmath.org, math, representations, understanding math

Syntactic and semantic thinkers

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A paper by Keith Weber

Reidar Mosvold’s math-ed blog recently provided a link to an article by Keith Weber (Reference [2]) 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 eq0001MP

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.

3) I suspect the syntactic reasoners also prefer to read proofs step by step, as I described in my post Grasshoppers and linear proofs, rather than skipping around like a grasshopper.

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 [3]).

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 [1]) 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.

Notes

  1. Handwaving is now called kinesthetic communication.  Just to keep you au courant.
  2. I took Joe Shoenfield’s course in logic when his book  Mathematical Logic [3] was still purple.
  3. Clockwise for left action, counterclockwise for right action.  Not.

References

  1. Leone L. Burton, Mathematicians as Enquirers: Learning about Learning Mathematics.  Springer, 2004.
  2. Keith Weber, How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Proof copy available from Science Direct.
  3. Joseph Shoenfield, Mathematical logic, Addison-Wesley 1967, reprinted 2001 by the Association for Symbolic Logic.
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language

Goodnight, Irene

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Look at this list:

Antigone
Aphrodite
Chloe
Hermione
Irene
Kalliope
Nike
Penelope
Phoebe
Zoe

All these are originally Greek names of supernatural beings (except Antigone?). The e is a feminine ending. Most of them are used now as women’s names. When Americans pronounce these names, with one exception they usually pronounce the final e.

The exception is “Irene”. I have heard British people say “I-reenie” but never an American. Is this because of “Good Night Irene”?

At one point when I was maybe eleven years old I bought a 45 of the Weavers singing Good Night Irene. It was my favorite song. The record had Tsena Tsena on the other side. I fell in love with Tsena Tsena which I had never heard before, but I still liked GNI too. For some time after that I looked for other records by the Weavers but I never saw one. Perhaps that was about the time the McCarthyites blacklisted them?

I was also attracted by the harmonies of some pieces by Bach. Now I think that the thing TT and Bach (and others of my favorite music, like some Procol Harum) have in common is the existence of both major and minor chords in the same piece. But when I asked my music teacher what was so wonderful about Bach she said she had never understood Bach.

Oh well.

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category theory, computer science, math, Sketches and forms

Addenda to the 1993 Sketches paper

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I have uploaded here a version of my 1993 sketches paper with an addendum listing a few relevant papers written since then.  I have not kept up with the field well enough to contemplate a complete revision of the 1993 paper.

I recommend that more people update their papers this way.  I did it by making a new PDF file with the added references and then using Acrobat to combine it with the old paper into one file.  That way I didn’t have to re-TeX the old paper, which is a good thing, since I don’t know where some of the .sty files are.

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math, proof, understanding math

Mastering a proof

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In response to Grasshoppers and linear proofs, Avery Andrews said:

Maybe a related question is how much time people do/ought spend on really mastering the proofs of theorems in textbooks, ‘mastering’ being, say, able to explain it in any desired amount of detail at least 2 weeks after last looking at it.

There are two different goals:

  1. Mastering the proof of a theorem in a textbook so that you can explain it in any desired amount of detail…
  2. Mastering a proof of the theorem so that you can explain it in any desired amount of detail…

My observation is that most research mathematicians don’t attempt (1); they are satisfied with (2).  Trying to understand a written proof in detail can be quite difficult:

  • The author may use misleading language.
  • The author may jump over a piece of reasoning that to them is obvious but not to you.
  • The author may mention a previous step or a theorem that justifies the current step, but get the reference wrong.

And so on.

In my observation the typical mathematician will look at the proof, perhaps getting some idea of the overall strategy of the whole proof or a particular part, and then think about it independently until they come up with a proof or part of it.  This may or may not be what the author had in mind.  But by thinking through it the reader will solidify their understanding of the proof in a way that reading and rereading step by step is unlikely to do.

When you construct your knowledge like that you are likely to have it in a permanent, well semi-permanent, way.

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language

Steven Brust commits a zeugma

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“…she immediately spurred her horse, yet the horse had hardly moved when Wadre moved his arm in an indication that she was not to advance beyond him, wherefore she drew rein, her sword, and the conclusion that the time was not yet quite at hand to charge.”

Steven Brust, The paths of the dead, p. 335.   New York: Tom Doherty Associates (2002), p. 335.

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math, language and other things that may show up in the wabe