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help with abstract math

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Mathematics is the tinkertoy of metaphor.--Ellis D. Cooper

Half this game is 90% mental.–Yogi Berra

The language that nature speaks is mathe­matics. The language that ordinary human beings speak is metaphor. --Freeman Dyson

# IMAGES AND METAPHORS

In this chapter, I say something about mental represen­tations (metaphors and images) in general, and provide examples of how metaphors and images help us understand math – and how they can confuse us.

Pay special attention to the section called two levels!  The distinction made there is vital but is often not made explicit.

Mental represen­tations and other kinds of represen­tations used in math are also discussed in in many other places, including other articles in abstractmath.org.

## Images and metaphors in general

We think and talk about our experiences of the world in terms of images and metaphors that are ultimately derived from immediate physical experience.  They are mental represen­tations of our experiences.

### Examples

#### Images

We know what a pyramid looks like. But when we refer to the government’s food pyramid we are not talking about actual food piled up to make a pyramid.  We are talking about a visual image of the pyramid.

#### Metaphors

We know by direct physical experience what it means to be warm or cold. We use these words as metaphors in many ways:

• We refer to a person as having a warm or cold personality. This has nothing to do with their body temperature.
• When someone is on a treasure hunt we may tell them they are “getting warm”, even if they are hunting outside in the snow.
• Children don’t always sort meta­phors out correctly. Father: “We are all going to fly to Saint Paul to see your cousin Petunia.” Child: “But Dad, I don’t know how to fly!”

### Other terminology

• My use of the word "image" means mental image. In the study of literature, the word “image” is used in a more general way, to refer to an expression that evokes a mental image.
• I use "metaphor" in the sense of conceptual metaphor. The word metaphor in literary studies is related to my use but is defined in terms of how it is expressed (see The word "metaphor").
• The metaphors mentioned above involving “warm” and “cold” evoke a sensory experience, and so could be called an image as well.
• In math education, the phrase concept image means the mental structure associated with a concept. There may be no direct connection with visual or other sensory experience.
• In abstractmath.org, I use the phrase metaphors and images to talk about all our mental represen­tations, without trying for fine distinctions.

### Mental represen­tations are imperfect

One basic fact about metaphors and images is that they apply only to certain aspects of the situation.

• When someone is getting physically warm we would expect them to start sweating.
• But if they are getting warm in a treasure hunt we don’t expect them to start sweating.
• We don’t expect the food pyramid to have a pharaoh buried underneath it, either.

Our brains handle these aspects of mental represen­tations easily and usually without our being conscious of them.  They are one of the primary ways we understand the world.

It is not possible to imagine mathematics without its computational and formal aspects, but to focus exclusively on them destroys the subject.William Byers

## Images and metaphors in math

### Types of represen­tations

Mathe­maticians who work with a particular kind of mathe­matical object have mental represen­tations of that type of object that help them understand it.  These mental represen­tations come in many forms.  Most of them fit into one of the types below, but the list shouldn't be taken too seriously: Some represen­tations fit more that of these types, and some may not fit into any of them except awkwardly.

Below I list some examples. Many of them refer to the arch function, the function defined by $h(t):=25-{{(t-5)}^{2}}$.

### Visual image

#### The arch function

• You can picture the arch function in terms of its graph, which is a parabola. This visualization suggests that the function has a single maximum point that appears to occur at $t=5$. That is an example of how metaphors can suggest (but not prove) theorems.
• You can think of the arch function more physically, as like the Gateway Arch. This metaphor is suggested by the graph.

#### Interior of a shape

• The interior of a circle or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house.
• Note that you can't draw the interior of a circle without drawing the circle.
• Sometimes, the interior can be described using analytic geometry. For example, the interior of the circle $x^2+y^2=1$ is the set of points $\{(x,y)|x^2+y^2\lt1\}$
• But the "interior" metaphor is imperfect: The boundary of a real-life container such as a bucket has thickness, in contrast to the boundary of a closed curve or a sphere.
• This observation illustrates my description of a metaphor as identifying part of one situation with part of another. One aspect is emphasized; another aspect, where they may differ, is ignored.

#### Real number line

• You may think of the real numbers as lying along a straight line (the real line) that extends infinitely far in both directions.  This is both visual and a metaphor (a real number “is” a place on the real line).
• This metaphor is imperfect because you can't draw the whole real line, but only part of it. But you can't draw the whole graph of the curve $y=25-(t-5)^2$, either.

#### Continuous functions

##### No gaps

“Continuous functions don’t have gaps in the graph”. This is a visual image, and it is usually OK in calculus class.

But consider the curve defined by $y=25-(t-5)^2$ for every real $x$ except $x=1$. It is not defined at $x=1$ (and so the function is discontinuous there) but its graph looks exactly like the graph in the figure above because no matter how much you magnify it you can't see the gap.

So is there a gap or not?

This is a typical math example that teachers make up to raise your consciousness.

##### No lifting

"Continuous functions can be drawn without lifting the chalk." This is true in most familiar cases (provided you draw the graph only on a finite interval). But consider the graph of the function defined by $f(0)=0$ and $f(t)=t\sin\frac{1}{t}\ \ \ \ \ \ \ \ \ \ (0\lt t\lt 0.16)$ (see Split Definition). This curve is continuous and is infinitely long even though it is defined on a finite interval, so you can't draw it with a chalk at all, picking up the chalk or not. Note that it has no gaps.

#### Keeping concepts separate by using mental "space"

I personally use visual images to remember relationships between abstract objects, as well.  For example, if I think of three groups, two of which are isomorphic (for example $\mathbb{Z}/3$ and $\text{Alt}(3)$), I picture them as in three different places in my head with a connection between the two isomorphic ones.

### Notation

Here I give some examples of thinking of math objects in terms of the notation used to name them. There is much more about notation as mathe­matical represen­tation in these sections of abmath:

Notation is both something you visualize in your head and also a physical represen­tation of the object.  In fact notation can also be thought of as a mathe­matical object in itself: this point of view is worked out in detail in mathe­matical logic and in theoretical computing science.   If you think about what notation “really is” a lot,  you can easily get a headache…

#### Symbols

• When I think of the square root of $2$, I visualize the symbol “$\sqrt{2}$”. That is both a typographical object and a mathe­matically defined symbolic represen­tation of the square root of $2$.
• Another symbolic represen­tation of the square root of $2$ is "$2^{1/2}$". I personally don't visualize that when I think of the square root of $2$, but there is nothing wrong with visualizing it that way.
• What is dangerous is thinking that the square root of $2$ is the symbol “$\sqrt{2}$” (or "$2^{1/2}$" for that matter). The square root of $2$ is an abstract mathe­matical object given by a precise mathe­matical definition.
• One precise definition of the square root of $2$ is "the positive real number $x$ for which $x^2=2\,$". See the discussion of this in the chapter on definitions.

#### Integers

• If I mention the number "two thousand, six hundred forty six" you may visualize it as "$2646$". That is its decimal represen­tation.
• But $2646$ also has a prime factorization, namely $2\times3^3\times7^2$.
• It is wrong to think of this number as being the notation "$2646$". Some notations are useful in one way and others are useful in another way.
• For example, the prime factorization of $2646$ tells you immediately that it is divisible by $49$.
• On the other hand, the decimal representation "$2646$" tells you immediately that it is bigger than two thousand.
• See Represen­tations of natural numbers.

#### Set notation

You can think of the set containing $1$, $3$ and $5$ and nothing else as represented by its common list notation $\{1, 3, 5\}$.  But remember that $\{5, 1,3\}$ is another notation for the same set. In other words the list notation has irrelevant features – the order in which the elements are listed in this case.

### Kinetic

#### Shoot a ball straight up

• The arch function could model the height over time of a physical object, perhaps a ball shot vertically upwards on a planet with no atmosphere.
• The ball starts upward at time $t=0$ at elevation $0$, reaches an elevation of (for example) $16$ units at time $t=2$, and lands at $t=10$.
• The parabola is not the path of the ball. The ball goes up and down along the $y$-axis. The second coordinate of a point on the parabola shows the location of the ball on the $y$-axis at time $t$.
• When you think about this event, you may imagine a physical event continuing over time, not just as a picture but as a feeling of going up and down.
• This feeling of the ball going up and down is created in your mind presumably using a mirror neuron. It is connected in your mind by a physical connection to the understanding of the function that has been created as connections among some of your neurons.
• Although $h(t)$ models the height of the ball, it is not the same thing as the height of the ball.  A mathe­matical object may have a relationship in our mind to physical processes or situations but it is distinct from them.
##### Remarks
1. This example involves a picture (graph of a function). Kinetic understanding can also help with learning math that does not involve pictures.  For example, when I think of evaluating the function ${{x}^{2}}+1$ at $3$, I visualize $3$ as moving into the $x$ slot and then the resulting formula $3^2+1$ transforming itself into $10$. (Not all mathematicians visualize it this way.)
2. I make the point of emphasizing the physical existence in your brain of kinetic feelings (and all other metaphors and images) to make it clear that this whole section on images and metaphors is about objects that have a physical existence; they are not abstract ideals in some imaginary ideal space not in our world. See Thinking about thought.

### Process

It is common to think of a function as a process: you put in a number (or other object) and the process produces another number or other object. There are examples in Images and metaphors for functions.

#### Long division

Let's divide $66$ by $7$ using long division. The process consists of writing down the decimal places one by one.

1. You guess at or count on your fingers to find the largest integer $n$ for which $7n\lt66$. That integer is $9$.
2. Write down $9.$ ($9$ followed by a decimal point).
3. $66-9\times7=3$, so find the largest integer $n$ for which $7n\lt3\times10$, which is $4$.
4. Adjoin $4$ to your answer, getting $9.4$
5. $3\times10-7\times4=2$, so find the largest integer $n$ for which $7n\lt2\times10$, which is $2$.
6. Adjoin $2$ to your answer, getting $9.42$.
7. $2\times10-7\times2=6$, so find the largest integer for which $7n\lt6\times10$, which is $8$.
8. Adjoin $8$ to your answer, getting $9.428$.
9. $6\times10-7\times8=4$, so find the largest integer for which $7n\lt4\times10$, which is $5$.
10. Adjoin $5$ to your answer, getting $9.4285$.

You can continue with the procedure to get as many decimal places as you wish of $\frac{66}{7}$.

##### Remark

The sequence of actions just listed is quite difficult to follow. What is difficult is not understanding what they say to do, but where did they get the numbers? So do this exercise!

##### Exercise worth doing:

Check that the procedure above is exactly what you do to divide $66$ by $7$ by the usual method taught in grammar school:

##### Remarks
• The long division process produces as many decimal places as you have stamina for. It is likely for most readers that when you do long division by hand you have done it so much that you know what to do next without having to consult a list of instructions.
• It is a process or procedure but not what you might want to call a function. The process recursively constructs the successive integers occurring in the decimal expansion of $\frac{66}{7}$.
• When you carry out the grammar school procedure above, you know at each step what to do next. That is why is it a process.
• Instructions (5) through (10) could be written in a programming language as a while loop, grouping the instructions in pairs of commands ((5) and (6), (7) and (8), and so on). However many times you go through the while loop determines the number of decimal places you get.
• The process of long division can also be described as a formally defined computable function $F$ for which $F(n)$ is the $n$th digit in the answer.
• Each of the answers to the exercises is then a mathematical object, and that brings us to the next type of metaphor...

### Object

A particular kind of metaphor or image for a mathematical concept is that of a mathematical object that represents the concept.

#### Examples

• The number $10$ is a mathematical object. The expression "$3^2+1$" is also a mathematical object. It encapsulates the process of squaring $3$ and adding $1$, and so its value is $10$.
• The long division process above finds the successive decimal places of a fraction of integers. A program that carries out the algorithm encapsulates the process of long division as an algorithm. The result of the calculation is a mathematical object, and so is the program if it is written in a formal programming language.
• The expression "$1958$" is a mathematical object, namely the decimal represen­tation of the number $1958$. The expression "$7A6$" is the hexadecimal represen­tation of $1958$. Both represen­tations are mathematical objects with precise definitions.

Represen­tations as math objects is discussed primarily in Represen­tations and Models. The difference between represen­tations as math objects and other kinds of mental represen­tations (images and metaphors) is primarily that a math object has a precise mathematical definition. Even so, they are also mental represen­tations.

## Uses of mental represen­tations

Mental represen­tations of a concept make up what is arguably the most important part of the mathe­matician's understanding of the concept.

### Many represen­tations

Different mental represen­tations of the same kind of object help you understand different aspects of the object.

 Every important mathe­matical object has many different kinds of represen­tations and mathe­maticians typically keep more that one of them in mind at once.

But images and metaphors are also dangerous (see below).

### New concepts and old ones

We especially depend on metaphors and images to understand a math concept that is new to us .  But if we work with it for awhile, finding lots of examples, and eventually proving theorems and providing counterexamples to conjectures, we begin to understand the concept in its own terms and the images and metaphors tend to fade away from our awareness.

Then, when someone asks us about this concept that we are now experts with, we trundle out our old images and metaphors – and are often surprised at how difficult and misleading our listener finds them!

Some mathe­maticians retreat from images and metaphors because of this and refuse to do more than state the definition and some theorems about the concept. They are wrong to do this. That behavior encourages the attitude of many people that

• Mathe­maticians can’t explain things.
• Math concepts are incomprehensible or bizarre.
• You have to have a mathe­matical mind to understand math.

All three of these statements are half-truths.

There is no doubt that a lot of abstract math is hard to understand, but understanding is certainly made easier with the use of images and metaphors.

### Images and metaphors on this website

This website has many examples of useful mental represen­tations.  Usually, when a chapter discusses a particular type of mathe­matical object, say rational numbers, there will be a subhead entitled "Images and metaphors for rational numbers".  This will suggest ways of thinking about them that many have found useful.

## Two levels of images and metaphors

Images and metaphors have to be used at two different levels, depending on your purpose.

• You should expect to use rich view for understanding, applications, and coming up with proofs.
• You must limit yourself to the rigorous view when constructing and checking proofs.

Math teachers and texts typically do not make an explicit distinction between these views, and you have to learn about it by osmosis. In practice, teachers and texts do make the distinction implicitly.  They will say things like, “You can think about this theorem as …” and later saying, “Now we give a rigorous proof of the theorem.”  Abstractmath.org makes this distinction explicit in many places throughout the site.

### The rich view

The kind of metaphors and images discussed in the mental represen­tations section above make math rich, colorful and intriguing to think about.  This is the rich view of math.  The rich view is vitally important.

• It is what makes math useful and interesting.
• It helps us to understand the math we are working with.
• It suggests applications.
• It suggests approaches to proofs.
##### Example

You expect the ball whose trajectory is modeled by the function h(t) above  to slow down as it rises, so the derivative of h must be smaller at t = 4  than it is at t = 2.  A mathe­matician might even say that that is an “informal proof” that $h'(4)<h'(2)$.  A rigorous proof is given below.

### The rigorous view: inertness

When we are constructing a definition or proof, we cannot trust all those wonderful images and metaphors

• Definitions must not use metaphors.
• Definitions must be taken literally.
• Proofs must use only logical reasoning based on definitions and previously proved theorems.

For the point of view of doing proofs, math objects must be thought of as inert (or static), like your pet rock. This means they

• don’t move or change over time, and
• don't interact with other objects, even other mathe­matical objects.

• When mathe­maticians say things like, “Now we give a rigorous proof…”, part of what they mean is that they have to forget about all the color and excitement of the rich view and think of math objects as totally inert. Like, put the object under an anesthetic when you are proving something about it.
• As I wrote previously, when you are trying to understand the arch function $h(t)=25-{{(t-5)}^{2}}$, it helps to think of it as representing a ball thrown directly upward, or as a graph describing the height of the ball at time $t$ which bends over like an arch at the time when the ball stops going upward and begins to fall down.
• When you proving something about it, you must be in the frame of mind that says the function (or the graph) is all laid out in front of you, unmoving. That is what the rigorous mode requires. Note that the rigorous mode is a way of thinking, not a claim about what the arch function "really is".
• When in rigorous mode,  a mathe­matician will think of $h$ as a complete mathe­matical object all at once, not changing over time. The function is the total relationship of the input values of the input parameter $t$ to the output values $h(t)$.  It consists of a bunch of interrelated information, but it doesn’t do anything and it doesn’t change.

#### Formal proof that $h'(4)<h'(2)$

Above, I gave an informal argument for this.   The rigorous way to see that $h'(4)\lt h'(2)$ for the arch function is to calculate the derivative $h'(t)=10-2t$ and plug in 4 and 2 to get $h'(4)=10-8=2$ which is less than $h'(2)=10-4=6$.

Note the embedded phrases.

This argument picks out particular data about the function that prove the statement.  It says nothing about anything slowing down as $t$ increases.  It says nothing about anything at all changing.

#### Other examples

• The rigorous way to say that “Integers go to infinity in both directions” is something like this:  “For every integer n there is an integer k such that k < n  and an integer m such that n < m.”
• The rigorous way to say that continuous functions don’t have gaps in their graph is to use this fact about continuity: If $f$ is continuous, then for any number $a$ in the domain of $f$, then $\lim_{t\to a}f(t)=f(a)$ for $t$ close enough to $a$.
• Conditional assertions are one important aspect of mathe­matical reasoning in which this concept of unchanging inertness clears up a lot of misunderstanding.   “If… then…” in our intuition contains an idea of causation and of one thing happening before another.  But if objects are inert they don’t cause anything and if they are unchanging then “when” is meaningless. (If $n$ is even, then $n^2$ is even, but $n^2$ doesn't "become even when $n$ is even".)

The rigorous view does not apply to all abstract objects, but only to mathe­matical objects.  See abstract objects for examples.

## Metaphors and images are dangerous

The price of metaphor is eternal vigilance.--Norbert Wiener

Every mental represen­tation has flaws. Each one provides a way of thinking about an $A$ as a kind of $B$ in some respects. But the represen­tation can have irrelevant features.  People new to the subject will be tempted to think about $A$ as a kind of $B$ in inappropriate respects as well.  This is a form of cognitive dissonance. The metaphors inherent in representations can also contradict each other. That is also a form of cognitive dissonance.

It may be that most difficulties students have with abstract math are based on not knowing which aspects of a given represen­tation are applicable in a given situation.  Indeed, on not being consciously aware that in general you must restrict the applicability of the mental pictures that come with a represen­tation.

In abstractmath.org you will sometimes see this statement:  “What is wrong with this metaphor:”  (or image, or represen­tation) to warn you about the flaws of that particular represen­tation.

#### Example

The graph of the arch function $h(t)$ makes it look like the two arms going downward become so nearly vertical that the curve has vertical asymptotes.  But it does not have asymptotes. The function is defined at every point of the $x$-axis. For example, there is a point on the curve underneath the point $(999,0)$, namely $(999, -988011)$.

That graph is a picture that sits on your screen without moving. It gives a static picture of the set of points $(t,25-{{(t-5)}^{2}})$ The function $h(t)$ can also be thought of as the path of a rocket, which is moving. So the two points of view contradict each other, which I think makes it hard for some students to "see" the movement when looking at the graph.

#### Example

A set is sometimes described as a container. But consider:  the integer 3 is "in" the set of all odd integers, and it is also "in" the set $\left\{ 1,\,2,\,3 \right\}$.  How could something be in two containers at once?  (More about this Images and Metaphors for Sets and in Some Specific Sets.)

#### Example

Mathe­maticians think of the real numbers as constituting a line infinitely long in both directions, with each number as a point on the line. But this does not mean that you can think of the line as a row of points. See density of the real line.

#### Example

We commonly think of functions as machines that turn one number into another.  But this does not mean that, given any such function, we can construct a machine (or a program) that can calculate it.  For many functions, it is not only impractical to do, it is theoretically impossible to do it. They are not computable. In other words, the machine picture of a function does not apply to all functions.

### Summary

 The images and metaphors you use to think about a mathe­matical object are limited in how they apply.
 The images and metaphors you use to think about a math object cannot be safely used in a proof.

 Only definitions and previously proved theorems can be used in a proof.

## Final remarks

### Mental represen­tations are physical represen­tations

It seems likely that cognitive phenomena such as images and metaphors are physically represented in the brain as collec­tions of neurons connected in specific ways.  Research on this topic is pro­ceeding rapidly.  Perhaps someday we will learn things about how we think physi­cally that actually help us learn things about math.

In any case, thinking about mathe­matical objects as physi­cally represented in your brain (not neces­sarily completely or correctly!) wipes out a lot of the dualistic talk about ideas and physical objects as separate kinds of things.  Ideas, in partic­ular math objects, are emergent constructs in the physical brain.

“Metaphor” is used in abstractmath.org to describe a type of thought configuration.  It is an implicit conceptual identification of part of one type of situation with part of another.

Metaphors are a fundamental way we understand the world. In particular, they are a fundamental way we understand math.

#### The word “metaphor”

The word “metaphor” is also used in rhetoric as the name of a type of figure of speech.  Authors often refer to metaphor in the meaning of  thought configuration as a conceptual metaphor.  Other figures of speech, such as analogy, simile and synecdoche, correspond to conceptual metaphors as well. The quora article provides a useful explanation of the way "analogy", "simile" and "metaphor" ae used.

In the figure-of-speech literary usage of the word metaphor, "My love is like a red red rose" is a simile, not a metaphor; "My love is a red red rose" is a metaphor.

## References

### References for metaphors in general cognition:

Michael Chorost, Your brain on metaphors. Chronicle of Higher Education, September 1, 2014.

Fauconnier, G. and Turner, M., The Way We Think: Conceptual Blending And The Mind's Hidden Complexities . Basic Books, 2008.

Lakoff, G., Women, Fire, and Dangerous Things. The University of Chicago Press, 1986.

Lakoff, G. and Mark Johnson, Metaphors We Live By.  The University of Chicago Press, 1980.

Wikipedia, Conceptual metaphors.

### References for metaphors and images in math:

Byers, W., How mathe­maticians Think.  Princeton University Press, 2007.

Lakoff, G. and R. E. Núñez, Where mathe­matics Comes From. Basic Books, 2000.

Math Stack Exchange list of explanatory images in math.

Núñez, R. E., “Do Real Numbers Really Move?”  Chapter in 18 Unconventional Essays on the Nature of mathe­matics, Reuben Hersh, Ed. Springer, 2006.

Vinner, S. and Tall, D. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity.

Charles Wells, Handbook of mathe­matical Discourse.

### Other articles in abstractmath.org and gyre&gimble

Conceptual and computational

Conceptual blending

Images and metaphors for functions

Literalism

Real numbers: images and metaphors

Represen­tations and models

Sets: Metaphors and images