This type of argument is, in modern times, called "antinomy." An antinomy is a proof that, since two contradictory propositions equally follow from a given assumption, that assumption must be false. Zeno is also called by Aristotle the inventor of dialectic. Dialectic originally meant simply discussion, but it has come to be a technical term in philosophy, and is used for that type of reasoning which seeks to develop the truth by making the false refute and contradict itself. The conception of dialectic is especially important in Zeno, Plato, Kant, and Hegel. All the arguments which Zeno uses against multiplicity and motion are in reality merely variations of one argument. That argument is as follows. It applies equally to space, to time, or to anything which can be quantitatively measured. For simplicity we will consider it only in its spatial significance. Any quantity of space, say the space enclosed within a circle, must either be composed of ultimate indivisible units, or it must be divisible _ad infinitum_. If it is composed of indivisible units, these must have magnitude, and we are faced with the contradiction of a magnitude which cannot be divided. If it is divisible _ad infinitum_, we are faced with the contradiction of supposing that an infinite number of parts can be added up and make a finite sum-total. It is thus a great mistake to suppose that Zeno's stories of Achilles and the tortoise, and of the flying arrow, are merely childish puzzles. On the contrary, Zeno was the first, by means of these stories, to bring to light the essential contradictions which lie in our ideas of space and time, and thus to set an important problem for all subsequent philosophy. {56}