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?