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A new kind of introduction to category theory

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About this article

  • This post is an alpha version of the first part of the intended article.
  • People who are beginners in learning abstract math concepts have many misunderstandings about the definitions and early theorems of category theory.
  • This article introduces a few basic concepts of category theory. It goes into detail in Purple Prose about the misunderstandings that can arise with each of the concepts. The article is not at all a complete introduction to categories.
  • My blog post Introducing abstract topics describes some of the strategies needed in teaching a new abstract math concept.
  • This article also introduces a few examples of categories that are primarily chosen to cause the reader to come up against some of those misunderstandings. The first example is completely abstract.
  • Math students usually see categories after considerable exposure to abstract math, but students in computing science and other fields may see it without having much background in abstraction. I hope teachers in such courses will include explanations of the sort of misunderstandings mentioned in this article.
  • Like all posts in Gyre&Gimble and all posts in abstractmath.org, this article is licensed under a Creative Commons Attribution-ShareAlike 2.5 License. If you are teaching a class involving category theory, feel free to hand it out, and to modify it (in which case you should include a link to this post).
  • You could also use the article as a source of remarks you make in the class about the topics.

About categories

To be written.

Definition of category

A category is a type of Mathematical structure consisting of two types of data, whose relationships are entirely determined by some axioms. After the definition is complete, I will introduce several categories with a detailed discussion of each one, explaining how they fit the definition of category.

Axiom 1: Data

  1. A category consists of two types of data: objects and arrows.
  2. No object can be an arrow and no arrow can be an object.

Notes for Axiom 1

  • An object of a category can be any kind of mathematical object. It does not have to be a set and it does not have to have elements.
  • Arrows of a category are also called morphisms. You may be familiar with “homomorphisms”, “homeomorphisms” or “isomorphisms”, all of which are functions. This does not mean that a “morphism” in an arbitrary category is a function.

Axiom 2: Domain and codomain

  1. Each arrow has a domain and a codomain, each of which is an object of the category.
  2. The domain and the codomain of an arrow may or may not be the same object.
  3. Each arrow has only one domain and only one codomain.

Notes for Axiom 2

  • If $f$ is an arrow with domain $A$ and codomain $B$, that fact is typically shown either by the notation “$f:A\to B$” or by a diagram like this:

  • The notation “$f:A\to B$” is like that used for functions. This notation may be used in any category, but it does not imply that $f$ is a function or that $A$ and $B$ have elements.
  • For such an arrow, the notation “$\text{dom}(f)$” refers to $A$ and “$\text{cod}(f)$” refers to $B$.
  • For a given category $\mathsf{C}$, the collection of all the arrows with domain $A$ and codomain $B$ may be denoted by
    • “$\text{Hom}(A,B)$” or
    • “$\text{Hom}_\mathsf{C}(A,B)$” or
    • “$\mathsf{C}(A,B)$”.
  • Some newer books and articles in category theory use the name source for domain and target for codomain. This usage has the advantage that a newcomer to category theory will be less likely to think of an arrow as a function.

Axiom 3: Composition

  1. If $f$ and $g$ are arrows in a category for which $\text{cod}(f)=\text{dom}(g)$, as in this diagram:

    then there is a unique arrow with domain $A$ and codomain $C$ called the composite of $f$ and $g$.

Notes for Axiom 3

    diagra

  • An important metaphor for composition is: Every path of length 2 has exactly one composite.
  • The unique arrow required by Axiom 3 may be denoted by “$g\circ f$” or “$gf$”. “$g\circ f$” is more explicit, but “$gf$” is much more commonly used by category theorists.
  • Many constructions in categories may be shown by diagrams, like the one used just above.
  • The diagram

    is said to commute if $h=g\circ f$. The idea is that going along $f$ and then $g$ is the same as going along $h$.

  • It is customary in some texts in category theory to indicate that a diagram commutes by putting a gyre in the middle:
  • The concept of category is an abstraction of the idea of function, and the composition of arrows is an abstraction of the composition of functions. It uses the same notation, “$g\circ f$”. If $f$ and $g$ are set functions, then for an element $x$ in the domain of $f$, \[(g\circ f)(x)=g(f(x))\]
  • But in arbitrary category, it may make no sense to evaluate an arrow $f$ at some element $x$; indeed, the domain of $f$ may not have elements at all, and then the statement “$(g\circ f)(x)=g(f(x))$” is meaningless.

Axiom 4: Identity arrows

  1. For each object $A$ of a category, there is a unique arrow denoted by $\textsf{id}_A$.
  2. $\textsf{dom}(\textsf{id}_A)=A$ and $\textsf{cod}(\textsf{id}_A)=A$.
  3. For any object $B$ and any arrow $f:B\to A$, the diagram

    commutes.

  4. For any object $C$ and any arrow $g:A\to C$, the diagram

    commutes.

Notes for Axiom 4

  • The fact stated in Axiom 4(b) could be shown diagrammatically either as

    or as

  • Facts (c) and (d) can be written in algebraic notation: For any arrow $f$ going to $A$,\[\textsf{id}_A\circ f=f\]and for any arrow $g$ coming from $A$,\[g\circ \textsf{id}_A=g\]

Axiom 5: Associativity

  1. If $f$, $g$ and $h$ are arrows in a category for which $\text{cod}(f)=\text{dom}(g)$ and $\text{cod}(g)=\text{dom}(h)$, as in this diagram:

    then there is a unique arrow $k$ with domain $A$ and codomain $C$ called the composite of $f$, $g$ and $h$.

  2. In the diagram below, the two triangles containing $k$ must both commute.

Notes for Axiom 5

  • Axiom 5b requires that \[h\circ(g\circ f)=(h\circ g)\circ f\](which both equal $k$), which is the usual formula for associativity.
  • Note that the top two triangles commute by Axiom 3.
  • The associativity axiom means that we can get rid of parentheses and write \[k=h\circ f\circ g\]just as we do for addition and multiplication of numbers.
  • In my opinion the notation using categorical diagrams communicates information much more clearly than algebraic notation does. In particular, you don’t have to remember the domains and codomains of the functions — they appear in the picture. I admit that diagrams take up much more space, but now that we read math stuff on a computer screen instead of on paper, space is free.

Examples of categories

For the first three examples, I will give a detailed explanation about how they fit the definition of category.

Example 1: MyFin

This first example is a small, finite category which I have named $\mathsf{MyFin}$ (my very own finite category). It is not at all an important category, but it has advantages as a first example.

  • It’s small enough that you can see all the objects and arrows on the screen at once, but big enough not to be trivial.
  • The objects and arrows have no properties other than being objects and arrows. (The other examples involve familiar math objects.)
  • So in order to check that $\mathsf{MyFin}$ really obeys the axioms for a category, you can use only the skeletal information given here. As a result, you must really understand the axioms!

A correct proof will be based on axioms and theorems. The proof can be suggested by your intuitions, but intuitions are not enough. When working with $\mathsf{MyFin}$ you won’t have any intuitions!

A diagram for $\mathsf{MyFin}$

This diagram gives a partial description of $\mathsf{MyFin}$.

Now let’s see how to make the diagram above into a category.

Axiom 1

  • The objects of $\mathsf{MyFin}$ are $A$, $B$, $C$ and $D$.
  • The arrows are $f$, $g$, $h$, $j$, $k$, $r$, $s$, $u$, $v$, $w$ and $x$.
  • You can regard the letters just listed as names of the objects and arrows. The point is that at this stage all you know about the objects and arrows are their names.
  • If you prefer, you can think of the arrows as the actual arrows shown in the $\mathsf{MyFin}$ diagram.
  • Our definition of $\mathsf{MyFin}$ is an abstract definition. You may have seen multiplication tables of groups given in terms of undefined letters. (If you haven’t, don’t worry.) Those are also abstract definitions.
  • Most of our other definitions of categories involve math objects you actually know something about. They are like the definition of division, for example, where the math objects are integers.

Axiom 2

  • The domains and codomains of the arrows are shown by the diagram above.
  • For example, $\text{dom}(r)=A$ and $\text{cod}(r)=C$, and $\text{dom}(v)=\text{cod}(v)=B$.

Axiom 3

Showing the $\mathsf{MyFin}$ diagram does not completely define $\mathsf{MyFin}$. We must say what the composites of all the paths of length 2 are.

  • In fact, most of them are forced, but two of them are not.
  • We must have $g\circ f=r$ because $r$ is the only arrow possible for the composite, and Axiom 3 requires that every path of length 2 must have a composite.
  • For the same reason, $h\circ g=s$.
  • All the paths involving $u$, $v$, $w$ and $x$ are forced:

  • (p1) $u\circ u=u$, $v\circ v=v$, $w\circ w=w$ and $x\circ x=x$.
  • (p2) $f\circ u=f$, $r\circ u=r$, $j\circ u=j$ and $k\circ u=k$. You can see that, for example, $f\circ u=f$ by opening up the loop on $f$ like this:

    There is only one arrow going from $A$ to $B$, namely$f$, so $f$ has to be the composite $f\circ u$.

  • (p3) $v\circ f=f$, $g\circ v=g$ and $s\circ v=s$.
  • (p4) $w\circ g=g$, $w\circ r=r$ and $h\circ w=h$.
  • (p5) $x\circ h=h$, $x\circ s=s$, $x\circ j=j$ and $x\circ k=k$.

  • For $s\circ f$ and $h\circ r$, we have to choose between $j$ and $k$ as composites. Since $s\circ f=(h\circ g)\circ f$ and $h\circ r=h\circ (g\circ f)$, Axiom 3 requires that we must chose one of $j$ and $k$ to be both composites.

    Definition: $s\circ f=h\circ r=j$.

    If we had defined $s\circ f=h\circ r=k$ we would have a different category, although one that is “isomorphic” to $\mathsf{MyFin}$ (you have to define “isomorphic” or look it up.)

    • Axiom 4

    • It is clear from the $\mathsf{MyFin}$ diagram that for each object there is just one arrow that has that object both as domain and as codomain, as required by Axiom 4a.
    • The requirements in Axiom 4b and 4c are satisfied by statements (p1) through (p5).

    Axiom 5

    • Since we have already required both $(h\circ g)\circ f$ and $h\circ(g\circ f)$ to be $k$, composition is associative.

    Example 2: Set

    To be written.

    This will be a very different example, because it involves known mathematical objects — sets and functions. But there are still issues, for example the fact that the inclusion of $\{1,2\}$ into $\{1,2,3\}$ and the identity map on $\{1,2\}$ are two different arows in the category of sets.

    Example 3: IntegerDiv

    To be written.

    The objects are all the positive integers and there is an arrow from $m$ to $n$ if and only if $m$ divides $n$. So this example involves familiar objects and predicates, but the arrows are nevertheless not functions that take elements to elements. Integers don’t have elements. I would expect to show how the GCD of two integers is a limit.

    References

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    Representations of functions I

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    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.

    Segments posted so far

    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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    Insights into mathematical definitions

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    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

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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

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    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.

    Comments

    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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    abstractmath.org beta

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    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.

    Previous posts about the evolution of abstractmath.org

    Books by Michael Barr and Charles Wells

    Toposes, triples and theories

    Category theory for computing science

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    Recent revisions to abstractmath.org

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    For the last six months or so I have been systematically going through the abstractmath.org files, editing them for consistency, updating them, and in some cases making major revisions.

    In the past I have usually posted revised articles here on Gyre&Gimble, but WordPress makes it difficult to simply paste the HTML into the WP editor, because the editor modifies the HTML and does things such as recognizing line breaks and extra spaces which an HTML interpreters is supposed to ignore.

    Here are two lists of articles that I have revised, with links.

    Major revisions

    Other revised articles

    Other recent changes

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    Notation for sets

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    This is a revision of the section of abstractmath.org on notation for sets.

    Sets of numbers

    The following notation for sets of numbers is fairly standard.

    Remarks

    • Some authors use $\mathbb{I}$ for $\mathbb{Z}$, but $\mathbb{I}$ is also used for the unit interval.
    • Many authors use $\mathbb{N}$ to denote the nonnegative integers instead
      of the positive ones.
    • To remember $\mathbb{Q}$, think “quotient”.
    • $\mathbb{Z}$ is used because the German word for “integer” is “Zahl”.

    Until the 1930’s, Germany was the world center for scientific and mathematical study, and at least until the 1960’s, being able to read scientific German was was required of anyone who wanted a degree in science. A few years ago I was asked to transcribe some hymns from a German hymnbook — not into English, but merely from fraktur (the old German alphabet) into the Roman alphabet. I sometimes feel that I am the last living American to be able to read fraktur easily.

    Element notation

    The expression “$x\in A$” means that $x$ is an element of the set $A$. The expression “$x\notin A$” means that $x$ is not an element of $A$.

    “$x\in A$” is pronounced in any of the following ways:

    • “$x$ is in $S$”.
    • “$x$ is an element of $S$”.
    • “$x$ is a member of $S$”.
    • “$S$ contains $x$”.
    • “$x$ is contained in $S$”.

    Remarks

    • Warning: The math English phrase “$A$ contains $B$” can mean either “$B\in A$” or “$B\subseteq A$”.
    • The Greek letter epsilon occurs in two forms in math, namely $\epsilon$ and $\varepsilon$. Neither of them is the symbol for “element of”, which is “$\in$”. Nevertheless, it is not uncommon to see either “$\epsilon$” or “$\varepsilon$” being used to mean “element of”.
    Examples
    • $4$ is an element of all the sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$.
    • $-5\notin \mathbb{N}$ but it is an element of all the others.

    List notation

    Definition: list notation

    A set with a small number of elements may be denoted by listing the elements inside braces (curly brackets). The list must include exactly all of the elements of the set and nothing else.

    Example

    The set $\{1,\,3,\,\pi \}$ contains the numbers $1$, $3$ and $\pi $ as elements, and no others. So $3\in \{1,3,\pi \}$ but $-3\notin \{1,\,3,\,\pi \}$.

    Properties of list notation

    List notation shows every element and nothing else

    If $a$ occurs in a list notation, then $a$ is in the set the notation defines.  If it does not occur, then it is not in the set.

    Be careful

    When I say “$a$ occurs” I don’t mean it necessarily occurs using that name. For example, $3\in\{3+5,2+3,1+2\}$.

    The order in which the elements are listed is irrelevant

    For example, $\{2,5,6\}$ and $\{5,2,6\}$ are the same set.

    Repetitions don’t matter

    $\{2,5,6\}$, $\{5,2,6\}$, $\{2,2,5,6 \}$ and $\{2,5,5,5,6,6\}$ are all different representations of the same set. That set has exactly three elements, no matter how many numbers you see in the list notation.

    Multisets may be written with braces and repeated entries, but then the repetitions mean something.

    When elements are sets

    When (some of) the elements in list notation are themselves sets (more about that here), care is required.  For example, the numbers $1$ and $2$  are not elements of the set \[S:=\left\{ \left\{ 1,\,2,\,3 \right\},\,\,\left\{ 3,\,4 \right\},\,3,\,4 \right\}\]The elements listed include the set $\{1, 2, 3\}$ among others, but not the number $2$.  The set $S$ contains four elements, two sets and two numbers. 

    Another way of saying this is that the element relation is not transitive: The facts that $A\in B$ and $B\in C$ do not imply that $A\in C$. 

    Sets are arbitrary

    • Any mathematical object can be the element of a set.
    • The elements of a set do not have to have anything in common.
    • The elements of a set do not have to form a pattern.
    Examples
    • $\{1,3,5,6,7,9,11,13,15,17,19\}$ is a set. There is no point in asking, “Why did you put that $6$ in there?” (Sets can be arbitrary.)
    • Let $f$ be the function on the reals for which $f(x)=x^3-2$. Then \[\left\{\pi^3,\mathbb{Q},f,42,\{1,2,7\}\right\}\] is a set. Sets do not have to be homogeneous in any sense.


    Setbuilder notation

    Definition:

    Suppose $P$ is an assertion. Then the expression “$\left\{x|P(x) \right\}$” denotes the set of all objects $x$ for which $P(x)$ is true. It contains no other elements.

    • The notation “$\left\{ x|P(x) \right\}$” is called setbuilder notation.

    • The assertion $P$ is called the defining condition for the set.

    • The set $\left\{ x|P(x) \right\}$ is called the truth set of the assertion $P$.
    Examples

    In these examples, $n$ is an integer variable and $x$ is a real variable..

    • The expression “$\{n| 1\lt n\lt 6 \}$” denotes the set $\{2, 3, 4, 5\}$. The defining condition is “$1\lt n\lt 6$”.  The set $\{2, 3, 4, 5\}$ is the truth set of the assertion “n is an integer and $1\lt n\lt 6$”.
    • The notation $\left\{x|{{x}^{2}}-4=0 \right\}$ denotes the set $\{2,-2\}$.
    • $\left\{ x|x+1=x \right\}$ denotes the empty set.
    • $\left\{ x|x+0=x \right\}=\mathbb{R}$.
    • $\left\{ x|x\gt6 \right\}$ is the infinite set of all real numbers bigger than $6$.  For example, $6\notin \left\{ x|x\gt6 \right\}$ and $17\pi \in \left\{ x|x\gt6 \right\}$.
    • The set $\mathbb{I}$ defined by $\mathbb{I}=\left\{ x|0\le x\le 1 \right\}$ has among its elements $0$, $1/4$, $\pi /4$, $1$, and an infinite number of
      other numbers. $\mathbb{I}$ is fairly standard notation for this set – it is called the unit interval.

    Usage and terminology

    • A colon may be used instead of “|”. So $\{x|x\gt6\}$ could be written $\{x:x\gt6\}$.
    • Logicians and some mathematicians called the truth set of $P$ the extension of $P$. This is not connected with the usual English meaning of “extension” as an add-on.
    • When the assertion $P$ is an equation, the truth set of $P$ is usually called the solution set of $P$. So $\{2,-2\}$ is the solution set of $x^2=4$.
    • The expression “$\{n|1\lt n\lt6\}$” is commonly pronounced as “The set of integers such that $1\lt n$ and $n\lt6$.” This means exactly the set $\{2,3,4,5\}$. Students whose native language is not English sometimes assume that a set such as $\{2,4,5\}$ fits the description.

    Setbuilder notation is tricky

    Looking different doesn’t mean they are different.

    A set can be expressed in many different ways in setbuilder notation. For example, $\left\{ x|x\gt6 \right\}=\left\{ x|x\ge 6\text{ and }x\ne 6 \right\}$. Those two expressions denote exactly the same set. (But $\left\{x|x^2\gt36 \right\}$ is a different set.)

    Russell’s Paradox

    In certain areas of math research, setbuilder notation can go seriously wrong. See Russell’s Paradox if you are curious.

    Variations on setbuilder notation

    An expression may be used left of the vertical line in setbuilder notation, instead of a single variable.

    Giving the type of the variable

    You can use an expression on the left side of setbuilder notation to indicate the type of the variable.

    Example

    The unit interval $I$ could be defined as \[\mathbb{I}=\left\{x\in \mathrm{R}\,|\,0\le x\le 1 \right\}\]making it clear that it is a set of real numbers rather than, say rational numbers.  You can always get rid of the type expression to the left of the vertical line by complicating the defining condition, like this:\[\mathbb{I}=\left\{ x|x\in \mathrm{R}\text{ and }0\le x\le 1 \right\}\]

    Other expressions on the left side

    Other kinds of expressions occur before the vertical line in setbuilder notation as well.

    Example

    The set\[\left\{ {{n}^{2}}\,|\,n\in \mathbb{Z} \right\}\]consists of all the squares of integers; in other words its elements are 0,1,4,9,16,….  This definition could be rewritten as $\left\{m|\text{ there is an }n\in \mathrm{}\text{ such that }m={{n}^{2}} \right\}$.

    Example

    Let $A=\left\{1,3,6 \right\}$.  Then $\left\{ n-2\,|\,n\in A\right\}=\left\{ -1,1,4 \right\}$.

    Warning

    Be careful when you read such expressions.

    Example

    The integer $9$ is an element of the set \[\left\{{{n}^{2}}\,|\,n\in \text{ Z and }n\ne 3 \right\}\]It is true that $9={{3}^{2}}$ and that $3$ is excluded by the defining condition, but it is also true that $9={{(-3)}^{2}}$ and $-3$ is not an integer ruled out by the defining condition.

    Reference

    Sets. Previous post.

    Acknowledgments

    Toby Bartels for corrections.

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    Math majors attacked by cognitive dissonance

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    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.

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

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    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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