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Writing math for the web

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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.

But I had other problems.

  • 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.

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Abstracting algebra

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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.

 

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1.000… and 0.999…

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Note: 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.
 
Recently Julian Wilson sent me this letter:
It is well known that students often have trouble accepting that $0.999\ldots$ is the same number as $1.000\ldots$.  However, there is at least one context in which these could be regarded as in some sense as being distinct. In a discrete dynamical system where the next iterate is formed by multiplying the current value by 10 and dropping the leading digit, and where you make a note at each iteration of the first digit after the decimal point, then 0.9999… generates a sequence of 9s, whereas 1.0000… generates a sequence of 0s. The imagery is of a stretching a circle, wrapping it ten times around itself and recording in which sector (labeled 0 to 9) you end up.
 
From the dynamical systems perspective, being in state 9 (and remaining there after each iteration) is different from being in state 0.
The $0.9999\ldots =1.0000\ldots$ equation is associated with several conceptual difficulties that math students have, which I will describe here.

The decimal representation is not the number

Another way of describing the equation is to say that "$0.999\ldots$" and "$1.000\ldots$" are distinct decimal representations of the same number, namely $1$. Julian's proposal provides a different interpretation of the notation, in which "$0.999\ldots$" and "$1.000\ldots$" are strings of symbols generated by two different machines.  Of course, that is correct.  But they are both correct decimal notation that correspond to the same number.

Mathematical writing will sometimes use notation to mean the abstract mathematical object it refers to, and at other times the text is referring to the notation itself.  For example,

$x^2+1$ is always positive.

refers to the value of $x^2+1$, but

If you substitute $5$ for $x$ in $x^2+1$ you get $26$.

refers to the expression "$x^2+1$".  Careful authors would write,

If you substitute $5$ for $x$ in "$x^2+1$" you get $26$.

This ambiguity in using mathematical notation is an example of what philosophers call the "use-mention" distinction, but they apply the phrase to many other situations as well.  Mathematicians have an operational knowledge of this distinction but many of them are not consciously aware of it.

Definitions

A decimal representation of a number by definition represents the number that a certain power series converges to. The two power series corresponding to 1.000… and to 0.999… both converge to 1:

\[1+\sum_{i=1}^{\infty}\frac{0}{10^n}=1\]

and

\[0+\sum_{i=1}^{\infty}\frac{9}{10^n}=1\] 

They are different power series (mention) but converge (use) to the same number.

Most students new to abstract math are not aware of the importance of definition in math. As they learn more, they may still hold on to the idea that you have to discover or reason out what a math word or expression means.  In purple prose, THE DEFINITION IS A DICTATOR. 

This does not mean that you can understand the concept merely by reading the definition.  The definition usually does not mention most of the important things about the concept.

Completed Infinity

A common remark by newbies about $0.999\ldots$ is that it gets closer and closer to $1$ but does not get there. So it can't be equal to $1$.  This shows a lack of understanding of completed infinity.  The point is that the notation "$0.999\ldots$" refers to a string beginning with "$0.$" and followed by an infinite sequence of $9$'s.  Now "$s$ is an infinite sequence of $9$'s" means precisely that $s$ has an entry $s_n$ for every positive integer $n$, and that $s_n$ is $9$ for every positive integer $n$. 

  • The expression is gradually unrolling over time, and does not ever "get there". 
  • All the nines are already there.

Both the preceding sentences are metaphorical.  They are about how you should think about "$0.999\ldots$".  The first metaphor is bad, the second metaphor is good.  Neither statement is a formal mathematical statement.  Neither statement says anything about what the sequence really is.  They are not statements about reality at all, they are about how you should think about the sequence if you are going to understand what mathematicians say about it. 

Metaphors are crucial to understanding math.  Too many students use the wrong metaphors, but too often no one tells them about it.

We need a math ed text for teachers

I am thinking of precalculus through typical college math major courses.  The issues I have discussed in this post are occasionally written about in the math ed literature but I have had difficulty finding many articles (on the web and on JStor) about these specific ideas.  Anyway, articles are not what we need.  We need a modest paperback book specifically aimed at teachers, covering the kinds of cognitive difficulties math students have when faced with abstraction. 

What I have written in abstractmath.org and in the Handbook are examples of what I mean, but they don't cover all the problems and they suffer from lack of focus.  (Note that the material in abstractmath.org and in posts on this blog can be used freely under a Creative Commons license — click on "Permissions" in the blue banner at the top of this page). 

Among math ed researchers, I have learned a lot from papers by Anna Sfard and David Tall. 

References

 
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Algebra is a difficult foreign language

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Note: 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.

Algebra

In a previous post, I said that the symbolic language of mathematics is difficult to learn and that we don't teach it well. (The symbolic language includes as a subset the notation used in high school algebra, precalculus, and calculus.) I gave some examples in that post but now I want to go into more detail.  This discussion is an incomplete sketch of some aspects of the syntax of the symbolic language.  I will write one or more posts about the semantics later.

The languages of math

First, let's distinguish between mathematical English and the symbolic language of math. 

  • Mathematical English is a special register or jargon of English. It has not only its special vocabulary, like any jargon, but also used ordinary English words such as "If…then", "definition" and "let" in special ways. 
  • The symbolic language of math is a distinct, special-purpose written language which is not a dialect of the English language and can in fact be read by mathematicians with little knowledge of English.
    • It has its own symbols and rules that are quite different from spoken languages. 
    • Simple expressions can be pronounced, but complicated expressions may only be pointed to or referred to.
  • A mathematical article or book is typically written using mathematical English interspersed with expressions in the symbolic language of math.

Symbolic expressions

A symbolic noun (logicians call it a term) is an expression in the symbolic language that names a number or other mathematical object, and may carry other information as well.

  • "3" is a noun denoting the number 3.
  • "$\text{Sym}_3$" is a noun denoting the symmetric group of order 3.
  • "$2+1$" is a noun denoting the number 3.  But it contains more information than that: it describes a way of calculating 3 as a sum.
  • "$\sin^2\frac{\pi}{4}$" is a noun denoting the number $\frac{1}{2}$, and it also describes a computation that yields the number $\frac{1}{2}$.  If you understand the symbolic language and know that $\sin$ is a numerical function, you can recognize "$\sin^2\frac{\pi}{4}$" as a symbolic noun representing a number even if you don't know how to calculate it.
  • "$2+1$" and "$\sin^2\frac{\pi}{4}$" are said to be encapsulated computations.
    • The word "encapsulated" refers to the fact that to understand what the expressions mean, you must think of the computation not as a process but as an object.
    • Note that a computer program is also an object, not a process.
  • "$a+1$" and "$\sin^2\frac{\pi x}{4}$" are encapsulated computations containing variables that represent numbers. In these cases you can calculate the value of these computations if you give values to the variables.  

A symbolic statement is a symbolic expression that represents a statement that is either true or false or free, meaning that it contains variables and is true or false depending on the values assigned to the variables.

  • $\pi\gt0$ is a symbolic assertion that is true.
  • $\pi\lt0$ is a symbolic assertion that it is false.  The fact that it is false does not stop it from being a symbolic assertion.
  • $x^2-5x+4\gt0$ is an assertion that is true for $x=5$ and false for $x=1$.
  • $x^2-5x+4=0$ is an assertion that is true for $x=1$ and $x=4$ and false for all other numbers $x$.
  • $x^2+2x+1=(x+1)^2$ is an assertion that is true for all numbers $x$. 

Properties of the symbolic language

The constituents of a symbolic expression are symbols for numbers, variables and other mathematical objects. In a particular expression, the symbols are arranged according to conventions that must be understood by the reader. These conventions form the syntax or grammar of symbolic expressions. 

The symbolic language has been invented piecemeal by mathematicians over the past several centuries. It is thus a natural language and like all natural languages it has irregularities and often results in ambiguous expressions. It is therefore difficult to learn and requires much practice to learn to use it well. Students learn the grammar in school and are often expected to understand it by osmosis instead of by being taught specifically.  However, it is not as difficult to learn well as a foreign language is.

In the basic symbolic language, expressions are written as strings of symbols.

  • The symbolic language gives (sometimes ambiguous) meaning to symbols placed above or below the line of symbols, so the strings are in some sense more than one dimensional but less than two-dimensional.
  • Integral notation, limit notation, and others, are two-dimensional enough to have two or three levels of symbols. 
  • Matrices are fully two-dimensional symbols, and so are commutative diagrams.
  • I will not consider graphs (in both senses) and geometric drawings in this post because I am not sure what I want to write about them.

Syntax of the language

One of the basic methods of the symbolic language is the use of constructors.  These can usually be analyzed as functions or operators, but I am thinking of "constructor" as a linguistic device for producing an expression denoting a mathematical object or assertion. Ordinary languages have constructors, too; for example "-ness" makes a noun out of a verb ("good" to "goodness") and "and" forms a grouping ("men and women").

Special symbols

The language uses special symbols both as names of specific objects and as constructors.

  • The digits "0", "1", "2" are named by special symbols.  So are some other objects: "$\emptyset$", "$\infty$".
  • Certain verbs are represented by special symbols: "$=$", "$\lt$", "$\in$", "$\subseteq$".
  • Some constructors are infixes: "$2+3$" denotes the sum of 2 and 3 and "$2-3$" denotes the difference between them.
  • Others are placed before, after, above or even below the name of an object.  Examples: $a'$, which can mean the derivative of $a$ or the name of another variable; $n!$ denotes $n$ factorial; $a^\star$ is the dual of $a$ in some contexts; $\vec{v}$ constructs a vector whose name is "$v$".
  • Letters from other alphabets may be used as names of objects, either defined in the context of a particular article, or with more nearly global meaning such as "$\pi$" (but "$\pi$" can denote a projection, too).

This is a lot of stuff for students to learn. Each symbol has its own rules of use (where you put it, which sort of expression you may it with, etc.)  And the meaning is often determined by context. For example $\pi x$ usually means $\pi$ multiplied by $x$, but in some books it can mean the function $\pi$ evaluated at $x$. (But this is a remark about semantics — more in another post.)

"Systematic" notation

  • The form "$f(x)$" is systematically used to denote the value of a function $f$ at the input $x$.  But this usage has variations that confuse beginning students:
    • "$\sin\,x$" is more common than "$\sin(x)$".
    • When the function has just been named as a letter, "$f(x)$" is more common that "$fx$" but many authors do use the latter.
  • Raising a symbol after another symbol commonly denotes exponentiation: "$x^2$" denotes $x$ times $x$.  But it is used in a different meaning in the case of tensors (and elsewhere).
  • Lowering a symbol after another symbol, as in "$x_i$"  may denote an item in a sequence.  But "$f_x$" is more likely to denote a partial derivative.
  • The integral notation is quite complicated.  The expression \[\int_a^b f(x)\,dx\] has three parameters, $a$, $b$ and $f$, and a bound variable $x$ that specifies the variable used in the formula for $f$.  Students gradually learn the significance of these facts as they work with integrals. 

Variables

Variables have deep problems concerned with their meaning (semantics). But substitution for variables causes syntactic problems that students have difficulty with as well.

  • Substituting $4$ for $x$ in the expression $3+x$ results in $3+4$. 
  • Substituting $4$ for $x$ in the expression $3x$ results in $12$, not $34$. 
  • Substituting "$y+z$" in the expression $3x$ results in $3(y+z)$, not $3y+z$.  Some of my calculus students in preforming this substitution would write $3\,\,y+z$, using a space to separate.  The rules don't allow that, but I think it is a perfectly natural mistake. 

Using expressions and writing about them

  • If I write "If $x$ is an odd integer, then $3+x$ is odd", then I am using $3+x$ in a sentence. It is a noun denoting an unspecified number which can be constructed in a specified way.
  • When I mention substituting $4$ for $x$ in "$3+x$", I am talking about the expression $3+x$.  I am not writing about a number, I am writing about a string of symbols.  This distinction causes students major difficulties and teacher hardly ever talk about it.
  • In the section on variables, I wrote "the expression $3+x$", which shows more explicitly that I am talking about it as an expression.
    • Note that quotes in novels don't mean you are talking about the expression inside the quotes, it means you are describing the act of a person saying something.
  • It is very common to write something like, "If I substitute $4$ for $x$ in $3x$ I get $3 \times 4=12$".  This is called a parenthetic assertion, and it is literally nonsense (it says I get an equation).
  • If I pronounce the sentence "We know that $x\gt0$" we pronounce "$x\gt0$" as "$x$ is greater than zero",  If I pronounce the sentence "For any $x\gt0$ there is $y\gt0$ for which $x\gt y$", then I pronounce the expression "$x\gt0$" as "$x$ greater than zero$",  This is an example of context-sensitive pronunciation
  • There is a lot more about parenthetic assertions and context-sensitive pronunciation in More about the languages of math.

Conclusion

I have described some aspects of the syntax of the symbolic language of math. Learning that syntax is difficult and requires a lot of practice. Students who manage to learn the syntax and semantics can go on to learn further math, but students who don't are forever blocked from many rewarding careers. I heard someone say at the MathFest in Madison that about 25% of all high school students never really understand algebra.  I have only taught college students, but some students (maybe 5%) who get into freshman calculus in college are weak enough in algebra that they cannot continue. 

I am not proposing that all aspects of the syntax (or semantics) be taught explicitly.  A lot must be learned by doing algebra, where they pick up the syntax subconsciously just as they pick up lots of other behavior-information in and out of school. But teachers should explicitly understand the structure of algebra at least in some basic way so that they can be aware of the source of many of the students' problems. 

It is likely that the widespread use of computers will allow some parts of the symbolic language of math to be replaced by other methods such as using Excel or some visual manipulation of operations as suggested in my post Mathematical and linguistic ability.  It is also likely that the symbolic language will gradually be improved to get rid of ambiguities and irregularities.  But a deliberate top-down effort to simplify notation will not succeed. Such things rarely succeed.

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More about the definition of function

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Maya Incaand commented on my post Definition of "function":

Why did you decide against "two inequivalent descriptions in common use"?  Is it no longer true?

This question concerns [1], which is a draft article.  I have not promoted it to the standard article in abstractmath because I am not satisfied with some things in it. 

More specifically, there really are two inequivalent descriptions in common use.  This is stated by the article, buried in the text, but if you read the beginning, you get the impression that there is only one specification.  I waffled, in other words, and I expect to rewrite the beginning to make things clearer.

Below are the two main definitions you see in university courses taken by math majors and grad students.  A functional relation has the property that no two distinct ordered pairs have the same first element.

Strict definition: A function consists of a functional relation with specified codomain (the domain is then defined to be the set of first elements of pairs in the relation).  Thus if $A$ and $B$ are sets and $A\subseteq B$, then the identity function $1_A:A\to A$ and the inclusion function $i:A\to B$  are two different functions.

Relational definition: A function is a functional relation.  Then the identity and inclusion functions are the same function.  This means that a function and its graph are the same thing (discussed in the draft article).

These definitions are subject to variations:

Variations in the strict definition: Some authors use "range" for "codomain" in the definition, and some don't make it clear that two functions with the same functional relation but different codomains are different functions.

Variations in the relational definition: Most such definitions state explicitly that the domain and range are determined by the relation (the set of first coordinates and the set of second coordinates). 

Formalism

There are many other variations in the formalism used in the definition.  For example, the strict definition can be formalized (as in Wikipedia) as an ordered triple $(A, B, f)$ where $A$ and $B$ are sets and $f$ is a functional relation with the property thar every element of $A$ is the first element of an ordered pair in the relation.  

You could of course talk about an ordered triple $(A,f,B)$ blah blah.  Such definitions introduce arbitrary constructions that have properties irrelevant to the concept of function.  Would you ever say that the second element of the function $f(x)=x+1$ on the reals is the set of real numbers?  (Of course, if you used the formalism $(A,f,B)$ you would have to say the second element of the function is its graph! )

It is that kind of thing that led me to use a specification instead of a definition.  If you pay attention to such irrelevant formalism there seems to be many definitions of function.  In fact, at the university level there are only two, the strict definition and the relational definition.  The usage varies by discipline and age.  Younger mathematicians are more likely to use the strict definition.  Topologists use the strict definition more often than analysts (I think).

Usage

There is also variation in usage.

  • Most authors don't tell you which definition they use, and it often doesn't matter anyway. 
  • If an author defines a function using a formula, there is commonly an implicit assumption that the domain includes everything for which the formula is well-defined.  (The "everything" may be modified by referring to it as an integer, real, or complex function.)

Definitions of function on the web

Below are some definitions of function that appear on the web.  I have excluded most definitions aimed at calculus students or below; they often assume you are talking about numbers and formulas.  I have not surveyed textbooks and research papers.  That would have to be done for a proper scholarly article about mathematical usage of "function". But most younger people get their knowledge from the web anyway.

  1. Abstractmath draft article: Functions: Specification and Definition.  (Note:  Right now you can't get to this from the Table of Contents; you have to click the preceding link.) 
  2. Gyre&Gimble post: Definition of "function"
  3. Intmath discussion of function  Function as functional relation between numbers, with induced domain and range.
  4. Mathworld definition of function Functional-relation definition.  Defines $F:A\to B$ in a way that requires $B$ to be the image.
  5. Planet Math definition of function Strict definition.
  6. Prime Encyclopedia of Mathematics Functional-relation definition.
  7. Springer Encyclopedia of Math definition of function  Strict definition, except not clear if different codomains mean different functions.
  8. Wikipedia definition of function Discusses both definitions.
  9. Wisconsin Department of Public Instruction Definition of function  Function as functional relation.
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The meaning of the word “superposition”

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This is from the Wikipedia article on Hilbert's 13th Problem as it was on 31 March 2012:

[Hilbert’s 13th Problem suggests this] question: can every continuous function of three variables be expressed as a composition  of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of Andrey Kolmogorov. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question.  

In their paper A relation between multidimensional data compression and Hilbert’s 13th  problem,  Masahiro Yamada and Shigeo Akashi describe an example of Arnold's theorem this way: 

Let $f ( \cdot , \cdot, \cdot )$ be the function of three variable defined as \(f(x, y, z)=xy+yz+zx\), $x ,y , z\in \mathbb{C}$ . Then, we can easily prove that there do not exist functions of two variables $g(\cdot , \cdot )$ , $u(\cdot, \cdot)$ and $v(\cdot , \cdot )$ satisfying the following equality: $f(x, y, z)=g(u(x, y),v(x, z)) , x , y , z\in \mathbb{C}$ . This result shows us that $f$ cannot be represented any 1-time nested superposition constructed from three complex-valued functions of two variables. But it is clear that the following equality holds: $f(x, y, z)=x(y+z)+(yz)$ , $x,y,z\in \mathbb{C}$ . This result shows us that $f$ can be represented as a 2-time nested superposition.

The article about superposition in All about circuits says:

The strategy used in the Superposition Theorem is to eliminate all but one source of power within a network at a time, using series/parallel analysis to determine voltage drops (and/or currents) within the modified network for each power source separately. Then, once voltage drops and/or currents have been determined for each power source working separately, the values are all “superimposed” on top of each other (added algebraically) to find the actual voltage drops/currents with all sources active. 

Superposition Theorem in Wikipedia:

The superposition theorem for electrical circuits states that for a linear system the response (Voltage or Current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, while all other independent sources are replaced by their internal impedances.

Quantum superposition in Wikipedia:  

Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system — such as an electron — exists partly in all its particular, theoretically possible states (or, configuration of its properties) simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics).

Mathematically, it refers to a property of solutions to the Schrödinger equation; since theSchrödinger equation is linear, any linear combination of solutions to a particular equation will also be a solution of it. Such solutions are often made to be orthogonal (i.e. the vectors are at right-angles to each other), such as the energy levels of an electron. By doing so the overlap energy of the states is nullified, and the expectation value of an operator (any superposition state) is the expectation value of the operator in the individual states, multiplied by the fraction of the superposition state that is "in" that state

The CIO midmarket site says much the same thing as the first paragraph of the Wikipedia Quantum Superposition entry but does not mention the stuff in the second paragraph.

In particular, the  Yamada & Akashi article describes the way the functions of two variables are put together as "superposition", whereas the Wikipedia article on Hilbert's 13th calls it composition.  Of course, superposition in the sense of the Superposition Principle is a composition of multivalued functions with the top function being addition.  Both of Yamada & Akashi's examples have addition at the top.  But the Arnold theorem allows any continuous function at the top (and anywhere else in the composite).  

So one question is: is the word "superposition" ever used for general composition of multivariable functions? This requires the kind of research I proposed in the introduction of The Handbook of Mathematical Discourse, which I am not about to do myself.

The first Wikipedia article above uses "composition" where I would use "composite".  This is part of a general phenomenon of using the operation name for the result of the operation; for examples, students, even college students, sometimes refer to the "plus of 2 and 3" instead of the "sum of 2 and 3". (See "name and value" in abstractmath.org.)  Using "composite" for "composition" is analogous to this, although the analogy is not perfect.  This may be a change in progress in the language which simplifies things without doing much harm.  Even so, I am irritated when "composition" is used for "composite".

Quantum superposition seems to be a separate idea.  The second paragraph of the Wikipedia article on quantum superposition probably explains the use of the word in quantum mechanics.

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Offloading chunking

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In my previous post I wrote about the idea of offloading abstraction, the sort of things we do with geometric figures, diagrams (that post emphasized manipulable diagrams), drawing the tree of an algebraic expression, and so on.  This post describes a way to offload chunking.  

Chunking

I am talking about chunking in the sense of encapsulation, as some math ed. people use it.  I wrote about it briefly in [1], and [2] describes the general idea.  I don't have a good math ed reference for it, but I will include references if readers supply them.  

Chunking for some educators means breaking a complicated problem down into pieces and concentrating on them one by one.  That is not really the same thing as what I am writing about.  Chunking as I mean it enables you to think more coherently and efficiently about a complicated mathematical structure by objectifying some of the data in the structure.  

Project 

This project an example of how chunking could be made visible in interactive diagrams, so that the reader grasps the idea of chunking.  I guess I am chunking chunking.  

Here is a short version of an example of chunking worked out in ridiculous detail in reference [1]. 

Let \[f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\]  How do I know it is never negative?  Well, because it has the form (a positive number)(times)(something)$^6$.    Now (something)$^6$ is ((something)$^3)^2$ and a square is always nonnegative, so the function is (positive)(times)(nonnegative), so it has to be nonnegative.  

I recognized a salient fact about .0002, namely that it was positive: I grayed out (in my mind) its exact value, which is irrelevant.  I also noticed a salient fact about \[{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}\] namely that it was (a big mess that I grayed out)(to the 6th power).  And proceeded from there.  (And my chunking was inefficient; for example, it is more to the point that .0002 is nonnegative).

I believe you could make a movie of chunking like this using Mathematica CDF.  You would start with the formula, and then as the voiceover said "what's really important is that .0002 is nonnegative" the number would turn into a gray cloud with a thought balloon aimed at it saying "nonnegative".  The other part would turn into a gray cloud to the sixth, then the six would break into 3 times 2 as the voice comments on what is happening.  

It would take a considerable amount of work to carry this out.  Lots of decisions would need to be made.  

One problem is that Mathematica doesn't provide a way to do voiceovers directly (as far as I know).  Perhaps you could make a screen movie using screenshot software in real time while you talked and (offscreen) pushed buttons that made the various changes happen.

You could also do it with print instead of voiceover, as I did in the example in this post. In this case you need to arrange to have the printed part and the diagram simultaneously visible.  

I may someday try my hand at this.  But I would encourage others to attack this project if it interests them.  This whole blog is covered by the Creative Commons Attribution – ShareAlike 3.0 License", which means you may use, adapt and distribute the work freely provided you follow the requirements of the license.

I have other projects in mind that I will post separately.

References

  1. Abstractmath article on chunking.
  2. Wikipedia on chunking
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Abstract objects

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Some thoughts toward revising my article on mathematical objects.  

Mathematical objects are a kind of abstract object.  There are lots of abstract objects that are not mathematical objects,  For example, if you keep a calendar or schedule for appointments, that calendar is an abstract object.  (This example comes from [2]). 

It may be represented as a physical object or you may keep it entirely in your head.  I am not going to talk about the latter possibility, because I don't know what to say.

  1. If it is a paper calendar, that physical object represents the information that is contained in your calendar.  
  2. Same for a calendar on a computer, but that is stored as magnetic bits on a disk or in flash memory. A computer program (part of the operating system) is required to present it on the screen in such a way that you can read it.  Each time you open it, you get a new physical representation of the calendar.

Your brain contains a module (see [5], [7]) that interprets the representation in (1) or (2) and which has connections with other modules in your brain for dates, times, locations and whether the appointment is for a committee, a medical exam, or whatever.  

The calendar-interpreter module in your brain is necessary for the physical object to be a calendar.  The physical object is not in itself your calendar.  The calendar in this sense does not exist in the physical world.  It is abstract.  Since we think of it as a thing, it is an abstract object.

The abstract object "my calendar" affects the physical world (it causes you to go to the dentist next Tuesday).  The relation of the abstract object to the physical world is mediated by whatever physical object you call your calendar along with the modules in the brain that relate to it.  The modules in the brain are actions by physical objects, so this point of view does not involve Cartesian style dualism.

Note:  A module is a meme.  Are all memes modules?  This needs to be investigated.  Whatever they are, they exist as physical objects in people's brains.

Mathematical objects

A rigorous proof of a theorem about a mathematical object tends to refer to the object as if it were absolutely static and did not affect anything in the physical world.  I talked about this in [10], where I called it the dry bones representation of a mathematical object.  Mathematical objects don't have to be thought of this way, but (I suggest) what makes them mathematical objects is that they can be thought of in dry bones mode.  

If you use calculus to figure out how much fuel to use in a rocket to make it go a mile high, then actually use that amount in the rocket and send it off, your calculations have affected your physical actions, so you were thinking of the calculations as an abstract object.  But if you sit down to check your calculations, you concentrate on the steps one by one with the rules of algebra and calculus in mind.  You are looking at them as inert objects, like you would look at a bone of a dinosaur to see what species it belongs to. From that point of view your calculations form a mathematical object, because you are using the dry-bones approach.

Caveat

All this blather is about how you should think about mathematical objects.  It can be read as philosophy, but I have no intention of defending it as philosophy.  People learning abstract math at college level have a lot of trouble thinking about mathematical objects as objects, and my intention is to start clarifying some aspects of how you think about them in different circumstances.  (The operative word is "start" — there is a lot more to be said.)

About the exposition of this post (a commercial)

You will notice that I gave examples of abstract objects but did not define the word "abstract object".  I did the same with mathematical objects.  In both cases, I put the word "abstract object" or "mathematical object" in boldface at a suitable place in the exposition.

That is not the way it is done in math, where you usually make the definition of a word in a formal way, marking it as Definition, putting the word in bold or italics, and listing the attributes it must have.  I want to point out two things:

  • For the most part, that behavior is peculiar to mathematics.
  • This post is not a presentation of mathematical ideas.  

This gives me an opportunity for a commercial:  Read what we have written about definitions in References [1], [3] and [4].

References

  1. Atish Bagchi and Charles Wells, Varieties of Mathematical Prose, 1998.
  2. Reuben Hersh, What is mathematics, really? Oxford University Press, 1997. 
  3. Charles Wells, Handbook of Mathematical Discourse.
  4. Charles Wells, Mathematical objects in abstractmath.org
  5. Math and modules of the mind (previous post)
  6. Mathematical Concepts (previous post)
  7. Thinking about abstract math (previous post)
  8. Terrence W. Deacon, Incomplete Nature.  W. W. Norton, 2012. [I have read only a little of this book so far, but I think he is talking about abstract objects in the sense I have described above.]
  9. Gideon Rosen, Abstract Objects.  Stanford Encyclopedia of Philosophy.
  10. Representations II: Dry Bones (previous post)

 

http://plato.stanford.edu/entries/abstract-objects/

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Mathematical usage

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Comments about mathematical usage, extending those in my post on abuse of notation.

Geoffrey Pullum, in his post Dogma vs. Evidence: Singular They, makes some good points about usage that I want to write about in connection with mathematical usage.  There are two different attitudes toward language usage abroad in the English-speaking world. (See Note [1])

  • What matters is what people actually write and say.   Usage in this sense may often be reported with reference to particular dialects or registers, but in any case it is based on evidence, for example citations of quotations or a linguistic corpus.  (Note [2].)  This approach is scientific.
  • What matters is what a particular writer (of usage or style books) believes about  standards for speaking or writing English.  Pullum calls this "faith-based grammar".  (People who think in this way often use the word "grammar" for usage.)  This approach is unscientific.

People who write about mathematical usage fluctuate between these two camps.

My writings in the Handbook of Mathematical Discourse and abstractmath.org are mostly evidence based, with some comments here and there deprecating certain usages because they are confusing to students.  I think that is about the right approach.  Students need to know what is actual mathematical usage, even usage that many mathematicians deprecate.

Most math usage that is deprecated (by me and others) is deprecated for a reason.  This reason should be explained, and that is enough to stop it being faith-based.  To make it really scientific you ought to cite evidence that students have been confused by the usage.  Math education people have done some work of this sort.  Most of it is at the K-12 level, but some have worked with college students observing the way the solve problems or how they understand some concepts, and this work often cites examples.

Examples of usage to be deprecated

 

Powers of functions

f^n(x) can mean either iterated composition or multiplication of the values.  For example, f^2(x) can mean f(x)f(x) or f(f(x)).  This is exacerbated by the fact that in undergrad calculus texts,  \sin^{-1}x refers to the arcsine, and \sin^2 x refers to \sin x\sin x.  This causes innumerable students trouble.  It is a Big Deal.

In

Set "in" another set.  This is discussed in the Handbook.  My impression is that for students the problem is that they confuse "element of" with "subset of", and the fact that "in" is used for both meanings is not the primary culprit.  That's because most sets in practice don't have both sets and non-sets as elements.  So the problem is a Big Deal when students first meet with the concept of set, but the notational confusion with "in" is only a Small Deal.

Two

This is not a Big Deal.  But I have personally witnessed students (in upper level undergrad courses) that were confused by this.

Parentheses

The many uses of parentheses, discussed in abstractmath.  (The Handbook article on parentheses gives citations, including one in which the notation "(a,b)" means open interval once and GCD once in the same sentence!)  I think the only part that is a Big Deal, or maybe Medium Deal, is the fact that the value of a function f at an input x can be written either  "f\,x" or as "f(x)".  In fact, we do without the parentheses when the name of the function is a convention, as in \sin x or \log x, and with the parentheses when it is a variable symbol, as in "f(x)".  (But a substantial minority of mathematicians use f\,x in the latter case.  Not to mention xf.)  This causes some beginning calculus students to think "\sin x" means "sin" times x.

More

The examples given above are only a sampling of troubles caused by mathematical notation.   Many others are mentioned in the Handbook and in Abstractmath, but they are scattered.  I welcome suggestions for other examples, particularly at the college and graduate level. Abstractmath will probably have a separate article listing the examples someday…

Notes

[1] The situation Pullum describes for English is probably different in languages such as Spanish, German and French, which have Academies that dictate usage for the language.  On the other hand, from what I know about them most speakers of those languages ignore their dictates.

[2] Actually, they may use more than one corpus, but I didn't want to write "corpuses" or "corpora" because in either way I would get sharp comments from faith-based usage people.

References on mathematical usage

Bagchi, A. and C. Wells (1997), Communicating Logical Reasoning.

Bagchi, A. and C. Wells (1998)  Varieties of Mathematical Prose.

Bullock, J. O. (1994), ‘Literacy in the language of mathematics’. American Mathematical Monthly, volume 101, pages 735743.

de Bruijn, N. G. (1994), ‘The mathematical vernacular, a language for mathematics with typed sets’. In Selected Papers on Automath, Nederpelt, R. P., J. H. Geuvers, and R. C. de Vrijer, editors, volume 133 of Studies in Logic and the Foundations of Mathematics, pages 865  935.  

Epp, S. S. (1999), ‘The language of quantification in mathematics instruction’. In Developing Mathematical Reasoning in Grades K-12. Stiff, L. V., editor (1999),  NCTM Publications.  Pages 188197.

Gillman, L. (1987), Writing Mathematics Well. Mathematical Association of America

Higham, N. J. (1993), Handbook of Writing for the Mathematical Sciences. Society for Industrial and Applied Mathematics.

Knuth, D. E., T. Larrabee, and P. M. Roberts (1989), Mathematical Writing, volume 14 of MAA Notes. Mathematical Association of America.

Krantz, S. G. (1997), A Primer of Mathematical Writing. American Mathematical Society.

O'Halloran, K. L.  (2005), Mathematical Discourse: Language, Symbolism And Visual Images.  Continuum International Publishing Group.

Pimm, D. (1987), Speaking Mathematically: Communications in Mathematics Classrooms.  Routledge & Kegan Paul.

Schweiger, F. (1994b), ‘Mathematics is a language’. In Selected Lectures from the 7th International Congress on Mathematical Education, Robitaille, D. F., D. H. Wheeler, and C. Kieran, editors. Sainte-Foy: Presses de l’Université Laval.

Steenrod, N. E., P. R. Halmos, M. M. Schiffer, and J. A. Dieudonné (1975), How to Write Mathematics. American Mathematical Society.

Wells, C. (1995), Communicating Mathematics: Useful Ideas from Computer Science.

Wells, C. (2003), Handbook of Mathematical Discourse

Wells, C. (ongoing), Abstractmath.org.

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Two

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The post Are these questions unambiguous? in the blog Explaining Mathematics concerns the funny way mathematicians use the number “two” (Note [3]).  This is discussed in Abstractmath.org, based on usage quotations (see Note [1]) in the Handbook of Mathematical Discourse. They are citations  54, 119, 220, 229, 260, 322, 323 and 338.  The list is in the online version of the Handbook (see Note [2]) which takes forever to load.  (There is a separate file for users of the paperback book but it is currently trashed.)

The usage quirk concerning “two” is exemplified by statements such as these:

  1. The sum of any two even integers is even.
  2. Courant gives Leibniz’ rule for finding the Nth derivative of the product of two functions.  (This is from Citation 323.)
  3. Are there two positive integers m and n, both greater than 1, satisfying mn=9? (This is from Explaining Mathematics.)

Statements 1 and 2 are of course true.  They are still true if the “two” things are the same.  Mathematicians generally assume that such a statement includes the case where the two things are the same.  If the case that they are the same is excluded, the statement becomes an unnecessarily weak assertion.

Statement 3, in my opinion, is badly written.  If the two positive integers have to be distinct, the answer is “no”.   I think any competent mathematical writer would write something like, “There are not two distinct integers m and n both greater than 1 for which mn = 9″.

It is fair to say that when mathematicians refer to “two integers” in statements like these, they are allowed to be the same.  If they can’t be the same for the sentence to remain true, they will (or at least should) insert a word such as “distinct”.

Of course, in some sentences the two integers can’t be the same because of some condition imposed in the context.  That doesn’t happen in the citations I have listed.  Maybe someone can contribute an example.

Notes

[1] In the Handbook, usage quotations are called “citations”.  It appears to me that the commonest name for citations among lexicographers is “usage quotations”, so I will start calling them that.

[2] I created the online version of the Handbook hastily in 2006.  It needs work, since it has TeX mistakes (which may irritate you but should not interfere with readability) and omits the quotations, illustrations, and some backlinks, including backlinks for the citations.  Some Day When I Get A Round Tuit…

[3] This funny property of “two” was discussed many years ago by Steenrod or Knuth or someone, and is mentioned in a paper by Susanna Epp, but I don’t currently have access to any of the references.

 

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