The Principles of Mathematics (1903)

§ 7

It is customary in mathematics to regard our variables as restricted to certain classes: in Arithmetic, for instance, they are supposed to stand for numbers. But this only means that if they stand for numbers, they satisfy some formula, i.e. the hypothesis that they are numbers implies the formula. This, then, is what is really asserted, and in this proposition it is no longer necessary that our variables should be numbers: the implication holds equally when they are not so. Thus, for example, the proposition x and y are numbers implies (x+y)2 = x2 + 2xy + y2 still holds equally if for x and y we substitute Socrates and Plato[2]: both hypothesis and consequent, in this case, will be false, but the implication will still be true. Thus in every proposition of pure mathematics, when fully stated, the variables have an absolutely unrestricted field: any conceivable entity may be substituted for any one of our variables without impairing the truth of our proposition.(§ 7 ¶ 1)

§ 7 n. 1. It is necessary to suppose arithmetical addition and multiplication defined (as may be easily done) so that the above formula remains significant when x and y are not numbers.