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Posted 12 February 2009

MORE ABOUT THE LANGUAGES OF MATH

Mathematical writing consists of a mixture of mathematical English and the symbolic language.  Fragments of each language are embedded in each other and refer to each other.  The relationship between the two languages can be subtle, and both languages contain explicit and implicit conventions concerning the words and notation used and also concerning the embeddings and references.  This chapter shows you some of that behavior. 

       Spoken mathematics is usually a combination of spoken math English, pronouncing simple symbolic expressions and pointing at complicated expressions that are essentially impossible to pronounce in a comprehensible way.

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:

Example

In a discourse 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 n to be any integer, but after the occurrence the context requires n = 3.

 This concept of context is not what is usually meant by the word.  In particular, here I am referring not to the context of the whole discourse,  but to the context within the text of each symbol or name at each location of its occurrence.

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. 

a)     Each clause can change the meaning of or constraints on one or more symbols or names.

b)     The conventions in effect during the discourse can put constraints on the symbols and names.

Remarks

This is not a mathematical definition.  Both (a) and (b) are fuzzy.  In (a), do you include in the context the consequences of the statements made in the discourse?  In (b), what the conventions are may not be clear.  (In my experience, they are usually clear, at least to experienced mathematicians, but not always.)

“Before” and “after”

Note that the reference to “before” and “after” the phrase “Let n = 3” refer to the physical location in text and to actual time in spoken math.  More about this is in the Handbook, page 252, items (f) and (g).

 In particular, contextual changes of this sort take place using the pretense that you are reading the text in order, which most students and professionals do not do (they are grasshoppers).

Detailed example of math text

Here is a typical example of a theorem and its proof.  It is printed twice, the second time with comments in red 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 . The statement “m divides n” means that there is an integer q for which .

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  and . We must prove that there is an integer  for which . But , so let .  Then .

Second time, with analysis

Remark

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

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.

Embedding       

Statements in mathematical English may contain embedded expressions in the symbolic language.  The opposite can happen, too: sometimes symbolic expressions contains an embedded statement in math English.

Examples

¨  “The inequality   holds for all real numbers x.”  This sentence contains two embedded symbolic expressions.   “  ” is a symbolic assertion and “x” is a symbolic term.

¨  “Let S = {n | n is the product of two different primes}.”  (See setbuilder notation.)  Here the English sentence “n is the product of two different primes” appears inside the symbolic definition of S, which itself is embedded in the sentence beginning “Let S…”.

Displayed symbolic expressions

A symbolic expression may be written in line with the text it is embedded in, as in the two examples just above.  It may also be displayed, as in this sentence:

“The inequality

holds for all real numbers x.” 

The part of the sentence before the equation is displayed on one line, the equation is displayed (usually centered, as here) on the next line, and the rest of the sentence is in the line after the equation.  Most printed texts put extra vertical space between the equation and the two parts of the sentence.

Parenthetic assertions

The examples of embedding just given are not hard to understand.  However, some common practices by math writers (and speakers) can cause real problems for those new to a subject.  One phenomenon that causes problems is the parenthetic assertion. 

A symbolic assertion is parenthetic if it is embedded in a sentence in a natural language in such a way that it becomes a phrase (not a clause) embedded in the sentence.

 Example

(S) "For any  there is a  such that ."

The assertions “  " and "  " are parenthetic but "  " is a full clause.

This would commonly be pronounced this way:

(S’) “For any x greater than 0 there is a y greater than 0 such that y is less than x.”

 The sentence (S) could be written as

“For any x (that is greater than 0), there is a y (that is greater than 0) such that y is less than x.”

From this you can see where the name “parenthetic assertion” came from.

See other examples below.

Context-sensitive interpretation

In isolation, the assertion "  " is a complete sentence, typically pronounced “x is greater than  0". In sentence (S), however, it might be pronounced

¨  "x that is greater than 0" or just

¨  "x greater than 0"

This example shows that:  

Example

In the article on semantic contamination, I wrote,

“This infinite series converges to , which is , which is approximately 1.65.” 

Many authors would have written

“This infinite series converges to  ”.   (PA)

This is an example of a parenthetic assertion containing another parenthetic assertion.  It could be written this way (but no one ever would):

“This infinite series converges to  ”

If you read it literally, the statement (PA) appears to say that the infinite series (a mathematical object) is converging to a statement in the symbolic language.  Of course, you are not supposed to read (PA) literally, and no one who has gotten very far in abstract math would read it that way.  It is a parenthetic assertion, an instance of mathematical slang.  Many many authors use this technique.  (And some don’t.  See the Handbook article on parenthetic assertions for examples.) 

Example

In this example, a parenthetic assertion is used to define I:

“A closed subinterval of an interval  is a subset of I of the form  where  ”.

This example is therefore a definition containing a nested definition.  (The definition of I as [a, b] is contained inside the definition of “closed subinterval”.)  Readers inexperienced with notation may read this as saying that the closed subinterval is .  Your clue that this is not correct is that if it were the rest of the sentence wouldn’t make sense. 

Example

"Consider the circle  ”.  Again, the parenthetic remark contains another parenthetic remark inside it.  Notice the kinds of difficulties this can cause: 

¨  You are supposed to recognize that the symbol  means a circle as a topological space (pretty standard notation).

¨  The parenthetic assertion  uses the symbol “  ” for set inclusion, but it means a topological embedding.  (This is common usage and certainly justifiable but writers almost never tell you this.)

¨  How are you supposed to read “  ”?   Is it a definition of ?  Probably not: the author is thinking of  as the set of complex numbers and reminding you that as spaces,  is homeomorphic to .  It is common mathematical usage (and justifiable in many circumstances) to use the equal sign to mean “is isomorphic to”, “is homeomorphic to” and so on. 

More

See arrow notation for another example.

The meaning of a word can change depending on where it is in a sentence.  A notorious example is that the meaning of if is different depending on whether it occurs in a definition or a theorem.

Preconditions and postconditions

Preconditions

A precondition is a preliminary statement that puts a constraint on variables in the discussion that follows, and thus has a temporary effect on the contest.  Example:  “Suppose x is a real number.  Then .”  This is a common way of wording a conditional sentence.

Postconditions

A postcondition is a statement at the end of a context that puts a constraint on the variables occuring in the context, which are therefore retroactively constrained.

 Example:  “ , for any real number x.” 

Postconditions in the symbolic language

Postconditions in the symbolic language are not used in a consistent way.   You frequently have to have considerable knowledge of the subject being discussed to understand what is meant. 

Examples

Except for the first one, there examples are quotations from the research literature, modified in some cases.

(a) “By factoring, we see that

                                               ”

This means that the equation holds for all x except 0.   

       (b) “Consider an integral equation of the form

with a given symmetric kernel .”

The intent of this equation is to solve for g for a given function f defined on the interval (0,a).   How did I know this?  Because it is called an integral equation.

(c) “The largest region leaves in one iteration and is bounded by the curves

which satisfy .”

The phrase  here means that you are looking at two equations in two real variables x and a.  It would be legitimate to regard it instead as one equation in three variables, x and a real and i varying over the set {1, 2}. 

(d) “Taking x = 2 in the preceding equation, we obtain , so that

and we see that all the roots lie in the disk .”

Here, the constraint means that  for all n > 2, where n is by convention an integer.  This example is also discussed under parameter.

(e) “By applying the Mean Value Theorem to f on [a, d] and [d, b] respectively, we obtain

 ”

This does not mean  for all .  It means there exists a number  that makes the first equation true (similar remark for the second equation).  Compare this to the occurrence of “(n > 2)” in (d), which means for all n > 2.

More examples of preconditions and postconditions may be found under constraints, universally true statements, conditional statements and definitions.

Incomplete notation

Suppression of parameters

Math notation may omit a parameter on which the meaning of the notation depends.  See also default.

Example

A group with underlying set G and binary operation * may be called  G, so that the notation omits the binary operation. This is also an example of synecdoche.

Example

A text may, by the expression log x, refer to the logarithm with base e, and may or may not announce this fact explicitly. This is not an example of synecdoche.  More about this.

Synecdoche

Synecdoche is naming something by naming a part of it.  Referring to a car as “wheels” is an example of synecdoche. 

Example

 Naming a mathematical structure by its underlying set. This is also a case of suppression of parameters.

Example

Naming an equivalence class by a member of the class. Note that this is not an example of suppression of parameters. See well-defined.

Overloaded notation

A symbol or name for a function that takes on different meanings depending on which type of element it is evaluated at is said to be overloaded or polymorphic.

Example 1

The identity function is a polymorphic name, since there is technically a different identity function for each set.

Example 2

A text on linear algebra will very likely use + for addition of vectors in every vector space. Similarly, some texts on group theory will use e or 1 for the identity element for any group and juxtaposition for the binary operation of any group.

Example 3

The symbol  is overloaded in college math courses. When a and b are numbers,  is their product. When A and B are matrices,  is the matrix product. When v and w are 3-vectors,  is their vector product.

Other examples are given under equivalence relation and under  image.

Remarks

¨  Example 2 illustrates the common case of using the same symbol in every model for a particular operation in an axiomatically defined mathematical structure.

¨  The operation  does not use the same algorithm on matrices as it does on 3-vectors. This is the sort of phenomenon computer scientists call ad-hoc polymorphism. It is contrasted with parametric polymorphism, an example of which is the algorithm "swap the two entries in an ordered pair", which applies to ordered pairs of any type of element. (The parameter that gives rise to the name "parametric" is the type of element.)  The identity function provides a trivial example of parametric polymorphism.  More

¨  Many mathematicians think and speak informally of a parametrically polymorphic function as one single function. ("... the identity function is injective").

¨  The concept “overloaded” is natural in computing science because operations on different data types are typically implemented differently. For example, addition of integers is implemented differently from addition of floating point numbers in most computer languages.

Redundancy

Redundancy in discourse

A given discourse is redundant if it contains words and expressions that could be omitted without changing the meaning.  Redundancy in some circumstances is annoying but in others it is helpful for understanding.

Example

“The counting function of primes defined by  satisfies the formula .”

The phrase "the counting function of primes" is redundant, since the definition of  just following that phrase says it is the counting function for primes. This is in no way bad writing: the redundancy adds much to the reader's understanding (for this reader, anyway).

Type labeling

Type labeling is another commonly occurring systematic form of redundancy.  This is discussed in more detail in type of a variable.   

Example

Suppose a text defines G to be a group on page 42 and defines a certain subgroup H of G on page 46.  Then on page 48 it says “The subgroup H of the group G is normal in G”.  The phrases “the subgroup” and “the group” are redundant since H and G have already been defined as such.  Nevertheless this redundancy can be very helpful to the reader since the sentence is separated by a couple of pages from the definitions.

Redundancy in definitions

I have heard mathematicians say that a definition is not a proper definition if it is redundant.  In fact, definitions are very commonly redundant, including the usual definition of group (here).

Appendices

Grasshoppers

A grasshopper is a reader who starts reading a book or article at the point where it discusses what she is interested in, then jumps back and forth through the text finding information about the ideas she meets. This is contrasted with someone who starts at the beginning and reads straight through.

The terminology is due to Steenrod, who calls the reader who starts at the beginning and reads straight through a normal reader, a name which this particular grasshopper resents.  Since Steenrod also mentions the difficulty causes by “global” terminology, particularly terminology defined near the beginning of a book and used without comment in the rest of the text, I suspect him of having been  a grasshopper.

Garden path sentences

Garden path sentences have  the property that an initial part of the sentence looks like a complete sentence but is not. 

Examples

This sentence from an old popular song is a garden-path sentence:

“Throw Mama from the train a kiss.”

The example given above on this page is

“A closed subinterval of an interval  is a subset of I of the form  where  ”.

The first part of the sentence could be read to mean “A closed subinterval of an interval I  is [a,b].”  This is not what it means because the rest of the sentence then would be nonsense.

More about log

You could argue that when you write “log x” and announce that the base for your logs is always e, then you have not suppressed any parameter because the way you defined it, “log” doesn’t have a parameter.  But it is very common usage to write  with the parameter explicit.  You can change the way you define a symbol but you can’t change what the reader expects. 

Intent of assertions

Parameters

Variations in meaning

Definition: Divides 

 [Changes the status of the word “divides” so that it becomes the definiendum.  The scope is the following paragraph.] 

Let m and n be integers

[m and n are new symbols in this discourse, constrained to be integers]  

with  

[another constraint on m]  

The statement “m divides n” means that there is an integer q

 [another new symbol constrained to be an integer]

for which .

[Now we know that m, n and q must satisfy a certain equation.  Note that this occurs as the conclusion of a conditional statement in a definition, so we do not know that m divides n.].

Theorem

[This announces that the next paragraph is a mathematical statement and it claims that the 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 nonzero integers,

[Now we know that m, n and p are all nonzero integers.  This ostensibly changes that status of m and n, which were variables used in the preceding paragraph, but now all previous constraints are discarded.  We are starting over with m, n, and p.  In particular, n is now nonzero, which it was not in the previous paragraph.

and suppose m divides n and n divides p. 

[Now we know 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 n divides p.]

Proof

By definition of divides, there are integers q and q’ for which  and .

[q and q’ are new symbols that we are to assume satisfy the equations  and .   The phrase “by definition of divides” tells us (implicitly, not explicitly) that there are such integers, so in effect this sentence chooses q and q’ so that  and .  The reader probably knows that there is only one choice for each of q and q’ but in fact that claim is not being made here.  Note that m, n and p are not new symbols  they still fall within the scope of the previous paragraph, so we will know that  m divides n and n divides p] 

We must prove that there is an integer  

[Another new variable, which is an integer]

for which  

[q’’ is constrained by this equation.  At this point we ostensibly do not know that any integer q” satisfying the equation exists.].

But ,

[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.]

so let  

[We have already introduced q” and have put the constraint  on it.  Now we put another constraint on it, namely  ].

Then  

[This is an assertion about p, q” and n, justified (but not explicitly) by the claim .]