About this post
This post is the new revision of the chapter on Images and Metaphors in abstractmath.org.
Images and metaphors in math
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.
Besides mental representations, there are other kinds of representations used in math, discussed in the chapter on representations and models.
Mathematics is the tinkertoy of metaphor. –Ellis D. Cooper
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.
See Thinking about thought.
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.
- 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 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
Half this game is 90% mental. –Yogi Berra
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.
- Visual
- Notation
- Kinetic
- Process
- Object
All mental representations are conceptual metaphors. Metaphors are treated in detail in this chapter and in the chapter on images and metaphors for functions. See also literalism and Proofs without dry bones on Gyre&Gimble.
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
Geometric figures
- Most people (even non-mathematicians!) can easily form a mental image of a square or a circle.
- Many people can also draw a reasonable approximation of a square or a circle, and in my opinion if you can draw it, you have a mental image of it.
- Perhaps you would say you have a good mental image of a triangle and could draw one easily. But that is a very different task from the case of squares and circles. Think about that.
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 closed curve or a sphere is called that because it is like the interior in the everyday sense of a bucket or a house.
- 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.
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 (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…
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 $x^2=2$”. Another definition is that $\sqrt{2}=\frac{1}{2}\log2$.
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 “$2646$”. Different notations have different values, and there is no mathematical reason to make “$2646$” the “genuine” representation. See representations and Models.
For example, the prime factorization of $2646$ tells you immediately that it is divisible by $49$.
When I was in high school in the 1950’s, I was taught that it was incorrect to say “two thousand, six hundred and forty six”. Being naturally rebellious I used that extra “and” in the early 1960’s in dictating some number in a telegraph message. The Western Union operator corrected me. Of course, the “and” added to the cost. (In case you are wondering, I was in the middle of a postal Diplomacy game in Graustark.)
Formulas
- You can think of the arch function as its formula $25-{{(t-5)}^{2}}.$ 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 $t=5$ (see ratchet
effect).
- Another
formula for the same function is $-{{t}^{2}}+10t$. 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 $h(t)=25-{{(t-5)}^{2}}$
and $k(t)=-{{t}^{2}}+10t$ are the same function; in other words, $h = k$.
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 $x$-axis. A point on the parabola shows it locaion on the $x$ 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 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
- This example involves a picture (graph of a function). According to this report, 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 moving into the x slot and then the formula $9^2+1$ transforming
itself into $10$. (Not all mathematicians visualize it this way.)
- 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.
I remember visualizing algebra I this way even before I had ever heard of the Transformers.
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.
- You guess at or count on your fingers to find the largest integer $n$ for which $7n\lt66$. That integer is $9$.
- Write down $9.$ ($9$ followed by a decimal point).
- $66-9\times7=3$, so find the largest integer $n$ for which $7n\lt3\times10$, which is $4$.
- Adjoin $4$ to your answer, getting $9.4$
- $3\times10-7\times4=2$, so find the largest integer $n$ for which $7n\lt2\times10$, which is $2$.
- Adjoin $2$ to your answer, getting $9.42$.
- $2\times10-7\times2=6$, so find the largest integer for which $7n\lt6\times10$, which is $8$.
- Adjoin $8$ to your answer, getting $9.428$.
- $6\times10-7\times8=4$, so find the largest integer for which $7n\lt4\times10$, which is $5$.
- 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. But do you have the procedure in your head all at once?
- Well, 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.
- It can also be described as a formally defined recursive function $F$ for which $F(n)$ is the $n$th digit in the answer.
- Each of the program and the recursive definition mentioned in the last two bullets are exercises worth doing.
- 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 is a mathematical object.
- 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.
- 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.)
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.
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But images and metaphors are also dangerous (see below).
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.
In my opinion the third statement is only about 10 percent true.
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 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.
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.
When we are constructing a definition or proof we cannot
trust all those wonderful images and metaphors.
- Definitions must
not use metaphors.
- 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).
- 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 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 the $\varepsilon-\delta $ 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
The price of metaphor is eternal vigilance.–Norbert Wiener
Every
mental representation has flaws. Each oneprovides 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 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 arms going down are underneath every point of the $x$-axis. For example, there is a point on the curve underneath the point $(999,0)$, namely $(999, -988011)$.
Example
A set is sometimes described as analogous to 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 here.)
An analogy can be helpful, but it isn’t the same thing as the same thing. – The Economist
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 href=”http://en.wikipedia.org/wiki/Recursive_function_theory#Turing_computability”>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 the subject
cannot be directly 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.
About metaphors
The language that nature speaks is mathematics. The language that ordinary human beings speak is metaphor. Freeman Dyson
“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 simile and synecdoche, correspond to conceptual metaphors as well.
References for metaphors in general cognition:
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.
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.
Charles Wells,
Handbook of mathematical Discourse.
Charles Wells, Conceptual blending. Post in Gyre&Gimble.
Other articles in abstractmath.org
Conceptual and computational
Functions: images and metaphors
Real numbers: images and metaphors
representations and models
Sets: metaphors and images
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