Category Archives: language of math

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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Representations of functions III

Introduction to this post

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

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

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

Segments posted so far

Graphs of finite functions

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

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

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

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

Introductory Example

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

Graph

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

Table

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

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

Rule

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

\[(a\mapsto b,b\mapsto a,c\mapsto c,d\mapsto b,e\mapsto d)\]

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

\[(a\to b,b\to a,c\to c,d\to b,e\to d)\]

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

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

Two-line notation

Two-line notation is a kind of horizontal table.

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

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

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

Cograph

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

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

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

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

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

Rearrange the cograph

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

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

Endograph

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

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

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

Example: A permutation

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

Cograph

Endograph

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

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

Disjoint cycle notation

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

\[(0)(1,2)(3,4,5)(6,7,8,9)\]

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

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

\[(0)(1,2)(4,5,3)(6,7,8,9)\]
\[(0)(1,2)(8,9,6,7)(3,4,5)\]
\[(1,2)(3,4,5)(0)(6,7,8,9)\]
\[(2,1)(5,3,4)(9,6,7,8)\]
\[(5,3,4)(1,2)(6,7,8,9)\]

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

Example: A tree

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

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

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

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

The endograph

shown here is also a tree.

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

The general form of a finite function

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

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

This illustrates a general fact about finite functions:

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

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

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

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


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Insights into mathematical definitions

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

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

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

I hope math ed people will read the final remarks.

Peculiarities of math definitions

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

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

List of properties

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

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

Imposed by decree

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

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

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

Student difficulties

Giving the math definition low priority

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

Arbitrary bothers them

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

What do you DO with a definition?

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

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

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

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

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

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

What the operability of definitions and theorems means is:

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

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

Final remarks

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

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

References


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

    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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    Names of mathematical objects

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

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

    Sources of names

    Suggestive English words

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

    Examples

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

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

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

      Groups

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

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

      Fields

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

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

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

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

      Person’s name

      A concept may be named after a person.

      Examples

      • L’Hôpital’s Rule
      • Hausdorff space
      • Turing machine
      • Riemann surface
      • Riemannian manifold
      • Pythagorean Theorem
      • I have no idea why “Riemann” gets an ending when it is a manifold but not when it is a surface.

        Made-up name

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

        Named after notation

        Symbols

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

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

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

        Expressions

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

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

          Examples

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

        Names from other languages

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

        Examples

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

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

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

      Names made up from other languages’ roots

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

      Examples

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

      Mathematical names cause problems for students

      The name may suggest the wrong meaning

      This is discusses in detail in the article cognitive dissonance.

      The name may not suggest any meaning

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

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

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

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

      Pronunciation

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

      Most students have little knowledge of other languages

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

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

      Pronunciation of words from other languages has become unpredictable

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

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

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

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

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

      German spelling

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

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

      Transliterations from Cyrillic

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

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

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

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

    Plurals

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

    Plurals ending in a vowel

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

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

      singular

      plural

      matrix

      matrices

      simplex

      simplices

      vertex

      vertices

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

      Remarks

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

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

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    Recent revisions to abstractmath.org

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

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