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The most confusing notation in number theory

This is an observation in abstractmath that I think needs to be publicized more:

Two symbols used in the study of integers are notorious for their confusing similarity.

  • The expression “$m/n$” is a term denoting the number obtained by dividing $m$ by $n$. Thus “$12/3$” denotes $4$ and “$12/5$” denotes the number $2.4$.
  • The expression “$m|n$” is the assertion that “$m$ divides $n$ with no remainder”. So for example “$3|12$”, read “$3$ divides $12$” or “$12$ is a multiple of $3$”, is a true statement and “$5|12$” is a false statement.

Notice that $m/n$ is an integer if and only if $n|m$. Not only is $m/n$ a number and $n|m$ a statement, but the statement “the first one is an integer if and only if the second one is true” is correct only after the $m$ and $n$ are switched!

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Definition of function

Note: This is a revision of the article on specification and definition of functions from abstractmath.org. Many of the links in this article take you to other articles in abstractmath.org.

A function is a mathematical object.

To deal with functions as a math object, you need a precise definition of “function”. That is what this article gives you.

  • The article starts by giving a specification of “function”.
  • After that, we get into the technicalities of the
    definitions of the general concept of function.
  • Things get complicated because there are several inequivalent definitions of “function” in common use.

Specification of “function”

A function $f$ is a mathematical object which determines and is completely determined by the following data:


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

  • The value of $f$ at $a$ is most cohttp://www.abstractmath.org/MM/MMonly written $f(a)$, but see Functions: Notation and Terminology.
  • To evaluate $f$ at $a$ means to determine $f(a)$. The two examples of functions below show that different functions may have different strategies for evaluating them.
  • In the expression “$f(a)$”, $a$ is called the input or (old-fashioned) argument of $f$.
  • “FP” means functional property.
  • “VIC” means “value in codomain”.

Examples

I give two examples here. The examples of functions chapter contains many other examples.

A finite function

Let $F$ be the function defined on the set $\left\{\text{a},\text{b},\text{c},\text{d}\right\}$ as follows: \[F(\text{a})=\text{a},\,\,\,F(\text{b})=\text{c},\,\,\,F(\text{c})=\text{c},\,\,\,F(\text{d})=\text{b}\]In this definition, $\text{a},\text{b},\text{c},\text{d}$ are letters of the alphabet, not variables. This is the function called “Finite” in the chapter on examples of functions.

  • The definition of $F$ says “$F$ is defined on the set $\left\{\text{a},\,\text{b},\,\text{c},\,\text{d} \right\}$”. The phrase “is defined on”
    means that the domain is that set. That is standard terminology.
  • The value of $F$ at each element of the domain is given explicitly. The value at
    $\text{b}$, for example, is $\text{c}$, because the definition says that $F(\text{b}) = \text{c}$. No other reason needs to be given. Mathematical definitions can be arbitrary.
  • The codomain of $F$ is not specified, but must include the set $\{\text{a},\text{b},\text{c}\}$. The codomain of a function is often not specified when it is not important, which is most of the time in freshman calculus (for example).
  • The diagram below shows how $F$ obeys the rule that the value of an element $x$ in the domain is completely determined by $x$ and $F$.
  • If two arrows had started from the same element of the domain, then $F$ would not be a function. (It would be a multivalued function).
  • If there were an element of the domain that no arrow started from, it $F$ would not be a function. (It would be a partial function.)
  • In this example, to evaluate $F$ at $b$ (to determine the value of $F$ at $b$) means to look at the definition of $F$, which says among other things that the value is $c$ (or alternatively, look at the diagram above and see what letter the arrow starting at $b$ points to). In this case, “evaluation” does not imply calculating a formula.

A real-valued function

Let $G$ be the real-valued function defined by the formula $G(x)={{x}^{2}}+2x+5$.

  • The definition of $G$ gives the value at each element of the domain by a formula. The value at $3$, for example, is obtained by calculating \[G(3)=3^2+2\cdot3+5=20\]
  • The definition of $G$
    does not specify the domain. The convention in the case of functions defined on the real numbers by a formula is to take the domain to be all real numbers at which the formula is defined. In this case, that is every real number, so the domain is $\mathbb{R}$.
  • The definition of $G$ does not specify the codomain, either. However, the codomain must include all real numbers greater than or equal to $4$. (Why?)
  • So if an author wrote, “Let $H(x)=\frac{1}{x}$”, the domain would be the set of all real numbers except $0$. But a careful author would write, “Let $H(x)=\frac{1}{x}$ ($x\neq0$).”

What the specification means

  • The specification guarantees that a function satisfies all five of the properties listed.
  • The specification does not define a mathematical structure in the way mathematical structures have been defined in the past: In particular, it does not require a function to be one or more sets with structure.
  • Even so, it is useful to have the specification, because:

    Many mathematical definitions
    introduce extraneous technical elements
    which clutter up your thinking
    about the object they define.

History

The discussion below is an over­simpli­fication of the history of mathe­matics, which many people have written thick books about. A book relevant to these ideas is Plato’s Ghost, by Jeremy Gray.

Until late in the nineteenth century, functions were usually thought of as defined by formulas (including infinite series). Problems arose in the theory of harmonic analysis which made mathematicians require a more general notion of function. They came up with the concept of function as a set of ordered pairs with the functional property (discussed below), and that understanding revolutionized our understanding of math.

In particular, this definition, along with the use of set theory, enabled abstract math (ahem) to become a cohttp://www.abstractmath.org/MM/MMon tool for understanding math and proving theorems. It is conceivable that some readers may wish it hadn’t. Well, tough.

The modern definition of function given here (which builds on the ordered pairs with functional property definition) came into use beginning in the 1950’s. The modern definition became necessary in algebraic topology and is widely used in many fields today.

The concept of function as a formula never disappeared entirely, but was studied mostly by logicians who generalized it to the study of function-as-algorithm. Of course, the study of algorithms is one of the central topics of modern computing science, so the notion of function-as-formula (updated to function-as-algorithm) has achieved a new importance in recent years.

To state both the definition, we need a preliminary idea.


The functional property

A set $P$ of ordered pairs has the functional property if two pairs in $P$ with the same first coordinate have to have the same second coordinate (which means they are the same pair). In other words, if $(x,a)$ and $(x,b)$ are both in $P$, then $a=b$.

How to think about the functional property

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

Examples

  • The set $\{(1,2), (2,4), (3,2), (5,8)\}$ has the functional property, since no two different pairs have the same first coordinate. Note that there are two different pairs with the same second coordinate. This is irrelevant to the functional property.
  • The set $\{(1,2), (2,4), (3,2), (2,8)\}$ does not have the functional property. There are two different pairs with first coordinate 2.
  • The empty set $\emptyset$ has the function property vacuously.

Example: graph of a function defined by a formula


In calculus books, a picture like this one (of part of $y=x^2+2x+5$) is called a graph. Here I use the word “graph” to denote the set of ordered pairs
\[\left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in \mathbb{R } \right\}\]
which is a mathematical object rather than some ink on a page or pixels on a screen.

The graph of any function studied in beginning calculus has the functional property. For example, the set of ordered pairs above has the functional property because if $x$ is any real number, the formula ${{x}^{2}}+2x+5$ defines a specific real number.

  • if $x = 0$, then ${{x}^{2}}+2x+5=5$, so the pair $(0, 5)$ is an element of the graph of $G$. Each time you plug in $0$ in the formula you get 5.
  • if $x = 1$, then ${{x}^{2}}+2x+5=8$.
  • if $x = -2$, then ${{x}^{2}}+2x+5=5$.

You can measure where the point $\{-2,5\}$ is on the (picture of) the graph and see that it is on the blue curve as it should be. No other pair whose first coordinate is $-2$ is in the graph of $G$, only $(-2, 5)$. That is because when you plug $-2$ into the formula ${{x}^{2}}+2x+5$, you get $5$ and nothing else. Of course, $(0, 5)$ is in the graph, but that does not contradict the functional property. $(0, 5)$ and $(-2, 5)$ have the same second coordinate, but that is OK.



Mathematical definition of function

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

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

Using arrow notation, this implies that $f:\text{dom}f\to\text{cod} f$.

Remark

The main difference between the specification of function given previously and this definition is that the definition replaces the statement “$f$ has a value at $a$” by introducing a set of ordered pairs (the graph) with the functional property.

  • This set of ordered pairs is extra structure introduced by the definition mainly in order to make the definition a classical sets-with-structure.
  • This makes the graph, which should be a concept derived from the concept of function, appear to be a necessary part of the function.
  • That suggests incorrectly that the graph is more of a primary intuition that other intuitions such as function as map, function as transformer, and other points of view discussed in the article Images and meta­phors for functions.
  • The concept of graph of a function is indeed an important intuition, and is discussed with examples in the articles Graphs of continuous functions and Graphs of finite functions.
  • Nevertheless, the fact that the concept of graph appears in the definition of function does not make it the most important intuition.

Examples

  • Let $F$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ and define $A = \{1, 2, 3, 5\}$ and $B = \{2, 4, 8\}$. Then $F:A\to B$ is a function. In speaking, we would usually say, “$F$ is a function from $A$ to $B$.”
  • Let $G$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$ (same as above), and define $A = \{1, 2, 3, 5\}$ and $C = \{2, 4, 8, 9, 11, \pi, 3/2\}$. Then $G:A\to C$ is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in the codomain $C$, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
  • Let $H$ have graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Then $H:A\to \mathbb{R}$ is a function, since $2$, $4$ and $8$ are all real numbers.
  • Let $D = \{1, 2, 5\}$ and $E = \{1, 2, 3, 4, 5\}$. Then there is no function $D\to A$ and no function $E\to A$ with graph $\{(1,2), (2,4), (3,2), (5,8)\}$. Neither $D$ nor $E$ has exactly the same elements as the first coordinates of the graph.

Identity and inclusion

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

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

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

Other definitions of function

Original abstract definition of function

Definition

  • A function $f$ is a set of ordered pairs with the functional property.
  • If $f$ is a function according to this definition, the domain of $f$ is the set of first coordinates of all the pairs in $f$.
  • If $x\in \text{dom} f$, then we define the value of $f$ at $x$, denoted by $f(x)$, to be the second coordinate of the only ordered pair in $f$ whose first coordinate is $x$.

Remarks

  • This definition is still widely used in mathematical writing.
  • Many authors do not tell you which definition they are using.
  • For many purposes (including freshman calculus for the most part) it does not matter which definition is used.
  • In some branches of math, the modern definition adds great clarity to many complicated situations; using the older definition can even make it difficult to describe some important constructions. There is more about this in New Approaches below.

Possible confusion

Some confusion can result because of the presence of these two different definitions.

  • For example, since the identity function ${{\operatorname{id}}_{A}}:A\to A$ and the inclusion function ${{i}_{A,\,B}}:A\to B$ have the same graph, users of the older definition are required in theory to say they are the same function.
  • Also it requires you to say that the graph of a function is the same thing as the function.
  • In my observation, this does not make a problem in practice, unless there is a very picky person in the room.
  • It also appears to me that the modern definition is (quite rightly) winning and the original abstract definition is disappearing.

Multivalued function

The phrase multivalued function refers to an object that is like a function $f:S\to T$ except that for $s\in S$, $f(s)$ may denote more than one value.

Examples

  • Multivalued functions arose in considering complex functions. In cohttp://www.abstractmath.org/MM/MMon practice, the symbol $\sqrt{4}$ denoted $2$, although $-2$ is also a square root of $4$. But in complex function theory, the square root function takes on both the values $2$ and $-2$. This is discussed in detail in Wikipedia.
  • The antiderivative is an example of a multivalued operator. For any constant $C$, $\frac{x^3}{3}+C$ is an antiderivative of $x^2$, so that $\frac{x^3}{3}$, $\frac{x^3}{3}+42$, $\frac{x^3}{3}-1$ and $\frac{x^3}{3}+2\pi$ are among the infinitely many antiderivatives of $x^2$.

A multivalued function $f:S\to T$ can be modeled as a function with domain $S$ and codomain the set of all subsets of $T$. The two meanings are equivalent in a strong sense (naturally equivalent). Even so, it seems to me that they represent two differ­ent ways of thinking about
multivalued functions. (“The value may be any of these things…” as opposed to “The value is this whole set of things.”)

Some older mathematical papers in com­plex func­tion theory do not tell you that their functions are multi­valued. There was a time when com­plex func­tion theory was such a Big Deal in research mathe­matics that the phrase “func­tion theory” meant complex func­tion theory and every mathe­ma­tician with a Ph. D. knew that complex functions were multi­valued.

Partial function

A partial function $f:S\to T$ is just like a function except that its input may be defined on only a subset of $S$. For example, the function $f(x):=\frac{1}{x}$ is a partial function from the real numbers to the real numbers.

This models the behavior of computer programs (algorithms): if you consider a program with one input and one output as a function, it may not be defined on some inputs because for them it runs forever (or gives an error message).

In some texts in computing science and mathematical logic, a function is by
convention a partial function, and this fact may not be mentioned explicitly, especially in research papers.

The phrases “multivalued function” and “partial function” upset some picky types who say things like, “But a multi­valued func­tion is not a func­tion!”. A hot dog is not a dog, either. I once had a Russian teacher who was Polish and a German teacher who was Hungarian. So what? See the Hand­book (click on
radial category).

New approaches to functions

All the definitions of function given here produce mathematical structures, using the traditional way to define mathematical objects in terms of sets. Such definitions have disadvantages.

Mathematicians have many ways to think about functions. That a function is a set of ordered pairs with a certain property (functional) and possibly some ancillary ideas (domain, codomain, and others) is not the way we usually think about them$\ldots$Except when we need to reduce the thing we are studying to its absolutely most abstract form to make sure our proofs are correct.
That most abstract form is what I have called the rigorous view or the dry bones and it is when that reasoning is needed that the sets-with-structure approach has succeeded.

Our practice of abstraction has led us to new approaches to talking about functions. The most important one currently is category theory. Roughly, a category is a bunch of objects together with some arrows going between them that can be composed head to tail. Functions between sets are examples of this: the sets are the objects and the functions the arrows. But arrows in a category do not have to be functions; in that way category theory is an abstraction of functions.

This abstracts the idea of function in a way that brings out common ideas in various branches of math. Research papers in many branches of mathematics now routinely use the language of category theory. Categories now appear in some undergraduate math courses, meaning that Someone needs to write a chapter on category theory for abstractmath.org.

Besides category theory, computing scientists have come up with other abstract ways of dealing with functions, for example type theory. It has not come as far along as category theory, but has shown recent signs of major progress.

Both category theory and type theory define math objects in terms of their effect on and relationship with other math objects. This makes it possible to do abstract math entirely without using sets-with-structure as a means of defining concepts.

References

  • Functions in Wikipedia. This is an extensive and mostly well-done description of the use of functions in mathematics.

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Variable mathematical objects


VARIABLE MATHEMATICAL OBJECTS

In many mathematical texts, the variable $x$ may denote a real number, although which real number may not be specified. This is an example of a variable mathematical object. This point of view and terminology is not widespread, but I think it is worth understanding because it provides a deeper understanding of some aspects about how math is done.

Specific and variable mathematical objects


It is useful to distinguish between specific math objects and variable math objects.

Examples of specific math objects

  • The number $42$ (the math object represented as “42” in base $10$, “2A” in hexadecimal and “XLII” as a Roman numeral) is a specific math object. It is an abstract math object. It is not any of the representations just listed — they are just strings of letters and numbers.
  • The ordered pair $(4,3)$ is a specific math object. It is not the same as the ordered pair $(7,-2)$, which is another specific math object.
  • The sine function $\sin:\mathbb{R}\to\mathbb{R}$ is a specific math object. You may know that the sine function is also defined for all complex numbers, which gives another specific math object $\sin:\mathbb{C}\to\mathbb{C}$.
  • The group of symmetries of a square is a specific math object. (If you don’t know much about groups, the link gives a detailed description of this particular group.)

Variable math objects

Math books are full of references to math objects, typically named by a letter or a name, that are not completely specified. Some mathematicians call these variable objects (not standard terminology). The idea of a variable mathe­mati­cal object is not often taught as such in under­graduate classes but it is worth pondering. It has certainly clari­fied my thinking about expres­sions with variables.

Examples

  • If an author or lecturer says “Let $x$ be a real variable”, you can then think of $x$ as a variable real number. In a proof you can’t assume that $x$ is any particular real number such as $42$ or $\pi$.
  • If a lecturer says, “Let $(a,b)$ be an ordered pair of integers”, then all you know is that $a$ and $b$ are integers. This makes $(a,b)$ a variable ordered pair, specifically a pair of integers. The lecturer will not say it is a variable ordered pair since that terminology is not widely used. You have to understand that the phrase “Let $(a,b)$ be an ordered pair of integers” implies that it is a variable ordered pair just because “a” and “b” are letters instead of numbers.
  • If you are going to prove a theorem about functions, you might begin, "Let $f$ be a continuous function", and in the proof refer to $f$ and various objects connected to $f$. This makes $f$ a variable mathematical object. When you are proving things about $f$ you may use the fact that it is continuous. But you cannot assume that it is (for example) the sine function or any other particular function.
  • If someone says, “Let $G$ be a group” you can think of $G$ as a variable group. If you want to prove something about $G$ you are free to use the definition of “group” and any theorems you know of that apply to all groups, but you can’t assume that $G$ is any specific group.

Terminology

A logician would refer to the symbol $f$, thought of as denoting a function, as a vari­able, and likewise the symbol $G$, thought of as denoting a group. But mathe­maticians in general would not use the word “vari­able” in those situa­tions.

How to think about variable objects

The idea that $x$ is a variable object means thinking of $x$ as a genuine mathematical object, but with limitations about what you can say or think about it. Specifically,

Some assertions about a variable math object
may be neither true nor false.

Example

The statement, “Let $x$ be a real number” means that $x$ is to be regarded as a variable real number (usually called a “real variable”). Then you know the following facts:

  • The statement “${{x}^{2}}$ is not negative” is true.
  • The assertion “$x=x+1$” is false.
  • The assertion “$x\gt 0$” is neither true nor false.
Example

Suppose you are told that $x$ is a real number and that ${{x}^{2}}-5x=-6$.

  • You know (because it is given) that the statement “${{x}^{2}}-5x=-6$” is true.
  • By doing some algebra, you can discover that the statement “$x=2$ or $x=3$” is true.
  • The statement “$x=2$ and $x=3$” is false, because $2\neq3$.
  • The statement “$x=2$” is neither true nor false, and similarly for “$x=3$”.
  • This situation could be described this way: $x$ is a variable real number varying over the set $\{2,3\}$.
Example

This example may not be easy to understand. It is intended to raise your consciousness.

A prime pair is an ordered pair of integers $(n,n+2)$ with the property that both $n$ and $n+2$ are prime numbers.

Definition: $S$ is a PP set if $S$ is a set of pairs of integers with the property that every pair is a prime pair.

  • “$\{(3,5),(11,13)\}$ is a PP set” is true.
  • “$\{(5,7),(111,113),(149,151)\}$ is a PP set” is false.

Now suppose $SS$ is a variable PP set.

  • “$SS$ is a set” is true by definition.
  • “$SS$ contains $(7,9)$” is false.
  • “$SS$ contains $(3,5)$” is neither true nor false, as the examples just above show.
  • “$SS$ is an infinite set”:
    • This is certainly not true (see finite examples above).
    • This claim may be neither true nor false, or it may be plain false, because no one knows whether there is an infinite number of prime pairs.
    • The point of this example is to show that “we don’t know” doesn’t mean the same thing as “neither true nor false”.

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Power

I have rewritten the entry to “power” in the abstractmath.org Glossary:

POWER

Here are three variant phrases that say that $125=5^3$:

  • “$125$ is a power of $5$ with exponent $3$”.
  • “$125$ is the third power of $5$”.
  • “$125$ is $5$ to the third power”.

Some students are confused by such statements, and conclude that $3$ is the “power”. This usage appears in print in Wikipedia in its entry on Exponentiation (as it was on 22 November 2016):

“…$b^n$ is the product of multiplying $n$ bases:

\[b^n = \underbrace{b \times \cdots \times b}_n\]

In that case, $b^n$ is called the $n$-th power of $b$, or $b$ raised to the power $n$.”

As a result, students (and many mathematicians) refer to $n$ as the “power” in any expression of the form “$a^n$”. The number $n$ should be called the “exponent”. The word “power” should refer only to the result $a^n$. I know mathematical terminology is pretty chaotic, but it is silly to refer both to $n$ and to $a^n$ as the “power”.

Almost as silly as using $(a,b)$ to refer to an open interval, an ordered pair and the GCD. (See The notation $(a,b)$.)

Suggestion for lexicographical research: How widespread does referring to $n$ as the “power” come up in math textbooks or papers? (See usage.)

Thanks to Tomaz Cedilnik for comments on the first version of this entry.

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A slow introduction to category theory

Category theory turns math inside-out. Definitions depend on nothing inside, but on everything outside. — John Cook

About this post

This is a draft of the first part of an article on category theory that will be posted on abstractmath.org. It replaces an earlier version that was posted in June, 2016.

During the last year or so, I have been monitoring the category theory questions on Math Stack Exchange. Some of the queries are clearly from people who do not have enough of a mathematical background to understand basic abstract reasoning, for example the importance of definitions and the difficulties described in the abmath artice on Dysfunctional attitudes and behaviors. Category theory has become important in several fields outside mathematics, for example computer science and database theory.

This article is intended to get people started in category theory by giving a very detailed definition of “category” and some examples described in detail with an emphasis on how the example fits the definition of category. That’s all the present version does, but I intend to add some examples of constructions and properties such as the dual category, product, and other concepts that some of the inquirers on Math Stack Exchange had great difficulty with.

There is no way in which this article is a proper introduction to category theory. It is intended only to give beginners some help over the initial steps of understanding the subject, particularly the aspects of understanding that cause many hopeful math majors to fall off the Abstraction Cliff.

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 introduce several example categories with a detailed discussion of each one, explaining how they fit the definition of category.


Axiom 1: Data

A category consists of two types of data: objects and arrows.

Notes for Axiom 1

  • You will see in the section on Examples of categories that every definition of a category $\mathsf{C}$ starts by specifying what the objects of $\mathsf{C}$ are and what the arrows of $\mathsf{C}$ are. That is what Axiom I requires.
  • 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

Each arrow of a category has a domain and a codomain, each of which is an object of the category.

Notes for Axiom 2

  • The domain and the codomain of an arrow may or may not be the same object.
  • Each arrow has only one domain and only one codomain.
  • 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:
  • Warning: 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 an arrow $f:A\to B$, 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

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

  • 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 $h=g\circ f$.

  • It is useful to think of $f$ followed by $g$ as a path in the diagram. Then a metaphor for composition is: Every path of length 2 has exactly one composite.
  • It is customary in some texts in category theory to indicate that a diagram commutes by putting a gyre in the middle:
  • Note that the composite of the path that I described as “$f$ followed by $g$” is written as “$g\circ f$” or “$gf$”, which seems backward. Nevertheless, the most common notation in category theory for the composite of “$f$ followed by $g$” is $gf$. Some authors in computer science write “$f;g$” for “$gf$” to get around this problem.
  • 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

Note: WordpPress does not recognize the html command

    . Axiom 1 should be 4a, Axiom 2 4b Axiom 3 4c and Axiom 4 4d.

  1. For each object $A$ of a category, there is an arrow denoted by $\mathsf{id}_A$.
  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 $A$,\[g\circ \textsf{id}_A=g\]
  • There may be many arrows with domain and codomain both equal to $A$ (for example in the category $\mathsf{Set}$), but only one of them is $\textsf{id}_A$. It can be proved that $\textsf{id}_A$ is the unique arrow satisfying both (c) and (d) of the axiom.

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 $D$ 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 algebraic notation 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 these examples, I 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 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. (Some of 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: Data

  • 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.
  • Our other definitions of categories involve math objects you actually know something about.

Axiom 2: Domain and Codomain

  • 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: Composition

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: Identity arrows

Axiom 5: Associativity

  • 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: IntegerDiv

  • This example uses familiar mathematical objects — positive integers.
  • The arrows are not functions that can be applied to elements, since integers do not have elements.

Axiom 1: Data

  • The objects of IntegerDiv are all the positive integers.
  • Suppose $m$ and $n$ are positive integers:
  • If $m$ divides $n$, there is exactly one arrow from $m$ to $n$. I will call this arrow $\textsf{mdn}$. (This is my notation. There is no standard notation for this category.) For example there is one arrow from $2$ to $6$, denoted by $\textsf{2d6}$.
  • If $m$ does not divide $n$, there is no arrow from $m$ to $n$.

Axiom 2: Domain and codomain

The arrow denoted by $\textsf{mdn}$ has domain $m$ and codomain $n$.

Example

Example

Example

which may also be shown as

Axiom 3: Composition

The composite of

must be $\textsf{rdt}$, since that is the only arrow with domain $r$ and codomain $t$.

This fact can also be written this way: \[\mathsf{sdt}\circ\textsf{rds}=\textsf{rdt}\]

Axiom 4: Identity arrows

The composites

and

must commute since the arrows shown are the only possible arrows with the domains and codomains shown. In other words, $\textsf{id}_\textsf{r}=\textsf{rdr}$ and $\textsf{id}_\textsf{s}=\textsf{sds}$.

Axiom 5: Associativity

In the diagram below,

there is only one arrow from one integer to another, so $\textsf{k}$ must be both \[\textsf{tdu}\circ(\textsf{sdt}\circ\textsf{rds})\] and \[(\textsf{tdu}\circ\textsf{sdt})\circ\textsf{rds}\] as required.

Example 3: The category of Sets

In this section, I define the category $\mathsf{Set}$ (that is standard terminology in category theory.) This example will be very different from $\mathsf{MyFin}$, because it involves known mathematical objects — sets and functions.

Axiom 1: Data

  • Every set is an object of $\mathsf{Set}$ and nothing else is.
  • Every function between sets is an arrow of $\mathsf{Set}$ and nothing else is an arrow of $\mathsf{Set}$.

Axiom 2: Domain and codomain

For a given function $f$, $\text{dom}(f)$ is the domain of the function $f$ in the usual sense, and $\text{cod}(f)$ is the codomain of $f$ in the usual sense. (See Functions: specification and definition for more about domain and codomain.)

Examples

  • Let $f:\mathbb{R}\to\mathbb{R}$ be the function defined by $f(x):=x^2$. Then the arrow $f$ in $\mathsf{Set}$ satisfies $\text{dom}(f)= \mathbb{R}$ and also $\text{cod}(f)=\mathbb{R}$.
  • Let $j:\{1,2,3\}\to\{1,2,3,4\}$ be defined by $j(1):=1$, $j(2):=4$ and $j(3):=3$. Then $\text{dom}(j)=\{1,2,3\}$ and $\text{cod}(j)=\{1,2,3,4\}$.

Axiom 3: Composition

The composite of $f:A\to B$ and $g:B\to C$ is the function $g\circ f:A\to C$ defined by \[\text{(DC)}\,\,\,\,\,\,\,\,\,\,(g\circ f)(a):=g(f(a))\]

Note

Many other categories have a similar definition of composition, including categories whose objects are math structures with underlying sets and whose arrows are structure-preserving functions between the underlying sets. But be warned: There are many useful categories whose arrows do not evaluate at an element of an object because the objects don’t have elements. In that case, (DC) is meaningless. This is true of $\mathsf{MyFin}$ and $\mathsf{IntegerDiv}$.

Axiom 4: Identity arrows

For a set $A$, the identity arrow $\textsf{id}_A:A\to A$ is, as you might expect, the identity function defined by $\textsf{id}_A(a)=a$ for every $a\in A$. We must prove that these diagrams commute:

The calculations below show that they commute. They use the definition of composite given by (DC).

  • For any $b\in B$, \[(\textsf{id}_A\circ f)(b)=\textsf{id}_A(f(b))=f(b)\]
  • For any $a\in A$, \[(g\circ \textsf{id}_A)(a)=g(\textsf{id}_A(a))=g(a)\]

Note: In $\mathsf{Set}$, there are generally many arrows from a particular set $S$ to itself (for example there are $4$ from $\{1,2\}$ to itself), but only one is the identity arrow.

Axiom 5: Associativity

Composition of arrows in $\mathsf{Set}$ is associative because function composition is associative. Suppose we have functions as in this diagram:

We must show that the two triangles containing $k$ in this diagram commute:

In algebraic notation, this requires showing that for every element $a\in A$,\[(h\circ(g\circ f))(a))=((h\circ g)\circ f)(a)\]

The calculation below does that. It makes repeated use of Definition (DC) of composition. For any $a\in A$,\[\begin{equation}
\begin{split}
\big(h\circ (g\circ f)\big)(a)
& = h\big((g\circ f)(a)\big) \\
& = h\big(g(f(a))\big) \\
& = (h\circ g)(f(a)) \\
& = \big((h\circ g)\circ f\big)(a)
\end{split}
\end{equation}\]

Example 4: The category of Monoids


  • This definition makes repeated use of the fact that a homomorphism of monoids is a set function. Specifically, if $(S,\Delta)$ and $(T,\nabla)$ are monoids with identities $e_S$ and $e_{T}$, a homomorphism $h:S\to T$ must be a set function that satisfies the following two axioms: \[\text{(ME)}\,\,\,\,\,\,\,\,h(e_S)=e_T\] and for all elements $s, s’$ of $S$, \[\text{(MM)}\,\,\,\,\,\,\,\,h(s\Delta s’)=h(s)\nabla h(s’)\]
  • The category of monoids may be denoted by $\mathsf{Mon}$.

Axiom 1: Data

  • Every monoid is an object of the category of monoids, and nothing else is.
  • If $f$ is a homomorphism of monoids, then $f$ is an arrow of the category of monoids, and nothing else is.

Axiom 2: Domain and codomain

If $(S,\Delta)$ and $(T,\nabla)$ are monoids and $f:(S,\Delta)\to(T,\nabla)$ is a homomorphism of monoids, then the domain of $f$ is $(S,\Delta)$ and the codomain of $f$ is $(T,\nabla)$.

Notes

  • Since $f$ takes elements of the set $S$ to elements of the set $T$, it is also an arrow in the category $\mathsf{Set}$. In general, very few functions from $S$ to $T$ will be monoid homomorphisms from $(S,\Delta)$ to $(T,\nabla)$.
  • Many authors do not distinguish between $f$ regarded as an arrow of $\mathsf{Mon}$ and $f$ regarded as an arrow of $\mathsf{Set}$. Others may write $U(f)$ for the arrow in $\mathsf{Set}$. “$U$” stands for “underlying functor“, also called “forgetful functor”.

Axiom 3: Composition

The composite of

is the composite $g\circ f$ as set functions:

It is necessary to check that $g\circ f$ is a monoid homomorphism. The following calculation shows that it preserves the monoid operation; it makes repeated use of equations (DC) and (MM).

The calculation: For elements $r$ and $r’$ of $R$,\[\begin{align*}
(g\circ f)(r\,{\scriptstyle \square}\, r’)
&=g\left(f(r\, {\scriptstyle \square}\, r’)\right)\,\,\,\,\,\text{(DC)}\\ &=g\left(f(r) {\scriptstyle\, \Delta}\, f(r’)\right)\,\,\,\,\,\text{(MM)}\\
&=g(f(r)){\scriptstyle \,\nabla}\, g(f(r’))\,\,\,\,\text{(MM)}\\
&=(g\circ f)(r){\scriptstyle \,\nabla}\,(g\circ f)(r’)\,\,\,\,\,\text{(DC)}
\end{align*}\]

The fact that $g\circ f$ preserves the identity of the monoid is shown in the next section.

Axiom 4: Identity arrows

For a monoid $(S,\Delta)$, the identity function $\text{id}_S:S\to S$ preserves the monoid operation $\Delta$, because $\text{id}_S(s\Delta s’)=s\Delta s’$ by definition of the identity function, and that is $\text{id}_S(s)\Delta \text{id}_S(s’)$ for the same reason.

The required diagrams below must commute because the set functions commute and, by Axiom 3, the set composition of a monoid homomorphism is a monoid homomorphism.

We also need to show that $g\circ f$ as in

preserves identities. This calculation proves that it does; it uses (DC) and (ME)

\[\begin{align*}
(g\circ f)(\text{id}_R)
&=g(f((\text{id}_R))\\
&=g(\text{id}_S)\\
&=\text{id}_T
\end{align*}\]

Axiom 5: Associativity

The diagram

in the category $\mathsf{Set}$ commutes, so the diagram

must also commute.

References

All these references are available on line.

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Introducing abstract topics

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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Insights into mathematical definitions

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

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

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

I hope math ed people will read the final remarks.

Peculiarities of math definitions

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

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

List of properties

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

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

Imposed by decree

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

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

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

Student difficulties

Giving the math definition low priority

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

Arbitrary bothers them

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

What do you DO with a definition?

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

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

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

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

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

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

What the operability of definitions and theorems means is:

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

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

Final remarks

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

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

References


  • Bills, L., & Tall, D. (1998). Operable definitions in advanced mathematics: The case of the least upper bound. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 104-111). Stellenbosch, South Africa: University of Stellenbosch.
  • B. S. Edwards, and M. B. Ward, Surprises from mathematics education research: Student (mis) use of mathematical definitions (2004). American Mathematical Monthly, 111, 411-424.
  • G. Lakoff, Women, Fire and Dangerous
    Things
    . University of Chicago Press, 1990. See his discussion of concepts and prototypes.
  • J. Selden and A. Selden, Proof Construction Perspectives: Structure, Sequences of Actions, and Local Memory, Extended Abstract for KHDM Conference, Hanover, Germany, December 1-4, 2015. This paper may be downloaded from Academia.edu.
  • A Handbook of mathematical discourse, by Charles Wells. See concept, definition, and prototype.
  • Definitions, article in abstractmath.org. (Some of the ideas in this post have now been included in this article, but it is due for another revision.)
  • Definitions in logic and mathematics in Wikipedia.
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    This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.

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    Very early difficulties II

    Very early difficulties II

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

    Math StackExchange

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

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

    There are two caveats:

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

    Connotations of English words

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

    Infinite cardinals

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

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

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

    Morphism in category theory

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

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

    Example

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

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

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

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

    A MORPHISM DOESN’T HAVE TO BE A FUNCTION!

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

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

    So we need a more elaborate poster in the classroom:

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

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

    English words implementing logic

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

    A definition is a totalitarian dictator.

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

    Formula and term

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

    Exclusive or

    Question 804250 in MathSE says:

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

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

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

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

    Only if

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

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

    Consider these two pairs of examples.

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

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

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

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

    References

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    Very early difficulties in studying abstract math

    Introduction

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

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

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

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

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

    Rewrite according to the definition

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

    Example

    Definition

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

    Definition

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

    Theorem

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

    Proof

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

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

    “This proof is trivial”

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

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

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

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

    “This proof does not describe how mathematicians think”

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

    More examples

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

    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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    Problems caused for students by the two languages of math

    The two languages of math

    Mathematics is communicated using two languages: Mathematical English and the symbolic language of math (more about them in two languages).

    This post is a collection of examples of the sorts of trouble that the two languages cause beginning abstract math students. I have gathered many of them here since they are scattered throughout the literature. I would welcome suggestions for other references to problems caused by the languages of math.

    In many of the examples, I give links to the literature and leave you to fish out the details there. Almost all of the links are to documents on the internet.

    There is an extensive list of references.

    Conjectures

    Scattered through this post are conjectures. Like most of my writing about difficulties students have with math language, these conjectures are based on personal observation over 37 years of teaching mostly computer engineering and math majors. The only hard research of any sort I have done in math ed consists of the 426 citations of written mathematical writing included in the Handbook of Mathematical Discourse.

    Disclaimer

    This post is an attempt to gather together the ways in which math language causes trouble for students. It is even more preliminary and rough than most of my other posts.

    • The arrangement of the topics is unsatisfactory. Indeed, the topics are so interrelated that it is probably impossible to give a satisfactory linear order to them. That is where writing on line helps: Lots of forward and backward references.
    • Other people and I have written extensively about some of the topics, and they have lots of links. Other topics are stubs and need to be filled out. I have probably missed important points about and references to many of them.
    • Please note that many of the most important difficulties that students have with understanding mathematical ideas are not caused by the languages of math and are not represented here.

    I expect to revise this article periodically as I find more references and examples and understand some of the topics better. Suggestions would be very welcome.

    Intricate symbolic expressions

    I have occasionally had students tell me that have great difficulty understanding a complicated symbolic expression. They can’t just look at it and learn something about what it means.

    Example

    Consider the symbolic expression \[\displaystyle\left(\frac{x^3-10}{3 e^{-x}+1}\right)^6\]

    Now, I could read this expression aloud as if it were text, or more precisely describe it so that someone else could write it down. But if I am in math mode and see this expression I don’t “read” it, even to myself.

    I am one of those people who much of the time think in pictures or abstractions without words. (See references here.)

    In this case I would look at the expression as a structured picture. I could determine a number of things about it, and when I was explaining it I would point at the board, not try to pronounce it or part of it:

    • The denominator is always positive so the expression is defined for all reals.
    • The exponent is even so the value of the expression is always nonnegative. I would say, “This (pointing at the exponent) is an even power so the expression is never negative.”
    • It is zero in exactly one place, namely $x=\sqrt[3]{10}$.
    • Its derivative is also $0$ at $\sqrt[3]{10}$. You can see this without calculating the formula for the derivative (ugh).

    There is much more about this example in Zooming and Chunking.

    Algebra in high school

    There are many high school students stymied by algebra, never do well at it, and hate math as a result. I have known many such people over the years. A revealing remark that I have heard many times is that “algebra is totally meaningless to me”. This is sometimes accompanied by a remark that geometry is “obvious” or something similar. This may be because they think they have to “read” an algebraic expression instead of studying it as they would a graph or a diagram.

    Conjecture

    Many beginning abstractmath students have difficulty understanding a symbolic expression like the one above. Could this be cause by resistance to treating the expression as a structure to be studied?

    Context-sensitive pronunciation

    A symbolic assertion (“formula” to logicians) can be embedded in a math English sentence in different ways, requiring the symbolic assertion to be pronounced in different ways. The assertion itself is not modified in any way in these different situations.

    I used the phrase “symbolic assertion” in abstractmath.org because students are confused by the logicians’ use of “formula“.
    In everyday English, “$\text{H}_2\text{O}$” is the “formula” for water, but it is a term, not an assertion.

    Example

    “For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$.”

    • In the sentence above, the assertion “$x\gt0$” must be pronounced “$x$ that is greater than $0$” or something similar.
    • The standalone assertion “$x\gt0$” is pronounced “$x$ is greater than $0$.”
    • The sentence “Let $x\gt0$” must be pronounced “Let $x$ be greater than $0$”.

    The consequence is that the symbolic assertion, in this case “$x\gt0$”, does not reveal that role it plays in the math English sentence that it is embedded in.

    Many of the examples occurring later in the post are also examples of context-sensitive pronunciation.

    Conjectures

    Many students are subconsciously bothered by the way the same symbolic expression is pronounced differently in different math English sentences.

    This probably impedes some students’ progress. Teachers should point this phenomenon out with examples.

    Students should be discouraged from pronouncing mathematical expressions.

    For one thing, this could get you into trouble. Consider pronouncing “$\sqrt{3+5}+6$”. In any case, when you are reading any text you don’t pronounce the words, you just take in their meaning. Why not take in the meaning of algebraic expressions in the same way?

    Parenthetic assertions

    A parenthetic assertion is a symbolic assertion embedded in a sentence in math English in such a way that is a subordinate clause.

    Example

    In the math English sentence

    “For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$”

    mentioned above, the symbolic assertion “$x\gt0$” plays the role of a subordinate clause.

    It is not merely that the pronunciation is different compared to that of the independent statement “$x\gt0$”. The math English sentence is hard to parse. The obvious (to an experienced mathematician) meaning is that the beginning of the sentence can be read this way: “For every real number $x$, which is bigger than $0$…”.

    But new student might try to read it is “For every real number $x$ is greater than $0$ …” by literally substituting the standalone meaning of “$x\gt0$” where it occurs in the sentence. This makes the text what linguists call a garden path sentence. The student has to stop and start over to try to make sense of it, and the symbolic expression lacks the natural language hints that help understand how it should be read.

    Note that the other two symbolic expressions in the sentence are not parenthetic assertions. The phrase “real number” needs to be followed by a term, and it is, and the phrase “such that” must be followed by a clause, and it is.

    More examples

    • “Consider the circle $S^1\subseteq\mathbb{C}=\mathbb{R}^2$.” This has subordinate clauses to depth 2.
    • “The infinite series $\displaystyle\sum_{k=1}^\infty\frac{1}{k^2}$ converges to $\displaystyle\zeta(2)=\frac{\pi^2}{6}\approx1.65$”
    • “We define a null set in $I:=[a,b]$ to be a set that can be covered by a countable of intervals with arbitrarily small total length.” This shows a parenthetical definition.
    • “Let $F:A\to B$ be a function.”
      A type declaration is a function? In any case, it would be better to write this sentence simply as “Let $F:A\to B$”.

    David Butler’s post Contrapositive grammar has other good examples.

    Math texts are in general badly written. Students need to be taught how to read badly written math as well as how to write math clearly. Those that succeed (in my observation) in being able to read math texts often solve the problem by glancing at what is written and then reconstructing what the author is supposedly saying.

    Conjectures

    Some students are baffled, or at least bothered consciously or unconsciously, by parenthetic assertions, because the clues that would exist in a purely English statement are missing.

    Nevertheless, many if not most math students read parenthetic assertions correctly the first time and never even notice how peculiar they are.

    What makes the difference between them and the students who are stymied by parenthetic assertions?

    There is another conjecture concerning parenthetic assertions below.

    Context-sensitive meaning

    “If” in definitions

    Example

    The word “if” in definitions does not mean the same thing that it means in other math statements.

    • In the definition “An integer is even if it is divisible by $2$,” “if” means “if and only if”. In particular, the definition implies that a function is not even if it is not divisible by $2$.
    • In a theorem, for example “If a function is differentiable, then it is continuous”, the word “if” has the usual one-way meaning. In particular, in this case, a continuous function might not be differentiable.

    Context-sensitive meaning occurs in ordinary English as well. Think of a strike in baseball.

    Conjectures

    The nearly universal custom of using “if” to mean “if and only if” in definitions makes it a harder for students to understand implication.

    This custom is not the major problem in understanding the role of definitions. See my article Definitions.

    Underlying sets

    Example

    In a course in group theory, a lecturer may say at one point, “Let $F:G\to H$ be a homomorphism”, and at another point, “Let $g\in G$”.

    In the first sentence, $G$ refers to the group, and in the second sentence it refers to the underlying set of the group.

    This usage is almost universal. I think the difficulty it causes is subtle. When you refer to $\mathbb{R}$, for example, you (usually) are referring to the set of real numbers together with all its canonical structure. The way students think of it, a real number comes with its many relations and connections with the other real numbers, ordering, field properties, topology, and so on.

    But in a group theory class, you may define the Klein $4$-group to be $\mathbb{Z}_2\times\mathbb{Z}_2$. Later you may say “the symmetry group of a rectangle that is not a square is the Klein $4$-group.” Almost invariably some student will balk at this.

    Referring to a group by naming its underlying set is also an example of synecdoche.

    Conjecture

    Students expect every important set in math to have a canonical structure. When they get into a course that is a bit more abstract, suddenly the same set can have different structures, and math objects with different underlying sets can have the same structure. This catastrophic shift in a way of thinking should be described explicitly with examples.

    Way back when, it got mighty upsetting when the earth started going around the sun instead of vice versa. Remind your students that these upheavals happen in the math world too.

    Overloaded notation

    Identity elements

    A particular text may refer to the identity element of any group as $e$.

    This is as far as I know not a problem for students. I think I know why: There is a generic identity element. The identity element in any group is an instantiation of that generic identity element. The generic identity element exists in the sketch for groups; every group is a functor defined on that sketch. (Or if you insist, the generic identity element exists in the first order theory for groups.) I suspect mathematicians subconsciously think of identity elements in this way.

    Matrix multiplication

    Matrix multiplication is not commutative. A student may forget this and write $(A^2B^2=(AB)^2$. This also happens in group theory courses.

    This problem occurs because the symbolic language uses the same symbol for many different operations, in this case the juxtaposition notation for multiplication. This phenomenon is called overloaded notation and is discussed in abstractmath.org here.

    Conjecture

    Noncommutative binary operations written using juxtaposition cause students trouble because going to noncommutative operations requires abandoning some overlearned reflexes in doing algebra.

    Identity elements seem to behave the same in any binary operation, so there are no reflexes to unlearn. There are generic binary operations of various types as well. That’s why mathematicians are comfortable overloading juxtaposition. But to get to be a mathematician you have to unlearn some reflexes.

    Negation

    Sometimes you need to reword a math statement that contains symbolic expressions. This particularly causes trouble in connection with negation.

    Ordinary English

    The English language is notorious among language learners for making it complicated to negate a sentence. The negation of “I saw that movie” is “I did not see that movie”. (You have to put “d** not” (using the appropriate form of “do”) before the verb and then modify the verb appropriately.) You can’t just say “I not saw that movie” (as in Spanish) or “I saw not that movie” (as in German).

    Conjecture

    The method in English used to negate a sentence may cause problems with math students whose native language is not English. (But does it cause math problems with those students?)

    Negating symbolic expressions

    Examples

    • The negation of “$n$ is even and a prime” is “$n$ is either odd or it is not a prime”. The negation should not be written “$n$ is not even and a prime” because that sentence is ambiguous. In the heat of doing a proof students may sometimes think the negation is “$n$ is odd and $n$ is not a prime,” essentially forgetting about DeMorgan. (He must roll over in his grave a lot.)
    • The negation of “$x\gt0$” is “$x\leq0$”. It is not “$x\lt0$”. This is a very common mistake.

    These examples are difficulties caused by not understanding the math. They are not directly caused by difficulties with the languages of math.

    Negating expressions containing parenthetic assertions

    Suppose you want to prove:

    “If $f:\mathbb{R}\to\mathbb{R}$ is differentiable, then $f$ is continuous”.

    A good way to do this is by using the contrapositive. A mechanical way of writing the contrapositive is:

    “If $f$ is not continuous, then $f:\mathbb{R}\to\mathbb{R}$ is not differentiable.”

    That is not good. The sentence needs to be massaged:

    “If $f:\mathbb{R}\to\mathbb{R}$ is not continuous, then $f$ is not differentiable.”

    Even better would be to write the original sentence as:

    “Suppose $f:\mathbb{R}\to\mathbb{R}$. Then if $f$ is differentiable, then $f$ is continuous.”

    This is discussed in detail in David Butler’s post Contrapositive grammar.

    Conjecture

    Students need to be taught to understand parenthetic assertions that occur in the symbolic language and to learn to extract a parenthetic assertion and write it as a standalone assertion ahead of the statement it occurs in.

    Scope

    The scope of a word or variable consists of the part of the text for which its current definition is in effect.

    Examples

    • “Suppose $n$ is divisible by $4$.” The scope is probably the current paragraph or perhaps the current proof. This means that the properties of $n$ are constrained in that section of the text.
    • “In this book, all rings are unitary.” This will hold for the whole book.

    There are many more examples in the abstractmath.org article Scope.

    If you are a grasshopper (you like to dive into the middle of a book or paper to find out what it says), knowing the scope of a variable can be hard to determine. It is particularly difficult for commonly used words or symbols that have been defined differently from the usual usage. You may not suspect that this has happened since it might be define once early in the text. Some books on writing mathematics have urged writers to keep global definitions to a minimum. This is good advice.

    Finding the scope is considerably easier when the text is online and you can search for the definition.

    Conjecture

    Knowing the scope of a word or variable can be difficult. It is particular hard when the word or variable has a large scope (chapter or whole book.)

    Variables

    Variables are often introduced in math writing and then used in the subsequent discussion. In a complicated discussion, several variables may be referred to that have different statuses, some of them introduced several pages before. There are many particular ways discussed below that can cause trouble for students. This post is restricted to trouble in connection with the languages of math. The concept of variable is difficult in itself, not just because of the way the math languages represent them, but that is not covered here.

    Much of this part of the post is based on work of Susanna Epp, including three papers listed in the references. Her papers also include many references to other work in the math ed literature that have to do with understanding variables.

    See also Variables in abstractmath.org and Variables in Wikipedia.

    Types

    Students blunder by forgetting the type of the variable they are dealing with. The example given previously of problems with matrix multiplication is occasioned by forgetting the type of a variable.

    Conjecture

    Students sometimes have problems because they forget the data type of the variables they are dealing with. This is primarily causes by overloaded notation.

    Dependent and independent

    If you define $y=x^2+1$, then $x$ is an independent variable and $y$ is a dependent variable. But dependence and independence of variablesare more general than that example suggests.
    In an epsilon-delta proof of the limit of a function (example below,) $\varepsilon$ is independent and $\delta$ is dependent on $\varepsilon$, although not functionally dependent.

    Conjecture

    Distinguishing dependent and independent variables causes problems, particularly when the dependence is not clearly functional.

    I recently ran across a discussion of this on the internet but failed to record where I saw it. Help!

    Bound and free

    This causes trouble with integration, among other things. It is discussed in abstractmath.org in Variables and Substitution. I expect to add some references to the math ed literature soon.

    Instantiation

    Some of these variables may be given by existential instantiation, in which case they are dependent on variables that define them. Others may be given by universal instantiation, in which case the variable is generic; it is independent of other variables, and you can’t impose arbitrary restrictions on it.

    Existential instantiation

    A theorem that an object exists under certain conditions allows you to name it and use it by that name in further arguments.

    Example

    Suppose $m$ and $n$ are integers. Then by definition, $m$ divides $n$ if there is an integer $q$ such that $n=qm$. Then you can use “$q$” in further discussion, but $q$ depends on $m$ and $n$. You must not use it with any other meaning unless you start a new paragraph and redefine it.

    So the following (start of a) “proof” blunders by ignoring this restriction:

    Theorem: Prove that if an integer $m$ divides both integers $n$ and $p$, then $m$ divides $n+p$.

    “Proof”: Let $n = qm$ and $p = qm$…”

    Universal instantiation

    It is a theorem that for any integer $n$, there is no integer strictly between $n$ and $n+1$. So if you are given an arbitrary integer $k$, there is no integer strictly between $k$ and $k+1$. There is no integer between $42$ and $43$.

    By itself, universal instantiation does not seem to cause problems, provided you pay attention to the types of your variables. (“There is no integer between $\pi$ and $\pi+1$” is false.)

    However, when you introduce variables using both universal and existential quantification, students can get confused.

    Example

    Consider the definition of limit:

    Definition: $\lim_{x\to a} f(x)=L$ if and only if for every $\epsilon\gt0$ there is a $\delta\gt0$ for which if $|x-a|\lt\delta$ then $|f(x)-L|\lt\epsilon$.

    A proof for a particular instance of this definition is given in detail in Rabbits out of a Hat. In this proof, you may not put constraints on $\epsilon$ except the given one that it is positive. On the other hand, you have to come up with a definition of $\delta$ and prove that it works. The $\delta$ depends on what $f$, $a$ and $L$ are, but there are always infinitely many values of $\delta$ which fit the constraints, and you have to come up with only one. So in general, two people doing this proof will not get the same answer.

    Reference

    Susanna Epp’s paper Proof issues with existential quantification discusses the problems that students have with both existential and universal quantification with excellent examples. In particular, that paper gives examples of problems students have that are not hinted at here.

    References

    A nearly final version of The Handbook of Mathematical Discourse is available on the web with links, including all the citations. This version contains some broken links. I am unable to recompile it because TeX has evolved enough since 2003 that the source no longer compiles. The paperback version (without the citations) can be bought as a book here. (There are usually cheaper used versions on Amazon.)

    Abstractmath.org is a website for beginning students in abstract mathematics. It includes most of the material in the Handbook, but not the citations. The Introduction gives you a clue as to what it is about.

    Two languages

    My take on the two languages of math are discussed in these articles:

    The Language of Mathematics, by Mohan Ganesalingam, covers these two languages in more detail than any other book I know of. He says right away on page 18 that mathematical language consists of “textual sentences with symbolic material embedded like ‘islands’ in the text.” So for him, math language is one language.

    I have envisioned two separate languages for math in abstractmath.org and in the Handbook, because in fact you can in principle translate any mathematical text into either English or logical notation (first order logic or type theory), although the result in either case would be impossible to understand for any sizeable text.

    Topics in abstractmath.org

    Context-sensitive interpretation.

    “If” in definitions.

    Mathematical English.

    Parenthetic assertion.

    Scope

    Semantic contamination.

    Substitution.

    The symbolic language of math

    Variables.

    Zooming and Chunking.

    Topics in the Handbook of mathematical discourse.

    These topics have a strong overlap with the topics with the same name in abstractmath.org. They are included here because the Handbook contains links to citations of the usage.

    Context-sensitive.

    “If” in definitions.

    Parenthetic assertion.

    Substitution.

    Posts in Gyre&Gimble

    Names

    Naming mathematical objects

    Rabbits out of a Hat.

    Semantics of algebra I.

    Syntactic and semantic thinkers

    Technical meanings clash with everyday meanings

    Thinking without words.

    Three kinds of mathematical thinkers

    Variations in meaning in math.

    Other references

    Contrapositive grammar, blog post by David Butler.

    Proof issues with existential quantification, by Susanna Epp.

    The role of logic in teaching proof, by Susanna Epp (2003).

    The language of quantification in mathematics instruction, by Susanna Epp (1999).

    The Language of Mathematics: A Linguistic and Philosophical Investigation
    by Mohan Ganesalingam, 2013. (Not available from the internet.)

    On the communication of mathematical reasoning, by Atish Bagchi, and Charles Wells (1998a), PRIMUS, volume 8, pages 15–27.

    Variables in Wikipedia.

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