Actually an algebraic variable is named that way because you can change it every time you do the exercise.

So you don't change it during one exercise, but you can and certainly will change it when you change the exercise to another one with the same principle.

That behaves just like a function: the values of the variables change each time you call the function but not within. When you do a new exercise, it's really a bunch of new variables with the same name--variables cannot affect each other between exercises!

That's not analogous at all to mutation.

Right, but you're not mutating it in the middle of your calculation.

If you have a sum for example sum_{x=1}^{10}x^2, x definitely changes quite many times during your calculation.

No, that's an example of recursion. With recursion you aren't mutating the variable, you're shadowing its definition. This is a subtle, but important, distinction.

Recursion? The "function" is not calling itself at all. I'm quite sure that is not recursion, neither in the mathematical nor computer science sense.

Recursion is how you implement a sum without mutation (an imperative loop is how you'd implement it with mutation). You've merely given me the function written in a declarative notation.

I think we are talking about different things.

For example, if you integrate x^2 dx from x=0 to 10, x changes continuously.

It's clear that x is not fixed to any particular value.

Yeah, you're talking about the mathematical definition of a bound variable. Functional programming languages use this whenever you define a function:

   foo x = x * 2

Every time you call foo, x is bound to a new value (but only within the scope of foo). This is not the same as mutating some memory location associated with the name x.

x does not mutate at all. x is simply a placeholder for the values in the series of additions that your expression expands to, that is, 1² + 2² + 3² + ... + 10². Or stepwise, to 1² + sum_{x=2}^{10}x² etc. all three expressions are equivalent, and no mutation of x occurs anywhere

The series of additions can also be infinite, in which case it would be quite difficult to expand fully, but that doesn't prevent you from doing mathematics with it.

An algebraic "variable" is perhaps better called an "unknown"