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# VARIABLE MATHEMATICAL OBJECTS

In many mathematical texts, the variable $x$ may denote a real number, although which real number may not be specified. This is an example of a variable mathematical object. This point of view and terminology is not widespread, but I think it is worth understanding because it provides a deeper understanding of some aspects about how math is done.

## Specific and variable mathematical objects

It is useful to distinguish between specific math objects and variable math objects.

#### Examples of specific math objects

• The number $42$ (the math object represented as "42" in base $10$, "2A" in hexadecimal and "XLII" as a Roman numeral) is a specific math object. It is not any of the representations just listed -- they are just strings of letters and numbers.
• The ordered pair $(4,3)$ is a specific math object. It is not the same as the ordered pair $(7,-2)$, which is another specific math object.
• The sine function $\sin:\mathbb{R}\to\mathbb{R}$ is a specific math object. You may know that the sine function is also defined for all complex numbers, which gives another specific math object $\sin:\mathbb{C}\to\mathbb{C}$.
• The group of symmetries of a square is a specific math object. (If you don't know much about groups, the link gives a detailed description of this particular group.)

#### Variable math objects

Math books are full of references to math objects, typically named by a letter or a name, that are not completely specified. Some mathematicians call these variable objects (not standard terminology). The idea of a variable mathe­mati­cal object is not often taught as such in under­graduate classes but it is worth pondering. It has certainly clari­fied my thinking about expres­sions with variables.

#### Examples

• If an author or lecturer says "Let $x$ be a real variable", you can then think of $x$ as a variable real number. In a proof you can't assume that $x$ is any particular real number such as $42$ or $\pi$.
• If a lecturer says, "Let $(a,b)$ be an ordered pair of integers", then all you know is that $a$ and $b$ are integers. This makes $(a,b)$ a variable ordered pair, specifically a pair of integers. The lecturer will not say it is a variable ordered pair since that terminology is not widely used. You have to understand that the phrase "Let $(a,b)$ be an ordered pair of integers" implies that it is a variable ordered pair just because "a" and "b" are letters instead of numbers.
• If you are going to prove a theorem about functions, you might begin, "Let $f$ be a continuous function", and in the proof refer to $f$ and various objects connected to $f$. This makes $f$ a variable mathematical object. When you are proving things about $f$ you may use the fact that it is continuous. But you cannot assume that it is (for example) the sine function or any other particular function.

• If someone says, "Let $G$ be a group" you can think of $G$ as a variable group. If you want to prove something about $G$ you are free to use the definition of "group" and any theorems you know of that apply to all groups, but you can't assume that $G$ is any specific group.

#### Terminology

A logician would refer to the symbol $f$, thought of as denoting a function, as a vari­able, and likewise the symbol $G$, thought of as denoting a group. But mathe­maticians in general would not use the word "vari­able" in those situa­tions.

## How to think about variable objects

The idea that $x$ is a variable object means thinking of $x$ as a genuine mathematical object, but with limitations about what you can say or think about it. Specifically,

Some assertions about a variable math object
may be neither true nor false.

##### Example

The statement, "Let $x$ be a real number" means that $x$ is to be regarded as a variable real number (usually called a "real variable"). Then you know the following facts:

• The statement "${{x}^{2}}$ is not negative" is true.
• The assertion "$x=x+1$" is false.
• The assertion "$x\gt 0$" is neither true nor false.
##### Example

Suppose you are told that $x$ is a real number and that ${{x}^{2}}-5x=-6$.

• You know (because it is given) that the statement "${{x}^{2}}-5x=-6$" is true.
• By doing some algebra, you can discover that the statement "$x=2$ or $x=3$" is true.
• The statement "$x=2$ and $x=3$" is false, because $2\neq3$.
• The statement "$x=2$" is neither true nor false, and similarly for "$x=3$".
• This situation could be described this way: $x$ is a variable real number varying over the set $\{2,3\}$.
##### Example

This example may not be easy to understand. It is intended to raise your consciousness.

A prime pair is an ordered pair of integers $(n,n+2)$ with the property that both $n$ and $n+2$ are prime numbers.

Definition: $S$ is a PP set if $S$ is a set of pairs of integers with the property that every pair is a prime pair.

• "$\{(3,5),(11,13)\}$ is a PP set" is true.
• "$\{(5,7),(111,113),(149,151)\}$ is a PP set" is false.

Now suppose $SS$ is a variable PP set.

• "$SS$ is a set" is true by definition.
• "$SS$ contains $(7,9)$" is false.
• "$SS$ contains $(3,5)$" is neither true nor false, as the examples just above show.
• "$SS$ is an infinite set":
• This is certainly not true (see finite examples above).
• This claim may be neither true nor false, or it may be plain false, because no one knows whether there is an infinite number of prime pairs.
• The point of this example is to show that "we don't know" doesn't mean the same thing as "neither true nor false".