Category Archives: representations

Every post that talks about representation of mathematical objects in the most general sense.

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.

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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
\[\begin{pmatrix} 0&1&2&3&4&5&6&7&8&9\\0&2&1&4&5&3&7&8&9&6\end{pmatrix}\]

Cograph

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 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.

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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.

Graph of a function.

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
\[f(x):=\left\{
\begin{align}
2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\gt0) \\
1-x^2\,\,\,\,\,\,(-1\lt x\lt 0) \\
2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\lt-1)
\end{align}\right.\]

Graph of a discontinuous function.

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.

Discovering facts about a function
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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    Abstraction and axiomatic systems

    Abstraction and the axiomatic method

    This post will become an article in abstractmath.org.

    Abstraction

    An abstraction of a concept $C$ is a concept $C’$ with these properties:

    • $C’$ includes all instances of $C$ and
    • $C’$ is constructed by taking as axioms certain assertions that are true of all instances of $C$.

    There are two major situations where abstraction is used in math.

    • $C$ may be a familiar concept or property that has not yet been given a math definition.
    • $C$ may already have a mathe­matical definition using axioms. In that case the abstraction will be a generalization of $C$. 

    In both cases, the math definition may allow instances of $C’$ that were not originally thought of as being part of $C$.

    Example: Relations

    Mathematicians have made use of relations between math objects since antiquity.

    • For real numbers $r$ and $s$. “$r\lt x$” means that $r$ is less than $s$. So the statement “$5\lt 7$” is true, but the statement “$7\lt 5$” is false. We say that “$\lt$” is a relation on the real numbers. Other relations on real numbers denoted by symbols are “$=$” and “$\leq$”.
    • Suppose $m$ and $n$ are positive integers. $m$ and $n$ are said to be relatively prime if the greatest common divisor of $m$ and $n$ is $1$. So $5$ and $7$ are relatively prime, but $15$ and $21$ are not relatively prime. So being relatively prime is a relation on positive integers. This is a relation that does not have a commonly used symbol.
    • The concept of congruence of triangles has been used for a couple of millenia. In recent centuries it has been denoted by the symbol “$\cong$”. Congruence is a relation on triangles.

    One could say that a relation is a true-or-false statement that can be made about a pair of math objects of a certain type. Logicians have in fact made that a formal definition. But when set theory came to be used around 100 years ago as a basis for all definitions in math, we started using this definition:

    A relation on a set $S$ is a set $\alpha$ of ordered pairs of elements of $S$.

    “$\alpha$” is the Greek letter alpha.

    The idea is that if $(s,t)\in\alpha$, then $s$ is related by $\alpha$ to $t$, then $(s,t)$ is an element of $\alpha$, and if $s$ is not related by $\alpha$ to $t$, then $(s,t)$ is not an element of $\alpha$. That abstracts the everyday concept of relationship by focusing on the property that a relation either holds or doesn’t hold between two given objects.

    For example, the less-than relation on the set of all real numbers $\mathbb{R}$ is the set \[\alpha:=\{(r,s)|r\in\mathbb{R}\text{ and }s\in\mathbb{R}\text{ and }r\lt s\}\] In other words, $r\lt s$ if and only if $(r,s)\in \alpha$.

    Example

    A consequence of this definition is that any set of ordered pairs is a relation. Example: Let $\xi:=\{(2,3),(2,9),(9,1),(9,2)\}$. Then $\xi$ is a relation on the set $\{1,2,3,9\}$. Your reaction may be: What relation IS it? Answer: just that set of ordered pairs. You know that $2\xi3$ and $2\xi9$, for example, but $9\xi1$ is false. There is no other definition of $\xi$.

    Yes, the relation $\xi$ is weird. It is an arbitrary definition. It does not have any verbal description other than listing the element of $\xi$. It is probably useless. Live with it.

    The symbol “$\xi$” is a Greek letter. It looks weird, so I used it to name a weird relation. Its upper case version is “$\Xi$”, which is even weirder. I pronounce “$\xi$” as “ksee” but most mathematicians call it “si” or “zi” (rhyming with “pie”).

    Defining a relation as any old set of ordered pairs is an example of a reconstructive generalization.

    $n$-ary relations

    Years ago, mathematicians started coming up with things that were like relations but which involved more than two elements of a set.

    Example

    Let $r$, $s$ and $t$ be real numbers. We say that “$s$ is between $r$ and $t$” if $r\lt s$ and $s\lt t$. Then betweenness is a relation that is true or false about three real numbers.

    Mathematicians now call this a ternary relation. The abstract definition of a ternary relation is this: A ternary relation on a set $S$ is a set of ordered triple of elements of $S$. This is an reconstructive generalization of the concept of relation that allows ordered triples of elements as well as ordered pairs of elements.

    In the case of betweenness, we have to decide on the ordering. Let us say that the betweenness relation holds for the triple $(r,s,t)$ if $r\lt s$ and $s\lt t$. So $(4,5,7)$ is in the betweenness relation and $(4,7,5)$ is not.

    You could argue that in the sentence, “$s$ is between $r$ and $t$”, the $s$ comes first, so that we should say that the betweenness relation (meaning $r$ is between $s$ and $t$) holds for $(r,s,t)$ if $s\lt r$ and $r\lt t$. Well, when you write an article you can write it that way. But I am writing this article.

    Nowadays we talk about $n$-ary relations for any positive integer $n$. One consequence of this is that if we want to talk just about sets of ordered pairs we must call them binary relations.

    When I was a child there was only one kind of guitar and it was called “a guitar”. (My older cousin Junior has a guitar, but I had only a plastic ukelele.) Some time in the fifties, electrically amplified guitars came into being, so we had to refer to the original kind as “acoustic guitars”. I was a teenager when this happened, and being a typical teenager, I was completely contemptuous of the adults who reacted with extreme irritation at the phrase “acoustic guitar”.

    The axiomatic method

    The axiomatic method is a technique for studying math objects of some kind by formulating them as a type of math structure. You take some basic properties of the kind of structure you are interested in and set them down as axioms, then deduce other properties (that you may or may not have already known) as theorems. The point of doing this is to make your reasoning and all your assumptions completely explicit.

    Nowadays research papers typically state and prove their theorems in terms of math structures defined by axioms, although a particular paper may not mention the axioms but merely refer to other papers or texts where the axioms are given.  For some common structures such as the real numbers and sets, the axioms are not only not referenced, but the authors clearly don’t even think about them in terms of axioms: they use commonly-known properties (or real numbers or sets, for example) without reference.

    The axiomatic method in practice

    Typically when using the axiomatic method some of these things may happen:

    • You discover that there are other examples of this system that you hadn’t previously known about.  This makes the axioms more broadly applicable.
    • You discover that some properties that your original examples had don’t hold for some of the new examples.  Depending on your research goals, you may then add some of those properties to the axioms, so that the new examples are not examples any more.
    • You may discover that some of your axioms follow from others, so that you can omit them from the system.

    Example: Continuity

    A continuous function (from the set of real numbers to the set of real numbers) is sometimes described as a function whose graph you can draw without lifting your chalk from the board.  This is a physical description, not a mathe­matical definition.

    In the nineteenth century, mathe­ma­ticians talked about continuous functions but became aware that they needed a rigorous definition.  One possibility was functions given by formulas, but that didn’t work: some formulas give discontinuous functions and they couldn’t think of formulas for some continuous functions.

    This description of nineteenth century math is an oversimpli­fication.

    Cauchy produced the definition we now use (the epsilon-delta definition) which is a rigorous mathe­matical version of the no-lifting-chalk idea and which included the functions they thought of as continuous.

    To their surprise, some clever mathe­maticians produced examples of some weird continuous functions that you can’t draw, for example the sine blur function.  In the terminology in the discussion of abstraction above, the abstraction $C’$ (epsilon-delta continuous functions) had functions in it that were not in $C$ (no-chalk-lifting functions.) On the other hand, their definition now applied to functions between some spaces besides the real numbers, for example the complex numbers, for which drawing the graph without lifting the chalk doesn’t even make sense.

    Example: Rings

    Suppose you are studying the algebraic properties of numbers.  You know that addition and multiplication are both associative operations and that they are related by the distributive law:  $x(y+z)=xy+xz$. Both addition and multiplication have identity elements ($0$ and $1$) and satisfy some other properties as well: addition forms a commutative group for example, and if $x$ is any number, then $0\cdot x=0$.

    One way to approach this problem is to write down some of these laws as axioms on a set with two binary operations without assuming that the elements are numbers. In doing this, you are abstracting some of the properties of numbers.

    Certain properties such as those in the first paragraph of this example were chosen to define a type of math structure called a ring. (The precise set of axioms for rings is given in the Wikipedia article.)

    You may then prove theorems about rings strictly by logical deduction from the axioms without calling on your familiarity with numbers.

    When mathematicians did this, the following events occurred:

    • They discovered systems such as matrices whose elements are not numbers but which obey most of the axioms for rings.
    • Although multiplication of numbers is commutative, multiplication of matrices is not commutative.
    • Now they had to decide whether to require commutative of multiplication as an axioms for rings or not.  In this example, historically, mathe­maticians decided not to require multi­plication to be commutative, so (for example) the set of all $2\times 2$ matrices with real entries is a ring.
    • They then defined a commutative ring to be a ring in which multi­plication is commutative.
    • So the name “commutative ring” means the multiplication is commutative, because addition in rings is always commutative. Mathematical names are not always transparent.

    • You can prove from the axioms that in any ring, $0 x=0$ for all $x$, so you don’t need to include it as an axiom.

    Nowadays, all math structures are defined by axioms.

    Other examples

    • Historically, the first example of something like the axiomatic method is Euclid’s axiomatization of geometry.  The axiomatic method began to take off in the late nineteenth century and now is a standard tool in math.  For more about the axiomatic method see the Wikipedia article.
    • Partitions. and equivalence
      relations
      are two other concepts that have been axiomatized. Remarkably, although the axioms for the two types of structures are quite different, every partition is in fact an equivalence relation in exactly one way, and any equivalence relation is a partition in exactly one way.

    Remark

    Many articles on the web about the axiomatic method emphasize the representation of the axiom system as a formal logical theory (formal system). 
    In practice, mathematicians create and use a particular axiom system as a tool for research and understanding, and state and prove theorems of the system in semi-formal narrative form rather than in formal logic.


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    Functions: Metaphors, Images and Representations

    Please read this post at abstractmath.org. I originally posted the document here but some of the diagrams would not render, and I haven’t been able to figure out why. Sorry for having to redirect.

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    Proofs using diagrams

    Introduction

    This post gives a proof of an easy theorem in category theory using the graph-based logic approach of Graph based logic and sketches, (GBLS) by Atish Bagchi and me.

    Formal logic is typically defined in terms of formulas and terms, defined recursively as strings of characters, together with rules of inference. GBLS proposes a new approach to logic where diagrams are used instead of strings of characters. The exposition here spells out the proof in more detail than GBLS does and uses various experimental ways of drawing diagrams using Mathematica.

    To follow this proof, you need to be familiar with basic category theory. Most special definitions that are needed are defined in this post where they are first used. Section 1 of GBLS also gives the definitions you need with more context.

    The theorem

    The Theorem to be proved (it is Theorem 8.3.1 of GBLS) says that, in any category, if the triangles in the diagram below commute, then the outside square commutes. This is easy using the associative law: If $xf=h$ and $kx=g$, then $kh=k(xf)=(kx)f=gf$.

    Subject Diagram

    So what?

    This theorem is not interesting. The point of this post is to present a new approach to proving such theorems, using diagrams instead of strings. The reason that exhibiting the dig
    rammatic proof is interesting is that many different kinds of categories have a FL cattheory, including these:

    Essentially algebraic string-based logic is described in detail in Partial Horn logic and cartesian categories, by E. Palmgren and Steven Vickers.

    Commercial

    My concept of form in A generalization of the concept of sketch generalizes sketches to all the categories that can be defined as models of FL cattheories. So the method of proof using diagrams can be applied to theorems about the objects defined by forms.

    Concepts needed for the graph-based proof

    To prove the theorem, I will make use of $\mathbf{ThCat}$, the FL cattheory for categories.

    • An FL category is a category with all finite limits.
    • GLBS uses the word cattheory for what Category theory for computing science and Toposes, triples and theories call the theory of a sketch.
    • In many books and articles, and in nLab, a “sketch” is what we call the cattheory (or the theory) of a sketch. For us, the sketch is a generating collection of objects, arrows, diagrams, cones and cocones for the cattheory. The category of models of the sketch and the cattheory are equivalent.
    • $\mathbf{ThCat}$ is a category with finite limits freely generated by certain designated objects, arrows, commutative diagrams and limit cones, listed below.
    • A model of $\mathbf{ThCat}$ in $\mathbf{Set}$ (the category of sets, whichever one you like) is an FL functor $\mathfrak{C}:\mathbf{ThCat}\to\mathbf{Set}.$
    • Such a model $\mathfrak{C}$ is a small category, and every small category is such a model. If this statement worries you, read Section 3.4 of GBLS.
    • Natural transformations between models are FL-preserving functors that preserve the structure on the nose.
    • The category of models of $\mathbf{ThCat}$ in $\mathbf{Set}$ is equivalent to the category of small categories and morphisms, which, unlike the category of models, includes functors that don’t preserve things on the nose.
    • $\mathbf{ThCat}$ is an example of the theory of an FL sketch. Chapter 4 of GBLS describes this idea in detail. The theory has the same models as the sketch.
    • The sketch generating $\mathbf{ThCat}$ is defined in detail in section 7.2 of GBLS.

    Some objects and arrows of $\mathbf{ThCat}$

    I will make use of the following objects and arrows that occur in $\mathbf{ThCat}.$ A formal thing is a construction in $\mathbf{ThCat}$ that becomes an actual thing in a model. So for example a model $\mathfrak{C}$ of $\mathbf{ThCat}$ in $\mathbf{Set}$ is an actual (small) category, and $\mathfrak{C}(\mathsf{ar_2})$ is the set of all composable pairs of arrows in the category $\mathfrak{C}$.

    • $\mathsf{ob}$, the formal set of objects.
    • $\mathsf{ar}$, the formal set of arrows.
    • $\mathsf{ar}_2$, the formal set of composable pairs of arrows.
    • $\mathsf{ar}_3$, the formal set of composable triples of arrows.
    • $\mathsf{unit} : \mathsf{ob}\to \mathsf{ar}$ that formally picks out the identity arrow of an object.
    • $\mathsf{dom},\mathsf{cod} : \mathsf{ar}\to \mathsf{ob}$ that formally pick out the domain and codomain of an arrow.
    • $\mathsf{comp} : \mathsf{ar}_2\to \mathsf{ar}$ that picks out the composite of a composable pair.
    • $\mathsf{lfac}, \mathsf{rfac} :\mathsf{ar}_2\to \mathsf{ar}$ that pick out the left and right factors in a composable pair.
    • $\mathsf{lfac}, \mathsf{mfac},\mathsf{rfac} :\mathsf{ar}_3 \to\mathsf{ar}$ that pick out the left, middle and right factors in a composable triple of arrows.
    • $\mathsf{lass}, \mathsf{rass} : \mathsf{ar}_3 \to \mathsf{ar}_2$: $\mathsf{lass}$ formally takes $\langle{h,g,f}\rangle$ to $\langle{hg,f}\rangle$ and $\mathsf{rass}$ takes it to $\langle{h,gf}\rangle$.

    $\mathsf{ob}$, $\mathsf{ar}$, $\mathsf{unit}$, $\mathsf{dom}$, $\mathsf{cod}$ and $\mathsf{comp}$ are given primitives and the others are defined as limits of finite diagrams composed of those objects. This is spelled out in Chapter 7.2 of GBLS. The definition of $\mathbf{ThCat}$ also requires certain diagrams to be commutative. They are all provided in GBLS; the one enforcing associativity is shown later in this post.

    Color coding

    I will use color coding to separate syntax from semantics.

    • Syntax consists of constructions in $\mathbf{ThCat}.$ The description will always be a commutative diagram in black, with annotations as explained later.
    • The limit of the description will be an object in $\mathbf{ThCat}$ (the form) whose value in a model $\mathfrak{C}$ will be shown in green, because being an element of the value of a model makes it semantics.
    • When a limit cone is defined, the projections (which are arrows in $\mathbf{ThCat}$) will be shown in blue.

    Descriptions

    In graph-based logic, a type of construction that can be made in a category has a description, which (in the case of our Theorem) is a finite diagram in $\mathbf{ThCat}$. The value of the limit of the description in a model $\mathfrak{C}$ is the set of all instances of that type of construction in $\mathfrak{C}$.

    The Subject Diagram

    • This diagram is the subject matter of the Theorem. It is not assumed to be commutative.
    • As in most diagrams in category theory texts, the labels in this diagram are variables, so the diagram is implicitly universally quantified. The Subject Diagram is a generic diagram of its shape.
    • “Any diagram of its shape” includes diagrams in which some of the nodes may represent the same object. An extreme example is the graph in which every node is an object $\mathsf{E}$ and every arrow is its identity arrow. The diagram below is nevertheless an example of the Subject Diagram:
    • Shapes of diagrams are defined properly in Section 2.3 of
      GBLS and in Section 4.1 of Category Theory for Computing Science.

    The description of the Subject Diagram

    Diagram SDD below shows the Subject Diagram as the limit of its description. The description is the black diagram.


    Diagram SDD

    Definition of $\mathsf{ar}_2$

    The object $\mathsf{ar}_2$ of composable pairs of arrows is defined as a pullback:

    In the usual categorical notation this would be shown as

    This makes use of the fact that the unnamed blue arrow is induced by the other two projection arrows. In the rest of the post, projection arrows that are induced are normally omitted.

    An enrichment of the description

    Because $\mathsf{ar}_2$ is defined as a pullback, we can enrich the description of Diagram SDD by adjoining two pullbacks as shown below. This is Diagram 8.10 in GBLS. The enriched diagram has the same limit as the description of Diagram SDD.

    Enriched Diagram SDD

    Note that the projections from the limit to the two occurrences of $\mathsf{ar}_2$ induce all the other projections. This follows by diagram chasing; remember that the description must be a commutative diagram.

    Make the triangles commute

    To make the triangles commute, we add two comp arrows to the enriched diagram as shown below. These two arrows are not induced by the description; they are therefore additions to the description — they describe a more restrictive (green) diagram with commutative triangles and so are shown in black.

    Diagram TC: The triangles commute

    The left comp makes $xf=h$ and the right comp makes $kx=g$.

    The outside square commutes

    Now we enrich Diagram TC with four objects, <comp,id>, <id,comp> and three comp arrows as shown in bolder black. These objects and arrows already exist in $\mathbf{ThCat}$ and therefore do not change the limit, which must be the same as the limit of Diagram TC.

    The outside square commutes

    The diagram in bold black is exactly the commutative diagram that requires associativity for these particular objects and arrows, which immediately implies that $gf=kh$, as the Theorem requires.

    By the definitions of $\mathsf{ar_2}$ and $\mathsf{ar_3}$, the part of the description in bold black induces the rest of the diagram. Omitting the rest of the diagram would make $\mathsf{ar_2}$ and $\mathsf{ar_3}$ modules in the sense of GBLS, Chapter 7.4. Modules would be vital to deal with proofs more complicated than the one given here.

    References

    Creative Commons License

    This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

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    Math majors attacked by cognitive dissonance

    In some situations you may have conflicting information from different sources about a subject.   The resulting confusion in your thinking is called cognitive dissonance.

    It may happen that a person suffering cognitive dissonance suppresses one of the ways of understanding in order to resolve the conflict.  For example, at a certain stage in learning English, you (small child or non-native-English speaker) may learn a rule that the past tense is made from the present form by adding “-ed”. So you say “bringed” instead of “brought” even though you may have heard people use “brought” many times.  You have suppressed the evidence in favor of the rule.

    Some of the ways cognitive dissonance can affect learning math are discussed here

    Metaphorical contamination

    We think about math objects using metaphors, as we do with most concepts that are not totally concrete.  The metaphors are imperfect, suggesting facts about the objects that may not follow from the definition. This is discussed at length in the section on images and metaphors here.

    The real line

    Mathematicians think of the real numbers as constituting a line infinitely long in both directions, with each number as a point on the line. But this does not mean that you can think of the line as a row of points. Between any two points there are uncountably many other points. See density of the reals.

    Infinite math objects

    One of the most intransigent examples of metaphorical contamination occurs when students think about countably infinite sets. Their metaphor is that a sequence such as the set of natural numbers $\{0,1,2,3,4,\ldots\}$ “goes on forever but never ends”. The metaphor mathematicians have in mind is quite different: The natural numbers constitute the set that contains every natural number right now.

    Example

    An excruciating example of this is the true statement
    $.999\ldots=1.0$.” The notion that it can’t be true comes from thinking of “$0.999\ldots$” as consisting of the list of numbers \[0.9,0.99,0.999,0.9999,0.99999,\ldots\] which the student may say “gets closer and closer to $1.0$ but never gets there”.

    Now consider the way a mathematician thinks: The numbers are all already there, and they make a set.

    The proof that $.999\ldots=1.0$ has several steps. In the list below, I have inserted some remarks in red that indicate areas of abstract math that beginning students have trouble with.

    1. The elements of an infinite set are all in it at once. This is the way mathematicians think about infinite sets.
    2. By definition, an infinite decimal expansion represents the unique real number that is a limit point of its set of truncations.
    3. The problem that occurs with the word “definition” in this case is that a definition appears to be a dictatorial act. The student needs to know why you made this definition. This is not a stupid request. The act can be justified by the way the definition gets along with the algebraic and topological characteristic of the real numbers.

    4. It follows from $\epsilon-\delta$ machinations that the limit of the sequence $0.9,0.99,0.999,0.9999,0.99999,\ldots$ is $1.0$
    5. That means “$0.999\ldots$” represents $1.0$. (Enclosing a mathematical expression in quotes turns it into a string of characters.)
    6. The statement “$A$” represents $B$ is equivalent to the statement $A=B$. (Have you ever heard a teacher point this out?)
    7. It follows that that $0.999\ldots=1.0$.

    Each one of these steps should be made explicit. Even the Wikipedia article, which is regarded as a well written document, doesn’t make all of the points explicit.

    Semantic contamination

    Many math objects have names that are ordinary English words. 
    (See names.) So the person learning about them is faced with two inputs:

    • The definition of the word as a math object.
    • The meaning and connotations of the word in English.

    It is easy and natural to suppress the information given by the definition (or part of it) and rely only on the English meaning. But math does not work that way:

    If another source of understanding contradicts the definition
    THE DEFINITION WINS.

    “Cardinality”

    The connotations of a name may fit the concept in some ways and not others. Infinite cardinal numbers are a notorious example of this: there are some ways in which they are like numbers and other in which they are not. 

    For a finite set, the cardinality of the set is the number of elements in the set. Long ago, mathematicians started talking about the cardinality of an infinite set. They worked out a lot of facts about that, for example:

    • The cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers.
    • The cardinality of the number of points on the real line is the same as the cardinality of points in the real plane.

    The teacher may even say that there are just as many points on the real line as in the real points. And know-it-all math majors will say that to their friends.

    Many students will find that totally bizarre. Essentially, what has happened is that the math dictators have taken the phrase “cardinality” to mean what it usually means for finite sets and extend it to infinite sets by using a perfectly consistent (and useful) definition of “cardinality” which has very different properties from the finite case.

    That causes a perfect storm of cognitive dissonance.

    Math majors must learn to get used to situations like this; they occur in all branches of math. But it is bad behavior to use the phrase “the same number of elements” to non-mathematicians. Indeed, I don’t think you should use the word cardinality in that setting either: you should refer to a “one-to-one correspondence” instead and admit up front that the existence of such a correspondence is quite amazing.

    “Series”

    Let’s look at the word “series”in more detail. In ordinary English, a series is a bunch of things, one after the other.

    • The World Series is a series of up to seven games, coming one after another in time.
    • A series of books is not just a bunch of books, but a bunch of books in order.
    • In the case of the Harry Potter series the books are meant to be read in order.
    • A publisher might publish a series of books on science, named Physics, Chemistry,
      Astronomy, Biology,
      and so on, that are not meant to be read in order, but the publisher will still list them in order.(What else could they do? See Representing and thinking about sets.)

    Infinite series in math

    In mathematics an infinite series is an object expressed like this:

    \[\sum\limits_{k=1}^{\infty
    }{{{a}_{k}}}\]

    where the ${{a}_{k}}$ are numbers. It has partial sums

    \[\sum\limits_{k=1}^{n}{{{a}_{k}}}\]

    For example, if ${{a}_{k}}$ is defined to be $1/{{k}^{2}}$ for positive integers $k$, then

    \[\sum\limits_{k=1}^{6}{{{a}_{k}}}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}=\frac{\text{5369}}{\text{3600}}=\text{
    about }1.49\]

    This infinite series converges to $\zeta (2)$, which is about $1.65$. (This is not obvious. See the Zeta-function article in Wikipedia.) So this “infinite series” is really an infinite sum. It does not fit the image given by the English word “series”. The English meaning contaminates the mathematical meaning. But the definition wins.

    The mathematical word that corresponds to the usual meaning of “series” is “sequence”. For example, $a_k:=1/{{k}^{2}}$ is the infinite sequence $1,\frac{1}{4},\frac{1}{9},\frac{1}{16}\ldots$ It is not an infinite series.

    “Only if”

    “Only if” is also discussed from a more technical point of view in the article on conditional assertions.

    In math English, sentences of the form $P$ only if $Q$” mean exactly the same thing as “If $P$ then $Q$”. The phrase “only if” is rarely used this way in ordinary English discourse.

    Sentences of the form “$P$ only if $Q$” about ordinary everyday things generally do not mean the same thing as “If $P$ then $Q$”. That is because in such situations there are considerations of time and causation that do not come up with mathematical objects. Consider “If it is raining, I will carry an umbrella” (seeing the rain will cause me to carry the umbrella) and “It is raining only if I carry an umbrella” (which sounds like my carrying an umbrella will cause it to rain).   When “$P$ only if $Q$” is about math objects,
    there is no question of time and causation because math objects are inert and unchanging.

     Students sometimes flatly refuse to believe me when I tell them about the mathematical meaning of “only if”.  This is a classic example of semantic contamination.  Two sources of information appear to contradict each other, in this case (1) the professor and (2) a lifetime of intimate experience with the English language.  The information from one of these sources must be rejected or suppressed. It is hardly surprising that many students prefer to suppress the professor’s apparently unnatural and usually unmotivated claims.

    These words also cause severe cognitive dissonance

    • “If” causes notorious difficulties for beginners and even later. They are discussed in abmath here and here.
    • A, an
      and the implicitly signal the universal quantifier in certain math usages. They cause a good bit of trouble in the early days of some students.

    The following cause more minor cognitive dissonance.

    References for semantic contamination

    Besides the examples given above, you can find many others in these two works:

    • Pimm, D. (1987), Speaking Mathematically: Communications in Mathematics Classrooms.  Routledge & Kegan Paul.
    • Hersh, R. (1997),”Math lingo vs. plain English: Double entendre”. American Mathematical Monthly, vol 104,pages 48-51.
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    A very early satori that occurs with beginning abstract math students

    In the previous post Pattern recognition and me, I wrote about how much I enjoyed sudden flashes of understanding that were caused by my recognizing a pattern (or learning about a pattern). I have had several such, shall we say, Thrills in learning about math and doing research in math. This post is about a very early thrill I had when I first started studying abstract algebra. As is my wont, I will make various pronouncements about what these mean for teaching and understanding math.

    Cosets

    Early in any undergraduate course involving group theory, you learn about cosets.

    Basic facts about cosets

    1. Every subgroup of a group generates a set of left cosets and a set of right cosets.
    2. If $H$ is a subgroup of $G$ and $a$ and $b$ are elements of $G$, then $a$ and $b$ are in the same left coset of $H$ if and only if $a^{-1}b\in H$. They are in the same right coset of $H$ if and only if $ab^{-1}\in H$.
    3. Alternative definition: $a$ and $b$ are in the same left coset of $H$ if $a=bh$ for some $h\in H$ and are in the same right coset of $H$ if $a=hb$ for some $h\in H$
    4. One of the (left or right) cosets of $H$ is $H$ itself.
    5. The relations
      $a\underset{L}\sim b$ if and only if $a^{-1}b\in H$

      and

      $a\underset{R}\sim b$ if and only if $ab^{-1}\in H$

      are equivalence relations.

    6. It follows from (5) that each of the set of left cosets of $H$ and the set of right cosets of $H$ is a partition of $G$.
    7. By definition, $H$ is a normal subgroup of $G$ if the two sets of cosets coincide.
    8. The index of a subgroup in a group is the cardinal number of (left or right) cosets the subgroup has.

    Elementary proofs in group theory

    In the course, you will be asked to prove some of the interrelationships between (2) through (5) using just the definitions of group and subgroup. The teacher assigns these exercises to train the students in the elementary algebra of elements of groups.

    Examples:

    1. If $a=bh$ for some $h\in H$, then $b=ah’$ for some $h’\in H$. Proof: If $a=bh$, then $ah^{-1}=(bh)h^{-1}=b(hh^{-1})=b$.
    2. If $a^{-1}b\in H$, then $b=ah$ for some $h\in H$. Proof: $b=a(a^{-1}b)$.
    3. The relation “$\underset{L}\sim$” is transitive. Proof: Let $a^{-1}b\in H$ and $b^{-1}c\in H$. Then $a^{-1}c=a^{-1}bb^{-1}c$ is the product of two elements of $H$ and so is in $H$.
    Miscellaneous remarks about the examples
    • Which exercises are used depends on what is taken as definition of coset.
    • In proving Exercise 2 at the board, the instructor might write “Proof: $b=a(a^{-1}b)$” on the board and the point to the expression “$a^{-1}b$” and say, “$a^{-1}b$ is in $H$!”
    • I wrote “$a^{-1}c=a^{-1}bb^{-1}c$” in Exercise 3. That will result in some brave student asking, “How on earth did you think of inserting $bb^{-1}$ like that?” The only reasonable answer is: “This is a trick that often helps in dealing with group elements, so keep it in mind.” See Rabbits.
    • That expression “$a^{-1}c=a^{-1}bb^{-1}c$” doesn’t explicitly mention that it uses associativity. That, too, might cause pointing at the board.
    • Pointing at the board is one thing you can do in a video presentation that you can’t do in a text. But in watching a video, it is harder to flip back to look at something done earlier. Flipping is easier to do if the video is short.
    • The first sentence of the proof of Exercise 3 is, “Let $a^{-1}b\in H$ and $b^{-1}c\in H$.” This uses rewrite according to the definition. One hopes that beginning group theory students already know about rewrite according to the definition. But my experience is that there will be some who don’t automatically do it.
    • in beginning abstract math courses, very few teachers
      tell students about rewrite according to the definition. Why not?

    • An excellent exercise for the students that would require more than short algebraic calculations would be:
      • Discuss which of the two definitions of left coset embedded in (2), (3), (5) and (6) is preferable.
      • Show in detail how it is equivalent to the other definition.

    A theorem

    In the undergraduate course, you will almost certainly be asked to prove this theorem:

    A subgroup $H$ of index $2$ of a group $G$ is normal in $G$.

    Proving the theorem

    In trying to prove this, a student may fiddle around with the definition of left and right coset for awhile using elementary manipulations of group elements as illustrated above. Then a lightbulb appears:

    In the 1980’s or earlier a well known computer scientist wrote to me that something I had written gave him a satori. I was flattered, but I had to look up “satori”.

    If the subgroup has index $2$ then there are two left cosets and two right cosets. One of the left cosets and one of the right cosets must be $H$ itself. In that case the left coset must be the complement of $H$ and so must the right coset. So those two cosets must be the same set! So the $H$ is normal in $G$.

    This is one of the earlier cases of sudden pattern recognition that occurs among students of abstract math. Its main attraction for me is that suddenly after a bunch of algebraic calculations (enough to determine that the cosets form a partition) you get the fact that the left cosets are the same as the right cosets by a purely conceptual observation with no computation at all.

    This proof raises a question:

    Why isn’t this point immediately obvious to students?

    I have to admit that it was not immediately obvious to me. However, before I thought about it much someone told me how to do it. So I was denied the Thrill of figuring this out myself. Nevertheless I thought the solution was, shall we say, cute, and so had a little thrill.

    A story about how the light bulb appears

    In doing exercises like those above, the student has become accustomed to using algebraic manipulation to prove things about groups. They naturally start doing such calculations to prove this theorem. They presevere for awhile…

    Scenario I

    Some students may be in the habit of abandoning their calculations, getting up to walk around, and trying to find other points of view.

    1. They think: What else do I know besides the definitions of cosets?
    2. Well, the cosets form a partition of the group.
    3. So they draw a picture of two boxes for the left cosets and two boxes for the right cosets, marking one box in each as being the subgroup $H$.
    4. If they have a sufficiently clear picture in their head of how a partition behaves, it dawns on them that the other two boxes have to be the same.
    Remarks about Scenario I
    • Not many students at the earliest level of abstract math ever take a break and walk around with the intent of having another approach come to mind. Those who do Will Go Far. Teachers should encourage this practice. I need to push this in abstractmath.org.
    • In good weather, David Hilbert would stand outside at a shelf doing math or writing it up. Every once in awhile he would stop for awhile and work in his garden. The breaks no doubt helped. So did standing up, I bet. (I don’t remember where I read this.)
    • This scenario would take place only if the students have a clear understanding of what a partition is. I suspect that often the first place they see the connection between equivalence relations and partitions is in a hasty introduction at the beginning of a group theory or abstract algebra course, so the understanding has not had long to sink in.

    Scenario II

    Some students continue to calculate…

    1. They might say, suppose $a$ is not in $H$. Then it is in the other left coset, namely $aH$.
    2. Now suppose $a$ is not in the “other” right coset, the one that is not $H$. But there are only two right cosets, so $a$ must be in $H$.
    3. But that contradicts the first calculation I made, so the only possibility left is that $a$ is in the right coset $Ha$. So $aH\subseteq Ha$.
    4. Aha! But then I can use the same argument the other way around, getting $Ha\subseteq aH$.
    5. So it must be that $aH=Ha$. Aha! …indeed.
    Remarks about Scenario 2
    • In step (2), the student is starting a proof by contradiction. Many beginning abstract math students are not savvy enough to do this.
    • Step (4) involves recognizing that an argument has a dual. Abstractmath.org does not mention dual arguments and I can’t remember emphasizing the idea to my classes. Tsk.
    • Scenario 2 involves the student continuing algebraic calculations till the lightbulb strikes. The lightbulb could also occur in other places in the calculation.

    References

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    Improving abstractmath.org

    This post discusses some ideas I have for improving abstractmath.org.

    Handbook of mathematical discourse

    The Handbook was kind of a false start on abmath, and is the source of much of the material in abmath. It still contains some material not in abmath, parti­cularly the citations.

    By citations I mean lexicographical citations: examples of the usage from texts and scholarly articles.

    I published the Handbook of mathe­ma­tical discourse in 2003. The first link below takes you to an article that describes what the Handbook does in some detail. Briefly, the Handbook surveys the use of language in math (and some other things) with an emphasis on the problems it causes students. Its collection of citations of usage could some day could be the start of an academic survey of mathematical language. But don’t expect me to do it.

    Links

    The Handbook exists as a book and as two different web versions. I lost the TeX source of the Handbook a few years after I published the book, so none of the different web versions are perfect. Version 2 below is probably the most useful.

    1. Handbook of mathe­ma­tical discourse. Description.
    2. Handbook of mathe­ma­tical discourse. Hypertext version without pictures but with active internal links. Some links don’t work, but they won’t be repaired because I have lost the TeX input files.
    3. Handbook of mathe­ma­tical discourse. Paperback.
    4. Handbook of mathematical discourse. PDF version of the printed book, including illustrations and citations but without hyperlinks.
    5. Citations for the paperback version of the Handbook. (The hypertext version and the PDF version include the citations.)

    Abmath

    Soon after the Handbook was published, I started work on abstractmath.org, which I abbreviate as abmath. It is intended specifically for people beginning to study abstract math, which means roughly post-calculus. I hope their teachers will read it, too. I had noticed when I was teaching that many students hit a big bump when faced with abstraction, and many of them never recovered. They would typically move into another field, often far away from STEM stuff.

    Links

    These abmath articles give more detail about the purpose of this website and the thinking behind the way it is presented:

    Presentation of abmath

    Informal

    Abmath is written for students of abstract math and other beginners to tell them about the obstacles they may meet up with in learning abstract math. It is not a scholarly work and is not written in the style of a scholarly work. There is more detail about its style in my rant in Attitude.

    Scholarly works should not be written in the style of a scholarly work, either.

    Links

    To do:

    Every time I revise an article I find myself rewriting overly formal parts. Fifty years of writing research papers has taken its toll. I must say that I am not giving this informalization stuff very high priority, but I will continue doing it.

    No citations

    One major difference concerns the citations in the Handbook. I collected these in the late nineties by spending many hours at Jstor and looking through physical books. When I started abmath I decided that the website would be informal and aimed at students, and would contain few or no citations, simply because of the time it took to find them.

    Boxouts and small screens

    The Handbook had both sidebars on every page of the paper version containing a reference index to words on that page, and also on many pages boxouts with comments. It was written in TeX. I had great difficulty using TeX to control the placement of both the sidebars and especially the boxouts. Also, you couldn’t use TeX to let the text expand or contract as needed by the width of the user’s screen.

    Abmath uses boxouts but not sidebars. I wrote Abmath using HTML, which allows it to be presented on large or small screens and to have extensive hyperlinks.
    HTML also makes boxouts easy.

    The arrival of tablets and i-pods has made it desirable to allow an abmath page to be made quite narrow while still readable. This makes boxouts hard to deal with. Also I have gotten into the habit of posting revisions to articles on Gyre&Gimble, whose editor converts boxouts into inline boxes. That can probably be avoided.

    To do:

    I have to decide whether to turn all boxouts into inline small-print paragraphs the was you see them in this article. That would make the situation easier for people reading small screens. But in-line small-print paragraphs are harder to associate to the location you want them to refer, in contrast to boxouts.

    Abmath 2.0

    For the first few years, I used Microsoft Word with MathType, but was plagued with problems described in the link below. Then I switched to writing directly in HTML. The articles of abmath labeled “abstractmath.org 2.0” are written in this new way. This makes the articles much, much easier to update. Unfortunately, Word produces HTML that is extraordinarily complicated, so transforming them into abmath 2.0 form takes a lot of effort.

    Link

    Illustrations

    Abmath does not have enough illustrations and diagrams. Gyre&Gimble has many posts with static illustrations, some of them innovative. It also has some posts with interactive demos created with Mathematica. These demos require the reader to download the CDF Player, which is free. Unfortunately, it is available only for Windows, Mac and Linux, which precludes using them on many small devices.

    Links

    To do:

    • Create new illustrations where they might be useful, and mine Gyre&Gimble and other sources.
    • There are many animated GIFs out there in the math cloud. I expect many of them are licensed under Creative Commons so that I can use them.
    • I expect to experiment with converting some of the interactive CFD diagrams that are in Gyre&Gimble into animated GIFs or AVIs, which as far as I know will run on most machines. This will be a considerable improvement over static diagrams, but it is not as good as interactive diagrams, where you can have several sliders controlling different variables, move them back and forth, and so on. Look at Inverse image revisited. and “quintic with three parameters” in Demos for graph and cograph of calculus functions.

    Abmath content

    Language

    Abmath includes most of the ideas about language in the Handbook (rewritten and expanded) and adds a lot of new material.

    Links

    1. The languages of math. Article in abmath. Has links to the other articles about language.
    2. Syntactic and semantic thinkers. Gyre&Gimble post.
    3. Syntax trees in mathematicians’ brains. Gyre&Gimble post.
    4. A visualization of a computation in tree form.Gyre&Gimble post.
    5. Visible algebra I. Gyre&Gimble post.
    6. Algebra is a difficult foreign language. Gyre&Gimble post.
    7. Presenting binops as trees. Gyre&Gimble post.
    8. Moths to the flame of meaning. How linguistics students also have trouble with syntax.
    9. Varieties of mathematical prose, by Atish Bagchi and Charles Wells.

    To do:

    The language articles would greatly benefit from more illustrations. In parti­cular:

    • G&G contains several articles about using syntax trees (items 3, 4, 5 and 7 above) to understand algebraic expressions. A syntax tree makes the meaning of an algebraic expression much more transparent than the usual one-dimensional way of writing it.
    • Several items in the abmath article More about the language of math, for example the entries on parenthetic assertions and postconditions could benefit from a diagrammatic representation of the relation between phrases in a sentence and semantics (or how the phrases are spoken).
    • The articles on Names and Alphabets could benefit from providing spoken pronunciations of many words. But what am I going to do about my southern accent?
    • The boxed example of change in context as you read a proof in More about the language of math could be animated as you click through the proof. *Sigh* The prospect of animating that example makes me tired just thinking about it. That is not how grasshoppers read proofs anyway.

    Understanding and doing math

    Abmath discusses how we understand math and strategies for doing math in some detail. This part is based on my own observations during 35 years of teaching, as well as extensive reading of the math ed literature. The math ed literature is usually credited in footnotes.

    Links

    Math objects and math structures

    Understanding how to think about mathematical objects is, I believe, one of the most difficult hurdles newbies have to overcome in learning abstract math. This is one area that the math ed community has focused on in depth.

    The first two links below are take you to the two places in abmath that discuss this problem. The first article has links to some of the math ed literature.

    Links

    To do: Everything is a math object

    An important point about math objects that needs to be brought out more is that everything in math is a math object. Obviously math structures are math objects. But the symbol “$\leq$” in the statement “$35\leq45$” denotes a math object, too. And a proof is a math object: A proof written on a blackboard during a lecture does not look like it is an instance of a rigorously defined math object, but most mathe­maticians, including me, believe that in principle such proofs can be transformed into a proof in a formal logical system. Formal logics, such as first order logic, are certainly math objects with precise mathematical definitions. Definitions, math expressions and theorems are math objects, too. This will be spelled out in a later post.

    To do: Bring in modern ideas about math structure

    Classically, math structures have been presented as sets with structure, with the structure being described in terms of subsets and functions. My chapter on math structures only waves a hand at this. This is a decidedly out-of-date way of doing it, now that we have category theory and type theory. I expect to post about this in order to clarify my thinking about how to introduce categorical and type-theoretical ideas without writing a whole book about it.

    Particular math structures

    Abmath includes discussions
    of the problems students have with certain parti­cular types of structures. These sections talk mostly about how to think about these structure and some parti­cular misunder­standings students have at the most basic levels.

    These articles are certainly not proper intro­ductions to the structures. Abmath in general is like that: It tells students about some aspects of math that are known to cause them trouble when they begin studying abstract math. And that is all it does.

    Links

    To do:

    • I expect to write similar articles about groups, spaces and categories.
    • The idea about groups is to mention a few things that cause trouble at the very beginning, such as cosets, quotients and homomorphisms (which are all obviously related to each other), and perhaps other stumbling blocks.
    • With categories the idea is to stomp on misconceptions such as that the arrows have to be functions and to emphasize the role of categories in allowing us to define math structures in terms of their relations with other objects instead of in terms with sets.
    • I am going to have more trouble with spaces. Perhaps I will show how you can look at the $\epsilon$-$\delta$ definition of continuous functions on the reals and “discover” that they imply that inverse images of open sets are open, thus paving the way for the family-of-subsets definition of a topoogy.
    • I am not ruling out other particular structures.

    Proofs

    This chapter covers several aspects of proofs that cause trouble for students, the logical aspects and also the way proofs are written.

    It specifically does not make use of any particular symbolic language for logic and proofs. Some math students are not good at learning languages, and I didn’t see any point in introducing a specific language just to do rudimentary discussions about proofs and logic. The second link below discusses linguistic ability in connection with algebra.

    I taught logic notation as part of various courses to computer engineering students and was surprised to discover how difficult some students found using (for example) $p+q$ in one course and $p\vee q$ in another. Other students breezed through different notations with total insouciance.

    Links

    To do:

    Much of the chapter on proofs is badly written. When I get around to upgrading it to abmath 2.0 I intend to do a thorough rewrite, which I hope will inspire ideas about how to conceptually improve it.

    Creative Commons License

    This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.


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    More alphabets

    This post is the third and last in a series of posts containing revisions of the abstractmath.org article Alphabets. The first two were:

    Addition to the listings for the Greek alphabet

    Sigma: $\Sigma,\,\sigma$ or ς: sĭg'mɘ. The upper case $\Sigma $ is used for indexed sums.  The lower case $\sigma$ (don't call it "oh") is used for the standard deviation and also for the sum-of-divisors function. The ς form for the lower case has not as far as I know been used in math writing, but I understood that someone is writing a paper that will use it.

    Hebrew alphabet

    Aleph, א is the only Hebrew letter that is widely used in math. It is the cardinality of the set of integers. A set with cardinality א is countably infinite. More generally, א is the first of the aleph numbers $א_1$, $א_2$, $א_3$, and so on.

    Cardinality theorists also write about the beth (ב) numbers, and the gimel (ג) function. I am not aware of other uses of the Hebrew alphabet.

    If you are thinking of using other Hebrew letters, watch out: If you type two Hebrew letters in a row in HTML they show up on the screen in reverse order. (I didn't know HTML was so clever.)

    Cyrillic alphabet

    The Cyrillic alphabet is used to write Russian and many other languages in that area of the world. Wikipedia says that the letter Ш, pronounced "sha", is the only Cyrillic letter used in math. I have not investigated further.

    The letter is used in several different fields, to denote the Tate-Shafarevich group, the Dirac comb and the shuffle product.

    It seems to me that there are a whole world of possibillities for brash young mathematicians to name mathematical objects with other Cyrillic letters. Examples:

    • Ж. Use it for a ornate construction, like the Hopf fibration or a wreath product.
    • Щ. This would be mean because it is hard to pronounce.
    • Ъ. Guaranteed to drive people crazy, since it is silent. (It does have a name, though: "Yehr".)
    • Э. Its pronunciation indicates you are unimpressed (think Fonz).
    • ю. Pronounced "you". "ю may provide a counterexample". "I do?"

    Type styles

    Boldface and italics

    A typeface is a particular design of letters.  The typeface you are reading is Arial.  This is Times New Roman. This is Goudy. (Goudy may not render correctly on your screen if you don't have it installed.)

    Typefaces typically come in several styles, such as bold (or boldface) and italic.

    Examples



    Arial Normal Arial italic Arial bold
    Times Normal Times italic Times bold Goudy Normal Goudy italic Goudy bold

    Boldface and italics are used with special meanings (conventions) in mathematics. Not every author follows these conventions.

    Styles (bold, italic, etc.) of a particular typeface are supposedly called fonts.  In fact, these days “font” almost always means the same thing as “typeface”, so I  use “style” instead of “font”.

    Vectors

    A letter denoting a vector is put in boldface by many authors.

    Examples
    • “Suppose $\mathbf{v}$ be an vector in 3-space.”  Its coordinates typically would be denoted by $v_1$, $v_2$ and $v_3$.
    • You could also define it this way:  “Let $\mathbf{v}=({{v}_{1}},{{v}_{2}},{{v}_{3}})$ be a vector in 3-space.”  (See parenthetic assertion.)

    It is hard to do boldface on a chalkboard, so lecturers may use $\vec{v}$ instead of $\mathbf{v}$. This is also seen in print.

    Definitions

    The definiendum (word or phrase being defined) may be put in boldface or italics. Sometimes the boldface or italics is the only clue you have that the term is being defined. See Definitions.

    Example

     

    “A group is Abelian if its multiplication is commutative,” or  “A group is Abelian if its multiplication is commutative.”

    Emphasis

    Italics are used for emphasis, just as in general English prose. Rarely (in my experience) boldface may be used for emphasis.

    In the symbolic language

    It is standard practice in printed math to put single-letter variables in italics.   Multiletter identifiers are usually upright.

    Example

    Example: "$f(x)=a{{x}^{2}}+\sin x$".  Note that mathematicians would typically refer to $a$ as a “constant” or “parameter”, but in the sense we use the word “variable” here, it is a variable, and so is $f$.

    Example

    On the other hand, “e” is the proper name of a specific number, and so is “i”. Neither is a variable. Nevertheless in print they are usually given in italics, as in ${{e}^{ix}}=\cos x+i\sin x$.  Some authors would write this as ${{\text{e}}^{\text{i}x}}=\cos x+\text{i}\,\sin x$.  This practice is recommended by some stylebooks for scientific writing, but I don't think it is very common in math.

    Blackboard bold

     

    Blackboard bold letters are capital Roman letters written with double vertical strokes.   They look like this:

    \[\mathbb{A}\,\mathbb{B}\,\mathbb{C}\,\mathbb{D}\,\mathbb{E}\,\mathbb{F}\,\mathbb{G}\,\mathbb{H}\,\mathbb{I}\,\mathbb{J}\,\mathbb{K}\,\mathbb{L}\,\mathbb{M}\,\mathbb{N}\,\mathbb{O}\,\mathbb{P}\,\mathbb{Q}\,\mathbb{R}\,\mathbb{S}\,\mathbb{T}\,\mathbb{U}\,\mathbb{V}\,\mathbb{W}\,\mathbb{X}\,\mathbb{Y}\,\mathbb{Z}\]

    In lectures using chalkboards, they are used to imitate boldface.

    In print, the most common uses is to represent certain sets of numbers:

    Remarks

    • Mathe­ma­tica uses some lower case blackboard bold letters.
    • Many mathe­ma­tical writers disapprove of using blackboard bold in print.  I say the more different letter shapes that are available the better.  Also a letter in blackboard bold is easier to distinguish from ordinary upright letters than a letter in boldface is, particularly on computer screens.
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    The use of fraktur in math

    This post is a revision of the part of the abmath article on alphabets concerning the fraktur typeface, followed by some corrections and remarks. A revision of the section on the Greek alphabet was posted previously.

    Fraktur

    In some math subjects, a font tamily (typeface) called fraktur, formerly used for writing German, Norwegian, and some other languages, is used to name math objects.  The table below shows the upper and lower case fraktur letters. 








    $A,a$: $\mathfrak{A},\mathfrak{a}$ $H,h$: $\mathfrak{H},\mathfrak{h}$ $O,o$: $\mathfrak{O},\mathfrak{o}$ $V,v$: $\mathfrak{V},\mathfrak{v}$
    $B,b$: $\mathfrak{B},\mathfrak{b}$ $I,i$: $\mathfrak{I},\mathfrak{i}$ $P,p$: $\mathfrak{P},\mathfrak{p}$ $W,w$: $\mathfrak{W},\mathfrak{w}$
    $C,c$: $\mathfrak{C},\mathfrak{c}$ $J,j$: $\mathfrak{J},\mathfrak{j}$ $Q,q$: $\mathfrak{Q},\mathfrak{q}$ $X,x$: $\mathfrak{X},\mathfrak{x}$
    $D,d$: $\mathfrak{D},\mathfrak{d}$ $K,k$: $\mathfrak{K},\mathfrak{k}$ $R,r$: $\mathfrak{R},\mathfrak{r}$ $Y,y$: $\mathfrak{Y},\mathfrak{y}$
    $E,e$: $\mathfrak{E},\mathfrak{e}$ $L,l$: $\mathfrak{L},\mathfrak{l}$ $S,s$: $\mathfrak{S},\mathfrak{s}$ $Z,z$: $\mathfrak{Z},\mathfrak{z}$
    $F,f$: $\mathfrak{F},\mathfrak{f}$ $M,m$: $\mathfrak{M},\mathfrak{m}$ $T,t$: $\mathfrak{T},\mathfrak{t}$  
    $G,g$: $\mathfrak{G},\mathfrak{g}$ $N,n$: $\mathfrak{N},\mathfrak{n}$ $U,u$: $\mathfrak{U},\mathfrak{u}$  
    • Many of the forms are confusing and are commonly mispronounced by younger mathematicians.  (Ancient mathematicians like me have taken German classes in college that required learning fraktur.)  In particular the uppercase $\mathfrak{A}$ looks like $U$ but in fact is an $A$, and the uppercase $\mathfrak{I}$ looks like $T$ but is actually $I$.  
    • When writing on the board, some mathematicians use a cursive form when writing objects with names that are printed in fraktur.
    • Unicode regards fraktur as a typeface (font family) rather than as a different alphabet. However, unicode does provide codes for the fraktur letters that are used in math (no umlauted letters or ß). In TeX you type "\mathfrak{a}" to get $\mathfrak{a}$.
    • In my (limited) experience, native German speakers usually call this alphabet “Altschrift” instead of “Fraktur”.  It has also been called “Gothic”, but that word is also used to mean several other quite different typefaces (black­letter, sans serif and (gasp) the alphabet actually used by the Goths.
    • I have been doing mathematical research for around fifty years. It is clear to me that mathematicians' use of and familiarity with fraktur has declined a lot during that time. But it is not extinct. I have made a hasty and limited search of Jstor and found recent websites and research articles that use it in a variety of fields. There are also a few citations in the Handbook (search for "fraktur").

      • It is used in ring theory and algebraic number theory, in particular to denote ideals.
      • It is use in Lie algebra. In particular, the Lie algebra of a Lie group $G$ is commonly denoted by $\mathfrak{g}$.
      • The cardinality of the continuum is often denoted by $\mathfrak{c}$.
      • It is used occasionally in logic to denote models and other objects.
      • I remember that in the sixties and seventies fraktur was used in algebraic geometry, but I haven't found it in recent papers.

    Acknowledgements

    Thanks to Fernando Gouvêa for suggestions.

    Remarks about usage in abstractmath.org

    The Handbook has 428 citation for usages in the mathematical research literature. After finishing that book, I started abstractmath.org and decided that I would quote the Handbook for usages when I could but would not spend any more time looking for citations myself, which is very time consuming. Instead, in abmath I have given only my opinion about usage. A systematic, well funded project for doing lexicographical research in the math literature would undoubtedly show that my remarks were sometimes incorrect and very often, perhaps even usually, incomplete.

    Corrections to the post The Greek alphabet in math

    Willie Wong suggested some additional pronunciations for upsilon and omega:


    Upsilon: $\Upsilon ,\,\upsilon$  ŭp'sĭlŏn; (Br) ĭp'sĭlŏn. (Note: I have never heard anyone pronounce this letter, and various dictionaries suggest a ridiculous number of different pronunciations.) Rarely used in math; there are references in the Handbook.


    Omega: $\Omega ,\,\omega$: ōmā'gɘ, ō'māgɘ; (Br) ōmē'gɘ, ō'mēgɘ. $\Omega$ is often used as the name of a domain in $\mathbb{R}^n$. The set of natural numbers with the usual ordering is commonly denoted by $\omega$. Both forms have many other uses in advanced math.  

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