# Representations of functions III

## Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations. This post is a draft of the sections on representations of finite functions.

The diagrams in this post were created using the Mathematica Notebook Constructions for cographs and endographs of finite functions.nb.
You can access this notebook if you have Mathematica, which can be bought, but is available for free for faculty and students at many universities, or with Mathematica CDF Player, which is free for anyone and runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

## Graphs of finite functions

When a function is continuous, its graph shows up as a curve in the plane or as a curve or surface in 3D space. When a function is defined on a set without any notion of continuity (for example a finite set), the graph is just a set of ordered pairs and does not tell you much.

A finite function $f:S\to T$ may be represented in these ways:

• Its graph $\{(s,f(s))|s\in S\}$. This is graph as a mathematical object, not as a drawing or as a directed graph — see graph (two meanings)).
• A table, rule or two-line notation. (All three of these are based on the same idea, but differ in presentation and are used in different mathematical specialties.)
• By using labels with arrows between them, arranged in one of two ways:
• A cograph, in which the domain and the codomain are listed separately.
• An endograph, in which the elements of the domain and the codomain are all listed together without repetition.

All these techniques can also be used to show finite portions of infinite discrete functions, but that possibility will not be discussed here.

### Introductory Example

Let $\text{f}:\{a,b,c,d,e\}\to\{a,b,c,d\}$ be the function defined by requiring that $f(a)=c$, $f(b)=a$, $f(c)=c$, $f(d)=b$, and $f(e)=d$.

#### Graph

The graph of $f$ is the set
$(a,c),(b,a),(c,c),(d,b),(e,d)$
As with any set, the order in which the pairs are listed is irrelevant. Also, the letters $a$, $b$, $c$, $d$ and $e$ are merely letters. They are not variables.

#### Table

$\text{f}$ is given by this table:

This sort of table is the format used in databases. For example, a table in a database might show the department each employee of a company works in:

#### Rule

The rule determined by the finite function $f$ has the form

$(a\mapsto b,b\mapsto a,c\mapsto c,d\mapsto b,e\mapsto d)$

Rules are built in to Mathematica and are useful in many situations. In particular, the endographs in this article are created using rules. In Mathematica, however, rules are written like this:

$(a\to b,b\to a,c\to c,d\to b,e\to d)$

This is inconsistent with the usual math usage (see barred arrow notation) but on the other hand is easier to enter in Mathematica.

In fact, Mathematica uses very short arrows in their notation for rules, shorter than the ones used for the arrow notation for functions. Those extra short arrows don’t seems to exist in TeX.

#### Two-line notation

Two-line notation is a kind of horizontal table.

$\begin{pmatrix} a&b&c&d&e\\c&a&c&b&d\end{pmatrix}$

The three notations table, rule and two-line do the same thing: If $n$ is in the domain, $f(n)$ is shown adjacent to $n$ — to its right for the table and the rule and below it for the two-line.

Note that in contrast to the table, rule and two-line notation, in a cograph each element of the codomain is shown only once, even if the function is not injective.

#### Cograph

To make the cograph of a finite function, you list the domain and codomain in separate parallel rows or columns (even if the domain and codomain are the same set), and draw an arrow from each $n$ in the domain to $f(n)$ in the codomain.

This is the cograph for $\text{f}$, represented in columns

and in rows (note that $c$ occurs only once in the codomain)

Pretty ugly, but the cograph for finite functions does have its uses, as for example in the Wikipedia article composition of functions.

In both the two-line notation and in cographs displayed vertically, the function goes down from the domain to the codomain. I guess functions obey the law of gravity.

##### Rearrange the cograph

There is no expectation that in the cograph $f(n)$ will be adjacent to $n$. But in most cases you can rearrange both the domain and the codomain so that some of the structure of the function is made clearer; for example:

The domain and codomain of a finite function can be rearranged in any way you want because finite functions are not continuous functions. This means that the locations of points $x_1$ and $x_2$ have nothing to do with the locations of $f(x_1)$ and $f(x_2)$: The domain and codomain are discrete.

#### Endograph

The endograph of a function $f:S\to T$ contains one node labeled $s$ for each $s\in S\cup T$, and an arrow from $s$ to $s’$ if $f(s)=s’$. Below is the endograph for $\text{f}$.

The endograph shows you immediately that $\text{f}$ is not a permutation. You can also see that with whatever letter you start with, you will end up at $c$ and continue looping at $c$ forever. You could have figured this out from the cograph (especially the rearranged cograph above), but it is not immediately obvious in the cograph the way it in the endograph.

There are more examples of endographs below and in the blog post
A tiny step towards killing string-based math. Calculus-type functions can also be shown using endographs and cographs: See Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s, by Martin Flashman, and my blog posts Endographs and cographs of real functions and Demos for graph and cograph of calculus functions.

### Example: A permutation

Suppose $p$ is the permutation of the set $\{0,1,2,3,4,5,6,7,8,9\}$given in two-line form by
$\begin{pmatrix} 0&1&2&3&4&5&6&7&8&9\\0&2&1&4&5&3&7&8&9&6\end{pmatrix}$

#### Endograph

Again, the endograph shows the structure of the function much more clearly than the cograph does.

The endograph consists of four separate parts (called components) not connected with each other. Each part shows that repeated application of the function runs around a kind of loop; such a thing is called a cycle. Every permutation of a finite set consists of disjoint cycles as in this example.

#### Disjoint cycle notation

Any permutation of a finite set can be represented in disjoint cycle notation: The function $p$ is represented by:

$(0)(1,2)(3,4,5)(6,7,8,9)$

Given the disjoint cycle notation, the function can be determined as follows: For a given entry $n$, $p(n)$ is the next entry in the notation, if there is a next entry (instead of a parenthesis). If there is not a next entry, $p(n)$ is the first entry in the cycle that $n$ is in. For example, $p(7)=8$ because $8$ is the next entry after $7$, but $p(5)=3$ because the next symbol after $5$ is a parenthesis and $3$ is the first entry in the same cycle.

The disjoint cycle notation is not unique for a given permutation. All the following notations determine the same function $p$:

$(0)(1,2)(4,5,3)(6,7,8,9)$
$(0)(1,2)(8,9,6,7)(3,4,5)$
$(1,2)(3,4,5)(0)(6,7,8,9)$
$(2,1)(5,3,4)(9,6,7,8)$
$(5,3,4)(1,2)(6,7,8,9)$

Cycles such as $(0)$ that contain only one element are usually omitted in this notation.

### Example: A tree

Below is the endograph of a function $t:\{0,1,2,3,4,5,6,7,8,9\}\to\{0,1,2,3,4,5,6,7,8,9\}$

This endograph is a tree. The graph of a function $f$ is a tree if the domain has a particular element $r$ called the root with the properties that

• $f(r)=r$, and
• starting at any element of the domain, repreatedly applying $f$ eventually produces $r$.

In the case of $t$, the root is $4$. Note that $t(4)=4$, $t(t(7))=4$, $t(t(t(9)))=4$, $t(1)=4$, and so on.

The endograph

shown here is also a tree.

See the Wikipedia article on trees for the usual definition of tree as a special kind of graph. For reading this article, the definition given in the previous paragraph is sufficient.

### The general form of a finite function

This is the endograph of a function $t$ on a $17$-element set:

It has two components. The upper one contains one $2$-cycle, and no matter where you start in that component, when you apply $t$ over and over you wind up flipping back and forth in the $2$-cycle forever. The lower component has a $3$-cycle with a similar property.

This illustrates a general fact about finite functions:

• The endograph of any finite function contains one or more components $C_1$ through $C_k$.
• Each component $C_k$ contains exactly one $n_k$ cycle, for some integer $n_k\geq 1$, to which are attached zero or more trees.
• Each tree in $C_k$ is attached in such a way that its root is on the unique cycle contained in $C_k$.

In the example above, the top component has three trees attached to it, two to $3$ and one to $4$. (This tree does not illustrate the fact that an element of one of the cycles does not have to have any trees attached to it).

You can check your understanding of finite functions by thinking about the following two theorems:

• A permutation is a finite function with the property that its cycles have no trees attached to them.
• A tree is a finite function that has exactly one component whose cycle is a $1$-cycle.

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# Representations of functions II

## Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations.

This post includes a draft of the introduction to the entire new chapter (immediately below) and of the sections on graphs of continuous functions of one variable with values in the plane and in 3-space. Later posts will concern multivariable continuous functions and finite discrete functions.

## Introduction to the new Chapter

Functions can be represented visually in many different ways. There is a sharp difference between representing continuous functions and representing discrete functions.

For a continuous function $f$, $f(x)$ and $f(x’)$ tend to be close together when $x$ and $x’$ are close together. That means you can represent the values at an infinite number of points by exhibiting them for a bunch of close-together points. Your brain will automatically interpret the points nearby that are not represented.

Nothing like this works for discrete functions. Many different arrangements of the inputs and outputs can be made. Different arrangements may be useful for representing different properties of the function.

### Illustrations

The illustrations were created using these Mathematica Notebooks:

These notebooks contain many more examples of the ways functions can be represented than are given in this article. The notebooks also contain some manipulable diagrams which may help you understand the diagrams. In addition, all the 3D diagrams can be rotated using the cursor to get different viewpoints. You can access these tools if you have Mathematica, which is available for free for faculty and students at many universities, or with Mathematica CDF Player, which runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

## Functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}$

Suppose $F:\mathbb{R}\to\mathbb{R}\times\mathbb{R}$. That means you put in one number and get out a pair of numbers.

### The unit circle

An example is the unit circle, which is the graph of the function $t\mapsto(\cos t,\sin t)$. That has this parametric plot:

Because $\cos^2 t+\sin^2 t=1$, every real number $t$ produces a point on the unit circle. Four point are shown. For example,$(\cos\pi,\,\sin\pi)=(-1,0)$ and
$(\cos(5\pi/3),\,\sin(5\pi/3))=(\frac{1}{2},\frac{\sqrt3}{2})\approx(.5,.866)$

#### $t$ as time

In graphing functions $f:\mathbb{R}\to\mathbb{R}$, the plot is in two dimensions and consists of the points $(x,f(x))$: the input and the output. The parametric plot shown above for $t\mapsto(\cos^2 t+\sin^2)$ shows only the output points $(\cos t,\sin t)$; $t$ is not plotted on the graph at all. So the graph is in the plane instead of in three-dimensional space.

An alternative is to use time as the third dimension: If you start at some number $t$ on the real line and continually increase it, the value $f(t)$ moves around the circle counterclockwise, repeating every $2\pi$ times. If you decrease $t$, the value moves clockwise. The animated gif circlemovie.gif shows how the location of a point on the circle moves around the circle as $t$ changes from $0$ to $2\pi$. Every point is traversed an infinite number of times as $t$ runs through all the real numbers.

#### The unit circle with $t$ made explicit

Since we have access to three dimensions, we can show the input $t$ explicitly by using a three-dimensional graph, shown below. The blue circle is the function $t\mapsto(\cos t,\sin t,0)$ and the gold helix is the function $t\mapsto(\cos t,\sin t,.2t)$.

The introduction of $t$ as the value in the vertical direction changes the circle into a helix. The animated .gif covermovie.gif shows both the travel of a point on the circle and the corresponding point on the helix.

As $t$ changes, the circle is drawn over and over with a period of $2\pi$. Every point on the circle is traversed an infinite number of times as $t$ runs through all the real numbers. But each point on the helix is traversed exactly once. For a given value of $t$, the point on the helix is always directly above or below the point on the circle.

The helix is called the universal covering space of the circle, and the set of points on the helix over (and under) a particular point $p$ on the circle is called the fiber over $p$. The universal cover of a space is a big deal in topology.

### Figure-8 graph

This is the parametric graph of the function $t\mapsto(\cos t,\sin 2t)$.

Notice that it crosses itself at the origin, when $t$ is any odd multiple of $\frac{\pi}{2}$.

Below is the universal cover of the Figure-8 graph. As you can see, the different instances of crossing at $(0,0)$ are separated. The animated.gif Fig8movie shows the paths taken as $t$ changes on the figure 8 graph and on its universal cover

## Functions from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$

The graph of a function from a subset of $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ can also be drawn as a parametric graph in three-dimensional space, giving a three-dimensional curve. The trick that I used in the previous section of showing the input parameter so that you can see the universal cover won’t work in this case because it would require four dimensions.

### Universal covers

The gold curves in the figures for the universal covers of the circle and the figure 8 are examples of functions from $\mathbb{R}$ to $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$.

### The seven-pointed crown

Here are views from three different angles of the graph of the function $t\mapsto(\cos t, \sin t, \sin 7t)$:

The animated gif crownmovie.gif represents the parameter $t$ in time.

### Another curve in space

Below are two views of the curve defined by $t\mapsto({-4t^2+53t)/18,t,.4(-t^2+1-10t)}$.

The following plots the $x$-curve $-4t^2+53t)/18$ gold in the $yz$ plane and the $z$ curve $.4(-t^2+1-10t)$ in the $xy$ plane. The first and third views are arranged so that you see the curve just behind one of those two planes.

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# Representations of functions I

## Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations.

This post includes a draft of the introduction to the new chapter (immediately below) and of the section Graphs of continous functions of one variable. Later posts will concern multivariable continuous functions, probably in two or three sections, and finite discrete functions.

## Introduction to the new abstractmath chapter on representations of functions

Functions can be represented visually in many different ways. There is a sharp difference between representing continuous functions and representing discrete functions.

For a continuous function $f$, $f(x)$ and $f(x’)$ tend to be close together when $x$ and $x’$ are close together. That means you can represent the values at an infinite number of points by exhibiting them for a bunch of close-together points. Your brain will automatically interpret the points nearby that are not represented.

Nothing like this works for discrete functions. As you will see in the section on discrete functions, many different arrangements of the inputs and outputs can be made. In fact, different arrangements may be useful for representing different properties of the function.

### Illustrations

The illustrations were created using these Mathematica Notebooks:

These notebooks contain many more examples of the ways functions can be represented than are given in this article. The notebooks also contain some manipulable diagrams which may help you understand the diagrams. In addition, all the 3D diagrams can be rotated using the cursor to get different viewpoints. You can access these tools if you have Mathematica, which is available for free for faculty and students at many universities, or with Mathematica CDF Player, which runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

## Graphs of continous functions of one variable

The most familiar representations of continuous functions are graphs of functions with one real variable. Students usually first see these in secondary school. Such representations are part of the subject called Analytic Geometry. This section gives examples of such functions.

There are other ways to represent continuous functions, in particular the cograph and the endograph. These will be the subject of a separate post.

The graph of a function $f:S\to T$ is the set of ordered pairs $\{(x,f(x))\,|\,x\in S\}$. (More about this definition here.)

In this section, I consider continuous functions for which $S$ and $T$ are both subsets of the real numbers. The mathematical graph of such a function are shown by plotting the ordered pairs $(x,f(x))$ as points in the two-dimensional $xy$-plane. Because the function is continuous, when $x$ and $x’$ are close to each other, $f(x)$ and $f(x’)$ tend to be close to each other. That means that the points that have been plotted cause your brain to merge together into a nice curve that allows you to visualize how $f$ behaves.

#### Example

This is a representation of the graph of the curve $g(x):=2-x^2$ for approximately the interval $(-2,2)$. The blue curve represents the graph.

The brown right-angled line in the upper left side, for example, shows how the value of independent variable $x$ at $(0.5)$ is plotted on the horizontal axis, and the value of $g(0.5)$, which is $1.75$, is plotted on the vertical axis. So the blue graph contains the point $(0.5,g(0.5))=(0.5,1.75)$. The animated gif upparmovie.gif shows a moving version of how the curve is plotted.

#### Fine points

• The mathematical definition of the graph is that it is the set $\{(x,2-x^2)\,|\,x\in\mathbb{R}\}$. The blue curve is not, of course, the mathematical graph, it represents the mathematical graph.
• The blue curve consists of a large but finite collection of pixels on your screen, which are close enough together to appear to form a continuous curve which approximates the mathematical graph of the function.
• Notice that I called the example the “representation of the graph” instead of just “graph”. That maintains the distinction between the mathematical ordered pairs $(x,g(x))$ and the pixels you see on the screen. But in fact mathe­maticians and students nearly always refer to the blue line of pixels as the graph. That is like pointing to a picture of your grandmother and saying “this is my grandmother”. There is nothing wrong with saying things that way. But it is worth understanding that two different ideas are being merged.

### Discontinuous functions

A discontinuous function which is continuous except for a small finite number of breaks can also be represented with a graph.

#### Example

Below is the function $f:\mathbb{R}\to\mathbb{R}$ defined by
f(x):=\left\{ \begin{align} 2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\gt0) \\ 1-x^2\,\,\,\,\,\,(-1\lt x\lt 0) \\ 2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\lt-1) \end{align}\right.

##### Example

The Dirichlet function is defined by
$F(x):= \begin{cases} 1 & \text{if }x\text{ is rational}\\ \frac{1}{2} & \text{if }x\text{ is irrational}\\ \end{cases}$  for all real $x$.

The abmath article Examples of functions spells out in detail what happens when you try to draw this function.

### Graphs can fool you

The graph of a continuous function cannot usually show the whole graph, unless it is defined only on a finite interval. This can lead you to jump to conclusions.

##### Example

For example, you can’t tell from the the graph of the function $y=2-x^2$ whether it has a local minimum (because the graph does not show all of the function), although you can tell by using calculus on the formula that it does not have one. The graph looks like it might have a vertical asymptotes, but it doesn’t, again as you can tell from the formula.

by looking at its graph
is useful but dangerous.

##### Example

Below is the graph of the function
$f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1} \right)}^{6}}$

If you didn’t know the formula for the function (but know it is continuous), you could still see that it has a local maximum somewhere to the right of $x=1$. It looks like it has one or more zeroes around $x=-1$ and $x=2$. And it looks like it has an asymptote somewhere to the right of $x=2.5$.

If you do know the formula, you can find out many things about the function that you can’t depend on the graph to see.

• You can see immediately that $f$ has a zero at $x=\sqrt[3]{10}$, which is about $2.15$.
• If you notice that the denominator is positive for all $x$, you can figure out that
• $\sqrt[3]{10}$ is the only root.
• $f(x)\geq0$ for all $x$.
• $f$ has an asymptote as $x\to-\infty$ (use L’Hôpital).
• Numerical analysis (I used Mathematica) shows that $f'(x)$ has two zeros, at $\sqrt[3]{10}$ and at about $x=1.1648$. $f”(1.1648)$ is about $-10.67$ , which strongly suggests that $f$ has a local max near $1.1648$, consistent with the graph.
• Since $f$ is defined for every real number, it can’t have a vertical asymptote anywhere. The graph looks like it becomes vertical somewhere to the right of $x=2.4$, but that is simply an illustration of the unbelievably fast growth of any exponential function.
• The section on Zooming and Chunking gives other details.

## Acknowledgments

Sue VanHattum.

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# Insights into mathematical definitions

My general practice with abstractmath.org has been to write about the problems students have at the point where they first start studying abstract math, with some emphasis on the languages of math. I have used my own observations of students, lexicographical work I did in the early 2000’s, and papers written by workers in math ed at the college level.

A few months ago, I finished revising and updating abstractmath.org. This took rather more than a year because among other things I had to reconstitute the files so that the html could be edited directly. During that time I just about quit reading the math ed literature. In the last few weeks I have found several articles that have changed my thinking about some things I wrote in abmath, so now I need to go back and revise some more!

In this post I will make some points about definitions that I learned from the paper by Edwards and Ward and the paper by Selden and Selden

I hope math ed people will read the final remarks.

## Peculiarities of math definitions

When I use a word, it means just what I choose it to mean–neither more nor less.” — Humpty Dumpty

A mathematical definition is fundamentally different from other sorts of definitions in two different ways. These differences are not widely appreciated by students or even by mathematicians. The differences cause students a lot of trouble.

### List of properties

One of the ways in which a math definition is different from other kinds is that the definition of a math object is given by accumulation of attributes, that is, by listing properties that the object is required to have. Any object defined by the definition must have all those properties, and conversely any object with all the properties must be an example of the type of object being defined. Furthermore, there is no other criterion than the list of attributes.

Definitions in many fields, including some sciences, don’t follow this rule. Those definitions may list some properties the objects defined may have, but exceptions may be allowed. They also sometimes give prototypical examples. Dictionary definitions are generally based on observation of usage in writing and speech.

### Imposed by decree

One thing that Edwards and Ward pointed out is that, unlike definitions in most other areas of knowledge, a math definition is stipulated. That means that meaning of (the name of) a math object is imposed on the reader by decree, rather than being determined by studying the way the word is used, as a lexicographer would do. Mathematicians have the liberty of defining (or redefining) a math object in any way they want, provided it is expressed as a compulsory list of attributes. (When I read the paper by Edwards and Ward, I realized that the abstractmath.org article on math definitions did not spell that out, although it was implicit. I have recently revised it to say something about this, but it needs further work.)

An example is the fact that in the nineteenth century some mathe­maticians allowed $1$ to be a prime. Eventually they restricted the definition to exclude $1$ because including it made the statement of the Fundamental Theorem of Arithmetic complicated to state.

Another example is that it has become common to stipulate codomains as well as domains for functions.

## Student difficulties

### Giving the math definition low priority

Some beginning abstract math students don’t give the math definition the absolute dictatorial power that it has. They may depend on their understanding of some examples they have studied and actively avoid referring to the definition. Examples of this are given by Edwards and Ward.

### Arbitrary bothers them

Students are bothered by definitions that seem arbitrary. This includes the fact that the definition of “prime” excludes $1$. There is of course no rule that says definitions must not seem arbitrary, but the students still need an explanation (when we can give it) about why definitions are specified in the way they are.

### What do you DO with a definition?

Some students don’t realize that a definition gives a magic formula — all you have to do is say it out loud.
More generally, the definition of a kind of math object, and also each theorem about it, gives you one or more methods to deal with the type of object.

For example, $n$ is a prime by definition if $n\gt 1$ and the only positive integers that divide $n$ are $1$ and $n$. Now if you know that $p$ is a prime bigger than $10$ then you can say that $p$ is not divisible by $3$ because the definition of prime says so. (In Hogwarts you have to say it in Latin, but that is no longer true in math!) Likewise, if $n\gt10$ and $3$ divides $n$ then you can say that $n$ is not a prime by definition of prime.

The paper by Bills and Tall calls this sort of thing an operable definition.

The paper by Selden and Selden gives a more substantial example using the definition of inverse image. If $f:S\to T$ and $T’\subseteq T$, then by definition, the inverse image $f^{-1}T’$ is the set $\{s\in S\,|\,f(s)\in T’\}$. You now have a magic spell — just say it and it makes something true:

• If you know $x\in f^{-1}T’$ then can state that $f(x)\in T’$, and all you need to justify that statement is to say “by definition of inverse image”.
• If you know $f(x)\in T’$ then you can state that $x\in f^{-1}T’$, using the same magic spell.

Theorems can be operable, too. Wiles’ Theorem wipes out the possibility that there is an integer $n$ for which $n^{42}=365^{42}+666^{42}$. You just quote Wiles’ Theorem — you don’t have to calculate anything. It’s a spell that reveals impossibilities.

What the operability of definitions and theorems means is:

A definition or theorem is not just a static statement,it is a weapon for deducing truth.

Some students do not realize this. The students need to be told what is going on. They do not have to be discarded to become history majors just because they may not have the capability of becoming another Andrew Wiles.

## Final remarks

I have a wish that more math ed people would write blog posts or informal articles (like the one by Edwards and Ward) about what that have learned about students learning math at the college level. Math ed people do write scholarly articles, but most of the articles are behind paywalls. We need accessible articles and blog posts aimed at students and others aimed at math teachers.

And feel free to steal other math ed people’s ideas (and credit them in a footnote). That’s what I have been doing in abstractmath.org and in this blog for the last ten years.

## References

• Bills, L., & Tall, D. (1998). Operable definitions in advanced mathematics: The case of the least upper bound. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 104-111). Stellenbosch, South Africa: University of Stellenbosch.
• B. S. Edwards, and M. B. Ward, Surprises from mathematics education research: Student (mis) use of mathematical definitions (2004). American Mathematical Monthly, 111, 411-424.
• G. Lakoff, Women, Fire and Dangerous
Things
. University of Chicago Press, 1990. See his discussion of concepts and prototypes.
• J. Selden and A. Selden, Proof Construction Perspectives: Structure, Sequences of Actions, and Local Memory, Extended Abstract for KHDM Conference, Hanover, Germany, December 1-4, 2015. This paper may be downloaded from Academia.edu.
• A Handbook of mathematical discourse, by Charles Wells. See concept, definition, and prototype.
• Definitions, article in abstractmath.org. (Some of the ideas in this post have now been included in this article, but it is due for another revision.)
• Definitions in logic and mathematics in Wikipedia.

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# Very early difficulties II

This is the second part of a series of posts about certain difficulties math students have in the very early stages of studying abstract math. The first post, Very early difficulties in studying abstract math, gives some background to the subject and discusses one particular difficulty: Some students do not know that it is worthwhile to try starting a proof by rewriting what is to be proved using the definitions of the terms involved.

## Math StackExchange

The website Math StackExchange is open to any questions about math, even very easy ones. It is in contrast with Math OverFlow, which is aimed at professional mathematicians asking questions in their own field.

Math SE contains many examples of the early difficulties discussed in this series of posts, and I recommend to math ed people (not just RUME people, since some abstract math occurs in advanced high school courses) that they might consider reading through questions on Math SE for examples of misunderstanding students have.

There are two caveats:

• Most questions on Math SE are at a high enough level that they don’t really concern these early difficulties.
• Many of the questions are so confused that it is hard to pinpoint what is causing the difficulty that the questioner has.

## Connotations of English words

The terms(s) defined in a definition are often given ordinary English words as names, and the beginner automatically associates the connotations of the meaning of the English word with the objects defined in the definition.

### Infinite cardinals

If $A$ if a finite set, the cardinality of $A$ is simply a natural number (including $0$). If $A$ is a proper subset of another set $B$, then the cardinality of $A$ is strictly less than the cardinality of $B$.

In the nineteenth century, mathematicians extended the definition of cardinality for infinite sets, and for the most part cardinality has the same behavior as for finite sets. For example, the cardinal numbers are well-ordered. However, for infinite sets it is possible for a set and a proper subset of the set to have the same cardinality. For example, the cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers. This phenomenon causes major cognitive dissonance.

Question 1331680 on Math Stack Exchange shows an example of this confusion. I have also discussed the problem with cardinality in the abstractmath.org section Cardinality.

### Morphism in category theory

The concept of category is defined by saying there is a bunch of objects called objects (sorry bout that) and a bunch of objects called morphisms, subject to certain axioms. One requirement is that there are functions from morphisms to objects choosing a “domain” and a “codomain” of each morphism. This is spelled out in Category Theory in Wikibooks, and in any other book on category theory.

The concepts of morphism, domain and codomain in a category are therefore defined by abstract definitions, which means that any property of morphisms and their domains and codomains that is true in every category must follow from the axioms. However, the word “morphism” and the talk about domains and codomains naturally suggests to many students that a morphism must be a function, so they immediately and incorrectly expect to evaluate it at an element of its domain, or to treat it as a function in other ways.

#### Example

If $\mathcal{C}$ is a category, its opposite category $\mathcal{C}^{op}$ is defined this way:

• The objects of $\mathcal{C}^{op}$ are the objects of $\mathcal{C}$.
• A morphism $f:X\to Y$ of $\mathcal{C}^{op}$ is a morphism from $Y$ to $X$ of $\mathcal{C}$ (swap the domain and codomain).

In Question 980933 on Math SE, the questioner is saying (among other things) that in $\text{Set}^{op}$, this would imply that there has to be a morphism from a nonempty set to the empty set. This of course is true, but the questioner is worried that you can’t have a function from a nonempty set to the empty set. That is also true, but what it implies is that in $\text{Set}^{op}$, the morphism from $\{1,2,3\}$ to the empty set is not a function from $\{1,2,3\}$ to the empty set. The morphism exists, but it is not a function. This does not any any sense make the definition of $\text{Set}^{op}$ incorrect.

Student confusion like this tends to make the teacher want to have a one foot by six foot billboard in his classroom saying

A MORPHISM DOESN’T HAVE TO BE A FUNCTION!

However, even that statement causes confusion. The questioner who asked Question 1594658 essentially responded to the statement in purple prose above by assuming a morphism that is “not a function” must have two distinct values at some input!

That questioner is still allowing the connotations of the word “morphism” to lead them to assume something that the definition of category does not give: that the morphism can evaluate elements of the domain to give elements of the codomain.

So we need a more elaborate poster in the classroom:

The definition of “category” makes no requirement
that an object has elements
or that morphisms evaluate elements.

As was remarked long long ago, category theory is pointless.

### English words implementing logic

There are lots of questions about logic that show that students really do not think that the definition of some particular logical construction can possibly be correct. That is why in the abstractmath.org chapter on definitions I inserted this purple prose:

A definition is a totalitarian dictator.

It is often the case that you can explain why the definition is worded the way it is, and of course when you can you should. But it is also true that the student has to grovel and obey the definition no matter how weird they think it is.

#### Formula and term

In logic you learn that a formula is a statement with variables in it, for example “$\exists x((x+5)^3\gt2)$”. The expression “$(x+5)^3$” is not a formula because it is not a statement; it is a “term”. But in English, $H_2O$ is a formula, the formula for water. As a result, some students have a remarkably difficult time understanding the difference between “term” and “formula”. I think that is because those students don’t really believe that the definition must be taken seriously.

#### Exclusive or

Question 804250 in MathSE says:

“Consider $P$ and $Q$. Let $P+Q$ denote exclusive or. Then if $P$ and $Q$ are both true or are both false then $P+Q$ is false. If one of them is true and one of them is false then $P+Q$ is true. By exclusive or I mean $P$ or $Q$ but not both. I have been trying to figure out why the truth table is the way it is. For example if $P$ is true and $Q$ is true then no matter what would it be true?”

I believe that the questioner is really confused by the plus sign: $P+Q$ ought to be true if $P$ and $Q$ are both true because that’s what the plus sign ought to mean.

Yes, I know this is about a symbol instead of an English word, but I think the difficulty has the same dynamics as the English-word examples I have given.

If I have understood this difficulty correctly, it is similar to the students who want to know why $1$ is not a prime number. In that case, there is a good explanation.

#### Only if

The phrase “only if” simply does not mean the same thing in math as it does in English. In Question 17562 in MathSE, a reader asks the question, why does “$P$ only if $Q$” mean the same as “if $P$ then $Q$” instead of “if $Q$ then $P$”?

Many answerers wasted a lot of time trying to convince us that “$P$ only if $Q$” mean the same as “if $P$ then $Q$” in ordinary English, when in fact it does not. That’s because in English, clauses involving “if” usually connote causation, which does not happen in math English.

Consider these two pairs of examples.

1. “I take my umbrella only if it is raining.”
2. “If I take my umbrella, then it is raining.”
3. “I flip that switch only if a light comes on.”
4. “If I flip that switch, a light comes on.”

The average non-mathematical English speaker will easily believe that (1) and (4) are true, but will balk and (2) and (3). To me, (3) means that the light coming on makes me flip the switch. (2) is more problematical, but it does (to me) have a feeling of causation going the wrong way. It is this difference that causes students to balk at the equivalence in math of “$P$ only if $Q$” and “If $P$, then $Q$”. In math, there is no such thing as causation, and the truth tables for implication force us to live with the fact that these two sentences mean the same thing.

Henning Makholm’ answer to Question 17562 begins this way: “I don’t think there’s really anything to understand here. One simply has to learn as a fact that in mathematics jargon the words ‘only if’ invariably encode that particular meaning. It is not really forced by the everyday meanings of ‘only’ and’ if’ in isolation; it’s just how it is.” That is the best way to answer the question. (Other answerers besides Makholm said something similar.)

I have also discussed this difficulty (and other difficulties with logic) in the abmath section on “only if“.

## References

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# Very early difficulties in studying abstract math

## Introduction

There are a some difficulties that students have at the very beginning of studying abstract math that are overwhelmingly important, not because they are difficult to explain but because too many teachers don’t even know the difficulties exist, or if they do, they think they are trivial and the students should know better without being told. These difficulties cause too many students to give up on abstract math and drop out of STEM courses altogether.

I spent my entire career in math at Case Western Reserve University. I taught many calculus sections, some courses taken by math majors, and discrete math courses taken mostly by computing science majors. I became aware that some students who may have been A students in calculus essentially fell off a cliff when they had to do the more abstract reasoning involved in discrete math, and in the initial courses in abstract algebra, linear algebra, advanced calculus and logic.

That experience led me to write the Handbook of Mathematical Discourse and to create the website abstractmath.org. Abstractmath.org in particular grew quite large. It does describe some of the major difficulties that caused good students to fall of the abstraction cliff, but also describes many many minor difficulties. The latter are mostly about the peculiarities of the languages of math.

I have observed people’s use of language since I was like four or five years old. Not because I consciously wanted to — I just did. When I was a teenager I would have wanted to be a linguist if I had known what linguistics is.

I will describe one of the major difficulties here (failure to rewrite according to the definition) with an example. I am planning future posts concerning other difficulties that occur specifically at the very beginning of studying abstract math.

## Rewrite according to the definition

To prove that a statement
involving some concepts is true,
start by rewriting the statement
using the definitions of the concepts.

### Example

#### Definition

A function $f:S\to T$ is surjective if for any $t\in T$ there is an $s\in S$ for which $f(s)=t$.

#### Definition

For a function $f:S\to T$, the image of $f$ is the set $\{t\in T\,|\,\text{there is an }s\in S\text{ for which }f(s)=t\}$

#### Theorem

Let $f:S\to T$ be a function between sets. Then $f$ is surjective if and only if the image of $f$ is $T$.

#### Proof

If $f$ is surjective, then the statement “there is an $s\in S$ for which $f(s)=t$” is true for any $t\in T$ by definition of surjectivity. Therefore, by definition of image, the image of $f$ is $T$.

If the image of $f$ is $T$, then the definition of image means that there is an $s\in S$ for which $f(s)=t$ for any $t\in T$. So by definition of surjective, $f$ is surjective.

### “This proof is trivial”

The response of many mathematicians I know is that this proof is trivial and a student who can’t come up with it doesn’t belong in a university math course. I agree that the proof is trivial. I even agree that such a student is not a likely candidate for getting a Ph.D. in math. But:

• Most math students in an American university are not going to get a Ph.D. in math. They may be going on in some STEM field or to teach high school math.
• Some courses taken by students who are not math majors take courses in which simple proofs are required (particularly discrete math and linear algebra). Some of these students may simply be interested in math for its own sake!

A sizeable minority of students who are taking a math course requiring proofs need to be told the most elementary facts about how to do proofs. To refuse to explain these facts is a disfavor to the mathematics community and adds to the fear and dislike of math that too many people already have.

These remarks may not apply to students in many countries other than the USA. See When these problems occur.

### “This proof does not describe how mathematicians think”

The proof I wrote out above does not describe how I would come up with a proof of the statement, which would go something like this: I do math largely in pictures. I envision the image of $f$ as a kind of highlighted area of the codomain of $f$. If $f$ is surjective, the highlighting covers the whole codomain. That’s what the theorem says. I wouldn’t dream of writing out the proof I gave about just to verify that it is true.

### More examples

Abstractmath.org and Gyre&Gimble contain several spelled-out theorems that start by rewriting according to the definition. In these examples one then goes on to use algebraic manipulation or to quote known theorems to put the proof together.

### This post contains testable claims

Herein, I claim that some things are true of students just beginning abstract math. The claims are based largely on my teaching experience and some statements in the math ed literature. These claims are testable.

### When these problems occur

In the United States, the problems I describe here occur in the student’s first or second year, in university courses aimed at math majors and other STEM majors. Students typically start university at age 18, and when they start university they may not choose their major until the second year.

In much of the rest of the world, students are more likely to have one more year in a secondary school (sixth form in England lasts two years) or go to a “college” for a year or two before entering a university, and then they get their bachelor’s degree in three years instead of four as in the USA. Not only that, when they do go to university they enter a particular program immediately — math, computing science, etc.

These differences may mean that the abstract math cliff occurs early in a student’s university career in the USA and before the student enters university elsewhere.

In my experience at CWRU, some math majors fall of the cliff, but the percentage of computing science students having trouble was considerably greater. On the other hand, more of them survived the discrete math course when I taught it because the discrete math course contain less abstraction and more computation than the math major courses (except linear algebra, which had a balance similar to the discrete math course — and was taken by a sizeable number of non-math majors).

## References

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# Mathematical Information II

## Introduction

This is the second post about Mathematical Information inspired by talks the AMS meeting in Seattle in January, 2016. The first post was Mathematical Information I. That post covered, among other things, types of explanations.

In this post as in the previous one, footnotes link to talks at Seattle that inspired me to write about a topic. The speakers may not agree with what I say.

## The internet

### Publishing math on the internet

• Publishing on the internet is instantaneous, in the sense that once it is written (which of course may take a long time), it can be made available on the internet immediately.
• Publishing online is also cheap. It requires only a modest computer, an editor and LaTeX or MathJax, all of which are either free, one-time purchases, or available from your university. (These days all these items are required for publishing a math book on paper or submitting an article to a paper journal as well as for publishing on the internet.)
• Publishing online has the advantage that taking up more space does not cost more. I believe this is widely underappreciated. You can add comments explaining how you think about some type of math object, or about false starts that you had to abandon, and so on. If you want to refer to a diagram that occurs in another place in the paper, you can simply include a copy in the current place. (It took me much too long to realize that I could do things like that in abstractmath.org.)

### Online journals

Many new online journals have appeared in the last few years. Some of them are deliberately intended as a way to avois putting papers behind a paywall. But aside from that, online journals speed up publication and reduce costs (not necessarily to zero if the journal is refereed).

A special type of online journal is the overlay journalG. A paper published there is posted on ArXiv; the journal merely links to it. This provides a way of refereeing articles that appear on ArXiv. It seems to me that such journals could include articles that already appear on ArXiv if the referees deem them suitable.

## Types of mathematical communication

I wrote about some types of math communication in Mathematical Information I.

The paper Varieties of Mathematical Prose, by Atish Bagchi and me, describes other forms of communicating math not described here.

### What mathematicians would like to know

#### Has this statement been proved?G

• The internet has already made it easier to answer this query: Post it on MathOverflow or Math Stack Exchange.
• It should be a long-term goal of the math community to construct a database of what is known. This would be a difficult, long-term project. I discussed it in my article The Mathematical Depository: A Proposal, which concentrated on how the depository should work as a system. Constructing it would require machine reading and understanding of mathematical prose, which is difficult and not something I know much about (the article gives some references).
• An approach that would be completely different from the depository might be through a database of proved theorems that anyone could contribute to, like a wiki, but with editing to maintain consistency, avoid repetition, etc.

#### Known information about a conjecture

This information could include partial results.G An example would be Falting’s Theorem, which implies a partial result for Fermat’s Last Theorem: there is only a finite number of solutions of $x^n+y^n=z^n$ for integers $x, y, z, n$, $n\gt2$. That theorem became widely known, but many partial results never even get published.

#### Strategies for proofs

##### Strategies that are useful in a particular field.

The website Tricki is developing a list of such strategies.

It appears that Tricki should be referred to as “The Tricki”, like The Hague and The Bronx.

Note that there are strategies that essentially work just once, to prove some important theorem. For example, Craig’s Trick, to prove that a recursively enumerable theory is recursive. But of course, who can say that it will never be useful for some other theorem? I can’t think of how, though.

##### Strategies that don’t work, and whyG

The article How to discover for yourself the solution of the cubic, by Timothy Gowers, leads you down the garden path of trying to “complete the cubic” by copying the way you solve a quadratic, and then showing conclusively that that can’t possibly work.

Instructors should point out situations like that in class when they are relevant. A database of Methods That Work Here But Not There would be helpful, too. And, most important of all, if you run into a method that doesn’t work when you are trying to prove a theorem, when you do prove it, mention the failed method in your paper! (Remember: space is now free.)

### Examples and Counterexample

I discovered these examples in twenty minutes on the internet.

### Discussions

“Mathematical discussion is very useful and virtually unpublishable.”G But in the internet age they can take place online, and they do, in discussion lists for particular branches of math. That is not the same thing as discussing in person, but it is still useful.

#### PolymathG

Polymath sessions are organized attempts to use a kind of crowdsourcing to study (and hopefully prove) a conjecture. The Polymath blog and the Polymath wiki provide information about ongoing efforts.

### Videos

• Videos that teach math are used all over the world now, after the spectacular success of Khan Academy.
• Some math meetings produce videos of invited talks and make them available on You Tube. It would be wonderful if a systematic effort could be made to increase the number of such videos. I suppose part of the problem is that it requires an operator to operate the equipment. It is not impossible that filming an academic lecture could be automated, but I don’t know if anyone is doing this. It ought to be possible. After all, some computer games follow the motions of the player(s).
• There are some documentaries explaining research-level math to the general public, but I don’t know much about them. Documentaries about other sciences seem much more common.

## References

### The talks in Seattle

• List of all the talks.
• W. Timothy Gowers, How should mathe­matical knowledge be organized? Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.
• Mathematical discussions, links to pages by Timothy Gowers. “Often [these pages] contain ideas that I have come across in one way or another and wish I had been told as an undergraduate.”
• Colloquium notes

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# abstractmath.org beta

Around two years ago I began a systematic revision of abstractmath.org. This involved rewriting some of the articles completely, fixing many errors and bad links, and deleting some articles. It also involved changing over from using Word and MathType to writing directly in html and using MathJax. The changeover was very time consuming.

Before I started the revision, abstractmath.org was in alpha mode, and now it is in beta. That means it still has flaws, and I will be repairing them probably till I can’t work any more, but it is essentially in a form that approximates my original intention for the website.

I do not intend to bring it out of beta into “final form”. I have written and published three books, two of them with Michael Barr, and I found the detailed work necessary to change it into its final form where it will stay frozen was difficult and took me away from things I want to do. I had to do it that way then (the olden days before the internet) but now I think websites that are constantly updated and have live links are far more useful to people who want to learn about some piece of math.

My last book, the Handbook of Mathematical Discourse, was in fact published after the internet was well under way, but I was still thinking in Olden Days Paper Mode and never clearly realized that there was a better way to do things.

In any case, the entire website (as well as Gyre&Gimble) is published under a Creative Commons license, so if someone wants to include part or all of it in another website, or in a book, and revise it while they do it, they can do so as long as they publish under the terms of the license and link to abstractmath.org.

### Books by Michael Barr and Charles Wells

Toposes, triples and theories

Category theory for computing science

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# Mathematical Information I

## Introduction

The January, 2016 meeting of the American Mathematical Society in Seattle included a special session on Mathe­matical Information in the Digital Age of Science. Here is a link to the list of talks in that session (you have to scroll down a ways to get to the list).

Several talks at that session were about communi­cating math, to other mathe­maticians and to the general public. Well, that’s what I have been about for the last 20 years. Mostly.

### Overview

These posts discuss the ways we communi­cate math and (mostly in later posts) the revolution in math communication that the internet has caused. Parts of this discussion were inspired by the special session talks. When they are relevant, I include footnotes referring to the talks. Be warned that what I say about these ideas may not be the same as what the speakers had to say, but I feel I ought to give them credit for getting me to think about those concepts.

### Some caveats

• The distinctions between different kinds of math communi­cation are inevitably fuzzy.
• Not all kinds of communication are mentioned.
• Several types of communication normally occur in the same document.

## Articles published in journals

Until recently, math journals were always published on paper. Now many journals exist only on the internet. What follows is a survey of the types of articles published in journals.

### Refereed papers containing new results

These communications typically containing proofs of (usually new) theorems. Such papers are the main way that academic mathematicians get credit for their researchG for the purpose of getting tenure (at least in the USA), although some other types of credit are noted below.

Proofs published in refereed journals in the past were generally restricted to formal proofs, without very many comments intended to aid the reader’s under­standing. This restricted text was often enforced by the journal. In the olden days this would have been prompted by the expense of publishing on paper. I am not sure how much this restriction has relaxed in electronic journals.

I have been writing articles for abstractmath.org and Gyre&Gimble for many years, and it has taken me a very long time to get over unnecessarily restricting the space I use in what I write. If I introduce a diagram in an article and then want to refer to it later, I don’t have to link to it — I can copy it into the current location. If it makes sense for an informative paragraph to occur in two different articles, I can put it into both articles. And so on. Nowadays, that sort of thing doesn’t cost anything.

### Survey articles and invited addresses

You may also get credit for an invited address to a prestigious organi­zation, or for a survey of your field, in for example the Bulletin of the AMS. Invited addresses and surveys may contain considerably more explanatory asides. This was quite noticeable in the invited talks at the AMS Seattle meeting.

## Books

There is a whole spectrum of math books. The following list mentions some Fraunhofer lines on the spectrum, but the gamut really is as continuous as a large finite list of books could be. This list needs more examples. (This is a blog post, so it has the status of an alpha release.)

#### Research books that are concise and without much explanation.

The Bourbaki books that I have dipped into (mostly the algebra book and mostly in the 1970’s) are definitely concise and seem to strictly avoid explanation, diagrams, pictures, etc). I have heard people say they are unreadable, but I have not found them so.

#### Contain helpful explanations that will make sense to people in the field but probably would be formidable to someone in a substantially different area.

Toposes, triples and theories, by Michael Barr and Charles Wells. I am placing our book here in the spectrum because several non-category-theorists (some of them computer scientists) have remarked that it is “formidable” or other words like that.

#### Intended to introduce professional mathematicians to a particular field.

Categories for the working mathematician, by Saunders Mac Lane. I learned from this (the 1971 edition) in my early days as a category theorist, six years after getting my Ph.D. In fact, I think that this book belongs to the grad student level instead of here, but I have not heard any comments one way or another.

#### Intended to introduce math graduate students to a particular field.

There are lots of examples of good books in this area. Years ago (but well after I got my Ph.D.), I found Serge Lang’s Algebra quite useful and studied parts of it in detail.

But for grad students? It is still used for grad students, but perhaps Nathan Jacobson’s Basic Algebra would be a better choice for a first course in algebra for first-year grad students.

The post My early life as a mathematician discusses algebra texts in the olden days, among other things.

#### Intended to explain a part of math to a general audience.

Love and math: the heart of hidden reality. by Edward Frenkel, 2014. This is a wonderful book. After reading it, I felt that at last I had some clue as to what was going on with the Langlands Program. He assumes that the reader knows very little about math and gives hand-waving pictorial expla­nations for some of the ideas. Many of the concepts in the book were already familiar to me (not at an expert level). I doubt that someone who had had no college math courses that included some abstract math would get much out of it.

Symmetry: A Journey into the Patterns of Nature, by Marcus du Sautoy, 2009. He also produced a video on symmetry.

My post Explaining “higher” math to beginners, describes du Sautoy’s use of terminology (among others).

Secrets of creation: the mystery of the prime numbers (Volume 1) by Matthew Watkins (author) and Matt Tweed (Illustrator), 2015. This is the first book of a trilogy that explains the connection between the Riemann $\zeta$ function and the primes. He uses pictures and verbal descriptions, very little terminology or symbolic notation. This is the best attempt I know of at explaining deep math that might really work for non-mathe­maticians.

My post The mystery of the prime numbers: a review describes the first book.

#### Piper Harron’s Thesis

This is a remarkable departure from the usual dry, condensed, no-useful-asides Ph.D. thesis in math. Each chapter has three main parts, Layscape (explanations for nonspecialists — not (in my opinion) for nonmathe­maticians), Mathscape (most like what goes into the usual math paper but with much more explanation) and Weedscape (irrelevant stuff which she found helpful and perhaps the reader will too). The names of these three sections vary from chapter to chapter. This seems like a great idea, and the parts I have read are well-done.

## Types of explanations

Any explanation of math in any of the categories above will be of several different types. Some of them are considered here, and more will appear in Mathematical Information II.

The paper Varieties of Mathematical Prose, by Atish Bagchi and me, provides a more fine-grained description of certain types of math communication that includes some types of explanations and also other types of communication.

### Images and metaphors

#### In abstractmath.org

I have written about images and metaphors in abstractmath.org:

Abstractmath.org is aimed at helping students who are beginning their study of abstract math, and so the examples are mostly simple and not at a high level of abstraction. In the general literature, the images and metaphors that are written about may be much more sophisticated.

#### The User’s GuideW

Luke Wolcott edits a new journal called Enchiridion: Mathematics User’s Guides (this link allows you to download the articles in the first issue). Each article in this journal is written by a mathematician who has published a research paper in a refereed journal. The author’s article in Enchiridion provides information intended to help the reader to understand the research paper. Enchiridion and its rationale is described in more detail in the paper The User’s Guide Project: Giving Experential Context to Research Papers.

The guidelines for writing a User’s Guide suggest writing them in four parts, and one of the parts is to introduce useful images and metaphors that helped the author. You can see how the authors’ user’s guides carry this out in the first issue of Enchiridion.

#### Piper Harron’s thesis

Piper Harron’s explanation of integrals in her thesis is a description of integrals and measures using creative metaphors that I think may raise some mathematicians’ consciousness and others’ hackles, but I doubt it would be informative to a non-mathematician. I love “funky-summing” (p. 116ff): it communicates how integration is related to real adding up a finite bunch of numbers in a liberal-artsy way, in other words via the connotations of the word “funky”, in contrast to rigorous math which depends on every word have an accumulation-of-properties definition.

The point about “funky-summing” (in my opinion, not necessarily Harron’s) is that when you take the limit of all the Riemann sums as all meshes go to zero, you get a number which

• Is really and truly not a sum of numbers in any way
• Smells like a sum of numbers

Connotations communicate metaphors. Metaphors are a major cause of grief for students beginning abstract math, but they are necessary for understanding math. Working around this paradox is probably the most important problem for math teachers.

### Informal summaries of a proofW

The User’s Guide requires a “colloquial summary” of a paper as one of the four parts of the guide for that paper.

• Wolcott’s colloquial summary of his paper keeps the level aimed at non-mathematicians, starting with a hand-waving explanation of what a ring is. He uses many metaphors in the process of explaining what his paper does.
• The colloquial summary of another User’s Guide, by Cary Malkiewich, stays strictly at the general-public level. He uses a few metaphors. I liked his explanation of how mathematicians work first with examples, then finding patterns among the examples.
• The colloquial summary of David White’s paper stays at the general-public level but uses some neat metaphors. He also has a perceptive paragraph discussing the role of category theory in math.

The summaries I just mentioned are interesting to read. But I wonder if informal summaries aimed at math majors or early grad students might be more useful.

### Insights

The first of the four parts of the explanatory papers in Enchiridion is supposed to present the key insights and organizing principles that were useful in coming up with the proofs. Some of them do a good job with this. They are mostly very special to the work in question, but some are more general.

This suggests that when teaching a course in some math subject you make a point of explaining the basic techniques that have turned out very useful in the subject.

For example, a fundamental insight in group theory is:

Study the linear representations of a group.

That is an excellent example of a fundamental insight that applies everywhere in math:

Find a functor that maps the math objects you are studying to objects in a different branch of math.

The organizing principles listed in David White’s article has (naturally more specialized) insights like that.

### Proof stories

“Proof stories” tell in sequence (more or less) how the author came up with a proof. This means describing the false starts, insights and how they came about. Piper Harron’s thesis does that all through her work.

Some authors do more than that: their proof stories intertwine the mathe­matical events of their progress with a recount of life events, which sometimes make a mathe­matical difference and sometimes just produces a pause to let the proof stew in their brain. Luke Wolcott wrote a User’s Guide for one of his own papers, and his proof story for that paper involves personal experiences. (I recommend his User’s Guide as a model to learn from.)

Reports of personal experiences in doing math seem to add to my grasp of the math, but I am not sure I understand why.

## References

### The talks in Seattle

• List of all the talks.
• W. Timothy Gowers, How should mathe­matical knowledge be organized? Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.
• Colloquium notes. Gowers gave a series of invited addresses for which these are the notes. They have many instances of describing what sorts of problems obstruct a desirable step in the proof and what can be done about it.

• Luke Wolcott, The User’s Guide. Talk at the AMS Special Session on Mathe­matical Information in the Digital Age of Science, 6 January 2016.

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# My early life as a mathematician

## My early life as a mathematician.

Revised 22 January 2016.

In 1965, I received my Ph.D. at Duke University based on a dissertation about polynomials over finite fields. My advisor was Leonard Carlitz.

In Carlitz’s algebra course, the textbook was Van der Waerden’s Algebra. It is way too old-fashioned to be used nowadays, but it did indeed present post-Noether type abstract algebra. Carlitz also had me read large chunks of Martin Weber’s Lehrbuch der Algebra, written in German in 1895 (so totally not post-Noether) and published using Fraktur. A few years ago one of my sons asked me to retype the words to some of the songs written in Fraktur in a German-American shape note book in Roman type (but still in German), which I did. This was for German teachers in the Concordia Language Villages to use with their students. I sometimes wonder if I am the last person on earth able to read Fraktur fluently.

I learned mathematical logic from Joe Shoenfield from his dittoed notes that later became an excellent textbook. I rediscovered Craig’s Trick while working on problem he gave. That considerably strengthened my sense of self-worth.

I accepted a job at Western Reserve University, now Case Western Reserve University, where I stayed until I retired in 1999. In the few years after 1965, I wrote several papers about finite fields. They are all summarized in the book Finite Fields, by Rudolf Lidl and Harald Niederreiter.

I was almost immediately attracted to category theory and to computing science, both of which Carlitz hated. I did not let that stop me. (Now is the time to say, Follow The Beat of your Own Drum or some such cliché.)

Early on, Paul Dedecker was at CWRU briefly, and from him I learned about sheaves, cribles and the like. This inspired me to take part in an algebraic geometry summer school at Bowdoin College, where I learned from lectures by David Mumford and by reading his Red Book when it was still red.

Because one of the papers in finite fields showed that certain types of permutation polynomials formed wreath products of groups, I also pursued group theory, in particular by taking part in the finite group theory summer school at Bowdoin in 1970.

During that time I pored over Beck’s thesis on cohomology, which with the group theory I had learned resulted in my paper Automorphisms of group extensions. That paper has the most citations of all my research papers.

In the early days, I had several graduate students. All of them worked in group theory. One of them, Shair Ahmad, went on to produce several Ph.D. students, all in differential equations and dynamical systems.

One thing I can brag about is that I never ever told him I hated differential equations or dynamical systems. In fact, I didn’t hate either one. There were people in the department in both fields and they made me jealous the way they could model real life phenomena with those tools. One relevant point about that is that I was a liberal arts math major from Oberlin before going to Duke and had had very few courses in any kind of science. This made me very different from most people in the department, who has B.S. undergrad degrees.

In those days, John Isbell and Peter Hilton were in the math department at CWRU for awhile, which boosted my knowledge and interest in category theory. Hilton arranged for me to spend a year at the E.T.H. in Zürich, where I met Michael Barr. I eventually wrote two books on category theory with him. But that is getting away from Early Days, so I will stop here.