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.