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Variations in meaning in math

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Words in a natural language may have different meanings in different social groups or different places.  Words and symbols in both mathematical English and the symbolic language vary according to specialty and, occasionally, country (see convention, default).  And words and symbols can change their meanings from place to place within the same mathematical discourse (see scope).

This article mostly provides pointers to other articles in abstractmath.org that give more details about the ideas.

Conventions

A convention in mathematical discourse is notation or terminology used with a special meaning in certain contexts or in certain fields. Articles and books in a specialty do not always clue you in on these conventions.

Some conventions are nearly universal in math.

Example 1

The use of “if” to mean “if and only if” in a definition is a convention. More about this here. This is a hidden definition by cases. “Hidden” means that no one tells the students, except for Susanna Epp and me.

Example 2

Constants or parameters are conventionally denoted by a, b, … , functions by f, g, … and variables by x, y,…. More.

Example 3

Referring to a group (or other mathematical structure) and its underlying set by the same name is a convention.  This is an example of both synecdoche and context-sensitive.

Example 4

The meaning of ${{\sin }^{n}}x$ in many calculus books is:

  • The inverse sine (arcsin) if $n=-1$.
  • The mult­iplica­tive power for positive $n$; in other words, ${{\sin }^{n}}x={{(\sin x)}^{n}}$ if $n\ne -1$.

This, like Example 1, is a definition by cases. Unlike Example 1, calculus books often make it explicit. Explicit or not, this usage is an abomination.

Some conventions are pervasive among math­ematicians but different conventions hold in other subjects that use mathematics.

  • Scientists and engineers may regard a truncated decimal such as 0.252 as an approximation, but a mathematician is likely to read it as an exact rational number, namely $\frac{252}{1000}$.
  • In most computer languages a distinction is made between real numbers and integers;
    42 would be an integer but 42.0 would be a real number.  Older mathematicians may not know this.
  • Mathematicians use i to denote the imaginary unit. In electrical engineering it is commonly denoted j instead, a fact that many mathematicians are un­aware of. I first learned about it when a student asked me if i was the same as j.

Conventions may vary by country.

  • In France and possibly other countries schools may use “positive” to mean “nonnegative”, so that zero is positive. 
  • In the secondary schools in some places, the value of sin x may be computed clockwise starting at (0,1)  instead of counterclockwise starting at (1,0).  I have heard this from students. 

Conventions may vary by specialty within math.

“Field” and “log” are examples. 

Defaults

An interface to a computer program may have many possible choices for the user to make. In most cases, the interface will use certain choices automatically when the user doesn’t specify them.  One says the program defaults to those choices.  

Examples

  • A word processing program may default to justified paragraphs and insert mode, but allow you to pick ragged right or typeover mode.

    • I have spent a lot of time in both Minne­sota and Georgia and the remarks about skiing are based on my own observation. But these usages are not absolute. Some affluent Geor­gians may refer to snow skiing as “skiing”, for example, and this usage can result in a put-down if the hearer thinks they are talking about water skiing. One wonders where the boundary line is. Perhaps people in Kentucky are confused on the issue.

  • There is a sense in which the word “ski” defaults to snow skiing in Minnesota and to water skiing in Georgia.
  • “CSU” defaults to Cleveland State University in northern Ohio and to Colorado State University in parts of the west.

Math language behaves in this way, too.

Default usage in mathematical discourse

Symbols

  • In high school, $\pi$ refers by default to the ratio of the circumference of a circle to its diameter.  Students are often quite surprised when they get to abstract math courses and discover the many other meanings of $\pi $ (see here).
  • Recently authors in the popular literature seem to think that $\phi$ (phi) defaults to the golden ratio.  In fact, a search through the research literature shows very few hits for $\phi$ meaning the golden ratio: in other words, it usually means something else. 
  • The set $\mathbb{R}$ of real numbers has many different group structures defined on it but “The group $\mathbb{R}$” essentially always means that the group operation is ordinary addition.  In other words, “$\mathbb{R}$” as a group defaults to +.  Analogous remarks apply to “the field $\mathbb{R}$”. 
  • In informal conversation among many analysts, functions are continuous by default.
  • It used to be the case that in informal conversations among topologists, “group” defaulted to Abelian group. I don’t know whether that is still true or not.

Remark

This meaning of “default” has made it into dictionaries only since around 1960 (see the Wikipedia entry). This usage does not carry a derogatory connotation.   In abstractmath.org I am using the word to mean a special type of convention that imposes a choice of parameter, so that it is a special case of both “convention” and “suppression of parameters”.

Scope

Both mathematical English and the symbolic language have a feature that is uncommon in ordinary spoken or written English:  The meaning of a phrase or a symbolic expression can be different in different parts of the discourse.   The portion of the text in which a particular meaning is in effect is called the scope of the meaning.  This is accomplished in several ways.

Explicit statement

Examples

  • “In this paper, all groups are abelian”.  This means that every instance of the word “group” or any symbol denoting a group the group is constrained to be abelian.   The scope in this case is the whole paper.   See assumption.
  • “Suppose (or “let” or “assume”) $n$ is divisible by $4$”. Before this statement, you could not assume $n$ is divisible by $4$. Now you can, until the end of the current paragraph or section.

Definition

The definition of a word, phrase or symbol sets its meaning.  If the word definition is used and the scope is not given explicitly, it is probably the whole discourse.

Example

“Definition.  An integer is even if it is divisible by 2.”  This is marked as a definition, so it establishes the meaning of the word “even” (when applied to an integer) for the rest of the text. 

If

Used in modus ponens (see here) and (along with let, usually “now let…”) in proof by cases.

Example(modus ponens)

Suppose you want to prove that if an integer $n$ is divisible by $4$ then it is even. To show that it is even you must show that it is divisible by $2$. So you write:

  • “Let $n$ be divisible by $4$. That means $n=4k$ for some integer $k$. But then $n=2(2k)$, so $n$ is even by definition.”

Now if you start a new paragraph with something like “For any integer $n\ldots$” you can no longer assume $n$ is divisible by $4$.

Example (proof by cases)

Theorem: For all integers $n$, $n^2+n+1$ is odd.

Definitions:

  • “$n$ is even” means that $n=2s$ for some integer $s$.
  • “$n$ is odd” means that $n=2t+1$ for some integer $t$.

Proof:

  • Suppose $n$ is even. Then

    \[\begin{align*}
    n^2+n+1&=4s^2+2s+1\\
    &=2(2s^2+s)+1\\
    &=2(\text{something})+1
    \end{align*}\]

    so $n^2+n+1$ is odd. (See Zooming and Chunking.)

  • Now suppose $n$ is odd. Then

    \[\begin{align*}
    n^2+n+1&=(2t+1)^2+2t+1+1\\
    &=4t^2+4t+1+2t+1+1\\
    &=2(2t^2+3t)+3\\
    &=2(2t^2+3t+1)+1\\
    &=2(\text{something})+1
    \end{align*}\]

    So $n^2+n+1$ is odd.

Remark

The proof I just gave uses only the definition of even and odd and some high school algebra. Some simple grade-school facts about even and odd numbers are:

  • Even plus even is even.
  • Odd plus odd is even.
  • Even times even is even.
  • Odd times odd is odd.

Put these facts together and you get a nicer proof (I think anyway): $n^2+n$ is even, so when you add $1$ to it you must get an odd number.

Bound variables

A variable is bound if it is in the scope of an integral, quantifier, summation, or other binding operators.  More here.

Example

Consider this text:

Exercise: Show that for all real numbers $x$, it is true that $x^2\geq0$. Proof: Let $x=-2$. Then $x^2=(-2)^2=4$ which is greater than $0$. End of proof.”

The problem with that text is that in the statement, “For all real numbers $x$, it is true that $x^2\geq0$”, $x$ is a bound variable. It is bound by the universal quantifier “for all” which means that $x$ can be any real number whatever. But in the next sentence, the meaning of $x$ is changed by the assumption that $x=-2$. So the statement that $x\geq0$ only applies to $-2$. As a result the proof does not cover all cases.

Many students just beginning to learn to do proofs make this mistake. Fellow students who are a little further along may be astonished that someone would write something like that paragraph and might sneer at them. But this common mistake does not deserve a sneer, it deserves an explanation. This is an example of the ratchet effect.

Variable meaning in natural language

Meanings commonly vary in natural language because of conventions and defaults. But varying in scope during a conversation seems to me uncommon.

It does occur in games. In Skat and Bridge, the meaning of “trump” changes from hand to hand. The meaning of “strike” in a baseball game changes according to context: If the current batter has already had fewer than two strikes, a foul is a strike, but not otherwise.

I have not come up with non-game examples, and anyway games are played by rules that are suspiciously like mathematical axioms. Perhaps you can think of some non-game occasions in which meaning is determined by scoping that I have overlooked. 

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Dysfunctions in doing math I

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I am in the middle of revising the article in abstractmath.org on dysfunctional attitudes and behaviors in doing math. Here are three of the sections I have finished.

Misuse of analogy

When William Rowan Hamilton was trying to understand the new type of number called quaternions (MW, Wik) that he invented, he assumed by analogy that like other numbers, quaternion multiplication was commutative. It was a major revelation to him that they were not commutative.

Analogy may suggest new theorems or ways of doing things. But it is fallible. What happens particularly often in abstract math is applying a rule to a situation where it is not appropriate. This is an easy trap to fall into when the notation in two different cases has the same form; that is an example of formal analogy.

Matrix multiplication

Matrix multiplication is not commutative

If $r$ and $s$ are real numbers then the products $rs$ and $sr$ are always the same number. In other words, multiplication of real numbers is commutative : $rs = sr$ for all real numbers $r$ and $s$.

The product of two matrices $M $and $N$ is written $MN$, just as for numbers. But matrix multiplication is not commutative. For example,
\[\left(
\begin{array}{cc}
1 & 2 \\
3 & 4\\
\end{array}
\right)
\left(
\begin{array}{cc}
3 & 1 \\
3 &2\\
\end{array}
\right)
=
\left(
\begin{array}{cc}
9 & 5\\
21 & 11 \\
\end{array}
\right)\]
but
\[\left(
\begin{array}{cc}
3 & 1 \\
3 & 2\\
\end{array}
\right)
\left(\begin{array}{cc}
1 & 2 \\
3 & 4\\
\end{array}
\right)
=
\left(
\begin{array}{cc}
6 & 10\\
91 & 14 \\
\end{array}
\right)\]
Because $rs = sr$ for numbers, the formal similarity of the notation suggests $MN$ = $NM$, which is wrong.

This means you can’t blindly manipulate $MNM$ to become $M^2N$. More generally, a law such as $(MN)^n=M^nN^n$ is not correct when $M$ and $N$ are matrices.


You must understand the meanings
of the symbols you manipulate.

The product of two nonzero matrices can be 0

If the product of two numbers is 0, then one or both of the numbers is zero. But that is not true for matrix multiplication:
\[\left(
\begin{array}{cc}
-2 & 2 \\
-1 & 1\\
\end{array}
\right)
\left(
\begin{array}{cc}
1 & 1 \\
1 &1\\
\end{array}
\right)
=
\left(
\begin{array}{cc}
0 &0\\
0 & 0 \\
\end{array}
\right)\]

Canceling sine

  • Beginning calculus students have already learned algebra.
  • They have learned that an expression such as $xy$ means $x$ times $y$.
  • They have learned to cancel like terms in a quotient, so that for example \[\frac{3x}{3y}=\frac{x}{y}\]
  • They have learned to write the value of a function $f$ at the input $x$ by $f(x)$.
  • They have seen people write $\sin x$ instead of $\sin(x)$ but have never really thought about it.
  • So they write \[\frac{\sin x}{\sin y}=\frac{x}{y}\]

This happens fairly often in freshman calculus classes. But you wouldn’t do that, would you?

Boundary values of definitions

Definitions are usually inclusive

Definitions of math concepts usually include the special cases they generalize.

Examples

  • A square is a special case of rectangle. As far as I know texts that define “rectangle” include squares in the definition. Thus a square is a rectangle.
  • A straight line is a curve.
  • A group is a semigroup.
  • An integer is a real number. (But not always in computing languages — see here.)

But not always

  • The axioms of a field include a bunch of axioms that a one-element set satisfies, plus a special axiom that does nothing but exclude the one-element set. So a field has to have at least two elements, and that fact does not follow from the other axioms.
  • Boolean algebras are usually defined that way, too, but not always. MathWorld gives several definitions of Boolean algebra that disagree on this point.

When boundary values are not special cases

Definitions may or may not include other types of boundary values.

Examples

  • If $S$ is a set, it is a subset of itself. The empty set is also a subset of $S$.
  • Similarly the divisors of $6$ are $-6$, $-3$, $-2$, $-1$, $1$, $2$, $3$ and $6$, not just $2$ and $3$ and not just $1$, $2$, $3$ and $6$ (there are two different boundaries here).

But …

  • The positive real numbers include everything bigger than $0$, but not $0$. ( Note).

Blunders

A definition that includes such special cases may be called inclusive; otherwise it is exclusive. People new to abstract math very commonly use words defined inclusively as if their definition was exclusive.

  • They say things such as “That’s not a rectangle, it is a square!” and “Is that a group or a semigroup?”
  • They object if you say “Consider the complex number $\pi $.”

This appears to be natural linguistic behavior. Even so, math is picky-picky: a square is a rectangle, a group is a semigroup and $\pi$ is a complex number (of course, it is also a real number).

Co-intimidator

  • You attend a math lecture and the speaker starts talking about things you never heard of.
  • Your fellow students babble at you about manifolds and tensors and you thought they were car parts and lamps.
  • You suspect your professor is deliberately talking over your head to put you down.
  • You suspect your friends are trying to make you believe they are much smarter than you are.
  • You suspect your friends are smarter than you are.

There are two possibilities:

  • They are not trying to intimidate you (most common).
  • They are deliberately setting out to intimidate you with their arcane knowledge so you will know what a worm you are. (There are people like that.)

Another possibility, which can overlap with the two above, is:

  • You expect to be intimidated. You may be what might be called a co-intimidator, Similar to the way someone who is codependent wants some other person to be dependent on them. (This is not like the “co” in category theory: “product” and “coproduct” have a symmetric relationship with each other, but the co-intimidator relation is asymmetric.)

There are many ways to get around being intimidated.

  • Ask “What the heck is a manifold?”
  • (In a lecture where it might be imprudent or impractical to ask) Write down what they say, then later ask a friend or look it up.
  • Most teachers like to be asked to explain something. Yes, I know some professors repeatedly put down people. Change sections! If you can’t, live with it! Not knowing something says nothing bad about you.

And remember:


If you don’t know something
probably many other students don’t know it either.

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Extensional and Intensional

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This post uses the word intensional, which is not the word "intentional" and doesn't mean the same thing.

 

The connection between rich view/rigorous view and intensional/extensional

 

In the abmath article Images and Metaphors I wrote about the rigorous view of math, in contrast to the rich view which allows metaphors, images and intuition. F. Kafi has proposed the following thesis:

The rigorous mode of thinking deals with the extensional meaning of mathematical objects while the metaphoric mode of thinking deals with the intensional meaning of mathematical objects.

This statement is certainly suggestive as an analogy. I have several confused and disjointed thoughts about it.

What does "intensional" mean?

Philosophy

Philosophers say that "the third largest planet in the solar system" has intensional meaning and "Neptune" has extensional meaning. Among other things we might discover a planet ridiculously far out that is bigger than Neptune. But the word "Neptune" denotes a specific object.

The intensional meaning of "the third largest planet in the solar system" has a hidden time dimension that, if made overt, makes the statement more nearly explicit. (Don't read this paragraph as a mathematical statement; it is merely thrashing about to inch towards understanding.)

Computing science

Computer languages are distinguishes as intensional or extensional, but their meaning there is technical, although clearly related to the philosophers' meaning.

I don't understand it very well, but in Type Theory and in Logic, an intensional language seems to make a distinction between declaring two math objects to be equal and proving that they are equal. In an extensional language there is no such distinction, with the effect that in a typed language typing would be undecidable.

Here is another point: If you define the natural numbers by the Peano axioms, you can define addition and then prove that addition is commutative. But for example a vector space is usually defined by axioms and one of the axioms is a declaration that addition of vectors is commutative. That is an imposed truth, not a deduced one. So is the difference between intensional and extensional languages really a big deal or just a minor observation?

What is "dry-bones rigor"?

Another problem is that I have never spelled out in more than a little detail what I mean by rigor, dry-bones rigor as I have called it. This is about the process mathematicians go through to prove a theorem, and I don't believe that process can be given a completely mathematical description. But I could go into much more detail than I have in the past.

Suppose you set out to prove that if $f(x)$ is a differentiable function and $f(a)=0$ and the graph going from left to right goes UP before $x$ reaches $a$ and then DOWN for $x$ to the right of $a$, then $a$ has to be a maximum of the function. That is a metaphorical description based on the solid physical experience of walking up to the top of a hill. But when you get into the proof you start using lots of epsilons and deltas. This abandons ideas of moving up and down and left to right and so on. As one of the members of Bourbaki said, rigorous math is when everything goes dead. That sounds like extensionality, but isn't their work really based on the idea that everything has to be reduced to sets and logic? (This paragraph was modified on 2013.11.07)

Many perfectly rigorous proofs are based on reasoning in category theory. You can define an Abelian group as a categorical diagram with the property that any product preserving functor to any category will result in a group. This takes you away from sets altogether, and is a good illustration of the axiomatic method. It is done by using nodes, arrows and diagrams. The group is an object and the binary operation is an arrow from the square of the object. Commutativity is required by stating that a certain diagram must commute. But when you prove that two elements in an Abelian group (an Abelian topological group, an Abelian group in the category of differentiable manifolds, or whatever) can be added in either order, then you find yourself staring at dead arrows and diagrams rather than dead collections of things and so you are still in rigor mortis mode.

I will write a separate post describing these examples in much more detail than you might want to think about.

Metaphors and intensionality

One other thing I won't go into now: How are thinking in metaphors and intensional descriptions related? It seems to me the two ideas are related somehow, but I don't know how to formulate it.

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“Trivial”

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This is an update of an article in the abstractmath glossary.

About theorems

A theorem is said to be trivial to prove or trivially true

Example:

Here is a scenario that exemplifies (a):

  • A textbook defines the image of a function $F:A\to B$ to be the set of all elements of $B$ of the form $F(a)$ for some $a\in A$.
  • It then goes on to say that $F$ is surjective if for every element $b$ of $B$ there is an element $a\in A$ with the property that $F(a) = b$.
  • It then states a theorem, or give an exercise, that says that a function $F:A\to B$ is surjective if and only if the image of $F$ is $B$.
  • The proof follows immediately by rewriting using definitions.
  • The instructor calls the proof trivial and goes on to the next topic.
  • Some students are totally baffled.

I have seen this happen many times with this and other theorems.  This sort of incident may be why many intelligent people feel they are "bad at math".

People are not born knowing the principle of rewriting by definitions. The principle needs to be TAUGHT.

  • When a class is first introduced to proof techniques the instructor should explicitly describe rewriting by definitions with several examples.
  • After that, the instructor can say that a proof follows by rewriting by definitions and make it clear that the students will have to do the work (then or later).
  • Such a proof is justly called "trival" but saying it is trivial is also a putdown if no one has pointed out the procedure of rewriting by definitions.

Example:

This example illustrates (b).

Theorem: Let $G$ be a finite group and $H$ a subgroup of index $2$ (meaning it has half the number of elements of the group).  Then $H$ is normal in $G$.

Basic facts about groups and subgroups learned in first semester abstract algebra:

  • A subgroup of a group determines a partition consisting of left cosets and another partition of right cosets, each (in the finite case) with the same number of elements as the subgroup.
  • A subgroup is a left coset of itself and also a right coset of itself.
  • If every left coset is also a right coset and vice versa (so the two partitions just mentioned are the same), then by definition the subgroup is normal in the group.

Now if $H$ has index $2$ that means that each partition consists of two cosets. In both cases, one of them has to be $H$, so the other one has to be $G\setminus H$, which must therefore be a left and right coset of $H$. So $H$ is normal in $G$.

So once you understand the basics about cosets and normal subgroups, the fact that $H$ has to be normal if it is of index $2$ is "obvious". I don't think you should call this "trivial". Best to say it is "obvious if you have a clear understanding of cosets of groups".

About mathematical objects

  • A function may be called trivial if it is the identity function or a constant function, and possibly in other circumstances. (If someone showed that the cosmo­logical constant is 0 that would not be called trivial.)
  • A solution to an equation may be said to be trivial if it is 0 or 1. There may be other situations in which a solution is called "trivial" as well.
  • A mathematical structure is said to be trivial if its underlying set is empty or a singleton set. In particular, a subset of a set is nontrivial if it is nonempty. I have not found an example where "nontrivial subset" means it is not a singleton. 

Note: "Trivial" and "degenerate" overlap in meaning but are not interchangeable.  What is called "degenerate" seems to depend on the mathematical specialty.

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Thinking about a function as a mathematical object

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A mathematician’s mental representation of a function is generally quite rich and may involve many different metaphors and images kept in mind simultaneously. The abmath article on metaphors and images for functions discusses many of these representations, although the article is incomplete. This post is a fairly thorough rewrite of the discussion in that article of the representation of the concept of “function” as a mathematical object. You must think of functions as math objects when you are taking the rigorous view, which happens when you are trying to prove something about functions (or large classes of functions) in general.

What often happens is that you visualize one of your functions in many of the ways described in this article (it is a calculation, it maps one space to another, its graph is bounded, and so on) but those images can mislead you. So when you are completely stuck, you go back to thinking of the function as an axiomatically-defined mathe­matical structure of some sort that just sits there, like a complicated machine where you can see all the parts and how they relate to each other. That enables you to prove things by strict logical deduction. (Mathematicians mostly only go this far when they are desperate. We would much rather quote somebody’s theorem.) This is what I have called the dry bones approach.

The “mathematical structure” is most commonly a definition of function in terms of sets and axioms. The abmath article Specification and definition of “function” discusses the usual definitions of “function” in detail.

Example

This example is intended to raise your consciousness about the possibilities for functions as objects.

Consider the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=2{{\sin }^{2}}x-1$. Its value can be computed at many different numbers but it is a single, static math object.

You can apply operators to it

  • Just as you can multiply a number by $2$, you can multiply $f$ by $2$.   You can say “Let $g(x)=2f(x)$” or “Let $g=2f$”. Multiplying a numerical function by $2$ is an operator that take the function $f$ to $2f$. Its input is a function and its output is another function. Then the value of $g$ (which is $2f$) at any real $x$ is $g(x)=2f(x)=4{{\sin }^{2}}x-2$. The notation  “$g=2f$” reveals that mathematicians think of $f$ as a single math object just as the $3$ in the expression “$2\times 3$” represents the number $3$ as a single object.
  • But you can’t do arithmetic operations to functions that don’t have numerical output, such as the function $\text{FL}$ that takes an English word to its first letter, so $\text{FL}(`\text{wolf’})=`\text{w’}$. (The quotes mean that I am writing about the word ‘wolf’ and the letter ‘w’.) The expression $2\times \text{FL}(`\text{wolf’})$ doesn’t make sense because ‘w’ is a letter, not a number.
  • You can find the derivative.  The derivative operator is a function from differentiable functions to functions. Such a thing is usually called an operator.  The derivative operator is sometimes written as $D$, so $Df$ is the function defined by: “$(Df)(x)$ is the slope of the tangent line to $f$ at the point $(x,f(x)$.” That is a perfectly good definition. In calculus class you learn formulas that allow you to calculate $(Df)(x)$ (usually called “$f'(x)$”) to be $4 \sin (x) \cos (x)$.

Like all math objects, functions may have properties

  • The function defined by $f(x)=2{{\sin}^{2}}x-1$ is differentiable, as noted above. It is also continuous.
  • But $f$ is not injective. This means that two different inputs can give the same output. For example,$f(\frac{\pi}{3})=f(\frac{4\pi}{3})=\frac{1}{2}$. This is a property of the whole function, not individual values. It makes no sense to say that $f(\frac{\pi}{3})$ is injective.
  • The function $f$ is periodic with period $2\pi$, meaning that for any $x$, $f(x+2\pi)=f(x)$.     It is the function itself that has period $2\pi$, not any particular value of it.  

As a math object, a function can be an element of a set

  • For example,$f$ is an element of the set ${{C}^{\infty }}(\mathbb{R})$ of real-valued functions that have derivatives of all orders.
  • On ${{C}^{\infty }}(\mathbb{R})$, differentiation is an operator that takes a function in that set to another function in the set.   It takes $f(x)$ to the function $4\sin x\cos x$.
  • If you restrict $f$ to the unit interval, it is an element of the function space ${{\text{L}}^{2}}[0,1]$.   As such it is convenient to think of it as a point in the space (the whole function is the point, not just values of it).    In this particular space, you can think of the points as vectors in an uncountably-infinite-dimensional space. (Ideas like that weird some people out. Do not worry if you are one of them. If you keep on doing math, function spaces will seem ordinary. They are OK by me, except that I think they come in entirely too many different kinds which I can never keep straight.) As a vector, $f$ has a norm, which you can think of as its length. The norm of $f$ is about $0.81$.

The discussion above shows many examples of thinking of a function as an object. You are thinking about it as an undivided whole, as a chunk, just as you think of the number $3$ (or $\pi$) as just a thing. You think the same way about your bicycle as a whole when you say, “I’ll ride my bike to the library”. But if the transmission jams, then you have to put it down on the grass and observe its individual pieces and their relation to each other (the chain came off a gear or whatever), in much the same way as noticing that the function $g(x)=x^3$ goes through the origin and looks kind of flat there, but at $(2,8)$ it is really rather steep. Phrases like “steep” and “goes through the origin” are a clue that you are thinking of the function as a curve that goes left to right and levels off in one place and goes up fast in another — you are thinking in a dynamic, not a static way like the dry bones of a math object.

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The definition of “function”

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This is the new version of the abstractmath article on the definition of function. I had to adapt the formatting and some of it looks weird, but legible. It is prettier on abstractmath.org.

I expect to announce new revisions of other abmath articles on this blog, with links, but not to publish them here. This article brings out a new point of view about defining functions that I wanted to call attention to, so I am publishing it here, as well.

 

FUNCTIONS: SPECIFICATION AND DEFINITION

It is essential that you understand many of the images, metaphors and terminology that mathe­maticians use when they think and talk about functions. For many purposes, the precise mathematical definition of "function" does not play much of a role when you are trying to understand particular kinds of functions. But there is one point of view about functions that has resulted in fundamental progress in math:

 

 

A function is a mathematical object.

To deal with functions in that way 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 $a$, denoted by $f(a)$.

(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 operation of finding $f(a)$ given $f$ and $a$ is called evaluation.
  • "FP" means functional property.
  • "VIC" means "value in codomain".

Examples

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

A finite function

Let $F$ be the function defined on the set $\left\{1,\,2,3,6 \right\}$ as follows: $F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1$. This is the function called "Finite'' in the chapter on examples of functions.

  • The definition of $F$ says "$F$ is defined on the set $\left\{1,\,2,\,3,\,6 \right\}$". That phrase means that the domain is that set.
  • The value of $F$ at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that $F(2) = 3$. No other reason needs to be given. Mathematical definitions can be arbitrary.
  • The codomain of $F$ is not specified, but must include the set $\{1,2,3\}$. 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).

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 $G(3)=3^2+2\cdot3+5=20$.
  • 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 does not specify the codomain, either. However, must include all real numbers greater than or equal to 4. (Why?)

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.

     

     

    I will say more about this when I give the various definitions that are in use.

History

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.

This discussion 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.

In particular, this definition, along with the use of set theory, enabled abstract math (ahem) to become a common tool for understanding math and proving theorems. It is conceivable that some of you may wish it hadn't. Well, tough.

The more modern definition of function given here (which builds on the older definition) came into use beginning in the 1950's. The strict version 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 old abstract definition and the modern one, 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. 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.

Graph of a function.

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.

Modern 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:
  • $\text{dom} f$ is 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 $f:A\to B$.

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, which makes the graph, which should be a concept derived from the concept of function, into an apparently necessary part of the function.
  • That suggests incorrectly that the graph is more of a primary intuition that other intuitions such as function as relocator, function as transformer, and other points of view discussed in the article Intuitions and metaphors for functions.

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 $C$, 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

Remarks

Possible confusion

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

Multivalued function

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 all the cogno­scenti knew that their functions were multi­valued.

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 common 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 $x^2$.

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.")

The phrases "multivalued function" and "partial function" upset some picky types who say things like, "But a multi­valued func­tion is not a func­tion!". A step­mother is not a mother, either. See the Hand­book article on radial category.

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.

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.

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.

 

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Conceptual and Computational

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I have posted a revision of the article Conceptual and Computational on abstractmath.org.

  • It is the result of my first adventure in revising abstractmath.org in accordance with the ideas in my recent Gyre&Gimble post Writing math for the web.
  • One part of the new article incorporates some of the ideas of my post
    The power of being naive
  • I did not use the manipulable diagrams in the Naive post in the abstractmath post. It’s not clear to me how many one time drop-ins (which is what I mostly get in abstractmath) will be willing to install Wolfram CDF Player to fiddle with one or two diagrams.
  • I have been pleased at the way many of the topics covered in abstractmath come up high when you search for them in Google (including Conceptual Computational, but also things like Mathematical Object and Language of Math (where I even beat Wikipedia)). However, it may be that the high rank occurs because Google knows who I am. I will investigate next time I am in a library!
  • I expect to post pieces of Abstracting Algebra on abstractmath when they become decently finished enough.

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