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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".

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