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Representations of functions III

Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations. This post is a draft of the sections on representations of finite functions.

The diagrams in this post were created using the Mathematica Notebook Constructions for cographs and endographs of finite functions.nb.
You can access this notebook if you have Mathematica, which can be bought, but is available for free for faculty and students at many universities, or with Mathematica CDF Player, which is free for anyone and runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

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Graphs of finite functions

When a function is continuous, its graph shows up as a curve in the plane or as a curve or surface in 3D space. When a function is defined on a set without any notion of continuity (for example a finite set), the graph is just a set of ordered pairs and does not tell you much.

A finite function $f:S\to T$ may be represented in these ways:

  • Its graph $\{(s,f(s))|s\in S\}$. This is graph as a mathematical object, not as a drawing or as a directed graph — see graph (two meanings)).
  • A table, rule or two-line notation. (All three of these are based on the same idea, but differ in presentation and are used in different mathematical specialties.)
  • By using labels with arrows between them, arranged in one of two ways:
  • A cograph, in which the domain and the codomain are listed separately.
  • An endograph, in which the elements of the domain and the codomain are all listed together without repetition.

All these techniques can also be used to show finite portions of infinite discrete functions, but that possibility will not be discussed here.

Introductory Example

Let \[\text{f}:\{a,b,c,d,e\}\to\{a,b,c,d\}\] be the function defined by requiring that $f(a)=c$, $f(b)=a$, $f(c)=c$, $f(d)=b$, and $f(e)=d$.

Graph

The graph of $f$ is the set
\[(a,c),(b,a),(c,c),(d,b),(e,d)\]
As with any set, the order in which the pairs are listed is irrelevant. Also, the letters $a$, $b$, $c$, $d$ and $e$ are merely letters. They are not variables.

Table

$\text{f}$ is given by this table:

This sort of table is the format used in databases. For example, a table in a database might show the department each employee of a company works in:

Rule

The rule determined by the finite function $f$ has the form

\[(a\mapsto b,b\mapsto a,c\mapsto c,d\mapsto b,e\mapsto d)\]

Rules are built in to Mathematica and are useful in many situations. In particular, the endographs in this article are created using rules. In Mathematica, however, rules are written like this:

\[(a\to b,b\to a,c\to c,d\to b,e\to d)\]

This is inconsistent with the usual math usage (see barred arrow notation) but on the other hand is easier to enter in Mathematica.

In fact, Mathematica uses very short arrows in their notation for rules, shorter than the ones used for the arrow notation for functions. Those extra short arrows don’t seems to exist in TeX.

Two-line notation

Two-line notation is a kind of horizontal table.

\[\begin{pmatrix} a&b&c&d&e\\c&a&c&b&d\end{pmatrix}\]

The three notations table, rule and two-line do the same thing: If $n$ is in the domain, $f(n)$ is shown adjacent to $n$ — to its right for the table and the rule and below it for the two-line.

Note that in contrast to the table, rule and two-line notation, in a cograph each element of the codomain is shown only once, even if the function is not injective.

Cograph

To make the cograph of a finite function, you list the domain and codomain in separate parallel rows or columns (even if the domain and codomain are the same set), and draw an arrow from each $n$ in the domain to $f(n)$ in the codomain.

This is the cograph for $\text{f}$, represented in columns

and in rows (note that $c$ occurs only once in the codomain)

Pretty ugly, but the cograph for finite functions does have its uses, as for example in the Wikipedia article composition of functions.

In both the two-line notation and in cographs displayed vertically, the function goes down from the domain to the codomain. I guess functions obey the law of gravity.

Rearrange the cograph

There is no expectation that in the cograph $f(n)$ will be adjacent to $n$. But in most cases you can rearrange both the domain and the codomain so that some of the structure of the function is made clearer; for example:

The domain and codomain of a finite function can be rearranged in any way you want because finite functions are not continuous functions. This means that the locations of points $x_1$ and $x_2$ have nothing to do with the locations of $f(x_1)$ and $f(x_2)$: The domain and codomain are discrete.

Endograph

The endograph of a function $f:S\to T$ contains one node labeled $s$ for each $s\in S\cup T$, and an arrow from $s$ to $s’$ if $f(s)=s’$. Below is the endograph for $\text{f}$.

The endograph shows you immediately that $\text{f}$ is not a permutation. You can also see that with whatever letter you start with, you will end up at $c$ and continue looping at $c$ forever. You could have figured this out from the cograph (especially the rearranged cograph above), but it is not immediately obvious in the cograph the way it in the endograph.

There are more examples of endographs below and in the blog post
A tiny step towards killing string-based math. Calculus-type functions can also be shown using endographs and cographs: See Mapping Diagrams from A(lgebra) B(asics) to C(alculus) and D(ifferential) E(quation)s, by Martin Flashman, and my blog posts Endographs and cographs of real functions and Demos for graph and cograph of calculus functions.

Example: A permutation

Suppose $p$ is the permutation of the set \[\{0,1,2,3,4,5,6,7,8,9\}\]given in two-line form by
\[\begin{pmatrix} 0&1&2&3&4&5&6&7&8&9\\0&2&1&4&5&3&7&8&9&6\end{pmatrix}\]

Cograph

Endograph

Again, the endograph shows the structure of the function much more clearly than the cograph does.

The endograph consists of four separate parts (called components) not connected with each other. Each part shows that repeated application of the function runs around a kind of loop; such a thing is called a cycle. Every permutation of a finite set consists of disjoint cycles as in this example.

Disjoint cycle notation

Any permutation of a finite set can be represented in disjoint cycle notation: The function $p$ is represented by:

\[(0)(1,2)(3,4,5)(6,7,8,9)\]

Given the disjoint cycle notation, the function can be determined as follows: For a given entry $n$, $p(n)$ is the next entry in the notation, if there is a next entry (instead of a parenthesis). If there is not a next entry, $p(n)$ is the first entry in the cycle that $n$ is in. For example, $p(7)=8$ because $8$ is the next entry after $7$, but $p(5)=3$ because the next symbol after $5$ is a parenthesis and $3$ is the first entry in the same cycle.

The disjoint cycle notation is not unique for a given permutation. All the following notations determine the same function $p$:

\[(0)(1,2)(4,5,3)(6,7,8,9)\]
\[(0)(1,2)(8,9,6,7)(3,4,5)\]
\[(1,2)(3,4,5)(0)(6,7,8,9)\]
\[(2,1)(5,3,4)(9,6,7,8)\]
\[(5,3,4)(1,2)(6,7,8,9)\]

Cycles such as $(0)$ that contain only one element are usually omitted in this notation.

Example: A tree

Below is the endograph of a function \[t:\{0,1,2,3,4,5,6,7,8,9\}\to\{0,1,2,3,4,5,6,7,8,9\}\]

This endograph is a tree. The graph of a function $f$ is a tree if the domain has a particular element $r$ called the root with the properties that

  • $f(r)=r$, and
  • starting at any element of the domain, repreatedly applying $f$ eventually produces $r$.

In the case of $t$, the root is $4$. Note that $t(4)=4$, $t(t(7))=4$, $t(t(t(9)))=4$, $t(1)=4$, and so on.

The endograph

shown here is also a tree.

See the Wikipedia article on trees for the usual definition of tree as a special kind of graph. For reading this article, the definition given in the previous paragraph is sufficient.

The general form of a finite function

This is the endograph of a function $t$ on a $17$-element set:

It has two components. The upper one contains one $2$-cycle, and no matter where you start in that component, when you apply $t$ over and over you wind up flipping back and forth in the $2$-cycle forever. The lower component has a $3$-cycle with a similar property.

This illustrates a general fact about finite functions:

  • The endograph of any finite function contains one or more components $C_1$ through $C_k$.
  • Each component $C_k$ contains exactly one $n_k$ cycle, for some integer $n_k\geq 1$, to which are attached zero or more trees.
  • Each tree in $C_k$ is attached in such a way that its root is on the unique cycle contained in $C_k$.

In the example above, the top component has three trees attached to it, two to $3$ and one to $4$. (This tree does not illustrate the fact that an element of one of the cycles does not have to have any trees attached to it).

You can check your understanding of finite functions by thinking about the following two theorems:

  • A permutation is a finite function with the property that its cycles have no trees attached to them.
  • A tree is a finite function that has exactly one component whose cycle is a $1$-cycle.


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Representations of functions I

Introduction to this post

I am writing a new abstractmath chapter called Representations of Functions. It will replace some of the material in the chapter Functions: Images, Metaphors and Representations.

This post includes a draft of the introduction to the new chapter (immediately below) and of the section Graphs of continous functions of one variable. Later posts will concern multivariable continuous functions, probably in two or three sections, and finite discrete functions.

Introduction to the new abstractmath chapter on representations of functions

Functions can be represented visually in many different ways. There is a sharp difference between representing continuous functions and representing discrete functions.

For a continuous function $f$, $f(x)$ and $f(x’)$ tend to be close together when $x$ and $x’$ are close together. That means you can represent the values at an infinite number of points by exhibiting them for a bunch of close-together points. Your brain will automatically interpret the points nearby that are not represented.

Nothing like this works for discrete functions. As you will see in the section on discrete functions, many different arrangements of the inputs and outputs can be made. In fact, different arrangements may be useful for representing different properties of the function.

Illustrations

The illustrations were created using these Mathematica Notebooks:

These notebooks contain many more examples of the ways functions can be represented than are given in this article. The notebooks also contain some manipulable diagrams which may help you understand the diagrams. In addition, all the 3D diagrams can be rotated using the cursor to get different viewpoints. You can access these tools if you have Mathematica, which is available for free for faculty and students at many universities, or with Mathematica CDF Player, which runs on Windows, Mac and Linux.

Like everything in abstractmath.org, the notebooks are covered by a Creative Commons ShareAlike 3.0 License.

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Graphs of continous functions of one variable

The most familiar representations of continuous functions are graphs of functions with one real variable. Students usually first see these in secondary school. Such representations are part of the subject called Analytic Geometry. This section gives examples of such functions.

There are other ways to represent continuous functions, in particular the cograph and the endograph. These will be the subject of a separate post.

The graph of a function $f:S\to T$ is the set of ordered pairs $\{(x,f(x))\,|\,x\in S\}$. (More about this definition here.)

In this section, I consider continuous functions for which $S$ and $T$ are both subsets of the real numbers. The mathematical graph of such a function are shown by plotting the ordered pairs $(x,f(x))$ as points in the two-dimensional $xy$-plane. Because the function is continuous, when $x$ and $x’$ are close to each other, $f(x)$ and $f(x’)$ tend to be close to each other. That means that the points that have been plotted cause your brain to merge together into a nice curve that allows you to visualize how $f$ behaves.

Example

This is a representation of the graph of the curve $g(x):=2-x^2$ for approximately the interval $(-2,2)$. The blue curve represents the graph.

Graph of a function.

The brown right-angled line in the upper left side, for example, shows how the value of independent variable $x$ at $(0.5)$ is plotted on the horizontal axis, and the value of $g(0.5)$, which is $1.75$, is plotted on the vertical axis. So the blue graph contains the point $(0.5,g(0.5))=(0.5,1.75)$. The animated gif upparmovie.gif shows a moving version of how the curve is plotted.

Fine points

  • The mathematical definition of the graph is that it is the set $\{(x,2-x^2)\,|\,x\in\mathbb{R}\}$. The blue curve is not, of course, the mathematical graph, it represents the mathematical graph.
  • The blue curve consists of a large but finite collection of pixels on your screen, which are close enough together to appear to form a continuous curve which approximates the mathematical graph of the function.
  • Notice that I called the example the “representation of the graph” instead of just “graph”. That maintains the distinction between the mathematical ordered pairs $(x,g(x))$ and the pixels you see on the screen. But in fact mathe­maticians and students nearly always refer to the blue line of pixels as the graph. That is like pointing to a picture of your grandmother and saying “this is my grandmother”. There is nothing wrong with saying things that way. But it is worth understanding that two different ideas are being merged.

Discontinuous functions

A discontinuous function which is continuous except for a small finite number of breaks can also be represented with a graph.

Example

Below is the function $f:\mathbb{R}\to\mathbb{R}$ defined by
\[f(x):=\left\{
\begin{align}
2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\gt0) \\
1-x^2\,\,\,\,\,\,(-1\lt x\lt 0) \\
2-x^2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(x\lt-1)
\end{align}\right.\]

Graph of a discontinuous function.

Example

The Dirichlet function is defined by
\[F(x):=
\begin{cases}
1 &
\text{if }x\text{ is rational}\\
\frac{1}{2} &
\text{if }x\text{ is irrational}\\ \end{cases}\]  for all real $x$.

The abmath article Examples of functions spells out in detail what happens when you try to draw this function.

Graphs can fool you

The graph of a continuous function cannot usually show the whole graph, unless it is defined only on a finite interval. This can lead you to jump to conclusions.

Example

For example, you can’t tell from the the graph of the function $y=2-x^2$ whether it has a local minimum (because the graph does not show all of the function), although you can tell by using calculus on the formula that it does not have one. The graph looks like it might have a vertical asymptotes, but it doesn’t, again as you can tell from the formula.

Discovering facts about a function
by looking at its graph
is useful but dangerous.

Example

Below is the graph of the function
\[f(x)=.0002{{\left( \frac{{{x}^{3}}-10}{3{{e}^{-x}}+1}
\right)}^{6}}\]

If you didn’t know the formula for the function (but know it is continuous), you could still see that it has a local maximum somewhere to the right of $x=1$. It looks like it has one or more zeroes around $x=-1$ and $x=2$. And it looks like it has an asymptote somewhere to the right of $x=2.5$.

If you do know the formula, you can find out many things about the function that you can’t depend on the graph to see.

  • You can see immediately that $f$ has a zero at $x=\sqrt[3]{10}$, which is about $2.15$.
  • If you notice that the denominator is positive for all $x$, you can figure out that
    • $\sqrt[3]{10}$ is the only root.
    • $f(x)\geq0$ for all $x$.
    • $f$ has an asymptote as $x\to-\infty$ (use L’Hôpital).
  • Numerical analysis (I used Mathematica) shows that $f'(x)$ has two zeros, at $\sqrt[3]{10}$ and at about $x=1.1648$. $f”(1.1648)$ is about $-10.67$ , which strongly suggests that $f$ has a local max near $1.1648$, consistent with the graph.
  • Since $f$ is defined for every real number, it can’t have a vertical asymptote anywhere. The graph looks like it becomes vertical somewhere to the right of $x=2.4$, but that is simply an illustration of the unbelievably fast growth of any exponential function.
  • The section on Zooming and Chunking gives other details.

    Acknowledgments

    Sue VanHattum.


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    Functions: Metaphors, Images and Representations

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

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    Math majors attacked by cognitive dissonance

    In some situations you may have conflicting information from different sources about a subject.   The resulting confusion in your thinking is called cognitive dissonance.

    It may happen that a person suffering cognitive dissonance suppresses one of the ways of understanding in order to resolve the conflict.  For example, at a certain stage in learning English, you (small child or non-native-English speaker) may learn a rule that the past tense is made from the present form by adding “-ed”. So you say “bringed” instead of “brought” even though you may have heard people use “brought” many times.  You have suppressed the evidence in favor of the rule.

    Some of the ways cognitive dissonance can affect learning math are discussed here

    Metaphorical contamination

    We think about math objects using metaphors, as we do with most concepts that are not totally concrete.  The metaphors are imperfect, suggesting facts about the objects that may not follow from the definition. This is discussed at length in the section on images and metaphors here.

    The real line

    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. Between any two points there are uncountably many other points. See density of the reals.

    Infinite math objects

    One of the most intransigent examples of metaphorical contamination occurs when students think about countably infinite sets. Their metaphor is that a sequence such as the set of natural numbers $\{0,1,2,3,4,\ldots\}$ “goes on forever but never ends”. The metaphor mathematicians have in mind is quite different: The natural numbers constitute the set that contains every natural number right now.

    Example

    An excruciating example of this is the true statement
    $.999\ldots=1.0$.” The notion that it can’t be true comes from thinking of “$0.999\ldots$” as consisting of the list of numbers \[0.9,0.99,0.999,0.9999,0.99999,\ldots\] which the student may say “gets closer and closer to $1.0$ but never gets there”.

    Now consider the way a mathematician thinks: The numbers are all already there, and they make a set.

    The proof that $.999\ldots=1.0$ has several steps. In the list below, I have inserted some remarks in red that indicate areas of abstract math that beginning students have trouble with.

    1. The elements of an infinite set are all in it at once. This is the way mathematicians think about infinite sets.
    2. By definition, an infinite decimal expansion represents the unique real number that is a limit point of its set of truncations.
    3. The problem that occurs with the word “definition” in this case is that a definition appears to be a dictatorial act. The student needs to know why you made this definition. This is not a stupid request. The act can be justified by the way the definition gets along with the algebraic and topological characteristic of the real numbers.

    4. It follows from $\epsilon-\delta$ machinations that the limit of the sequence $0.9,0.99,0.999,0.9999,0.99999,\ldots$ is $1.0$
    5. That means “$0.999\ldots$” represents $1.0$. (Enclosing a mathematical expression in quotes turns it into a string of characters.)
    6. The statement “$A$” represents $B$ is equivalent to the statement $A=B$. (Have you ever heard a teacher point this out?)
    7. It follows that that $0.999\ldots=1.0$.

    Each one of these steps should be made explicit. Even the Wikipedia article, which is regarded as a well written document, doesn’t make all of the points explicit.

    Semantic contamination

    Many math objects have names that are ordinary English words. 
    (See names.) So the person learning about them is faced with two inputs:

    • The definition of the word as a math object.
    • The meaning and connotations of the word in English.

    It is easy and natural to suppress the information given by the definition (or part of it) and rely only on the English meaning. But math does not work that way:

    If another source of understanding contradicts the definition
    THE DEFINITION WINS.

    “Cardinality”

    The connotations of a name may fit the concept in some ways and not others. Infinite cardinal numbers are a notorious example of this: there are some ways in which they are like numbers and other in which they are not. 

    For a finite set, the cardinality of the set is the number of elements in the set. Long ago, mathematicians started talking about the cardinality of an infinite set. They worked out a lot of facts about that, for example:

    • The cardinality of the set of natural numbers is the same as the cardinality of the set of rational numbers.
    • The cardinality of the number of points on the real line is the same as the cardinality of points in the real plane.

    The teacher may even say that there are just as many points on the real line as in the real points. And know-it-all math majors will say that to their friends.

    Many students will find that totally bizarre. Essentially, what has happened is that the math dictators have taken the phrase “cardinality” to mean what it usually means for finite sets and extend it to infinite sets by using a perfectly consistent (and useful) definition of “cardinality” which has very different properties from the finite case.

    That causes a perfect storm of cognitive dissonance.

    Math majors must learn to get used to situations like this; they occur in all branches of math. But it is bad behavior to use the phrase “the same number of elements” to non-mathematicians. Indeed, I don’t think you should use the word cardinality in that setting either: you should refer to a “one-to-one correspondence” instead and admit up front that the existence of such a correspondence is quite amazing.

    “Series”

    Let’s look at the word “series”in more detail. In ordinary English, a series is a bunch of things, one after the other.

    • The World Series is a series of up to seven games, coming one after another in time.
    • A series of books is not just a bunch of books, but a bunch of books in order.
    • In the case of the Harry Potter series the books are meant to be read in order.
    • A publisher might publish a series of books on science, named Physics, Chemistry,
      Astronomy, Biology,
      and so on, that are not meant to be read in order, but the publisher will still list them in order.(What else could they do? See Representing and thinking about sets.)

    Infinite series in math

    In mathematics an infinite series is an object expressed like this:

    \[\sum\limits_{k=1}^{\infty
    }{{{a}_{k}}}\]

    where the ${{a}_{k}}$ are numbers. It has partial sums

    \[\sum\limits_{k=1}^{n}{{{a}_{k}}}\]

    For example, if ${{a}_{k}}$ is defined to be $1/{{k}^{2}}$ for positive integers $k$, then

    \[\sum\limits_{k=1}^{6}{{{a}_{k}}}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}=\frac{\text{5369}}{\text{3600}}=\text{
    about }1.49\]

    This infinite series converges to $\zeta (2)$, which is about $1.65$. (This is not obvious. See the Zeta-function article in Wikipedia.) So this “infinite series” is really an infinite sum. It does not fit the image given by the English word “series”. The English meaning contaminates the mathematical meaning. But the definition wins.

    The mathematical word that corresponds to the usual meaning of “series” is “sequence”. For example, $a_k:=1/{{k}^{2}}$ is the infinite sequence $1,\frac{1}{4},\frac{1}{9},\frac{1}{16}\ldots$ It is not an infinite series.

    “Only if”

    “Only if” is also discussed from a more technical point of view in the article on conditional assertions.

    In math English, sentences of the form $P$ only if $Q$” mean exactly the same thing as “If $P$ then $Q$”. The phrase “only if” is rarely used this way in ordinary English discourse.

    Sentences of the form “$P$ only if $Q$” about ordinary everyday things generally do not mean the same thing as “If $P$ then $Q$”. That is because in such situations there are considerations of time and causation that do not come up with mathematical objects. Consider “If it is raining, I will carry an umbrella” (seeing the rain will cause me to carry the umbrella) and “It is raining only if I carry an umbrella” (which sounds like my carrying an umbrella will cause it to rain).   When “$P$ only if $Q$” is about math objects,
    there is no question of time and causation because math objects are inert and unchanging.

     Students sometimes flatly refuse to believe me when I tell them about the mathematical meaning of “only if”.  This is a classic example of semantic contamination.  Two sources of information appear to contradict each other, in this case (1) the professor and (2) a lifetime of intimate experience with the English language.  The information from one of these sources must be rejected or suppressed. It is hardly surprising that many students prefer to suppress the professor’s apparently unnatural and usually unmotivated claims.

    These words also cause severe cognitive dissonance

    • “If” causes notorious difficulties for beginners and even later. They are discussed in abmath here and here.
    • A, an
      and the implicitly signal the universal quantifier in certain math usages. They cause a good bit of trouble in the early days of some students.

    The following cause more minor cognitive dissonance.

    References for semantic contamination

    Besides the examples given above, you can find many others in these two works:

    • Pimm, D. (1987), Speaking Mathematically: Communications in Mathematics Classrooms.  Routledge & Kegan Paul.
    • Hersh, R. (1997),”Math lingo vs. plain English: Double entendre”. American Mathematical Monthly, vol 104,pages 48-51.
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    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 using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook GraphCograph.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work properly.

    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.

    Line segment

    $y=a x+b$

    Cubic

    $y=a x^3-b x$

    Sine

    $y=\sin a x$.

    Sine and its derivative

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

    Quintic with three parameters

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

    Thanks to Martin Flashman for corrections.

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    Modules for mathematical objects

    Notes on viewing.

    A recent article in Scientific American mentions discusses the idea that concepts are represented in the brain by clumps of neurons.  Other neuroscientists have proposed that each concept is distributed among millions of neurons, or that each concept corresponds to one neuron.  

    I have written many posts about the idea that:  

    • Each mathematical concept is embodied in some kind of module in the brain.
    • This idea is a useful metaphor for understanding how we think about mathematical objects.
    • You don't have to know the details of the method of storage for this metaphor to be useful.  
    • The metaphor clears up a number of paradoxes and conundrums that have agitated philosophers of math.

    The SA article inspired me to write about just how such a module may work in some specific cases.  

    Integers

    Mathematicians normally thinks of a particular integer, say $42$, as some kind of abstract object, and the decimal representation "42" as a representation of the integer, along with XLII and 2A$_{16}$.  You can visualize the physical process like this: 

    • The mathematician has a module Int (clump of neurons or whatever) that represents integers, and a module FT that represents the particular integer $42$. 
    • There is some kind of asymmetric three-way connection from FT to Int and a module EO (for "element of" or "IS_A"). 
    • That the connection is "asymmetric" means that the three modules play different roles in the connection, meaning something like "$42$ IS_A Integer"
    • The connection is a physical connection, not a sentence, and when  FT is alerted ("fired"?), Int and EO are both alerted as well. 
    • That means that if someone asks the mathematician, "Is $42$ an integer?", they answer immediately without having to think about it — it is a random access concept like (for many people) knowing that September has 30 days.
    • The module for $42$ has many other connections to other modules in the brain, and these connections vary among mathematicians.

    The preceding description gives no details about how the modules and interconnections are physically processed.  Neuroscientists probably would have lots of ideas about this (with no doubt considerable variation) and would criticize what I wrote as misrepresenting the physical details in some ways.  But the physical details are their job, not mine.  What I claim is that this way of thinking makes it plausible to view abstract objects and their properties and relationships as physical objects in the brain.  You don't have to know the details any more than you have to know the details of how a rainbow works to see it (but you know that a rainbow is a physical phenomenon).

    This way of thinking provides a metaphor for thinking about math objects, a metaphor that is plausibly related to what happens in the real world.

    Students

    A student may have a rather different representation of $42$ in the brain.  For one thing, their module for $42$ may not distinguish the symbol "42" from the number $42$, which is an abstract object.   As a result they ask questions such as, "Is $42$ composite in hexadecimal?"  This phenomenon reveals a complicated situation. 

    • People think they are talking about the same thing when in fact their internal modules for that thing may be very differently connected to other concepts in their brain.
    • Mathematicians generally share many more similarities in their modules for $42$ than people in general do.  When they differ, the differences may be of the sort that one of them is a number theorist, so knows more about $42$ (for example, that it is a Catalan number) than another mathematician does.  Or has read The Hitchhiker's Guide to the Galaxy.
    • Mathematicians also share a stance that there are right and wrong beliefs about mathematical objects, and that there is a received method for distinguishing correct from erroneous statements about a particular kind of object. (I am not saying the method always gives an answer!).
    • Of course, this stance constitutes a module in the brain. 
    • Some philosophers of education believe that this stance is erroneous, that the truth or falsity of statements are merely a matter of social acceptance.
    • In fact, the statements in purple are true of nearly all mathematicians.  
    • The fact that the truth or falsity of statements is merely a matter of social acceptance is also true, but the word "merely" is misleading.
    • The fact is that overwhelming evidence provided by experience shows that the "received method" (proof) for determining the truth of math statements works well and can be depended on. Teachers need to convince their students of this by examples rather that imposing the received method as an authority figure.

    Real numbers

    A mathematician thinks of a real number as having a decimal representation.

    • The representation is an infinitely long list of decimal digits, together with a location for the decimal point. (Ignoring conventions about infinite strings of zeroes.)
    • There is a metaphor that you can go along the list from left to right and when you do you get a better approximation of the "value" of the real number. (The "value" is typically thought of in terms of the metaphor of a point on the real line.)
    • Mathematicians nevertheless think of the entries in the decimal expansion of a real number as already in existence, even though you may not be able to say what they all are.
    • There is no contradiction between the points of view expressed in the last two bullets.
    • Students frequently do not believe that the decimal entries are "already there".  As a result they may argue fiercely that $.999\ldots$ cannot possibly be the same number as $1$.  (The Wikipedia article on this topic has to be one of the most thoroughly reworked math articles in the encyclopedia.)

    All these facts correspond to modules in mathematicians' and students' brains.  There are modules for real number, metaphor, infinite list, decimal digit, decimal expansion, and so on.  This does not mean that the module has a separate link to each one of the digits in the decimal expansion.  The idea that there is an entry at every one of the infinite number of locations is itself a module, and no one has ever discovered a contradiction resulting from holding that belief.

    References

    • Brain cells for Grandmother, by Rodrigo Quian Quiroga, Itzhak Fried and Christof Koch.  Scientific American, February 2013, pages 31ff.

    Gyre&Gimble posts on modules

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    Abstracting algebra

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    Representations of mathematical objects

    This is a long post. Notes on viewing.

    About this post

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

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

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

    Types of representations

    Using language

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

    Mathematical objects

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

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

    Visual representations

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

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

    Mental representations

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

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

    Metaphors

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

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

    Properties of representations

    A representation of a math object may or may not

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

    Examples of representations

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

    Functions

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

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

    Diatribe 

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

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

    The positive integers

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

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

    Graph of a function

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

    Graph of a cubic function

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

    Continuity

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

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

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

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

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

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

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

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

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

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

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

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

    References

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

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    Representing and thinking about sets

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

    Representations of sets

    Sets are represented in the math literature in several different ways, some mentioned here.  Also mentioned are some other possibilities.  Introducing a variety of representations of any type of math object is desirable because students tend to assume that the representation is the object.

    Curly bracket notation

    The standard representation for a finite set is of the form "$\{1,3,5,6\}$". This particular example represents the unique set containing the integers $1$, $3$, $5$ and $6$ and nothing else. This means precisely that the statement "$n$ is an element of $S$" is true if $n=1$, $n=3$, $n=5$ or $n=6$, and it is false if $n$ represents any other mathematical object. 

    In the way the notation is usually used, "$\{1,3,5,6\}$", "$\{3,1,5,6\}$", "$\{1,5,3,6\}$",  "$\{1,6,3,5,1\}$" and $\{ 6,6,3,5,1,5\}$ all represent the same set. Textbooks sometimes say "order and repetition don't matter". But that is a statement about this particular representation style for sets. It is not a statement about sets.

    It would be nice to come up with a representation for sets that doesn't involve an ordering. Traditional algebraic notation is essentially one-dimensional and so automatically imposes an ordering (see Algebra is a difficult foreign language).    

    Let the elements move

    In Visible Algebra II, I experimented with the idea of putting the elements at random inside a circle and letting them visibly move around like goldfish in a bowl.  (That experiment was actually for multisets but it applies to sets, too.)  This is certainly a representation that does not impose an ordering, but it is also distracting.  Our visual system is attracted to movement (but not as much as a cat's visual system).  

    Enforce natural ordering

    One possibility would be to extend the machinery in a visible algebra system that allows you to make a box you could drag elements into. 

    This box would order the elements in some canonical order (numerical order for numbers, alphabetical order for strings of letters or words) with the property that if you inserted an element in the wrong place it would rearrange itself, and if you tried to insert an element more than once the representation would not change.  What you would then have is a unique representation of the set.

    An example is the device below.  (If you have Mathematica, not just CDF player, you can type in numbers as you wish instead of having to use the buttons.) 

    This does not allow a representation of a heterogenous set such as $\{3,\mathbb{R},\emptyset,\left(\begin{array}{cc}1&2\\0&1\\ \end{array}\right)\}$.  So what?  You can't represent every function by a graph, either.

    Hanger notation

    The tree notation used in my visual algebra posts could be used for sets as well, as illustrated below. The system allows you to drag the elements listed into different positions, including all around the set node. If you had a node for lists, that would not be possible.

    This representation has the pedagogical advantage of shows that a set is not its elements.

    • A set is distinct from its elements
    • A set is completely determined by what the elements are.

    Pattern recognition

    Infinite sets are sometimes represented using the curly bracket notation using a pattern that defines the set.  For example, the set of even integers could be represented by $\{0,2,4,6,\ldots\}$.  Such a representation is necessarily a convention, since any beginning pattern can in fact represent an infinite number of different infinite sets.  Personally, I would write, "Consider the even integers $\{0,2,4,6,\ldots\}$", but I would not write,  "Consider the set $\{0,2,4,6,\ldots\}$".

    By the way, if you are writing for newbies, you should say,"Consider the set of even integers $\{0,2,4,6,\ldots\}$". The sentence "Consider the even integers $\{0,2,4,6,\ldots\}$" is unambiguous because by convention a list of numbers in curly brackets defines a set. But newbies need lots of redundancy.

    Representation by a sentence

    Setbuilder notation is exemplified by $\{x|x>0\}$, which denotes the positive reals, given a convention or explicit statement that $x$ represents a real number.  This allows the representation of some infinite sets without depending on a possibly ambiguous pattern. 

    A Visible Algebra system needs to allow this, too. That could be (necessarily incompletely) done in this way:

    • You type in a sentence into a Setbuilder box that defines the set.
    • You then attach a box to the Setbuilder box containing a possible element.
    • The system then answers Yes, No, or Can't Tell.

    The Can't Tell answer is a necessary requirement because the general question of whether an element is in a set defined by a first order sentence is undecidable. Perhaps the system could add some choices:

    • Try for a second.
    • Try for an hour.
    • Try for a year.
    • Try for the age of the universe.

    Even so, I'll bet a system using Mathematica could answer many questions like this for sentences referring to a specific polynomial, using the Solve or NSolve command.  For example, the answer to the question, "Is $3\in\{n|n\lt0 \text{ and } n^2=9\}$?" (where $n$ ranges over the integers) would be "No", and the answer to  "Is $\{n|n\lt0 \text{ and } n^2=9\}$ empty?" would also be "No". [Corrected 2012.10.24]

    References

    1. Explaining “higher” math to beginners (previous post)
    2. Algebra is a difficult foreign language (previous post)
    3. Visible Algebra II (previous post)
    4. Sets: Notation (abstractmath article)
    5. Setbuilder notation (Wikipedia)
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    Conceptual blending

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

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

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

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

    A particular function

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

    FORMULA:

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

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

    TREE: 

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

    GRAPH:

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

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

    GEOMETRIC

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

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

    HEIGHT:

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

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

    KINETIC:

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

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

    OBJECT:

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

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

    DEFINITION

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

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

    SPECIFICATION

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

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

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

    Notes

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

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

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

    References

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

     

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