The unsatisfactoriness of such descriptions is no new discovery: the logical difficulties connected with the attempt to describe change in terms of series of successive things or events have been familiar since the time when Zeno invented the famous dilemma of Achilles’ race with the tortoise. Mathematicians have been in the habit of telling us that these difficulties depend simply on the fact that we imagine the series of positions at which Achilles and the tortoise find themselves from moment to moment as finite: the device of the infinite series, they say, satisfies all the requirements needed for representing change and solves all the logical difficulties which arise from it. Bergson’s difficulties, however, cannot be solved in this way for they are not based upon the discovery of logical absurdities but upon the discrepancy between the description and the fact. What he maintains is that the description of change in terms of an infinite series of stages leaves out the change altogether. Zeno’s logical dilemma as to how Achilles could ever catch up with the tortoise provided the tortoise was given a start, however small, may be countered by the ingenuity of the mathematicians’ infinite series. Bergson’s difficulty turns on a question of fact, not of logic, and cannot be so met. He solves the problem simply by denying that Achilles or the tortoise ever are at particular points at particular moments. Such a description of change, he says, leaves out the real changing. And the introduction of the notion of an infinite series only makes the matter worse. For stages do not change, and so, if there is to be any change, it must, presumably, take place in between one stage and the next. But in between any two stages of an infinite series there are supposed to be an infinite number of other stages, so that to any given stage there _is_ no next stage. Change, therefore, cannot take place between one stage and the next one, there being no next one, and since it is equally impossible that it should take place at any one of the stages themselves it follows that an infinite series of stages leaves out change altogether. Similarly a series of instants before and after one another leaves out of time just the element of passage, becoming, which is its essence. The truth, Bergson says, is that with fixed stages, no matter how many you take, and no matter in what relation you arrange them, you cannot reproduce the change and time which actually occur as facts directly known. If Achilles or the tortoise are ever at different places at different moments then neither of them really moves at all. Change and time, as represented by abstractions, according to the intellectual method, consist of stages in relations of succession, but the fact does not happen by stages and is not held together by relations: if we compare the representation with the fact we find that they differ profoundly in their form.