The intent of mathematical assertions

An assertion in mathematical writing can be a claim, a definition or a constraint.  It may be difficult to determine the intent of the author.  That is discussed briefly here.

Assertions in math texts can play many different roles.

English sentences can state facts, ask question, give commands, and other things.  The intent of an English sentence is often obvious, but sometimes it can be unexpectedly different from what is apparent in the sentence.  For example, the statement “Could you turn the TV down?” is apparently a question expecting a yes or no answer, but in fact it may be a request. (See the Wikipedia article on speech acts.) Such things are normally understood by people who know each other, but people for whom English is a foreign language or who have a different culture have difficulties with them.

There are some problems of this sort in math English and the symbolic language, too.  An assertion can have the intent of being a claim, a definition, or a constraint.

Most of the time the intent of an assertion in math is obvious. But there are conventions and special formats that newcomers to abstract math may not recognize, so they misunderstand the point of the assertion. This section takes a brief look at some of the problems.

Terminology

The way I am using the words “assertion”, “claim”, and “constraint” is not standard usage in math, logic or linguistics.

Claims

In most circumstances, you would expect that if a lecturer or author makes a math assertion, they are claiming that it is a true statement, and you would be right.

Examples
1. “The $240$th digit of $\pi$ after the decimal point is $4$.”
2. “If a function is differentiable, it must be continuous.”
3. “$7\gt3$”

Remarks

• You don’t have to know whether these statements are true or not to recognize them as claims. An incorrect claim is still a claim.
• The assertion in (a) is a statement, in this case a false one.  If it claimed the googolth digit was $4$ you would never be able to tell whether it is true or not, but it
still would be an assertion intended as a claim.
• The assertion in (b) uses the standard math convention that an indefinite noun phrase (such as “a widget”) in the subject of a sentence is universally quantified (see also the article about “a” in the Glossary.) In other words, “An integer divisible by $4$ must be even” claims that any integer divisible by $4$ is even. This statement is claim, and it is true.
• (c) is a (true) claim in the symbolic language. (Note that “$3 + 4$” is not an assertion at all, much less a claim.)

Definitions

Definitions are discussed primarily in the chapter on definitionsA definition is not the same thing as a claim.

Example

The definition

“An integer is even if it is divisible by $2$”

makes the claim

“An
integer is even if and only if it is
divisible by $2$”

true.

(If you are surprised that the definition uses “if” but the claim uses “if and only if”, see the Glossary article on “if”.)

Unmarked definitions

Math texts sometimes define something without saying that it is a definition. Because of that, students may sometimes think a claim is a definition.

Example

Suppose that the concept of “even integer” was new to you and the book said, “A number is even if it is divisible by $4$.” Perhaps you thought that this was a definition. Later the book refers to $6$ as even and you pull your hair out wondering why. The statement is a correct claim but an incorrect definition. A good writer would write something like “Recall that a number is even if it is divisible by $2$, so that in particular it is even if it is divisible by $4$.”

On the other hand, you may think a definition is only a claim.

Example

A lecturer may say “By definition, an integer is even if it is divisible by $2$”, and you write down: “An integer is even if it is divisible by $2$”. Later, you get all panicky wondering How did she know that?? (This has happened to me.)

The confusion in the preceding example can also occur if a books says, “An integer is even if it is divisible by $2$” and you don’t know about the convention that when an author puts a word or phrase in boldface or italics it may mean that they are defining it.

A good writer always labels definitions

Constraints

Here are two assertions that contain variables.

• “$n$ is even.”
• “$x\gt1$”.

Such an assertion is a constraint (or a condition) if the intent is
that the assertion will hold in that part of the text (the scope of the constraint). The part of the text in which it holds is usually the immediate vicinity unless the authors explicitly says it will hold in a larger part of the text such as “this chapter” or “in the rest of the book”.

Examples
• Sometimes the wording makes it clear that the phrase is a constraint. So a statement such as “Suppose $3x^2-2x-5\geq0$” is a constraint on the possible values of $x$.
• The statement “Suppose $n$ is even” is an explicit requirement that $n$ be even and an implicit requirement that $n$ be an integer.
• A condition for which you are told to find the solution(s) is a constraint. For example: “Solve the equation $3x^2-2x-5=0$”. This equation is a constraint on the variable $x$. “Solving” the equation means saying explicitly which numbers make the equation true.

Postconditions

The constraint may appear in parentheses after the assertion as a postcondition on an assertion.

Example

“$x^2\gt x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\text{all }x\gt1)$”

which means that if the constraint “$x\gt1$” holds, then “$x^2\gt x$” is true. In other words, for all $x\gt1$, the statement $x^2\gt x$ is true. In this statement, “$x^2\gt x$” is not a constraint, but a claim which is true when the constraint is true.

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Context

This is a revised draft of the abstractmath.org article on context in math texts. Note: WordPress changed double primes into quotes. Tsk.

Context

Written and especially spoken language depends heavily on the context – the physical surroundings, the preceding conversation, and social and cultural assumptions.  Mathematical statements are produced in such contexts, too, but here I will discuss a special thing that happens in math conversation and writing that does not seem to happen much in other sorts of discourse:

The meanings of expressions
in both the symbolic language and math English
change from phrase to phrase
as the speaker or writer changes the constraints on them.

Example

In a math text, before the occurrence of a phrase such as “Let $n=3$”, $n$ may be known only as an integer variable.  After the phrase, it means specifically $3$.  So this phrase changes the meaning of $n$ by constraining $n$
to be $3$.  We say the context of occurrences of “$n$” before the phrase requires only that $n$ be an integer, but after the occurrence the context requires $n=3$.

Definition

In this article, the context at a particular location in mathematical discourse is the sum total of what the reader or listener can know about the symbols and names used in the discourse when they have read everything up to that location.

Remarks

• Each clause can change the meaning of or constraints on one or more symbols or names. The conventions in effect during the discourse can also put constraints on the symbols and names.
• Chierchia and McConnell-Ginet give a mathematical definition of context in the sense described here.
• The references to “before” and “after” the phrase “Let $3$” refer to the physical location in text and to actual time in spoken math. There is more about this phenomenon in the Handbook of Mathematical Discourse, page 252, items (f) and (g).
• Contextual changes of this sort take place using the pretense that you are reading the text in order, which many students and professionals do not do (they are “grasshoppers”).
• I am not aware of much context-changing in everyday speech. One place it does occur is in playing games. For example, during some card games the word “trumps” changes meaning from time to time.
• In symbolic logic, the context at a given place may be denoted by “$\Gamma$”.

Detailed example of a math text

Here is a typical example of a theorem and its proof.  It is printed twice, the second time with comments about the changes of context.  This is the same proof that is already analyzed practically to death in the chapter on presentation of proofs.

First time through

Definition: Divides

Let $m$ and $n$ be integers with $m\ne 0$. The statement “$m$ divides $n$” means that there is an integer $q$ for which $n=qm$

Theorem

Let $m$, $n$ and $p$ be integers, with $m$ and $n$ nonzero, and suppose $m$ divides $n$ and $n$ divides $p$.  Then $m$ divides $p$.

Proof

By definition of divides, there are integers $q$ and $q’$ for which $n=qm$ and $p=q’n$. We must prove that there is an integer $q”$ for which $p=q”m$. But $p=q’n=q’qm$, so let $q”=q’q$.  Then $p=q”m$.

Second time, with analysis

 Definition: Divides Begins a definition. The word “divides” is the word being defined. The scope of the definition is the following paragraph. Let $m$ and $n$ be integers $m$ and $n$ are new symbols in this discourse, constrained to be integers. with $m\ne 0$ Another constraint on $m$. The statement “$m$ divides $n$ means that” This phrase means that what follows is the definition of “$m$ divides $n$” there is an integer $q$ “There is” signals that we are beginning an existence statement and that $q$ is the bound variable within the existence statement. for which $n=qm$ Now we know that “$m$ divides $n$” and “there is an integer $q$ for which $m=qn$” are equivalent statements.  Notes: (1) The first statement would only have implied the second statement if this had not been in the context of a definition. (2) After the conclusion of the definition, $m$, $n$ and $q$ are undefined variables. Theorem This announces that the next paragraph is a statement has been proved. In fact, in real time the statement was proved long before this discourse was written, but in terms of reading the text in order, it has not yet been proved. Let $m$, $n$ and $p$ be integers, “Let” tells us that the following statement is the hypothesis of an implication, so we can assume that $m$, $n$ and $p$ are all integers.  This changes the status of $m$ and $n$, which were variables used in the preceding paragraph, but whose constraints disappeared at the end of the paragraph.  We are starting over with $m$ and $n$. with $m$ and $n$ nonzero. This clause is also part of the hypothesis. We can assume $m$ and $n$ are constrained to be nonzero. and suppose $m$ divides $n$ and $n$ divides $p$. This is the last clause in the hypothesis. We can assume that $m$ divides $n$ and $n$ divides $p$. Then $m$ divides $p$. This is a claim that $m$ divides $p$. It has a different status from the assumptions that $m$ divides $n$ and $n$ divides $p$. If we are going to follow the proof we have to treat $m$ and $n$ as if they divide $n$ and $p$ respectively. However, we can’t treat $m$ as if it divides $p$. All we know is that the author is claiming that $m$ divides $p$, given the facts in the hypothesis. Proof An announcement that a proof is about to begin, meaning a chain of math reasoning. The fact that it is a proof of the Theorem just stated is not explicitly stated. By definition of divides, there are integers $q$ and $q’$ for which $n=qm$ and $p=q’n$. The proof uses the direct method (rather than contradiction or induction or some other method) and begins by rewriting the hypothesis using the definition of “divides”. The proof does not announce the use of these techniques, it just starts in doing it. So $q$ and $q$’ are new symbols that satisfy the equations $n=qm$ and $p=q’n$. The phrase “by definition of divides” justifies the introduction of $q$ and $q’$. $m$, $n$ and $p$ have already been introduced in the statement of the Theorem. We must prove that there is an integer $q”$ for which $p=q”m$. Introduces a new variable $q”$ which has not been given a value. We must define it so that $p=q”m$; this requirement is justified (without saying so) by the definition of “divides”. But $p=q’n=q’qm$, This is a claim about $p$, $q$, $q’$, $m$ and $n$.  It is justified by certain preceding sentences but this justification is not made explicit. Note that “$p=q’n=q’qm$” pivots on $q’n$, in other words makes two claims about it. so let $q”=q’q$. We have already introduced $q”$; now we give it the value $q”=q’q$. Then $p=q”m$ This is an assertion about $p$, $q”$ and $n$, justified (but not explicitly — note the hidden use of associativity) by the previous claim that $p=q’n=q’qm$. The proof is now complete, although no statement asserts that it is.

Remark

If you have some skill in reading proofs, all the stuff in the right hand column happens in your brain without, for the most part, your being conscious of it.

Acknowledgment

Thanks to Chris Smith for correcting errors.

References for “context”

Chierchia, G. and S. McConnell-Ginet
(1990), Meaning and Grammar. The MIT Press.

de Bruijn, N. G. (1994), “The mathematical vernacular, a
language for mathematics with typed sets”. In Selected Papers on Automath,
Nederpelt, R. P., J. H. Geuvers, and R. C. de Vrijer, editors, volume 133 of
Studies in Logic and the Foundations of Mathematics, pages 865 – 935. Elsevier

Steenrod, N. E., P. R. Halmos, M. M. Schif­fer,
and J. A. Dieudonné (1975), How to Write Mathematics.
American Mathematical Society.

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The two languages of math

Mathematics is communicated using two languages: Mathematical English and the symbolic language of math (more about them in two languages).

This post is a collection of examples of the sorts of trouble that the two languages cause beginning abstract math students. I have gathered many of them here since they are scattered throughout the literature. I would welcome suggestions for other references to problems caused by the languages of math.

In many of the examples, I give links to the literature and leave you to fish out the details there. Almost all of the links are to documents on the internet.

There is an extensive list of references.

Conjectures

Scattered through this post are conjectures. Like most of my writing about difficulties students have with math language, these conjectures are based on personal observation over 37 years of teaching mostly computer engineering and math majors. The only hard research of any sort I have done in math ed consists of the 426 citations of written mathematical writing included in the Handbook of Mathematical Discourse.

Disclaimer

This post is an attempt to gather together the ways in which math language causes trouble for students. It is even more preliminary and rough than most of my other posts.

• The arrangement of the topics is unsatisfactory. Indeed, the topics are so interrelated that it is probably impossible to give a satisfactory linear order to them. That is where writing on line helps: Lots of forward and backward references.
• Other people and I have written extensively about some of the topics, and they have lots of links. Other topics are stubs and need to be filled out. I have probably missed important points about and references to many of them.
• Please note that many of the most important difficulties that students have with understanding mathematical ideas are not caused by the languages of math and are not represented here.

I expect to revise this article periodically as I find more references and examples and understand some of the topics better. Suggestions would be very welcome.

Intricate symbolic expressions

I have occasionally had students tell me that have great difficulty understanding a complicated symbolic expression. They can’t just look at it and learn something about what it means.

Example

Consider the symbolic expression $\displaystyle\left(\frac{x^3-10}{3 e^{-x}+1}\right)^6$

Now, I could read this expression aloud as if it were text, or more precisely describe it so that someone else could write it down. But if I am in math mode and see this expression I don’t “read” it, even to myself.

I am one of those people who much of the time think in pictures or abstractions without words. (See references here.)

In this case I would look at the expression as a structured picture. I could determine a number of things about it, and when I was explaining it I would point at the board, not try to pronounce it or part of it:

• The denominator is always positive so the expression is defined for all reals.
• The exponent is even so the value of the expression is always nonnegative. I would say, “This (pointing at the exponent) is an even power so the expression is never negative.”
• It is zero in exactly one place, namely $x=\sqrt[3]{10}$.
• Its derivative is also $0$ at $\sqrt[3]{10}$. You can see this without calculating the formula for the derivative (ugh).

Algebra in high school

There are many high school students stymied by algebra, never do well at it, and hate math as a result. I have known many such people over the years. A revealing remark that I have heard many times is that “algebra is totally meaningless to me”. This is sometimes accompanied by a remark that geometry is “obvious” or something similar. This may be because they think they have to “read” an algebraic expression instead of studying it as they would a graph or a diagram.

Conjecture

Many beginning abstractmath students have difficulty understanding a symbolic expression like the one above. Could this be cause by resistance to treating the expression as a structure to be studied?

Context-sensitive pronunciation

A symbolic assertion (“formula” to logicians) can be embedded in a math English sentence in different ways, requiring the symbolic assertion to be pronounced in different ways. The assertion itself is not modified in any way in these different situations.

I used the phrase “symbolic assertion” in abstractmath.org because students are confused by the logicians’ use of “formula“.
In everyday English, “$\text{H}_2\text{O}$” is the “formula” for water, but it is a term, not an assertion.

Example

“For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$.”

• In the sentence above, the assertion “$x\gt0$” must be pronounced “$x$ that is greater than $0$” or something similar.
• The standalone assertion “$x\gt0$” is pronounced “$x$ is greater than $0$.”
• The sentence “Let $x\gt0$” must be pronounced “Let $x$ be greater than $0$”.

The consequence is that the symbolic assertion, in this case “$x\gt0$”, does not reveal that role it plays in the math English sentence that it is embedded in.

Many of the examples occurring later in the post are also examples of context-sensitive pronunciation.

Conjectures

Many students are subconsciously bothered by the way the same symbolic expression is pronounced differently in different math English sentences.

This probably impedes some students’ progress. Teachers should point this phenomenon out with examples.

Students should be discouraged from pronouncing mathematical expressions.

For one thing, this could get you into trouble. Consider pronouncing “$\sqrt{3+5}+6$”. In any case, when you are reading any text you don’t pronounce the words, you just take in their meaning. Why not take in the meaning of algebraic expressions in the same way?

Parenthetic assertions

A parenthetic assertion is a symbolic assertion embedded in a sentence in math English in such a way that is a subordinate clause.

Example

In the math English sentence

“For every real number $x\gt0$ there is a real number $y$ such that $x\gt y\gt0$”

mentioned above, the symbolic assertion “$x\gt0$” plays the role of a subordinate clause.

It is not merely that the pronunciation is different compared to that of the independent statement “$x\gt0$”. The math English sentence is hard to parse. The obvious (to an experienced mathematician) meaning is that the beginning of the sentence can be read this way: “For every real number $x$, which is bigger than $0$…”.

But new student might try to read it is “For every real number $x$ is greater than $0$ …” by literally substituting the standalone meaning of “$x\gt0$” where it occurs in the sentence. This makes the text what linguists call a garden path sentence. The student has to stop and start over to try to make sense of it, and the symbolic expression lacks the natural language hints that help understand how it should be read.

Note that the other two symbolic expressions in the sentence are not parenthetic assertions. The phrase “real number” needs to be followed by a term, and it is, and the phrase “such that” must be followed by a clause, and it is.

More examples

• “Consider the circle $S^1\subseteq\mathbb{C}=\mathbb{R}^2$.” This has subordinate clauses to depth 2.
• “The infinite series $\displaystyle\sum_{k=1}^\infty\frac{1}{k^2}$ converges to $\displaystyle\zeta(2)=\frac{\pi^2}{6}\approx1.65$”
• “We define a null set in $I:=[a,b]$ to be a set that can be covered by a countable of intervals with arbitrarily small total length.” This shows a parenthetical definition.
• “Let $F:A\to B$ be a function.”
A type declaration is a function? In any case, it would be better to write this sentence simply as “Let $F:A\to B$”.

David Butler’s post Contrapositive grammar has other good examples.

Math texts are in general badly written. Students need to be taught how to read badly written math as well as how to write math clearly. Those that succeed (in my observation) in being able to read math texts often solve the problem by glancing at what is written and then reconstructing what the author is supposedly saying.

Conjectures

Some students are baffled, or at least bothered consciously or unconsciously, by parenthetic assertions, because the clues that would exist in a purely English statement are missing.

Nevertheless, many if not most math students read parenthetic assertions correctly the first time and never even notice how peculiar they are.

What makes the difference between them and the students who are stymied by parenthetic assertions?

There is another conjecture concerning parenthetic assertions below.

Context-sensitive meaning

“If” in definitions

Example

The word “if” in definitions does not mean the same thing that it means in other math statements.

• In the definition “An integer is even if it is divisible by $2$,” “if” means “if and only if”. In particular, the definition implies that a function is not even if it is not divisible by $2$.
• In a theorem, for example “If a function is differentiable, then it is continuous”, the word “if” has the usual one-way meaning. In particular, in this case, a continuous function might not be differentiable.

Context-sensitive meaning occurs in ordinary English as well. Think of a strike in baseball.

Conjectures

The nearly universal custom of using “if” to mean “if and only if” in definitions makes it a harder for students to understand implication.

This custom is not the major problem in understanding the role of definitions. See my article Definitions.

Underlying sets

Example

In a course in group theory, a lecturer may say at one point, “Let $F:G\to H$ be a homomorphism”, and at another point, “Let $g\in G$”.

In the first sentence, $G$ refers to the group, and in the second sentence it refers to the underlying set of the group.

This usage is almost universal. I think the difficulty it causes is subtle. When you refer to $\mathbb{R}$, for example, you (usually) are referring to the set of real numbers together with all its canonical structure. The way students think of it, a real number comes with its many relations and connections with the other real numbers, ordering, field properties, topology, and so on.

But in a group theory class, you may define the Klein $4$-group to be $\mathbb{Z}_2\times\mathbb{Z}_2$. Later you may say “the symmetry group of a rectangle that is not a square is the Klein $4$-group.” Almost invariably some student will balk at this.

Referring to a group by naming its underlying set is also an example of synecdoche.

Conjecture

Students expect every important set in math to have a canonical structure. When they get into a course that is a bit more abstract, suddenly the same set can have different structures, and math objects with different underlying sets can have the same structure. This catastrophic shift in a way of thinking should be described explicitly with examples.

Way back when, it got mighty upsetting when the earth started going around the sun instead of vice versa. Remind your students that these upheavals happen in the math world too.

Identity elements

A particular text may refer to the identity element of any group as $e$.

This is as far as I know not a problem for students. I think I know why: There is a generic identity element. The identity element in any group is an instantiation of that generic identity element. The generic identity element exists in the sketch for groups; every group is a functor defined on that sketch. (Or if you insist, the generic identity element exists in the first order theory for groups.) I suspect mathematicians subconsciously think of identity elements in this way.

Matrix multiplication

Matrix multiplication is not commutative. A student may forget this and write $(A^2B^2=(AB)^2$. This also happens in group theory courses.

This problem occurs because the symbolic language uses the same symbol for many different operations, in this case the juxtaposition notation for multiplication. This phenomenon is called overloaded notation and is discussed in abstractmath.org here.

Conjecture

Noncommutative binary operations written using juxtaposition cause students trouble because going to noncommutative operations requires abandoning some overlearned reflexes in doing algebra.

Identity elements seem to behave the same in any binary operation, so there are no reflexes to unlearn. There are generic binary operations of various types as well. That’s why mathematicians are comfortable overloading juxtaposition. But to get to be a mathematician you have to unlearn some reflexes.

Negation

Sometimes you need to reword a math statement that contains symbolic expressions. This particularly causes trouble in connection with negation.

Ordinary English

The English language is notorious among language learners for making it complicated to negate a sentence. The negation of “I saw that movie” is “I did not see that movie”. (You have to put “d** not” (using the appropriate form of “do”) before the verb and then modify the verb appropriately.) You can’t just say “I not saw that movie” (as in Spanish) or “I saw not that movie” (as in German).

Conjecture

The method in English used to negate a sentence may cause problems with math students whose native language is not English. (But does it cause math problems with those students?)

Negating symbolic expressions

Examples

• The negation of “$n$ is even and a prime” is “$n$ is either odd or it is not a prime”. The negation should not be written “$n$ is not even and a prime” because that sentence is ambiguous. In the heat of doing a proof students may sometimes think the negation is “$n$ is odd and $n$ is not a prime,” essentially forgetting about DeMorgan. (He must roll over in his grave a lot.)
• The negation of “$x\gt0$” is “$x\leq0$”. It is not “$x\lt0$”. This is a very common mistake.

These examples are difficulties caused by not understanding the math. They are not directly caused by difficulties with the languages of math.

Negating expressions containing parenthetic assertions

Suppose you want to prove:

“If $f:\mathbb{R}\to\mathbb{R}$ is differentiable, then $f$ is continuous”.

A good way to do this is by using the contrapositive. A mechanical way of writing the contrapositive is:

“If $f$ is not continuous, then $f:\mathbb{R}\to\mathbb{R}$ is not differentiable.”

That is not good. The sentence needs to be massaged:

“If $f:\mathbb{R}\to\mathbb{R}$ is not continuous, then $f$ is not differentiable.”

Even better would be to write the original sentence as:

“Suppose $f:\mathbb{R}\to\mathbb{R}$. Then if $f$ is differentiable, then $f$ is continuous.”

This is discussed in detail in David Butler’s post Contrapositive grammar.

Conjecture

Students need to be taught to understand parenthetic assertions that occur in the symbolic language and to learn to extract a parenthetic assertion and write it as a standalone assertion ahead of the statement it occurs in.

Scope

The scope of a word or variable consists of the part of the text for which its current definition is in effect.

Examples

• “Suppose $n$ is divisible by $4$.” The scope is probably the current paragraph or perhaps the current proof. This means that the properties of $n$ are constrained in that section of the text.
• “In this book, all rings are unitary.” This will hold for the whole book.

There are many more examples in the abstractmath.org article Scope.

If you are a grasshopper (you like to dive into the middle of a book or paper to find out what it says), knowing the scope of a variable can be hard to determine. It is particularly difficult for commonly used words or symbols that have been defined differently from the usual usage. You may not suspect that this has happened since it might be define once early in the text. Some books on writing mathematics have urged writers to keep global definitions to a minimum. This is good advice.

Finding the scope is considerably easier when the text is online and you can search for the definition.

Conjecture

Knowing the scope of a word or variable can be difficult. It is particular hard when the word or variable has a large scope (chapter or whole book.)

Variables

Variables are often introduced in math writing and then used in the subsequent discussion. In a complicated discussion, several variables may be referred to that have different statuses, some of them introduced several pages before. There are many particular ways discussed below that can cause trouble for students. This post is restricted to trouble in connection with the languages of math. The concept of variable is difficult in itself, not just because of the way the math languages represent them, but that is not covered here.

Much of this part of the post is based on work of Susanna Epp, including three papers listed in the references. Her papers also include many references to other work in the math ed literature that have to do with understanding variables.

Types

Students blunder by forgetting the type of the variable they are dealing with. The example given previously of problems with matrix multiplication is occasioned by forgetting the type of a variable.

Conjecture

Students sometimes have problems because they forget the data type of the variables they are dealing with. This is primarily causes by overloaded notation.

Dependent and independent

If you define $y=x^2+1$, then $x$ is an independent variable and $y$ is a dependent variable. But dependence and independence of variablesare more general than that example suggests.
In an epsilon-delta proof of the limit of a function (example below,) $\varepsilon$ is independent and $\delta$ is dependent on $\varepsilon$, although not functionally dependent.

Conjecture

Distinguishing dependent and independent variables causes problems, particularly when the dependence is not clearly functional.

I recently ran across a discussion of this on the internet but failed to record where I saw it. Help!

Bound and free

This causes trouble with integration, among other things. It is discussed in abstractmath.org in Variables and Substitution. I expect to add some references to the math ed literature soon.

Instantiation

Some of these variables may be given by existential instantiation, in which case they are dependent on variables that define them. Others may be given by universal instantiation, in which case the variable is generic; it is independent of other variables, and you can’t impose arbitrary restrictions on it.

Existential instantiation

A theorem that an object exists under certain conditions allows you to name it and use it by that name in further arguments.

Example

Suppose $m$ and $n$ are integers. Then by definition, $m$ divides $n$ if there is an integer $q$ such that $n=qm$. Then you can use “$q$” in further discussion, but $q$ depends on $m$ and $n$. You must not use it with any other meaning unless you start a new paragraph and redefine it.

So the following (start of a) “proof” blunders by ignoring this restriction:

Theorem: Prove that if an integer $m$ divides both integers $n$ and $p$, then $m$ divides $n+p$.

“Proof”: Let $n = qm$ and $p = qm$…”

Universal instantiation

It is a theorem that for any integer $n$, there is no integer strictly between $n$ and $n+1$. So if you are given an arbitrary integer $k$, there is no integer strictly between $k$ and $k+1$. There is no integer between $42$ and $43$.

By itself, universal instantiation does not seem to cause problems, provided you pay attention to the types of your variables. (“There is no integer between $\pi$ and $\pi+1$” is false.)

However, when you introduce variables using both universal and existential quantification, students can get confused.

Example

Consider the definition of limit:

Definition: $\lim_{x\to a} f(x)=L$ if and only if for every $\epsilon\gt0$ there is a $\delta\gt0$ for which if $|x-a|\lt\delta$ then $|f(x)-L|\lt\epsilon$.

A proof for a particular instance of this definition is given in detail in Rabbits out of a Hat. In this proof, you may not put constraints on $\epsilon$ except the given one that it is positive. On the other hand, you have to come up with a definition of $\delta$ and prove that it works. The $\delta$ depends on what $f$, $a$ and $L$ are, but there are always infinitely many values of $\delta$ which fit the constraints, and you have to come up with only one. So in general, two people doing this proof will not get the same answer.

Reference

Susanna Epp’s paper Proof issues with existential quantification discusses the problems that students have with both existential and universal quantification with excellent examples. In particular, that paper gives examples of problems students have that are not hinted at here.

References

A nearly final version of The Handbook of Mathematical Discourse is available on the web with links, including all the citations. This version contains some broken links. I am unable to recompile it because TeX has evolved enough since 2003 that the source no longer compiles. The paperback version (without the citations) can be bought as a book here. (There are usually cheaper used versions on Amazon.)

Abstractmath.org is a website for beginning students in abstract mathematics. It includes most of the material in the Handbook, but not the citations. The Introduction gives you a clue as to what it is about.

Two languages

My take on the two languages of math are discussed in these articles:

The Language of Mathematics, by Mohan Ganesalingam, covers these two languages in more detail than any other book I know of. He says right away on page 18 that mathematical language consists of “textual sentences with symbolic material embedded like ‘islands’ in the text.” So for him, math language is one language.

I have envisioned two separate languages for math in abstractmath.org and in the Handbook, because in fact you can in principle translate any mathematical text into either English or logical notation (first order logic or type theory), although the result in either case would be impossible to understand for any sizeable text.

Topics in abstractmath.org

Context-sensitive interpretation.

Mathematical English.

Scope

Semantic contamination.

The symbolic language of math

Variables.

Zooming and Chunking.

Topics in the Handbook of mathematical discourse.

These topics have a strong overlap with the topics with the same name in abstractmath.org. They are included here because the Handbook contains links to citations of the usage.

Posts in Gyre&Gimble

Names

Naming mathematical objects

Rabbits out of a Hat.

Semantics of algebra I.

Syntactic and semantic thinkers

Technical meanings clash with everyday meanings

Three kinds of mathematical thinkers

Other references

Contrapositive grammar, blog post by David Butler.

Proof issues with existential quantification, by Susanna Epp.

The role of logic in teaching proof, by Susanna Epp (2003).

The language of quantification in mathematics instruction, by Susanna Epp (1999).

The Language of Mathematics: A Linguistic and Philosophical Investigation
by Mohan Ganesalingam, 2013. (Not available from the internet.)

On the communication of mathematical reasoning, by Atish Bagchi, and Charles Wells (1998a), PRIMUS, volume 8, pages 15–27.

Variables in Wikipedia.

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More alphabets

This post is the third and last in a series of posts containing revisions of the abstractmath.org article Alphabets. The first two were:

Addition to the listings for the Greek alphabet

Sigma: $\Sigma,\,\sigma$ or ς: sĭg'mɘ. The upper case $\Sigma$ is used for indexed sums.  The lower case $\sigma$ (don't call it "oh") is used for the standard deviation and also for the sum-of-divisors function. The ς form for the lower case has not as far as I know been used in math writing, but I understood that someone is writing a paper that will use it.

Hebrew alphabet

Aleph, א is the only Hebrew letter that is widely used in math. It is the cardinality of the set of integers. A set with cardinality א is countably infinite. More generally, א is the first of the aleph numbers $א_1$, $א_2$, $א_3$, and so on.

Cardinality theorists also write about the beth (ב) numbers, and the gimel (ג) function. I am not aware of other uses of the Hebrew alphabet.

If you are thinking of using other Hebrew letters, watch out: If you type two Hebrew letters in a row in HTML they show up on the screen in reverse order. (I didn't know HTML was so clever.)

Cyrillic alphabet

The Cyrillic alphabet is used to write Russian and many other languages in that area of the world. Wikipedia says that the letter Ш, pronounced "sha", is the only Cyrillic letter used in math. I have not investigated further.

The letter is used in several different fields, to denote the Tate-Shafarevich group, the Dirac comb and the shuffle product.

It seems to me that there are a whole world of possibillities for brash young mathematicians to name mathematical objects with other Cyrillic letters. Examples:

• Ж. Use it for a ornate construction, like the Hopf fibration or a wreath product.
• Щ. This would be mean because it is hard to pronounce.
• Ъ. Guaranteed to drive people crazy, since it is silent. (It does have a name, though: "Yehr".)
• Э. Its pronunciation indicates you are unimpressed (think Fonz).
• ю. Pronounced "you". "ю may provide a counterexample". "I do?"

Type styles

Boldface and italics

A typeface is a particular design of letters.  The typeface you are reading is Arial.  This is Times New Roman. This is Goudy. (Goudy may not render correctly on your screen if you don't have it installed.)

Typefaces typically come in several styles, such as bold (or boldface) and italic.

Examples

 Arial Normal Arial italic Arial bold Times Normal Times italic Times bold Goudy Normal Goudy italic Goudy bold

Boldface and italics are used with special meanings (conventions) in mathematics. Not every author follows these conventions.

Styles (bold, italic, etc.) of a particular typeface are supposedly called fonts.  In fact, these days “font” almost always means the same thing as “typeface”, so I  use “style” instead of “font”.

Vectors

A letter denoting a vector is put in boldface by many authors.

Examples
• “Suppose $\mathbf{v}$ be an vector in 3-space.”  Its coordinates typically would be denoted by $v_1$, $v_2$ and $v_3$.
• You could also define it this way:  “Let $\mathbf{v}=({{v}_{1}},{{v}_{2}},{{v}_{3}})$ be a vector in 3-space.”  (See parenthetic assertion.)

It is hard to do boldface on a chalkboard, so lecturers may use $\vec{v}$ instead of $\mathbf{v}$. This is also seen in print.

Definitions

The definiendum (word or phrase being defined) may be put in boldface or italics. Sometimes the boldface or italics is the only clue you have that the term is being defined. See Definitions.

Example

“A group is Abelian if its multiplication is commutative,” or  “A group is Abelian if its multiplication is commutative.”

Emphasis

Italics are used for emphasis, just as in general English prose. Rarely (in my experience) boldface may be used for emphasis.

In the symbolic language

It is standard practice in printed math to put single-letter variables in italics.   Multiletter identifiers are usually upright.

Example

Example: "$f(x)=a{{x}^{2}}+\sin x$".  Note that mathematicians would typically refer to $a$ as a “constant” or “parameter”, but in the sense we use the word “variable” here, it is a variable, and so is $f$.

Example

On the other hand, “e” is the proper name of a specific number, and so is “i”. Neither is a variable. Nevertheless in print they are usually given in italics, as in ${{e}^{ix}}=\cos x+i\sin x$.  Some authors would write this as ${{\text{e}}^{\text{i}x}}=\cos x+\text{i}\,\sin x$.  This practice is recommended by some stylebooks for scientific writing, but I don't think it is very common in math.

Blackboard bold

Blackboard bold letters are capital Roman letters written with double vertical strokes.   They look like this:

$\mathbb{A}\,\mathbb{B}\,\mathbb{C}\,\mathbb{D}\,\mathbb{E}\,\mathbb{F}\,\mathbb{G}\,\mathbb{H}\,\mathbb{I}\,\mathbb{J}\,\mathbb{K}\,\mathbb{L}\,\mathbb{M}\,\mathbb{N}\,\mathbb{O}\,\mathbb{P}\,\mathbb{Q}\,\mathbb{R}\,\mathbb{S}\,\mathbb{T}\,\mathbb{U}\,\mathbb{V}\,\mathbb{W}\,\mathbb{X}\,\mathbb{Y}\,\mathbb{Z}$

In lectures using chalkboards, they are used to imitate boldface.

In print, the most common uses is to represent certain sets of numbers:

Remarks

• Mathe­ma­tica uses some lower case blackboard bold letters.
• Many mathe­ma­tical writers disapprove of using blackboard bold in print.  I say the more different letter shapes that are available the better.  Also a letter in blackboard bold is easier to distinguish from ordinary upright letters than a letter in boldface is, particularly on computer screens.
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The use of fraktur in math

This post is a revision of the part of the abmath article on alphabets concerning the fraktur typeface, followed by some corrections and remarks. A revision of the section on the Greek alphabet was posted previously.

Fraktur

In some math subjects, a font tamily (typeface) called fraktur, formerly used for writing German, Norwegian, and some other languages, is used to name math objects.  The table below shows the upper and lower case fraktur letters.

 $A,a$: $\mathfrak{A},\mathfrak{a}$ $H,h$: $\mathfrak{H},\mathfrak{h}$ $O,o$: $\mathfrak{O},\mathfrak{o}$ $V,v$: $\mathfrak{V},\mathfrak{v}$ $B,b$: $\mathfrak{B},\mathfrak{b}$ $I,i$: $\mathfrak{I},\mathfrak{i}$ $P,p$: $\mathfrak{P},\mathfrak{p}$ $W,w$: $\mathfrak{W},\mathfrak{w}$ $C,c$: $\mathfrak{C},\mathfrak{c}$ $J,j$: $\mathfrak{J},\mathfrak{j}$ $Q,q$: $\mathfrak{Q},\mathfrak{q}$ $X,x$: $\mathfrak{X},\mathfrak{x}$ $D,d$: $\mathfrak{D},\mathfrak{d}$ $K,k$: $\mathfrak{K},\mathfrak{k}$ $R,r$: $\mathfrak{R},\mathfrak{r}$ $Y,y$: $\mathfrak{Y},\mathfrak{y}$ $E,e$: $\mathfrak{E},\mathfrak{e}$ $L,l$: $\mathfrak{L},\mathfrak{l}$ $S,s$: $\mathfrak{S},\mathfrak{s}$ $Z,z$: $\mathfrak{Z},\mathfrak{z}$ $F,f$: $\mathfrak{F},\mathfrak{f}$ $M,m$: $\mathfrak{M},\mathfrak{m}$ $T,t$: $\mathfrak{T},\mathfrak{t}$ $G,g$: $\mathfrak{G},\mathfrak{g}$ $N,n$: $\mathfrak{N},\mathfrak{n}$ $U,u$: $\mathfrak{U},\mathfrak{u}$
• Many of the forms are confusing and are commonly mispronounced by younger mathematicians.  (Ancient mathematicians like me have taken German classes in college that required learning fraktur.)  In particular the uppercase $\mathfrak{A}$ looks like $U$ but in fact is an $A$, and the uppercase $\mathfrak{I}$ looks like $T$ but is actually $I$.
• When writing on the board, some mathematicians use a cursive form when writing objects with names that are printed in fraktur.
• Unicode regards fraktur as a typeface (font family) rather than as a different alphabet. However, unicode does provide codes for the fraktur letters that are used in math (no umlauted letters or ß). In TeX you type "\mathfrak{a}" to get $\mathfrak{a}$.
• In my (limited) experience, native German speakers usually call this alphabet “Altschrift” instead of “Fraktur”.  It has also been called “Gothic”, but that word is also used to mean several other quite different typefaces (black­letter, sans serif and (gasp) the alphabet actually used by the Goths.
• I have been doing mathematical research for around fifty years. It is clear to me that mathematicians' use of and familiarity with fraktur has declined a lot during that time. But it is not extinct. I have made a hasty and limited search of Jstor and found recent websites and research articles that use it in a variety of fields. There are also a few citations in the Handbook (search for "fraktur").

• It is used in ring theory and algebraic number theory, in particular to denote ideals.
• It is use in Lie algebra. In particular, the Lie algebra of a Lie group $G$ is commonly denoted by $\mathfrak{g}$.
• The cardinality of the continuum is often denoted by $\mathfrak{c}$.
• It is used occasionally in logic to denote models and other objects.
• I remember that in the sixties and seventies fraktur was used in algebraic geometry, but I haven't found it in recent papers.

Acknowledgements

Thanks to Fernando Gouvêa for suggestions.

The Handbook has 428 citation for usages in the mathematical research literature. After finishing that book, I started abstractmath.org and decided that I would quote the Handbook for usages when I could but would not spend any more time looking for citations myself, which is very time consuming. Instead, in abmath I have given only my opinion about usage. A systematic, well funded project for doing lexicographical research in the math literature would undoubtedly show that my remarks were sometimes incorrect and very often, perhaps even usually, incomplete.

Corrections to the post The Greek alphabet in math

Willie Wong suggested some additional pronunciations for upsilon and omega:

Upsilon: $\Upsilon ,\,\upsilon$  ŭp'sĭlŏn; (Br) ĭp'sĭlŏn. (Note: I have never heard anyone pronounce this letter, and various dictionaries suggest a ridiculous number of different pronunciations.) Rarely used in math; there are references in the Handbook.

Omega: $\Omega ,\,\omega$: ōmā'gɘ, ō'māgɘ; (Br) ōmē'gɘ, ō'mēgɘ. $\Omega$ is often used as the name of a domain in $\mathbb{R}^n$. The set of natural numbers with the usual ordering is commonly denoted by $\omega$. Both forms have many other uses in advanced math.

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Presenting binary operations

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

Before binary operations are introduced

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

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

In elementary school

In elementary school you see expressions such as

• $3+4$
• $3\times 4$
• $3-4$

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

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

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

Algebra

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

Example

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

The expressions “$3+5$” and “$8$”

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

Another example is “$3+5=6+2$”.

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

Note

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

Binary operations

Early examples

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

Common operations

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

Extensional definitions

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

Properties

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

Binary operation as function

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

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

This is abstraction twice.

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

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

Other representations of binary operations.

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

Binary operations in high school and college algebra

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

Trees

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

Example

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

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

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

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

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

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The two languages of math

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

Introduction

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

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

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

Contents

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

Notes

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

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Representations of mathematical objects

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This is a long post. Notes on viewing.

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

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

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

Types of representations

Using language

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

Mathematical objects

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

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

Visual representations

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

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

Mental representations

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

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

Metaphors

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

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

Properties of representations

A representation of a math object may or may not

• determine it completely
• exhibit some of its properties
• suggest easy proofs of some theorems
• provide a useful way of thinking about it

Examples of representations

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

Functions

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

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

Diatribe

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

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

The positive integers

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

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

Graph of a function

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

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

Continuity

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

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

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

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

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

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

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

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

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

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

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

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

References

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

Notes on Viewing

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

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Visible algebra II

MathJax.Hub.Config({ jax: ["input/TeX","output/NativeMML"], extensions: ["tex2jax.js"], tex2jax: { inlineMath: [ ['$','$'] ], processEscapes: true } }); The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Wolfram website. The code for the demos is in the Mathematica notebook algebra2.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

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.

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Semantics of algebra I

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

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.

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