Introduction

In a recent post I began a discussion of the mental, physical and mathematical representations of a mathematical object. The discussion continues here. Mathematicians, linguists, cognitive scientists and math educators have investigate some aspects of this topic, but there are many subtle connections between the different ideas which need to be studied.

I don’t have any overall theoretical grasp of these relationships. What I will do here is grope for an overall theory by mentioning a whole bunch of fine points. Some of these have been discussed in the literature and some (as far as I know) have not been discussed.  Many of them (I hope)  can be described as “an obvious fact about representations but no one has pointed it out before”.  Such fine points could be valuable; I think some scholars who have written about mathematical discourse and math in the classroom are not aware of many of these facts.

I am hoping that by thrashing around like this here (for graphs of functions) and for other concepts (set, function, triangle, number …) some theoretical understanding may emerge of what it means to understand math, do math, and talk about math.

The graph of a function

Let’s look at the graph of the function $latex {y=x^3-x}&fg=000000$.

What you are looking at is a physical representation of the graph of the function. The graph creates in your brain a mental representation of the graph of the function. These are subtly related to each other and to the mathematical definition of the graph.

Fine points

1. The mathematical definition [2] of the graph of this function is: The set of ordered pairs of numbers $latex {(x,x^3-x)}&fg=000000$ for all real numbers $latex {x}&fg=000000$.
2. In the physical representation, each point $latex {(x,x^3-x)}&fg=000000$ is shown in a location determined by the conventional $latex {x-y}&fg=000000$ coordinate system, which uses a straight-line representation of the real numbers with labels and ticks.
• The physical representation makes use of the fact that the function is continuous. It shows the graph as a curving line rather than a bunch of points.
• The physical representation you are looking at is not the physical representation I am looking at. They are on different computer screens or pieces of paper. We both expect that the representations are very similar, in some sense physically isomorphic.
• “Location” on the physical representation is a physical idea. The mathematical location on the mathematical graph is essentially the concept of the physical location refined as the accuracy goes to infinity. (This last statement is a metaphor attached to a genuine mathematical construction, for example Cauchy sequences.)
3. The mathematical definition of “graph” and the physical representation are related by a metaphor. (See Note 1.)
• The physical curve in blue in the picture corresponds via the metaphor to the graph in the mathematical sense: in this way, each location on the physical curve corresponds to an ordered pair of the form $latex {(x,x^3-x)}&fg=000000$.
• The correspondence between the locations and the pairs is imperfect. You can’t measure with infinite accuracy.
• The set of ordered pairs $latex {(x,x^3-x)}&fg=000000$ form a parametrized curve in the mathematical sense. This curve has zero thickness. The curve in the physical representation has positive thickness.
• Not all the points in the mathematical graph actually occur on the physical curve: The physical curve doesn’t show the left and right infinite tails.
• The physical curve is drawn to show some salient characteristics of the curve, such as its extrema and inflection points. This is expected by convention in mathematical writing. If the graph had left out a maximum, for example, the author would be constrained (by convention!) to say so.
• An experienced mathematician or advanced student understands the fine points just listed. A newbie may not, and may draw false conclusions about the function from the graph. (Note 2.)
4. If you are a mathematician or at least a math student, seeing the physical graph shown above produces a mental image(see Note 3.) of the graph in your mind.
5. The mathematical definition and the mental image are connected by a metaphor. This is not the same metaphor as the one that connects the physical representation and the mathematical definition.
• The curve I visualize in my mental representation has an S shape and so does the physical representation. Or does it? Isn’t the S-ness of the shape a fact I construct mentally (without consciously intending to do so!)?
• Does the curve in the mental rep have thickness? I am not sure this is a meaningful question. However, if you are a sufficiently sophisticated mathematician, your mental image is annotated with the fact that the curve has zero thickness. (See Note 4.)
• The curve in your mental image of the curve may very well be blue (just because you just looked at my picture) but you must have an annotation to the effect that that is irrelevant! That is the essence of metaphor: Some things are identified with each other and others are emphatically not identified.
• The coordinate axes do exist in the physical representation and they don’t exist in the mathematical definition of the graph. Of course they are implied by the definition by the properties of the projection functions from a product. But what about your mental image of the graph? My own image does not show the axes, but I do “know” what the coordinates of some of the points are (for example, $latex {(-1,0)}&fg=000000$) and I “see” some points (the local maximum and the local minimum) whose coordinates I can figure out.

Notes

1. This is metaphor in the sense lately used by cognitive scientists, for example in [6]. A metaphor can be described roughly as two mental images in which certain parts of one are identified with certain parts of another, in other words a pushout. The rhetorical use of the word “metaphor” requires it to be a figure of speech expressed in a certain way (the identification is direct rather than expressed by “is like” or some such thing.)  In my use in this article a metaphor is something that occurs in your brain.  The form it takes in speech or writing is not relevant.

2. I have noticed, for example, that some students don’t really understand that the left and right tails go off to infinity horizontally as well as vertically.   In fact, the picture above could mislead someone into thinking the curve has vertical asymptotes: The right tail looks like it goes straight up.  How could it get to x equals a billion if it goes straight up?

3. The “mental image” is of course a physical structure in your brain.  So mental representations are physical representations.

4. I presume this “annotation” is some kind of physical connection between neurons or something.  It is clear that a “mental image” is some sort of physical construction or event in the brain, but from what little I know about cognitive science, the scientists themselves are still arguing about the form of the construction.  I would appreciate more information on this. (If the physical representation of mental images is indeed still controversial, this says nothing bad about cognitive science, which is very new.)

References

[1] Mental Representations in Math (previous post).

[2] Definitions (in abstractmath).

[3] Lakoff, G. and R. E. Núñez (2000), Where Mathematics Comes From. Basic Books.

The Mathematical Definition of Function

Introduction

This post is a completely rewritten version of the abstractmath article on the definition of function. Like every part of abstractmath, the chapter on functions is designed to get you started thinking about functions. It is no way complete. Wikipedia has much more complete coverage of mathematical functions, but be aware that the coverage is scattered over many articles.

The concept of function in mathematics is as important as any mathematical idea. The mathematician’s concept of function includes the kinds of functions you studied in calculus but is much more abstract and general. If you are new to abstract math you need to know:

• The precise meaning of the word “function” and other concepts associated with functions. That’s what this section is about.
• Notation and terminology for functions. (That will be a separate section of abstractmath.org which I will post soon.)
• The many different kinds of functions there are. (See Examples of Functions in abmath).
• The many ways mathematicians think about functions. The abmath article Images and Metaphors for Functions is a stub for this.

I will use two running examples throughout this discussion:

• $latex {F}&fg=000000$ is the function defined on the set $latex {\left\{1,\,2,3,6 \right\}}&fg=000000$ as follows: $latex {F(1)=3,\,\,\,F(2)=3,\,\,\,F(3)=2,\,\,\,F(6)=1}&fg=000000$. This is a function defined on a finite set by explicitly naming each value.
• $latex {G}&fg=000000$ is the real-valued function defined by the formula $latex {G(x)={{x}^{2}}+2x+5}&fg=000000$.

Specification of function

We start by giving a specification of “function”. (See the abstractmath article on specification.) After that, we get into the technicalities of the definitions of the general concept of function.

Specification: A function $latex {f}&fg=000000$ is a mathematical object which determines and is completely determined bythe following data:

• $latex {f}&fg=000000$ has a domain, which is a set. The domain may be denoted by $latex {\text{dom }f}&fg=000000$.
• $latex {f}&fg=000000$ has a codomain, which is also a set and may be denoted by $latex {\text{cod }f}&fg=000000$.
• For each element $latex {a}&fg=000000$ of the domain of $latex {f}&fg=000000$, $latex {f}&fg=000000$ has a value at $latex {a}&fg=000000$, denoted by $latex {f(a)}&fg=000000$.
• The value of $latex {f}&fg=000000$ at $latex {a}&fg=000000$ is completely determined by $latex {a}&fg=000000$ and $latex {f}&fg=000000$ .
• The value of $latex {f}&fg=000000$ at $latex {a}&fg=000000$ must be an element of the codomain of $latex {f}&fg=000000$.

The operation of finding $latex {f(a)}&fg=000000$ given $latex {f}&fg=000000$ and $latex {a}&fg=000000$ is called evaluation.

Examples

• The definition above of the finite function $latex {F}&fg=000000$ specifies that the domain is the set $latex {\left\{1,\,2,\,3,\,6 \right\}}&fg=000000$. The value of $latex {F}&fg=000000$ at each element of the domain is given explicitly. The value at 3, for example, is 2, because the definition says that $latex {F(2) = 3}&fg=000000$. The codomain of $latex {F}&fg=000000$ is not specified, but must include the set $latex {\{1,2,3\}}&fg=000000$.
• The definition of $latex {G}&fg=000000$ above gives the value at each element of the domain by a formula. The value at 3, for example, is $latex {G(3)=3^2+2\cdot3+5=20}&fg=000000$. The definition does not specify the domain or the codomain. 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 $latex {{\mathbb R}}&fg=000000$. The codomain must include all real numbers greater than or equal to 4. (Why?)

Comment: The formula above that defines the function $latex G$ in fact defines a function of complex numbers (even quaternions).

Definition of function

In the nineteenth century, mathematicians realized that it was necessary for some purposes (particularly harmonic analysis) to give a mathematical definition of the concept of function. A stricter version of this definition turned out to be necessary in algebraic topology and other fields, and that is the one I give here.

To state this definition we need a preliminary idea.

The functional property

A set R of ordered pairs has the functional property if two pairs in R with the same first coordinate have to have the same second coordinate (which means they are the same pair).

Examples

• The set $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ has the functional property, since no two different pairs have the same first coordinate. It is true that two of them have the same second coordinate, but that is irrelevant.
• The set $latex {\{(1,2), (2,4), (3,2), (2,8)\}}&fg=000000$ does not have the functional property. There are two different pairs with first coordinate 2.
• The graphs of functions in beginning calculus have the functional property.
• The empty set $latex {\emptyset}&fg=000000$ has the functional property .

Example: Graph of a function defined by a formula

The graph of the function $latex {G}&fg=000000$ given above has the functional property. The graph is the set

$latex \displaystyle \left\{ (x,{{x}^{2}}+2x+5)\,\mathsf{|}\,x\in {\mathbb R} \right\}.&fg=000000$

If you repeatedly plug in one real number over and over, you get out the same real number every time. Example:

• if $latex {x = 0}&fg=000000$, then $latex {{{x}^{2}}+2x+5=5}&fg=000000$.  You get 5 every time you plug in 0.
• if $latex {x = 1}&fg=000000$, then $latex {{{x}^{2}}+2x+5=8}&fg=000000$.
• if $latex {x =-2}&fg=000000$, then $latex {{{x}^{2}}+2x+5=5}&fg=000000$.

This set has the functional property because if $latex {x}&fg=000000$ is any real number, the formula $latex {{{x}^{2}}+2x+5}&fg=000000$ defines a specific real number. (This description of the graph implicitly assumes that $latex {\text{dom } G={\mathbb R}}&fg=000000$.)  No other pair whose first coordinate is $latex {-2}&fg=000000$ is in the graph of $latex {G}&fg=000000$, only $latex {(-2, 5)}&fg=000000$. That is because when you plug $latex {-2}&fg=000000$ into the formula $latex {{{x}^{2}}+2x+5}&fg=000000$, you get $latex {5}&fg=000000$ every time. Of course, $latex {(0, 5)}&fg=000000$ is in the graph, but that does not contradict the functional property. $latex {(0, 5)}&fg=000000$ and $latex {(-2, 5)}&fg=000000$ have the same second coordinate, but that is OK.

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 “$latex {G(x)}&fg=000000$” for any $latex {x }&fg=000000$ in the domain of $latex {G}&fg=000000$ and not be ambiguous.

Mathematical definition of function

A function$latex {f}&fg=000000$ is a mathematical structure consisting of the following objects:

• A set called the domain of $latex {f}&fg=000000$, denoted by $latex {\text{dom } f}&fg=000000$.
• A set called the codomain of $latex {f}&fg=000000$, denoted by $latex {\text{cod } f}&fg=000000$.
• A set of ordered pairs called the graph of $latex { f}&fg=000000$, with the following properties:
• $latex {\text{dom } f}&fg=000000$ is the set of all first coordinates of pairs in the graph of $latex {f}&fg=000000$.
• Every second coordinate of a pair in the graph of $latex {f}&fg=000000$ is in $latex {\text{cod } f}&fg=000000$ (but $latex {\text{cod } f}&fg=000000$ may contain other elements).
• The graph of $latex {f}&fg=000000$ has the functional property. Using arrow notation, this implies that $latex {f:A\rightarrow B}&fg=000000$.

Examples

• Let $latex {F}&fg=000000$ have graph $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ and define $latex {A = \{1, 2, 3, 5\}}&fg=000000$ and $latex {B = \{2, 4, 8\}}&fg=000000$. Then $latex {F:A\rightarrow B}&fg=000000$ is a function.
• Let $latex {G}&fg=000000$ have graph $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ (same as above), and define $latex {A = \{1, 2, 3, 5\}}&fg=000000$ and $latex {C = \{2, 4, 8, 9, 11, \pi, 3/2\}}&fg=000000$. Then $latex {G:A\rightarrow C}&fg=000000$ is a (admittedly ridiculous) function. Note that all the second coordinates of the graph are in $latex {C}&fg=000000$, along with a bunch of miscellaneous suspicious characters that are not second coordinates of pairs in the graph.
• Let $latex {H}&fg=000000$ have graph $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$. Then $latex {H:A\rightarrow {\mathbb R}}&fg=000000$ is a function.

According to the definition of function, $latex {F}&fg=000000$, $latex {G}&fg=000000$ and $latex {H}&fg=000000$ are three different functions.

Identity and inclusion

Suppose we have two sets A and B with $latex {A\subseteq B}&fg=000000$.

• The identity function on A is the function $latex {{{\text{id}}_{A}}:A\rightarrow A}&fg=000000$ defined by $latex {{{\text{id}}_{A}}(x)=x}&fg=000000$ for all$latex {x\in A}&fg=000000$. (Many authors call it $latex {{{1}_{A}}}&fg=000000$).
• The inclusion function from A to B is the function $latex {i:A\rightarrow B}&fg=000000$ defined by $latex {i(x)=x}&fg=000000$ for all $latex {x\in A}&fg=000000$. Note that there is a different function for each pair of sets A and B for which $latex {A\subseteq B}&fg=000000$. Some authors call it $latex {{{i}_{A,\,B}}}&fg=000000$ or $latex {\text{in}{{\text{c}}_{A,\,B}}}&fg=000000$.

Remark The identity function and an inclusion function for the same set A have exactly the same graph, namely $latex {\left\{ (a,a)|a\in A \right\}}&fg=000000$.

Graphs and functions

• If $latex {f}&fg=000000$ is a function, the domain of $latex {f}&fg=000000$ is the set of first coordinates of all the pairs in $latex {f}&fg=000000$.
• If $latex {x\in \text{dom } f}&fg=000000$, then $latex {f(x)}&fg=000000$ is the second coordinate of the only ordered pair in $latex {f}&fg=000000$ whose first coordinate is $latex {x}&fg=000000$.

Examples

The set $latex {\{(1,2), (2,4), (3,2), (5,8)\}}&fg=000000$ has the functional property, so it is the graph of a function. Call the function $latex {H}&fg=000000$. Then its domain is $latex {\{1,2,3,5\}}&fg=000000$ and $latex {H(1) = 2}&fg=000000$ and $latex {H(2) = 4}&fg=000000$. $latex {H(4)}&fg=000000$ is not defined because there is no ordered pair in H beginning with $latex {4}&fg=000000$ (hence $latex {4}&fg=000000$ is not in $latex {\text{dom } H}&fg=000000$.)

I showed above that the graph of the function $latex {G}&fg=000000$, ordinarily described as “the function $latex {G(x)={{x}^{2}}+2x+5}&fg=000000$”, has the functional property. The specification of function requires that we say what the domain is and what the value is at each point. These two facts are determined by the graph.

Other definitions of function

Because of the examples above, many authors define a function as a graph with the functional property. Now, the graph of a function $latex {G}&fg=000000$ may be denoted by $latex {\Gamma(G)}&fg=000000$.  This is an older, less strict definition of function that doesn’t work correctly with the concepts of algebraic topology, category theory, and some other branches of mathematics.

For this less strict definition of function, $latex {G=\Gamma(G)}&fg=000000$, which causes a clash of our mental images of “graph” and “function”. In every important way except the less-strict definition, they ARE different!

A definition is a device for making the meaning of math technical terms precise. When a mathematician think of “function” they think of many aspects of functions, such as a map of one shape into another, a graph in the real plane, a computational process, a renaming, and so on. One of the ways of thinking of a function is to think about its graph. That happens to be the best way to define the concept of function.  (It is the less strict definition and it is a necessary concept in the modern definition given here.)

The occurrence of the graph in either definition doesn’t make thinking of a function in terms of its graph the most important way of visualizing  it. I don’t think it is even in the top three.