or synthetical, we must change the predicate into its contradictory opposite, and apply this to the subject without a copula. If this gives a _contradictio in adjecto_, then the judgment was analytical; otherwise it was synthetical. That Arithmetic rests on the pure intuition or perception of time is not so evident as that Geometry is based upon that of space.(15) It can be proved, however, in the following manner. All counting consists in the repeated affirmation of unity. Only for the purpose of always knowing how often we have already affirmed unity do we mark it each time with another word: these are the numerals. Now repetition is only possible through succession. But succession, that is, being after one another, depends directly upon the intuition or perception of _time_. It is a conception which can only be understood by means of this; and thus counting also is only possible by means of time. This dependence of all counting upon time is also betrayed by the fact that in all languages multiplication is expressed by “time,” thus by a time-concept: _sexies_, ἑξακις, _six fois_, _sex mal_. But simple counting is already a multiplication by one, and for this reason in Pestalozzi’s educational establishment the children are always made to multiply thus: “Two times two is four times one.” Aristotle already recognised the close relationship of number and time, and expounded it in the fourteenth chapter of the fourth book of the “Physics.” Time is for him “the number of motion” (“ὁ χρονος αριθμος εστι κινησεως”). He very profoundly suggests the question whether time could be if the soul were not, and answers it in the negative. If arithmetic had not this pure intuition or perception of time at its foundation, it would be no science _a priori_, and therefore its propositions would not have infallible certainty. Although time, like space, is the form of knowledge of the subject, yet, just like space, it presents itself as independent of the subject and completely objective. Against our will, or without our knowledge, it goes fast or slow. We ask what o’clock it is; we investigate time, as if it were something quite objective. And what is this objective existence? Not the progress of the stars, or of the clocks, which merely serve to measure the course of time itself, but it is something different from all things, and yet, like them, independent of our will and knowledge. It exists only in the heads of percipient beings, but the uniformity of its course and its independence of the will give it the authority of objectivity.