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Posted 12 August 2008

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

Before a phrase such as “Let n = 3”, n may be known only as an integer variable, or it may not have been used at all. After the phrase, it means specifically 3.  So this phrase changes the context by constraining n to be 3.

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

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.

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 nonzero integers, 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

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, 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.  In fact n is still an integer as it was before, and so is m, but at the moment we can no longer assume that  -- although that won’t last long!]

and suppose m divides n and n divides p. 

[Now we know that m divides n and n divides p.  In particular, we know that m and n are nonzero by definition of “divides”.  However, that is a conclusion we may draw based on our knowledge of the definition of “divides”; it is not explicitly stated in the discourse.]

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 .]

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

¨  “If x is any real number, then .”  This sentence contains two symbolic expressions.  The first x.  It is a symbolic description.  The second one is .  It is a symbolic assertion.

¨  “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.  

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