The problem of the origin of the truths of formal sciences is not so difficult as it is sometimes represented. The theorems of higher mathematics are the products of certain _operations_ performed with the elementary forms described in the definition with which the mathematician starts. These operations are not arbitrary; they are not merely verbal processes; they are realities of highest importance. Not material realities, but realities, nevertheless. They are functions, and mathematics deals with the products of functions. It is true that we might call twice two by any other name than four; we might call it _vier_, or _quatre_, but the operation 1 + 1 + 1 + 1 would remain the same, by whatever name we call its result. Mathematical truths, accordingly, are not empty in the sense that they are meaningless; for they are significant in the highest degree. They give real information, not about things, but about certain relations that obtain among things. They describe certain operations in which formal relations are traced. And they describe them exhaustively, so that the result is, as the Germans call it, _eindeutig bestimmt_, and the result will, under all circumstances, be the same. Twice two _will always be_ the same as 1 + 1 + 1 + 1. This “it will always be,” is called necessary. There is nothing dreadful about it, nor is there any mystery connected with it. It is not an awful fate that decrees it, but it is the nature of sameness, that the same is and will be the same, so long as it remains the same. It is often overlooked that every number in arithmetic is the result of an operation which is symbolised by a certain figure. Numbers are not concrete things; and as soon as we forget that they are products of a function, we are liable to lapse into mistakes. This happens most frequently with the numbers “zero” and “infinite.” The latter of these two symbols is often looked upon as a concrete thing; and because the infinite, with actual reality, is, in its completeness, inconceivable, it has made, of every one who stumbled over this stone of offence, a mystic, and many a radical, fearless thinker bows down to worship before the idea of infinitude-function as it would be if it were a real thing. Says Mr. Dixon, “Our reason cannot inform us about the form of existence, unless it is first given.” This is very true. The form is given, and formal systems such as the numerical system and the lines and figures of mathematics are mental constructions built of the stones quarried out of the relational given in experience.