The Principles of Mathematics (1903)

§ 6

Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain variables. The notion of the variable is one of the most difficult with which Logic has to deal, and in the present work a satisfactory theory as to its nature, in spite of much discussion, will hardly be found. For the present, I only wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. Elementary Arithmetic might be thought to form an exception: 1+1=2 appears neither to contain variables nor to assert an implication. But as a matter of fact, as will be shown in Part II, the true meaning of this proposition is: If x is one and y is one, and x differs from y, then x and y are two. And this proposition both contains variables and asserts an implication. We shall find always, in all mathematical propositions, that the words any or some occur; and these words are the marks of a variable and a formal implication. Thus the above proposition may be expressed in the form: Any unit and any other unit are two units. The typical proposition of mathematics is of the form φ(x, y, z, ...) implies ψ(x, y, z, ...), whatever values x, y, z, ... may have; where φ(x, y, z, ...) and ψ(x, y, z, ...), for every set of values x, y, z, ..., are propositions. It is not asserted that φ is always true, nor yet that ψ is always true, but merely that, in all cases, when φ is false as much as when φ is true, ψ follows from it.(§ 6 ¶ 1)

The distinction between a variable and a constant is somewhat obscured by mathematical usage. It is customary, for example, to speak of parameters as in some sense constants, but this is a usage which we shall have to reject. A constant is to be something absolutely definite, concerning which there is no ambiguity whatever. Thus, 1, 2, 3, e, π, Socrates, are constants; and so are man, and the human race, past, present and future, considered collectively. Proposition, implication, class, etc. are constants; but a proposition, any proposition, some proposition are not constants, for these phrases do not denote one definite object. And thus what are called parameters are simply variables. Take, for example, the quation ax + by + c = 0, considered as the equation for a straight line in a plane. Here we say that x and y are variables, while a, b, c are constants. But unless we are dealing with one absolutely particular line, say the line from a particular point in London to a particular point in Cambridge, our a, b, c are not definite numbers, but stand for any numbers, and are thus also variables. And in Geometry nobody does deal with actual particular lines; we always discuss any line. The point is that we collect the various couples x, y into classes of classes, each class being defined as those couples that have a certain fixed relation to one triad (a, b, c). But from class to class, a, b, c also vary, and are therefore properly variables.(§ 6 ¶ 2)