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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

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.



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.


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:


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


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.

Now suppose $SS$ is a variable PP set.

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