Variables and Language

Question

I've been thinking lately about the kind of language we use when doing math involving variables. Consider a typical variable defining statement:

"Let x = 2."

If we try to parse this statement literally, 'x' is the name of some object we are letting equal 2. However, if 'x' is already the name of something (even if we don't know the exact object it is the name of), we get nonsense. For example, if 'x' is the name of the country Canada, then the let statement becomes

"Let Canada = 2,"

which is nonsense. We could also try to interpret 'x' as the pronoun 'it,' which leads to

"Let it = 2,"

but this is also problematic because pronouns in the English language (to the best of my knowledge) are never bound explicitly (for example, in the sentence "I cannot remember this equation because it has too many terms," the pronoun 'it' gets bound by context to 'this equation').

There are numerous other places in mathematics where trying to interpret a variable literally as a noun or pronoun leads to problems. For instance, the sentence

"If $x$ increases, then $x^{-1}$ decreases,"

suggests that 'x' is the name of some object that is truly capable of change. However, objects like this do not exist in mathematics, e.g. the number 1 is always 1, has always been 1, and never ceases to be 1.

Further examples arise when trying to parse statements involving the quantifiers $\forall$ and $\exists$.

My question: (1) How exactly should we understand variables? Can we formalize any further our intuition that they are symbols that may stand for any object? (2) Given (1), how do we properly incorporate variables into our everyday language so that all of our mathematical statements have clear, unambiguous meanings?

Answer 1

(1) natural language doesn't really have variables. The nearest you can get to "let x = 2" is to say something like "let's think of the number 2 and, for convenience, let's refer to it as x". So after you've said that "x" acts as a noun denoting 2.

(2) In statements like "if $x$ increases then $x^{-1}$" decreases, we are using a standard mathematical convention whereby a formula like $x$ or $x^{-1}$ is interpreted as a function of the variable $x$ ($x \mapsto x$ or $x \mapsto x^{-1}$). It then makes sense to talk about the function as increasing or decreasing.

Answer 2

Variables are a way for us to abstract over things. If we can make a statement for all humans, instead of repeating the statement for each human, we can abstract over them with a variable and say "Let $x$ be a human, then [...]" in a succinct and clear way.

From the context in which one uses a variable it is clear what that variable ranges over. That range constitutes the domain of a variable. For example, in the previous statement it was clear (explicitly stated) that $x$ ranges over all humans ("Let $x$ be a human"). Sometimes, though, the domain is not explicitly stated. In those cases you can assume the most general domain that you can guess. In your example ("Let $x = 2$"), one guesses that $x$ is a number and can not be a cookie.

Regarding your 2nd question, in math we use nominal representation of variables: we represent variables with names, like "$x$", "$y$" or "$it$". In the nominal representation, there is no other way to avoid the problem you describe than to keep track of all the names of the variables that we have introduced in the context. Things can get complicated when variables have local scopes, which means that you can introduce a variable $x$ and later discharge it, allowing you to re-use the name $x$ for another variable later on.

Using names to represent variables, makes it easy for us to refer to variables: we just use their names. In the context of formal languages, another way to represent variables is by numbers (de Bruijn indices, nameless representation). If we would use such a representation in natural language (totally not recommended), referring to a variable would translate to something like "the first/second/... variable I introduced". This representation ensures that no common name between two different variables can occur, but it takes a lot of counting to refer to them and is not recommended for humans.