High school math definition of a variable: the first step from the concrete into the abstract...

I don't like the word "variable" in math. When we're solving an equation, $x$ is nothing more than the name of a number whose value we do not yet know, $x$ is not in any sense "variable". It's not as if the value of $x$ can change.

And if we are defining a function $f$, we might say something like, if $x$ is a number, then $f(x) = x^2 + 7$ . Even here $x$ is not "variable". We are just saying that if $x$ is a (specific) number, then $f(x)$ is the number $x^2 + 7$.

Now, in computer programming, you have variables whose value can actually change. That's different.

There probably should be a strict distinction made between variables and constants. For example in a quadratic equation $f(x) = Ax^2+Bx+C,$ the letters $A,B$, and $C$ are constants in the sense that they are standing in for specific values. The variable in the equation (rather the independent variable) is $x$. The dependent variable is $y=f(x)$. This equation, which represents a parabola, is different than $0 =Ax^2+Bx+C$. Here $A$, $B$, and $C$ still represent constants, but $x$ is an unknown quantity. It does not vary. Still the nature of the solution set to that equation varies if the values of the constants $A$, $B$, and $C$ vary.

To say that the ``constants may vary'' does sound oxymoronic and counter-intuitive. I am sure that we all are guilty of having uttered such a sentence. One point of algebraic expressions is that any of the letters may be considered as a variable. Once specific values are chosen, the express becomes more definite. Indeed, all algebraic laws (at the level of high school mathematics) can be considered to be universal truths about the expression for any specific values of the variables.

I am sure that there are other real-word instances of such ambiguity in the language.

The duality embodied in the definition of a variable as:

"A symbol used to represent one or more numbers."

Is such that we don't have to make the distinction between an "unknown specific quantity" and a "varying" quantity.