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VARIABLE MATHEMATICAL OBJECTS

Specific and variable mathematical objects

It is useful to distinguish between specific math objects and variable math objects . The number $42$ (the math object called "42", not the representation "42") is a specific math object. So is the sine function.

Math books are full of references to math objects, typically named by a letter, that are not completely specified. I 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. But in a proof you can't assume that $x$ is any particular real number.
• If they say, "Let $G$ be a group" you can think of $G$ as a variable group.
• 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.

A logician would refer to the symbol $f$, thought of as denoting a function, as a vari­able, but mathe­maticians in general would not use the word "vari­able" in that situa­tion.

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. There are two related points of view:

View 1:

Some statements about the object are neither true nor false.

Example

Given the statement, "Let $x$ be a real number", you know these things:

• The assertion "${{x}^{2}}=-1$" is false.
• The assertion "$x>0$" is neither true nor false.
• That the assertion "$x$ is a real number" is true is in a certain sense the most general true statement you can make about $x$.

So View 1 means that you are thinking of $x$ as a variable real number. So suppose you set out to solve the equation ${{x}^{2}}-5x=-6$ for real $x$. Before you do anything, you know that $x$ is real but you don't know what it is. After you solve the equation, you know that $x=2$ or $x=3$, but you still don't know which it is. It is still a variable real number, but with only two possible values.

View 2:

You don't know whether some statements about the object are true or not.

Example

Suppose the statement is "Let $x$ be a real number".

• You know that $x$ is a real number (duh).
• You know $x$ is either positive or nonnegative.
• You know ${{x}^{2}}$ is not negative.
• You don't know whether $x$ is positive or not.

So view 2 involves thinking of x as a particular real number, but you have incomplete information about it.

Opinion

Suppose you set out to solve the equation ${{x}^{2}}-5x=-6$ for real $x$. It is a specific number, but at the start you know only that it is real (because you assumed it). After you finish calculating, you know that $x=2$ or $x=3$. Either of those values makes the equation ${{x}^{2}}-5x=-6$ true, but which one were you thinking of?

It is that phenomenon that makes me like View 1 better. According to View 1, $x$ was a variable real number all through the process of your calculation, and it still was when you finished, but you had narrowed the possibilities to two numbers, both of which fit the equation.

The notion that you were thinking of a particular number while you were doing the calculation is incoherent.

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