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Representations 2

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Introduction

In a recent post I began a discussion of the mental, physical and mathematical representations of a mathematical object. The discussion continues here. Mathematicians, linguists, cognitive scientists and math educators have investigate some aspects of this topic, but there are many subtle connections between the different ideas which need to be studied.

I don’t have any overall theoretical grasp of these relationships. What I will do here is grope for an overall theory by mentioning a whole bunch of fine points. Some of these have been discussed in the literature and some (as far as I know) have not been discussed.  Many of them (I hope)  can be described as “an obvious fact about representations but no one has pointed it out before”.  Such fine points could be valuable; I think some scholars who have written about mathematical discourse and math in the classroom are not aware of many of these facts.

I am hoping that by thrashing around like this here (for graphs of functions) and for other concepts (set, function, triangle, number …) some theoretical understanding may emerge of what it means to understand math, do math, and talk about math.

The graph of a function

Let’s look at the graph of the function {y=x^3-x}.

What you are looking at is a physical representation of the graph of the function. The graph creates in your brain a mental representation of the graph of the function. These are subtly related to each other and to the mathematical definition of the graph.

Fine points

  1. The mathematical definition [2] of the graph of this function is: The set of ordered pairs of numbers {(x,x^3-x)} for all real numbers {x}.
  2. In the physical representation, each point {(x,x^3-x)} is shown in a location determined by the conventional {x-y} coordinate system, which uses a straight-line representation of the real numbers with labels and ticks.
    • The physical representation makes use of the fact that the function is continuous. It shows the graph as a curving line rather than a bunch of points.
    • The physical representation you are looking at is not the physical representation I am looking at. They are on different computer screens or pieces of paper. We both expect that the representations are very similar, in some sense physically isomorphic.
    • “Location” on the physical representation is a physical idea. The mathematical location on the mathematical graph is essentially the concept of the physical location refined as the accuracy goes to infinity. (This last statement is a metaphor attached to a genuine mathematical construction, for example Cauchy sequences.)
  3. The mathematical definition of “graph” and the physical representation are related by a metaphor. (See Note 1.)
    • The physical curve in blue in the picture corresponds via the metaphor to the graph in the mathematical sense: in this way, each location on the physical curve corresponds to an ordered pair of the form {(x,x^3-x)}.
    • The correspondence between the locations and the pairs is imperfect. You can’t measure with infinite accuracy.
    • The set of ordered pairs {(x,x^3-x)} form a parametrized curve in the mathematical sense. This curve has zero thickness. The curve in the physical representation has positive thickness.
    • Not all the points in the mathematical graph actually occur on the physical curve: The physical curve doesn’t show the left and right infinite tails.
    • The physical curve is drawn to show some salient characteristics of the curve, such as its extrema and inflection points. This is expected by convention in mathematical writing. If the graph had left out a maximum, for example, the author would be constrained (by convention!) to say so.
    • An experienced mathematician or advanced student understands the fine points just listed. A newbie may not, and may draw false conclusions about the function from the graph. (Note 2.)
  4. If you are a mathematician or at least a math student, seeing the physical graph shown above produces a mental image(see Note 3.) of the graph in your mind.
  5. The mathematical definition and the mental image are connected by a metaphor. This is not the same metaphor as the one that connects the physical representation and the mathematical definition.
    • The curve I visualize in my mental representation has an S shape and so does the physical representation. Or does it? Isn’t the S-ness of the shape a fact I construct mentally (without consciously intending to do so!)?
    • Does the curve in the mental rep have thickness? I am not sure this is a meaningful question. However, if you are a sufficiently sophisticated mathematician, your mental image is annotated with the fact that the curve has zero thickness. (See Note 4.)
    • The curve in your mental image of the curve may very well be blue (just because you just looked at my picture) but you must have an annotation to the effect that that is irrelevant! That is the essence of metaphor: Some things are identified with each other and others are emphatically not identified.
    • The coordinate axes do exist in the physical representation and they don’t exist in the mathematical definition of the graph. Of course they are implied by the definition by the properties of the projection functions from a product. But what about your mental image of the graph? My own image does not show the axes, but I do “know” what the coordinates of some of the points are (for example, {(-1,0)}) and I “see” some points (the local maximum and the local minimum) whose coordinates I can figure out.

Notes

1. This is metaphor in the sense lately used by cognitive scientists, for example in [6]. A metaphor can be described roughly as two mental images in which certain parts of one are identified with certain parts of another, in other words a pushout. The rhetorical use of the word “metaphor” requires it to be a figure of speech expressed in a certain way (the identification is direct rather than expressed by “is like” or some such thing.)  In my use in this article a metaphor is something that occurs in your brain.  The form it takes in speech or writing is not relevant.

2. I have noticed, for example, that some students don’t really understand that the left and right tails go off to infinity horizontally as well as vertically.   In fact, the picture above could mislead someone into thinking the curve has vertical asymptotes: The right tail looks like it goes straight up.  How could it get to x equals a billion if it goes straight up?

3. The “mental image” is of course a physical structure in your brain.  So mental representations are physical representations.

4. I presume this “annotation” is some kind of physical connection between neurons or something.  It is clear that a “mental image” is some sort of physical construction or event in the brain, but from what little I know about cognitive science, the scientists themselves are still arguing about the form of the construction.  I would appreciate more information on this. (If the physical representation of mental images is indeed still controversial, this says nothing bad about cognitive science, which is very new.)

References

[1] Mental Representations in Math (previous post).

[2] Definitions (in abstractmath).

[3] Lakoff, G. and R. E. Núñez (2000), Where Mathematics Comes From. Basic Books.

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Grasshoppers and linear proofs

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Below, I give an detailed example of how the context of a proof changes as you read the proof line by line. This example comes from the abstractmath article on context.  I mean something like verbal context or  context in the computer science sense (see also Reference [1]): the values of all the relevant variables as specified up to the current statement in the proof.  For example, if the proof says “Suppose x = 3″, then when you read succeeding statements you know that x has the value 3, as long as it is not changed in some later statement.

Here is the text I will analyze:

Definition: Divides

Let m and n be integers with m\ne 0. The statement “m divides n” means that there is an integer q for which n=qm.

Theorem

Let m, n and p be integers, with m and n nonzero, and suppose m divides n and n divides p .  Then m divides p.

Proof

By definition of divides, there are integers q and q’ for which n=qm and p=q'n. We must prove that there is an integer q'' for which p=q''n. But p=q'n=q'qm, so let q''=q'q.  Then p=q''n.

0) Definition: Divides Changes the status of the word “divides” so that it becomes the definiendum. The scope is the following paragraph.
1) Let m and n be integers m and n are new symbols in this discourse, constrained to be integers
2) with m\neq 0 another constraint on m
3) The statement “m divides n” means that This sentence fragment gives the rest of the sentence (in the box below it) a special status.
4) there is an integer q for which n = qm. This clause introduces q, another new symbol constrained to be an integer.  The clause imposes a restraint on m, n and q, that they satisfy the equation n = qm. But we know this only in the scope of the word Definition, which ends at the end of the sentence.  Once we read the word Theorem we no longer know that q exists, much less that it satisfies the constraint.  Indeed, the statement of the definition means that one way to prove the theorem is to find an integer q for which n = qm. This is not stated explicitly, and indeed the reader would be wrong to draw the conclusion that in what follows the theorem will be proved in this way. (In fact it will in this example, but the author could have done some other kind of proof. )
5) Theorem The placement of the word “Theorem” here announces that the next paragraph is a mathematical statement and that the statement has been proved.  In real time the statement was proved long before this discourse was written, but in terms of reading the text in order, it has not yet been proved.
6) Let m, n and p be integers, We are starting a new context, in which we know that m, n and p are all  integers.  This changes that status of m and n, which were variables used in the preceding paragraph, but now all previous constraints are discarded. We are starting over with m, n, and p.  We are also starting what the reader must recognize as the hypotheses of a conditional sentence, since that affects the context in a very precise way.
7) with m and n nonzero. Now m and n are nonzero.  Note that in the previous paragraph n was not constrained to be nonzero.  Between the words “Let” and “with” in the current sentence, neither were constrained to be nonzero.
8 ) and n divides p.  More new constraints:  m divides n and n divides p.
9) Then m divides p.   The word “then” signals that we are starting the conclusion of the conditional sentence.  It makes a claim that m divides p whenever the conditions in the hypothesis are true.  Because it is the conclusion, it has a different status from the assumptions that m divides n and n divides p.   We can’t treat m as if it divides p even though this sentence says it does.  All we know is that the author is claiming that m divides p if the hypotheses are true, and we expect (because the next word is “Proof”) that this claim will shortly be proved.
10) Proof

 This starts a new paragraph.  It does not necessarily wipe out the context.  If the proof is going to be by the direct method (assume hypothesis, prove conclusion) — as it does — then it will still be true that m and n are nonzero integers,  m divides n and n divides p.
11) By definition of divides, there are integers q and q′ for which n = qm and p = q’n .

 Since this proof starts by stating the hypothesis of the definition of “divides”, we now know that we are using the direct method, and that q and q’ are new symbols that we are to assume satisfy the equations  n = qm and p = q’n.   The phrase “by definition of divides” tells us (because the definition was given previously) that there are such integers, so in effect this sentence chooses q and q′ so that  n = qm and p = q’n.  The reader probably knows that there is only one choice for each of q and q′ but in fact that claim is not being made here.  Note that m, n and p are not new symbols – they still fall within the scope of the previous paragraph, so we still know that  m divides n and n divides p. If the proof were by contradiction, we would not know that.
12) We must prove that there is an integer  q” for which p = q”n q’’ is introduced by this sentence and is constrained by the equation. The scope of this sentence is just this sentence. The existence of  q’’ and the constraint on it do not exist in the context after the sentence is finished.  However, the constraints previously imposed on m, n, p, q and q’ do continue.
13) But  p = q’n = q’qm This is a claim about p, q, q′, m and n.  The equations are justified by certain preceding sentences but this justification is not made explicit.
14) so let q” = q’q We are establishing a new variable q″ in the context.   Now we put another constraint on it, namely q” = q’q.  It is significant that a variable named q″ was introduced once before, in the reference to the definition of divides.  A convention of mathematical discourse tells you to expect the author to establish that it fits the requirement of the definition. This condition is triggered by using the same symbol q″ both here and in the definition.
15) Then p = q”n This is an assertion about p, q″ and n, justified (but not explicitly) by the claim that p = q’n = q’qm.
16) The proof is now complete, although no statement asserts that.

I have several comments to make about this kind of analysis that are (mostly) not included in the abstractmath article.

a) This is supposed to be what goes through an experienced mathematician’s head while they are reading the proof.  Mostly subconsciously.  Linguists (as in Reference [1]) seem to think something like this takes place in your mind when you read any text, but it gets much denser in mathematical text.  Computer scientists analyze the operation of subprograms in this way, too.

b) Comment (a) is probably off the mark.  With a short proof like that, I get a global picture of the proof as my eyes dart back and forth over the various statements in the proof.  Now, I am a grasshopper: I read math stuff by jumping back and forth trying to understand the structure of the argument.  I do this both locally in a short proof and also globally when reading a long article or book:  I page through to find the topic I want and then jump back and forth finding the meanings of words and phrases I don’t understand.

c) I think most mathematicians are either grasshoppers or they are not good readers and they simply do not learn math by reading text.  I would like feedback on this.

d) If (a) is incorrect, should I omit this example from abstractmath?  I don’t think so.  My experience in teaching tells me that

  1. some students think this is perfectly obvious and why would I spend time constructing the example?,
  2. others are not aware that this is going on in their head and they are amazed to realize that it is really happening,
  3. and still others do not understand how to read proofs and when you tell them this sort of thing goes on in your head they are terminally intimidated.  (“Terminally” in the sense that they dye their hair black and become sociology majors.  They really do.)  Is that bad?  Well, I don’t think so.  I would like to hear arguments on the other side.

e) Can you figure out why item 8 of the analysis is labeled as “8 )” instead of “8)”?

Time is running out. I have other comments to make which must wait for a later post.

References

G. Chierchia and S. McConnell-Ginet (1990), Meaning and Grammar. The MIT Press.

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