without, by the very act, repeating the scientific experiment which establishes the contrary. Since, then, the axioms and the misnamed definitions are but inductions from experience, and since the definitions are only hypothetically true, the deductive or demonstrative sciences--of which these axioms and definitions form together the first principles--must really be themselves inductive and hypothetical. Indeed, it is to the fact that the results are thus only conditionally true, that the necessity and certainty ascribed to demonstration are due. It is so even with the Science of Number, i.e. arithmetic and algebra. But here the truth has been hidden through the errors of two opposite schools; for while many held the truths in this science to be _à priori_, others paradoxically considered them to be merely verbal, and every process to be simply a succession of changes in terminology, by which equivalent expressions are substituted one for another. The excuse for such a theory as this latter was, that in arithmetic and algebra we carry no ideas with us (not even, as in a geometrical demonstration, a mental diagram) from the beginning, when the premisses are translated into signs, till the end, when the conclusion is translated back into things. But, though this is so, yet in every step of the calculation, there is a real inference of facts from facts: but it is disguised by the comprehensive nature of the induction, and the consequent generality of the language. For numbers, though they must be numbers of something, may be numbers of anything; and therefore, as we need not, when using an algebraical symbol (which represents all numbers without distinction), or an arithmetical number, picture to ourselves all that it stands for, we may picture to ourselves (and this not as a sign of things, but as being itself a thing) the number or symbol itself as conveniently as any other single thing. That we are conscious of the numbers or symbols, in their character of things, and not of mere signs, is shown by the fact that our whole process of reasoning is carried on by predicating of them the properties of things.