## Abstractmath

I built my website abstractmath.org during the years 2002 through 2006. After that I made sporadic changes, but medical operations and then teaching courses as an adjunct for a couple of years kept me from making much progress until 2010.

This post is an explanation of the tools I used for abstractmath, what went right and what went wrong, and my plans for redoing the website.

## Methodology

My previous experience in publishing math was entirely with TeX. When I began work on abstractmath, I wanted to produce html files, primarily because they refloated the text when the window width changed. I was thinking of small screens and people wanting to look at several windows at once.

In those days, there was no method of starting with a LaTeX input file and producing an html file that preserved all the math and all the formatting. I have over the years spent many hours trying out various systems that claimed to do it and not found one that did not require major massaging to get the look I wanted. Most of them can cannot implement all LaTeX commands, or even most of the LaTeX formatting commands. (I have not looked at any of these since 2011.)

In contrast, systems such as PDFTeX turn even very complicated (in formatting and in math) LaTeX files into nearly perfect PDF files. Unfortunately, PDF files are a major impediment to having several windows open at once.

### Word and MathType

My solution was to write abstractmath articles using Microsoft Word with MathType, which provides a plugin for Word.

The MathType interface was a very useful expansion of the Equation Editor in Word, and it produced little .gif files that were automatically inserted into the text. MathType also provided a command to create an html file. This file was produced with the usual “_files” folder that contained all the illustrations I had included as well as all the .gif files that MathType created. The html file contained code that put each .gif file in the right place in the typeset text.

That combination worked well. Using Word allowed me tight control over formatting and allowed floating textboxes, which I used freely. They very nicely moved around when you changed the width of the window.

I had used textboxes in my book A Handbook of Mathematical Discourse for apt quotations, additional comments, and (very clever if I say so myself) page indexes. The Handbook is available in several ways:

• Amazon. The citations are not included.
• The Handbook in paper form. A pdf file showing the book as it appears on paper (all the illos, textboxes and page indexes, no hyperlinks), plus all the citations. (This paragraph was modified on 2013-05-02).
• A version with hyperlinks, This includes the citations but omits the boxes and the illustrations, and it has hyperlinks to the citations. The page indexes are replaced by internal hyperlinks.
• The citations.

That book was written in TeX with much massaging using AWK commands. Boxes are much easier to do in Word than they are in TeX, and the html files produced by MathType preserved them quite well. The abmath article on definitions shows boxes used both for side comments and for quotations.

There were some problems with using MathType and Word together. In particular, a longish article would have dozens or hundreds of .gif files, which greatly slowed down uploading via ftp. I now have WebDrive (thanks to CWRU) and that may make it quicker.

## Rot sets in

Without my doing anything at all, the articles on abstractmath began deteriorating. This had several main causes.

• Html was revised over time. Currently it is HTML5.0.
• Browsers changed way they rendered the html. And they had always differed among themselves in some situations.
• Microsoft Word changed the way it generated html.

Two of the more discouraging instances of rot were:

• Many instances of math formulas are now out of line with the surrounding text. This happened without my doing anything. It varies by browser and by when I last revised the article.
• Some textboxes deteriorated. In particular, textboxes generated by newer versions of Word were sometimes nearly illegible. Part of the reason for this is that Word started saving them as images.

## Failed Forays

The main consequence of all this was that I was afraid of trying to revise articles (or complete them) because it would make them harder to read or ugly. So I set out to find new ways to produce abmath articles. This has taken a couple of years, while abmath is a big mess sprawling there on its website. A mostly legible big mess, and most of the links work, but frustrating to its appearance-sensitive author.

### Automatically convert to a new system

My first efforts were to find another system with the property that I could convert my present Word files or html files to the new system without much hand massaging.

I tried converting the Word files to LaTeX input. This was made easier (I thought) because MathType now provided a means for turning all the MathType itty bitty .gif files into LaTeX expressions. I wrote Word macros to convert much of the formatting (italics, bold, subheads, purple prose, and so on) into LaTeX formatting — although I did have to go through the Word text, select each specially formatted piece, and apply the correct macro.

• Converting the Mathype images files to LaTeX caused problems because it messed up the spaces before and after the formulas.
• I worked with great sweat and tears to write a macro to extract the addresses from the links — and failed. If I had presevered I probably would have learned how to do it, and learned a lot of Word macros programming in the process.

The automatic conversion process appeared to require more and more massaging.

I made some attempts at automatically converting the html files that Word generates (instead of the doc files), but they are an enormous mess. They insert a huge amount of code (especialy spans) into the text, making it next to impossible to read the code or find anything.

It was beginning to look like I would have to go to an entirely new system and rewrite all the articles from scratch. This was attractive in one respect: in writing this blog my style has changed and I was seeing lots of things I would say or do differently. I have also changed my mind about the importance of some things, and abmath now has stubs and incomplete articles that ought to be eliminated with references to Wikipedia.

## Go for rewriting

Meanwhile, I was having trouble with Gyre&Gimble. The WordPress editor works pretty well, but two new products came along:

• MathJax was introduced, providing a much better way to use TeX to insert formulas. (Note: MathType recently implemented the use of MathJax into its html output.)

• Mathematica CDF files, which are interactive diagrams that can be inserted directly into html. (My post Improved Clouds has examples.)

Both MathJax and CDF Player require entering links directly in the html code the WordPress editor produces. The WordPress editor trashed the html code I had entered every time I switched back and forth between “visual” (wysiwyg) and html.

I switched to CKEdit, which preserved the html but has a lot of random behavior. I learned to understand some of the behavior but finally gave up. I started writing my blogs in html using the Coffee Cup HTML Editor — that is how I am writing this. Then I paste it into the WordPress editor.

My current plan is to start revising each abmath article in this way:

• Write html code for the special formatting I want, mostly the code that produces the header, but also purple prose and other things. Once done I can use this code for all the abmath articles with little massaging.
• Start with the Word doc file for an article and use MathType to toggle all the MathType-generated gif files into TeX.
• Generate the html file in a way that preserves the TeX code with dollar signs. (There are two ways to do this and I have not made up my mind which to use.)
• Start revising!

I have already begun doing this. My intention is to revise each abstractmath article, post it, and announce the posting on Gyre&Gimble or on Google+. If an article is heavily revised I expect to post it (or parts of it) on Gyre&Gimble. Some of these things will be ready soon.

## Last minute notes

• I used WinEdt, a text editor, to write the Handbook of Mathematical Discourse. It is a powerful html editor, with an extensive macro language that in particular allows rearranging the menus and adding new code to call other applications. It is especially designed for TeX, so is not as convenient as it stands for html. However, its macro language would allow me to convert it to a system that will do most of what Coffee Cup can do. I might do this because Coffee Cup has no macro language and (as far as I can tell) has no way to revise or add to menus.
• It is early days yet, but I am thinking of including pieces of Abstracting Algebra into abstractmath.org.

Posted in abstractmath.org, exposition. Tags:. No Comments »

## Abstracting algebra

This post has been turned into a page on WordPress, accessible in the upper right corner of the screen.  The page will be referred to by all topic posts for Abstracting Algebra.

## Election systems

This post begins a new thread in Gyre&Gimble: Election systems.  In 1968 I created a game called Parlement and ran a few games by mail before being absorbed by family and career.  In it you ran a political party in a parliament in the style of many European countries: The parliament forms a government, votes on bills & budgets, then an election is held where each party runs on its record.

My interest in election systems has continued sporadically since them.  Since the nineties I have been programming in Mathematica and made stabs at implementing various systems for achieving proportionality. Now I expect to devote several posts to Mathematica demos of election and apportionment systems.

## Proportional representation

An elected body is chosen by a list method of proportional representation this way:

• The election is by districts, each electing several members.  In most cases the number of seats each district has in the body is fixed ahead of the election.
• Each voter votes for a party list.
• The proportion of the integer number of representatives that party will have out of the total number of seats alloted to the district is chosen to be near to the proportion of votes the party list receives out of the total votes.
• The method for choosing the number of seats for the party can be any one of many methods proposed in the past 200 or so years.  Only a few methods are actually used in practice.
• Once the number of members for the party is determined, that many persons on the list are chosen according to some method.  There are quite a few different methods used for this.  I will not write about this aspect at all.

This post looks at the method of equal proportions. This method may also be used to apportion the legislature of countries such as the USA that has states or provinces so that each state has a suitably proportionate number of seats.  For most of the history of the USA, that method has been the method of equal proportions, but in the early days other methods were used.

My impression is that the equal proportion method is the most common method used in legislatures elected by the list system, and is also the most common method used for apportioning legislative seats among states or provinces.  There is much information about these things scattered over many articles in Wikipedia, and a close study may prove me wrong about this.

Note: The summary above is oversimplified and leaves out many details.  The references list more details than most people would ever want to know.

## The equal proportions method

Several sites listed in the references describe the equal proportions method in detail.  The method for calculation used in the demo (there are others) works this way:

• Create a list $V$ of weights indexed by the party number.
• For proportional representation for parties, each party starts with $0$ seats and  $V_p$ is initially the number of votes party $p$ has for the district.
• For state representation, each state starts with $1$ seat and the the initial $V_s$ is the number of votes state $s$ receives divided by $\sqrt{2}$.
• Suppose $S$ seats are to be assigned .
• Assign the first one to the party $m$ for which $V_m$ is the maximum of the list $V$.  (In the case of states, this is the first seat after the initial one.)
• Change the list by setting $V_m:=\frac{V_m}{\sqrt{2}}$ ($V_m:=\frac{V_m}{\sqrt{6}}$ in the case of states).
• At the $k$th step, if $n$ is the party for which $V_n$ is the max of the list $V$ in its current state, assign the $k+1$st seat to party $n$ and set $V_n:= \frac{V_n}{\sqrt{u(u+1)}}$ (the geometric mean), where $u$ is the number of seats party $n$ had before the new one was assigned.
• Stop which $k=S$.

## Interactive demos of the method of equal proportions

To manipulate these demos, click on "Enable Dynamics".  At present the demo has a bug that makes the table pink (Mathematica's way of giving an error message). Move any slider and the pink will disappear forever.  The demos are also available on my website: PartySimple.cdf and PartyComparison.cdf.  If you download them onto a machine with CDF player installed and run them the pink table does not happen.  I can't imagine what could cause an error like that only when run embedded in html.

Both demos have the same controls.

• Five sliders labeled n1 through n5 control the number of votes they get in the election.
• The bottom slider controls the number of seats that district gets to elect.
• By moving a slider you control the information it represents.
• By clicking on the plus sign to the right of the slider, you toggle a list of controls below it.  The party vote sliders begin with the controls invisible and the seats slider begins with them visible.

### How the EP method works

The demo below assumes that five parties, numbered 1 through 5, are running to get seats in the elected body.  You may change the votes (Column Votes) received by any party, and also the total number of representatives to be chosen (Seats).

#### Data

• Seats is the total number of representatives assigned to the elected body. It can be changed using the Seats slider.
• Quota is Total votes divided by Seats.  Some systems use slight variations on this quota.
• The Name of the party is given just as a number.  In a suitably fancy demo you might give each party a real name.
• Votes is the number of votes the party receives in the election, controlled by the relevant slider.
• SeatsAsgd  is the number of seats the equal proportions method assigns to that party.
• Weight: This number is $\frac{\text{Votes}}{\sqrt{\text{SeatsAsgd}\times(\text{SeatsAsgd}+1)}}$

#### Playing with the demo

• Start with any settings for the sliders, and press the $+$ button underneath the seat slider. This allots one more seat to the representatives.
• You will see that the party with the largest weight gets the new seat.  That is how the method works: The algorithm starts with no seats and adds one at a time until the correct number of seats is reached.
• The weight is a function of the number of seats allotted to the party.
• Try changing the Votes for a single party, letting the total number of seats remain the same.  Which parts of the data get changed by doing this?  Do you understand why?
• Try changing the votes for all the parties, so that one has most of the votes and the others have only a few votes apiece.  Start with Seats at zero and step the seats up one at a time.  Notice what happens in column SeatsAsgd and what happens to Weight.

### How close to an accurate apportionment does it get?

• Quota is Total votes divided by Seats.  Note how it changes when you move any slider.
• The Name of the party is given just as a number.  In a suitably fancy demo you might give each party a real name.
• Votes is the number of votes the party receives in the election, controlled by the relevant slider.
• SeatsAsgd is the number of seats the equal proportions method assigns to that party, given the total number of Seats allotted.
• SeatsIdeal is the party's Votes divided by Quota.  Note that this is generally close to SeatsAsgd, which is usually but not always either the floor or the ceiling of SeatsIdeal. Try to find a setting where that is not true (hint: Give one party most of the votes.)
• VotesPerSeat is Votes divided by SeatsAsgd.  Compare it to SeatsIdeal.  They are usually close, but can  be quite far off if only a few Seats are assigned.
• Deviation is VotesPerSeat divided by Quota.  This is a relative measure of how far away from exactness the representations of the parties are.

### Playing with the demo

• Move Seats down to 2 or 3.  Notice that the deviations are quite bad, even $40\%$ off sometimes. Move Seats  up and see that deviations get much better.  Can you understand why that happens?
• Note that usually Seats is either the integer just below SeatsIdeal (the floor) or the integer just above it (the ceiling).  This is reasonable and is called "keeping quota".
• Make one or two parties large and the others small, then move the Seats slider around.  You find examples where Seats busts through the ceiling, "breaking quota".  Sometimes Seats is several units bigger than the ceiling.
• Note that if you step Seats up one at a time, the only thing that ever happens is that one party's seats goes up one unit.  Some other common systems cause some other party's seats to go down occasionally when the total seats is incremented.  That obviously never happens with EP.

These demos were designed for people to learn about a concept by experimenting with them.  Such demo should be fairly simple with only choices and displays relevant to what you are trying to show.

You can also build elaborate CDF's. RiemannSums.nb contains a command PlotRiemann which allows for many options showing different kinds of Riemann sums with different options, and you could design a single demo with many buttons, sliders and other gadgets that allow for all sorts of possibilities (but I have not designed such a monster).

I do expect to eventually design a command such as PlotRiemann that does for voting systems something like what PlotRiemann does for Riemann Sums, but the way to do that is to create one feature or option at a time.  I will be doing that and they will result in other election systems demos that I will post here from time to time.  (Promises, promises).

## References

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

To manipulate the demo in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demo is in the Mathematica notebook Equal Proportions.nb.

## Representations of mathematical objects

### MathJax.Hub.Config({ jax: ["input/TeX","output/NativeMML"], extensions: ["tex2jax.js"], tex2jax: { inlineMath: [ ['$','$'] ], processEscapes: true } });

This is a long post. Notes on viewing.

A mathematical object, or a type of math object, is represented in practice in a great variety of ways, including some that mathematicians rarely think of as "representations".

In this post you will find examples and comments about many different types of representations as well as references to the literature. I am not aware that anyone has considered all these different ideas of representation in one place before. Reading through this post should raise your consciousness about what is going on when you do math.

This is also an experiment in exposition.  The examples are discussed in a style similar to the way a Mathematica command is discussed in the Documentation Center, using mostly nonhierarchical bulleted lists. I find it easy to discover what I want to know when it is written in that way.  (What is hard is discovering the name of a command that will do what I want.)

## Types of representations

### Using language

• Language can be used to define a type of object.
• A definition is intended to be precise enough to determine all the properties that objects of that type all have.  (Pay attention to the two uses of the word "all" in that sentence; they are both significant, in very different ways.)
• Language can be used to describe an object, exhibiting properties without determining all properties.
• It can also provide metaphors, making use of one of the basic tools of our brain to understand the world.
• The language used is most commonly mathematical English, a special dialect of English.
• The symbolic language of mathematics (distinct from mathematical English) is used widely in calculations. Phrases from the symbolic language are often embedded in a statement in math English. The symbolic language includes among others algebraic notation and logical notation.
• The language may also be a formal language, a language that is mathematically defined and is thus itself a mathematical object. Logic texts generally present the first order predicate calculus as a formal language.
• Neither mathematical English nor the symbolic language is a formal language. Both allow irregularities and ambiguities.

### Mathematical objects

The representation itself may be a mathematical object, such as:

• A linear representation of a group. Not only are the groups mathematical objects, so is the representation.
• An embedding of a manifold into Euclidean space. A definition given in a formal language of the first order predicate calculus of the property of commutativity of binary operations. (Thus a property can be represented as a math object.)

### Visual representations

A math object can be represented visually using a physical object such as a picture, graph (in several senses), or diagram.

• The visual processing of our brain is our major source of knowledge of the world and takes about a fifth of the brain's processing power.  We can learn many things using our vision that would take much longer to learn using verbal descriptions.  (Proofs are a different matter.)
• When you look at a graph (for example) your brain creates a mental representation of the graph (see below).

### Mental representations

If you are a mathematician, a math object such as "$42$", "the real numbers" or "continuity" has a mental representation in your brain.

• In the math ed literature, such a representation is called "mental image", "concept image", "procept", or "schema".   (The word "image" in these names is not thought of as necessarily visual.)
• The procept or schema describe all the things that come to mind when you think about a particular math object: The definition, important theorems, visual images, important examples, and various metaphors that help you understand it.
• The visual images occuring in a mental schema for an object may themselves be mental representations of physical objects. The examples and theorems may be mental representations of ideas you learned from language or pictures, and so on.  The relationships between different kinds of representations get quite convoluted.

### Metaphors

Conceptual metaphors are a particular kind of mental representation of an object which involve mentally associating some aspects of the objects with some aspects of something else — a physical object, an image, an action or another abstract object.

• A conceptual metaphor may give you new insight into the object.
• It may also mislead you because you think of properties of the other object that the math object doesn't have.
• A graph of a function is a conceptual metaphor.
• When you say that a point on a graph "rises as it goes from left to right" your metaphor is an action.
• When you say that the cosets of a normal subgroup of a group "get along" with the group multiplication, your metaphor identifies a property they have with an aspect of human behavior.

## Properties of representations

A representation of a math object may or may not

• determine it completely
• exhibit some of its properties
• suggest easy proofs of some theorems
• provide a useful way of thinking about it

## Examples of representations

This list shows many of the possibilities of representation.  In each case I discuss the example in terms of the two bulleted lists above. Some of the examples are reused from my previous publications.

### Functions

Example (F1) "Let $f(x)$ be the function defined by $f(x)=x^3-x$."

• This is an expression in mathematical English that a fluent reader of mathematical English will recognize gives a definition of a specific function.
• (F1) is therefore a representation of that function.
• The word "representation" is not usually used in this way in math.  My intention is that it should be recognized as the same kind of object as many other representations.
• The expression contains the formula $x^3-x$.  This is an encapsulated computation in the symbolic language of math. It allows someone who knows basic algebra and calculus to perform calculations that find the roots, extrema and inflection points of the function $f$.
• The word "let" suggests to the fluent reader of mathematical English that (F1) is a definition which is probably going to hold for the next chunk of text, but probably not for the whole article or book.
• Statements in mathematical English are generally subject to conventions.  In a calculus text (F1) would automatically mean that the function had the real numbers as domain and codomain.
• The last two remarks show that a beginner has to learn to read mathematical English.
• Another convention is discussed in the following diatribe.

#### Diatribe

You would expect $f(x)$ by itself to mean the value of $f$ at $x$, but in (F1) the $x$ has the property of a bound variable.  In mathematical English, "let" binds variables. However, after the definition, in the text the "$x$" in the expression "$f(x)$" will be free, but the $f$ will be bound to the specific meaning.  It is reasonable to say that the term "$f(x)$" represents the expression "$x^3-x$" and that $f$ is the (temporary) name of the function. Nevertheless, it is very common to say "the function $f(x)$" to mean $f$.

A fluent reader of mathematical English knows all this, but probably no one has ever said it explicitly to them.  Mathematical English and the symbolic language should be taught explicitly, including its peculiarities such as "the function $f(x)$".  (You may want to deprecate this usage when you teach it, but students deserve to understand its meaning.)

### The positive integers

You have a mental representation of the positive integers $1,2,3,\ldots$.  In this discussion I will assume that "you" know a certain amount of math.  Non-mathematicians may have very different mental representations of the integers.

• You have a concept of "an integer" in some operational way as an abstract object.
• "Abstract object" needs a post of its own. Meanwhile see Mathematical Objects (abstractmath) and the Wikipedia articles on Mathematical objects and Abstract objects.
• You have a connection in your brain between the concept of integer and the concept of listing things in order, numbering them by $1,2,3,\ldots$.
• You have a connection in your brain between the concept of an integer and the concept of counting a finite number of objects.  But then you need zero!
• You understand how to represent an integer using the decimal representation, and perhaps representations to other bases as well.
• Your mental image has the integer "$42"$ connected to but not the same as the decimal representation "42". This is not true of many students.
• The decimal rep has a picture of the string "42" associated to it, and of course the picture of the string may come up when you think of the integer $42$ as well (it does for me — it is a an icon for the number $42$.)
• You have a concept of the set of integers.
• Students need to be told that by convention "the set of integers" means the set of all integers.  This particularly applies to students whose native language does not have articles, but American students have trouble with this, too.
• Your concept of  "the set of integers" may have the icon "$\mathbb{N}$" associated with it.  If you are a mathematician, the icon and the concept of the set of integers are associated with each other but not identified with each other.
• For me, at least, the concept "set of integers" is mentally connected to each integer by the "element of" relation. (See third bullet below.)
• You have a mental representation of the fact that the set of integers is infinite.
• This does not mean that your brain contains an infinite number of objects, but that you have a representation of infinity as a concept, it is brain-connected to the concept of the set of integers, and also perhaps to a proof of the fact that $\mathbb{N}$ is infinite.
• In particular, the idea that the set of integers is mentally connected to each integer does not mean that the whole infinite number of integers is attached in your brain to the concept of the set of integers.  Rather, the idea is a predicate in your brain.  When it is connected to "$42$", it says "yes".  To "$\pi$" it says "No".
• Philosophers worry about the concept of completed infinity.  It exists as a concept in your brain that interacts as a meme with concepts in other mathematicians' brains. In that way, and in that way only (as far as I am concerned) it is a physical object, in particular an object that exists in scattered physical form in a social network.

### Graph of a function

This is a graph of the function $y=x^3-x$:

• The graph is a physical object, either on a screen or on paper
• It is processed by your visual system, the most powerful sensory management system in your brain
• It also represents the graph in the mathematical sense (set of ordered pairs) of the function $y=x^3-x$
• Both the mathematical graph and the physical graph are represented by modules in your brain, which associates the two of them with each other by a conceptual metaphor
• The graph shows some properties of the function: inflection point, going off to infinity in a specific way, and so on.
• These properties are made apparent (if you are knowledgeable) by means of the powerful pattern recognition system in your brain. You see them much more quickly than you can discover them by calculation.
• These properties are not proved by the graph. Nevertheless, the graph communicates information: for example, it suggests that you can prove that there is an inflection point near $(0,0)$.
• The graph does not determine or define the function: It is inaccurate and it does not (cannot) show all of the graph.

### Continuity

Example (C1) The $\epsilon-\delta$ definition of the continuity of a function $f:\mathbb{R}\to\mathbb{R}$ may be given in the symbolic language of math:

A function $f$ is continuous at a number $c$ if $\forall\epsilon(\epsilon\gt0\implies(\forall x(\exists\delta(|x-c|\lt\delta\implies|f(x)-f(c)|\lt\epsilon)))$

• To understand (C1), you must be familiar with the notation of first order logic.  For most students, getting the notation right is quite a bit of work.
• You must also understand  the concepts, rules and semantics of first order logic.
• Even if you are familiar with all that, continuity is still a difficult concept to understand.
• This statement does show that the concept is logically complicated. I don't see how it gives any other intuition about the concept.

Example (C2) The definition of continuity can also be represented in mathematical English like this:

A function $f$ is continuous at a number $c$ if for any $\epsilon\gt0$ and for any $x$ there is a $\delta$ such that if $|x-c|\lt\delta$, then $|f(x)-f(c)|\lt\epsilon$.

• This definition doesn't give any more intuition that (C1) does.
• It is easier to read that (C1) for most math students, but it still requires intimate familiarity with the quirks of math English.
• The fact that "continuous" is in boldface signals that this is a definition.  This is a convention.
• The phrase "For any $\epsilon\gt0$" contains an unmarked parenthetic insertion that makes it grammatically incoherent.  It could be translated as: "For any $\epsilon$ that is greater than $0$".  Most math majors eventually understand such things subconsciously.  This usage is very common.
• Unless it is explicitly pointed out, most students won't notice that  if you change the phrase "for any $x$ there is a $\delta$"  to "there is a $\delta$ for any $x$" the result means something quite different.  Cauchy never caught onto this.
• In both (C1) and (C2), the "if" in the phrase "A function $f$ is continuous at a number $c$ if…" means "if and only if" because it is in a definition.  Students rarely see this pointed out explicitly.

Example (C3) The definition of continuity can be given in a formally defined first order logical theory

• The theory would have to contain function symbols and axioms expressing the algebra of real numbers as an ordered field.
• I don't know that such a definition has ever been given, but there are various semi-automated and automated theorem-proving systems (which I know little about) that might be able to state such a definition.  I would appreciate information about this.
• Such a definition would make the property of continuity a mathematical object.
• An automated theorem-proving system might be able to prove that $x^3-x$ is continuous, but I wonder if the resulting proof would aid your intuition much.

Example (C4) A function from one topological space to another is continuous if the inverse of every open set in the codomain is an open set in the domain.

• This definition is stated in mathematical English.
• In definitions (C1) – (C3), the primitive data are real numbers and the statement uses properties of an ordered field.
• In (C4), the data are real numbers and the arithmetic operations of a topological field, along with the open sets of the field. The ordering is not mentioned.
• This shows that a definition need not mention some important aspects of the structure.
• One marvelous example of this is that  a partition of a set and an equivalence relation on a set are based on essentially disjoint sets of data, but they define exactly the same type of structure.

Example (C4) "The graph of a continuous function can be drawn without picking up the chalk".

• This is a metaphor that associates an action with the graph.
• It is incorrect: The graphs of some continuous functions cannot be drawn.  For example, the function $x\mapsto x^2\sin(1/x)$ is continuous on the interval $[-1,1]$ but cannot be drawn at $x=0$.
• Generally speaking, if the function can be drawn then it can be drawn without picking up the chalk, so the metaphor provides a useful insight, and it provides an entry into consciousness-raising examples like the one in the preceding bullet.

## References

1. 1.000… and .999… (post)
2. Conceptual blending (post)
3. Conceptual blending (Wikipedia)
4. Conceptual metaphors (Wikipedia)
5. Convention (abstractmath)
6. Definitions (abstractmath)
7. Embodied cognition (Wikipedia)
8. Handbook of mathematical discourse (see articles on conceptual blendmental representationrepresentationmetaphor, parenthetic assertion)
9. Images and Metaphors (abstractmath).
10. The interplay of text, symbols and graphics in math education, Lin Hammill
11. Math and the modules of the mind (post)
12. Mathematical discourse: Language, symbolism and visual images, K. L. O’Halloran.
13. Mathematical objects (abmath)
14. Mathematical objects (Wikipedia)
15. Mathematical objects are “out there?” (post)
16. Metaphors in computing science ​(post)
17. Procept (Wikipedia)
18. Representations 2 (post)
19. Representations and models (abstractmath)
20. Representations II: dry bones (post)
21. Representation theorems (Wikipedia) Concrete representations of abstractly defined objects.
22. Representation theory (Wikipedia) Linear representations of algebraic structures.
23. Semiotics, symbols and mathematical visualization, Norma Presmeg, 2006.
24. The transition to formal thinking in mathematics, David Tall, 2010
25. Theory in mathematical logic (Wikipedia)
26. What is the object of the encapsulation of a process? Tall et al., 2000.
27. Where mathematics comes from, by George Lakoff and Rafael Núñez, Basic Books, 2000.
28. Where mathematics comes from (Wikipedia) This is a review of the preceding book.  It is a permanent link to the version of 04:23, 25 October 2012.  The review is opinionated, partly wrong, not well written and does not fit the requirements of a Wikipedia entry.  I recommend it anyway; it is well worth reading.  It contains links to three other reviews.

### Notes on Viewing

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

## Representations of sets

Sets are represented in the math literature in several different ways, some mentioned here.  Also mentioned are some other possibilities.  Introducing a variety of representations of any type of math object is desirable because students tend to assume that the representation is the object.

### Curly bracket notation

The standard representation for a finite set is of the form "$\{1,3,5,6\}$". This particular example represents the unique set containing the integers $1$, $3$, $5$ and $6$ and nothing else. This means precisely that the statement "$n$ is an element of $S$" is true if $n=1$, $n=3$, $n=5$ or $n=6$, and it is false if $n$ represents any other mathematical object.

In the way the notation is usually used, "$\{1,3,5,6\}$", "$\{3,1,5,6\}$", "$\{1,5,3,6\}$",  "$\{1,6,3,5,1\}$" and $\{ 6,6,3,5,1,5\}$ all represent the same set. Textbooks sometimes say "order and repetition don't matter". But that is a statement about this particular representation style for sets. It is not a statement about sets.

It would be nice to come up with a representation for sets that doesn't involve an ordering. Traditional algebraic notation is essentially one-dimensional and so automatically imposes an ordering (see Algebra is a difficult foreign language).

### Let the elements move

In Visible Algebra II, I experimented with the idea of putting the elements at random inside a circle and letting them visibly move around like goldfish in a bowl.  (That experiment was actually for multisets but it applies to sets, too.)  This is certainly a representation that does not impose an ordering, but it is also distracting.  Our visual system is attracted to movement (but not as much as a cat's visual system).

### Enforce natural ordering

One possibility would be to extend the machinery in a visible algebra system that allows you to make a box you could drag elements into.

This box would order the elements in some canonical order (numerical order for numbers, alphabetical order for strings of letters or words) with the property that if you inserted an element in the wrong place it would rearrange itself, and if you tried to insert an element more than once the representation would not change.  What you would then have is a unique representation of the set.

An example is the device below.  (If you have Mathematica, not just CDF player, you can type in numbers as you wish instead of having to use the buttons.)

This does not allow a representation of a heterogenous set such as $\{3,\mathbb{R},\emptyset,\left(\begin{array}{cc}1&2\\0&1\\ \end{array}\right)\}$.  So what?  You can't represent every function by a graph, either.

### Hanger notation

The tree notation used in my visual algebra posts could be used for sets as well, as illustrated below. The system allows you to drag the elements listed into different positions, including all around the set node. If you had a node for lists, that would not be possible.

This representation has the pedagogical advantage of shows that a set is not its elements.

• A set is distinct from its elements
• A set is completely determined by what the elements are.

### Pattern recognition

Infinite sets are sometimes represented using the curly bracket notation using a pattern that defines the set.  For example, the set of even integers could be represented by $\{0,2,4,6,\ldots\}$.  Such a representation is necessarily a convention, since any beginning pattern can in fact represent an infinite number of different infinite sets.  Personally, I would write, "Consider the even integers $\{0,2,4,6,\ldots\}$", but I would not write,  "Consider the set $\{0,2,4,6,\ldots\}$".

By the way, if you are writing for newbies, you should say,"Consider the set of even integers $\{0,2,4,6,\ldots\}$". The sentence "Consider the even integers $\{0,2,4,6,\ldots\}$" is unambiguous because by convention a list of numbers in curly brackets defines a set. But newbies need lots of redundancy.

### Representation by a sentence

Setbuilder notation is exemplified by $\{x|x>0\}$, which denotes the positive reals, given a convention or explicit statement that $x$ represents a real number.  This allows the representation of some infinite sets without depending on a possibly ambiguous pattern.

A Visible Algebra system needs to allow this, too. That could be (necessarily incompletely) done in this way:

• You type in a sentence into a Setbuilder box that defines the set.
• You then attach a box to the Setbuilder box containing a possible element.
• The system then answers Yes, No, or Can't Tell.

The Can't Tell answer is a necessary requirement because the general question of whether an element is in a set defined by a first order sentence is undecidable. Perhaps the system could add some choices:

• Try for a second.
• Try for an hour.
• Try for a year.
• Try for the age of the universe.

Even so, I'll bet a system using Mathematica could answer many questions like this for sentences referring to a specific polynomial, using the Solve or NSolve command.  For example, the answer to the question, "Is $3\in\{n|n\lt0 \text{ and } n^2=9\}$?" (where $n$ ranges over the integers) would be "No", and the answer to  "Is $\{n|n\lt0 \text{ and } n^2=9\}$ empty?" would also be "No". [Corrected 2012.10.24]

### References

1. Explaining “higher” math to beginners (previous post)
2. Algebra is a difficult foreign language (previous post)
3. Visible Algebra II (previous post)
4. Sets: Notation (abstractmath article)
5. Setbuilder notation (Wikipedia)

### Notes on Viewing

• This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
• To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demos is in the Mathematica notebook Representing sets.nb.

## Visible algebra II

### MathJax.Hub.Config({ jax: ["input/TeX","output/NativeMML"], extensions: ["tex2jax.js"], tex2jax: { inlineMath: [ ['$','$'] ], processEscapes: true } }); Notes on viewing:

• This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
• To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demos is in the Mathematica notebook algebra2.nb.

I have written about visible algebra in previous posts (see References). My ideas about the interface are constantly changing. Some new ideas are described here.

In the first place I want to make it clear that what I am showing in these posts is a simulation of a possible visual algebra system.  I have not constructed any part of the system; these posts only show something about what the interface will look like.  My practice in the last few years is to throw out ideas, not construct completed documents or programs.  (I am not saying how long I will continue to do this.)  All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.

## Commutative and associative operations

Times and Plus are commutative and associative operations.  They are usually defined as binary operations.  A binary operation $*$ is said to be commutative if for all $x$ and $y$ in the underlying set of the operation, $x*y=y*x$, and it is associative if for all $x$,$y$ and $z$ in the underlying set of the operation, $(x*y)*z=x*(y*z)$.

It is far better to define a commutative and associative operation $*$ on some underlying set $S$ as an operation on any multiset of elements of $S$.  A multiset is like a set, in particular elements can be rearranged in any way, but it is not like a set in that elements can be repeated and a different number of repetitions of an element makes a different multiset.  So for any particular multiset, the number of repetitions of each element is fixed.  Thus $\{a,a,b,b,c\} = \{c,b,a,b,a\}$ but $\{a,a,b,b,c\}\neq\{c,b,a,b,c\}$. This means that the function (operation) Plus, for example, is defined on any multiset of numbers, and $\mathbf{Plus}\{a,a,b,b,c\}=\mathbf{Plus} \{c,b,a,b,a\}$ but $\mathbf{Plus}\{a,a,b,b,c\}$ might not be equal to $\mathbf{Plus} \{c,b,a,b,c\}$.

This way of defining (any) associative and commutative operation comes from the theory of monads.  An operation defined on all the multisets drawn from a particular set is necessarily commutative and associative if it satisfies some basic monad identities, the main one being it commutes with union of multisets (which is defined in the way you would expect, and if this irritates you, read the Wikipedia article on multisets.). You don't have to impose any conditions specifically referring to commutativity or associativity.  I expect to write further about monads in a later post.

The input process for a visible algebra system should allow the full strength of this fact. You can attach as many inputs as you want to Times or Plus and you can move them around.  For example, you can click on any input and move it to a different place in the following demo.

Other input notations might be suitable for different purposes.  The example below shows how the inputs can be placed randomly in two dimensions (but preserving multiplicity).  I experimented with making it show the variables slowly moving around inside the circle the way the fish do in that screensaver (which mesmerizes small children, by the way — never mind what it does to me), but I haven't yet made it work.

A visible algebra system might well allow directly input tables to be added up (or multiplied), like the one below. Spreadsheets have such an operation In particular, the spreadsheet operation does not insist that you apply it only as a binary operation to columns with two entries.  By far the most natural way to define addition of numbers is as an operation on multisets of numbers.

## Other operations

Operations that are associative but not commutative, such as matrix multiplication, can be defined the monad way as operations on finite lists (or tuples or vectors) of numbers.  The operation is automatically associative if you require it to preserve concatenation of lists and some other monad requirements.

Some binary operations are neither commutative nor associative.  Two such operations on numbers are Subtract and Power.  Such operations are truly binary operations; there is no obvious way to apply them to other structures.  They are only binary because the two inputs have different roles.  This suggests that the inputs be given names, as in the examples below.

Later, I will write more about simplifying trees, solving the max area problem for rectangles surmounted by semicircles, and other things concerning this system of doing algebra.

## Generating a Collatz tree

I have written a short Mathematica program that generates the function tree of the Collatz function.  The code is in the document collatz.nb on the abmath website.

### Examples

Here are some examples.  The first one is generated by the integers between 1 and 26.  (27 is to be avoided because it makes a shoot that is 111 nodes high.)  The primes from 1 to 26 generates the same tree.

This one is generated by the odd numbers from 1 to 26:

This is generated by the even numbers from 1 to 50:

### Remark

This program is not of great import, but it was fun doing it and I learned more Mathematica. In particular, I learned that you cannot assign to a parameter in a function definition. For example, I had to write fv[gl_List, n_Integer] := (nn := n; ggl := {nn}; (Sow[1]; While[! MemberQ[ggl, cf[nn]], (Sow[nn]; Sow[cf[nn]]); nn = cf[nn]]) // Reap) instead of fv[gl_List, n_Integer] := (Sow[1]; While[! MemberQ[gl, cf[n]], (Sow[n]; Sow[cf[n]]); n = cf[n]]) // Reap) (where n:=cf[n] wouldn't have worked either).

Posted in math, Mathematica. Tags:. No Comments »

## Mathematical and linguistic ability

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

## Some personal history

When I was young, I was your typical nerdy geek.  (Never mind what I am now that I am old.)

In high school, I was fascinated by languages, primarily by their structure.  I would have wanted to become a linguist if I had known there was such a thing.  I was good at grasping the structure of a language and read grammars for fun. I was only pretty good at picking up vocabulary. I studied four different languages in high school and college and Turkish when I was in the military.  I know a lot about their structure but am not fluent in any of them (possibly including English).

After college, I decided to go to math grad school.  This was soon after Sputnik and jobs for PhD's were temporarily easy to get.

I always found algebra easy.  When I had to learn other symbolic languages, for example set theory, first order logic, and early programming languages, I found them easy too.  I had enough geometric insight that I did well in all my math courses, but my real strength was in learning languages.

When I got a job at (what is now) Case Western Reserve University, I began learning category theory and a bit of cohomology of groups. I wrote a paper about group automorphisms that got into Transactions of the AMS.  (Full disclosure: I am bragging).

The way Saunders Mac Lane did cohomology, he used "$+$" as a noncommutative operation.  No problem with that, I did lots of calculations in his notation.  In reading category theory I learned how to reason using commutative diagrams.  That is radically different from other math — it isn't strings of symbols — but I caught on. I read Beck's thesis in detail.  Beck wrote functions on the right (unlike Mac Lane) which I adapted to with no problem.  In fact my automorphisms paper and many others in those days was written with functions on the right.

Later on in my career, I learned to program in Forth reasonably well. It is a reverse Polish language. Then (by virtue of summer grants in the 1990's) to use Mathematica, which I now use a lot:  I am an "experienced" user but not an "expert".

## Learning foreign languages in studying math

I taught mostly engineering students during my 35 years at CWRU (especially computer engineering). When I used a text (including my own discrete math class notes) some students pleaded with me not to use $P\wedge Q$ and $P \vee Q$ but let them use $PQ$ and $P+Q$ like they did in their CS courses.  Likewise $1$ and $0$ instead of T and F.  Many of them simply could not switch easily between different codes.  Similar problems occurred in classes in first order logic.

In the early days of calculators when most of them were reverse Polish, some students never mastered their use.

These days, a common complaint about Mathematica is that it is a difficult language to learn; at the MAA meeting in Madison (where I am as I write this) they didn't even staff a booth.  Apparently too many of the professors can't handle Mathematica.

I gave up writing papers with functions on the right because several professional mathematicians complained that they found them too hard to read. I guess not all professional mathematicians can switch code easily, either.

There are many great mathematicians whose main strength is geometric understanding, not linguistic understanding.  Nevertheless, to become a mathematician you have to have enough linguistic ability to learn…

## Algebra

The big elephant in the room is ordinary symbolic algebra as is used in high school algebra and precalculus.  This of course causes difficulty among first year calculus students, too, but college profs are spared the problem that high school teachers have with a large percentage of the students never really grasping how algebra works.  We don't see those students in STEM courses.

It is surely the case that algebra is a difficult and unintuitive foreign language.  I have carried on about this in my stuff about the languages of math in my abstractmath site.

Some students already in college don't really understand expressions such as $x^2$.  You still get some who sporadically think it means $2x$.  (They don't always think that, but it happens when they are off guard.)  Lots of them don't understand the difference between $x^2$ and $2^x$.

In complicated situations, students don't grasp the difference between an expression such as $x^2+2x+1$ and a statement like $x^2+2x+1=0$.  Not to mention the difference between the way $x^2+2x+1=0$ and $x^2+2x+1=(x+1)^2$ are different kinds of statements even though the difference is not indicated in the syntax.

There are many irregularities and ambiguities (just like any natural language — the symbolic language of math is a natural language!): consider $\sin xy$, $\sin x + y$, $\sin x/y$.  (Don't squawk to me about order of operators.  That's as bad as aus, außer, bei, mit, zu.  German can't help it, but mathematical notation could.)

One monstrous ambiguity is $(x,y)$, which could be an ordered pair, the GCD, or an open interval.  I found an example of two of those in the same sentence in the Handbook of Mathematical Discourse, and today in a lecture I saw someone use it with two meanings about three inches apart on a transparency.

Anyway, the symbolic language of math is difficult and we don't teach it well.

## Structuring calculations

There are other ways to structure calculations that are much more transparent.  Most of them use two or three dimensions.

• Spreadsheets: It is easy to approximate the zeros of a function using a spreadsheet and changing the input till you get the value near zero. Why can't middle school students be taught that?
• Bret Victor has made suggestions for easy ways to calculate things.
• My post Visible Algebra I suggest a two-dimensional approach to putting together calculations.  (There are several more posts coming about that idea.)
• Mathematica interactive demos could maybe be provided in a way that would allow them to be joined together to make a complicated calculation. (Modules such as an inverse image constructor.)  I have not tried to do this.

A lot of these alternatives work better because they make full use of two dimensions.  Toolkits could be made for elementary school students (there are some already but I am not familiar with them).

It is impractical to expect that every high school student master basic algebraic notation.  It is difficult and we don't know how to teach it to everyone. With the right toolkits, we could provide everyone, not just students, to put together usable calculations on their computer and experiment with them.  This includes working out the effect of different payment periods on loans, how much paint you need for a room, and many other things.

STEM students will still have to learn algebraic notation as we use it now.  It should be taught as a foreign language with explicit instruction in its syntax (sentences and terms, scope of an operator, and so on), ambiguities and peculiarities.

## Making visible the abstraction in algebraic notation

To manipulate the demos below, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website.

### Algebraic notation

Algebraic notation contains a hidden abstract structure coded by apparently arbitrary conventions that many college calculus students don't understand completely. This very simple example shows one of the ways in which calc students may be confused:

1. $x+2y$
2. $(x+2)y$

Students often mean to express formula 2 when they write something like $x\!\!+\!\!2\,\,\,\,y$ (with a space).  This is a perfectly natural way to write it. But it is against the rules, I presume because in handwriting it is not clear when you mean a space and when you don't.

Formula 1 can also be written as $x+(2y)$, and if it were usually written that way students (I predict) would be less confused.   Always writing it this way would exacerbate the clutter of parentheses but would allow a simple rule:

Evaluate every expression inside parentheses first, starting with the innermost.

### Using trees for algebra

Writing algebraic expressions as a tree (as in computing science)

• makes it obvious what gets evaluated first
• uses no parentheses at all.

An example of using the tree of an expression to do calculations is available in Expressions.nb (requires Mathematica) and Expressions.cdf (requires CDF player only) on my Mathematica website.  I could imagine using tree expressions instead of standard notation as the normal way of doing things. That would require working on Ipads or some such and would take a big amount of investment in software making it intuitive and easy to use.  No, I am not going to embark on such an adventure, but I think it ought to be attempted.  (Brett Victor has many ideas like this.)

### Transforming algebraic notation into trees

The two manipulable diagrams below show the algebraic notation being transformed into tree form.  I expect that this will make the abstract structure more concrete for many students and I encourage others to show it to their students.  Note that the tree form makes everything explicit.  The code for these diagrams is in Handmade Exp Tree.nb

After I return from a ten-day trip I will explore the possibility of making the expression-to-tree transformer turn the expression into an evaluable tree as in Expressions.nb and Expressions.cdf.  In the I hope not to distant future students should have access to many transformers that morph expressions from one form into another.  Such transformers are much more politically correct than Optimus Prime.

Offloading chunking and Computable algebraic expressions in tree form are earlier posts related to this post.

## Metaphors in computing science I

Michael Barr recently told me of a transcription of a talk by Edsger Dijkstra dissing the use of metaphors in teaching programming and advocating that every program be written together with a proof that it works.  This led me to think about the metaphors used in computing science, and that is what this post is about.  It is not a direct answer to what Dijkstra said.

We understand almost anything by using metaphors.  This is a broader sense of metaphor than that thing in English class where you had to say "my love is a red red rose" instead of "my love is like a red red rose".  Here I am talking about conceptual metaphors (see references at the end of the post).

### Metaphor: A program is a set of instructions

You can think of a program as a list of instructions that you can read and, if it is not very complicated, understand how to carry them out.  This metaphor comes from your experience with directions on how to do something (like directions from Google Maps or for assembling a toy).   In the case of a program, you can visualize doing what the program says to do and coming out with the expected output. This is one of the fundamental metaphors for programs.

Such a program may be informal text or it may be written in a computer language.

#### Example

A description of how to calculate $n!$ in English could be:  "Multiply the integers $1$ through $n$".  In Mathematica, you could define the factorial function this way:

fac[n_] := Apply[Times, Table[i, {i, 1, n}]]

This more or less directly copies the English definition, which could have been reworded as "Apply the Times function to the integers from $1$ to $n$ inclusive."  Mathematica programmers customarily use the abbreviation "@@" for Apply because it is more convenient:

Fac[n_]:=Times @@ Table[i, {i, 1, 6}]

As far as I know, C does not have list operations built in.  This simple program gives you the factorial function evaluated at $n$:

j=1;  for (i=2; i<=n; i++)   j=j*i; return j;

This does the calculation in a different way: it goes through the numbers $1, 2,\ldots,n$ and multiplies the result-so-far by the new number.  If you are old enough to remember Pascal or Basic, you will see that there you could use a DO loop to accomplish the same thing.

#### What this metaphor makes you think of

Every metaphor suggests both correct and incorrect ideas about the concept.

• If you think of a list of instructions, you typically think that you should carry out the instructions in order.  (If they are Ikea instructions, your experience may have taught you that you must carry out the instructions in order.)
• In fact, you don't have to "multiply the numbers from $1$ to $n$" in order at all: You could break the list of numbers into several lists and give each one to a different person to do, and they would give their answers to you and you would multiply them together.
• The instructions for calculating the factorial can be translated directly into Mathematica instructions, which does not specify an order.   When $n$ is large enough, Mathematica would in fact do something like the process of giving it to several different people (well, processors) to speed things up.
• I had hoped that Wolfram alpha would answer "720" if I wrote "multiply the numbers from $1$ to $6$" in its box, but it didn't work.  If it had worked, the instruction in English would not be translated at all. (Note added 7 July 2012:  Wolfram has repaired this.)
• The example program for C that I gave above explicitly multiplies the numbers together in order from little to big.  That is the way it is usually taught in class.  In fact, you could program a package for lists using pointers (a process taught in class!) and then use your package to write a C program that looks like the  "multiply the numbers from $1$ to $n$" approach.  I don't know much about C; a reader could probably tell me other better ways to do it.

So notice what happened:

• You can translate the "multiply the numbers from $1$ to $n$" directly into Mathematica.
•  For C, you have to write a program that implements multiplying the numbers from $1$ to $n$. Implementation in this sense doesn't seem to come up when we think about instruction sets for putting furniture together.  It is sort of like: Build a robot to insert & tighten all the screws.

Thus the concept of program in computing science comes with the idea of translating the program instruction set into another instruction set.

• The translation provided above for Mathematica resembles translating the instruction set into another language.
• The two translations I suggested for C (the program and the definition of a list package to be used in the translation) are not like translating from English to another language.  They involve a conceptual reconstruction of the set of instructions.

Similarly, a compiler translates a program in a computer language into machine code, which involves automated conceptual reconstruction on a vast scale.

#### Other metaphors

• C or Mathematica as like a natural language in some ways
• Compiling (or interpreting) as translation

Computing science has used other VIM's (Very Important Metaphors) that I need to write about later:

• Semantics (metaphor: meaning)
• Program as text – this allows you to treat the program as a mathematical object
• Program as machine, with states and actions like automata and Turing machines.
• Specification of a program.  You can regard  "the product of the numbers from $1$ to $n$" as a specification.  Notice that saying "the product" instead of "multiply" changes the metaphor from "instruction" to "specification".

#### References

Conceptual metaphors (Wikipedia)

Images and Metaphors (article in abstractmath)

Images and Metaphors for Sets (article in abstractmath)

Images and Metaphors for Functions (incomplete article in abstractmath)