Mathematics

How can was say that a variable such as x exists as a number or at all in an equation when by using a variable we claim to know nothing of what "goes in there" to complete the equation?

Comments

There are a couple different attitudes towards this kind of question. One takes the idea of variables very seriously: There really are things that are called "variables". It's not that we don't know various things about them. A variable number, for example, reallly is intrinsically not any number in particular.

That, however, is a minority view nowadays, though it has a distinguished history. See Kit Fine's Arbitrary Objects for a recent defense of it.

The more common view, which originates with Gottlob Frege, is that variables are like pronouns. Consider "Everyone who met someone liked her". Here, the word "her" does not stand for anyone in particular: Rather, it stands for the person each person in question met. So, if we're talking about the people on the team, and those people are Bill, Dick, and Harry, and Bill met Betty, Dick met Jane, and Harry met Sally, then what "her" refers to "v aries" as we consider the different people: It refers to Betty when we consider Bill; to Jane when we consider Dick; and to Sally when we consider Harry. But there isn't really any thing that varies or changes. It's just that the relation between "her" and the thing it denotes can vary.

This analysis is less obvious when we consider equations like "x + y = y + x". But such an equation is typically taken to mean "For all x and y, x + y = y + x", that is, to mean something like: The result of adding a number to any number is the same as the result of adding the first number to the second number. Here, the phrases "the first number" and "the second number" act as pronouns, and they behave the same way "her" does in the preceding example. You might have noticed, however, that there's no pronoun corresponding to the first "x" and "y". But we can fix that if we say, somewhat more awkwardly: Every number is such that every number is such that the result of adding the first one to the second one is the same as the result of adding the second one to the first one. Awkward indeed!

The same kind of thing can be said about other kinds of equations, like "x2 - 1 = 0". But exactly what one should say depends upon how, exactly, one is thinking of the equation as being used. Sometimes it's used as a kind of question: Which number has a square that is one greater than zero? I.e.: Which number is such that its square is one greater than zero?

The usual way to analyze expressions of generality, like "every number" in theoretical semantics takes the true structure of the sentence to be something like that I just illustrated. Such expressions are called "quantifiers" and they are said to "govern" other expressions called "traces" that act like pronouns, or variables. There are similar things to be said about expressions like "which number". So the correspondence between pronouns and variables, from the standpoint of theoretical linguistics, runs very deep.

I would add that it is important to distinguish between a variable and what the variable stands for. You ask how we can say that "a variable such as x exists as a number." I would say the variable is a letter, not a number, but it stands for a number. As Richard has explained, what number it stands for, or whether there is a single number that it stands for, may depend on the context.

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