M. Bergson, if I am rightly informed, came into philosophy through the gateway of mathematics. The old antinomies of the infinite were, I imagine, the irritant that first woke his faculties from their dogmatic slumber. You all remember Zeno's famous paradox, or sophism, as many of our logic books still call it, of Achilles and the tortoise. Give that reptile ever so small an advance and the swift runner Achilles can never overtake him, much less get ahead of him; for if space and time are infinitely divisible (as our intellects tell us they must be), by the time Achilles reaches the tortoise's starting-point, the tortoise has already got ahead of _that_ starting-point, and so on _ad infinitum_, the interval between the pursuer and the pursued growing endlessly minuter, but never becoming wholly obliterated. The common way of showing up the sophism here is by pointing out the ambiguity of the expression 'never can overtake.' What the word 'never' falsely suggests, it is said, is an infinite duration of time; what it really means is the inexhaustible number of the steps of which the overtaking must consist. But if these steps are infinitely short, a finite time will suffice for them; and in point of fact they do rapidly converge, whatever be the original interval or the contrasted speeds, toward infinitesimal shortness. This proportionality of the shortness of the times to that of the spaces required frees us, it is claimed, from the sophism which the word 'never' suggests. But this criticism misses Zeno's point entirely. Zeno would have been perfectly willing to grant that if the tortoise can be overtaken at all, he can be overtaken in (say) twenty seconds, but he would still have insisted that he can't be overtaken at all. Leave Achilles and the tortoise out of the account altogether, he would have said--they complicate the case unnecessarily. Take any single process of change whatever, take the twenty seconds themselves elapsing. If time be infinitely divisible, and it must be so on intellectualist principles, they simply cannot elapse, their end cannot be reached; for no matter how much of them has already elapsed, before the remainder, however minute, can have wholly elapsed, the earlier half of it must first have elapsed. And this ever re-arising need of making the earlier half elapse _first_ leaves time with always something to do _before_ the last thing is done, so that the last thing never gets done. Expressed in bare numbers, it is like the convergent series 1/2 plus 1/4 plus 1/8..., of which the limit is one. But this limit, simply because it is a limit, stands outside the series, the value of which approaches it indefinitely but never touches it. If in the natural world there were no other way of getting things save by such successive addition of their logically involved fractions, no complete units or whole things would ever come into being, for the fractions' sum would always leave a remainder.