It is true, as Mr. Dixon says, that “Any language which seems to imply that there is some dread necessity about mathematical truths is very misleading.” But to say, as Mr. Dixon does in another passage, that the truisms of mathematics are arbitrary truths, is more misleading still. The theorems of the formal sciences are not “assertions whose truth depends solely on the definition of their terms.” They are “real assertions which convey some real subjective or objective information.” Mr. Dixon objects to Kant’s assertion that 7 + 5 = 12 is not only _a priori_ but also synthetic. He declares, in contradistinction to Kant, that it is deduced from definitions alone; that therefore it is empty, and cannot give any information about things. This latter proposition, which is a phrase of Kant’s, appears in this context as an inconsistency of Kant’s. And it would be an inconsistency, if it had to be understood in the sense in which Mr. Dixon quotes it. We construe Kant’s phrase that “the _a priori_ is empty, and cannot give information about things,” in a different way. We think that Kant intends to say that the _a priori_ imparts real information concerning relations and forms; but that it does not impart real information concerning substances or the materiality of things. It is apparent that the assertion 7 + 5 = 12 cannot be derived from the definitions of 7 or 5. Similarly, the ideas of higher mathematics are not deduced from the few definitions of elementary mathematics that tell what points, lines, parallels, etc., are. Of what, then, are these complex theorems of mathematics, products? They are not derived from sense-experience, nor from the definitions of their ultimate elements. Is their origin mysterious? Here Kant leaves us in the lurch; he simply declares that formal truth is _a priori_ and transcendental; and those of his disciples who call themselves, with preference, transcendentalists, have ample occasion to introduce in this _lacuna_ of Kantian thought, all the mysticism they please.