Modules for mathematical objects

2013/01/15 — SixWingedSeraph

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

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.

 

 

 
Posted in exposition, math, proof, representations, understanding math. Tags: , , , , , , , , , . No Comments »

Abstracting algebra

2012/12/21 — SixWingedSeraph

This post has been turned into a page on WordPress, accessible in the upper right corner of the screen.  The page will be referred to by all topic posts for Abstracting Algebra.

 

Posted in abstractmath.org, exposition, math, Mathematica, Uncategorized, understanding math. Tags: , , , , , , , , , , , , , , , , , , , , , , , . No Comments »

Representations of mathematical objects

2012/11/02 — SixWingedSeraph

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)
  3. Conceptual blending (post)
  4. Conceptual blending (Wikipedia)
  5. Conceptual metaphors (Wikipedia)
  6. Convention (abstractmath)
  7. Definitions (abstractmath)
  8. Embodied cognition (Wikipedia)
  9. Handbook of mathematical discourse (see articles on conceptual blend, mental representation, representation, metaphor, parenthetic assertion)
  10. Images and Metaphors (abstractmath).
  11. The interplay of text, symbols and graphics in math education, Lin Hammill
  12. Math and the modules of the mind (post)
  13. Mathematical discourse: Language, symbolism and visual images, K. L. O’Halloran.
  14. Mathematical objects (abmath)
  15. Mathematical objects (Wikipedia)
  16. Mathematical objects are “out there?” (post)
  17. Metaphors in computing science ​(post)
  18. Procept (Wikipedia)
  19. Representations 2 (post)     
  20. Representations and models (abstractmath)
  21. Representations II: dry bones (post)
  22. Representation theorems (Wikipedia) Concrete representations of abstractly defined objects.
  23. Representation theory (Wikipedia) Linear representations of algebraic structures.
  24. Semiotics, symbols and mathematical visualization, Norma Presmeg, 2006.
  25. The transition to formal thinking in mathematics, David Tall, 2010
  26. Theory in mathematical logic (Wikipedia)
  27. What is the object of the encapsulation of a process? Tall et al., 2000.
  28. Where mathematics comes from, by George Lakoff and Rafael Núñez, Basic Books, 2000. 
  29. 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.

Posted in exposition, math, representations, understanding math. Tags: , , , , , , , , , , , , , , , , , , , , , , . 2 Comments »

Representing and thinking about sets

2012/10/12 — SixWingedSeraph

Notes on viewing

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)

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.
  • To manipulate the demos in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website. The code for the demos is in the Mathematica notebook Representing sets.nb.  
Posted in exposition, language, language of math, math, representations, understanding math. Tags: , , , , , , , , , . No Comments »

Conceptual blending

2012/06/18 — SixWingedSeraph

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

A 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 blend, mental representation, representation, 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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Thinking about abstract math

2011/11/01 — SixWingedSeraph

 

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).

 

 

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Endograph and cograph of real functions

2011/06/09 — Charles Wells

This post is covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the work provided you follow the requirements of the license.

Introduction

In the article Functions: Images and Metaphors in abstractmath I list a bunch of different images or metaphors for thinking about functions. Some of these metaphors have realizations in pictures, such as a graph or a surface shown by level curves. Others have typographical representations, as formulas, algorithms or flowcharts (which are also pictorial). There are kinetic metaphors — the graph of $latex {y=x^2}&fg=000000$ swoops up to the right.

Many of these same metaphors have realizations in actual mathematical representations.

Two images (not mentioned in the abstractmath article) are the cograph and the endograph of a real function of one variable. Both of these are visualizations that correspond to mathematical representations. These representations have been used occasionally in texts, but are not used as much as the usual graph of a continuous function. I think they would be useful in teaching and perhaps even sometimes in research.

A rough and unfinished Mathematica notebook is available that contains code that generate graphs and cographs of real-valued functions. I used it to generate most of the examples in this post, and it contains many other examples. (Note [1].)

The endograph of a function

In principle, the endograph (Note [2]) of a function $latex {f}&fg=000000$ has a dot for each element of the domain and of the codomain, and an arrow from $latex {x}&fg=000000$ to $latex {f(x)}&fg=000000$ for each $latex {x}&fg=000000$ in the domain. For example, this is the endograph of the function $latex {n\mapsto n^2+1 \pmod 11}&fg=000000$ from the set $latex {\{0,1,\ldots,10\}}&fg=000000$ to itself:


“In principle” means that the entire endograph can be shown only for small finite functions. This is analogous to the way calculus books refer to a graph as “the graph of the squaring function” when in fact the infinite tails are cut off.

Real endographs

I expect to discuss finite endographs in another post. Here I will concentrate on endographs of continuous functions with domain and codomain that are connected subsets of the real numbers. I believe that they could be used to good effect in teaching math at the college level.

Here is the endograph of the function $latex {y=x^2}&fg=000000$ on the reals:

I have displayed this endograph with the real line drawn in the usual way, with tick marks showing the location of the points on the part shown.

The distance function on the reals gives us a way of interpreting the spacing and location of the arrowheads. This means that information can be gleaned from the graph even though only a finite number of arrows are shown. For example you see immediately that the function has only nonnegative values and that its increase grows with $latex {x}&fg=000000$.(See note [3]).

I think it would be useful to show students endographs such as this and ask them specific questions about why the arrows do what they do.

For the one shown, you could ask these questions, probably for class discussion rather that on homework.

  • Explain why most of the arrows go to the right. (They go left only between 0 and 1 — and this graph has such a coarse point selection that it shows only two arrows doing that!)
  • Why do the arrows cross over each other? (Tricky question — they wouldn’t cross over if you drew the arrows with negative input below the line instead of above.)
  • What does it say about the function that every arrowhead except two has two curves going into it?

Real Cographs

The cograph (Note [4] of a real function has an arrow from input to output just as the endograph does, but the graph represents the domain and codomain as their disjoint union. In this post the domain is a horizontal representation of the real line and the codomain is another such representation below the domain. You may also represent them in other configurations (Note [5]).

Here is the cograph representation of the function $latex {y=x^2}&fg=000000$. Compare it with the endograph representation above.

Besides the question of most arrows going to the right, you could also ask what is the envelope curve on the left.

More examples

Absolute value function

Arctangent function

Notes

[1] This website contains other notebooks you might find useful. They are in Mathematica .nb, .nbp, or .cdf formats, and can be read, evaluated and modified if you have Mathematica 8.0. They can also be made to appear in your browser with Wolfram CDF Player, downloadable free from Wolfram site. The CDF player allows you to operate any interactive demos contained in the file, but you can’t evaluate or modify the file without Mathematica.

The notebooks are mostly raw code with few comments. They are covered by the Creative Commons Attribution – ShareAlike 3.0 License, which means you may use, adapt and distribute the code following the requirements of the license. I am making the files available because I doubt that I will refine them into respectable CDF files any time soon.

[2] I call them “endographs” to avoid confusion with the usual graphs of functions — – drawings of (some of) the set of ordered pairs $latex {x,f(x)}&fg=000000$ of the function.

[3] This is in contrast to a function defined on a discrete set, where the elements of the domain and codomain can be arranged in any old way. Then the significance of the resulting arrangement of the arrows lies entirely in which two dots they connect. Even then, some things can be seen immediately: Whether the function is a cycle, permutation, an involution, idempotent, and so on.

Of course, the placement of the arrows may tell you more if the finite sets are ordered in a natural way, as for example a function on the integers modulo some integer.

[4] The text [1] uses the cograph representation extensively. The word “cograph” is being used with its standard meaning in category theory. It is used by graph theorists with an entirely different meaning.

[5] It would also be possible to show the domain codomain in the usual $latex {x-y}&fg=000000$ plane arrangement, with the domain the $latex {x}&fg=000000$ axis and the codomain the $latex {y}&fg=000000$ axis. I have not written the code for this yet.

References

[1] Sets for Mathematics, by F. William Lawvere and Robert Rosebrugh. Cambridge University Press, 2003.

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Representations 2

2011/05/25 — Charles Wells

Introduction

In a recent post I began a discussion of the mental, physical and mathematical representations of a mathematical object. The discussion continues here. Mathematicians, linguists, cognitive scientists and math educators have investigate some aspects of this topic, but there are many subtle connections between the different ideas which need to be studied.

I don’t have any overall theoretical grasp of these relationships. What I will do here is grope for an overall theory by mentioning a whole bunch of fine points. Some of these have been discussed in the literature and some (as far as I know) have not been discussed.  Many of them (I hope)  can be described as “an obvious fact about representations but no one has pointed it out before”.  Such fine points could be valuable; I think some scholars who have written about mathematical discourse and math in the classroom are not aware of many of these facts.

I am hoping that by thrashing around like this here (for graphs of functions) and for other concepts (set, function, triangle, number …) some theoretical understanding may emerge of what it means to understand math, do math, and talk about math.

The graph of a function

Let’s look at the graph of the function $latex {y=x^3-x}&fg=000000$.

What you are looking at is a physical representation of the graph of the function. The graph creates in your brain a mental representation of the graph of the function. These are subtly related to each other and to the mathematical definition of the graph.

Fine points

  1. The mathematical definition [2] of the graph of this function is: The set of ordered pairs of numbers $latex {(x,x^3-x)}&fg=000000$ for all real numbers $latex {x}&fg=000000$.
  2. In the physical representation, each point $latex {(x,x^3-x)}&fg=000000$ is shown in a location determined by the conventional $latex {x-y}&fg=000000$ coordinate system, which uses a straight-line representation of the real numbers with labels and ticks.
    • The physical representation makes use of the fact that the function is continuous. It shows the graph as a curving line rather than a bunch of points.
    • The physical representation you are looking at is not the physical representation I am looking at. They are on different computer screens or pieces of paper. We both expect that the representations are very similar, in some sense physically isomorphic.
    • “Location” on the physical representation is a physical idea. The mathematical location on the mathematical graph is essentially the concept of the physical location refined as the accuracy goes to infinity. (This last statement is a metaphor attached to a genuine mathematical construction, for example Cauchy sequences.)
  3. The mathematical definition of “graph” and the physical representation are related by a metaphor. (See Note 1.)
    • The physical curve in blue in the picture corresponds via the metaphor to the graph in the mathematical sense: in this way, each location on the physical curve corresponds to an ordered pair of the form $latex {(x,x^3-x)}&fg=000000$.
    • The correspondence between the locations and the pairs is imperfect. You can’t measure with infinite accuracy.
    • The set of ordered pairs $latex {(x,x^3-x)}&fg=000000$ form a parametrized curve in the mathematical sense. This curve has zero thickness. The curve in the physical representation has positive thickness.
    • Not all the points in the mathematical graph actually occur on the physical curve: The physical curve doesn’t show the left and right infinite tails.
    • The physical curve is drawn to show some salient characteristics of the curve, such as its extrema and inflection points. This is expected by convention in mathematical writing. If the graph had left out a maximum, for example, the author would be constrained (by convention!) to say so.
    • An experienced mathematician or advanced student understands the fine points just listed. A newbie may not, and may draw false conclusions about the function from the graph. (Note 2.)
  4. If you are a mathematician or at least a math student, seeing the physical graph shown above produces a mental image(see Note 3.) of the graph in your mind.
  5. The mathematical definition and the mental image are connected by a metaphor. This is not the same metaphor as the one that connects the physical representation and the mathematical definition.
    • The curve I visualize in my mental representation has an S shape and so does the physical representation. Or does it? Isn’t the S-ness of the shape a fact I construct mentally (without consciously intending to do so!)?
    • Does the curve in the mental rep have thickness? I am not sure this is a meaningful question. However, if you are a sufficiently sophisticated mathematician, your mental image is annotated with the fact that the curve has zero thickness. (See Note 4.)
    • The curve in your mental image of the curve may very well be blue (just because you just looked at my picture) but you must have an annotation to the effect that that is irrelevant! That is the essence of metaphor: Some things are identified with each other and others are emphatically not identified.
    • The coordinate axes do exist in the physical representation and they don’t exist in the mathematical definition of the graph. Of course they are implied by the definition by the properties of the projection functions from a product. But what about your mental image of the graph? My own image does not show the axes, but I do “know” what the coordinates of some of the points are (for example, $latex {(-1,0)}&fg=000000$) and I “see” some points (the local maximum and the local minimum) whose coordinates I can figure out.

Notes

1. This is metaphor in the sense lately used by cognitive scientists, for example in [6]. A metaphor can be described roughly as two mental images in which certain parts of one are identified with certain parts of another, in other words a pushout. The rhetorical use of the word “metaphor” requires it to be a figure of speech expressed in a certain way (the identification is direct rather than expressed by “is like” or some such thing.)  In my use in this article a metaphor is something that occurs in your brain.  The form it takes in speech or writing is not relevant.

2. I have noticed, for example, that some students don’t really understand that the left and right tails go off to infinity horizontally as well as vertically.   In fact, the picture above could mislead someone into thinking the curve has vertical asymptotes: The right tail looks like it goes straight up.  How could it get to x equals a billion if it goes straight up?

3. The “mental image” is of course a physical structure in your brain.  So mental representations are physical representations.

4. I presume this “annotation” is some kind of physical connection between neurons or something.  It is clear that a “mental image” is some sort of physical construction or event in the brain, but from what little I know about cognitive science, the scientists themselves are still arguing about the form of the construction.  I would appreciate more information on this. (If the physical representation of mental images is indeed still controversial, this says nothing bad about cognitive science, which is very new.)

References

[1] Mental Representations in Math (previous post).

[2] Definitions (in abstractmath).

[3] Lakoff, G. and R. E. Núñez (2000), Where Mathematics Comes From. Basic Books.

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Mental, Physical and Mathematical Representations

2011/05/17 — Charles Wells

For a given mathematical object, a mathematician may have:

  • A mental representation of the object. This can be a metaphor, a mental image, or a kinesthetic understanding of the object.
  • A physical representation of the object. This may be a (physical) picture or drawing or three-dimensional model of the object.
  • A mathematical definition and one or more mathematical representations for the object. Such a representation is itself a mathematical object.

The boldface things in this list are related to each other in lots of ways, and they are fuzzy and overlap and don’t include every phenomenon connected with a math object.

I have written about these things ([1], [2], [3], [4]). So have lots of other people. In this post I summarize these ideas. I expect to write about particular examples later on and will use this as a reference.

Two Examples

The following examples point out a few of the relationships between the ideas in boldface above. There is much more to understand.

Function as black box

The idea that a function is a black box or machine with input and output is a metaphor for a function.

A is a metaphor for B means that A and B are cognitively pasted together in such a way that the behavior of A is in many ways like the behavior of B. Such a thing is both useful and dangerous, dangerous because there will be ways in which A behaves that suggest inappropriate ideas about B.

The function as machine is a good metaphor: for example functional composition involves connecting the output of one machine to the input of another, and the inverse function is like running the machine backward.

The function as machine is a bad metaphor: For example, it is wrong to think you could build a machine to calculate any given function exactly. But you can still imagine such a machine, given by a specification (it outputs the value of the function at a given input) and then, in your imagination, connecting the input of one to the output of another must perforce calculate the composite of the corresponding functions.

Like any metaphor, this is a mental representation. That means the metaphor has a physical instantiation in your brain. So a metaphor has a physical representation.

Different people won’t have quite the same concept of a particular metaphor. So a metaphor will have lots of slightly different physical representations, but mathematicians form a community, and communication between mathematicians fine-tunes the different physical instantiations so that they correspond more closely to each other. This is the sense in which mathematical objects have a shared existence in a community as Reuben Hersh has suggested.

A function is a mathematical object, which can be rigorously specified as a set of ordered pairs together with a domain and a codomain. There is a cognitive relationship between the concepts of function as math object and function as black box with input and output.

Triangle

A triangle can be drawn, or created on a computer and a physical image printed out. You may also have a mental image of the triangle.

The physical and the mental images are not the same thing, but they are definitely related. The relationship is mediated by the neuronal circuitry behind your retinas, which performs a highly sophisticated transformation of the pixels on your retina into an organized physical structure in your brain, connected to various other neurons.

This circuitry exists because it helps us get a useful understanding of the world through our eyes. So a picture of a triangle takes advantage of pre-existing neuron structure to generate a useful mental representation that helps us understand and prove things about triangles.

This mental representation also lives in a community of mathematician. Like any community, it has subgroups with “dialects” — varying understanding of representation.

For example, a mathematician who looks at the triangle below sees a triangle that looks like a right triangle. A student sees a triangle that is a right triangle.

This is “sees” in the sense of what their brain reports after all that processing. The mathematician’s brain connects the “triangle I am seeing” module (in their brain) to the “looks like a right triangle” module, but does not connect it to the “is a right triangle” module because they don’t see any statement in the surrounding text that it is a right triangle. The student, on the other hand, fallaciously makes the connection to “is a right triangle” directly.

In some sense, a student who does not make that connection directly is already a mathematician.

A triangle also exists as a mathematical object in your and my brain. It is described by a formal mathematical definition. The pictures of triangles you see above do not fit this definition. For one thing, the line segments in the pictures have thickness. But the pictures trigger a reaction in your neurons that causes your brain to cognitively paste together the line segments in the drawing to the segments required by the formal definition. This is a kind of metaphor of concrete-to-abstract that connects drawings to math objects that mathematicians use all the time.

Note that this “concrete-to-abstract metaphor” itself has a physical existence in your brain.  It drops, for example, the property of thickness that the line segments in the drawing have when matching them (in the metaphor) with the line segments in the corresponding abstract triangle.  On the other hand, it preserves the sense the all three angles in the triangle are acute.  The abstract mathematical concept of triangle (the generic triangle) has no requirement on the angles except that they add up to pi.

Summary

The discussions above describe a few of the complex and subtle relationships that exist between

  • Mental representations of math objects
  • Physical representations of math objects
  • Formally defined math objects and their formally defined representations.

I have purported to discuss how mathematics is understood (especially in connection with language) in several articles and a book but only a few of the relationships I just described are mentioned in any of those articles. Perhaps one or two things I said caused you to react: “Actually, that’s obviously true but I never thought of it before”. (Much the way I had mathematicians in the ’60′s tell me, “I see what you mean that addition is a function of two variables, but I never thought of it that way before”.) (I was a brash category theorist wannabe then.)

A lot of research has been done on understanding math, and some research has been done on mathematical discourse. But what has been done has merely exposed the fin of the shark.

References

[1] Images and metaphors (in abstractmath).

[2] Representations and Models (in abstractmath).

[3] Mathematical Concepts (previous blog).

[4] Mental Representations in Math (previous blog).

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Mathematical concepts

2009/06/10 — Charles Wells

This post was triggered by John Armstrong’s comment on my last post.

We need  to distinguish two ideas: representations of a mathematical concept and the total concept.  (I will say more about terminology later.)

Example: We can construct the quotient of the kernel of a group homomorphism by taking its cosets and defining a multiplication on them.  We can construct the image of the homomorphism by take the set of values of the homomorphism and using the multiplication induced by the codomain group.   The quotient group and the image are the same mathematical structure in the sense that anything useful you can say about one is true of the other.   For example, it may be useful to know the cardinality of the quotient (image) but it is not useful to know what its elements are.

But hold on, as the Australians say, if we knew that the codomain was an Abelian group then we would know that the quotient group was abelian because the elements of the image form a subgroup of the codomain. (But the Australians I know wouldn’t say that.)

Now that kind of thinking is based on the idea that the elements of the image are “really” elements of the codomain whereas elements of the quotients are “really” subsets of the domain.  That is outmoded thinking.  The image and the quotient are the same in all important aspects because they are naturally isomorphic.   We should think of the quotient as just as much as subgroup of the codomain as the image is.  John Baez (I think) would say that to ask whether the subgroup embedding is the identity on elements or not is an evil question.

Let’s step back and look at what is going on here.  The definition of the quotient group is a construction using cosets.  The definition of the image is a construction using values of the homomorphism.  Those are two different specific  representations of the same concept.

But what is the concept, as distinct from its representations?  Intuitively, it is

  • All the constructions made possible by the definition of the concept.
  • All the statements that are true about the concept.

(That is not precise.)

The total concept is like the clone plus the equational theory of a specific type of algebra in the sense of universal algebra.  The clone is all the operations you can construct knowing the given signature and equations and the equational theory is the set of all equations that follow from them.  That is one way of describing it.  Another is the monad in Set that gives the type of algebra — the operations are the arrows and the equations are the commutative diagrams.

Note: The preceding description of the monad is not quite right.  Also the whole discussion omits mention of the fact that we are in the world (doctrine) of universal algebra.  In the world of first order logic, for example, we need to refer to the classifying topos of the category of algebras of that type (or to its first order theory).

Terminology

We need better terminology for all this.  I am not going to propose better terminology, so this is a shaggy dog story.

Math ed people talk about a particular concept image of a concept as well as the total schema of the concept.

In categorical logic, we talk about the sketch or presentation of the concept vs. the theory. The theory is a category (of the kind appropriate to the doctrine) that contains all the possible constructions and commutative diagrams that follow from the presentation.

In this post I have used “total concept” to refer to the schema or theory.  I have referred the particular things as  “representations” (for example construct the image of a homomorphism by cosets or by values of the homomorphism).

“Representation” does not have the same connotations as “presentation”.  Indeed a presentation of a group and a representation of a group are mathematically  two different things.  But I suspect they are two different aspects of the same idea.

All this needs to be untangled.  Maybe we should come up with two completely arbitrary words, like “dostak” and “dosh”.

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