What does “twice two makes four” mean? Two means 1 + 1, and twice two means 1 + 1 plus 1 + 1. This sum is called “four”; and what we call five is 4 + 1. To maintain that the operation 2 × 2 might produce the result 5, is to admit conditions which have been excluded. Pure forms are not like animals which multiply; they are and remain such as they have been posited. When we put two amœbas into a glass and then add two other amœbas, it is quite possible that in the mean time one of the first set has divided into two. In that case we would have five amœbas. But the operation 2 × 2 cannot breed any additional units, so as to produce a greater number than the sum of 1 + 1 + 1 + 1. Nor can we let any of these units disappear into naught, so as to produce the result 3, without committing some inconsistency in our thought-operations, for the products of our thought-operations are rigid and must remain such as they have been posited. They are not animals blessed with fertility but pure forms and nothing but pure forms. How could Mr. John Stuart Mill overlook so palpable a contrast as that between formal knowledge and sense-experience? He was apparently prejudiced against the term “a priori,” which as we freely confess is a very poor and inadequate expression. Mr. Mill himself states the cause of his prejudice in his autobiography. He says: “There is not any idea, feeling, or power, in the human mind, which, in order to account for it, requires that its origin should be referred to any other source than experience.” Mr. Mill was justly exasperated against anything _a priori_, for in his time, it had become customary among certain philosophers to classify all pet superstitions which could not be proved by experience as _a priori_. Mr. Mill continues: “Whatever may be the practical value of a true philosophy of these matters, it is hardly possible to exaggerate the mischiefs of a false one. The notion that truths external to the mind may be known by intuition or consciousness, independently of observation and experience, is, I am persuaded, in these times the great intellectual support of false doctrines and bad institutions. By the aid of this theory every inveterate belief and every intense feeling, of which the origin is not remembered, is enabled to dispense with the obligation of justifying itself by reason, and is erected into its own all-sufficient voucher and justification. There never was such an instrument devised for consecrating all deep-seated prejudices.”