There are a number of distinct questions which are apt to be confused when the mathematical continuum is said to be inadequate to the facts of sense. We may state these, in order of diminishing generality, as follows:-- (a) Are series possessing mathematical continuity logically possible? (b) Assuming that they are possible logically, are they not impossible as applied to actual sense-data, because, among actual sense-data, there are no such fixed mutually external terms as are to be found, _e.g._, in the series of fractions? (c) Does not the assumption of points and instants make the whole mathematical account fictitious? (d) Finally, assuming that all these objections have been answered, is there, in actual empirical fact, any sufficient reason to believe the world of sense continuous? Let us consider these questions in succession. (a) The question of the logical possibility of the mathematical continuum turns partly on the elementary misunderstandings we considered at the beginning of the present lecture, partly on the possibility of the mathematical infinite, which will occupy our next two lectures, and partly on the logical form of the answer to the Bergsonian objection which we stated a few minutes ago. I shall say no more on this topic at present, since it is desirable first to complete the psychological answer. (b) The question whether sense-data are composed of mutually external units is not one which can be decided by empirical evidence. It is often urged that, as a matter of immediate experience, the sensible flux is devoid of divisions, and is falsified by the dissections of the intellect. Now I have no wish to argue that this view is _contrary_ to immediate experience: I wish only to maintain that it is essentially incapable of being _proved_ by immediate experience. As we saw, there must be among sense-data differences so slight as to be imperceptible: the fact that sense-data are immediately given does not mean that their differences also _must_ be immediately given (though they _may_ be). Suppose, for example, a coloured surface on which the colour changes gradually--so gradually that the difference of colour in two very neighbouring portions is imperceptible, while the difference between more widely separated portions is quite noticeable. The effect produced, in such a case, will be precisely that of "interpenetration," of transition which is not a matter of discrete units. And since it tends to be supposed that the colours, being immediate data, must _appear_ different if they _are_ different, it seems easily to follow that "interpenetration" must be the ultimately right account. But this does not follow. It is unconsciously assumed, as a premiss for a _reductio ad absurdum_ of the analytic view, that, if A and B are immediate data, and A differs from B, then the fact that they differ must also be an immediate datum. It is difficult to say how this assumption arose, but I