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Very early difficulties in studying abstract math

Introduction

There are a some difficulties that students have at the very beginning of studying abstract math that are overwhelmingly important, not because they are difficult to explain but because too many teachers don’t even know the difficulties exist, or if they do, they think they are trivial and the students should know better without being told. These difficulties cause too many students to give up on abstract math and drop out of STEM courses altogether.

I spent my entire career in math at Case Western Reserve University. I taught many calculus sections, some courses taken by math majors, and discrete math courses taken mostly by computing science majors. I became aware that some students who may have been A students in calculus essentially fell off a cliff when they had to do the more abstract reasoning involved in discrete math, and in the initial courses in abstract algebra, linear algebra, advanced calculus and logic.

That experience led me to write the Handbook of Mathematical Discourse and to create the website abstractmath.org. Abstractmath.org in particular grew quite large. It does describe some of the major difficulties that caused good students to fall of the abstraction cliff, but also describes many many minor difficulties. The latter are mostly about the peculiarities of the languages of math.

I have observed people’s use of language since I was like four or five years old. Not because I consciously wanted to — I just did. When I was a teenager I would have wanted to be a linguist if I had known what linguistics is.

I will describe one of the major difficulties here (failure to rewrite according to the definition) with an example. I am planning future posts concerning other difficulties that occur specifically at the very beginning of studying abstract math.

Rewrite according to the definition

To prove that a statement
involving some concepts is true,
start by rewriting the statement
using the definitions of the concepts.

Example

Definition

A function $f:S\to T$ is surjective if for any $t\in T$ there is an $s\in S$ for which $f(s)=t$.

Definition

For a function $f:S\to T$, the image of $f$ is the set \[\{t\in T\,|\,\text{there is an }s\in S\text{ for which }f(s)=t\}\]

Theorem

Let $f:S\to T$ be a function between sets. Then $f$ is surjective if and only if the image of $f$ is $T$.

Proof

If $f$ is surjective, then the statement “there is an $s\in S$ for which $f(s)=t$” is true for any $t\in T$ by definition of surjectivity. Therefore, by definition of image, the image of $f$ is $T$.

If the image of $f$ is $T$, then the definition of image means that there is an $s\in S$ for which $f(s)=t$ for any $t\in T$. So by definition of surjective, $f$ is surjective.

“This proof is trivial”

The response of many mathematicians I know is that this proof is trivial and a student who can’t come up with it doesn’t belong in a university math course. I agree that the proof is trivial. I even agree that such a student is not a likely candidate for getting a Ph.D. in math. But:

  • Most math students in an American university are not going to get a Ph.D. in math. They may be going on in some STEM field or to teach high school math.
  • Some courses taken by students who are not math majors take courses in which simple proofs are required (particularly discrete math and linear algebra). Some of these students may simply be interested in math for its own sake!

A sizeable minority of students who are taking a math course requiring proofs need to be told the most elementary facts about how to do proofs. To refuse to explain these facts is a disfavor to the mathematics community and adds to the fear and dislike of math that too many people already have.

These remarks may not apply to students in many countries other than the USA. See When these problems occur.

“This proof does not describe how mathematicians think”

The proof I wrote out above does not describe how I would come up with a proof of the statement, which would go something like this: I do math largely in pictures. I envision the image of $f$ as a kind of highlighted area of the codomain of $f$. If $f$ is surjective, the highlighting covers the whole codomain. That’s what the theorem says. I wouldn’t dream of writing out the proof I gave about just to verify that it is true.

More examples

Abstractmath.org and Gyre&Gimble contain several spelled-out theorems that start by rewriting according to the definition. In these examples one then goes on to use algebraic manipulation or to quote known theorems to put the proof together.

Comments

This post contains testable claims

Herein, I claim that some things are true of students just beginning abstract math. The claims are based largely on my teaching experience and some statements in the math ed literature. These claims are testable.

When these problems occur

In the United States, the problems I describe here occur in the student’s first or second year, in university courses aimed at math majors and other STEM majors. Students typically start university at age 18, and when they start university they may not choose their major until the second year.

In much of the rest of the world, students are more likely to have one more year in a secondary school (sixth form in England lasts two years) or go to a “college” for a year or two before entering a university, and then they get their bachelor’s degree in three years instead of four as in the USA. Not only that, when they do go to university they enter a particular program immediately — math, computing science, etc.

These differences may mean that the abstract math cliff occurs early in a student’s university career in the USA and before the student enters university elsewhere.

In my experience at CWRU, some math majors fall of the cliff, but the percentage of computing science students having trouble was considerably greater. On the other hand, more of them survived the discrete math course when I taught it because the discrete math course contain less abstraction and more computation than the math major courses (except linear algebra, which had a balance similar to the discrete math course — and was taken by a sizeable number of non-math majors).

References

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Functions: Metaphors, Images and Representations

Please read this post at abstractmath.org. I originally posted the document here but some of the diagrams would not render, and I haven’t been able to figure out why. Sorry for having to redirect.

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Images and metaphors in math

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



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.

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

  • So is there a gap or not?
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

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

  1. 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.)
  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.

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.

  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. 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.) 

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.

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
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.
  • 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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This work is licensed under a Creative Commons Attribution-ShareAlike 2.5 License.


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Inverse image demo revisited

This post is an update of the post Demonstrating the inverse image of a function.

To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. CDF Player works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome.

The code for the demos, with some explanatory remarks, is in the file InverseImage.nb on my ,Mathematica website. That website also includes some other examples as .cdf files.

If the diagrams don’t appear, or appear but show pink diagrams, or if the formulas in the text are too high or too low, refresh the screen.

  • The vertical red interval has the horizontal green interval(s) as inverse image.
  • You can move the sliders back and forth to move to different points on the curve. The sliders control the vertical red interval. $a$ is the lower point of the vertical red line and $b$ is the upper point.
  • As you move the sliders back and forth you will see the inverse image breaking up into a disjoint union in intervals, merging into a single interval, or disappearing entirely.
  • The arrow at the upper right makes it run automatically.
  • If you are using Mathematica, you can enter values into the boxes, but if you are using CDF Player, you can only change the number using the slider or the plus and minus incrementers.

 

This is the graph of $y=x^2-1$.

The graph of $-.5 + .5 x + .2 x^2 – .19 x^3 – .015 x^4 + .01 x^5 $

The graph of the rational function $0.5 x+\frac{1.5 \left(x^4-1\right)}{x^4+1}$

The graph of a straight line whose slope can be changed. You can design demos of other functions with variable parameters.

The graph of the sine function. The other demos were coded using the Mathematica Reduce function to get the inverse image. This one had to be done in an ad hoc way as explained in the InverseImage.nb file.

 

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Extensional and Intensional

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

 

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

 

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

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

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

What does "intensional" mean?

Philosophy

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

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

Computing science

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

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

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

What is "dry-bones rigor"?

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

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

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

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

Metaphors and intensionality

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

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Thinking about a function as a mathematical object

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

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

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

Example

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

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

You can apply operators to it

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

Like all math objects, functions may have properties

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

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

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

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

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Representations of mathematical objects

This is a long post. Notes on viewing.

About this post

A mathematical object, or a type of math object, is represented in practice in a great variety of ways, including some that mathematicians rarely think of as "representations".  

In this post you will find examples and comments about many different types of representations as well as references to the literature. I am not aware that anyone has considered all these different ideas of representation in one place before. Reading through this post should raise your consciousness about what is going on when you do math.  

This is also an experiment in exposition.  The examples are discussed in a style similar to the way a Mathematica command is discussed in the Documentation Center, using mostly nonhierarchical bulleted lists. I find it easy to discover what I want to know when it is written in that way.  (What is hard is discovering the name of a command that will do what I want.)

Types of representations

Using language

  • Language can be used to define a type of object.
  • A definition is intended to be precise enough to determine all the properties that objects of that type all have.  (Pay attention to the two uses of the word "all" in that sentence; they are both significant, in very different ways.)
  • Language can be used to describe an object, exhibiting properties without determining all properties.
  • It can also provide metaphors, making use of one of the basic tools of our brain to understand the world. 
  • The language used is most commonly mathematical English, a special dialect of English.
  • The symbolic language of mathematics (distinct from mathematical English) is used widely in calculations. Phrases from the symbolic language are often embedded in a statement in math English. The symbolic language includes among others algebraic notation and logical notation. 
  • The language may also be a formal language, a language that is mathematically defined and is thus itself a mathematical object. Logic texts generally present the first order predicate calculus as a formal language. 
  • Neither mathematical English nor the symbolic language is a formal language. Both allow irregularities and ambiguities.

Mathematical objects

The representation itself may be a mathematical object, such as:

  • A linear representation of a group. Not only are the groups mathematical objects, so is the representation.
  • An embedding of a manifold into Euclidean space. A definition given in a formal language of the first order predicate calculus of the property of commutativity of binary operations. (Thus a property can be represented as a math object.)

Visual representations

A math object can be represented visually using a physical object such as a picture, graph (in several senses), or diagram.  

  • The visual processing of our brain is our major source of knowledge of the world and takes about a fifth of the brain's processing power.  We can learn many things using our vision that would take much longer to learn using verbal descriptions.  (Proofs are a different matter.)
  • When you look at a graph (for example) your brain creates a mental representation of the graph (see below).

Mental representations

If you are a mathematician, a math object such as "$42$", "the real numbers" or "continuity" has a mental representation in your brain.  

  • In the math ed literature, such a representation is called "mental image", "concept image", "procept", or "schema".   (The word "image" in these names is not thought of as necessarily visual.) 
  • The procept or schema describe all the things that come to mind when you think about a particular math object: The definition, important theorems, visual images, important examples, and various metaphors that help you understand it. 
  • The visual images occuring in a mental schema for an object may themselves be mental representations of physical objects. The examples and theorems may be mental representations of ideas you learned from language or pictures, and so on.  The relationships between different kinds of representations get quite convoluted.

Metaphors

Conceptual metaphors are a particular kind of mental representation of an object which involve mentally associating some aspects of the objects with some aspects of something else — a physical object, an image, an action or another abstract object.

  • A conceptual metaphor may give you new insight into the object.
  • It may also mislead you because you think of properties of the other object that the math object doesn't have.
  • A graph of a function is a conceptual metaphor.
  • When you say that a point on a graph "rises as it goes from left to right" your metaphor is an action. 
  • When you say that the cosets of a normal subgroup of a group "get along" with the group multiplication, your metaphor identifies a property they have with an aspect of human behavior.

Properties of representations

A representation of a math object may or may not

  • determine it completely
  • exhibit some of its properties
  • suggest easy proofs of some theorems
  • provide a useful way of thinking about it
  • mislead you about the object's properties
  • mislead you about what is significant about the object

Examples of representations

This list shows many of the possibilities of representation.  In each case I discuss the example in terms of the two bulleted lists above. Some of the examples are reused from my previous publications.

Functions

Example (F1) "Let $f(x)$ be the function defined by $f(x)=x^3-x$."

  • This is an expression in mathematical English that a fluent reader of mathematical English will recognize gives a definition of a specific function.
  • (F1) is therefore a representation of that function.  
  • The word "representation" is not usually used in this way in math.  My intention is that it should be recognized as the same kind of object as many other representations.
  • The expression contains the formula $x^3-x$.  This is an encapsulated computation in the symbolic language of math. It allows someone who knows basic algebra and calculus to perform calculations that find the roots, extrema and inflection points of the function $f$.  
  • The word "let" suggests to the fluent reader of mathematical English that (F1) is a definition which is probably going to hold for the next chunk of text, but probably not for the whole article or book.
  • Statements in mathematical English are generally subject to conventions.  In a calculus text (F1) would automatically mean that the function had the real numbers as domain and codomain.
  • The last two remarks show that a beginner has to learn to read mathematical English. 
  • Another convention is discussed in the following diatribe.

Diatribe 

You would expect $f(x)$ by itself to mean the value of $f$ at $x$, but in (F1) the $x$ has the property of a bound variable.  In mathematical English, "let" binds variables. However, after the definition, in the text the "$x$" in the expression "$f(x)$" will be free, but the $f$ will be bound to the specific meaning.  It is reasonable to say that the term "$f(x)$" represents the expression "$x^3-x$" and that $f$ is the (temporary) name of the function. Nevertheless, it is very common to say "the function $f(x)$" to mean $f$.  

A fluent reader of mathematical English knows all this, but probably no one has ever said it explicitly to them.  Mathematical English and the symbolic language should be taught explicitly, including its peculiarities such as "the function $f(x)$".  (You may want to deprecate this usage when you teach it, but students deserve to understand its meaning.)

The positive integers

You have a mental representation of the positive integers $1,2,3,\ldots$.  In this discussion I will assume that "you" know a certain amount of math.  Non-mathematicians may have very different mental representations of the integers.

  • You have a concept of "an integer" in some operational way as an abstract object.
  • "Abstract object" needs a post of its own. Meanwhile see Mathematical Objects (abstractmath) and the Wikipedia articles on Mathematical objects and Abstract objects.
  • You have a connection in your brain between the concept of integer and the concept of listing things in order, numbering them by $1,2,3,\ldots$.
  • You have a connection in your brain between the concept of an integer and the concept of counting a finite number of objects.  But then you need zero!
  • You understand how to represent an integer using the decimal representation, and perhaps representations to other bases as well. 
  • Your mental image has the integer "$42"$ connected to but not the same as the decimal representation "42". This is not true of many students.
  • The decimal rep has a picture of the string "42" associated to it, and of course the picture of the string may come up when you think of the integer $42$ as well (it does for me — it is a an icon for the number $42$.)
  • You have a concept of the set of integers. 
  • Students need to be told that by convention "the set of integers" means the set of all integers.  This particularly applies to students whose native language does not have articles, but American students have trouble with this, too.
  • Your concept of  "the set of integers" may have the icon "$\mathbb{N}$" associated with it.  If you are a mathematician, the icon and the concept of the set of integers are associated with each other but not identified with each other.
  • For me, at least, the concept "set of integers" is mentally connected to each integer by the "element of" relation. (See third bullet below.)
  • You have a mental representation of the fact that the set of integers is infinite.  
  • This does not mean that your brain contains an infinite number of objects, but that you have a representation of infinity as a concept, it is brain-connected to the concept of the set of integers, and also perhaps to a proof of the fact that $\mathbb{N}$ is infinite.
  • In particular, the idea that the set of integers is mentally connected to each integer does not mean that the whole infinite number of integers is attached in your brain to the concept of the set of integers.  Rather, the idea is a predicate in your brain.  When it is connected to "$42$", it says "yes".  To "$\pi$" it says "No".
  • Philosophers worry about the concept of completed infinity.  It exists as a concept in your brain that interacts as a meme with concepts in other mathematicians' brains. In that way, and in that way only (as far as I am concerned) it is a physical object, in particular an object that exists in scattered physical form in a social network.

Graph of a function

This is a graph of the function $y=x^3-x$:

Graph of a cubic function

  • The graph is a physical object, either on a screen or on paper
  • It is processed by your visual system, the most powerful sensory management system in your brain
  • It also represents the graph in the mathematical sense (set of ordered pairs) of the function $y=x^3-x$
  • Both the mathematical graph and the physical graph are represented by modules in your brain, which associates the two of them with each other by a conceptual metaphor
  • The graph shows some properties of the function: inflection point, going off to infinity in a specific way, and so on.
  • These properties are made apparent (if you are knowledgeable) by means of the powerful pattern recognition system in your brain. You see them much more quickly than you can discover them by calculation.
  • These properties are not proved by the graph. Nevertheless, the graph communicates information: for example, it suggests that you can prove that there is an inflection point near $(0,0)$.
  • The graph does not determine or define the function: It is inaccurate and it does not (cannot) show all of the graph.
  • More subtle details about this graph are discussed in my post Representations 2.

Continuity

Example (C1) The $\epsilon-\delta$ definition of the continuity of a function $f:\mathbb{R}\to\mathbb{R}$ may be given in the symbolic language of math:

A function $f$ is continuous at a number $c$ if \[\forall\epsilon(\epsilon\gt0\implies(\forall x(\exists\delta(|x-c|\lt\delta\implies|f(x)-f(c)|\lt\epsilon)))\]

  • To understand (C1), you must be familiar with the notation of first order logic.  For most students, getting the notation right is quite a bit of work.  
  • You must also understand  the concepts, rules and semantics of first order logic.  
  • Even if you are familiar with all that, continuity is still a difficult concept to understand.
  • This statement does show that the concept is logically complicated. I don't see how it gives any other intuition about the concept. 

Example (C2) The definition of continuity can also be represented in mathematical English like this:

A function $f$ is continuous at a number $c$ if for any $\epsilon\gt0$ and for any $x$ there is a $\delta$ such that if $|x-c|\lt\delta$, then $|f(x)-f(c)|\lt\epsilon$. 

  • This definition doesn't give any more intuition that (C1) does.
  • It is easier to read that (C1) for most math students, but it still requires intimate familiarity with the quirks of math English.
  • The fact that "continuous" is in boldface signals that this is a definition.  This is a convention.
  • The phrase "For any $\epsilon\gt0$" contains an unmarked parenthetic insertion that makes it grammatically incoherent.  It could be translated as: "For any $\epsilon$ that is greater than $0$".  Most math majors eventually understand such things subconsciously.  This usage is very common.
  • Unless it is explicitly pointed out, most students won't notice that  if you change the phrase "for any $x$ there is a $\delta$"  to "there is a $\delta$ for any $x$" the result means something quite different.  Cauchy never caught onto this.
  • In both (C1) and (C2), the "if" in the phrase "A function $f$ is continuous at a number $c$ if…" means "if and only if" because it is in a definition.  Students rarely see this pointed out explicitly.  

Example (C3) The definition of continuity can be given in a formally defined first order logical theory

  • The theory would have to contain function symbols and axioms expressing the algebra of real numbers as an ordered field. 
  • I don't know that such a definition has ever been given, but there are various semi-automated and automated theorem-proving systems (which I know little about) that might be able to state such a definition.  I would appreciate information about this.
  • Such a definition would make the property of continuity a mathematical object.
  • An automated theorem-proving system might be able to prove that $x^3-x$ is continuous, but I wonder if the resulting proof would aid your intuition much.

Example (C4) A function from one topological space to another is continuous if the inverse of every open set in the codomain is an open set in the domain.

  • This definition is stated in mathematical English.
  • All definitions start with primitive data. 
  • In definitions (C1) – (C3), the primitive data are real numbers and the statement uses properties of an ordered field.
  • In (C4), the data are real numbers and the arithmetic operations of a topological field, along with the open sets of the field. The ordering is not mentioned.
  • This shows that a definition need not mention some important aspects of the structure. 
  • One marvelous example of this is that  a partition of a set and an equivalence relation on a set are based on essentially disjoint sets of data, but they define exactly the same type of structure.

Example (C4) "The graph of a continuous function can be drawn without picking up the chalk".

  • This is a metaphor that associates an action with the graph.
  • It is incorrect: The graphs of some continuous functions cannot be drawn.  For example, the function $x\mapsto x^2\sin(1/x)$ is continuous on the interval $[-1,1]$ but cannot be drawn at $x=0$. 
  • Generally speaking, if the function can be drawn then it can be drawn without picking up the chalk, so the metaphor provides a useful insight, and it provides an entry into consciousness-raising examples like the one in the preceding bullet.

References

  1. 1.000… and .999… (post)
  2. Conceptual blending (post)
  3. Conceptual blending (Wikipedia)
  4. Conceptual metaphors (Wikipedia)
  5. Convention (abstractmath)
  6. Definitions (abstractmath)
  7. Embodied cognition (Wikipedia)
  8. Handbook of mathematical discourse (see articles on conceptual blendmental representationrepresentationmetaphor, parenthetic assertion)
  9. Images and Metaphors (abstractmath).
  10. The interplay of text, symbols and graphics in math education, Lin Hammill
  11. Math and the modules of the mind (post)
  12. Mathematical discourse: Language, symbolism and visual images, K. L. O’Halloran.
  13. Mathematical objects (abmath)
  14. Mathematical objects (Wikipedia)
  15. Mathematical objects are “out there?” (post)
  16. Metaphors in computing science ​(post)
  17. Procept (Wikipedia)
  18. Representations 2 (post)     
  19. Representations and models (abstractmath)
  20. Representations II: dry bones (post)
  21. Representation theorems (Wikipedia) Concrete representations of abstractly defined objects.
  22. Representation theory (Wikipedia) Linear representations of algebraic structures.
  23. Semiotics, symbols and mathematical visualization, Norma Presmeg, 2006.
  24. The transition to formal thinking in mathematics, David Tall, 2010
  25. Theory in mathematical logic (Wikipedia)
  26. What is the object of the encapsulation of a process? Tall et al., 2000.
  27. Where mathematics comes from, by George Lakoff and Rafael Núñez, Basic Books, 2000. 
  28. Where mathematics comes from (Wikipedia) This is a review of the preceding book.  It is a permanent link to the version of 04:23, 25 October 2012.  The review is opinionated, partly wrong, not well written and does not fit the requirements of a Wikipedia entry.  I recommend it anyway; it is well worth reading.  It contains links to three other reviews.

Notes on Viewing  

This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

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Conceptual blending

This post uses MathJax.  If you see formulas in unrendered TeX, try refreshing the screen.

A conceptual blend is a structure in your brain that connects two concepts by associating part of one with part of another.  Conceptual blending is a major tool used by our brain to understand the world.

The concept of conceptual blend includes special cases, such as representations, images and conceptual metaphors, that math educators have used for years to understand how mathematics is communicated and how it is learned.  The Wikipedia article is a good starting place for understanding conceptual blending. 

In this post I will illustrate some of the ways conceptual blending is used to understand a function of the sort you meet with in freshman calculus.  I omit the connections with programs, which I will discuss in a separate post.

A particular function

Consider the function $h(t)=4-(t-2)^2$. You may think of this function in many ways.

FORMULA:

$h(t)$ is defined by the formula $4-(t-2)^2$.

  • The formula encapsulates a particular computation of the value of $h$ at a given value $t$.
  • The formula defines the function, which is a stronger statement than saying it represents the function.
  • The formula is in standard algebraic notation. (See Note 1)
  • To use the formula requires one of these:
    • Understand and use the rules of algebra
    • Use a calculator
    • Use an algebraic programming language. 
  • Other formulas could be used, for example $4t-t^2$.
    • That formula encapsulates a different computation of the value of $h$.

TREE: 

$h(t)$ is also defined by this tree (right).
  • The tree makes explicit the computation needed to evaluate the function.
  • The form of the tree is based on a convention, almost universal in computing science, that the last operation performed (the root) is placed at the top and that evaluation is done from bottom to top.
  • Both formula and tree require knowledge of conventions.
  • The blending of formula and tree matches some of the symbols in the formula with nodes in the tree, but the parentheses do not appear in the tree because they are not necessary by the bottom-up convention.
  • Other formulas correspond to other trees.  In other words, conceptually, each tree captures not only everything about the function, but everything about a particular computation of the function.
  • More about trees in these posts:

GRAPH:

$h(t)$ is represented by its graph (right). (See note 2.)

  • This is the graph as visual image, not the graph as a set of ordered pairs.
  • The blending of graph and formula associates each point on the (blue) graph with the value of the formula at the number on the x-axis directly underneath the point.
  • In contrast to the formula, the graph does not define the function because it is a physical picture that is only approximate.
  • But the formula does represent the function.  (This is "represents" in the sense of cognitive psychology, but not in the mathematical sense.)
  • The blending requires familiarity with the conventions concerning graphs of functions. 
  • It sets into operation the vision machinery of your brain, which is remarkably elaborate and powerful.
    • Your visual machinery allows you to see instantly that the maximum of the curve occurs at about $t=2$. 
  • The blending leaves out many things.
    • For one, the graph does not show the whole function.  (That's another reason why the graph does not define the function.)
    • Nor does it make it obvious that the rest of the graph goes off to negative infinity in both directions, whereas that formula does make that obvious (if you understand algebraic notation).  

GEOMETRIC

The graph of $h(t)$ is the parabola with vertex $(2,4)$, directrix $x=2$, and focus $(2,\frac{3}{4})$. 

  • The blending with the graph makes the parabola identical with the graph.
  • This tells you immediately (if you know enough about parabolas!) that the maximum is at $(2,4)$ (because the directrix is vertical).
  • Knowing where the focus and directrix are enables you to mechanically construct a drawing of the parabola using a pins, string, T-square and pencil.  (In the age of computers, do you care?)

HEIGHT:

$h(t)$ gives the height of a certain projectile going straight up and down over time.

  • The blending of height and graph lets you see instantly (using your visual machinery) how high the projectile goes. 
  • The blending of formula and height allows you to determing the projectile's velocity at any point by taking the derivative of the function.
  • A student may easily be confused into thinking that the path of the projectile is a parabola like the graph shown.  Such a student has misunderstood the blending.

KINETIC:

You may understand $h(t)$ kinetically in various ways.

  • You can visualize moving along the graph from left to right, going, reaching the maximum, then starting down.
    • This calls on your experience of going over a hill. 
    • You are feeling this with the help of mirror neurons.
  • As you imagine traversing the graph, you feel it getting less and less steep until it is briefly level at the maximum, then it gets steeper and steeper going down.
    • This gives you a physical understanding of how the derivative represents the slope.
    • You may have seen teachers swooping with their hand up one side and down the other to illustrate this.
  • You can kinetically blend the movement of the projectile (see height above) with the graph of the function.
    • As it goes up (with $t$ increasing) the projectile starts fast but begins to slow down.
    • Then it is briefly stationery at $t=2$ and then starts to go down.
    • You can associate these feelings with riding in an elevator.
      • Yes, the elevator is not a projectile, so this blending is inaccurate in detail.
    • This gives you a kinetic understanding of how the derivative gives the velocity and the second derivative gives the acceleration.

OBJECT:

The function $h(t)$ is a mathematical object.

  • Usually the mental picture of function-as-object consists of thinking of the function as a set of ordered pairs $\Gamma(h):=\{(t,4-(t-2)^2)|t\in\mathbb{R}\}$. 
  • Sometimes you have to specify domain and codomain, but not usually in calculus problems, where conventions tell you they are both the set of real numbers.
  • The blend object and graph identifies each point on the graph with an element of $\Gamma(h)$.
  • When you give a formal proof, you usually revert to a dry-bones mode and think of math objects as inert and timeless, so that the proof does not mention change or causation.
    • The mathematical object $h(t)$ is a particular set of ordered pairs. 
    • It just sits there.
    • When reasoning about something like this, implication statements work like they are supposed to in math: no causation, just picking apart a bunch of dead things. (See Note 3).
    • I did not say that math objects are inert and timeless, I said you think of them that way.  This post is not about Platonism or formalism. What math objects "really are" is irrelevant to understanding understanding math [sic].

DEFINITION

definition of the concept of function provides a way of thinking about the function.

  • One definition is simply to specify a mathematical object corresponding to a function: A set of ordered pairs satisfying the property that no two distinct ordered pairs have the same second coordinate, along with a specification of the codomain if that is necessary.
  • A concept can have many different definitions.
    • A group is usually defined as a set with a binary operation, an inverse operation, and an identity with specific properties.  But it can be defined as a set with a ternary operation, as well.
    • A partition of a set is a set of subsets of a set with certain properties. An equivalence relation is a relation on a set with certain properties.  But a partition is an equivalence relation and an equivalence relation is a partition.  You have just picked different primitives to spell out the definition. 
    • If you are a beginner at doing proofs, you may focus on the particular primitive objects in the definition to the exclusion of other objects and properties that may be more important for your current purposes.
      • For example, the definition of $h(t)$ does not mention continuity, differentiability, parabola, and other such things.
      • The definition of group doesn't mention that it has linear representations.

SPECIFICATION

A function can be given as a specification, such as this:

If $t$ is a real number, then $h(t)$ is a real number, whose value is obtained by subtracting $2$ from $t$, squaring the result, and then subtracting that result from $4$.

  • This tells you everything you need to know to use the function $h$.
  • It does not tell you what it is as a mathematical object: It is only a description of how to use the notation $h(t)$.

Notes

1. Formulas can be give in other notations, in particular Polish and Reverse Polish notation. Some forms of these notations don't need parentheses.

2. There are various ways to give a pictorial image of the function.  The usual way to do this is presenting the graph as shown above.  But you can also show its cograph and its endograph, which are other ways of representing a function pictorially.  They  are particularly useful for finite and discrete functions. You can find lots of detail in these posts and Mathematica notebooks:

3. See How to understand conditionals in the abstractmath article on conditionals.

References

  1. Conceptual blending (Wikipedia)
  2. Conceptual metaphors (Wikipedia)
  3. Definitions (abstractmath)
  4. Embodied cognition (Wikipedia)
  5. Handbook of mathematical discourse (see articles on conceptual blendmental representationrepresentation, and metaphor)
  6. Images and Metaphors (article in abstractmath)
  7. Links to G&G posts on representations
  8. Metaphors in Computing Science (previous post)
  9. Mirror neurons (Wikipedia)
  10. Representations and models (article in abstractmath)
  11. Representations II: dry bones (article in abstractmath)
  12. The transition to formal thinking in mathematics, David Tall, 2010
  13. What is the object of the encapsulation of a process? Tall et al., 2000.

 

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Freezing a family of functions

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook algebra1.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Some background

  • Generally, I have advocated using all sorts of images and metaphors to enable people to think about particular mathematical objects more easily.
  • In previous posts I have illustrated many ways (some old, some new, many recently using Mathematica CDF files) that you can provide such images and metaphors, to help university math majors get over the abstraction cliff.
  • When you have to prove something you find yourself throwing out the images and metaphors (usually a bit at a time rather than all at once) to get down to the rigorous view of math [1], [2], [3], to the point where you think of all the mathematical objects you are dealing with as unchanging and inert (not reacting to anything else).  In other words, dead.
  • The simple example of a family of functions in this post is intended to give people a way of thinking about getting into the rigorous view of the family.  So this post uses image-and-metaphor technology to illustrate a way of thinking about one of the basic proof techniques in math (representing the object in rigor mortis so you can dissect it).  I suppose this is meta-math-ed.  But I don’t want to think about that too much…
  • This example also illustrates the difference between parameters and variables. The bottom line is that the difference is entirely in how we think about them. I will write more about that later.

 A family of functions

This graph shows individual members of the family of functions \( y=a\sin\,x\) for various values of a. Let’s look at some of the ways you can think about this.

  • Each choice of  “shows the function for that value of the parameter a“.  But really, it shows the graph of the function, in fact only the part between x=-4 and x= 4.
  • You can also think of it as showing the function changing shape as a changes over time (as you slide the controller back and forth).

Well, you can graph something changing over time by introducing another axis for time.  When you graph vertical motion of a particle over time you use a two-dimensional picture, one axis representing time and the other the height of the particle. Our representation of the function y=a\sin\,x is a two-dimensional object (using its graph) so we represent the function in 3-space, as in this picture, where the slider not only shows the current (graph of the) function for parameter value a but also locates it over a on the z axis.

The picture below shows the surface given by y=a\sin\,x as a function of both variables a and x. Note that this graph is static: it does not change over time (no slide bar!). This is the family of functions represented as a rigorous (dead!) mathematical object.

If you click the “Show Curves” button, you will see a selection of the curves in middle diagram above drawn as functions of x for certain values of a. Each blue curve is thus a sine wave of amplitude a. Pushing that button illustrates the process going on in your mind when you concentrate on one aspect of the surface, namely its cross-sections in the x direction.

Reference [4] gives the code for the diagrams in this post, as well as a couple of others that may add more insight to the idea. Reference [5] gives similar constructions for a different family of functions.

References

  1. Rigorous view in abstractmath.org 
  2. Representations II: Dry Bones (post)
  3. Representations III: Rigor and Rigor Mortis (post)
  4. FamiliesFrozen.nb.
  5. AnotherFamiliesFrozen.nb (Mathematica file showing another family of functions)
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Case Study in Exposition: Secant

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code comes from several Mathematica notebooks lists in the References. The notebooks are available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Pictures, metaphors and etymology

Math texts and too many math teachers do not provide enough pictures and metaphors to help students understand a concept.  I suspect that the etymology of the technical terms might also be useful. This post is an experimental exposition of the math concept of “secant” that use pictures, metaphors and etymology to describe the concept.

The exposition is interlarded with comments about what I am doing and why.  An exposition directly aimed at students would be slimmer — but some explanations of why you are doing such and such in an exposition are not necessarily out of place every time!

Secant Line

The word “secant” is used in various related ways in math.  To start with, a secant line on a curve is the unique line determined by two distinct points on the curve, like this:


The word “secant” comes from the Latin word for “cut”, which came from the Indo-European root “sek”, meaning “cut”.  The IE root also came directly into English via various Germanic sound changes to give us “saw” and “sedge”.

The picture

Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept.  The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points.  You also get a very strong understanding of how the secant line is a function of the two given points.  I don’t think that is obvious to someone without some experience with such things.

This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects.  (Math books are full of such pictures.)  So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds.  This is the sort of claim that is amenable to field testing.

The metaphor

Most metaphors are based on a physical phenomenon.  The mathematical meanings of “secant” use the metaphor of cutting.  When the word “secant” was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor.   In those days essentially every European scholar read Latin. To them “secant” would transparently mean “cutting”.  This is not transparent to many of us these days, so the metaphor may be hidden.

If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.

  • The straight line does not really cut the curve.  Indeed, the curve itself is both an abstract object that is not physical, so can’t be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it?  Cut the screen?  The line can’t do that.
  • You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
  • The metaphor is restricted further by saying that it is determined by two points on the curve.   This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines.  You could define such a family by using one point on the curve and a slope, for example.  This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.

Secant on circle

Another use of the word “secant” is the red line in this picture:


This is the secant line on the unit circle determined by the origin and one point on the circle, with one difference: The secant of the angle is the line segment between the origin and the point on the curve.  This means it corresponds to a number, and that number is what we mean by “secant” in trigonometry.

To the ancient Greeks, a (positive) number was the length of a line segment.

The Definition

The secant of an angle $\theta$ is usually defined as $\frac{1}{\cos\theta}$, which you can see by similar triangles is the length of the red line in the picture above.

In many parts of the world, trig students don’t learn the word “secant”. They simply use $\frac{1}{\cos\theta}$.

This illustrates important facts about definitions:

  • Different equivalent definitions all make the same theorems true.
  • Different equivalent definitions can give you a very different understanding of the concept.

The red-line-segment-in-picture definition gives you a majorly important visual understanding of the concept of “secant”.  You can tell a lot from its behavior right off (it goes to infinity near $\pi/2$, for example).

The definition $\sec\theta=\frac{1}{\cos\theta}$ gives you a way of computing $\sec\theta$.  It also reduces the definition of $ \sec\theta$ to a previously known concept.

It used to be common to give only the $ \frac{1}{\cos\theta}$ definition of secant, with no mention of the geometric idea behind it.  That is a crime.  Yes, I know many students don’t want to “understand” stuff, they only want to know how to do the problems.  Teachers need to talk them out of that attitude.  One way to do that in this case is to test them on the geometric definition.

Etymology

This idea was known to the Arabs, and brought into European view in the 16th century by Danish mathematician Thomas Fincke in “Geometria Rotundi” (1583), where the first known use of the word “secant” occurs.  I have not checked, but I suspect from the title of the book that the geometric definition was the one he used in the book.

It wold be interesting to know the original Arabic name for secant, and what physical metaphor it is based on.  A cursory search of the internet gave me the current name in Arabic for secant but nothing else.

Graph of the secant function

The familiar graph of the secant function can be seen as generated by the angle sweeping around the curve, as in the picture below. The two red line segments always have the same length.


References

Mathematica notebooks used in this post:

 

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