## Representations of mathematical objects

### MathJax.Hub.Config({ jax: ["input/TeX","output/NativeMML"], extensions: ["tex2jax.js"], tex2jax: { inlineMath: [ ['$','$'] ], processEscapes: true } });

This is a long post. Notes on viewing.

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

## 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 $y=x^3-x$:

• 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.

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

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

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

## References

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

### Notes on Viewing

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

## A visualization of a computation in tree form

To manipulate the demo below, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website.

This demonstration shows the step by step computation of the value of the expression $3x^2+2(1+y)$ shown as a tree.  By moving the first slider from right to left, you go through the six steps of the computation. You may select the values of $x$ and $y$ with the second and third sliders.  If you click on the plus sign next to a slide, a menu opens up that allows you to make the slider move automatically, shows the values, and other things.

Note that subtrees on the same level are evaluate left to right.  Parallel processing would save two steps.

The code for this demo is in the file  Live evaluation of expressions in TreeForm 3.  The code is ad-hoc.  It might be worthwhile for someone to design a package that produces this sort of tree for any expression.

A previous post related to this post is Making visible the abstraction in algebraic notation.

Note: To manipulate the diagrams in this post and in most of the files it links to, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website.

The diagram above shows you the tangent line to the curve $y=x^3-x$ at a specific point.  The slider allows you to move the point around, and the tangent line moves with it. You can click on one of the plus signs for options about things you can do with the slider.  (Note: This is not new.  Many other people have produced diagrams like this one.)

A diagram showing a tangent line drawn on the board or in a paper book requires you visualize how the tangent line would look at other points.  This imposes a burden of visualization on you.  Even if you are a new student you won't find that terribly hard (am I wrong?) but you might miss some things at first:

• There are places where the tangent line is horizontal.
• There are places where some of the tangent lines cross the curve at another point. Many calculus students believe in the myth that the tangent line crosses the curve at only one point.  (It is not really a myth, it is a lie.  Any decent myth contains illuminating stories and metaphors.)
• You may not envision (until you have some experience anyway) how when you move the tangent line around it sort of rocks like a seesaw.

You see these things immediately when you manipulate the slider.

Manipulating the slider reduces the load of abstract thinking in your learning process.     You have less to keep in your memory; some of the abstract thinking is offloaded onto the diagram.  This could be described as contracting out (from your head to the picture) part of the visualization process.  (Visualizing something in your head is a form of abstraction.)

Of course, reading and writing does that, too.  And even a static graph of a function lowers your visualization load.  What interactive diagrams give the student is a new tool for offloading abstraction.

You can also think of it as providing external chunking.  (I'll have to think about that more…)

### Simple manipulative diagrams vs. complicated ones

The diagram above is very simple with no bells and whistles.  People have come up with much more complicated diagrams to illustrate a mathematical point.  Such diagrams:

• May give you buttons that give you a choice of several curves that show the tangent line.
• May give a numerical table that shows things like the slope or intercept of the current tangent line.
• May also show the graph of the derivative, enabling you to see that it is in fact giving the value of the slope.

Such complicated diagrams are better suited for the student to play with at home, or to play with in class with a partner (much better than doing it by yourself).  When the teacher first explains a concept, the diagrams ought to be simple.

### Examples

• The Definition of derivative demo (from the Wolfram Demonstration Project) is an example that provides a table that shows the current values of some parameters that depend on the position of the slider.
• The Wolfram demo Graphs of Taylor Polynomials is a good example of a demo to take home and experiment extensively with.  It gives buttons to choose different functions, a slider to choose the expansion point, another one to choose the number of Taylor polynomials, and other things.
• On the other hand, the Wolfram demo Tangent to a Curve is very simple and differs from the one above in one respect: It shows only a finite piece of the tangent line.  That actually has a very different philosophical basis: it is representing for you the stalk of the tangent space at that point (the infinitesimal vector that contains the essence of the tangent line).
• Brian Hayes wrote an article in American Scientist containing a moving graph (it moves only  on the website, not in the paper version!) that shows the changes of the population of the world by bars representing age groups.  This makes it much easier to visualize what happens over time.  Each age group moves up the graph — and shrinks until it disappears around age 100 — step by step.  If you have only the printed version, you have to imagine that happening.  The printed version requires more abstract visualization than the moving version.
• Evaluating an algebraic expression requires seeing the abstract structure of the expression, which can be shown as a tree.  I would expect that if the students could automatically generate the tree (as you can in Mathematica)  they would retain the picture when working with an expression.  In my post computable algebraic expressions in tree form I show how you could turn the tree into an evaluation aid.  See also my post Syntax trees.

This blog has a category "Mathematica" which contains all the graphs (many of the interactive) that are designed as an aid to offloading abstraction.

### The abstraction cliff

In universities in the USA, a math major typically starts with calculus, followed by courses such as linear algebra, discrete math, or a special intro course for math majors (which may be taken simultaneously with calculus), then go on to abstract algebra, analysis, and other courses involving abstraction and proofs.

At this point, too many of them hit a wall; their grades drop and they change majors.  They had been getting good grades in high school and in calculus because they were strong in algebra and geometry, but the sudden increase in abstraction in the newer courses completely baffles them. I believe that one big difficulty is that they can't grasp how to think about abstract mathematical objects.  (See Reference [9] and note [a].)   They have fallen off the abstraction cliff.  We lose too many math majors this way. (Abstractmath.org is my major effort to address the problems math majors have during or after calculus.)

This post is a summary of the way I see how mathematicians and students think about math.  I will use it as a reference in later posts where I will write about how we can communicate these ways of thinking.

### Concept Image

In 1981, Tall and Vinner  [5] introduced the notion of the concept image that a person has about a mathematical concept or object.   Their paper's abstract says

The concept image consists of all the cognitive structure in the individual's mind that is associated with a given concept. This may not be globally coherent and may have aspects which are quite different from the formal concept definition.

The concept image you may have of an abstract object generally contains many kinds of constituents:

• visual images of the object
• metaphors connecting the object to other concepts
• descriptions of the object in mathematical English
• descriptions and symbols of the object in the symbolic language of math
• kinetic feelings concerning certain aspects of the object
• how you calculate parameters of the object
• how you prove particular statements about the object

This list is incomplete and the items overlap.  I will write in detail about these ideas later.

The name "concept image" is misleading [b]), so when I have written about them, I have called them metaphors or mental representations as well as concept images, for example in [3] and [4].

### Abstract mathematical concepts

This is my take on the notion of concept image, which may be different from that of most researchers in math ed. It owes a lot to the ideas of Reuben Hersh [7], [8].

• An abstract mathematical concept is represented physically in your brain by what I have called "modules" [1] (physical constituents or activities of the brain [c]).
• The representation generally consists of many modules.  They correspond to the list of constituents of a concept image given above.  There is no assumption that all the modules are "correct".
• This representation exists in a semi-public network of mathematicians' and students' brains. This network exercises (incomplete) control over your personal representation of the abstract structure by means of conversation with other mathematicians and reading books and papers.  In this sense, an abstract concept is a social object.  (This is the only point of view in the philosophy of math that I know of that contains any scientific content.)

### Notes

[a]  Before you object that abstraction isn't the only thing they have trouble with, note that a proof is an abstract mathematical object. The written proof is a representation of the abstract structure of the proof.  Of course, proofs are a special kind of abstract structure that causes special problems for students.

[b] Cognitive science people use "image" to include nonvisual representations, but not everyone does.  Indeed, cognitive scientists use "metaphor" as well with a broader meaning than your high school English teacher.  A metaphor involves the cognitive merging of parts of two concepts (specifically with other parts not merged). See [6].

[c] Note that I am carefully not saying what the modules actually are — neurons, networks of neurons, events in the brain, etc.   From the point of view of teaching and understanding math, it doesn't matter what they are, only that they exist and live in a society where they get modified by memes  (ideas, attitudes, styles physically transmitted from brain to brain by speech, writing, nonverbal communication, appearance, and in other ways).

### References

1. Math and modules of the mind (previous post)
2. Mathematical Concepts (previous post)
3. Mental, physical and mathematical representations (previous post)
4. Images and Metaphors (abstractmath.org)
5. David Tall and Schlomo Vinner, Concept Image and Concept Definition in Mathematics with particular reference to limits and continuity, Journal Educational Studies in Mathematics, 12 (May, 1981), no. 2, 151–169.
6. Conceptual metaphor (Wikipedia article).
7. What is mathematics, really? by Reuben Hersh, Oxford University Press, 1999.  Read online at Questia.
8. 18 Unconventional Essays on the Nature of Mathematics, by Reuben Hersh. Springer, 2005.
9. Mathematical objects (abstractmath.org).

## Demonstrating the inverse image of an interval

To manipulate the graph below, you must have Wolfram CDF Player installed on your computer. It is available free from their website.

You may find abstract math difficult because the symbolism hides an elaborate algebraic structure which must be visualized mentally to understand the meaning.  Several recent posts here ([1], [2], [3],[4]) have described controllable, explicit visualizations of some part of this abstract structure using Wolfram CDF Player. Such visualizations should help understand the structures, so that the visualizations will come to mind when you work with them later. Not only that, but a more sophisticated packaging of this code might conceivably be worked with directly.  Jason Dyer’s blog has examples of how this might work. Maybe twenty years from now people using math would expect to work directly with an interactive visualization of (some of) the abstract structure.

The graph below allows you to visualize the inverse image of an interval under the function $latex f(x)=x^3-2x$.  Before now, a teacher may illustrate one instance of this picture by drawing it hastily on a whiteboard.  Now they can project it on a screen and point out various phenomena, for example noticing that the inverse image of different intervals may have one, two or three components.  Not only that, the student can access this picture at home and experiment with it.