We saw that, accepting the view that a length is composed of points, the existence of incommensurables proves that every finite length must contain an infinite number of points. In other words, if we were to take away points one by one, we should never have taken away all the points, however long we continued the process. The number of points, therefore, cannot be _counted_, for counting is a process which enumerates things one by one. The property of being unable to be counted is characteristic of infinite collections, and is a source of many of their paradoxical qualities. So paradoxical are these qualities that until our own day they were thought to constitute logical contradictions. A long line of philosophers, from Zeno[28] to M. Bergson, have based much of their metaphysics upon the supposed impossibility of infinite collections. Broadly speaking, the difficulties were stated by Zeno, and nothing material was added until we reach Bolzano's _Paradoxien des Unendlichen_, a little work written in 1847-8, and published posthumously in 1851. Intervening attempts to deal with the problem are futile and negligible. The definitive solution of the difficulties is due, not to Bolzano, but to Georg Cantor, whose work on this subject first appeared in 1882.
[28] In regard to Zeno and the Pythagoreans, I have derived much valuable information and criticism from Mr P. E. B. Jourdain.
In order to understand Zeno, and to realise how little modern orthodox metaphysics has added to the achievements of the Greeks, we must consider for a moment his master Parmenides, in whose interest the paradoxes were invented.[29] Parmenides expounded his views in a poem divided into two parts, called "the way of truth" and "the way of opinion"--like Mr Bradley's "Appearance" and "Reality," except that Parmenides tells us first about reality and then about appearance. "The way of opinion," in his philosophy, is, broadly speaking, Pythagoreanism; it begins with a warning: "Here I shall close my trustworthy speech and thought about the truth. Henceforward learn the opinions of mortals, giving ear to the deceptive ordering of my words." What has gone before has been revealed by a goddess, who tells him what really _is_. Reality, she says, is uncreated, indestructible, unchanging, indivisible; it is "immovable in the bonds of mighty chains, without beginning and without end; since coming into being and passing away have been driven afar, and true belief has cast them away." The fundamental principle of his inquiry is stated in a sentence which would not be out of place in Hegel:[30] "Thou canst not know what is not--that is impossible--nor utter it; for it is the same thing that can be thought and that can be." And again: "It needs must be that what can be thought and spoken of is; for it is possible for it to be, and it is not possible for what is nothing to be." The impossibility of change follows from this principle; for what is past can be spoken of, and therefore, by the principle, still is.