Category Archives: proof

exposition, math, proof, representations, understanding math

Modules for mathematical objects

2013/01/15 SixWingedSeraph Leave a comment

A recent article in Scientific American mentions discusses the idea that concepts are represented in the brain by clumps of neurons. Other neuroscientists have proposed that each concept is distributed among millions of neurons, or that each concept corresponds to one neuron.

I have written many posts about the idea that:

Each mathematical concept is embodied in some kind of module in the brain.
This idea is a useful metaphor for understanding how we think about mathematical objects.
You don't have to know the details of the method of storage for this metaphor to be useful.
The metaphor clears up a number of paradoxes and conundrums that have agitated philosophers of math.

The SA article inspired me to write about just how such a module may work in some specific cases.

Integers

Mathematicians normally thinks of a particular integer, say $42$, as some kind of abstract object, and the decimal representation "42" as a representation of the integer, along with XLII and 2A$_{16}$. You can visualize the physical process like this:

The mathematician has a module Int (clump of neurons or whatever) that represents integers, and a module FT that represents the particular integer $42$.
There is some kind of asymmetric three-way connection from FT to Int and a module EO (for "element of" or "IS_A").
That the connection is "asymmetric" means that the three modules play different roles in the connection, meaning something like "$42$ IS_A Integer"
The connection is a physical connection, not a sentence, and when FT is alerted ("fired"?), Int and EO are both alerted as well.
That means that if someone asks the mathematician, "Is $42$ an integer?", they answer immediately without having to think about it — it is a random access concept like (for many people) knowing that September has 30 days.
The module for $42$ has many other connections to other modules in the brain, and these connections vary among mathematicians.

The preceding description gives no details about how the modules and interconnections are physically processed. Neuroscientists probably would have lots of ideas about this (with no doubt considerable variation) and would criticize what I wrote as misrepresenting the physical details in some ways. But the physical details are their job, not mine. What I claim is that this way of thinking makes it plausible to view abstract objects and their properties and relationships as physical objects in the brain. You don't have to know the details any more than you have to know the details of how a rainbow works to see it (but you know that a rainbow is a physical phenomenon).

This way of thinking provides a metaphor for thinking about math objects, a metaphor that is plausibly related to what happens in the real world.

Students

A student may have a rather different representation of $42$ in the brain. For one thing, their module for $42$ may not distinguish the symbol "42" from the number $42$, which is an abstract object. As a result they ask questions such as, "Is $42$ composite in hexadecimal?" This phenomenon reveals a complicated situation.

People think they are talking about the same thing when in fact their internal modules for that thing may be very differently connected to other concepts in their brain.
Mathematicians generally share many more similarities in their modules for $42$ than people in general do. When they differ, the differences may be of the sort that one of them is a number theorist, so knows more about $42$ (for example, that it is a Catalan number) than another mathematician does. Or has read The Hitchhiker's Guide to the Galaxy.
Mathematicians also share a stance that there are right and wrong beliefs about mathematical objects, and that there is a received method for distinguishing correct from erroneous statements about a particular kind of object. (I am not saying the method always gives an answer!).
Of course, this stance constitutes a module in the brain.
Some philosophers of education believe that this stance is erroneous, that the truth or falsity of statements are merely a matter of social acceptance.
In fact, the statements in purple are true of nearly all mathematicians.
The fact that the truth or falsity of statements is merely a matter of social acceptance is also true, but the word "merely" is misleading.
The fact is that overwhelming evidence provided by experience shows that the "received method" (proof) for determining the truth of math statements works well and can be depended on. Teachers need to convince their students of this by examples rather that imposing the received method as an authority figure.

Real numbers

A mathematician thinks of a real number as having a decimal representation.

The representation is an infinitely long list of decimal digits, together with a location for the decimal point. (Ignoring conventions about infinite strings of zeroes.)
There is a metaphor that you can go along the list from left to right and when you do you get a better approximation of the "value" of the real number. (The "value" is typically thought of in terms of the metaphor of a point on the real line.)
Mathematicians nevertheless think of the entries in the decimal expansion of a real number as already in existence, even though you may not be able to say what they all are.
There is no contradiction between the points of view expressed in the last two bullets.
Students frequently do not believe that the decimal entries are "already there". As a result they may argue fiercely that $.999\ldots$ cannot possibly be the same number as $1$. (The Wikipedia article on this topic has to be one of the most thoroughly reworked math articles in the encyclopedia.)

All these facts correspond to modules in mathematicians' and students' brains. There are modules for real number, metaphor, infinite list, decimal digit, decimal expansion, and so on. This does not mean that the module has a separate link to each one of the digits in the decimal expansion. The idea that there is an entry at every one of the infinite number of locations is itself a module, and no one has ever discovered a contradiction resulting from holding that belief.

References

Brain cells for Grandmother, by Rodrigo Quian Quiroga, Itzhak Fried and Christof Koch. Scientific American, February 2013, pages 31ff.

Gyre&Gimble posts on modules

Notes on Viewing

This post uses Mat hJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

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Freezing a family of functions

2011/11/11 SixWingedSeraph 1 Comment

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra1.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Some background

Generally, I have advocated using all sorts of images and metaphors to enable people to think about particular mathematical objects more easily.
In previous posts I have illustrated many ways (some old, some new, many recently using Mathematica CDF files) that you can provide such images and metaphors, to help university math majors get over the abstraction cliff.
When you have to prove something you find yourself throwing out the images and metaphors (usually a bit at a time rather than all at once) to get down to the rigorous view of math [1], [2], [3], to the point where you think of all the mathematical objects you are dealing with as unchanging and inert (not reacting to anything else). In other words, dead.
The simple example of a family of functions in this post is intended to give people a way of thinking about getting into the rigorous view of the family. So this post uses image-and-metaphor technology to illustrate a way of thinking about one of the basic proof techniques in math (representing the object in rigor mortis so you can dissect it). I suppose this is meta-math-ed. But I don’t want to think about that too much…
This example also illustrates the difference between parameters and variables. The bottom line is that the difference is entirely in how we think about them. I will write more about that later.

A family of functions

This graph shows individual members of the family of functions $ y=a\sin\,x$ for various values of $a$ . Let’s look at some of the ways you can think about this.

Each choice of “shows the function for that value of the parameter $a$ “. But really, it shows the graph of the function, in fact only the part between $x=-4$ and $x= 4$ .
You can also think of it as showing the function changing shape as $a$ changes over time (as you slide the controller back and forth).

Well, you can graph something changing over time by introducing another axis for time. When you graph vertical motion of a particle over time you use a two-dimensional picture, one axis representing time and the other the height of the particle. Our representation of the function $y=a\sin\,x$ is a two-dimensional object (using its graph) so we represent the function in 3-space, as in this picture, where the slider not only shows the current (graph of the) function for parameter value $a$ but also locates it over $a$ on the $z$ axis.

The picture below shows the surface given by $y=a\sin\,x$ as a function of both variables $a$ and $x$ . Note that this graph is static: it does not change over time (no slide bar!). This is the family of functions represented as a rigorous (dead!) mathematical object.

If you click the “Show Curves” button, you will see a selection of the curves in middle diagram above drawn as functions of $x$ for certain values of $a$ . Each blue curve is thus a sine wave of amplitude $a$ . Pushing that button illustrates the process going on in your mind when you concentrate on one aspect of the surface, namely its cross-sections in the $x$ direction.

Reference [4] gives the code for the diagrams in this post, as well as a couple of others that may add more insight to the idea. Reference [5] gives similar constructions for a different family of functions.

References

Rigorous view in abstractmath.org
Representations II: Dry Bones (post)
Representations III: Rigor and Rigor Mortis (post)
FamiliesFrozen.nb.
AnotherFamiliesFrozen.nb (Mathematica file showing another family of functions)

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Templates in mathematical practice

2010/03/11 Charles Wells 4 Comments

This post is a first pass at what will eventually be a section of abstractmath.org. It’s time to get back to abstractmath; I have been neglecting it for a couple of years.

What I say here is based mainly on my many years of teaching discrete mathematics at Case Western Reserve University in Cleveland and more recently at Metro State University in Saint Paul.

Beginning abstract math

College students typically get into abstract math at the beginning in such courses as linear algebra, discrete math and abstract algebra. Certain problems that come up in those early courses can be grouped together under the notion of (what I call) applying templates [note 0]. These are not the problems people usually think about concerning beginners in abstract math, of which the following is an incomplete list:

Reasoning from axioms
Encapsulating processes (Tall et al, 2000).
Semantic contamination (abstractmath)
Translating between mathematical English and logic (abstractmath)

The students’ problems discussed here concern understanding what a template is and how to apply it.

Templates can be formulas, rules of inference, or mini-programs. I’ll talk about three examples here.

The template for quadratic equations

The solution of a real quadratic equation of the form ${ax^2+bx+c=0}$ is given by the formula

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

This is a template for finding the roots of the equations. It has subtleties.

For example, the numerator is symmetric in ${a}$ and ${c}$ but the denominator isn’t. So sometimes I try to trick my students (warning them ahead of time that that’s what I’m trying to do) by asking for a formula for the solution of the equation ${a+bx+cx^2=0}$ . The answer is

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2c}$

I start writing it on the board, asking them to tell me what comes next. When we get to the denominator, often someone says “ ${2a}$ ”.

The template is telling you that the denominator is 2 times the coefficient of the square term. It is not telling you it is “ ${a}$ ”. Using a template (in the sense I mean here) requires pattern matching, but in this particular example, the quadratic template has a shallow incorrect matching and a deeper correct matching. In detail, the shallow matching says “match the letters” and the deep matching says “match the position of the letters”.

Most of the time the quadratic being matched has particular numbers instead of the same letters that the template has, so the trap I just described seldom occurs. But this makes me want to try a variation of the trick: Find the solution of ${3+5x+2x^2=0}$ . Would some students match the textual position (getting ${a=3}$ ) instead of the functional position (getting ${a=5}$ )? [Note [0]). If they did they would get the solutions ${(-1,-\frac{2}{3})}$ instead of ${(-1,-\frac{3}{2})}$ .

Substituting in algebraic expressions have other traps, too. What sorts of mistakes would students have solving ${3x^2+b^2x-5=0}$ ?

Most students on the verge of abstract math don’t make mistakes with the quadratic formula that I have described. The thing about abstract math is that it uses more sophisticated templates

subject to conditions
with variations
with extra levels of abstraction

The template for proof by induction

This template gives a method of proof of a statement of the form ${\forall{n}\mathcal{P}(n)}$ , where ${\mathcal{P}}$ is a predicate (presumably containing ${n}$ as a variable) and ${n}$ varies over positive integers. The template says:

Goal: Prove ${\forall{n}\mathcal{P}(n)}$ .

Method:

Prove ${\mathcal{P}(1)}$
For an arbitrary integer ${n>1}$ , assume ${\mathcal{P}(n)}$ and deduce ${\mathcal{P}(n+1)}$ .

For example, to prove ${\forall n (2^n+1\geq n^2)}$ using the template, you have to prove that ${2^2+1\geq 1^1}$ , and that for any ${n>1}$ , if ${2^n+1\geq n^2}$ , then ${2^{n+1}+1\geq (n+1)^2}$ . You come up with the need to prove these statements by substituting into the template. This template has several problems that the quadratic formula does not have.

Variables of different types

The variable ${n}$ is of type integer and the variable ${\mathcal{P}}$ is of type predicate [note 0]. Having to deal with several types of variables comes up already in multivariable calculus (vectors vs. numbers, cross product vs. numerical product, etc) and they multiply like rabbits in beginning abstract math classes. Students sometimes write things like “Let ${\mathcal{P}=n+1}$ ”. Multiple types is a big problem that math ed people don’t seem to discuss much (correct me if I am wrong).

Free and bound

The variable ${n}$ occurs as a bound variable in the Goal and a free variable in the Method. This happens in this case because the induction step in the Method originates as the requirement to prove ${\forall n(\mathcal{P}(n)\rightarrow\mathcal{P}(n+1))}$ , but as I have presented it (which seems to be customary) I have translated this into a requirement based on modus ponens. This causes students problems, if they notice it. (“You are assuming what you want to prove!”) Many of them apparently go ahead and produce competent proofs without noticing the dual role of ${n}$ . I say more power to them. I think.

The template has variations

You can start the induction at other places.
You may have to have two starting points and a double induction hypothesis (for ${n-1}$ and ${n}$ ). In fact, you will have to have two starting points, because it seems to be a Fundamental Law of Discrete Math Teaching that you have to talk about the Fibonacci function ad nauseam.
Then there is strong induction.

It’s like you can go to the store and buy one template for quadratic equations, but you have to by a package of templates for induction, like highway engineers used to buy packages of plastic French curves to draw highway curves without discontinuous curvature.

The template for row reduction

I am running out of time and won’t go into as much detail on this one. Row reduction is an algorithm. If you write it up as a proper computer program there have to be all sorts of if-thens depending on what you are doing it for. For example if want solutions to the simultaneous equations

2x+4y+z	=	1
x+2y	=	0
x+2y+4z	=	5

you must row reduce the matrix

2	4	1	1
1	2	0	0
1	2	4	5

(I haven’t yet figured out how to wrap this in parentheses) which gives you

1	2	0	0
0	0	1	0
0	0	0	1

This introduces another problem with templates: They come with conditions. In this case the condition is “a row of three 0s followed by a nonzero number means the equations have no solutions”. (There is another condition when there is a row of all 0’s.)

It is very easy for the new student to get the calculation right but to never sit back and see what they have — which conditions apply or whatever.

When you do math you have to repeatedly lean in and focus on the details and then lean back and see the Big Picture. This is something that has to be learned.

What to do, what to do

I have recently experimented with being explicit about templates, in particular going through examples of the use of a template after explicitly stating the template. It is too early to say how successful this is. But I want to point out that even though it might not help to be explicit with students about templates, the analysis in this post of a phenomenon that occurs in beginning abstract math courses

may still be accurate (or not), and
may help teachers teach such things if they are aware of the phenomenon, even if the students are not.

Notes

Many years ago, I heard someone use the word “template” in the way I am using it now, but I don’t recollect who it was. Applied mathematicians sometimes use it with a meaning similar to mine to refer to soft algorithms–recipes for computation that are not formal algorithms but close enough to be easily translated into a sufficiently high level computer language.
In the formula ${ax^2+bx+c}$ , the “ ${a}$ ” has the first textual position but the functional position as the coefficient of the quadratic term. This name “functional position” has nothing to do with functions. Can someone suggest a different name that won’t confuse people?
I am using “variable” the way logicians do. Mathematicians would not normally refer to “ ${\mathcal{P}}$ ” as a variable.
I didn’t say anything about how templates can involve extra layers of abstract. That will have to wait.

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Naive proofs

2009/10/23 Charles Wells 3 Comments

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

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.
(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.
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.)
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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Mastering a proof

2009/09/12 Charles Wells 3 Comments

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:

Mastering the proof of a theorem in a textbook so that you can explain it in any desired amount of detail…
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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Grasshoppers and linear proofs

2009/09/09 Charles Wells 7 Comments

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

some students think this is perfectly obvious and why would I spend time constructing the example?,
others are not aware that this is going on in their head and they are amazed to realize that it is really happening,
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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How "math is logic" ruined math for a generation

2009/04/03 Charles Wells 3 Comments

Mark Meckes responded to my statement

But it seems to me that this sort of thinking has mostly resulted in people thinking philosophy of math is merely a matter of logic and set theory. That point of view has been ruinous to the practice of math.

with this comment:

I may be misreading your analysis of the second straw man, but you seem to imply that “people thinking philosophy of math is merely a matter of logic and set theory” has done great damage to mathematics. I think that’s quite an overstatement. It means that in practice, mathematicians find philosophy of mathematics to be irrelevant and useless. Perhaps philosophers of mathematics could in principle have something to say that mathematicians would find helpful but in practice they don’t; however, we’re getting along quite well without their help.

On the other hand, maybe you only meant that people who think “philosophy of math is merely a matter of logic and set theory” are handicapped in their own ability to do mathematics. Again, I think most mathematicians get along fine just not thinking about philosophy.

Mark is right that at least this aspect of philosophy of math is irrelevant and useless to mathematicians. But my remark that the attitude that “philosophy of math is merely a matter of logic and set theory” is ruinous to math was sloppy, it was not what I should have said. I was thinking of a related phenomenon which was ruinous to math communication and teaching.

By the 1950’s many mathematicians adopted the attitude that all math is is theorem and proof. Images, metaphors and the like were regarded as misleading and resulting in incorrect proofs. (I am not going to get into how this attitude came about). Teachers and colloquium lecturers suppressed intuitive insights and motivations in their talks and just stated the theorem and went through the proof.

I believe both expository and research papers were affected by this as well, but I would not be able to defend that with citations.

I was a math student 1959 through 1965. My undergraduate calculus (and advanced calculus) teacher was a very good teacher but he was affected by this tendency. He knew he had to give us intuitive insights but he would say things like “close the door” and “don’t tell anyone I said this” before he did. His attitude seemed to be that that was not real math and was slightly shameful to talk about. Most of my other undergrad teachers simply did not give us insights.

In graduate school I had courses in Lie Algebra and Mathematical Logic from the same teacher. He was excellent at giving us theorem-proof lectures, much better than most teachers, but he never gave us any geometric insights into Lie Algebra (I never heard him say anything about differential equations!) or any idea of the significance of mathematical logic. We went through Killing’s classification theorem and Gödel’s incompleteness theorem in a very thorough way and I came out of his courses pleased with my understanding of the subject matter. But I had no idea what either one of them had to do with any other part of math.

I had another teacher for several courses in algebra and various levels of number theory. He was not much for insights, metaphors, etc, but he did do well in explaining how you come up with a proof. My teacher in point set topology was absolutely awful and turned me off the Moore Method forever. The Moore method seems to be based on: don’t give the student any insights whatever. I have to say that one of my fellow students thought the Moore method was the best thing since sliced bread and went on to get a degree from this teacher.

These dismal years in math teaching lasted through the seventies and perhaps into the eighties. Apparently now younger professors are much more into insights, images and metaphors and to some extent into pointing out connections with the rest of math and science. Since I have been retired since 1999 I don’t have much exposure to the newer generation and I am not sure how thoroughly things have changed.

One noticeable phenomenon was that category theorists (I got into category theory in the mid seventies) were very assiduous in lectures and to some extent in papers in giving motivation and insight. It may be that attitudes varied a lot between different disciplines.

This Dark Ages of math teaching was one of the motivations for abstractmath.org. My belief is that not only should we give the students insights, images and metaphors to think about objects, and so on, but that we should be upfront about it: Tell them what we are doing (don’t just mutter the word “intuitive”) and point out that these insights are necessary for understanding but are dangerous when used in proofs. Tell them these things with examples. In every class.

My other main motivation for abstractmath.org was the way math language causes difficulties. But that is another story.

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math, proof, representations, understanding math

Proofs without dry bones

2009/03/24 Charles Wells 3 Comments

I have discussed images, metaphors and proofs in math in two ways:

(A) A mathematical proof

This example comes from Fauconnier, Mappings in Thought and Language, Cambridge Univ. Press, 1997. I discuss it in the Handbook, pages 46 and 153. See the Wikipedia article on conceptual blending.

(B) Rigor and rigor mortis

The following is quoted from a previous post here. See also the discussion in abstractmath.

When we are trying to understand or explain math, we may use various kinds of images and metaphors about the subject matter to construct a colorful and rich representation of the mathematical objects and processes involved. I described some of these briefly here. They can involve thinking of abstract things moving and changing and affecting each other.

When we set out to prove some math statement, we go into what I have called “rigorous mode”. We feel that we have to forget some of the color and excitement of the rich view. We must think of math objects as inert and static. They don’t move or change over time and they don’t interact with other objects or the real world. In other words, pretend that all math objects are dead.

We don’t always go all the way into this rigorous mode, but if we use an image or metaphor in a proof and someone challenges us about it, we may rewrite that part to get rid of the colorful representation and replace it by a calculation or line of reasoning that refers to the math objects as if they were inert and static – dead.

I didn’t contradict myself.
I want to clear up some tension between these two ideas.

The argument in (A) is a genuine mathematical proof, just as it is written. It contains hidden assumptions (enthymemes), but all math proofs contain hidden assumptions. My remarks in (B) do not mean that a proof is not a proof until everything goes dead, but that when challenged you have to abandon some of the colorful and kinetic reasoning to make sure you have it right. (This is a standard mathematical technique (note 1).)

One of the hidden assumptions in (A) is that two monks walking the opposite way on the path over the same interval of time will meet each other. This is based on our physical experience. If someone questions this we have several ways to get more rigorous. One many mathematicians might think of is to model the path as a curve in space and consider two different parametrizations by the unit interval that go in opposite directions. This model can then appeal to the intermediate value theorem to assert that there is a point where the two parametrizations give the same value.

I suppose that argument goes all the way to the dead. In the original argument the monk is moving. But the parametrized curve just sits there. The parametrizations are sets of ordered pairs in R x (R x R x R). Nothing is moving. All is dry bones. Ezekiel has not done his thing yet.

This technique works, I think, because it allows classical logic to be correct. It is not correct in everyday life when things are moving and changing and time is passing.

Avoid models; axiomatize directly
But it certainly is not necessary to rigorize this argument by using parametrizations involving the real numbers. You could instead look at the situation of the monk and make some axioms the events being described. For example, you could presumably make axioms on locations on the path that treat the locations as intervals rather than as points.

The idea is to make axioms that state properties that intervals have but doesn’t say they are intervals. For example that there is a relation “higher than” between locations that must be reflexive and transitive but not antisymmetric. I have not done this, but I would propose that you could do this without recreating the classical real numbers by the axioms. (You would presumably be creating the intuitionistic real numbers.)

Of course, we commonly fall into using the real numbers because methods of modeling using real numbers have been worked out in great detail. Why start from scratch?

About the heading on this section: There is a sense in which “axiomatizing directly” is a way of creating a model. Nevertheless there is a distinction between these two approaches, but I am to confused to say anything about this right now.

First order logic.
It is commonly held that if you rigorize a proof enough you could get it all the way down to a proof in first order logic. You could do this in the case of the proof in (A) but there is a genuine problem in doing this that people don’t pay enough attention to.

The point is you replace the path and the monks by mathematical models (a curve in space) and their actions by parametrizations. The resulting argument calls on well known theorems in real analysis and I have no doubt can be turned into a strict first order logic argument. But the resulting argument is no longer about the monk on the path.

The argument in (A) involves our understanding of a possibly real physical situation along with a metaphorical transference in time of the two walks (a transference that takes place in our brain using techniques (conceptual blending) the brain uses every minute of every day). Changing over to using a mathematical model might get something wrong. Even if the argument using parametrized curves doesn’t have any important flaws (and I don’t believe it does) it is still transferring the argument from one situation to another.

Conclusion:
Mathematical arguments are still mathematical arguments whether they refer to mathematical objects or not. A mathematical argument can be challenged and tested by uncovering hidden assumptions and making them explicit as well as by transferring the argument to a classical mathematical situation.

Note 1. Did you ever hear anyone talking about rigor requiring making images and metaphors dead? This is indeed a standard mathematical technique but it is almost always suppressed, or more likely unnoticed. But I am not claiming to be the first one to reveal it to the world. Some of the members of Bourbaki talked this way. (I have lost the reference to this.)

They certainly killed more metaphors than most mathematicians.

Note 2. This discussion about rigor and dead things is itself a metaphor, so it involves a metametaphor. Metaphors always have something misleading about them. Metametaphorical statements have the potential of being far worse. For example, the notion that mathematics contains some kind of absolute truth is the result of bad metametaphorical thinking.

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Lincoln and proof

2009/02/13 Charles Wells Leave a comment

This is a good day to read John Armstrong’s post on Lincoln and proof.

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abstractmath.org, exposition, Handbook, language of math, math, proof, understanding math

Abstractmath.org after four years

2009/01/23 Charles Wells Leave a comment

I have been working on the abstractmath website for about four years now (with time off for three major operations). Much has been written, but there are still lots of stubs that need to be filled in. Also much of it needs editing for stylistic uniformity, and for filling in details and providing more examples in some hastily written sections that read like outlines. Not to mention correcting errors, which seem to multiply when I am not looking. The website consists of four main parts and some ancillary chapters. I will go into more detail about some of the parts in later articles.

The languages of math. This is a description of mathematical English and the symbolic language of math (which are two different languages!) with an emphasis on the problems they cause people new to abstract math (roughly, math after calculus). At this point, I have completed a fairly thorough edit of the whole chapter that makes it almost presentable. Start with the Introduction.

Proofs. Mathematical proofs are a central problem for abstract math newbies. People interested in abstract math must learn to read and understand proofs. A proof is narrated in mathematical English. A proof has a logical structure. The reader must extract the logical structure from the narrative form. The chapter on proofs gives examples of proofs and discusses the logical structure and its relationship with the narration. The introduction to the chapter on proofs tells more about it.

Understanding math. There are certain barriers to understanding math that are difficult to get over. Mathematicians, math educators and philosophers work on various aspects of these problems and this chapter draws on their work and my own observations as a mathematician and a teacher.

All true statements about a math object must follow from the definition. That sounds clear enough. But in fact there are subtleties about definitions teachers may not tell students about because they are not aware of them themselves. For example, a definition can really mislead you about how to think about a math object.

The section on math objects breaks new ground (in my opinion) about how to think about them. I also discuss representations and models and images and metaphors (which I think is especially important), and in shorter articles about other topics such as abstraction and pattern recognition.

Doing math. This chapter points out useful behaviors and dysfunctional behaviors in doing math, with concrete examples. Beginners need to be told that when proving an elementary theorem they need to rewrite what is to be proved according to the definitions. Were you ever told that? (If you went to a Jesuit high school, you probably were.) Beginners need to be told that they should not try the same computational trick over and over even though it doesn’t work. That they need to look at examples. That they need to zoom in and out, looking at a detail and then the big picture. We need someone to make movies illustrating these things.

These other articles are outside the main organization:
Topic articles. Sets, real numbers, functions, and so on. In each case I talk just a bit about the topic to get the newbie over the initial hump.
Diagnostic examples. Examples chosen to evoke a misunderstanding, with a link to where it is explained. This needs to be greatly expanded.
Attitudes. This explains my point of view in doing abstractmath.org. I expect to rewrite it.

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Integers

Students

Real numbers

References

Some background

A family of functions

References

References

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