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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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    Introducing abstract topics

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    I have been busy for the past several years revising abstractmath.org (abmath). Now I believe, perhaps foolishly, that most of the articles in abmath have reached beta, so now it is time for something new.

    For some time I have been considering writing introductions to topics in abstract math, some typically studied by undergraduates and some taken by scientists and engineers. The topics I have in mind to do first include group theory and category theory.

    The point of these introductions is to get the student started at the very beginning of the topic, when some students give up in total confusion. They meet and fall off of what I have called the abstraction cliff, which is discussed here and also in my blog posts Very early difficulties and Very early difficulties II.

    I may have stolen the phrase “abstraction cliff” from someone else.

    Group theory

    Group theory sets several traps for beginning students.

    Multiplication table

    • A student may balk when a small finite group is defined using a set of letters in a multiplication table.
      “But you didn’t say what the letters are or what the multiplication is?”
    • Such a definition is an abstract definition, in contrast to the definition of “prime”, for example, which is stated in terms of already known entities, namely the integers.
    • The multiplication table of a group tells you exactly what the binary operation is and any set with an operation that makes such a table correct is an example of the group being defined.
    • A student who has no understanding of abstraction is going to be totally lost in this situation. It is quite possible that the professor has never even mentioned the concept of abstract definition. The professor is probably like most successful mathematicians: when they were students, they understood abstraction without having to have it explained, and possibly without even noticing they did so.

    Cosets

    • Cosets are a real killer. Some students at this stage are nowhere near thinking of a set as an object or a thing. The concept of applying a binary operation on a pair of sets (or any other mathematical objects with internal structure) is completely foreign to them. Did anyone ever talk to them about mathematical objects?
    • The consequence of this early difficulty is that such a student will find it hard to understand what a quotient group is, and that is one of the major concepts you get early in a group theory course.
    • The conceptual problems with multiplication of cosets is similar to those with pointwise addition of functions. Given two functions $f,g:\mathbb{R}\to\mathbb{R}$, you define $f+g$ to be the function \[(f+g)(x):=f(x)+g(x)\] Along with pointwise multiplication, this makes the space of functions $\mathbb{R}\to\mathbb{R}$ a ring with nice properties.
    • But you have to understand that each element of the ring is a function thought of as a single math object. The values of the function are properties of the function, but they are not elements of the ring. (You can include the real numbers in the ring as constant functions, but don’t confuse me with facts.)
    • Similarly the elements of the quotient group are math objects called cosets. They are not elements of the original group. (To add to the confusion, they are also blocks of a congruence.)

    Isomorphic groups

    • Many books, and many professors (including me) regard two isomorphic groups as the same. I remember getting anguished questions: “But the elements of $\mathbb{Z}_2$ are equivalence classes and the elements of the group of permutations of $\{1,2\}$ are functions.”
    • I admit that regarding two isomorphic groups as the same needs to be treated carefully when, unlike $\mathbb{Z}_2$, the group has a nontrivial automorphism group. ($\mathbb{Z}_3$ is “the same as itself” in two different ways.) But you don’t have to bring that up the first time you attack that subject, any more than you have to bring up the fact that the category of sets does not have a set of objects on the first day you define categories.

    Category theory

    Category theory causes similar troubles. Beginning college math majors don’t usually meet it early. But category theory has begun to be used in other fields, so plenty of computer science students, people dealing with databases, and so on are suddenly trying to understand categories and failing to do so at the very start.

    The G&G post A new kind of introduction to category theory constitutes an alpha draft of the first part of an article introducing category theory following the ideas of this post.

    Objects and arrows are abstract

    • Every once in a while someone asks a question on Math StackExchange that shows they have no idea that an object of a category need not have elements and that morphisms need not be functions that take elements to elements.
    • One questioner understood that the claim that a morphism need not be a function meant that it might be a multivalued function.

    Duality

    • That misunderstanding comes up with duality. The definition of dual category requires turning the arrows around. Even if the original morphism takes elements to elements, the opposite morphism does not have to take elements to elements. In the case of the category of sets, an arrow in $\text{Set}^{op}$ cannot take elements to elements — for example, the opposite of the function $\emptyset\to\{1,2\}$.
    • The fact that there is a concrete category equivalent to $\text{Set}^{op}$ is a red herring. It involves different sets: the function corresponding to the function just mentioned goes from a four-element set to a singleton. But in the category $\text{Set}^{op}$ as defined it is simply an arrow, not a function.

    Not understanding how to use definitions

    • Some of the questioners on Math Stack Exchange ask how to prove a statement that is quite simple to prove directly from the definitions of the terms involved, but what they ask and what they are obviously trying to do is to gain an intuition in order to understand why the statement is true. This is backward — the first thing you should do is use the definition (at least in the first few days of a math class — after that you have to use theorems as well!
    • I have discussed this in the blog post Insights into mathematical definitions (which gives references to other longer discussions by math ed people). See also the abmath section Rewrite according to the definitions.

    How an introduction to a math topic needs to be written

    The following list shows some of the tactics I am thinking of using in the math topic introductions. It is quite likely that I will conclude that some tactics won’t work, and I am sure that tactics I haven’t mentioned here will be used.

    • The introductions should not go very far into the subject. Instead, they should bring an exhaustive and explicit discussion of how to get into the very earliest part of the topic, perhaps the definition, some examples, and a few simple theorems. I doubt that a group theory student who hasn’t mastered abstraction and what proofs are about will ever be ready to learn the Sylow theorems.
    • You can’t do examples and definitions simultaneously, but you can come close by going through an example step by step, checking each part of the definition.

      • There is a real split between students who want the definitions first
      (most of whom don’t have the abstraction problems I am trying to overcome)
      and those who really really think they need examples first (the majority)
      because they don’t understand abstraction.

    • When you introduce an axiom, give an example of how you would prove that some binary operation satisfies the axiom. For example, if the axiom is that every element of a group must have an inverse, right then and there prove that addition on the integers satisfies the axiom and disprove that multiplication on integers satisies it.
    • When the definition uses some undefined math objects, point out immediately with examples that you can’t have any intuition about them except what the axioms give you. (In contrast to definition of division of integers, where you and the student already have intuitions about the objects.)
    • Make explicit the possible problems with abstractmath.org and Gyre&Gimble) will indeed find it difficult to become mathematical researchers — but not impossible!
    • But that is not the point. All college math professors will get people who will go into theoretical computing science, and therefore need to understand category theory, or into particle physics, and need to understand groups, and so on.
    • By being clear at the earliest stages of how mathematicians actually do math, they will produce more people in other fields who actually have some grasp of what is going on with the topics they have studied in math classes, and hopefully will be willing to go back and learn some more math if some type of math rears its head in the theories of their field.
    • Besides, why do you want to alienate huge numbers of people from math, as our way of teaching in the past has done?
    • “Our” means grammar school teachers, high school teachers and college professors.

    Acknowledgment

    Thanks to Kevin Clift for corrections.

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    Representations of functions III

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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 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.

    Segments posted so far

    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

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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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    Mathematical Information I

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    Introduction

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

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

    Overview

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

    Some caveats

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

    Articles published in journals

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

    Refereed papers containing new results

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

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

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

    Survey articles and invited addresses

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

    Books

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

    Research books that are concise and without much explanation.

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

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

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

    Intended to introduce professional mathematicians to a particular field.

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

    Intended to introduce math graduate students to a particular field.

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

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

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

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

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

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

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

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

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

    Piper Harron’s Thesis

    The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: an Artist’s Rendering, Ph.D. thesis by Piper Harron.

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

    These blog posts have useful comments about her thesis:

    Types of explanations

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

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

    Images and metaphors

    In abstractmath.org

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

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

    The User’s GuideW

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

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

    Piper Harron’s thesis

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

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

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

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

    Informal summaries of a proofW

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

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

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

    Insights

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

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

    For example, a fundamental insight in group theory is:

    Study the linear representations of a group.

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

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

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

    Proof stories

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

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

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

    References

    The talks in Seattle

    • List of all the talks.

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

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

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    Names of mathematical objects

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    This is a revision of the abstractmath.org article on names.

    The name of a mathematical object is a word or phrase in math English used to identify an object. A name plays the same role that symbolic terms play in the symbolic language.

    Sources of names


    Suggestive English words

    A suggestive name is a a common English word or phrase, chosen to suggest its meaning. This means it is a type of metaphor.

    Examples


    In none of these examples is
    the metaphorical meaning
    exactly suitable to be
    the mathe­matical definition.

    • “Curve”, “point”, “line”, “slope“, “circle” and many other English words are used in elementary math with precise meanings that more or less fit their everyday meanings.
    • Connected subspace (of a topological space). When you draw a picture of a connected set it looks “connected”.
    • “Set” suggests a collection of things and provides a reasonable metaphor for its mathe­matical meaning. Both the abstractmath article on sets and the Wikipedia article on sets give you insight on why this metaphor cannot be entirely accurate.

      • Random English words

      Most English words used in math are not suggestive. They are either chosen at random or were intended to suggest something but misfired in some way.

      Groups

      A group is a collection of math objects with a binary operation defined on it subject to certain constraints. The binary operation is much more impor­tant than the underlying set! To many non-mathe­maticians, a “group” sounds like essentially what a mathe­matician calls a “set”.

      The concept of group was one of the earliest mathe­matical concepts des­cribed as a set-with-structure. I believe that a group was origi­nally referred to as a “group of trans­forma­tions”. May­be that phrase got shortened to “group” without anyone realizing what a disas­trous met­a­phor it caused.

      Fields

      A field in the algebraic sense is a structure which is not in any way suggested by the word “field”. The German word for field in this sense is “Körper”, which means “body”. That is about as bad as “group”, and I suspect it was motivated in much the same way. The name “Körper” may be due to Dedekind. I don’t know who to blame for “field”.

      A field in the sense of an assignment of a scalar or a vector to every point in a space is a completely separate notion than that of field as an algebra. The concept was invented in the nineteenth century by physicists, but any math student is likely to see fields in this sense in several different courses.

      Perhaps the second meaning of field was suggested by contour plowing.

      The word “field” is also discussed in the Glossary.

      Person’s name

      A concept may be named after a person.

      Examples

      • L’Hôpital’s Rule
      • Hausdorff space
      • Turing machine
      • Riemann surface
      • Riemannian manifold
      • Pythagorean Theorem

        • I have no idea why “Riemann” gets an ending when it is a manifold but not when it is a surface.

        Made-up name

        Some names are made up in a random way, not based on any oter language. Googol is an example.

        Named after notation

        Symbols

        A mathematical object may be named by the typographical symbol(s) used to denote it. This is used both formally and in on-the-fly references.  

        Some objects have standard names that are single letters (Greek or Roman), such as $e$, $i$ and $\pi$. There is much more about this in Alphabets.

        Be warned that any letter can be given another definition. $\pi$ is also used to name a projection, $i$ is commonly used as an index, and $e$ means energy in physics.

        Expressions

        • The multiplication in a Lie Algebra is called the “Lie bracket”. It is written “$[v,w]$”.
        • In quantum mechanics, a vector $\vec{w}$ may be notated “$|w\rangle$” and called a “ket”. Another vector $\vec{v}$ induces a linear operator on vectors that is denoted by “$\langle v|$”, which is called a “bra”. The action of $\langle v|$ on $|w\rangle$ is the inner product $\langle v|w\rangle$, which suggested the “bra” and “ket” terminology (from “bracket”). You can blame Paul Dirac for this stuff.
        • In 1985, Michael Barr and I published a book in category theory called Triples, Toposes and Theories. Immediately after that everyone in category theory started saying “monad” for what had been called “triple”. (The notation for a triple, er, monad, is of the form “$(T,\eta,\mu)$”.)

          • Synecdoche

          A synecdoche is a name of part of something that is used as a name for the whole thing.

          Examples

        The Tochar­ians appear to have called a cart by their word for wheel several thousand years ago. See the blog post by Don Ringe.

        Names from other languages

        In English, many technical names are borrowed from other languages. It may be difficult to determine what the meaning in the old language has to do with the mathematical meaning.

        Examples

      • Matrix. This is the Latin word for “uterus”. I suppose the analogy is with “container”.
      • Parabola. “Parabola” is a word borrowed from Greek in late Latin, meaning something like “comparison”. The parabola $y=x^2$ “compares” a number with its square: it curves upward because the area of a square grows faster than the length of its side. “Parable” is from the same word.
      • Algebra. This comes from an Arabic word meaning the art of setting joints, or more generally “restore”. It came through Spanish where it once meant “surgical procedure” but that meaning is now obsolete.

      Much of this information comes from The On-Line Etymological Dictionary. (Read its article about “sine”.) See also my articles on secant and tangent.

      I enjoy finding out about etymol­ogies, but I concede that knowing an ety­mol­ogy doesn’t help you very much in under­standing the math.

      Names made up from other languages’ roots

      A name may be a new word made out of (usually) Greek or Latin roots.

      Examples

      • Homomorphism. “Homo” in Greek is a root meaning “same” and “morphism” comes from a root referring to shape.
      • Quasiconformal. “Quasi” is a Latin word meaning something like “as if”. It is a prefix mathematicians use a bunch. It usually implies a weakening of the constraints that define the word it is attached to. A map is conformal if it preserves angles in a certain sense, and it is quasiconformal then it does not preserve angles but it does take circles into ellipses in a certain restricted sense (which conformal maps also do). So it replaces a constraint by a weaker constraint.


      Mathematical names cause problems for students

      The name may suggest the wrong meaning

      This is discusses in detail in the article cognitive dissonance.

      The name may not suggest any meaning

      English is unusual among major languages in the number of technical words borrowed from other languages instead of being made up from native roots.  We have some, listed under suggestive names.  But how can you tell from looking at them what “parabola” or “homomorphism” mean?   This applies to concepts named after people, too: The fact that “Hausdorff” is German for a village near an estate doesn’t tell me what a Hausdorff space is.

      The English word “carnivore” (from Latin roots) can be translated as “Fleischfresser” in German; to a German speaker, that word means literally “meat eater”.  So a question such as “What does a carnivore eat” translates into something like, “What does a meat-eater eat?” 

      Chinese is another language that forms words in that way: see the discussion of “diagonal” in Julia Lan Dai’s blog.  (I stole the carnivore example from her blog, too.)

      The result is that many technical words in English do not suggest their meaning at all to a reader not familiar with the subject.  Of course, in the case of “carnivore” if you know Latin, French or Spanish you are likely to guess the meaning, but it is nevertheless true that English has a kind of elitist stratum of technical words that provide little or no clue to their meaning and Chinese and German do not, at least not so much. This is a problem in all technical fields, not just in math.

      Pronunciation

      There are two main reasons math students have difficulties in pronouncing technical words in math.

      Most students have little knowledge of other languages

      Forty years ago nearly all Ph.D. students had to show mastery in reading math in two foreign languages; this included pronunciation, although that was not emphasized. Today the language requirements in the USA are much weaker, and younger educated Americans are generally weak in foreign languages. As a result, graduate students pronounce foreign names in a variety of ways, some of which attract ridicule from older mathematicians.

      Example: the graduate student at a blackboard who came to the last step of a long proof and announced, “Viola!”, much to the hilarity of his listeners.

      Pronunciation of words from other languages has become unpredictable

      In English-speaking countries until the early twentieth century, the practice was to pronounce a name from another language as if it were English, following the rules of English pronunciation.

      We still pronounce many common math words this way: “Euclid” is pronounced “you-clid” and “parabola” with the second syllable rhyming with “dab”.

      But other words (mostly derived from people’s names) are pronounced using the pronunciation of the language they came from, or what the speaker thinks is the foreign pronunciation. This particularly involves pronouncing “a” as “ah”, “e” like “ay”, and “i” like “ee”.

      Examples
      • Euler (oiler)
      • Fourier (foo-ree-ay)
      • Lagrange (second a pronounced “ah”)
      • Lie (lee)
      • Riemann (ree-monn)

      The older practice of pronunciation is explained by history: In 1100 AD, the rules of pronunciation of English, Ger­man and French, in particular, were remarkably similar. Over the centuries, the sound systems changed, and Eng­lish­men, for example, changed their pronunciation of “Lagrange” so that the second syllable rhymes with “range”, whereas the French changed it so that the second vowel is nasalized (and the “n” is not otherwise pronounced) and rhymes with the “a” in “father”.

      German spelling

      The German letters “ä”, “ö” and “ü” may also be spelled “ae”, “oe” and “ue” respectively. It is far better to spell “Möbius” as “Moebius” than to spell it “Mobius”.

      The German letter “ß” may be spelled “ss” and often is by the Swiss. Thus Karl Weierstrass spelled his last name “Weierstraß”. Students sometimes confuse the letter “ß” with “f” or “r”. In English language documents it is probably better to use “ss” than “ß”.

      Transliterations from Cyrillic

       The name of the Russian mathematician mot commonly spelled “Chebyshev” in English is also spelled Chebyshov, Chebishev, Chebysheff, Tschebischeff, Tschebyshev, Tschebyscheff and Tschebyschef. (Also Tschebyschew in papers written in German.) The only spelling in the list above that could be said to have some official sanction is “Chebyshev”, which is used by the Library of Congress.

      The correct spelling of his name is “Чебышев” since he was Russian and the Russian language uses the Cyrillic alphabet.

      In spite of the fact that most of the transliterations show the last vowel to be an “e”, the name in Russian is pronounced approximately “chebby-SHOFF”, accent on the last syllable.  Now, that is a ridiculous situation, and it is the transliterators who are ridiculous, not Russian spelling, which in spite of that peculiarity about the Cyrillic letter “e” is much more nearly phonetic than English spelling.

      Some other Russian names have variant spellings (Tychonov, Vinogradov) but Chebyshev probably wins the prize for the most.

    Plurals

    Many authors form the plural of certain technical words using endings from the language from which the words originated. Students may get these wrong, and may sometimes meet with ridicule for doing so.

    Plurals ending in a vowel

    Here are some of the common mathematical terms with vowel plurals.

    singular plural
    automaton automata
    polyhedron polyhedra
    focus foci
    locus loci
    radius radii
    formula formulae
    parabola parabolae
    • Linguists have noted that such plurals seem to be processed differently from s-plurals.  In particular, when used as adjectives, most nouns appear in the singular, but vowel-plural nouns appear in the plural: Compare “automata theory” with “group theory”.  No one says groups theory.  I used to say “automaton theory” but people looked at me funny.
    • The plurals that end in a (of Greek and Latin neuter nouns) are often not recognized as plurals and are therefore used as singulars.  That is how “data” became singular.  This does not seem to happen with my students with the -i plurals and the -ae plurals.
    • In the written literature, the -ae plural appears to be dying, but the -a and -i plurals are hanging on. The commonest -ae plural is “formulae”; other feminine Latin nouns such as “parabola” are usually used with the English plural. In the 1990-1995 issues of six American mathematics journals, I found 829 occurrences of “formulas” and 260 occurrences of “formulae”, in contrast with 17 occurrences of “parabolas” and and no occurrences of “parabolae”. (There were only three occurrences of “parabolae” after 1918.)  In contrast, there were 107 occurrences of “polyhedra” and only 14 of “polyhedrons”.

      • Plurals in s with modified roots

      singular

      plural

      matrix

      matrices

      simplex

      simplices

      vertex

      vertices

      Students recognize these as plurals but produce new singulars for the words as back formations. For example, one hears “matricee” and “verticee” as the singular for “matrix” and “vertex”. I have also heard “vertec”.

      Remarks

      It is not unfair to say that some scholars insist on using foreign plurals as a form of one-upmanship. Students and young professors need to be aware of these plurals in their own self interest.

      It appears to me that ridicule and put-down for using standard English plurals instead of foreign plurals, and for mispronouncing foreign names, is much less common than it was thirty years ago. However, I am assured by students that it still happens. 

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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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