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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, occasionally, country (see convention, default).  And words and symbols can change their meanings from place to place within the same mathe­matical discourse (see scope).

This article mostly provides pointers to other articles in abstractmath.org that give more details about the ideas.

Conventions

A convention in mathematical discourse is notation or terminology used with a special meaning in certain contexts or in certain fields. Articles and books in a specialty do not always clue you in on these conventions.

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 in Definitions and in the Glossary.

People new to abstract math
are rarely aware of this usage of "if" in definitions.

Example 2

Constants or parameters are conventionally denoted by $a$, $b$, $c$... , functions by $f$, $g$, $h$, ..., and variables by $x$, $y$, $z$,.... There is more detail about this in Variables and substitution.

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 synecdoche.

Example 4

The meaning of "${{\sin }^{n}}x$" in many calculus books is:

• The inverse sine (arcsin) if $n=-1$.
• The mult­iplica­tive power of $\sin x$ for positive $n$; in other words, ${{\sin }^{n}}x={{(\sin x)}^{n}}$ if $n\ne -1$.

This, like Example 1, is a definition by cases. Unlike Example 1, calculus books often make it explicit. Explicit or not, this usage is an abomination.

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

• 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 $\frac{252}{1000}$.
• 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, unless, like me, they have taught engineering students.

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 $(1,0)$.

Conventions may vary by specialty within math

"Field”, "graph" 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.  

Non-math Examples

• A word processing program may default to justified paragraphs and insert mode, but allow you to pick ragged right or typeover mode.
• "CSU" defaults to Cleveland State University in northern Ohio and to Colorado State University in parts of the west.
• There is a sense in which the word "ski" defaults to snow skiing in Minnesota and to water skiing in Georgia.

I have spent a lot of time in both Minne­sota and Georgia and the remarks about skiing are based on my own observation. But these usages are not absolute. Some affluent Geor­gians may refer to snow skiing as "skiing", for example, and this usage can result in a put-down if the hearer thinks they are talking about water skiing. One wonders where the boundary line is. Perhaps people in Kentucky are confused on the issue.

Default usage in mathematical discourse

Math language has defaults, too.

Symbols

• In high school, "$\pi$" 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 $\pi $ (see here).
• Recently authors in the popular literature seem to think that $\phi$ (phi) defaults to the golden ratio.  In fact, a search through the research literature shows very few hits for $\phi$ meaning the golden ratio: in other words, it usually means something else. 
• The set $\mathbb{R}$ of real numbers has many different group structures defined on it but "The group $\mathbb{R}$” essentially always means that the group operation is ordinary addition.  In other words, "$\mathbb{R}$” as a group defaults to +.  Analogous remarks apply to "the field $\mathbb{R}$”. 
• In informal conversation among many analysts, functions are continuous by default.
• It used to be the case that in informal conversations among topologists, "group" defaulted to Abelian group. I don't know whether that is still true or not.

Remark

This meaning of "default" has made it into dictionaries only since around 1960 (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 imposes a choice of parameter, so that it is a special case of both " convention ” and "suppression of parameters”.

Both mathematical English and the symbolic language have a feature that is uncommon in ordinary spoken or written English:

The meaning of a phrase or a symbolic expression
can be different in different parts of the same text.

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

Examples

• "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.
• "Suppose (or "let" or "assume") $n$ is divisible by $4$". Before this statement, you could not assume $n$ is divisible by $4$. Now you can, until the end of the current paragraph or section.

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. 

If

Used in modus ponens (see here) and (along with let, usually "now let…”) in proof by cases.

Example(modus ponens)

Suppose you want to prove that if an integer $n$ is divisible by $4$ then it is even. To show that it is even you must show that it is divisible by $2$. So you write:

Now if you start a new paragraph with something like "For any integer $n\ldots$" you can no longer assume $n$ is divisible by $4$.

Example (proof by cases)

Theorem: For all integers $n$, $n^2+n+1$ is odd.

Definitions:

• "$n$ is even" means that $n=2s$ for some integer $s$.
• "$n$ is odd" means that $n=2t+1$ for some integer $t$.

Proof:

• Suppose $n$ is even. Then \[\begin{align*} n^2+n+1&=4s^2+2s+1\\ &=2(2s^2+s)+1\\ &=2(\text{something})+1 \end{align*}\] so $n^2+n+1$ is odd. (See Zooming and Chunking.)
• Now suppose $n$ is odd. Then \[\begin{align*} n^2+n+1&=(2t+1)^2+2t+1+1\\ &=4t^2+4t+1+2t+1+1\\ &=2(2t^2+3t)+3\\ &=2(2t^2+3t+1)+1\\ &=2(\text{something})+1 \end{align*}\] So $n^2+n+1$ is odd.

Remark

The proof I just gave uses only the definition of even and odd and some high school algebra. Some simple grade-school facts about even and odd numbers are:

• Even plus even is even.
• Odd plus odd is even.
• Even times even is even.
• Odd times odd is odd.

Put these facts together and you get a nicer proof (I think anyway): $n^2+n$ is even, so when you add $1$ to it you must get an odd number.

Bound variables

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

Example

Consider this text:

"Exercise: Show that for all real numbers $x$, it is true that $x^2\geq0$. Proof: Let $x=-2$. Then $x^2=(-2)^2=4$ which is greater than $0$. End of proof."

The problem with that text is that in the statement, "For all real numbers $x$, it is true that $x^2\geq0$", $x$ is a bound variable. It is bound by the universal quantifier "for all" which means that $x$ can be any real number whatever. But in the next sentence, the meaning of $x$ is changed by the assumption that $x=-2$. So the statement that $x\geq0$ only applies to $-2$. As a result the proof does not cover all cases.

Many students just beginning to learn to do proofs make this mistake. Fellow students who are a little further along may be astonished that someone would write something like that paragraph and might sneer at them. But this common mistake does not deserve a sneer, it deserves an explanation. This is an example of the ratchet effect.

Variable meaning in natural language

Meanings commonly vary in natural language because of conventions and defaults. But varying in scope during a conversation seems to me uncommon.

It does occur in games.

• In Skat and Bridge, the meaning of "trump" changes from hand to hand.
• The meaning of "strike" in a baseball game changes according to context: If the current batter has already had fewer than two strikes, a foul is a strike, but not otherwise.

I have not come up with non-game examples, and anyway games are played by rules that are suspiciously like mathematical axioms. Perhaps you can think of some non-game occasions in which meaning is determined by scoping that I have overlooked.

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