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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.

There are other kinds of representations used in math, discussed here.

### Contents

Images and metaphors

# Images and metaphors

We think and talk about our experiences of the world in terms of images and metaphors based on immediate physical experience.  These are our mental representations of our experiences.

#### Example of 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 constituting a pyramidal pile.  We are talking about a visual image of the pyramid.  In the study of literature, the word “image” is used in a more general way, to refer to an evocation of a sensory experience (more).

#### Examples of 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.

## Terminology

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, so there may be no direct connection with sensory experience.   In abstractmath.org, I will usually 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.

# 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.  Some of them are listed here, but don’t take the classification too seriously.

¨  Visual images, for example of what a right triangle looks like.

¨  Notation, for example visualizing the square root of 2 by the symbol ”.

¨  Kinetic understanding, for example thinking of the value of a function as approaching a limit by imagining yourself moving along the graph of the function.

¨  Metaphorical understanding, for example thinking of a function such as  as a machine that turns one number into another: for example, when you put in 3 out comes 9.

### Example

Consider the function .

#### Visual images

You can picture this function in terms of its graph, which is a parabola.   You can think of it more physically, as like the Gateway Arch.  The graph visualization suggests that the function has a single maximum point that appears to occur at .

#### Notation

You can think of the function as its formula .  The formula tells you that its graph will be a parabola (if you know that quadratics give parabolas) and it tells you instantly without calculus that its maximum will be at  (see ratchet effect).

Another formula for the same function is .   The formula is only another representation of the function.  It is not the same thing as the function.  The functions h(t) and k(t) defined on   by  and  are the same function;  in other words, h = k.

#### Kinetic

The function h(t)  could model the height over time of a physical object, perhaps a ball shot vertically upwards on a planet with no atmosphere. You could think of the ball starting at time t = 0 at elevation 0, reaching an elevation of (for example) 16 units at time t = 2, and landing at t = 10.  You are imagining a physical event continuing over time, not just as a picture but as a feeling of going up and down (see mirror neuron).  This feeling of the ball going up and down is attached in your brain to your understanding of the function h(t).

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 is distinct from them.

#### Metaphor

One metaphor for functions is that it is a machine that turns one number into another.  For example, the function h(t)  turns 0 into 0 (which is therefore a fixed point) and 5 into 25.  It also turns 10 into 0, so once you have the output you can’t necessarily tell what the input was (the function is not injective).

### More about types of representations

#### Visual images

The graph of a function is a visual image, and so is a picture of a rectangle.

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  and  ), I picture them as in three different places with a connection between the two isomorphic ones.  I know of no published research on this kind of imagery.

#### Notation

It is not wrong to think of a particular notation of a math object when you think of the object.  When I think of the square root of 2,  I think of the symbol “  ”.  But it is a very bad idea to think of the symbol as the same thing as the object. A mathematical object may be referred to by many different notations.   For example the square root of 2 can be represented as .  See a function example here and set examples here and here

Of course, in a sense 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 (common in mathematical logic and in theoretical computing science.)   If you think about what notation “really is” a lot,  you can easily get a headache…

#### Kinetic

Our example above involved a picture (graph of a function).  According to this report, kinetic understanding can also help with learning math that does not involve pictures.  I know that when I think of evaluating the function  at 3, I visualize 3 moving into the x slot and then the formula  transforming itself into 10.  I remember doing this even before I had ever heard of the Transformers.

#### Metaphor

Metaphors are treated in detail below and in the chapter on images and metaphors for functions.   See also literalism and this post on Gyre&Gimble.

### More examples

¨  “Continuous functions don’t have gaps in the graph”.   This is a visual image.

¨  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).

¨  You can think of the set containing 1, 3 and 5 and nothing else in terms of its list notation {1, 3, 5}.  But remember that {5, 1,3} is the same set.  In other words the list notation has irrelevant features  the order in which the elements are listed in this case.

¨  The interior of a closed curve or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house.  Note 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.

## Uses of mental representations

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

¨  Mental representations of mathematical objects using metaphors and images are necessary for understanding and communicating about them (especially with types of objects that are new to us).

¨  They are necessary for seeing how the theory can be applied.

¨  They are useful for coming up with proofs. (See example below.)

### 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 representations

and skilled mathematicians generally have several 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.

¨  The rich view is for understanding, applications, and coming up with proofs.

¨  You must limit yourself to the rigorous view when constructing and reading proofs.

Math teachers and texts typically do not make an explicit distinction between these views, and you have to learn about it by osmosis.  Teachers and texts do make the distinction implicitly in practice.  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.  More about this.

¨  It suggests applications.

¨  It suggests approaches to proving things about math.

##### 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 .  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 and 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.

(See also abstract object).

#### Rigorous

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.

#### Example

The ball function h(t) does not move or change.

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 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 :  Above, I gave an informal argument for this.  The rigorous way to see that  for the ball function h(t) is to calculate the derivative

and plug in 4 and 2 to get , which is less than .  (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 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 the definition of continuity.

¨  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 (see also here).  But if objects are inert they don’t cause anything and if they are unchanging then “when” is meaningless.

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

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.

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 one 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 function h(t) makes it look like the two arms going downward become so nearly vertical that the curve has vertical asymptotes.  But it deos not have asymptotes.  The arms going down spread out underneath every point of the x-axis.  For example, there is a point on the curve underneath the point (999,0), namely (999, -988,011).

#### 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 .  How could something be in two containers at once?  (More about this here.)

#### 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 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.  In other words, the machine picture of a function does not apply to all functions.

I have been told that the book by Cutland is a good starting place to learn about this, but I have not yet examined the book.

The images and metaphors you use to think about a mathematical object

are limited in how they apply

Mental representations can suggest proofs, but they cannot be used in proofs. Only definitions and previously proved theorems can be used in a proof.

The images and metaphors you use to think about the subject
cannot be directly used in a proof.

# Appendices

## 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.

## About metaphors

“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, and 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 simile and synecdoche, correspond to conceptual metaphors as well.

#### References for metaphors in general cognition:

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.

Turner, M., Reading Minds: The Study of English in the Age of Cognitive Science. Princeton University Press.  1991

#### References for metaphors in math:

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

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

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

### Instantly?

is a square, so its smallest possible value is 0, which happens when .  This means the largest possible value of  occurs at