Welcome to the new site for Gyre&Gimble. It now resides on my own website, abstractmath.org. This will enable me to post interactive documents using Mathematica CDF files, and gives me more control in other ways as well.
Warning: The links to other G&G posts in pre-2010 posts are wrong. They still take you to the old site. The old site will remain available until I can repair this problem.
If you are using an RSS feed, you need to change it (see META to the right). Keep in touch!
Send to KindleIn my post Playing with Riemann Sums I showed a couple of clouds of points, each representing a particular Riemann sum for a particular function. I have extended the code in a couple of ways.
The new code is in the Mathematica notebook and CDF file called MoreRiemann in the Mathematica section of abstractmath. The .nb form is a Mathematica Notebook, which requires Mathematica to run and allows you to manipulate the objects and change the code in the notebook as you wish. In particular, you can rerun the commands generating the clouds to get a new random result. The .cdf file contains the same material and can be viewed using Mathematica CDF Player, which is available free here. Both files have several other examples besides the ones shown below.
As always, my code is one-time code to show the ideas, but it is available freely via the Creative Commons Attribution – ShareAlike 3.0 License. I hope people will feel free to develop it further for use in teaching or for their own purposes.
Below is a cloud for 
The next cloud shows random blue dots with the same meaning as above. The red dots are Riemann sums with uniform subintervals evaluated at midpoints. 
Possible discussion question for both of the clouds above:
The following cloud is like the cloud above with the addition of green dots representing uniform partitions evaluated at the left endpoint or right endpoint. (But the mesh scale is extended, giving different proportions to the picture.)
Of course the left endpoint gives the upper sums and the right endpoint gives the lower sums.
(These are idle questions I haven't thought about myself, but I'll bet they could be turned into good projects in analysis classes.)
Here is a cloud for 
The cloud below is for the same integral but uses uniform subintervals for the midpoint and adds green points for both the left endpoint and the right endpoint of uniform subinterval.
This is a similar picture for 
And finally, here is 
Send to KindleI 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 
Here is another permutation:
You can see that to get 
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 
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
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 
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 
The following function is not idempotent even though it has only trivial loops. But the digraph does tell you easily that it satisfies 
[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.
Send to KindleThis is a rant about technical problems with creating abstractmath.org. You will not get great new insights into mathematical language. You will not get any purty pictures, either. But if you read the following anyway and have suggestions, I would appreciate them no end.
I have long been frustrated with the process I used to create articles for abstractmath. The process has been this: I write the article in Word using MathType, then use their facility for generating an html file that uses pictures (stored in separate files) for the more complicated math expressions, then load them into the abstractmath website.
There are many good and bad things about this. Two of the most aggravating:
For the past few weeks I have experimented with generating PDF documents using PDFLaTeX. I did this with the section called Functions: Notation and Terminology, the only one that is posted so far. Posting it required 45 minutes of fixing links in other articles by hand.
Creating that section in TeX was a pain. I used GrindEQ Math to convert the original document to TeX, which required a great deal of preprocessing (mostly to recover the links, but for some formatting things too) and postprocessing to fix many many things. I also had to recreate the sidebars by hand; I used wrapfig. Wrapfig does not work well.
In the process of converting this and some other files that I may post later, I created a bunch of macros in the Word macro language (horrible, although in principle I believe in OOP) and the WindEdt macro language, which is pretty good. Even with the macros, it is a lot of labor to do the conversion.
I have decided to abandon the effort. I may post a few more articles that I have already transferred to PDF, but I doubt I will revise any more from scratch.
One thing that has changed since I started doing the conversions was that a recent revision of MathType allows you to type the equations directly in TeX and to toggle back and forth between MathType form and TeX form. Before that you had to select symbols from a palette. That was the single most frustrating thing about using Word with MathType. I type fluently in TeX but I couldn’t use it. Now that I can type the TeX in directly the prospect of editing articles and writing new ones using Word is much less painful.
So one direction I will go in is to revise the articles already on the web using Word and MathType. I also expect to kill a good many of the incomplete or less well thought out ones in favor of links to Wikipedia.
But there is another direction, opened up by Mathematica’s Computable Document files. I have published some experiments with this in the last four posts here. I expect to be able to turn some of the articles in abstractmath.org into computable documents. The reader will have to have to have the free CDF Player on their computer, but if you download it once you have it forever.
Send to KindleThis post has been superseded by the post Demos for graph and cograph of calculus functions.
New Version 6 July 2011
This is a new version of a post originally created on 30 June, 2011. –Charles Wells
When you put the graph and cograph of a function with parameters side by side, interesting things may happen. I have created the files CographExample.nb and CographExample.cdf to illustrate this.
The .nb form is a Mathematica Notebook, which requires Mathematica to run and allows you to manipulate the objects and change the code in the notebook as you wish. The .cdf file contains the same material and can be viewed using Mathematica CDF Player, which is available free here. The CDF Player allows you to change the parameters with the slidebars, so that you can experience the phenomena discusses in the example, but you cannot otherwise modify the file.
You cannot include a Mathematica computable document directly into a Word Press document, so here are screenshots of the cograph example with several different settings of its parameters.
Some Screenshots and sample questions follow. These not in the original version. I added these thanks to encouragement by Sam Alexander.
My idea is that students will manipulate the slider to see what happens and do some algebra on the relations between a, b and x to explain the phenomena that occur.
Questions about the graph (left figure).
G1. Prove that the two straight lines are parallel for any choice of a and b.
G2. When do the red and blue straight lines in the graph coincide? Answer: The lines coincide when 
G3. Find a formula in terms of a and b for the distance between the two straight lines in the graph. Answer: 
Questions about the cograph.
C1. When do the red and blue arrows in the cograph coincide? Answer: Same as answer to Question G1.
C2. In the cograph, for what aand b are the arrow targets for a given choice of x closer together than the arrow sources? Answer: $latex |a|>1$. b is irrelevant.
C3. Manipulate a and b. For some values the blue arrows all cross each other at the same point. (Same question for red arrows.) When does this happen? Answer: When a is negative.
C4. Describe the common point where all the red arrows cross when a is negative. (Same question for blue arrows.) Answer: The point is 
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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.
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.
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.
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.
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.
Send to KindleWhen 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:
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.
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 
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.
[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.
[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.)
Send to KindleThis 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.
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 
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].)
In principle, the endograph (Note [2]) of a function 
“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.
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
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
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.
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 
Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.
Arctangent function 
[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 
[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 
[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.
Send to KindleLearning 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.
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 
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 
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
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.
Send to KindleIn 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.
Let’s look at the graph of the function 
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
for all real numbers . is shown in a location determined by the conventional coordinate system, which uses a straight-line representation of the real numbers with labels and ticks.
. form a parametrized curve in the mathematical sense. This curve has zero thickness. The curve in the physical representation has positive thickness.
) and I “see” some points (the local maximum and the local minimum) whose coordinates I can figure out.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.)
[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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