Category Archives: understanding math

Demos for graph and cograph of calculus functions

The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers but not necessarily on many smaller devices.

This post provides interactive examples of the endograph and cograph of real functions. Those two concepts were defined and discussed in the previous post Endograph and cograph of real functions.

Such representations of functions, put side by side with the conventional graph, may help students understand how to interpret the usual graph representation. For example: What does it mean when the arrows slant to the left? spread apart? squeeze together? flip over? Going back and forth between the conventional graph and the cograph or engraph for a particular function should make you much more in tune to the possibilities when you see only the conventional graph of another function.

This is not a major advance for calculus teachers, but it may be a useful tool. The source code is the Mathematica Notebook GraphCograph.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License.

Line segment

$y=a x+b$


Quadratic

$y=ax^2+b$


Quadratic and its derivative

$y=a x^2$ (blue) and $y=2 a x$ (red)

Cubic

$y=a x^3-b x$

Sine

$y=a \sin b x$

Sine and its derivative

$y=\sin a x$ (blue) and $y=a\cos x$ (red)

Quintic with three parameters

$y=a x^5-b
x^4-0.21 x^3+0.2 x^2+0.5 x-c$

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.

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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 #mentalrepresen­tations>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.


Pattern recognition in understanding math

Abstract patterns

This post is a revision of the article on pattern recognition in abstractmath.org.

When you do math, you must recognize abstract patterns that occur in

  • Symbolic expressions
  • Geometric figures
  • Relations between different kinds of math structures.
  • Your own mental representations of mathematical objects

This happens in high school algebra and in calculus, not just in the higher levels of abstract math.

Examples

Most of these examples are revisited in the section called Laws and Constraints.

At most

For real numbers $x$ and $y$, the phrase “$x$ is at most $y$” means by definition $x\le y$. To understand this definition requires recognizing the pattern “$x$ is at most $y$” no matter what expressions occur in place of $x$ and $y$, as long as they evaluate to real numbers.

Examples

  • “$\sin x$ is at most $1$” means that $\sin x\le 1$. This happens to be true for all real $x$.
  • “$3$ is at most $7$” means that $3\leq7$. You may think that “$3$ is at most $7$” is a silly thing to say, but it nevertheless means that $3\leq7$ and so is a correct statement.
  • “$x^2+(y-1)^2$ is at most $5$” means that
    $x^2+(y-1)^2\leq5$. This is true for some pairs $(x,y)$ and false for others, so it is a constraint. It defines the disk below:

The product rule for derivatives

The product rule for differentiable functions $f$ and $g$ tells you that the derivative of $f(x)g(x)$ is \[f'(x)\,g(x)+f(x)\,g'(x)\]

Example

You recognize that the expression ${{x}^{2}}\sin x$ fits the pattern $f(x)g(x)$ with $f(x)={{x}^{2}}$ and $g(x)=\sin x$. Therefore you know that the derivative of ${{x}^{2}}\,\sin x$ is \[2x\sin x+{{x}^{2}}\cos x\]

The quadratic formula

The quadratic formula for the solutions of an equation of the form $a{{x}^{2}}+bx+c=0$ is usually given as\[r=\frac{-b\pm
\sqrt{{{b}^{2}}-4ac}}{2a}\]

Example

If you are asked for the roots of $3{{x}^{2}}-2×-1=0$, you recognize that the polynomial on the left fits the pattern $a{{x}^{2}}+bx+c$ with

  • $a\leftarrow3$ (“$a$ replaced by $3$”)
  • $b\leftarrow-2$
  • and $c\leftarrow-1$.

Then
substituting those values in the quadratic formula gives you the roots $-1/3$ and $1$.

Difficulties with the quadratic formula

A little problem

The quadratic formula is easy to use but it can still cause pattern recognition problems. Suppose you are asked to find the solutions of $3{{x}^{2}}-7=0$. Of course you can do this by simple algebra — but pretend that the first thing you thought of was using the quadratic formula.

  • Then you got upset because you have to apply it to $a{{x}^{2}}+bx+c$
  • and $3{{x}^{2}}-7$ has only two terms
  • but $a{{x}^{2}}+bx+c$ has three terms…
  • (Help!)
  • Do Not Be Anguished:
  • Write
    $3{{x}^{2}}-7$ as $3{{x}^{2}}+0\cdot x-7$, so $a=3$, $b=0$ and $c=-7$.
  • Then put those values into the quadratic formula and you get $x=\pm \sqrt{\frac{7}{3}}$.   
  • This is an example of the following useful principle:


    Write zero cleverly.

    I suspect that most people reading this would not have had the problem with $3{{x}^{2}}-7$ that I have just described. But before you get all insulted, remember:


    The thing about really easy examples is that they give you the point without getting you lost in some complicated stuff you don’t understand very well.

    A fiendisher problem

      Even college students may have trouble with the following problem (I know because I have tried it on them):

    What are the solutions of the equation $a+bx+c{{x}^{2}}=0$?

    The answer

             

    \[r=\frac{-b\pm
    \sqrt{{{b}^{2}}-4ac}}{2a}\]

    is wrong. The correct answer is

                                     \[r=\frac{-b\pm
    \sqrt{{{b}^{2}}-4ac}}{2c}\]


    When you remember a pattern with particular letters in it and an example has some of the same letters in it, make sure they match the pattern!

    The substitution rule for integration

    The chain rule says that the derivative of a function of the form $f(g(x))$ is $f'(g(x))g'(x)$. From this you get the substitution rule for finding indefinite integrals:

                                      \[\int{f'(g(x))g'(x)\,dx}=f(g(x))+C\]

    Example

    To find $\int{2x\,\cos
    ({{x}^{2}})\,dx}$, you recognize that you can take $f(x)=\sin x$and $g(x)={{x}^{2}}$ in the formula, getting \[\int{2x\,\cos ({{x}^{2}})\,dx}=\sin ({{x}^{2}})\]    Note that in the way I wrote the integral, the functions occur in the opposite order from the pattern. That kind of thing happens a lot.

    Laws and constraints

    • The statement “$(x+1)^2=x^2+2x+1$” is a pattern that is true for all numbers $x$. $3^2=2^2+2\times2+1$ and $(-2)^2=(-1)^2+2\times(-1)+1$, and so on. Such a pattern is a universal assertion, so it is a theorem. When the statement is an equation, as in this case, it is also called a law.
    • The statement “$\sin x\leq 1$” is also true for all $x$, and so is a theorem.
    • The statement “$x^2+(y-1)^2$ is at most $5$” is true for some real numbers and not others, so it is not a theorem, although it is a constraint.
    • The quadratic formula says that:
      The solutions of an equation
      of the form $a{{x}^{2}}+bx+c=0$ is
      given by\[r=\frac{-b\pm
      \sqrt{{{b}^{2}}-4ac}}{2a}\]

      This is true for all complex numbers $a$, $b$, $c$.
      The $x$ in the equation is not a free variable, but a “variable to be solved for” and does not appear in the quadratic formula. Theorems like the quadratic formula are usually called “formulas” rather than “laws”.

    • The product rule for derivatives

      The derivative of $f(x)g(x)$ is $f'(x)\,g(x)+f(x)\,g'(x)$

      is true for all differentiable functions $f$ and $g$. That means it is true for both of these choices of $f$ and $g$:

      • $f(x)=x$ and $g(x)=x\sin x$
      • $f(x)=x^2$ and $g(x)=\sin x$

      But both choices of $f$ and $g$ refer to the same function $x^2\sin x$, so if you apply the product rule in either case you should get the same answer. (Try it).

    Some bothersome types of pattern recognition

    Dependence on conventions

    Definition: A quadratic polynomial in $x$is an expression of the form $a{{x}^{2}}+bx+c$.   

    Examples

    • $-5{{x}^{2}}+32×-5$ is a quadratic polynomial: You have to recognize that it fits the pattern in the definition by writing it as $(-5){{x}^{2}}+32x+(-5)$
    • So is ${{x}^{2}}-1$: You have to recognize that it fits the definition by writing it as ${{x}^{2}}+0\cdot x+(-1)$ (I wrote zero cleverly).

    Some authors would just say, “A quadratic polynomial is an expression of the form $a{{x}^{2}}+bx+c$” leaving you to deduce from conventions on variables that it is a polynomial in $x$ instead of in $a$ (for example).

    Note also that I have deliberately not mentioned what sorts of numbers $a$, $b$, $c$ and $x$ are. The authors may assume that you know they are using real numbers.

    An expression as an instance of substitution

    One particular type of pattern recognition that comes up all the time in math is recognizing that a given expression is an instance of a substitution into a known expression.

    Example

    Students are sometimes baffled when a proof uses the fact that ${{2}^{n}}+{{2}^{n}}={{2}^{n+1}}$ for positive integers $n$. This requires the recognition of the patterns $x+x=2x$ and $2\cdot
    \,{{2}^{n}}={{2}^{n+1}}$.

    Similarly ${{3}^{n}}+{{3}^{n}}+{{3}^{n}}={{3}^{n+1}}$.

    Example

    The assertion

    \[{{x}^{2}}+{{y}^{2}}\ge 0\ \ \ \ \ \text{(1)}\]

    has as a special case

    \[(-x^2-y^2)^2+(y^2-x^2)^2\ge
    0\ \ \ \ \ \text{(2)}\]

    which involves the substitutions $x\leftarrow -{{x}^{2}}-{{y}^{2}}$ and $y\leftarrow
    {{y}^{2}}-{{x}^{2}}$.

    Remarks
    • If you see (2) in a text and the author blithely says it is “never negative”, that is because it is of the form \[{{x}^{2}}+{{y}^{2}}\ge 0\] with certain expressions substituted for $x$ and $y$. (See substitution and The only axiom for algebra.)
    • The fact that there are minus signs in (2) and that $x$ and $y$ play different roles in (1) and in (2) are red herrings. See ratchet effect and variable clash.
    • Most people with some experience in algebra would see quickly that (2) is correct by using chunking. They would visualize (2) as

      \[(\text{something})^2+(\text{anothersomething})^2\ge0\]
      This shows that in many cases


      chunking is a psychological inverse to substitution

    • Note that when you make these substitutions you have to insert appropriate parentheses (more here). After you make the substitution, the expression of course can be simplified a whole bunch, to

      \[2({{x}^{4}}+{{y}^{4}})\ge0\]

    • A common cause of error in doing this (a mistake I make sometimes) is to try to substitute and simplify at the same time. If the situation is complicated, it is best to

      substitute as literally as possible and then simplify

    Integration by Parts

    The rule for integration by parts says that

                             \[\int{f(x)\,g'(x)\,dx=f(x)\,g(x)-\int{f'(x)\,g(x)\,dx}}\]

    Suppose you need to find $\int{\log x\,dx}$.(In abstractmath.org, “log” means ${{\log }_{e}}$).  Then we can recognize this integral as having the pattern for the left side of the parts formula with $f(x)=1$ and $g(x)=\log \,x$. Therefore

    \[\int{\log x\,dx=x\log x-\int{\frac{1}{x}dx=x\log \,x-x+c}}\]

    How on earth did I think to recognize $\log x$ as $1\cdot \log x$??  
    Well, to tell the truth because some nerdy guy (perhaps I should say some other nerdy guy) clued me in when I was taking freshman calculus. Since then I have used this device lots of times without someone telling me — but not the first time.

    This is an example of another really useful principle:


    Write $1$ cleverly.

    Two different substitutions give the same expression

    Some proofs involve recognizing that a symbolic expression or figure fits a pattern in two different ways. This is illustrated by the next two examples. (See also the remark about the product rule above.) I have seen students flummoxed by Example ID, and Example ISO is a proof that is supposed to have flummoxed medieval geometry students.

    Example ID

    Definition: In a set with an associative binary operation and an identity element $e$, an element $y$ is the inverse of an element $x$ if

    \[xy=e\ \ \ \ \text{and}\ \ \ \ yx=e \ \ \ \ (1)\]

    In this situation, it is easy to see that $x$ has only one inverse: If $xy=e$ and $xz=e$ and $yx=e$ and $zx=e$, then \[y=ey=(zx)y=z(xy)=ze=z\]

    Theorem: ${{({{x}^{-1}})}^{-1}}=x$.

    Proof: I am given that ${{x}^{-1}}$ is the inverse of $x$, By definition, this means that

    \[x{{x}^{-1}}=e\ \ \ \text{and}\ \ \ {{x}^{-1}}x=e \ \ \ \ (2)\]

    To prove the theorem, I must show that $x$ is the inverse of ${{x}^{-1}}$. Because $x^{-1}$ has only one inverse, all we have to do is prove that

    \[{{x}^{-1}}x=e\ \ \ \text{and}\ \ \ x{{x}^{-1}}=e\ \ \ \ (3)\]  

    But (2) and (3) are equivalent! (“And” is commutative.)

    Example ISO

    This sort of double substitution occurs in geometry, too.

    Theorem: If a triangle has two equal angles, then it has two equal sides.

    Proof: In the figure, assume $\angle ABC=\angle ACB$. Then triangle $ABC$ is congruent to triangle $ACB$ since the sides $BC$ and $CB$ are equal (they are the same line segment!) and the adjoining angles are equal by hypothesis.

    The point is that although triangles $ABC$ and $ACB$ are the same triangle, and sides $BC$ and $CB$ are the same line segment, the proof involves recognizing them as geometric figures in two different ways.

    This proof (not Euclid’s origi­nal proof) is hundreds of years old and is called the pons asinorum (bridge of donkeys). It became famous as the first theorem in Euclid’s books that many medi­eval stu­dents could not under­stand. I con­jecture that the name comes from the fact that the triangle as drawn here resembles an ancient arched bridge. These days, isos­ce­les tri­angles are usually drawn taller than they are wide.

    Technical problems in carrying out pattern matching

    Parentheses

    In matching a pattern you may have to insert parentheses. For example, if you substitute $x+1$ for $a$, $2y$ for
    $b$ and $4$ for $c$ in the expression \[{{a}^{2}}+{{b}^{2}}={{c}^{2}}\] you get \[{{(x+1)}^{2}}+4{{y}^{2}}=16\]
    If you did the substitution literally without editing the expression so that it had the correct meaning, you would get \[x+{{1}^{2}}+2{{y}^{2}}={{4}^{2}}\] which is not the result of performing the substitution in the expression ${{a}^{2}}+{{b}^{2}}={{c}^{2}}$.   

    Order switching

    You can easily get confused if the patterns involve a switch in the order of the variables.

    Notation for integer division

    • For integers $m$ and $n$, the phrase “$m$ divides $n$” means there is an integer $q$ for which $n=qm$.
    • In number theory (which in spite of its name means the theory of positive integers) the vertical bar is used to denote integer division. So $3|6$ because $6=2\times 3$ ($q$ is $2$ in this case). But “$3|7$” is false because there is no integer $q$ for which $7=q\times 3$.
    • An equivalent definition of division says that $m|n$ if and only if $n/m$ is an integer. Note that $6/3=2$, an integer, but $7/3$ is not an integer.
    • Now look at those expressions:
    • “$m|n$” means that there is an integer $q$ for which $n=qm$.In these two expressions, $m$ and $n$ occur in opposite order.
    • “$m|n$” is true only if $n/m$ is an integer. Again, they are in opposite order. Another way of writing $n/m$ is $\frac{n}{m}$. When math people pronounce “$\frac{n}{m}$” they usually say, “$n$ over $m$” using the same order.
  • I taught these notation in courses for computer engineering and math majors for years. Some of the students stayed hopelessly confused through several lectures and lost points repeatedly on homework and exams by getting these symbols wrong.
  • The problem was not helped by the fact that “$|$” and “$/$” are similar but have very different syntax:

    Math notation gives you no clue which symbols are operators (used to form expressions) and which are verbs (used to form assertions).

  • A majority of the students didn’t have so much trouble with this kind of syntax. I have noticed that many people have no sense of syntax and other people have good intuitive understanding of syntax. I suspect the second type of people find learning foreign languages easy.
  • Many of the articles in the references below concern syntax.
  • References

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


    The only axiom of algebra

    This is one of a series of posts I am writing to help me develop my thoughts about how particular topics in my book Abstracting Algebra (“AbAl“) should be organized. This post concerns the relation between substitution and evaluation that essentially constitutes the definition of algebra. The Mathematica code for the diagrams is in Subs Eval.nb.

    Substitution and evaluation

    This post depends heavily on your understanding of the ideas in the post Presenting binary operations as trees.

    Notation for evaluation

    I have been denoting evaluation of an expression represented as a tree like this:



    In standard algebra notation this would be written:\[(6-4)-1=2-1=1\]

    Comments

    This treatment of evaluation is intended to give you an intuition about evaluation that is divorced from the usual one-dimensional (well, nearly) notation of standard algebra. So it is sloppy. It omits fine points that will have to be included in AbAl.

    • The evaluation goes from bottom up until it reaches a single value.
    • If you reach an expression with an empty box, evaluation stops. Thus $(6-3)-a$ evaluates only to $3-a$.
    • $(6-a)-1$ doesn’t evaluate further at all, although you can use properties peculiar to “minus” to change it to $5-a$.
    • I used the boxed “1” to show that the value is represented as a trivial tree, not a number. That’s so it can be substituted into another tree.

    Notation for substitution

    I will use a configuration like this

    to indicate the data needed to substitute the lower tree into the upper one at the variable (blank box). The result of the substitution is the tree

    In standard algebra one would say, “Substitute $3\times 4$ for $a$ in the expression $a+5$.” Note that in doing this you have to name the variable.

    Example

    “If you substitute $12$ for $a$ in $a+5$ you get $12+5$”:

    results in

    Example

    “If you substitute $3\times 4$ for $a$ in $a+b$ you get $3\times4+b$”:

    results in

    Comments

    Like evaluation, this treatment of substitution omits details that will have to be included in AbAl.

    • You can also substitute on the right side.
    • Substitution in standard algebraic notation often requires sudden syntactic changes because the standard notation is essentially two-dimensional. Example: “If you substitute $3+ 4$ for $a$ in $a\times b$ you get $(3+4)\times b$”.
    • The allowed renaming of free variables except when there is a clash causes students much trouble. This has to be illustrated and contrasted with the “binop is tree” treatment which is context-free. Example: The variable $b$ in the expression $(3\times 4)+b$ by itself could be changed to $a$ or $c$, but in the sentence “If you substitute $3+ 4$ for $a$ in $a\times b$ you get $(3+4)\times b$”, the $b$ is bound. It is going to be difficult to decide how much of this needs explaining.

    The axiom

    The Axiom for Algebra says that the operations of substitution and evaluation commute: if you apply them in either order, you get the same resulting tree. That says that for the current example, this diagram commutes:

    The Only Axiom for Algebra

    In standard algebra notation, this becomes:

    • Substitute, then evaluate: If $a=3\times 4$, then $a+5=3\times 4+5=12+5$.
    • Evaluate, then substitute: If $a=3\times 4$, then $a=12$, so $a+5=12+5$.

    Well, how underwhelming. In ordinary algebra notation my so-called Only Axiom amounts to a mere rewording. But that’s the point:


    The Only Axiom of Algebra is what makes algebraic manipulation work.

    Miscellaneous comments

    • In functional notation, the Only Axiom says precisely that $\text{eval}∘\text{subst}=\text{subst}∘(\text{eval},\text{id})$.
    • The Only Axiom has a symmetric form: $\text{eval}∘\text{subst}=\text{subst}∘(\text{id},\text{eval})$ for the right branch.
    • You may expostulate: “What about associativity and commutativity. They are axioms of algebra.” But they are axioms of particular parts of algebra. That’s why I include examples using operations such as subtraction. The Only Axiom is the (ahem) only one that applies to all algebraic expressions.
    • You may further expostulate: Using monads requires the unitary or oneidentity axiom. Here that means that a binary operation $\Delta$ can be applied to one element $a$, and the result is $a$. My post Monads for high school III. shows how it is used for associative operations. The unitary axiom is necessary for representing arbitrary binary operations as a monad, which is a useful way to give a theoretical treatment of algebra. I don’t know if anyone has investigated monads-without-the-unitary-axiom. It sounds icky.
    • The Only Axiom applies to things such as single valued functions, which are unary operations, and ternary and higher operations. They also apply to algebraic expressions involving many different operations of different arities. In that sense, my presentation of the Only Axiom only gives a special case.
    • In the case of unary operations, evaluation is what we usually call evaluation. If you think about sets the way I do (as a special kind of category), evaluation is the same as composition. See “Rethinking Set Theory”, by Tom Leinster, American Mathematical Monthly, May, 2014.
    • Calculus functions such as sine and the exponential are unary operations. But not all of calculus is algebra, because substitution in the differential and integral operators is context-sensitive.

    References

    Preceding posts in this series

    Remarks concerning these posts
    • Each of the posts in this series discusses how I will present a small part of AbAl.
    • The wording of some parts of the posts may look like a first draft, and such wording may indeed appear in the text.
    • In many places I will talk about how I should present the topic, since I am not certain about it.

    Other references

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


    Presenting binops as trees

    Binary operations as trees

    This is one of a series of posts I am writing to help me develop my thoughts about how particular topics in my book Abstracting Algebra (“AbAl“) should be organized. In some parts, I present various options that I have not decided between.

    This post concerns the presen­ta­tion of binary operations as trees. The Mathematica code for the diagrams is in Substitution in algebra.nb

    Binary operations as functions

    A binary operation or binop $\Delta$ is a function of two variables whose value at $(a,b)$ is traditionally denoted by $a\Delta b$. Most commonly, the function is restricted to having inputs and outputs in the same set. In other words, a binary operation is a function $\Delta:S\times S\to S$ defined on some set $S$. $S$ is the underlying set of the operation. For now, this will be the definition, although binops may be generalized to multiple sets later in the book.

    In AbAl:

    • Binops will be defined as functions in the way just described.
    • Algebraic expressions will be represented
      as trees, which exhibit more clearly the structure of the expressions that is encoded in algebraic notation.
    • They will also be represented using the usual infix expressions such as “$3\times 5$” and “$3-5$”,

    Fine points

    The definition of a binop as a function has termi­no­logical consequences. The correct point of view concerning a function is that it determines its domain and its codomain. In particular:


    A binary operation determines its underlying set.

    Thus if we talk about an arbitrary binop $\Delta$, we don’t have to give a name to its underlying set. We can just say “the underlying set of $\Delta$” or “$U(\Delta)$”.

    Examples

    “$+$” is not one binary operation.

    • $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a binary operation.
    • $+:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is another binary operation.

    Mathematicians commonly refer to these particular binops as “addition on the integers” and “addition on the reals”.

    Remark

    You almost never see this attitude in textbooks on algebra. It is required by both category theory and type theory, two Waves flooding into math. Category theory is a middle-aged Wave and type theory, in the version of homo­topy type theory, is a brand new baby Wave. Both Waves have changed and will change our under­standing of math in deep ways.

    Trees

    An arbitrary binop $\Delta$ can be represented as a binary tree in this way:

    generic binop

    This tree represents the expression that in standard algebraic notation is “$a\Delta b$”.

    In more detail, the tree is an ordered rooted binary tree. The “ordered” part means that the leaves (nodes with no descendants) are in a specific left to right order. In AbAl, I will define trees in some detail, with lots of pictures.

    The root shows the operation and the two leaves show elements of the underlying set. I follow the custom in computing science to put the root at the top.

    Metaphors should not dictate your life by being taken literally.

    Remark

    The Wikipedia treatment of trees is scat­tered over many articles and they almost always describe things mostly in words, not pictures. Describing math objects in words when you could use pictures is against my religion. Describing is not the same as defining, which usually requires words.

    Some concrete examples:



        
        

    3trees

    These are represen­ta­tions of the expressions “$3+5$”, “$3\times5$”, and “$3-5$”.

    Just as “$5+3$” is a different expression from “$3+5$”, the left tree in 3trees above is a different expression from this one:



        

    switch

    They have the same value, but they are distinct as expressions — otherwise, how could you state the commutative law?

    Fine points

    I regard an expression as an abstract math object that can have many repre­sentations. For example “$3+5$” and the left tree in 3trees are two different represen­ta­tions of the same (abstract) expression. This deviates from the usual idea that “expression” refers to a typographical construction.

    In previous posts, when the operation is not commutative, I have sometimes labeled the legs like this:


    I have thought about using this notation consistently in AbAl, but I suspect it would be awkward in places.

    Evaluation and substitution


    The two basic operations on algebraic expressions
    are evaluation and substitution.

    They and the Only Axiom of Algebra, which I will discuss in a later post, are all that is needed to express the true nature of algebra.

    Evaluation

    • If you evaluate $3+5$ you get $8$.
    • If you evaluate $3\times 5$ you get $15$.
    • If you evaluate $3-5$ you get $-2$.

    I will show evaluation on trees like this:




    Evaluation with trace

    A more elaborate version, valuation with trace, would look like this. This allows you to keep track of where the valuations come from.




    You could also keep track of the operation used at each node. An interactive illustration of this is in the post Visible algebra I supplement. That illustration requires CDF Player to be installed on your computer. You can get it free from the Mathematica website.

    Variables

    In the tree above, the $a$ and $b$ are variables, just as they are in the equivalent expression $a\Delta b$. Algebra beginners have a hard time understanding variables.

    • You can’t evaluate an expression until you substitute numbers for the letters, which produces an instance of expression. (“Instance” is the preferable name for this, but I often refer to such a thing as an “example”.)
    • If a variable is repeated you have to substitute the same value for each occurrence. So $a\Delta b$ is a different expression from $a\Delta a$: $2+3$ is an instance of $a+b$ but it is not an instance of $a+a$. But $a\Delta a$ and $b\Delta b$ are the same expression: any instance of one is an instance of the other.
    • Substitute $a\Delta b$ for $a$ in $a\Delta b$ and you get $(a\Delta b)\Delta b$. You may have committed variable clash. You might have meant $(a\Delta b)\Delta c$. (Somebody please tell me a good link that describes variable clash.)
    • Later, you will deal with multiplication tables for algebraic structures. There the elements are denoted by letters of the alphabet. They can’t be substituted for.

    Empty boxes

    A straightforward way to denote variables would be to use empty boxes:

    The idea is that a number (element of the underlying set) can be inserted in each box. If $3$ (left) and $5$ (right) are placed in the boxes, evaluation would place the value of $3\Delta5$ in the root. Each empty box represents a separate variable.

    Empty boxes could also be used in the standard algebraic notation: $\Delta$ or $+$ or $-$.
    I have seen that notation in texts explaining variables, but I don’t know a reference. I expect to use this notation with trees in AbAl.

    To achieve the effect of one variable in two different places, as in

    we can cause it to repeat, as below, where “$\text{id}$” denotes the identity function on the underlying set:

    To evaluate at a number (member of the underlying set) you insert a number into the only empty box

    which evaluates to

    which of course evaluates to $3\Delta3$.

    This way of treating repeated variables exhibits the nature of repeated variables explicitly and naturally, putting the values automatically in the correct places. This process, like everything in this section, comes from monad theory. It also reminds me of linear logic in that it shows that if you want to use a value more than once you have to copy it.

    Substitution

    Given two binary trees



          

    you could attach the root of the first one to one of the leaves of the second one, in two different ways, to get these trees:



          


    2trees

    which in standard algebra notation would be written $(a-b)-c)$ and $a-(b-c)$ respectively. Note that this tree



    would be represented in algebra as $(a-b)-b$.

    In general, substituting a tree for an input (variable or empty box) consists of replacing the empty box by the whole tree, identifying the root of the new tree with the empty box. In graph theorem, “substitution” may be called “grafting”, which is a good metaphor.

    You can evaluate the left tree in 2trees at particular numbers to evaluate it in two stages:



    Of course, evaluating the right one at the same values would give you a different answer, since subtraction is not associative. Here is another example:


    Binary trees in general

    By repeated substitution, you can create general binary trees built up of individual trees of this form:

    In AbAl I will give examples of such things and their counterparts in algebraic notation. This will include binary trees involving more than one binop, as well. I showed an example in the previous post, which example I repeat here:

    It represents the precise unsimplified expression

    \[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\]

    Some of the operations in that tree are associative and commutative, which is why the expression can be simplified. The collection of all (finite) binary trees built out of a single binop with no assumption that it satisfies laws (associative, commutative and so on) is the free algebra on that binary operation. It is the mother of all binary operations, so it plays the same role for an arbitrary binop that the set of lists plays for associative operations, as described in Monads for High School III: Algebras. All this will be covered in later chapters of AbAl.

    References

    Preceding posts in this series

    Other references

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    Presenting binary operations

    This is the first of a set of notes I am writing to help me develop my thoughts about how particular topics in my book Abstracting algebra should be organized. This article describes my plan for the book in some detail. The present post has some thoughts about presenting binary operations.

    Before binary operations are introduced

    Traditionally, an abstract algebra book assumes that the student is familiar with high school algebra and will then proceed with an observation that such operations as $+$ and $\times$ can be thought of as functions of two variables that take a number to another number. So the first abstract idea is typically the concept of binary operation, although in another post I will consider whether that really should be the first abstract concept.

    The Abstracting Algebra book will have a chapter that presents concrete examples of algebraic operations and expressions on numbers as in elementary school and as in high school algebra. This section of the post outlines what should be presented there. Each subsection needs to be expanded with lots of examples.

    In elementary school

    In elementary school you see expressions such as

    • $3+4$
    • $3\times 4$
    • $3-4$

    The student invariably thinks of these expressions as commands to calculate the value given by the expression.

    They will also see expressions such as
    \[\begin{equation}
    \begin{array}[b]{r}
    23\\
    355\\
    + 96\\
    \hline
    \end{array}
    \end{equation}\]
    which they will take as a command to calculate the sum of the whole list:
    \[\begin{equation}
    \begin{array}[b]{r}
    23\\
    355\\
    + 96\\
    \hline
    474
    \end{array}
    \end{equation}\]

    That uses the fact that addition is associative, and the format suggests using the standard school algorithm for adding up lists. You don’t usually see the same format with more than two numbers for multiplication, even though it is associative as well. In some elementary schools in recent years students are learning other ways of doing arithmetic and in particular are encouraged to figure out short cuts for problems that allow them. But the context is always “do it”, not “this represents a number”.

    Algebra

    In algebra you start using letters for numbers. In algebra, “$a\times b$” and “$a+b$” are expressions in the symbolic language of math, which means they are like noun phrases in English such as “My friend” and “The car I bought last week and immediately totaled” in that both are used semantically as names of objects. English and the symbolic language are both languages, but the symbolic language is not a natural language, nor is it a formal language.

    Example

    In beginning algebra, we say “$3+5=8$”, which is a (true) statement.

    Basic facts about this equation:

    The expressions “$3+5$” and “$8$”

    • are not the same expression
    • but in the standard semantics of algebra they have the same meaning
    • and therefore the equation communicates information that neither “$3+5$” nor “$8$” communicate.

    Another example is “$3+5=6+2$”.

    Facts like this example need to be communicated explicitly before binary operations are introduced formally. The students in a college abstract algebra class probably know the meaning of an equation operationally (subconsciously) but they have never seen it made explicit. See Algebra is a difficult foreign language.

    Note

    The equation “$3+5=6+2$” is an expression just as much as “$3+5$” and “$6+2$” are. It denotes an object of type “equation”, which is a mathematical object in the same way as numbers are. Most mathematicians do not talk this way, but they should.

    Binary operations

    Early examples

    Consciousness-expanding examples should appear early and often after binary operations are introduced.

    Common operations

    • The GCD is a binary operation on the natural numbers. This disturbs some students because it is not written in infix form. It is associative. The GCD can be defined conceptually, but for computation purposes needs (Euclid’s) algorithm. This gives you an early example of conceptual definitions and algorithms.
    • The maximum function is another example of this sort. This is a good place to point out that a binary operation with the “same” definition cen be defined on different sets. The max function on the natural numbers does not have quite the same conceptual definition as the max on the integers.

    Extensional definitions

    In order to emphasize the arbitrariness of definitions, some random operations on a small finite sets should be given by a multiplication table, on sets of numbers and sets represented by letters of the alphabet. This will elicit the common reaction, “What operation is it?” Hidden behind this question is the fact that you are giving an extensional definition instead of a formula — an algorithm or a combination of familiar operations.

    Properties

    The associative and commutative properties should be introduced early just for consciousness-raising. Subtraction is not associative or commutative. Rock paper scissors is commutative but not associative. Groups of symmetries are associative but not commutative.

    Binary operation as function

    The first definition of binary operation should be as a function. For example, “$+$” is a function that takes pairs of numbers to numbers. In other words, $+:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ is a function.

    We then abstract from that example and others like it from specific operations to arbitrary functions $\Delta:S\times S\to S$ for arbitrary sets $S$.

    This is abstraction twice.

    • First we replace the example operations by an arbitrary operation. such as multiplication, subtraction, GCD and MAX on $\mathbb{Z}$, or something complicated such as \[(x,y)\mapsto 3(xy-1)^2(x^2+xy^3)^3\].
    • Then we replace sets of numbers by arbitrary sets. An example would be the random multiplication on the set $\{1,2,5\}$ given by the table
      \[
      \begin{array}{c|ccc}
      \Delta& 1&2&5\\
      \hline
      1&2&2&1\\
      2&5&2&1\\
      5&2&1&5
      \end{array}
      \]
      This defines a function $\Delta:\{1,2,5\}\times\{1,2,5\}\to\{1,2,5\}$ for which for example $\Delta(2,1)=5$, or $2\Delta 1=5$. This example uses numbers as elements of the set and is good for eliciting the “What operation is it?” question.
    • I will use examples where the elements are letters of the alphabet, as well. That sort of example makes the students think the letters are variables they can substitute for, another confusion to be banished by the wise professor who know the right thing to say to make it clear. (Don’t ask me; I taught algebra for 35 years and I still don’t know the right thing to say.)

    It is important to define prefix notation and infix notation right away and to use both of them in examples.

    Other representations of binary operations.

    The main way of representing binary operations in Abstracting Algebra will be as trees, which I will cover in later posts. Those posts will be much more interesting than this one.

    Binary operations in high school and college algebra

    • Some binops are represented in infix notation: “$a+b$”, “$a-b$”, and “$a\times b$”.
    • “$a\times b$” is usually written “$ab$” for letters and with the “$\times$” symbol for numbers.
    • Some binops have idiosyncratic representation: “$a^b$”, “${a}\choose{b}$”.
    • A lot of binops such as GCD and MAX are given as functions of two variables (prefix notation) and their status as binary operations usually goes unmentioned. (That is not necessarily wrong.)
    • The symbol “$(a,b)$” is used to denote the GCD (a binop) and is also used to denote a point in the plane or an open interval, both of which are not strictly binops. They are binary operations in a multisorted algebra (a concept I expect to introduce later in the book.)
    • Some apparent binops are in infix notation but have flaws: In “$a/b$”, the second entry can’t be $0$, and the expression when $a$ and $b$ are integers is often treated as having good forms ($3/4$) and bad forms ($6/8$).

    Trees

    The chaotic nature of algebraic notation I just described is a stumbling block, but not the primary reason high school algebra is a stumbling block for many students. The big reason it is hard is that the notation requires students to create and hold complicated abstract structures in their head.

    Example

    This example is a teaser for future posts on using trees to represent binary operations. The tree below shows much more of the structure of a calculation of the area of a rectangle surmounted by a semicircle than the expression

    \[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\]
    does.

    The tree explicitly embodies the thought process that leads to the formula:

    • You need to add the area of the rectangle and the area of the semicircle.
    • The area of the rectangle is width times height.
    • The area of the semicircle is $\frac{1}{2}(\pi r^2)$.
    • In this case, $r=\frac{1}{2}w$.

    Any mathematician will extract the same abstract structure from the formula\[A=wh+\frac{1}{2}\left(\pi(\frac{1}{2}w)^2\right)\] This is difficult for students beginning algebra.

    References

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    The two languages of math

    I am revising the (large) section of abstractmath.org that concerns the languages of math. Below is most of the the introduction to that section, which contains in particular detailed links to its contents. All of these are now available, but only a few of them have been revised. They are the ones that say “Abstractmath 2.0″ in the header.

    Introduction

    Mathematics in the English-speaking world is communicated using two languages:

    • Mathematical English is a special form of English.
    • It uses ordinary words with special meanings.
    • Some of its structural words (“if”, “or”) have different meanings from those of ordinary English.
    • It is both written and spoken.
    • Other languages also have special mathematical forms.
    • The symbolic language of math is a distinct, special-purpose language.
    • It has its own symbols and rules that are rather unlike those that spoken languages have.
    • It is not a dialect of English.
    • It is largely a written language.
    • Simple expressions can be pronounced, but complicated expressions may only be pointed to or referred to.
    • It is used by all mathematicians, not just those who write math in English.

    Math in writing and in lectures involve both mathematical English and the symbolic language. They are embedded in each other and refer back and forth to each other.

    Contents

    The languages of math are covered in three chapters, each with several parts. Some things are not covered; see Notes.

    Links to other sites


    Notes

    Math communication also uses pictures, graphs and diagrams, which abstractmath.org doesn’t discuss. Also not covered is the history and etymology of mathematical notation.

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    Dysfunctions in doing math III

    This post concludes the work begun in Dysfunctions in doing math I and Dysfunctions in doing math II, with more revisions to the article in abstractmath on dysfunctions.

    False symmetry

    Bases of vector spaces

    In a finite dimensional vector space $V$ with subspace $W$, every basis of $W$ can be extended to a basis of $V$. But in general there are bases of $V$ that do not contain a subset that is a basis of $W$. A tragic lack of symmetry that causes innocent students to lose points in linear algebra.

    Example

    The plane $P$ defined by $x=y$ is a two-dimensional subspace of the three dimensional Euclidean space with axes $x,y,z$. One basis of $P$ is $\{(1,1,0),(0,0,1)\}$. It can be extended to the basis $\{(1,1,0),(0,0,1),(0,1,0)\}$ of $\mathbb{R}^3$. But the basis $\{(1,0,0),(0,1,0),(0,0,1)\}$ of $\mathbb{R}^3$ does not contain a subset that is a basis of $P$.

    Normal subgroups

    Every subgroup $B$ of a commutative group $A$ is a normal subgroup of $A$. But if $B$ is an commutative subgroup of a non-commutative group $S$, then $B$ may not be a normal subgroup of $S$. For example, $\text{Sym}_3$ (the group of symmetries of an equilateral triangle) has three subgroup with two elements each. Each subgroup is commutative, but is not a normal subgroup of $\text{Sym}_3$.

    Jump the fence

    If you are working with an expression whose variables are constrained to certain values, and you substitute a value in the expression that violates the constraint, you jump the fence
    (also called a fencepost error).

    Example

    The Fibonacci numbers (MW, Wi) are usually defined inductively like this:

    \[F(n)=\left\{ \begin{align}
    & 0\text{ if }n=0 \\
    & 1\text{ if }n=1 \\
    & F(n-1)+F(n-2)\text{ if }n\gt 1 \\
    \end{align} \right.\]

    In calculating a sum of Fibonacci numbers, you might write

    \[\sum_{k=0}^{n}{F(k)=}\sum_{k=0}^{n}{F(k-1)+}\sum_{k=0}^{n}{F(k-2)}\]
    This contains errors : the sums on the right involve $F(-1)$ and $F(-2)$, which are not defined by the definition above. You could add
    \[F(n)=0\text{ if }n\lt 0\]
    to the definition to get around this, or keep better track of the fence by writing

    \[\sum_{k=0}^{n}{F(k)=}\sum_{k=1}^{n}{F(k-1)+}\sum_{k=2}^{n}{F(k-2)}\,\,\,\,\,\,\,\,\,\text{
    }(n>1)\]

    (The notation “$(n \gt 1)$” means “for all $n$ greater than $1$.” See here )

    Literalism

    Every type of math object has to have a definition. In giving a definition, a few of the many ingredients that are involved in that type of object are selected as a basis for the definition. They are not necessarily the most important parts. People who make definitions try to use as little as possible in the definition so that it is easier to verify that something is an example of the thing being defined.

    A definitional literalist is someone who insists on thinking about a type of math object primarily in terms of what the definition says it is.


    Definitional literalism inhibits your understanding of abstract math.

    Ordered pairs

    One of the major tools in the study of the foundations of mathematics is to try to define all mathematical objects in terms of as few as possible objects. The most common form this takes is to define everything in terms of sets. For example, the ordered pair $(a,b)$ can be defined to be the set $\{a, \{a, b\}\}$.
    (See Wi). A definitional literalist will conclude that the ordered pair $(a,b)$ is the set $\{a, \{a, b\}\}$.

    This would mean that it makes sense to say that $a\in(a,b)$ but $b\notin(a,b)$.
    No mathematician would ever think of saying such things.

    What is important about an ordered pair is its specification:

    • An ordered pair has a first coordinate and a second coordinate.
    • What the first and second coordinates are completely determine the ordered pair.

    It is ludicrous to say something like “$a\in (a,b)$”. The “definition” that $(a,b)$ is the set $\{a,\{\{a,b\}\}$ is done purely for the purpose of showing that the study of ordered pairs can be reduced to the study of sets. It is not a fact about ordered pairs that we can use.

    Equivalence relations

    An equivalence relation on a set S is a relation on S with certain properties. A partition on S is a set of subsets with certain properties. The two definitions can be proven to give the same structure (that is done here).

    I have personally heard literalists say,
    “How can they give the same structure? One is a relation and one is a partition.” The point is that an equivalence relation/partition has a total structure which can be described either by starting with a relation and imposing axioms, or by giving a set of subsets and imposing axioms. Each set of axioms describes exactly the same structure; every theorem that can be deduced from the axioms for an equivalence relation can be deduced from the axioms for a partition.

    Functions

    The
    (less strict) definition of function says that a function is a set of ordered pairs with the functional property.

    This does not mean that if your function is $F ( x ) = 2 x + 1$, then you would say “$\left( 3,\,7 \right)\in F$” . The most common practice is to say that “$F (3) = 7$” or “the value of $F$ at $3$ is $7$” or something of the sort.

    I do know mathematicians who tell me that they really do think of a function as a set of ordered pairs and would indeed say “$\left( 3,\,7 \right)\in F$”.

    Vanishing

    Many years ago I had a math professor who hated it with a purple passion if anyone said a function vanishes at some number $a$, meaning its value at $a$ is $0$. If you said, “The function $x^2-1$ vanishes at $1$”, he would say, “Pah! The function is still there isn’t it?”

    There are in fact two different points a literalist can make about such a statement.

    • The function’s value at $1$ is $0$. The function is not zero anywhere, it is $x^2-1$, or if you have other literalness attitudes, it is “the function $f(x)$ defined by $f(x)=x^2-1$”.
    • Even its value doesn’t literally “vanish”. The value is written as “$0$”. Look at it closely. You can see it. It has not vanished.

    The phrase “the function vanishes at $a$” is a metaphor. Mathematicians use metaphors in writing and talking about math all the time, just as people do in writing and talking about anything. Nevertheless, being occasionally the obnoxious literalist sometimes clears up misunderstanding. That is why mathematicians have a reputation for literalism.

    Method addiction

    Beginners at abstract math sometimes have the attitudes that a problem must be solved or a proof constructed by a specific procedure. They become quite uncomfortable when faced with problem solutions that involve guessing or conceptual proofs that involve little or no calculation.

    Example

    Once I gave a problem in my Theoretical Computer Science class that in order to solve it required finding the largest integer $n$ for which $n!\lt109$ Most students solved it correctly, but several wrote apologies on their paper for doing it by trial and error. Of course:


    Trial and error is a perfectly valid method.

    Example

    Students at a more advanced level may feel insecure in the case where they are faced with solving a problem for which they know there is no known feasible algorithm, a situation that occurs mostly in senior and graduate level classes. For example, there are no known feasible general algorithms for determining if two finite groups given by their multiplication tables are isomorphic, and there is no algorithm at all to determine if two presentations (generators and relations) give the same group. Even so, the question, “Are the dihedral group of order 8 and the quaternion group isomorphic?” is not hard. (Answer: No, they have different numbers of elements of order 2 and 4.)


    Sometimes you can solve special cases of unsolvable problems.

    See also look ahead and conceptual.

    Proof by Example

    Definition: An integer is even if it is divisible by 2.

    Theorem : Prove that if
    $n$ is an even integer then so is ${{n}^{2}}$.

    This is proved by universal generalization .

    One type of mistake made by beginners for proofs like this would be the following:

    “Proof: Let $n = 8$. Then ${{n}^{2}}=64$ and $64$ is even.”

    This violates the requirement of universal generalization that you have ” made no restrictions on $c$” – you have restricted it to being a particular even integer!

    It may be that some people who make this kind of mistake don’t understand universal generalization (see also bound variable). But for others, the mistake is caused by misreading the phrase “An integer is even if…” to read that you can prove the statement by picking an integer and showing that it is true for that integer. But in fact, “an” in a statement like this means “any”. See indefinite article.

    Reading variable names as labels

    An assertion such as “There are six times as many students as professors” is translated by some students as $6s = p$ instead of $6p = s$ (where $p$ and $s$ have the obvious meanings). This sort of thing can be avoided by plugging in numbers for the variables to see if the resulting equations make sense. You know it’s wrong to say that if you have $12$ professors then you have $2$ students!

    Math ed people have referred to this as the “student-professor problem”. But it is not the real student-professor problem.

    The representation is the object

    Many newbies at abstract mathematics firmly believe that the number $735$ is the expression “735”. In fact, the number $735$ is an abstract math object, not a string of symbols that represents the number. This attitude inhibits your ability to use whatever representation of an object is best for the purpose.

    Example

    Someone faced with a question such as “Does $21$ divide $3 \cdot5\cdot72$?” may immediately multiply the expression out to get $1080$ and then carry out long division to see if indeed $21$ divides $1080$. They will say things such as, “I can’t tell what the number is until I multiply it out.”

    In this example, it is easy to see that $21$ does not divide $3 \cdot5\cdot72$, because if it did, $7$ would be a prime factor, but $7$ does not divide $72$.

    Integers have many representations: decimal, binary, the prime factorization, and so on. Clearly the prime factorization is the best form for determining divisors, whereas for example the decimal notation is a good form for determining which of two integers is the larger. For example, is $3 \cdot5\cdot72$ bigger or smaller than $2\cdot 11\cdot49$?

    Unique

    By definition, a set $R$ of ordered pairs has the functional property if two pairs in $R$ with the same first coordinate have to have the same second coordinate

    It is wrong to rephrase the definition this way: “The first coordinate determines a unique second coordinate.” That use of “unique” is ambiguous. It could mean the set \[\{(1,2),
    (2,4), (3,2), (5,8)\}\] does not have the functional property because the first coordinate in $(1,2)$ determines $2$ and the first coordinate in $(3,2)$ determines $2$, so it is “not unique”. This statement is wrong. . The set does have the functional property.

    A related error is to reword the definition of injective by saying, “For each input there is a unique output.” It is easy to read this and think injectivity is merely the functional property.

    It seemed to me that during the 35 years I taught calculus and discrete math, students fell into this trap about 100,000 times. Of course, this could be a slight exaggeration.


    Avoid rewording any definition that does not use the word unique
    so that it DOES use the word unique.
    Such activity fries your brain and turns A’s into B’s.


    Unnecessarily weak assertion

    Examples

    • The statement “Either $x \gt 0$ or $x \lt 2$” is true (for real numbers). Yes, you could make a stronger statement, for example “Either $x\le 0$ or $x \gt 0$”. But the statement “Either $x \gt 0$ or $x \lt 2$” is still true.
    • Some students have problems with the true statements “$2\le 2$” and with “$2\le 3$” for a similar reason, since in fact $2 = 2$ and $2 \lt 3$.
    • You may get a twinge if someone says “Many primes are odd”, since in fact there is only one that is not
      odd. But it is still true that many primes are odd.

    An unnecessarily weak assertion may occur in math texts because it is the form your proof gives you, or it is the form you need for a proof. In the latter case you may feel the author has pulled a rabbit out of a hat.

    There is another example here.


    It is not wrong for an author to make an unnecessarily weak assertion.




    Rabbits

    Sometimes when you are reading or listening to a proof you will find yourself following each step but with no idea why these steps are going to give a proof. This can happen with the whole structure of the proof or with the sudden appearance of a step that seems like the prover pulled a rabbit out of a hat . You feel as if you are walking blindfolded.

    Example
    (mysterious proof structure)

    The lecturer says he will prove that for an integer $n$, if $n^2$ is even then $n$ is even. He begins the proof: Let $n^2$ be odd” and then continues to the conclusion, “Therefore $n$ is odd.”

    Why did he begin a proof about being even with the assumption that $n$ is odd?

    The answer is that in this case he is doing a proof by contrapositive . If you don’t recognize the pattern of the proof you may be totally lost. This can happen if you don’t recognize other forms, for example contradiction and induction.

    Example (rabbit)

    You are reading a proof that $\underset{x\to
    2}{\mathop{\lim }}{{x}^{2}}=4$. It is an $\varepsilon \text{-}\delta$ proof, so what must be proved is:

    • (*) For any positive real number $\varepsilon $,
    • there is a positive real number $\delta $ for which:
    • if $\left| x-2 \right|\lt\delta$ then
    • $\left| x^2-4 \right|\lt\varepsilon$.

    Proof

    Here is the proof, with what I imagine might be your agitated reaction to certain steps. Below is a proof with detailed explanations .

    1) Suppose $\varepsilon \gt0$ is given.

    2) Let $\delta =\text{min}\,(1,\,\frac{\varepsilon }{5})$ (the minimum of the two numbers 1 and $\frac{\varepsilon}{5}$ ).

    Where the *!#@! did that come from? They pulled it out of thin air! I can’t see where we are going with this proof!

    3) Suppose that $\left| x-2 \right|\lt\delta$.

    4) Then $\left| x-2 \right|\lt1$ by (2) and (3).

    5) By (4) and algebra, $\left|x+2 \right|\lt5$.

    Well, so what? We know that $\left| x+39
    \right|\lt42$ and lots of other things, too. Why did they do this?

    6) Also $\left| x-2 \right|\lt\frac{\varepsilon }{5}$ by (2).

    7) Then $\left| {{x}^{2}}-4
    \right|=\left| (x-2)(x+2) \right|\lt\frac{\varepsilon }{5}\cdot 5=\varepsilon$ by (5) and (6). End of Proof.

    Remarks

    This proof is typical of proofs in texts.

    • Steps 2) and 5) look like they were rabbits pulled out of a hat.
    • The author gives no explanation of where they came from.
    • Even so, each step of the proof follows from previous steps, so the proof is correct.
    • Whether you are surprised or not has nothing to do with whether it is correct.
    • In order to understand a proof, you do not have to know where the rabbits came from.
    • In general, the author did not think up the proof steps in the order they occur in the proof. (See this remark in the section on Forms of Proofs.)
    • See also look ahead.

    Acknowledgments

    Thanks to Robert Burns for corrections and suggestions

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    Variations in meaning in math

    Words in a natural language may have different meanings in different social groups or different places.  Words and symbols in both mathematical English and the symbolic language vary according to specialty and, occasionally, country (see convention, default).  And words and symbols can change their meanings from place to place within the same mathematical discourse (see scope).

    This article mostly provides pointers to other articles in abstractmath.org that give more details about the ideas.

    Conventions

    A convention in mathematical discourse is notation or terminology used with a special meaning in certain contexts or in certain fields. Articles and books in a specialty do not always clue you in on these conventions.

    Some conventions are nearly universal in math.

    Example 1

    The use of “if” to mean “if and only if” in a definition is a convention. More about this here. This is a hidden definition by cases. “Hidden” means that no one tells the students, except for Susanna Epp and me.

    Example 2

    Constants or parameters are conventionally denoted by a, b, … , functions by f, g, … and variables by x, y,…. More.

    Example 3

    Referring to a group (or other mathematical structure) and its underlying set by the same name is a convention.  This is an example of both synecdoche and context-sensitive.

    Example 4

    The meaning of ${{\sin }^{n}}x$ in many calculus books is:

    • The inverse sine (arcsin) if $n=-1$.
    • The mult­iplica­tive power for positive $n$; in other words, ${{\sin }^{n}}x={{(\sin x)}^{n}}$ if $n\ne -1$.

    This, like Example 1, is a definition by cases. Unlike Example 1, calculus books often make it explicit. Explicit or not, this usage is an abomination.

    Some conventions are pervasive among math­ematicians but different conventions hold in other subjects that use mathematics.

    • Scientists and engineers may regard a truncated decimal such as 0.252 as an approximation, but a mathematician is likely to read it as an exact rational number, namely $\frac{252}{1000}$.
    • In most computer languages a distinction is made between real numbers and integers;
      42 would be an integer but 42.0 would be a real number.  Older mathematicians may not know this.
    • Mathematicians use i to denote the imaginary unit. In electrical engineering it is commonly denoted j instead, a fact that many mathematicians are un­aware of. I first learned about it when a student asked me if i was the same as j.

    Conventions may vary by country.

    • In France and possibly other countries schools may use “positive” to mean “nonnegative”, so that zero is positive. 
    • In the secondary schools in some places, the value of sin x may be computed clockwise starting at (0,1)  instead of counterclockwise starting at (1,0).  I have heard this from students. 

    Conventions may vary by specialty within math.

    Field” and “log” are examples. 

    Defaults

    An interface to a computer program may have many possible choices for the user to make. In most cases, the interface will use certain choices automatically when the user doesn’t specify them.  One says the program defaults to those choices.  

    Examples

    • A word processing program may default to justified paragraphs and insert mode, but allow you to pick ragged right or typeover mode.
    • I have spent a lot of time in both Minne­sota and Georgia and the remarks about skiing are based on my own observation. But these usages are not absolute. Some affluent Geor­gians may refer to snow skiing as “skiing”, for example, and this usage can result in a put-down if the hearer thinks they are talking about water skiing. One wonders where the boundary line is. Perhaps people in Kentucky are confused on the issue.

    • There is a sense in which the word “ski” defaults to snow skiing in Minnesota and to water skiing in Georgia.
    • “CSU” defaults to Cleveland State University in northern Ohio and to Colorado State University in parts of the west.

    Math language behaves in this way, too.

    Default usage in mathematical discourse

    Symbols

    • In high school, $\pi$ refers by default to the ratio of the circumference of a circle to its diameter.  Students are often quite surprised when they get to abstract math courses and discover the many other meanings of $\pi $ (see here).
    • Recently authors in the popular literature seem to think that $\phi$ (phi) defaults to the golden ratio.  In fact, a search through the research literature shows very few hits for $\phi$ meaning the golden ratio: in other words, it usually means something else. 
    • The set $\mathbb{R}$ of real numbers has many different group structures defined on it but “The group $\mathbb{R}$” essentially always means that the group operation is ordinary addition.  In other words, “$\mathbb{R}$” as a group defaults to +.  Analogous remarks apply to “the field $\mathbb{R}$”. 
    • In informal conversation among many analysts, functions are continuous by default.
    • It used to be the case that in informal conversations among topologists, “group” defaulted to Abelian group. I don’t know whether that is still true or not.

    Remark

    This meaning of “default” has made it into dictionaries only since around 1960 (see the Wikipedia entry). This usage does not carry a derogatory connotation.   In abstractmath.org I am using the word to mean a special type of convention that imposes a choice of parameter, so that it is a special case of both “convention” and “suppression of parameters”.

    Scope

    Both mathematical English and the symbolic language have a feature that is uncommon in ordinary spoken or written English:  The meaning of a phrase or a symbolic expression can be different in different parts of the discourse.   The portion of the text in which a particular meaning is in effect is called the scope of the meaning.  This is accomplished in several ways.

    Explicit statement

    Examples

    • “In this paper, all groups are abelian”.  This means that every instance of the word “group” or any symbol denoting a group the group is constrained to be abelian.   The scope in this case is the whole paper.   See assumption.
    • “Suppose (or “let” or “assume”) $n$ is divisible by $4$”. Before this statement, you could not assume $n$ is divisible by $4$. Now you can, until the end of the current paragraph or section.

    Definition

    The definition of a word, phrase or symbol sets its meaning.  If the word definition is used and the scope is not given explicitly, it is probably the whole discourse.

    Example

    “Definition.  An integer is even if it is divisible by 2.”  This is marked as a definition, so it establishes the meaning of the word “even” (when applied to an integer) for the rest of the text. 

    If

    Used in modus ponens (see here) and (along with let, usually “now let…”) in proof by cases.

    Example(modus ponens)

    Suppose you want to prove that if an integer $n$ is divisible by $4$ then it is even. To show that it is even you must show that it is divisible by $2$. So you write:

    • “Let $n$ be divisible by $4$. That means $n=4k$ for some integer $k$. But then $n=2(2k)$, so $n$ is even by definition.”

    Now if you start a new paragraph with something like “For any integer $n\ldots$” you can no longer assume $n$ is divisible by $4$.

    Example (proof by cases)

    Theorem: For all integers $n$, $n^2+n+1$ is odd.

    Definitions:

    • “$n$ is even” means that $n=2s$ for some integer $s$.
    • “$n$ is odd” means that $n=2t+1$ for some integer $t$.

    Proof:

    • Suppose $n$ is even. Then

      \[\begin{align*}
      n^2+n+1&=4s^2+2s+1\\
      &=2(2s^2+s)+1\\
      &=2(\text{something})+1
      \end{align*}\]

      so $n^2+n+1$ is odd. (See Zooming and Chunking.)

    • Now suppose $n$ is odd. Then

      \[\begin{align*}
      n^2+n+1&=(2t+1)^2+2t+1+1\\
      &=4t^2+4t+1+2t+1+1\\
      &=2(2t^2+3t)+3\\
      &=2(2t^2+3t+1)+1\\
      &=2(\text{something})+1
      \end{align*}\]

      So $n^2+n+1$ is odd.

    Remark

    The proof I just gave uses only the definition of even and odd and some high school algebra. Some simple grade-school facts about even and odd numbers are:

    • Even plus even is even.
    • Odd plus odd is even.
    • Even times even is even.
    • Odd times odd is odd.

    Put these facts together and you get a nicer proof (I think anyway): $n^2+n$ is even, so when you add $1$ to it you must get an odd number.

    Bound variables

    A variable is bound if it is in the scope of an integral, quantifier, summation, or other binding operators.  More here.

    Example

    Consider this text:

    Exercise: Show that for all real numbers $x$, it is true that $x^2\geq0$. Proof: Let $x=-2$. Then $x^2=(-2)^2=4$ which is greater than $0$. End of proof.”

    The problem with that text is that in the statement, “For all real numbers $x$, it is true that $x^2\geq0$”, $x$ is a bound variable. It is bound by the universal quantifier “for all” which means that $x$ can be any real number whatever. But in the next sentence, the meaning of $x$ is changed by the assumption that $x=-2$. So the statement that $x\geq0$ only applies to $-2$. As a result the proof does not cover all cases.

    Many students just beginning to learn to do proofs make this mistake. Fellow students who are a little further along may be astonished that someone would write something like that paragraph and might sneer at them. But this common mistake does not deserve a sneer, it deserves an explanation. This is an example of the ratchet effect.

    Variable meaning in natural language

    Meanings commonly vary in natural language because of conventions and defaults. But varying in scope during a conversation seems to me uncommon.

    It does occur in games. In Skat and Bridge, the meaning of “trump” changes from hand to hand. The meaning of “strike” in a baseball game changes according to context: If the current batter has already had fewer than two strikes, a foul is a strike, but not otherwise.

    I have not come up with non-game examples, and anyway games are played by rules that are suspiciously like mathematical axioms. Perhaps you can think of some non-game occasions in which meaning is determined by scoping that I have overlooked.

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    Thinking about thought

    Modules of the brain

    Cognitive neuroscientists have taken the point of view that concepts, memories, words, and so on are represented in the brain by physical systems: perhaps they are individual neurons, or systems of structures, or even waves of discharges. In my previous writing I have referred to these as modules, and I will do that here. Each module is connected to many other modules that encode various properties of the concept, thoughts and memories that occur when you think of that concept (in other words stimulate the module), and so on.

    How these modules implement the way we think and perceive the world is not well understood and forms a major research task of cognitive neuroscience. The fact that they are implemented in physical systems in the brain gives us a new way of thinking about thought and perception.

    Examples

    The grandmother module

    There is a module in your brain representing the concept of grandmother. It is likely to be connected to other modules representing your actual grandmothers if you have any memory of them. These modules are connected to many others — memories (if you knew them), other relatives related to them, incidents in their lives that you were told about, and so on. Even if you don’t have any memory of them, you have a module representing the fact that you don’t have any memory of them, and maybe modules explaining why you don’t.

    Each different aspect related to “grandmother” belongs to a separate module somehow connected to the grandmother module. That may be hard to believe, but the human brain has over eighty billion neurons.

    A particular module connected with math

    There is a module in your brain connected with the number $42$. That module has many connections to things you know about it, such as its factorization, the fact that it is an integer, and so on. The module may also have connections to a module concerning the attitude that $42$ is the Answer. If it does, that module may have a connection with the module representing Douglas Adams. He was physically outside your body, but is the number $42$ outside your body?

    That has a decidedly complicated answer. The number $42$ exists in a network of brains which communicate with each other and share some ideas about properties of $42$. So it exists socially. This social existence occasionally changes your knowledge of the properties of $42$ and in particular may make you realize that you were wrong about some of its aspects. (Perhaps you once thought it was $7\times 8$.)

    This example suggests how I have been using the module idea to explain how we think about math.

    A new metaphor for understanding thinking

    I am proposing to use the idea of module as a metaphor for thinking about thinking. I believe that it clarifies a lot of the confusion people have about the relation between thinking and the real world. In particular it clarifies why we think of mathematical objects as if they were real-world objects (see Modules and math below.)

    I am explicitly proposing this metaphor as a successor to previous metaphors drawn from science to explain things. For example when machines became useful in the 18th century many naturalists used metaphors such as the Universe is a Machine or the Body is a Machine as a way of understanding the world. In the 20th century we fell heavily for the metaphor that the Mind Is A Computer (or Program). Both the 18th century and the 20th century metaphors (in my opinion) improved our understanding of things, even though they both fell short in many ways.

    In no way am I claiming that the ways of thinking I am pushing have anything but a rough resemblance to current neuroscientists’ thinking. Even so, further discoveries in neuroscience may give us even more insight into thinking that they do now. Unless at some point something goes awry and we have to, ahem, think differently again.

    For thousands of years, new scientific theories have been giving us new metaphors for thinking about life, the universe and everything. I am saying here is a new apple on the tree of knowledge; let’s eat it.

    The rest of this post elaborates my proposed metaphor. Like any metaphor, it gets some things right and some wrong, and my explanations of how it works are no doubt full of errors and dubious ideas. Nevertheless, I think it is worth thinking about thought using these ideas with the usual correction process that happens in society with new metaphors.

    Our theory of the world

    We don’t have any direct perception of the “real world”; we have only the sensations we get from those parts of our body which sense things in the world. These sensations are organized by our brain into a theory of the world.

    • The theory of the world says that the world is “out there” and that our sensory units give us information about it. We are directly aware of our experiences because they are a function of our brain. That the experiences (many of them) originate from outside our body is a very plausible theory generated by our brain on the bases of these experience.
    • The theory is generated by our brain in a way that we cannot observe and is out of our control (mostly). We see a table and we know we can see in in daytime but not when it is dark and we can bump into it, which causes experiences to occur via our touch and sound facilities. But the concept of “table” and the fact that we decide something is or is not a table takes place in our brain, not “out there”.
    • We do make some conscious amendments to the theory. For example, we “know” the sky is not a blue shell around our world, although it looks like it. That we think of the apparent blue surface as an artifact of our vision processing comes about through conscious reasoning. But most of how we understand the world comes about subconsciously.
    • Our brain (and the rest of our body) does an enormous amount of processing to create the view of the world that we have. Visual perception requires a huge amount of processing in our brain and the other sensory methods we use also undergo a lot of processing, but not as much as vision.
    • The theory of the world organizes a lot of what we experience as interaction with physical objects. We perceive physical objects as having properties such as persistence, changing with time, and so on. Our brains create the concept of physical object and the properties of persistence, changing, and particular properties an individual object might have.
    • We think of the Mississippi River as an object that is many years old even though none of its current molecules are the same as were in the river a decade ago. How is it one thing when its substance is constantly changing? This is a famous and ancient conundrum which becomes a non-problem if you realize that the “object” is created inside your brain and imposed by your thinking on your understanding of the world.
    • The notion that semantics is a connection between our brain and the outside world has also become a philosophical conundrum that vanishes if we understand that the connection with the outside world exists entirely inside our theory, which is entirely within our brain.

    Society

    Our brain also has a theory of society We are immersed in a world of people, that we have close connections with some of them and more distant connections with many other via speech, stories, reading and various kinds of long-distance communications.

    • We associate with individual people, in our family and with our friend. The communication is not just through speech: it involves vision heavily (seeing what The Other is thinking) and probably through pheromones, among other channels. For one perspective on vision, see The vision revolution, by Mark Changizi. (Review)
    • We consciously and unconsciously absorb ideas and attitudes (cultural and otherwise) from the people around us, especially including the adults and children we grow up with. In this way we are heavily embedded in the social world, which creates our point of view and attitudes by our observation and experience and presumably via memes. An example is the widespread recent changes in attitudes in the USA concerning gay marriage.
    • The theory of society seems to me to be a mechanism in our brain that is separate from our theory of the physical world, but which interacts with it. But it may be that it is better to regard the two theories as modules in one big theory.

    Modules and math

    The module associated with a math object is connected to many other modules, some of which have nothing to do with math.

    • For example, they may have have connections to our sensory organs. We may get a physical feeling that the parabola $y=x^2$ is going “up” as $x$ “moves to the right”. The mirror neurons in our brain that “feel” this are connected to our “parabola $y=x^2$” module. (See Constructivism and Platonism and the posts it links to.)
    • I tend to think of math objects as “things”. Every time I investigate the number $111$, it turns out to be $3\times37$. Every time I investigate the alternating group on $6$ letters it is simple. If I prove a new theorem it feels as if I have discovered the theorem. So math objects are out there and persistent.
    • If some math calculation does not give the same answer the second time I frequently find that I made a mistake. So math facts are consistent.
    • There is presumably a module that recognizes that something is “out there” when I have repeatable and consistent experiences with it. The feeling originates in a brain arranged to detect consistent behavior. The feeling is not evidence that math objects exist in some ideal space. In this way, my proposed new way of thinking about thought abolishes all the problems with Platonism.
    • If I think of two groups that are isomorphic (for example the cyclic group of order $3$ and the alternating group of rank $3$), I picture them as in two different places with a connection between the two isomorphic ones. This phenomenon is presumably connected with modules that respond to seeing physical objects and carrying with them a sense of where they are (two different places). This is a strategy my brain uses to think about objects without having to name them, using the mechanism already built in to think about two things in different places.

    Acknowledgments

    Many of the ideas in this post come from my previous writing, listed in the references. This post was also inspired by ideas from Chomsky, Jackendoff (particularly Chapter 9), the Scientific American article Brain cells for Grandmother by Quian Quiroga, Fried and Koch, and the papers by Ernest and Hersh.


    References

    Previous posts

    In reverse chronological order

    Abstractmath articles

    Other sources

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