Posted 24 June 2009

VARIATIONS IN MEANING

Words in a natural language may have different meanings in different social groups or different places.  Words and symbols in both mathematical English and the symbolic language vary according to specialty and, rarely, country (see convention, default).  And in contrast to natural languages, words and symbols can change their meanings from place to place within the same mathematical discourse (see scope).

Conventions

A convention in mathematical discourse is notation or terminology used with a special meaning in certain contexts or in certain fields.   Nice authors remind you of conventions, particular when writing for students. 

Some conventions are nearly universal in math

Example 1

The use of if to mean “if and only if” in a definition is a convention. More about this here.

Example 2

Constants or parameters are conventionally denoted by a, b, ... , functions by f, g, ... and variables by x, y,.... More.

Example 3

Referring to a group (or other mathematical structure) and its underlying set by the same name is a convention.  This is an example of both synecdoche and context-sensitive.

Example 4

The meaning of  is:

¨  The inverse sine (arcsin) if n = −1

¨  The multiplicative power for positive n (  ) if  

This is a common, often explicit, convention in calculus texts.

Some conventions are pervasive among mathematicians but different conventions hold in other subjects that use mathematics.

¨  Scientists and engineers may regard a truncated decimal such as 0.252 as an approximation, but a mathematician is likely to read it as an exact rational number, namely .

¨  In most computer languages a distinction is made between real numbers and integers; 42 would be an integer but 42.0 would be a real number.  Older mathematicians may not know this.

¨  Mathematicians use i to denote the imaginary unit. In electrical engineering it is commonly denoted j instead, a fact that many mathematicians are unaware of. I first learned about it when a student asked me if i was the same as j.

Conventions may vary by country

¨  In France and possibly other countries schools may use “positive” to mean “nonnegative”, so that zero is positive. 

¨  In the secondary schools in some places, the value of sin x may be computed clockwise starting at (0,1)  instead of counterclockwise starting at (1,0).  I have heard this from students. 

Conventions may vary by specialty within math

“Field” and “log” are examples. 

Defaults

An interface to a computer program may have many possible choices for the user to make. In most cases, the interface will use certain choices automatically when the user doesn't specify them.  One says the program defaults to those choices.  

Example

A word processing program may default to justified paragraphs and insert mode, but allow you to pick ragged right or typeover mode.

Language behaves in this way, too. 

Examples

¨  There is a sense in which the word "ski" defaults to snow skiing in Minnesota and to water skiing in Georgia.

¨  "CSU" defaults to Cleveland State University in northern Ohio and to Colorado State University in parts of the west.

Default usage in mathematical discourse

Symbols

¨  In high school,  evidently refers by default to the ratio of the circumference of a circle to its diameter.  Students are often quite surprised when they get to abstract math courses and discover the many other meanings of  (see here).

¨  Recently authors in the popular literature seem to think that  (phi) refers to the golden ratio.  In fact, a search through the research literature shows very few hits for  meaning the golden ratio: in other words, it usually means something else. 

¨  The set  has many different group structures defined on it but “The group  ” essentially always means that the group operation is ordinary addition.  In other words, “  ” as a group defaults to +.  Analogous remarks apply to
“the field  ” (and  and  ). 

¨  In informal conversation among many analysts, functions are continuous by default.

Remark

This meaning of "default" has made it into dictionaries only since around 1990 (see the Wikipedia entry). This usage does not carry a derogatory connotation.   In abstractmath.org I am using the word to mean a special type of convention that requires a choice of parameter, so that it is a special case of both “convention” and “suppression of parameters”.

 

Scope

Both mathematical English and the symbolic language have a feature that is essentially absent from ordinary spoken or written English:  The meaning of a phrase or a symbolic expression can be different in different parts of the discourse.   The portion of the text in which a particular meaning is in effect is called the scope of the meaning.  This is accomplished in several ways.

Explicit statement

Example

“In this paper, all groups are abelian”.  This means that every instance of the word “group” or any symbol denoting a group the group is constrained to be abelian.   The scope in this case is the whole paper.   See assumption.

Definition

The definition of a word, phrase or symbol sets its meaning.  If the word definition is used and the scope is not given explicitly, it is probably the whole discourse.

Example

“Definition.  An integer is even if it is divisible by 2.”  This is marked as a definition, so it establishes the meaning of the word “even” (when applied to an integer) for the rest of the text. 

Definitions whose scope is only part of the text are usually given using words such as “if” and “let” (see below).

If 

Used in conditional assertions (see here) and (along with let, usually “now let…”) in proof by cases.  

Bound variables

A variable is bound if it is in the scope of an integral, quantifier, summation, or other binding operators.  More here.