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

The Mathematical Definition of Function

2011/05/20 Charles Wells Leave a comment

Introduction

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

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

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

I will use two running examples throughout this discussion:

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

Specification of function

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

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

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

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

Examples

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

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

Definition of function

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

To state this definition we need a preliminary idea.

The functional property

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

Examples

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

Example: Graph of a function defined by a formula

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

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

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

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

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

How to think about the functional property

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

Mathematical definition of function

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

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

Examples

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

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

Identity and inclusion

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

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

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

Graphs and functions

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

Examples

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

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

Other definitions of function

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

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

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

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

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language of math, math, representations, understanding math

Mental, Physical and Mathematical Representations

2011/05/17 Charles Wells 2 Comments

For a given mathematical object, a mathematician may have:

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

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

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

Two Examples

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

Function as black box

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

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

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

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

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

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

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

Triangle

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

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

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

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

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

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

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

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

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

Summary

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

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

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

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

References

[1] Images and metaphors (in abstractmath).

[2] Representations and Models (in abstractmath).

[3] Mathematical Concepts (previous blog).

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

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abstractmath.org, category theory, computer science, language of math, math, representations, understanding math

Operation as metaphor in math

2011/01/17 Charles Wells 3 Comments

Operation: Is it just a name or is there a metaphor behind it?

A function of the form ${f:S\times S\rightarrow S}$ may be called a binary operation on ${S}$ . The main point to notice is that it takes pairs of elements of ${S}$ to the same set ${S}$ .

A binary operation is a special case of n-ary operation for any natural number ${n}$ , which is a function of the form ${f:S^n\rightarrow S}$ . A ${1}$ -ary (unary) operation on ${S}$ is a function from a set to itself (such as the map that takes an element of a group to its inverse), and a ${0}$ -ary (nullary) operation on ${S}$ is a constant.

It is useful at times to consider multisorted algebra, where a binary operation can be a function ${f:S_1\times S_2\rightarrow S_3}$ where the ${S_i}$ are possibly different sets. Then a unary operation is simply a function.

Calling a function a multisorted unary operation suggest a different way of thinking about it, but as far as I can tell the difference is only that the author is thinking of algebraic operations as examples. This does not seem to be a different metaphor the way “function as map” and “function as transformation” are different metaphors. Am I missing something?

In the 1960’s some mathematicians (not algebraists) were taken aback by the idea that addition of real numbers (for example) is a function. I observed this personally. I don’t think any mathematician would react this way today.

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The emergence of space as a character

2011/01/11 Charles Wells 2 Comments

This is an update of a post from a couple of years ago.

Until computers came along, there was no such thing as a space character. The space between printed words was simply a space. In computing, each letter is represented by a certain number, and starting in the early days the space was represented by the number 32 (in decimal notation). In that sense, the way we thought of the space between printed words shifted from empty space to an object represented by empty space.

In the late eighties, I was at a church service on the Sunday when they talk about the budget. After the talk, ten members of the congregation marched up front each carrying a sign with one letter on it. They arranged themselves to spell

This was concrete evidence that we had changed the way we think about spaces between words. The congregation of this upscale church included many engineers and other professional people.

The space character is used in Mathematica to denote multiplication: One writes “x y’’ to mean x times y. This allows multiletter variable names without ambiguity. “distance time’’ would be the product of distance and time. When you have some experience with Mathematica, you think of space between variables as a genuine symbol meaning multiplication.

Space is used in other places in math with a kind of positive meaning; for example, “sin x’’ means the result of evaluating the sine function at x. But I don’t believe most mathematicians think of that space as a symbol. I didn’t until I thought of writing this comment. I am not at all sure it is useful to think of it that way.

When lead type was used in hand typesetting, there were different sizes of blank lead slugs to put in between letters. With linotypes, a different technique was used: wedges were shoved down between the letters to force the line of type to be right justified.

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abstractmath.org, category theory, Handbook, language of math, math, understanding math

Multivalued Functions

2011/01/03 Charles Wells Leave a comment

Multivalued functions

I am reconstructing the abstractmath website and am currently working on the part on functions. This has generated some bloggable blustering.

The phrase multivalued function refers to an object that is like a function ${f:S\rightarrow T}$ except that for ${s\in S}$ , ${f(s)}$ may denote more than one value. Multivalued functions arose in considering complex functions such as ${\sqrt{z}}$ . Another example: the indefinite integral is a multivalued operator.

It is useful to think of a multivalued function as a function although it violates one of the requirements of being a function (being single-valued).

A multivalued function ${f:S\rightarrow T}$ can be modeled as a function with domain ${S}$ and codomain the set of all subsets of ${T}$ . The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two different ways of thinking about multivalued functions.: “The value may be any of these things…” as opposed to “The value is this whole set of things.”) The “value may be any of these…” idea has a perfectly good mathematical model: a relation (set of ordered pairs) from ${S}$ to ${T}$ which is the inverse of a surjective function.

Phrases such as “multivalued function” and “partial function” upset some uptight types who say things like, “But a multivalued function is not a function!”. A stepmother is not a mother, either.

I fulminated at length about this in the Handbook article on radial category. (This is conceptual category in the sense of Lakoff, Women, fire and dangerous things, University of Chicago, 1986.). The Handbook is on line, but it downloads very slowly, so I have extracted the particular page on radial categories here.

Functions generate a radial category of concepts in mathematics. There are lots of other concepts in math that have generated radial categories. Think of “incomplete proof” or “left identity”. Radial categories are a basic mechanism of the way we think and function in the world. They should not be banished from mathematics.

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Knowledge is a (pre)sheaf

2010/09/28 Charles Wells 2 Comments

Mathematical structures as metaphors

People understand aspects of life that they don’t have good words for. Math could supply them with some names for these concepts. Just as music theory explains how Mozart’s music, blues, and klezmer music are different from each other (part of the explanation is: different scales).

It would be convenient if everyone understood comments such as “Race and ethnicity are not Boolean concepts”. Well, they don’t. In the case of race, I think many people over 70 years old or so are hung up on the idea that a person is either black or white. They ask questions about a mixed race person like “What is he?” Younger people seem to know better, but they don’t have a way of expressing the idea that the concepts of Boolean and fuzzy set would give them. In a similar way, ethnicity is a function of (at least) two independent variables: ancestry and culture. Many people understand this without having a decent way to say it. But who outside of mathematicians knows from independent variables?

The Theory of Everything is a sheaf of theories

Reading The Grand Design, by Stephen Hawking and Leonard Mlodinow, led me to the idea that knowledge, at least scientific knowledge, is like a sheaf. Astronomy, biology, chemistry and physics are different systems of knowledge. In some sense Newton discovered a map that interpreted astronomy in physics, Linus Pauling did something like that with chemistry and physics (calculating chemical reactions using quantum mechanics), and Crick and Watson got hold of a basic fact that interprets biology in chemistry.

Now physicists are worried because (in terms of the metaphors of sheaves) physics seems to consist of two theories, quantum mechanics and large-scale physics, that may be different open sets in a sheaf that doesn’t have a global element, and possibly even worse, the restriction maps to their intersection may not be compatible. In other words, it not only doesn’t have a global element but it may be only a presheaf!

Now that will not sit well with scientists. Ordinary people go through life having different theories about love, religion, politics, when you kick a table it hurts your foot, and so on, and don’t seem to worry a bit about whether the restriction maps are compatible. Many scientists seem to me to believe that all the restriction maps are compatible, but we don’t know the details yet. And many of them want to throw out whole theories (astrology, ESP, and lately religion) because they can’t think how the restriction maps could be compatible.

There is evidence that the scientists are right: more and more overlaps between different theories have been shown compatible over the years. All different experiences can be connected by one sheaf of theories. That feeling is base on historical experience, but also it is intrinsic to the scientific method to assume that you can reconcile different aspects of whatever you are studying. It isn’t a matter solely of faith that there is one Theory (sheaf) of Everything; it is a matter of methodology. That knowledge forms a sheaf, not just a presheaf is the claim that all knowledge is compatible. That there may not be a global element, one Theory of Everything, is a separate idea and one that Hawking & Mlodinow seem to hint at. It is certainly worth considering the possibility that there is no global element in the Universal Sheaf of Theories.

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Presenting math on the web

2010/07/12 Charles Wells 2 Comments

This is a long post about ways to present math on the web, in the context of what I have done with The Handbook of Mathematical Discourse and abstractmath.org (Abmath). “Ways to present math” include both organization and production technology.

The post is motivated by and focused on my plans to reconstruct Abmath this fall, when I will not be teaching. During the last couple of years I have experimented with several possibilities for the reconstruction (while doing precious little on the actual website) and have come to a tentative conclusion about how I will do it. I am laying all this out here, past history and future plans, in the hope that readers will have suggestions that will help the process (or change my mind).

I set out to write both the Handbook and Abmath using ideas about how math should be presented on the web. They came out differently. Now I think I went wrong with some of the ways in which I organized Abmath and that I need to reconstruct it so that it is more like the Handbook. On the other hand, I have decided to stick with the production method I used for Abmath. I will explain.

Organization

My concept for both these works was that they would have these properties:

1) Each work would be a cloud of articles. They would have little or no hierarchy. They would consist of lots of short articles, not organized into chapters, sections and subsections.

2) The articles would be densely hyperlinked with each other and with the rest of the web. The reader would use the links to move from article to article. The articles might occur in alphabetical order in the production file but to the reader the order would be irrelevant.

I wanted the works to be organized that way because that is what I wanted from an information-presenting website. I want it that way because I am a grasshopper. Wikipedia and n-lab are each organized as a cloud of articles. I started writing the Handbook in the late nineties before Wikipedia began.

The Handbook exists in two forms. The web version is a hypertext PDF file that consists of short articles with extensive interlinking. The printed book has the same short articles arranged in alphabetical order. In the book form, the links are replaced by page indices (“paper hyperlinks”). In both forms some links are arranged as lists of related topics.

Abstractmath.org is a large, interlinked collection of html pages. They are organized in four large sections with many subsections.

Many entrances

For this cloud of articles arrangement to work, there must be many entrances into the website, so that a reader can find what they want. The Handbook has a list of entries in alphabetical order. Certain entries (for example the entries on attitudes, on behaviors, and on multiple meanings) have internal lists of links to examples of what that entry discusses. In addition, the paper version has an index that (in theory) provides links to all important occurrences of each concept in the book. This index is not included in the current hypertext version, although the LaTeX package hyperref would make it possible to include it. On the other hand, the hypertext version has the PDF search capability.

Abmath has a table of contents, listing articles in hierarchical form, as well as an index, which is different from the Handbook index in that it gives only one link from each word or phrase. In addition, it has header sections that briefly describe the contents of each main section and (in some cases) subsection, and also a Diagnostic Examples section (currently fragmentary)in which each entry provides a description of a particular problem that someone may have in understanding abstract math, with links to where it is discussed. The website currently has no search capability.

The Handbook is really a cloud of articles, and Abmath is not. I made a serious mistake imposing a hierarchy on Abmath, and that is the main thing I want to correct when I reconstruct it. Basically, I want to dissolve the hierarchy into a cloud of articles.

Production methods

The Handbook was composed using LaTeX. It originally existed in hypertext form (in a PDF file) and lived on the web for several years, generating many useful suggestions. I wrote a LaTeX header that could be set to produce PDF output with hyperlinks or PDF output formatted as a book with paper hyperlinks; that form was eventually published as a book.

I used a number of Awk programs to gather the various kinds of links. For example, every entry referring to a math word that has multiple meanings was marked and an Awk program gathered them into a list of links.

I generated the html pages for Abmath using Microsoft Word and MathType. MathType is very easy to use and has the capability (recently acquired) of converting all math entries that it generated into TeX. The method used for Abmath has several defects. You can’t apply Awk (or nowadays Python) programs to a Word document since it is in a proprietary format. Another problem is that the appearance of the result varies with browser.

But the Abmath method also has advantages. It produces html documents which can be read in windows that you can make narrower or wider and the text will adjust. PDF files are fixed width and rigid, and I find clicking on links requires you to be annoyingly precise with your fingers.

So my original thought was to go back to LaTeX for the new version of Abmath. There are several ways to produce html files from LaTeX, and converting the MathType entries to TeX provides a big headstart on converting the Word files into text files. Then I could use Awk to do a lot of bookkeeping and cut the hyperlink errors, the way I did with the Handbook.

So at first I was quite nostalgic about the wonderful time I had doing the Handbook in LaTeX — until I remembered all the fussing I did to include illustrations and marginal remarks. (I couldn’t just put the illo there and leave it.) Until I remembered how slowly the resulting PDF file loads because there seems to be no way to break it into individual article files without breaking the links.

And then I found that (as far as I could determine) there is no HTMLTeX that produces a reasonable HTML file from any TeX file the way PDFTeX produces a PDF file from any TeX file, using Knuth’s TeX program. In fact all the TeX to HTML systems I investigated don’t use Knuth’s program at all — they just have code in some programming language that reads a TeX file and interprets what the programmer felt like interpreting. I would love to be contradicted concerning this.

So now my thought is to stick with Word and MathType. And to do textual manipulation I will have to learn Word Basic. I just ordered two books on Word Basic. I would rather learn Python, but I have to work with what I have already done. Stay tuned.

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Making L'Hopital's Rule Go Crazy

2010/07/09 Charles Wells 3 Comments

Making L’Hôpital’s Rule Go Crazy

Calc II homework problem: Determine ${\lim_{n\rightarrow\infty} \dfrac{n}{\sqrt{n^2+1}}}$ , if it exists.

Answer: This is easy algebra: ${\dfrac{n}{\sqrt{n^2+1}}=\dfrac{\frac{1}{n}n}{\frac{1}{n}\sqrt{n^2+1}}= \dfrac{1}{\sqrt{1+\frac{1}{n^2}}}}$ , which goes to 1 as ${n\rightarrow\infty}$ .

This is nevertheless a Mean Problem. Usually, L’Hôpital’s Rule works on problems like this, but if you try it with this problem, you go into an infinite loop:

${\lim_{n\rightarrow\infty}\dfrac{n}{\sqrt{n^2+1}}= \lim_{n\rightarrow\infty}\dfrac{\sqrt{n^2+1}}{n}= \lim_{n\rightarrow\infty}\dfrac{n}{\sqrt{n^2+1}}= \lim_{n\rightarrow\infty}\dfrac{\sqrt{n^2+1}}{n}=\ldots}$ .

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Function as map

2010/05/24 Charles Wells Leave a comment

This is a first draft of an article to eventually appear in abstractmath.

Images and metaphors

To explain a math concept, you need to explain how mathematicians think about the concept. This is what in abstractmath I call the images and metaphors carried by the concept. Of course you have to give the precise definition of the concept and basic theorems about it. But without the images and metaphors most students, not to mention mathematicians from a different field, will find it hard to prove much more than some immediate consequences of the definition. Nor will they have much sense of the place of the concept in math and applications.

Teachers will often explain the images and metaphors with handwaving and pictures in a fairly vague way. That is good to start with, but it’s important to get more precise about the images and metaphors. That’s because images and metaphors are often not quite a good fit for the concept — they may suggest things that are false and not suggest things that are true. For example, if a set is a container, why isn’t the element-of relation transitive? (A coin in a coinpurse in your pocket is a coin in your pocket.)

“A metaphor is a useful way to think about something, but it is not the same thing as the same thing.” (I think I stole that from the Economist.) Here, I am going to get precise with the notion that a function is a map. I am acting like a mathematician in “getting precise”, but I am getting precise about a metaphor, not about a mathematical object.

A function is a map

A map (ordinary paper map) of Minnesota has the property that each point on the paper represents a point in the state of Minnesota. This map can be represented as a mathematical function from a subset of a 2-sphere to ${{\mathbb R}^2}$ . The function is a mathematical idealization of the relation between the state and the piece of paper, analogous to the mathematical description of the flight of a rocket ship as a function from ${{\mathbb R}}$ to ${{\mathbb R}^3}$ .

The Minnesota map-as-function is probably continuous and differentiable, and as is well known it can be angle preserving or area preserving but not both.

So you can say there is a point on the paper that represents the location of the statue of Paul Bunyan in Bemidji. There is a set of points that represents the part of the Mississippi River that lies in Minnesota. And so on.

A function has an image. If you think about it you will realize that the image is just a certain portion of the piece of paper. Knowing that a particular point on the paper is in the image of the function is not the information contained in what we call “this map of Minnesota”.

This yields what I consider a basic insight about function-as-map: The map contains the information about the preimage of each point on the paper map. So:

The map in the sense of a “map of Minnesota” is represented by the whole function, not merely by the image.

I think that is the essence of the metaphor that a function is a map. And I don’t think newbies in abstractmath always understand that relationship.

A morphism is a map

The preceding discussion doesn’t really represent how we think of a paper map of Minnesota. We don’t think in terms of points at all. What we see are marks on the map showing where some particular things are. If it is a road map it has marks showing a lot of roads, a lot of towns, and maybe county boundaries. If it is a topographical map it will show level curves showing elevation. So a paper map of a state should be represented by a structure preserving map, a morphism. Road maps preserve some structure, topographical maps preserve other structure.

The things we call “maps” in math are usually morphisms. For example, you could say that every simple closed curve in the plane is an equivalence class of maps from the unit circle to the plane. Here equivalence class meaning forget the parametrization.

The very fact that I have to mention forgetting the parametrization is that the commonest mathematical way to talk about morphisms is as point-to-point maps with certain properties. But we think about a simple closed curve in the plane as just a distorted circle. The point-to-point correspondence doesn’t matter. So this example is really talking about a morphism as a shape-preserving map. Mathematicians introduced points into talking about preserving shapes in the nineteenth century and we are so used to doing that that we think we have to have points for all maps.

Not that points aren’t useful. But I am analyzing the metaphor here, not the technical side of the math.

Groups are functors

People who don’t do category theory think the idea of a mathematical structure as a functor is weird. From the point of view of the preceding discussion, a particular group is a functor from the generic group to some category. (The target category is Set if the group is discrete, Top if it is a topological group, and so on.)

The generic group is a group in a category called its theory or sketch that is just big enough to let it be a group. If the theory is the category with finite products that is just big enough then it is the Lawvere theory of the group. If it is a topos that is just big enough then it is the classifying topos of groups. The theory in this sense is equivalent to some theory in the sense of string-based logic, for example the signature-with-axioms (equational theory) or the first order theory of groups. Johnstone’s Elephant book is the best place to find the translation between these ideas.

A particular group is represented by a finite-limit-preserving functor on the algebraic theory, or by a logical functor on the classifying topos, and so on; constructions which bring with them the right concept of group homomorphisms as well (they will be any natural transformations).

The way we talk about groups mimics the way we talk about maps. We look at the symmetric group on five letters and say its multiplication is noncommutative. “Its multiplication” tells us that when we talk about this group we are talking about the functor, not just the values of the functor on objects. We use the same symbols of juxtaposition for multiplication in any group, “ ${1}$ ” or “ ${e}$ ” for the identity, “ ${a^{-1}}$ ” for the inverse of ${a}$ , and so on. That is because we are really talking about the multiplication, identity and inverse function in the generic group — they really are the same for all groups. That is because a group is not its underlying set, it is a functor. Just like the map of Minnesota “is” the whole function from the state to the paper, not just the image of the function.

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exposition, Handbook, language, language of math, math

Expository writing in the future

2010/05/16 Charles Wells 3 Comments

I have written a lot about math exposition in the past. [Note 1.] Lately I have been thinking about the effect of technological change on exposition.

Texting

A lot of commentators have complained that their students’ writing style has “deteriorated” because of texting, specifically their use of abbreviations and acronyms.

Last January I resumed teaching mathematics after an exactly ten year lapse. My students and I email a lot, post on message boards, hand in homework, write up tests. I have seen very few “lol”s and “cu”s and the like, mostly in emails and almost entirely from students whose native language is not English. (See Note 1.)

As far as I can see the students’ written language has not deteriorated. In fact I think native English speakers write better English than they did ten years ago. (But Minnesota has a considerably better educational system than Ohio.)

Besides, if lol and cu become part of the written language, so what? Many Old Fogies may find it jarring, but Old Fogies die and their descendants talk however they want to.

Bulleted lists

I have been using Powerpoint part of the time in teaching (I had already given some talks using it). People complain about that affecting our style, too. But I think that in particular bulleted and numbered lists are great. I wish people would use them more often. Consider this passage from a recent version of Thomas’ Calculus [1]:

$\displaystyle \int_a^bx\,dx=\dfrac{b^2}{2}-\dfrac{a^2}{2}\quad (a< b)\quad\quad\quad(1)$

This computation gives the area of a trapezoid. Equation (1) remains valid when ${a}$ and ${b}$ are negative. When ${a<b<0}$ , the definite integral value … is a negative number, the negative of the area of the trapezoid dropping down to the line ${y=x}$ below the ${x}$ -axis. When ${a<0}$ and ${b>0}$ , Equation (1) is still valid and the definite integral gives the difference between two areas …

It would be much better to write something like this:

Equation (1) is valid for any ${a}$ and ${b}$ .

When ${a}$ and ${b}$ are positive, Equation (1) gives the area of a trapezoid.

When ${a}$ and ${b}$ are both negative, the result is negative and is the negative of the area…

When ${a<0}$ and ${b>0}$ , the result is the difference between two areas…

That is much easier to read than the first version, in which you have to parse through the paragraph detecting that it states parallel facts. That is not terribly difficult but it slows you down. Especially in this case where the sentences are not written in parallel and contain remarks about validity in scattered places when in fact the equation is valid for all cases.

This book does use numbered or lettered lists in many other places.

The future is upon us

Lots of lists and illustrations require more paper. This will go away soon. Some future edition of the book on an e-reader could contain this list of facts as a nicely spaced list, much easier to grasp, and could contain three graphs, with ${a}$ and ${b}$ respectively left of the ${x}$ -axis, straddling it, and to the right of it. This will cost some preparation time but no paper and computer memory at the scale of a book is practically free.

I use bulleted lists a lot in abstractmath, as here. Abstractmath is intended to be read on the computer. It is not organized linearly and a paper copy would not be particularly useful.

By the way, since the last time I looked at this page all the bullets have been replaced with copyright signs. (In three different browsers!) Somebody’s been Messing With Me. AArgH.

The Irish mystery writer Ken Bruen regularly uses lists, without bullets or numbers. Look at page 3 of The Killing of the Tinkers.

Some people find bulleted lists jarring simply because they are new. I think some are academic snobs who diss anything that sounds like something a business person would do. See my remarks at the end of the section on texting.

Notes

1. You can see much of what I have said on this blog about exposition by reading the posts labeled “exposition” (scroll down to the list of categories in the left column.) See also Varieties of Mathematical Prose by Atish Bagchi and me.

2. Foreign language speakers also write things like “Hi Charles” instead of “Dear Professor Wells” or using no greeting at all (which is probably the best thing to do). Dealing with a foreign language requires familiarity with the local social structure and customs of address, of being aware of levels of the various formal and informal registers, and so on. When we lived in Switzerland, how was I to know that “Ciao” went with “du” and “wiederluege” went with “Sie”? (If I remember correctly. Ye Gods, that was 35 years ago.)

References

1. Thomas’ Calculus, Early Transcendentals, Eleventh Edition, Media Upgrade. Pearson Education, 2008.

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Gyre&Gimble

All posts by Charles Wells

The Mathematical Definition of Function

Introduction

Specification of function

Examples

Definition of function

The functional property

Example: Graph of a function defined by a formula

Mathematical definition of function

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Two Examples

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