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Etymology

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Retire

I was recently asked about the etymology of the English word “retire”(in connection with quitting work).  It comes from Old French “retirer”, compounded from “re” (meaning “back”, a prefix used in Latin) and the Old French verb “tirer” meaning something like “pull” (which comes from a Germanic language, not Latin, and is related to “tier”, but not apparently to “tire”).

Its earliest citations in the Oxford English Dictionary show meanings such as

  • Pull back or retreat from the enemy.
  • To move back for safety or storage (“they retired to their houses”).
  • Leave office or work permanently.

All these meanings appear in print in the 16th century.

What good does it do to know this?  Not much.  You can’t explain the modern meaning of a word knowing the meaning of its ancient roots.

In the case of “retire”, I can make up a story of meanings changing using a chain of metaphors.

  1. “Retirer” in French meant literally “pull back” in the physical sense, for example pulling on a dog’s leash to drag it back so it won’t get into a fight with another dog. This literal meaning has not survived in the English word “retire” (nor, I think, in the French word “retirer”).
  2. In the 12th century (sez the OED without citation) the French word was used to refer to an army pulling back from a battle.  This is clearly a metaphor based on the literal meaning.  In a phrase such as “The Army retired from battle” it has become intransitive, but perhaps people once said things like “The General retired the Army from battle”.  Note that in modern English we could use the exact same metaphor with “pull back”: “The General pulled the Army back from battle”, although “withdrew” would be more common.
  3. Now someone comes along and uses the metaphor “going to work is like being in a battle”, and says things like “He retired from his job”.   This happened in English before 1533 and the usage has survived to this day.  It is probably the commonest meaning of the word “retire” now.

Now all that is a story I made up.  It is plausible, but it might have happened in a different way.  It is not at all likely we will discover the workings of metaphors in the minds of people who lived 600 years ago.  (Conceivably someone could have written down their thoughts about the word “retire” and it will be discovered in an odd subcrypt of Durham Cathedral and some linguist would get very excited, but I could win the lottery, too).

That’s why knowing the original literal meaning of the roots of a modern English word really means nothing about the modern meaning.  There could have been many steps along the way where a metaphorical usage became the standard meaning, then someone took the standard meaning and used it in another metaphor, maybe many times.  And metaphors aren’t the only method.  Words can change meaning because of misunderstanding, specialization, generalization, use in secret languages that become public, and so on.

I didn’t include etymology in the Handbook, mainly for this reason.  But there are certain mathematical words where knowing the metaphor or even the literal meaning can be of help.  I’ll write about that in a separate article.

 

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Unless

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Mark Meckes recently wrote (private communication):

I’m teaching a fairly new transition course at Case this term, which involves explicitly teaching students the basics of mathematical English along with the obvious things like logic and proof techniques.  I had a student recently ask about how to interpret “A unless B”.  After a fairly lively discussion in class today, we couldn’t agree on the truth table for this statement, and concluded in the end that “unless” is best avoided in mathematical writing.  I checked the Handbook of Mathematical Discourse to see if you had anything to say about it there, but there isn’t an entry for it.  So, are you aware of a standard interpretation of “unless” in mathematical English?

I did not consider  “unless” while writing HMD.   What should be done to approach a subject like this is to

  • think up examples  (preferably in a bull session with other mathematicians) and try to understand what they mean logically, then
  • do an extensive research of the mathematical literature to see if you can find examples that do and do not correspond  with your tentative understanding.  (Usually you find other uses besides the one you thought of, and sometimes you will discover that what you came up with is completely wrong.)  

What follows is an example of this process.

I can think of three possible meanings for “P unless Q”:

1.  “P if and only if not Q”,
2.  “not Q implies P”
3.  “not P implies Q”.

An example that satisfies (1) is “x^2-x is positive unless 0 \leq x \leq 1“.  I have said that specific thing to my classes — calculus students tend not to remember that the parabola is below the line y=x on that interval. (And that’s the way you should show them — draw a picture, don’t merely lecture.  Indeed, make them draw a picture.)

An example of (2) that is not an example of (1) is “x^2-x is positive unless x = 1/2“.  I don’t think anyone would say that, but they might say “x^2-x is positive unless, for example, x = 1/2“.  I would say that is a correct statement in mathematical English.  I guess the phrase “for example” translates into telling you that this is a statement of form “Q implies not P”, where Q is now “x = 1/2”.   Using the contrapositive, that is equivalent to “P implies not Q”, but that is neither (2) nor (3).

An example of (3) that is not an example of (1) is “x^2-x is positive unless -1 < x < 1“.  I think that any who said that (among math people) would be told that they are wrong, because for example (\frac{-1}{2})^2-\frac{-1}{2} = \frac{3}{4}.  That reaction amounts to saying that (3) is not a correct interpretation of “P unless Q”.

Because of examples like these, my conjecture is that “P unless Q” means “P if and only if not Q”.  But to settle this point requires searching for “unless” in the math literature and seeing if you can find instances where “P unless Q” is not equivalent to “P if and only if not Q”.  (You could also see what happens with searching for “unless” and “example” close together.)

Having a discussion such as the above where you think up examples can give you a clue, but you really need to search the literature.  What I did with the Handbook is to search JStor, available online at Case.  I have to say I had definite opinions about several usages that were overturned during the literature search. (What “brackets” means is an example.)

My proxy server at Case isn’t working right now but when I get it repaired I will look into this question.

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A tiny step towards killing string-based math

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I discussed endographs of real functions in my post  Endographs and cographs of real functions.  Endographs of finite functions also provide another way of thinking about functions, and I show some examples here.  This is not a new idea; endographs have appeared from time to time in textbooks, but they are not used much, and they have the advantage of revealing some properties of a function instantly that cannot be seen so easily in a traditional graph or cograph.

In contrast to endographs of functions on the real line, an endograph of a finite function from a set to itself contains all the information about the function.  For real functions, only some of the arrows can be shown; you are dependent on continuity to interpolate where the infinite number of intermediate arrows would be, and of course, it is easy to produce a function, with, say, small-scale periodicity, that the arrows would miss, so to speak.  But with an endograph of a finite function, WYSIATI (what you see is all there is).

Here is the endograph of a function.  It is one function.  The graph has four connected components.

You can see immediately that it is a permutation  of the set \{1,2,3,4,5,6\}, and that it is involution (a permutation f for which f f=\text{id}).  In cycle notation, it is the permutation (1 2)(5 6), and the connected components of the endograph correspond to the cycle structure.

Here is another permutation:

You can see that to get f^n=\text{id} you would have to have n=6, since you have to apply the 3-cycle 3 times and the transposition twice to get the identity.   The cycle structure (1 2 4)(0 3) tells you this, but you have to visualize it acting to see that.  The endograph gives the newbie a jumpstart on the visualization.  “The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols” (Brett Victor).   This is an argument for insisting that this permutation is the endograph, and the abstract string of symbols (1 2 4)(0 3) is a representation of secondary importance.  [See Note 1.]

Here is the cograph of the same function.  It requires a bit of visualization or tracing arrows around to see its cycle structure.

If I had rearranged the nodes like this

the cycle structure would be easier to see.  This does not indicate as much superiority of the endograph metaphor over the cograph metaphor as you might think:  My endograph code [Note 2] uses Mathematica’s graph-displaying algorithm, which automatically shows cycles clearly.   The cograph code that I wrote specifies the placement of the nodes explicitly, so I rearranged them to obtain the second cograph above using my knowledge of the cycle structure.

The following endographs of functions that are not permutations exhibit the general fact that the graph of a finite function consists of cycles with trees attached.   This structure is obvious from the endographs, and it is easy to come up with a proof of this property of finite functions by tracing your finger around the endographs.

This is the endograph of the polynomial 2 n^9+5 n^8+n^7+4 n^6+9 n^5+1 over the finite field of 11 elements.

Here is another endograph:

I constructed this explicitly by writing a list of rules, and then used Mathematica’s interpolating polynomial to determine that it is given by the polynomial

6 x^{16}+13 x^{15}+x^{14}+3 x^{13}+10 x^{12}+5 x^{11}\\ +14 x^{10}+4 x^9+9 x^8+x^7+14 x^6\\ +15 x^5+16 x^4+14 x^3+4 x^2+15 x+11

in GF[17].

Quite a bit is known about polynomials over finite fields that give permutations.  For example there is an easy proof using interpolating polynomials that a polynomial that gives a transposition must have degree q-2.  The best reference for this stuff is Lidl and Niederreiter, Introduction to Finite Fields and their Applications

The endographs above raise questions such as what can you say about the degree or coefficients of a polynomial that gives a digraph like the function f below that is idempotent (f f=f).  Students find idempotence vs. involution difficult to distinguish between.  Digraphs show you almost immediately what is going on.  Stare at the digraph below for a bit and you will see that if you follow f to a node and then follow  it again you stay where you are (the function is the identity on its image).  That’s another example of the insights you can get from a new metaphor for a mathematical object.

The following function is not idempotent even though it has only trivial loops.  But the digraph does tell you easily that it satisfies f^4=f^3.

Notes

[1] Atish Bagchi and I have contributed to this goal in Graph Based Logic and Sketches, which gives a bare glimpse of the possibility of considering that the real objects of logic are diagrams and their limits and morphisms between them, rather than hard-to-parse strings of letters and logical symbols.  Implementing this (and implementing Brett Victor’s ideas) will require sophisticated computer support.  But that support is coming into existence.  We won’t have to live with string-based math forever.

[2] The Mathematica notebook used to produce these pictures is here.  It has lots of other examples.

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Computable algebraic expressions in tree form

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Invisible algebra

  1. An  expression such as $4(x-2)=6$ has an invisible abstract structure.  In this simple case it is

using the style of presenting trees used in academic computing science.  The parentheses are a clue to the structure; omitting them results in  $4x-2=6$, which has the different structure

By the time students take calculus they supposedly have learned to perceive and work with this invisible structure, but many of them still struggle with it.  They have a lot of trouble with more complex expressions, but even something like $\sin x + y$ gives some of them trouble.

Make the invisible visible

The tree expression makes the invisible structure explicit. Some math educators such as Jason Dyer and Bret Victor have experimented with the idea of students working directly with a structured form of an algebraic expression, including making the structured form interactive.

How could the tree structure be used to help struggling algebra students?

1) If they are learning on the computer, the program could provide the tree structure at the push of a button. Lessons could be designed to present algebraic expressions that look similar but have different structure.

2) You could point out things such as:

a) “inside the parentheses pushes it lower in the tree”
b) “lower in the tree means it is calculated earlier”

3) More radically, you could teach algebra directly using the tree structure, with the intention of introducing the expression-as-a-string form later.  This is analogous to the use of the initial teaching alphabet for beginners at reading, and also the use of shape notes to teach sight reading of music for singing.  Both of these methods have been shown to help beginners, but the ITA didn’t catch on and although lots of people still sing from shape notes (See Note 1) they are not as far as I know used for teaching in school.

4) You could produce an interactive form of the structure tree that the student could use to find the value or solve the equation.  But that needs a section to itself.

Interactive trees

When I discovered the TreeForm command in Mathematica (which I used to make the trees above), I was inspired to use it and the Manipulate command to make the tree interactive.


This is a screenshot of what Mathematica shows you.  When this is running in Mathematica, moving the slide back and forth causes the dependent values in the tree also change, and when you slide to 3.5, the slot corresponding to $ 4(x-2)$ becomes 6 and the slot over “Equals” becomes “True”:

As seen in this post, these are just screen shots that you can’t manipulate.  The Mathematica notebook Expressions.nb gives the code for this and lets you experiment with it.  If you don’t have Mathematica available to you, you can still manipulate the tree with the slider if you download the CDF form of the notebook and open it in Mathematica CDF Player, which is available free here.  The abstractmath website has other notebooks you may want to look at as well.

Moving the slider back and forth constitutes finding the correct value of x by experiment.  This is a peculiar form of bottom-up evaluation.   With an expression whose root node is a value rather than an equation, wiggling the slider constitutes calculating various values with all the intermediate steps shown as you move it.  Bret Victor ‘s blog shows a similar system, though not showing the tree.

Another way to use the tree is to arrange to show it with the calculated values blank.  (The constants and the labels showing the operation would remain.)   The student could start at the top blank space (over Times)  and put in the required value, which would obviously have to be 6 to make the space over Equals change to “True”.  Then the blank space over Plus would have to be 1.5 in order to make multiplying it by 4 be 6.  Then the bottom left blank space would have to be 3.5 to make it equal to 1.5 when -2 is added.  This is top down evaluation.

You could have the student enter these numbers in the blank spaces on the computer or print out the tree with blank spaces and have them do it with a pencil.  Jason Dyer’s blog has examples.

Implementation

My example code in the notebook is a kludge.  If you defined a  special VertexRenderingFunction for TreeForm in Mathematica, you could create a function that would turn any algebraic expression into a manipulatable tree with a slider like the one above (or one with blank spaces to be filled in).  [Note 2]. I expect I will work on that some time soon but my main desire in this series of blog posts is to through out ideas with some Mathematica code attached that others might want to develop further. You are free to reuse all the Mathematica code and all my blog posts under the Creative Commons Attribution – ShareAlike 3.0 License.  I would like to encourage this kind of open-source behavior.

Notes

1. Including me every Tuesday at 5:30 pm in Minneapolis (commercial).

2. There is a problem with Equals.  In the hacked example above I set the increment the value jumps by when the slider is moved to 0.1, so that the correct value 3.5 occurs when you slide.  If you had an equation with an irrational root this would not work.  One thing that should work is to introduce a fuzzy form of Equals with the slide-increment smaller that the latitude allowed in the fuzzy Equals.

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Playing with Riemann Sums

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When I was a junior taking advanced calculus under Fuzzy Vance at Oberlin College, I sat one night staring at the book trying to understand how the Riemann Sum of a function “converged” to the value of the integral.  The book said that it converges as the mesh goes to 0, and that any Riemann Sum was included in the convergence theorem, with any choice of partition, even irregular, and with any choice of points-to-evaluate-at in each subinterval.  [Note 1.]

I had a satori [Note 2].  I felt like the guy in the ads who sits in front of his new ultrafast computer with the wind blowing his hair back and bracing himself by holding onto the desk.  (My hair was dark then but I certainly was not wearing a tie.)

That convergence theorem was talking about something BIG.

I visualized a Cloud of Riemann Sums floating around and swerving closer to the Right Answer as their meshes decreased.

A Riemann Sum has a lot of parameters:

  • Its mesh.  This can be any positive real number.
  • Its choice of subintervals. Any positive integer!  There can be billions of subintervals.
  • And, ye gods, the individual choice of each evaluation point for each subinterval in each Riemann Sum. 

Those are three independent parameters, except for the constraint imposed by the mesh on each choice of subintervals.  [Note 3].

I tell my students that we have to zoom in and zoom out [Reference 2] from a problem.  When we zoom out a complicated structure is thought of as a point in a certain relationship with other structures-as-points.  Then to understand something we zoom in (selectively) to see the details that make it work.  What I remember from my satori is that I didn’t visualize them as points but rather as little blurs, sort of like the blurs in Mumford’s red book [Reference 3], which I think was the first non-constipated math text I had ever seen.

Riemann Sums in Mathematica

In the nineties, I was on a grant to create Mathematica programs for students, and one of the notebooks I created allowed you to easily exhibit Riemann sums with various parameters.  I also included code that would show a cloud.

Below is a cloud.  It is a plot of the values of 300 Riemann sums for \int_0^{\pi} \sin x \,dx.  They have randomly chosen meshes from 0 to \pi/2 and the subintervals and individual evaluation points for each subinterval are also chosen randomly.

The cloud below is a plot of the values of 300 Riemann sums for the area of the upper right quarter circle of radius 2 with center at origin.  Its meshes range from 0 to 1, and other properties are similar to the one above.  The vertical spread of the points is considerably bigger,  presumably because of the vertical tangent line at the right hand end of the integral.

When you click on the code for either of these you get a different cloud with the same parameters.

You can access the notebook containing the code for this via Abmath Gate.    Be sure to read the ReadMe file.

Notes

[1] This was 1961.  Of course the book didn’t say things such as “with any choice of points-to-evaluate-at”.  It said what it had to say in stilted academic prose which required reading it two or three times before understanding it.  Academic prose is much better these days.  See Reference [1].

I was quite good at reading complicated prose. My ACT scores were a tad higher in English or Language or whatever it is called that they were in Math.  With the Internet, math exposition should do much more with pictures, interactive things, and lots of examples (which don’t waste paper now).  But that is another diatribe…

[2] This is a snooty word for lightbulb flashing over your head.  Every once in awhile I give in to the temptation to use some obscure word to impress people as to the variety of things I know about.  Teachers, don’t do this to your students.  Other professors are fair game.

[3] The same choice of subinterval can correspond to many different meshes, if your definition of mesh requires only that each subinterval be narrower than the mesh, rather than requiring that the mesh be the size of the biggest subinterval.  (I had never thought about that until I wrote this.)

[4] The Mathematica Demonstrations website has several other notebooks that exhibit Riemann Sums.

References

[1]  The Revolution in Technical Exposition II, post on this blog.

[2]  Zooming and Chunking in abmath.

[3] D. Mumford, The Red Book of Varieties and Schemes (second expanded ed.), Springer Lecture Notes in Math 1358, Springer-Verlag, Berlin, 1999.   (I have not seen this edition.  What I remember is the Red Book as it was in the 1967 Algebraic Geometry Summer School at Bowdoin.  I hope the smudges survive in the new version.  As I remember the smudges were bigger for points that were more generic or something like that.  Those smudges caused me a kind of sartori, too.)

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Endograph and cograph of real functions

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This post is covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the work provided you follow the requirements of the license.

Introduction

In the article Functions: Images and Metaphors in abstractmath I list a bunch of different images or metaphors for thinking about functions. Some of these metaphors have realizations in pictures, such as a graph or a surface shown by level curves. Others have typographical representations, as formulas, algorithms or flowcharts (which are also pictorial). There are kinetic metaphors — the graph of {y=x^2} swoops up to the right.

Many of these same metaphors have realizations in actual mathematical representations.

Two images (mentioned only briefly in the abstractmath article) are the cograph and the endograph of a real function of one variable. Both of these are visualizations that correspond to mathematical representations. These representations have been used occasionally in texts, but are not used as much as the usual graph of a continuous function. I think they would be useful in teaching and perhaps even sometimes in research.

A rough and unfinished Mathematica notebook is available that contains code that generate graphs and cographs of real-valued functions. I used it to generate most of the examples in this post, and it contains many other examples. (Note [1].)

The endograph of a function

In principle, the endograph (Note [2]) of a function {f} has a dot for each element of the domain and of the codomain, and an arrow from {x} to {f(x)} for each {x} in the domain. For example, this is the endograph of the function {n\mapsto n^2+1 \pmod 11} from the set {\{0,1,\ldots,10\}} to itself:


“In principle” means that the entire endograph can be shown only for small finite functions. This is analogous to the way calculus books refer to a graph as “the graph of the squaring function” when in fact the infinite tails are cut off.

Real endographs

I expect to discuss finite endographs in another post. Here I will concentrate on endographs of continuous functions with domain and codomain that are connected subsets of the real numbers. I believe that they could be used to good effect in teaching math at the college level.

Here is the endograph of the function {y=x^2} on the reals:

I have displayed this endograph with the real line drawn in the usual way, with tick marks showing the location of the points on the part shown.

The distance function on the reals gives us a way of interpreting the spacing and location of the arrowheads. This means that information can be gleaned from the graph even though only a finite number of arrows are shown. For example you see immediately that the function has only nonnegative values and that its increase grows with {x}.(See note [3]).

I think it would be useful to show students endographs such as this and ask them specific questions about why the arrows do what they do.

For the one shown, you could ask these questions, probably for class discussion rather that on homework.

  • Explain why most of the arrows go to the right. (They go left only between 0 and 1 — and this graph has such a coarse point selection that it shows only two arrows doing that!)
  • Why do the arrows cross over each other? (Tricky question — they wouldn’t cross over if you drew the arrows with negative input below the line instead of above.)
  • What does it say about the function that every arrowhead except two has two curves going into it?

Real Cographs

The cograph (Note [4] of a real function has an arrow from input to output just as the endograph does, but the graph represents the domain and codomain as their disjoint union. In this post the domain is a horizontal representation of the real line and the codomain is another such representation below the domain. You may also represent them in other configurations (Note [5]).

Here is the cograph representation of the function {y=x^2}. Compare it with the endograph representation above.

Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.

More examples

Absolute value function

Arctangent function

Notes

[1] This website contains other notebooks you might find useful. They are in Mathematica .nb, .nbp, or .cdf formats, and can be read, evaluated and modified if you have Mathematica 8.0. They can also be made to appear in your browser with Wolfram CDF Player, downloadable free from Wolfram site. The CDF player allows you to operate any interactive demos contained in the file, but you can’t evaluate or modify the file without Mathematica.

The notebooks are mostly raw code with few comments. They are covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the code following the requirements of the license. I am making the files available because I doubt that I will refine them into respectable CDF files any time soon.

[2] I call them “endographs” to avoid confusion with the usual graphs of functions — — drawings of (some of) the set of ordered pairs {x,f(x)} of the function.

[3] This is in contrast to a function defined on a discrete set, where the elements of the domain and codomain can be arranged in any old way. Then the significance of the resulting arrangement of the arrows lies entirely in which two dots they connect. Even then, some things can be seen immediately: Whether the function is a cycle, permutation, an involution, idempotent, and so on.

Of course, the placement of the arrows may tell you more if the finite sets are ordered in a natural way, as for example a function on the integers modulo some integer.

[4] The text [1] uses the cograph representation extensively. The word “cograph” is being used with its standard meaning in category theory. It is used by graph theorists with an entirely different meaning.

[5] It would also be possible to show the domain codomain in the usual {x-y} plane arrangement, with the domain the {x} axis and the codomain the {y} axis. I have not written the code for this yet.

References

[1] Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2003.

[2] Martin Flashman’s website contains many exampls of cographs of functions, which he calls mapping diagrams.

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Skills needed for learning languages and math

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Learning a language involves a variety of skills, and so does learning math. Some skills are apparently needed for both, but others are distinct.

Learning languages and learning math

Some years ago I sat in on a second year college Spanish class. Most of the other students were ages 18-24. The students showed a wide spectrum of ability.

  • Some were quite fluent and conversed easily. Others struggled to put a sentence together.
  • Some had trouble with basic grammar, for example adjective-noun agreement (number and gender). I would not have thought second year students would do that. Some also had trouble with verbs. Spanish verbs are generally difficult, but second year students shouldn’t have trouble with using “canta” with singular subjects and “cantan” with plural ones.
  • Some had trouble reading aloud, stumbling over pronunciation, such putting the accent in the right place in real time and pronouncing some letters correctly (“ll”, “e”, intervocalic “s”). The rules for accent and pronouncing letters are very easy in Spanish, and I was surprised that second year students would have difficulty with them. But the speech of most of them sounded good to me.

I can read Spanish pretty well, but have had very little practice speaking or writing it. I comprehend some of what they say on Univision (soap operas are particularly easy, but I still miss more than half of it), but then I am hard of hearing. I used to have a reasonable ability to speak and understand street German; judging from experience I think it would come back rapidly if we went to live in a German-speaking city again. I can easily read math papers written in Spanish or in German, but I couldn’t come close to giving a math lecture in either language.

Some find learning rules of pronunciation that are different from English very hard, like the Spanish students I mentioned above. I know that some people can’t keep “ei” and “ie” straight in German, and some Russian students find it hard to get used to the Cyrillic alphabet. I find that part of language learning easy. I also find learning grammar and using it in real time fairly easy. I have more difficulty remembering vocabulary.

Learning the new sounds of a language is an entirely different problem from learning the rules of pronunciation.

Mathematical ability

Some difficulties that students have with the symbolic language of math [1] are probably the same kind of difficulties that language students have with learning another language.

When I have taught elementary logic, I usually have a scattering of students who can’t keep the symbols {\land} and {\lor} separate. (See Note [a].) Some even have the same trouble with intersection and union of sets. This is sort of like differentiating “ie” and “ei” in German, except that the latter distinction runs into cognitive dissonance [2] caused by the usual English pronunciation.

Of course, both language students and math students have immense problems with cognitive dissonance in areas other than symbol-learning. For example, many technical words in math have meanings different from ordinary English usage, such as “if”, “group”, and “category”. Language students have difficulties with “false friends” such as “Gift”, which is the German word for “poison”, and very common words such as prepositions, which can have several different translations into English depending on context — and many prepositions in other European languages look like English prepositions. (Note [b]).

On the other hand, some types of mathematical learning seem to involve problems language students don’t run into.

Substitution, for example, appears to me to cause conceptual difficulties that are not like anything in learning language. But I would like to hear examples to the contrary.

If {f(x) = x^2+3x+1}, then {f(x+1)= (x+1)^2+3(x+1)+1}. Is there anything like this in natural languages? And simplifying this to {x^4+5 x^2+5} is not like anything in natural language either — is it?

Is there anything in learning natural languages that is like thinking of an element of a set? Or like the two-level quantification involved in understanding the definition of continuity?

Is there anything in learning math that involves the same kind of difficulty as learning to pronounce a new sound in another language? (Well, making a speech sound involves moving parts of your mouth in three dimensions, and some people find visualizing 3D shapes difficult. But that seems like a stretch to me).

A proposal for investigation

Students show a wide variety of conceptual skills. Some skills seem to be required both in learning mathematics and in learning a foreign language. Others are different. Also, there is a difference between learning school math and learning abstract math at the college level (Note [c]).

TOPIC FOR RESEARCH

  • Identify the types of concept formation that learning a foreign language and learning math have in common.
  • Determine if “being good at languages” and “being good at mathematics” are correlated at the high school level.
  • Ditto for college-level abstract math.

Undoubtedly math teachers and language teachers have written about certain specific issues of the sort I have discussed, but I think we need a systematic comparative investigation of skills involved in the tasks of learning languages and learning math.

I have made proposals for research concerning various other questions with math ed, particularly in connection with linguistics. I will install a new topic “Proposals for research” in my “List of categories” (on the left side of the screen under “Recent posts”) and mark this and other articles that contain such proposals.

Notes

[a]. That is why, in the mathematical reasoning sections of abmath, for example [3], I use the usual English wordings of mathematical assertions instead of systematically using logical symbolism. For many students, introducing symbols and then immediately using them to talk about the subtleties of meaning and usage puts a difficult burden on some of the students. (I do define the symbols in asides).

This may not be the right thing to do. If a student finds it hard to learn to use symbols easily and fluently, should they be studying math?

[b]. I once knew a teenage German who spoke pretty good English, but he could not bear to use the English possessive case. That’s because German young people (assuming I understand this correctly) hate to say things like “Das Auto meines Vaters” and instead say “Das Auto von meinem Vater”. Unfortunately this resulted in his saying in English “The car from my father”, “The girlfriend from my brother” and so on…

[c]. I have been concerned primarily with understanding the difficulties students have when starting to study abstract math after they have had calculus. I have seen many students ace calculus and flunk abstract algebra or logic. There is a wall to fall off of there. The only organization I know of concerned with this is RUME, although it is involved with college calculus as well as what comes after.

References

[1] The symbolic language of math.

[2] Cognitive dissonance.

[3] Conditional assertions.

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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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The Mathematical Definition of Function

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Introduction

This post is a completely rewritten version of the abstractmath article on the definition of function. Like every part of abstractmath, the chapter on functions is designed to get you started thinking about functions. It is no way complete. Wikipedia has much more complete coverage of mathematical functions, but be aware that the coverage is scattered over many articles.

The concept of function in mathematics is as important as any mathematical idea. The mathematician’s concept of function includes the kinds of functions you studied in calculus but is much more abstract and general. If you are new to abstract math you need to know:

  • The precise meaning of the word “function” and other concepts associated with functions. That’s what this section is about.
  • Notation and terminology for functions. (That will be a separate section of abstractmath.org which I will post soon.)
  • The many different kinds of functions there are. (See Examples of Functions in abmath).
  • The many ways mathematicians think about functions. The abmath article Images and Metaphors for Functions is a stub for this.

I will use two running examples throughout this discussion:

  • {F} is the function defined on the set {\left\{1,\,2,3,6 \right\}} as follows: {F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1}. This is a function defined on a finite set by explicitly naming each value.
  • {G} is the real-valued function defined by the formula {G(x)={{x}^{2}}+2x+5}.

Specification of function

We start by giving a specification of “function”. (See the abstractmath article on specification.) After that, we get into the technicalities of the definitions of the general concept of function.

Specification: A function {f} is a mathematical object which determines and is completely determined bythe following data:

  • {f} has a domain, which is a set. The domain may be denoted by {\text{dom }f}.
  • {f} has a codomain, which is also a set and may be denoted by {\text{cod }f}.
  • For each element {a} of the domain of {f}, {f} has a value at {a}, denoted by {f(a)}.
  • The value of {f} at {a} is completely determined by {a} and {f} .
  • The value of {f} at {a} must be an element of the codomain of {f}.

The operation of finding {f(a)} given {f} and {a} is called evaluation.

Examples

  • The definition above of the finite function {F} specifies that the domain is the set {\left\{1,\,2,\,3,\,6 \right\}}. The value of {F} at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that {F(2) = 3}. The codomain of {F} is not specified, but must include the set {\{1,2,3\}}.
  • The definition of {G} above gives the value at each element of the domain by a formula. The value at 3, for example, is {G(3)=3^2+2\cdot3+5=20}. The definition does not specify the domain or the codomain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is {{\mathbb R}}. The codomain must include all real numbers greater than or equal to 4. (Why?)

Comment: The formula above that defines the function G in fact defines a function of complex numbers (even quaternions).

Definition of function

In the nineteenth century, mathematicians realized that it was necessary for some purposes (particularly harmonic analysis) to give a mathematical definition of the concept of function. A stricter version of this definition turned out to be necessary in algebraic topology and other fields, and that is the one I give here.

To state this definition we need a preliminary idea.

The functional property

A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).

Examples

  • The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, since no two different pairs have the same first coordinate. It is true that two of them have the same second coordinate, but that is irrelevant.
  • The set {\{(1,2), (2,4), (3,2), (2,8)\}} does not have the functional property. There are two different pairs with first coordinate 2.
  • The graphs of functions in beginning calculus have the functional property.
  • The empty set {\emptyset} has the functional property .

Example: Graph of a function defined by a formula

The graph of the function {G} given above has the functional property. The graph is the set

\displaystyle \left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in {\mathbb R} \right\}.

If you repeatedly plug in one real number over and over, you get out the same real number every time. Example:

  • if {x = 0}, then {{{x}^{2}}+2x+5=5}.  You get 5 every time you plug in 0.
  • if {x = 1}, then {{{x}^{2}}+2x+5=8}.
  • if {x =-2}, then {{{x}^{2}}+2x+5=5}.

This set has the functional property because if {x} is any real number, the formula {{{x}^{2}}+2x+5} defines a specific real number. (This description of the graph implicitly assumes that {\text{dom } G={\mathbb R}}.)  No other pair whose first coordinate is {-2} is in the graph of {G}, only {(-2, 5)}. That is because when you plug {-2} into the formula {{{x}^{2}}+2x+5}, you get {5} every time. Of course, {(0, 5)} is in the graph, but that does not contradict the functional property. {(0, 5)} and {(-2, 5)} have the same second coordinate, but that is OK.

How to think about the functional property

The point of the functional property is that for any pair in the set of ordered pairs, the first coordinate determines what the second one is. That’s why you can write “{G(x)}” for any {x } in the domain of {G} and not be ambiguous.

Mathematical definition of function

A function{f} is a mathematical structure consisting of the following objects:

  • A set called the domain of {f}, denoted by {\text{dom } f}.
  • A set called the codomain of {f}, denoted by {\text{cod } f}.
  • A set of ordered pairs called the graph of { f}, with the following properties:
    • {\text{dom } f} is the set of all first coordinates of pairs in the graph of {f}.
    • Every second coordinate of a pair in the graph of {f} is in {\text{cod } f} (but {\text{cod } f} may contain other elements).
    • The graph of {f} has the functional property. Using arrow notation, this implies that {f:A\rightarrow B}.

Examples

  • Let {F} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} and define {A = \{1, 2, 3, 5\}} and {B = \{2, 4, 8\}}. Then {F:A\rightarrow B} is a function.
  • Let {G} have graph {\{(1,2), (2,4), (3,2), (5,8)\}} (same as above), and define {A = \{1, 2, 3, 5\}} and {C = \{2, 4, 8, 9, 11, \pi, 3/2\}}. Then {G:A\rightarrow C} is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in {C}, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
  • Let {H} have graph {\{(1,2), (2,4), (3,2), (5,8)\}}. Then {H:A\rightarrow {\mathbb R}} is a function.

According to the definition of function, {F}, {G} and {H} are three different functions.

Identity and inclusion

Suppose we have two sets A and B with {A\subseteq B}.

  • The identity function on A is the function {{{\text{id}}_{A}}:A\rightarrow A} defined by {{{\text{id}}_{A}}(x)=x} for all{x\in A}. (Many authors call it {{{1}_{A}}}).
  • The inclusion function from A to B is the function {i:A\rightarrow B} defined by {i(x)=x} for all {x\in A}. Note that there is a different function for each pair of sets A and B for which {A\subseteq B}. Some authors call it {{{i}_{A,\,B}}} or {\text{in}{{\text{c}}_{A,\,B}}}.

Remark The identity function and an inclusion function for the same set A have exactly the same graph, namely {\left\{ (a,a)|a\in A \right\}}.

Graphs and functions

  • If {f} is a function, the domain of {f} is the set of first coordinates of all the pairs in {f}.
  • If {x\in \text{dom } f}, then {f(x)} is the second coordinate of the only ordered pair in {f} whose first coordinate is {x}.

Examples

The set {\{(1,2), (2,4), (3,2), (5,8)\}} has the functional property, so it is the graph of a function. Call the function {H}. Then its domain is {\{1,2,3,5\}} and {H(1) = 2} and {H(2) = 4}. {H(4)} is not defined because there is no ordered pair in H beginning with {4} (hence {4} is not in {\text{dom } H}.)

I showed above that the graph of the function {G}, ordinarily described as “the function {G(x)={{x}^{2}}+2x+5}”, has the functional property. The specification of function requires that we say what the domain is and what the value is at each point. These two facts are determined by the graph.

Other definitions of function

Because of the examples above, many authors define a function as a graph with the functional property. Now, the graph of a function {G} may be denoted by {\Gamma(G)}.  This is an older, less strict definition of function that doesn’t work correctly with the concepts of algebraic topology, category theory, and some other branches of mathematics.

For this less strict definition of function, {G=\Gamma(G)}, which causes a clash of our mental images of “graph” and “function”. In every important way except the less-strict definition, they ARE different!

A definition is a device for making the meaning of math technical terms precise. When a mathematician think of “function” they think of many aspects of functions, such as a map of one shape into another, a graph in the real plane, a computational process, a renaming, and so on. One of the ways of thinking of a function is to think about its graph. That happens to be the best way to define the concept of function.  (It is the less strict definition and it is a necessary concept in the modern definition given here.)

The occurrence of the graph in either definition doesn’t make thinking of a function in terms of its graph the most important way of visualizing  it. I don’t think it is even in the top three.

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Mental, Physical and Mathematical Representations

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For a given mathematical object, a mathematician may have:

  • A mental representation of the object. This can be a metaphor, a mental image, or a kinesthetic understanding of the object.
  • A physical representation of the object. This may be a (physical) picture or drawing or three-dimensional model of the object.
  • A mathematical definition and one or more mathematical representations for the object. Such a representation is itself a mathematical object.

The boldface things in this list are related to each other in lots of ways, and they are fuzzy and overlap and don’t include every phenomenon connected with a math object.

I have written about these things ([1], [2], [3], [4]). So have lots of other people. In this post I summarize these ideas. I expect to write about particular examples later on and will use this as a reference.

Two Examples

The following examples point out a few of the relationships between the ideas in boldface above. There is much more to understand.

Function as black box

The idea that a function is a black box or machine with input and output is a metaphor for a function.

A is a metaphor for B means that A and B are cognitively pasted together in such a way that the behavior of A is in many ways like the behavior of B. Such a thing is both useful and dangerous, dangerous because there will be ways in which A behaves that suggest inappropriate ideas about B.

The function as machine is a good metaphor: for example functional composition involves connecting the output of one machine to the input of another, and the inverse function is like running the machine backward.

The function as machine is a bad metaphor: For example, it is wrong to think you could build a machine to calculate any given function exactly. But you can still imagine such a machine, given by a specification (it outputs the value of the function at a given input) and then, in your imagination, connecting the input of one to the output of another must perforce calculate the composite of the corresponding functions.

Like any metaphor, this is a mental representation. That means the metaphor has a physical instantiation in your brain. So a metaphor has a physical representation.

Different people won’t have quite the same concept of a particular metaphor. So a metaphor will have lots of slightly different physical representations, but mathematicians form a community, and communication between mathematicians fine-tunes the different physical instantiations so that they correspond more closely to each other. This is the sense in which mathematical objects have a shared existence in a community as Reuben Hersh has suggested.

A function is a mathematical object, which can be rigorously specified as a set of ordered pairs together with a domain and a codomain. There is a cognitive relationship between the concepts of function as math object and function as black box with input and output.

Triangle

A triangle can be drawn, or created on a computer and a physical image printed out. You may also have a mental image of the triangle.

The physical and the mental images are not the same thing, but they are definitely related. The relationship is mediated by the neuronal circuitry behind your retinas, which performs a highly sophisticated transformation of the pixels on your retina into an organized physical structure in your brain, connected to various other neurons.

This circuitry exists because it helps us get a useful understanding of the world through our eyes. So a picture of a triangle takes advantage of pre-existing neuron structure to generate a useful mental representation that helps us understand and prove things about triangles.

This mental representation also lives in a community of mathematician. Like any community, it has subgroups with “dialects” — varying understanding of representation.

For example, a mathematician who looks at the triangle below sees a triangle that looks like a right triangle. A student sees a triangle that is a right triangle.

This is “sees” in the sense of what their brain reports after all that processing. The mathematician’s brain connects the “triangle I am seeing” module (in their brain) to the “looks like a right triangle” module, but does not connect it to the “is a right triangle” module because they don’t see any statement in the surrounding text that it is a right triangle. The student, on the other hand, fallaciously makes the connection to “is a right triangle” directly.

In some sense, a student who does not make that connection directly is already a mathematician.

A triangle also exists as a mathematical object in your and my brain. It is described by a formal mathematical definition. The pictures of triangles you see above do not fit this definition. For one thing, the line segments in the pictures have thickness. But the pictures trigger a reaction in your neurons that causes your brain to cognitively paste together the line segments in the drawing to the segments required by the formal definition. This is a kind of metaphor of concrete-to-abstract that connects drawings to math objects that mathematicians use all the time.

Note that this “concrete-to-abstract metaphor” itself has a physical existence in your brain.  It drops, for example, the property of thickness that the line segments in the drawing have when matching them (in the metaphor) with the line segments in the corresponding abstract triangle.  On the other hand, it preserves the sense the all three angles in the triangle are acute.  The abstract mathematical concept of triangle (the generic triangle) has no requirement on the angles except that they add up to pi.

Summary

The discussions above describe a few of the complex and subtle relationships that exist between

  • Mental representations of math objects
  • Physical representations of math objects
  • Formally defined math objects and their formally defined representations.

I have purported to discuss how mathematics is understood (especially in connection with language) in several articles and a book but only a few of the relationships I just described are mentioned in any of those articles. Perhaps one or two things I said caused you to react: “Actually, that’s obviously true but I never thought of it before”. (Much the way I had mathematicians in the ’60’s tell me, “I see what you mean that addition is a function of two variables, but I never thought of it that way before”.) (I was a brash category theorist wannabe then.)

A lot of research has been done on understanding math, and some research has been done on mathematical discourse. But what has been done has merely exposed the fin of the shark.

References

[1] Images and metaphors (in abstractmath).

[2] Representations and Models (in abstractmath).

[3] Mathematical Concepts (previous blog).

[4] Mental Representations in Math (previous blog).

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