When once the above general doctrine is rejected, it is obvious that, where there is change, there must be a succession of states. There cannot be change--and motion is only a particular case of change--unless there is something different at one time from what there is at some other time. Change, therefore, must involve relations and complexity, and must demand analysis. So long as our analysis has only gone as far as other smaller changes, it is not complete; if it is to be complete, it must end with terms that are not changes, but are related by a relation of earlier and later. In the case of changes which appear continuous, such as motions, it seems to be impossible to find anything other than change so long as we deal with finite periods of time, however short. We are thus driven back, by the logical necessities of the case, to the conception of instants without duration, or at any rate without any duration which even the most delicate instruments can reveal. This conception, though it can be made to seem difficult, is really easier than any other that the facts allow. It is a kind of logical framework into which any tenable theory must fit--not necessarily itself the statement of the crude facts, but a form in which statements which are true of the crude facts can be made by a suitable interpretation. The direct consideration of the crude facts of the physical world has been undertaken in earlier lectures; in the present lecture, we have only been concerned to show that nothing in the crude facts is inconsistent with the mathematical doctrine of continuity, or demands a continuity of a radically different kind from that of mathematical motion. LECTURE VI THE PROBLEM OF INFINITY CONSIDERED HISTORICALLY It will be remembered that, when we enumerated the grounds upon which the reality of the sensible world has been questioned, one of those mentioned was the supposed impossibility of infinity and continuity. In view of our earlier discussion of physics, it would seem that no _conclusive_ empirical evidence exists in favour of infinity or continuity in objects of sense or in matter. Nevertheless, the explanation which assumes infinity and continuity remains incomparably easier and more natural, from a scientific point of view, than any other, and since Georg Cantor has shown that the supposed contradictions are illusory, there is no longer any reason to struggle after a finitist explanation of the world. The supposed difficulties of continuity all have their source in the fact that a continuous series must have an infinite number of terms, and are in fact difficulties concerning infinity. Hence, in freeing the infinite from contradiction, we are at the same time showing the logical possibility of continuity as assumed in science.