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.

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Discussions about algebra

2012/11/01 — SixWingedSeraph

I want to call your attention to the following items about ways to avoid algebra and why you should and shouldn't.

Don't kill math, by Evan Miller

Inventing on principle, video by Bret Victor

 

Posted in exposition, language of math, math, representations, understanding math. Tags: , , , , . No 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 »

Explaining “higher” math to beginners

2012/09/21 — SixWingedSeraph


Notes on viewing

Explaining math

I am in the process of writing an explanation of monads for people with not much math background.  In that article, I began to explain my ideas about exposition for readers at that level and after I had written several paragraphs decided I needed a separate article about exposition.  This is that article. It is mostly about language.

Who is it written for?

Interested laypeople

There are many recent books explaining some aspect of math for people who are not happy with high school algebra; some of them are listed in the references.  They must be smart readers who know how to concentrate, but for whom algebra and logic and definition-theorem-proof do not communicate.  They could be called interested laypeople, but that is a lousy name and I would appreciate suggestions for a better name. 

Math newbies

My post on monads is aimed at people who have some math, and who are interested in "understanding" some aspect of "higher math"; not understanding in the sense of being able to prove things about monads, but merely how to think about them.   I will call them math newbies.  Of course, I am including math majors, but I want to make it available to other people who are willing to tackle mathematical explanations and who are interested in knowing more about advanced stuff. 

These "other people" may include people (students and practitioners) in other science & technology areas as well as liberal-artsy people.  There are such people, I have met them.  I recall one theologian who asked me about what was the big deal about ruler-and-compass construction and who seemed to feel enlightened when I told him that those constructions preserve exactly the ideal nature of geometric objects.  (I later found out he was a famous theologian I had never heard of, just like Ngô Bảo Châu is a famous mathematician nonmathematicians have never heard of.)

Algebra and other foreign languages

If you are aiming at interested laypeople you absolutely must avoid algebra.  It is a foreign language that simply does not communicate to most of the educated people in the world.  Learning a foreign language is difficult. 

So how do you avoid algebra?  Well, you have to be clever and insightful.  The book by Matthew Watkins (below) has absolutely wonderful tricks for doing that, and I think anyone interested in math exposition ought to read it.  He uses metaphors, pictures and saying the same thing in different words. When you finish reading his book, you won't know how to prove statements related to the prime number theorem (unless you already knew how) but you have a good chance of understanding the statement of some theorem in that subject. See my review of the book for more details.

If your article is for math newbies, you don't have to avoid algebra completely.  But remember they are newbies and not as fluent as you are — they do things analogous to "Throw Mama from the train a kiss" and "I can haz cheeseburger?".  But if you are trying to give them some way of thinking about a concept, you need many other things (metaphors, illustrative applications, diagrams…)  You don't need the definition-theorem-proof style too common in "exposition".  (You do need that for math majors who want to become professional mathematicians.) 

Unfamiliar notation

In writing expositions for interested laypeople or math newbies, you should not introduce an unfamiliar notation system (which is like a minilanguage).  I expect to write the monad article without commutative diagrams.  Now, commutative diagrams are a wonderful invention, the best way of writing about categories, and they are widely used by other than category theorists.  But to explain monads to a newbie by introducing and then using commutative diagrams is like incorporating a short grammar of Spanish which you will then use in an explanation of Sancho Panza's relationship with Don Quixote. 

The abstractmath article on and, or and not does not use any of the several symbolic notations for logic that are in use.  The explanations simply use "and", "or" and "not".  I did introduce the notation, but didn't use it in the explanations.  When I rewrite the article I expect to put the notation at the end of the article instead of in the middle.  I expect to rewrite the other articles on mathematical reasoning to follow that practice, too.

Technical terminology

This is about the technical terminology used in math.  Technical terminology belongs to the math dialect (or register) of English, which is not a foreign language in the same sense as algebra.  One big problem is changing the meaning of ordinary English words to a technical meaning.  This requires a definition, and definitions are not something most people take seriously until they have been thoroughly brainwashed into using mathematical methodology.  Math majors have to be brainwashed in this way, but if you are writing for laypeople or newbies you cannot use the technology of formal definition.

Groups, simple groups

"You say the Monster Group is SIMPLE???  You must be a GENIUS!"  So Mark Ronan in his book (below) referred to simple groups as atoms.  Marcus du Sautoy calls them building blocks.  The mathematical meaning of "simple group" is not a transparent consequence of the meanings of "simple" and "group". Du Sautoy usually writes "group of symmetries" instead of just "group", which gives you an image of what he is talking about without having to go into the abstract definition of group. So in that usage, "group" just means "collection", which is what some students continue to think well after you give the definition.  

A better, but ugly, name for "group" might be "symmetroid". It sounds technical, but that might be an advantage, not a disadvantage. "Group" obviously means any collection, as I've known since childhood. "Symmetroid" I've never heard of so maybe I'd better find out what it means.

In beginning abstract math courses my students fervently (but subconsciously) believe that they can figure out what a word means by what it means already, never mind the "definition" which causes their eyes to glaze over. You have to be really persuasive to change their minds.

Prime factorization

Matthew Watkins referred to the prime factorization of an integer as a cluster. I am not sure why Watkins doesn't like "prime factorization", which usually refers to an expression such as  $p^{n_1}_1p^{n_2}_2\ldots p^{n_k}_k$.  This (as he says) has a spurious ordering that makes you have to worry about what the uniqueness of factorization means. The prime factorization is really a multiset of primes, where the order does not matter. 

Watkins illustrates a cluster of primes as a bunch of pingpong balls stuck together with glue, so the prime factorization of 90 would be four smushed together balls marked 2, 3, 3 and 5. Below is another way of illustrating the prime factorization of 90. Yes, the random movement programming could be improved, but Mathematica seduces you into infinite playing around and I want to finish this post. (Actually, I am beginning to think I like smushed pingpong balls better. Even better would be a smushed pingpong picture that I could click on and look at it from different angles.)

Metaphors, pictures, graphs, animation

Any exposition of math should use metaphors, pictures and graphs, especially manipulable pictures (like the one above) and graphs.  This applies to expositions for math majors as well as laypeople and newbies.  Calculus and other texts nowadays have begun doing this, more with pictures than with metaphors. 

I was turned on to these ideas as far back as 1967 (date not certain) when I found an early version of David Mumford's "Red Book", which I think evolved into the book The Red Book of Varieties and Schemes.  The early version was a revelation to me both about schemes and about exposition. I have lost the early book and only looked at the published version briefly when it appeared (1999).  I remember (not necessarily correctly) that he illustrated the spectrum as a graph whose coordinates were primes, and generic points were smudges.  Writing this post has motivated me to go to the University of Minnesota math library and look at the published version again.

References

Expositions for educated non-mathematicians

Previous posts in G&G

Relevant abmath articles

Notes on Viewing  

  • This post uses MathJax. If you see mathematical formulas 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 algebra2.nb.
Posted in exposition, language of math, math, representations, understanding math. Tags: , , , , , , , , , , , , . No Comments »

Visible algebra II

2012/09/10 — SixWingedSeraph

Notes on viewing:

  • This post uses MathJax. If you see mathematical formulas 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 algebra2.nb.

More about visible algebra

I have written about visible algebra in previous posts (see References). My ideas about the interface are constantly changing. Some new ideas are described here.

In the first place I want to make it clear that what I am showing in these posts is a simulation of a possible visual algebra system.  I have not constructed any part of the system; these posts only show something about what the interface will look like.  My practice in the last few years is to throw out ideas, not construct completed documents or programs.  (I am not saying how long I will continue to do this.)  All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.

Commutative and associative operations

Times and Plus are commutative and associative operations.  They are usually defined as binary operations.  A binary operation $*$ is said to be commutative if for all $x$ and $y$ in the underlying set of the operation, $x*y=y*x$, and it is associative if for all $x$,$y$ and $z$ in the underlying set of the operation, $(x*y)*z=x*(y*z)$. 

It is far better to define a commutative and associative operation $*$ on some underlying set $S$ as an operation on any multiset of elements of $S$.  A multiset is like a set, in particular elements can be rearranged in any way, but it is not like a set in that elements can be repeated and a different number of repetitions of an element makes a different multiset.  So for any particular multiset, the number of repetitions of each element is fixed.  Thus $\{a,a,b,b,c\} = \{c,b,a,b,a\}$ but $\{a,a,b,b,c\}\neq\{c,b,a,b,c\}$. This means that the function (operation) Plus, for example, is defined on any multiset of numbers, and \[\mathbf{Plus}\{a,a,b,b,c\}=\mathbf{Plus} \{c,b,a,b,a\}\] but $\mathbf{Plus}\{a,a,b,b,c\}$ might not be equal to $\mathbf{Plus} \{c,b,a,b,c\}$.

This way of defining (any) associative and commutative operation comes from the theory of monads.  An operation defined on all the multisets drawn from a particular set is necessarily commutative and associative if it satisfies some basic monad identities, the main one being it commutes with union of multisets (which is defined in the way you would expect, and if this irritates you, read the Wikipedia article on multisets.). You don't have to impose any conditions specifically referring to commutativity or associativity.  I expect to write further about monads in a later post. 

The input process for a visible algebra system should allow the full strength of this fact. You can attach as many inputs as you want to Times or Plus and you can move them around.  For example, you can click on any input and move it to a different place in the following demo.

Other input notations might be suitable for different purposes.  The example below shows how the inputs can be placed randomly in two dimensions (but preserving multiplicity).  I experimented with making it show the variables slowly moving around inside the circle the way the fish do in that screensaver (which mesmerizes small children, by the way — never mind what it does to me), but I haven't yet made it work.

A visible algebra system might well allow directly input tables to be added up (or multiplied), like the one below. Spreadsheets have such an operation In particular, the spreadsheet operation does not insist that you apply it only as a binary operation to columns with two entries.  By far the most natural way to define addition of numbers is as an operation on multisets of numbers.

Other operations

Operations that are associative but not commutative, such as matrix multiplication, can be defined the monad way as operations on finite lists (or tuples or vectors) of numbers.  The operation is automatically associative if you require it to preserve concatenation of lists and some other monad requirements.

Some binary operations are neither commutative nor associative.  Two such operations on numbers are Subtract and Power.  Such operations are truly binary operations; there is no obvious way to apply them to other structures.  They are only binary because the two inputs have different roles.  This suggests that the inputs be given names, as in the examples below.

Later, I will write more about simplifying trees, solving the max area problem for rectangles surmounted by semicircles, and other things concerning this system of doing algebra.

References

Previous posts about visible algebra

Other references

 

Posted in category theory, exposition, language of math, math, representations, understanding math. Tags: , , , , , , , , , , , , , , . No Comments »

Visible algebra I supplement

2012/09/08 — SixWingedSeraph

Note: This post uses MathJax. If you see mathematical formulas 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 demo in this post, you must have Wolfram CDF Player installed on your computer. It is available free from the Wolfram website.

Active calculation of area

In my previous post Visible algebra I constructed a computation tree for calculating the area of a window consisting of a rectangle surmounted by a semicircle. The visual algebra system described there constructs a computation by selecting operations and attaching them to a tree, which can then be used to calculate the area of the window. 

I promised to produce a live computation tree later; it is below.  The Mathematica code for this tree is in the notebook algebra1.nb.

Press the buttons from left to right to simulate the computation that would take place in a genuine algebra system.  Note that if you skip button 2 you get the effect of parallel computation (the only place in the calculation that can be parallelized).

In Visual Algebra I the tree was put together step by step by reasoning out how you would calculate the area of the window: (1) the area is the sum of the areas of the semidisk and the rectangle, (2) the rectangle is width times height, (3) the semidisk has half the area of a disk of radius half the width of the rectangle, and so on.  So the resulting tree is a transparent construction that lets you see the reasoning that created it.  

The resulting tree could obviously be simplified.  But if you were designing a few such windows, why should you simplify it?  You certainly don't need to simplify it to speed up the computation.  On the other hand, if you are going on to solve the problem of finding the maximum area you can get if the perimeter is fixed, you will have to do some algebraic manipulation and so you do want a simplified expression.    

Later, I will write more about simplifying trees, solving the max area problem, and other things concerning this system of doing algebra.

Remark

What I am showing in these posts is a simulation of a possible visible algebra system.  I have not constructed any part of the system; these posts only show something about what the interface will look like.  My practice in the last few years is to throw out ideas, not construct completed documents or programs.  (I am not saying how long I will continue to do this.)  All these posts, Mathematica programs and abstractmath.org are available to reuse under a Creative Commons license.

Posted in exposition, language of math, math, Mathematica, representations, understanding math. Tags: , , , , , , . No Comments »

1.000… and 0.999…

2012/08/29 — SixWingedSeraph

 

Note: This post uses MathJax. If you see mathematical formulas with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.
 
Recently Julian Wilson sent me this letter:
It is well known that students often have trouble accepting that $0.999\ldots$ is the same number as $1.000\ldots$.  However, there is at least one context in which these could be regarded as in some sense as being distinct. In a discrete dynamical system where the next iterate is formed by multiplying the current value by 10 and dropping the leading digit, and where you make a note at each iteration of the first digit after the decimal point, then 0.9999… generates a sequence of 9s, whereas 1.0000… generates a sequence of 0s. The imagery is of a stretching a circle, wrapping it ten times around itself and recording in which sector (labeled 0 to 9) you end up.
 
From the dynamical systems perspective, being in state 9 (and remaining there after each iteration) is different from being in state 0.
The $0.9999\ldots =1.0000\ldots$ equation is associated with several conceptual difficulties that math students have, which I will describe here.

The decimal representation is not the number

Another way of describing the equation is to say that "$0.999\ldots$" and "$1.000\ldots$" are distinct decimal representations of the same number, namely $1$. Julian's proposal provides a different interpretation of the notation, in which "$0.999\ldots$" and "$1.000\ldots$" are strings of symbols generated by two different machines.  Of course, that is correct.  But they are both correct decimal notation that correspond to the same number.

Mathematical writing will sometimes use notation to mean the abstract mathematical object it refers to, and at other times the text is referring to the notation itself.  For example,

$x^2+1$ is always positive.

refers to the value of $x^2+1$, but

If you substitute $5$ for $x$ in $x^2+1$ you get $26$.

refers to the expression "$x^2+1$".  Careful authors would write,

If you substitute $5$ for $x$ in "$x^2+1$" you get $26$.

This ambiguity in using mathematical notation is an example of what philosophers call the "use-mention" distinction, but they apply the phrase to many other situations as well.  Mathematicians have an operational knowledge of this distinction but many of them are not consciously aware of it.

Definitions

A decimal representation of a number by definition represents the number that a certain power series converges to. The two power series corresponding to 1.000… and to 0.999… both converge to 1:

\[1+\sum_{i=1}^{\infty}\frac{0}{10^n}=1\]

and

\[0+\sum_{i=1}^{\infty}\frac{9}{10^n}=1\] 

They are different power series (mention) but converge (use) to the same number.

Most students new to abstract math are not aware of the importance of definition in math. As they learn more, they may still hold on to the idea that you have to discover or reason out what a math word or expression means.  In purple prose, THE DEFINITION IS A DICTATOR. 

This does not mean that you can understand the concept merely by reading the definition.  The definition usually does not mention most of the important things about the concept.

Completed Infinity

A common remark by newbies about $0.999\ldots$ is that it gets closer and closer to $1$ but does not get there. So it can't be equal to $1$.  This shows a lack of understanding of completed infinity.  The point is that the notation "$0.999\ldots$" refers to a string beginning with "$0.$" and followed by an infinite sequence of $9$'s.  Now "$s$ is an infinite sequence of $9$'s" means precisely that $s$ has an entry $s_n$ for every positive integer $n$, and that $s_n$ is $9$ for every positive integer $n$. 

  • The expression is gradually unrolling over time, and does not ever "get there". 
  • All the nines are already there.

Both the preceding sentences are metaphorical.  They are about how you should think about "$0.999\ldots$".  The first metaphor is bad, the second metaphor is good.  Neither statement is a formal mathematical statement.  Neither statement says anything about what the sequence really is.  They are not statements about reality at all, they are about how you should think about the sequence if you are going to understand what mathematicians say about it. 

Metaphors are crucial to understanding math.  Too many students use the wrong metaphors, but too often no one tells them about it.

We need a math ed text for teachers

I am thinking of precalculus through typical college math major courses.  The issues I have discussed in this post are occasionally written about in the math ed literature but I have had difficulty finding many articles (on the web and on JStor) about these specific ideas.  Anyway, articles are not what we need.  We need a modest paperback book specifically aimed at teachers, covering the kinds of cognitive difficulties math students have when faced with abstraction. 

What I have written in abstractmath.org and in the Handbook are examples of what I mean, but they don't cover all the problems and they suffer from lack of focus.  (Note that the material in abstractmath.org and in posts on this blog can be used freely under a Creative Commons license — click on "Permissions" in the blue banner at the top of this page). 

Among math ed researchers, I have learned a lot from papers by Anna Sfard and David Tall. 

References

 
Posted in abstractmath.org, exposition, language of math, math, representations, understanding math. Tags: , , , , , , . 4 Comments »

Semantics of algebra I

2012/08/22 — SixWingedSeraph

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

In the post Algebra is a difficult foreign language  I listed some of the difficulties of the syntax of the symbolic language of math (which includes high school algebra and precalculus).  The semantics causes difficulties as well.  Again I will list some examples without any attempt at completeness.

The status of the symbolic language as a language

There is a sharp distinction between the symbolic language of math and mathematical English, which I have written about in The languages of math and in the Handbook of mathematical discourse. Other authors do not make this sharp distinction (see the list of references at the end of this post). The symbolic language occurs embedded in mathematical English and the embedding has its own semantics which may cause great difficulty for students.

The symbolic language of math can be described as a natural formal language. Pieces of it were invented by mathematicians and others over the course of the last several hundred years. Individual pieces (notation such as "$3x+1=2y$") can be given a strictly formal syntax, but the whole system is ambiguous, inconsistent, and context-sensitive.  When you get to the research level, it has many dialects: Research mathematicians in one field may not be able to read research articles in a very different field.

Examples

I think the examples below will make these claims plausible.  This should be the subject of deep research.

Superscripts and functions

  • A superscript, as in $5^2$ or $x^3$, has a pretty standard meaning denoting a power, at least until you get to higher level stuff such as tensors.  
  • A function can be denoted by a letter, symbol, or string, and the notation $f(x)$ refers to the value of the function at input $x$.  

For functions defined on numbers, it is common in precalculus and higher to write $f^2(x)$ to denoted $(f(x))^2=f(x)\,f(x)$.  Since the value of certain multiletter functions are commonly written without the parentheses (for example, $\sin\,x$), one writes $\sin^2x$ to mean $(\sin\,x)^2$.

The notation $f^n$ is also widely used to mean the $n$th iterate of $f$ (if it exists), so $f^3(x)=f(f(f(x)))$ and so on.  This leads naturally to writing $f^{-1}(x)$ for the inverse function of $f$; this is common notation whether the function $f$ is bijective or not (in which case $f^{-1}$ is set-valued).  Thus $\sin^{-1}x$ means $\arcsin\,x$.

It is notorious that words in mathematical English have different meanings in different texts.  This is an example in the symbolic language (and not just at the research level) of a systematic construction that can give expressions that have ambiguous meanings.

This phenomenon is an example of why I say the symbolic language of math is a natural formal language: I have described a natural extension of notation used with multiplication of values that has been extended to being used for the binary operation of composition.  And that leads to students thinking that $\sin^{-1}x$ means $\frac{1}{\sin\,x}$. 

History can overtake notation, too: Mathematicians probably took to writing $\sin\,x$ instead of $\sin(x)$ because it saves writing.  That was not very misleading in the old days when mathematical variables were always single symbols.  But students see multiletter variable names all the time these days (in programming languages, Excel and elsewhere), so of course some of them think $\sin\,x$ means $\sin$ times $x$. People who do this are not idiots.

Juxtaposition

Juxtaposition of two symbols means many different things.

  • If $m$ and $n$ are numbers, $mn$ denotes the product of the two numbers.
    • Multiplication is commutative, so $mn$ and $nm$ denote the same number, but they correspond to different calculations.  
  • If $M$ and $N$ are matrices, $MN$ denotes the matrix product of the two matrices.
    • This is a binary operation but it is not the same operation denoted by juxtaposition of numbers. (In fact it involves both addition and multiplication of numbers.)
    • Now $MN$ may not be the same matrix as $NM$.
  • If $A$ and $B$ are points in a geometric drawing, $AB$ denotes the line segment from $A$ to $B$.
    • This is a function of two variables denoting points whose value is a line segment.  
    • It is not what is usually called a binary operation, although as an opinionated category theorist I would call it a multisorted binary operation.
    • It is commutative, but it doesn't make sense to ask if it is associative.

This phenomenon is called overloaded notation.  

  • In order to understand the meaning of the juxtaposition of symbols, you have to know the type of the variables.
  • ​The surrounding text may tell you specifically the variables denote matrices or whatever. So this is an instance of context-sensitive semantics. 
    • Students tend to expect that they know what any formula means in isolation from the text.  It may make them very sad to discover that this doesn't work — once they believe it, which can take quite a while.
  • In many cases the problem is alleviated by the use of convention.
    • Matrices are usually denoted by capital letters, numbers by lower case letters.
    • But points in geometry are usually denoted by capital letters too.  So you have to know that referring to a geometric diagram is significant to understanding the notation. This is an indirect form of context-sensitivity.  Did any teacher every point this out to students?  Does it appear anywhere in print?

The earlier example of $\sin^{-1}x$ is a case which is not context-sensitive.  Knowing the types of the variables won't help.  Of course, if the author explains which meaning is meant, that explanation is within the context of the book!  That is not a lot of help for grasshoppers like me that look back and forth at different parts of a math book instead of reading it straight through..  

Equations

Consider the expressions

  1. $x^2-5x+4=0$
  2. $x^2+y^2=1$
  3. $x^2+2x+1=(x+1)^2$

They are assertions that two expressions have the same value. A strictly logical view of an equation containing variables is that it puts a constraint on the variables.  It is true of some numbers (or pairs of numbers) and false of others.  That is the defining property of an equation. Equation 1 requires that $x=1$ or $x=4$.  Equation 2 imposes a constraint which is satisfied by uncountably many pairs of real numbers, and is also not true of uncountably many pairs. But equation 3 puts no constraint on the variable.  It is true of every number $x$.

A strictly logical view of symbolic notation does math a disservice.  Here, the notion that an equation is by definition a symbolic statement that has a truth set and a falsity set may be correct but it is not the important thing about any particular equation. When we read and do math we have many different metaphors and images about a concept.  The definition of a kind of object is often in terms of things that may not be the most important things to know about it.  (One of the most important fact about groups is that it is an abstraction of symmetries, which the axioms don't mention at all.)

Equation 1. is something that would make most people set out to discover the truth set.  Equation 2. calls out for drawing its graph.  Equation 3. being an identity means that is useful in algebraic reasoning.  The images they call up are different and what you do with them is different.  The images and metaphors that cluster around a concept are an important part of the semantics of the symbolic language.

I expect to post separately about the semantics of variables and about the semantics of symbolic language embedded in mathematical English.

References

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Algebra is a difficult foreign language

2012/08/15 — SixWingedSeraph

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

Algebra

In a previous post, I said that the symbolic language of mathematics is difficult to learn and that we don't teach it well. (The symbolic language includes as a subset the notation used in high school algebra, precalculus, and calculus.) I gave some examples in that post but now I want to go into more detail.  This discussion is an incomplete sketch of some aspects of the syntax of the symbolic language.  I will write one or more posts about the semantics later.

The languages of math

First, let's distinguish between mathematical English and the symbolic language of math. 

  • Mathematical English is a special register or jargon of English. It has not only its special vocabulary, like any jargon, but also used ordinary English words such as "If…then", "definition" and "let" in special ways. 
  • The symbolic language of math is a distinct, special-purpose written language which is not a dialect of the English language and can in fact be read by mathematicians with little knowledge of English.
    • It has its own symbols and rules that are quite different from spoken languages. 
    • Simple expressions can be pronounced, but complicated expressions may only be pointed to or referred to.
  • A mathematical article or book is typically written using mathematical English interspersed with expressions in the symbolic language of math.

Symbolic expressions

A symbolic noun (logicians call it a term) is an expression in the symbolic language that names a number or other mathematical object, and may carry other information as well.

  • "3" is a noun denoting the number 3.
  • "$\text{Sym}_3$" is a noun denoting the symmetric group of order 3.
  • "$2+1$" is a noun denoting the number 3.  But it contains more information than that: it describes a way of calculating 3 as a sum.
  • "$\sin^2\frac{\pi}{4}$" is a noun denoting the number $\frac{1}{2}$, and it also describes a computation that yields the number $\frac{1}{2}$.  If you understand the symbolic language and know that $\sin$ is a numerical function, you can recognize "$\sin^2\frac{\pi}{4}$" as a symbolic noun representing a number even if you don't know how to calculate it.
  • "$2+1$" and "$\sin^2\frac{\pi}{4}$" are said to be encapsulated computations.
    • The word "encapsulated" refers to the fact that to understand what the expressions mean, you must think of the computation not as a process but as an object.
    • Note that a computer program is also an object, not a process.
  • "$a+1$" and "$\sin^2\frac{\pi x}{4}$" are encapsulated computations containing variables that represent numbers. In these cases you can calculate the value of these computations if you give values to the variables.  

A symbolic statement is a symbolic expression that represents a statement that is either true or false or free, meaning that it contains variables and is true or false depending on the values assigned to the variables.

  • $\pi\gt0$ is a symbolic assertion that is true.
  • $\pi\lt0$ is a symbolic assertion that it is false.  The fact that it is false does not stop it from being a symbolic assertion.
  • $x^2-5x+4\gt0$ is an assertion that is true for $x=5$ and false for $x=1$.
  • $x^2-5x+4=0$ is an assertion that is true for $x=1$ and $x=4$ and false for all other numbers $x$.
  • $x^2+2x+1=(x+1)^2$ is an assertion that is true for all numbers $x$. 

Properties of the symbolic language

The constituents of a symbolic expression are symbols for numbers, variables and other mathematical objects. In a particular expression, the symbols are arranged according to conventions that must be understood by the reader. These conventions form the syntax or grammar of symbolic expressions. 

The symbolic language has been invented piecemeal by mathematicians over the past several centuries. It is thus a natural language and like all natural languages it has irregularities and often results in ambiguous expressions. It is therefore difficult to learn and requires much practice to learn to use it well. Students learn the grammar in school and are often expected to understand it by osmosis instead of by being taught specifically.  However, it is not as difficult to learn well as a foreign language is.

In the basic symbolic language, expressions are written as strings of symbols.

  • The symbolic language gives (sometimes ambiguous) meaning to symbols placed above or below the line of symbols, so the strings are in some sense more than one dimensional but less than two-dimensional.
  • Integral notation, limit notation, and others, are two-dimensional enough to have two or three levels of symbols. 
  • Matrices are fully two-dimensional symbols, and so are commutative diagrams.
  • I will not consider graphs (in both senses) and geometric drawings in this post because I am not sure what I want to write about them.

Syntax of the language

One of the basic methods of the symbolic language is the use of constructors.  These can usually be analyzed as functions or operators, but I am thinking of "constructor" as a linguistic device for producing an expression denoting a mathematical object or assertion. Ordinary languages have constructors, too; for example "-ness" makes a noun out of a verb ("good" to "goodness") and "and" forms a grouping ("men and women").

Special symbols

The language uses special symbols both as names of specific objects and as constructors.

  • The digits "0", "1", "2" are named by special symbols.  So are some other objects: "$\emptyset$", "$\infty$".
  • Certain verbs are represented by special symbols: "$=$", "$\lt$", "$\in$", "$\subseteq$".
  • Some constructors are infixes: "$2+3$" denotes the sum of 2 and 3 and "$2-3$" denotes the difference between them.
  • Others are placed before, after, above or even below the name of an object.  Examples: $a'$, which can mean the derivative of $a$ or the name of another variable; $n!$ denotes $n$ factorial; $a^\star$ is the dual of $a$ in some contexts; $\vec{v}$ constructs a vector whose name is "$v$".
  • Letters from other alphabets may be used as names of objects, either defined in the context of a particular article, or with more nearly global meaning such as "$\pi$" (but "$\pi$" can denote a projection, too).

This is a lot of stuff for students to learn. Each symbol has its own rules of use (where you put it, which sort of expression you may it with, etc.)  And the meaning is often determined by context. For example $\pi x$ usually means $\pi$ multiplied by $x$, but in some books it can mean the function $\pi$ evaluated at $x$. (But this is a remark about semantics — more in another post.)

"Systematic" notation

  • The form "$f(x)$" is systematically used to denote the value of a function $f$ at the input $x$.  But this usage has variations that confuse beginning students:
    • "$\sin\,x$" is more common than "$\sin(x)$".
    • When the function has just been named as a letter, "$f(x)$" is more common that "$fx$" but many authors do use the latter.
  • Raising a symbol after another symbol commonly denotes exponentiation: "$x^2$" denotes $x$ times $x$.  But it is used in a different meaning in the case of tensors (and elsewhere).
  • Lowering a symbol after another symbol, as in "$x_i$"  may denote an item in a sequence.  But "$f_x$" is more likely to denote a partial derivative.
  • The integral notation is quite complicated.  The expression \[\int_a^b f(x)\,dx\] has three parameters, $a$, $b$ and $f$, and a bound variable $x$ that specifies the variable used in the formula for $f$.  Students gradually learn the significance of these facts as they work with integrals. 

Variables

Variables have deep problems concerned with their meaning (semantics). But substitution for variables causes syntactic problems that students have difficulty with as well.

  • Substituting $4$ for $x$ in the expression $3+x$ results in $3+4$. 
  • Substituting $4$ for $x$ in the expression $3x$ results in $12$, not $34$. 
  • Substituting "$y+z$" in the expression $3x$ results in $3(y+z)$, not $3y+z$.  Some of my calculus students in preforming this substitution would write $3\,\,y+z$, using a space to separate.  The rules don't allow that, but I think it is a perfectly natural mistake. 

Using expressions and writing about them

  • If I write "If $x$ is an odd integer, then $3+x$ is odd", then I am using $3+x$ in a sentence. It is a noun denoting an unspecified number which can be constructed in a specified way.
  • When I mention substituting $4$ for $x$ in "$3+x$", I am talking about the expression $3+x$.  I am not writing about a number, I am writing about a string of symbols.  This distinction causes students major difficulties and teacher hardly ever talk about it.
  • In the section on variables, I wrote "the expression $3+x$", which shows more explicitly that I am talking about it as an expression.
    • Note that quotes in novels don't mean you are talking about the expression inside the quotes, it means you are describing the act of a person saying something.
  • It is very common to write something like, "If I substitute $4$ for $x$ in $3x$ I get $3 \times 4=12$".  This is called a parenthetic assertion, and it is literally nonsense (it says I get an equation).
  • If I pronounce the sentence "We know that $x\gt0$" we pronounce "$x\gt0$" as "$x$ is greater than zero",  If I pronounce the sentence "For any $x\gt0$ there is $y\gt0$ for which $x\gt y$", then I pronounce the expression "$x\gt0$" as "$x$ greater than zero$",  This is an example of context-sensitive pronunciation
  • There is a lot more about parenthetic assertions and context-sensitive pronunciation in More about the languages of math.

Conclusion

I have described some aspects of the syntax of the symbolic language of math. Learning that syntax is difficult and requires a lot of practice. Students who manage to learn the syntax and semantics can go on to learn further math, but students who don't are forever blocked from many rewarding careers. I heard someone say at the MathFest in Madison that about 25% of all high school students never really understand algebra.  I have only taught college students, but some students (maybe 5%) who get into freshman calculus in college are weak enough in algebra that they cannot continue. 

I am not proposing that all aspects of the syntax (or semantics) be taught explicitly.  A lot must be learned by doing algebra, where they pick up the syntax subconsciously just as they pick up lots of other behavior-information in and out of school. But teachers should explicitly understand the structure of algebra at least in some basic way so that they can be aware of the source of many of the students' problems. 

It is likely that the widespread use of computers will allow some parts of the symbolic language of math to be replaced by other methods such as using Excel or some visual manipulation of operations as suggested in my post Mathematical and linguistic ability.  It is also likely that the symbolic language will gradually be improved to get rid of ambiguities and irregularities.  But a deliberate top-down effort to simplify notation will not succeed. Such things rarely succeed.

References

 

 

Posted in abstractmath.org, exposition, language, language of math, math, representations, understanding math. Tags: , , , , , , , , , , , , , . No Comments »

Mathematical and linguistic ability

2012/08/04 — SixWingedSeraph

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

Some personal history

When I was young, I was your typical nerdy geek.  (Never mind what I am now that I am old.)

In high school, I was fascinated by languages, primarily by their structure.  I would have wanted to become a linguist if I had known there was such a thing.  I was good at grasping the structure of a language and read grammars for fun. I was only pretty good at picking up vocabulary. I studied four different languages in high school and college and Turkish when I was in the military.  I know a lot about their structure but am not fluent in any of them (possibly including English).

After college, I decided to go to math grad school.  This was soon after Sputnik and jobs for PhD's were temporarily easy to get.

I always found algebra easy.  When I had to learn other symbolic languages, for example set theory, first order logic, and early programming languages, I found them easy too.  I had enough geometric insight that I did well in all my math courses, but my real strength was in learning languages. 

When I got a job at (what is now) Case Western Reserve University, I began learning category theory and a bit of cohomology of groups. I wrote a paper about group automorphisms that got into Transactions of the AMS.  (Full disclosure: I am bragging). 

The way Saunders Mac Lane did cohomology, he used "$+$" as a noncommutative operation.  No problem with that, I did lots of calculations in his notation.  In reading category theory I learned how to reason using commutative diagrams.  That is radically different from other math — it isn't strings of symbols — but I caught on. I read Beck's thesis in detail.  Beck wrote functions on the right (unlike Mac Lane) which I adapted to with no problem.  In fact my automorphisms paper and many others in those days was written with functions on the right. 

Later on in my career, I learned to program in Forth reasonably well. It is a reverse Polish language. Then (by virtue of summer grants in the 1990's) to use Mathematica, which I now use a lot:  I am an "experienced" user but not an "expert".

Learning foreign languages in studying math

I taught mostly engineering students during my 35 years at CWRU (especially computer engineering). When I used a text (including my own discrete math class notes) some students pleaded with me not to use $P\wedge Q$ and $P \vee Q$ but let them use $PQ$ and $P+Q$ like they did in their CS courses.  Likewise $1$ and $0$ instead of T and F.  Many of them simply could not switch easily between different codes.  Similar problems occurred in classes in first order logic. 

In the early days of calculators when most of them were reverse Polish, some students never mastered their use. 

These days, a common complaint about Mathematica is that it is a difficult language to learn; at the MAA meeting in Madison (where I am as I write this) they didn't even staff a booth.  Apparently too many of the professors can't handle Mathematica.

I gave up writing papers with functions on the right because several professional mathematicians complained that they found them too hard to read. I guess not all professional mathematicians can switch code easily, either. 

There are many great mathematicians whose main strength is geometric understanding, not linguistic understanding.  Nevertheless, to become a mathematician you have to have enough linguistic ability to learn…

Algebra

The big elephant in the room is ordinary symbolic algebra as is used in high school algebra and precalculus.  This of course causes difficulty among first year calculus students, too, but college profs are spared the problem that high school teachers have with a large percentage of the students never really grasping how algebra works.  We don't see those students in STEM courses.

It is surely the case that algebra is a difficult and unintuitive foreign language.  I have carried on about this in my stuff about the languages of math in my abstractmath site. 

Some students already in college don't really understand expressions such as $x^2$.  You still get some who sporadically think it means $2x$.  (They don't always think that, but it happens when they are off guard.)  Lots of them don't understand the difference between $x^2$ and $2^x$.

In complicated situations, students don't grasp the difference between an expression such as $x^2+2x+1$ and a statement like $x^2+2x+1=0$.  Not to mention the difference between the way $x^2+2x+1=0$ and $x^2+2x+1=(x+1)^2$ are different kinds of statements even though the difference is not indicated in the syntax.

There are many irregularities and ambiguities (just like any natural language — the symbolic language of math is a natural language!): consider $\sin xy$, $\sin x + y$, $\sin x/y$.  (Don't squawk to me about order of operators.  That's as bad as aus, außer, bei, mit, zu.  German can't help it, but mathematical notation could.)

One monstrous ambiguity is $(x,y)$, which could be an ordered pair, the GCD, or an open interval.  I found an example of two of those in the same sentence in the Handbook of Mathematical Discourse, and today in a lecture I saw someone use it with two meanings about three inches apart on a transparency.

Anyway, the symbolic language of math is difficult and we don't teach it well.

Structuring calculations

There are other ways to structure calculations that are much more transparent.  Most of them use two or three dimensions.

  • Spreadsheets: It is easy to approximate the zeros of a function using a spreadsheet and changing the input till you get the value near zero. Why can't middle school students be taught that?
  • Bret Victor has made suggestions for easy ways to calculate things.
  • My post Visible Algebra I suggest a two-dimensional approach to putting together calculations.  (There are several more posts coming about that idea.)
  • Mathematica interactive demos could maybe be provided in a way that would allow them to be joined together to make a complicated calculation. (Modules such as an inverse image constructor.)  I have not tried to do this.

A lot of these alternatives work better because they make full use of two dimensions.  Toolkits could be made for elementary school students (there are some already but I am not familiar with them).  

It is impractical to expect that every high school student master basic algebraic notation.  It is difficult and we don't know how to teach it to everyone. With the right toolkits, we could provide everyone, not just students, to put together usable calculations on their computer and experiment with them.  This includes working out the effect of different payment periods on loans, how much paint you need for a room, and many other things.

STEM students will still have to learn algebraic notation as we use it now.  It should be taught as a foreign language with explicit instruction in its syntax (sentences and terms, scope of an operator, and so on), ambiguities and peculiarities.

Posted in category theory, exposition, Handbook, language, language of math, math, understanding math. Tags: , , , , , , , . No Comments »
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