 Parmenides had argued that there were strict metaphysical requirements on any object of knowledge; the later Eleatics (named for following Parmenidean doctrines rather than for strictly geographical reasons), Zeno of Elea (born ca. 490) and Melissus of Samos (fl. ca. 440), extended and explored the consequences of his arguments. Zeno paid particular attention to the contrast between the requirements of logical argument and the evidence of the senses (Vlastos 1967 is a masterly treatment of Zeno; see also McKirahan 1999 and 2005). The four famous paradoxes of motion, for which he is now and in antiquity best known, purported to show that, despite the evidence all around us, the ordinary motion of everyday experience is impossible. The paradoxes claim that motions can never be begun (the Achilles) or be completed (the Dichotomy), entail contradictions (the Moving Blocks), or are altogether impossible (the  Arrow).[6]  Recent philosophers of space and time (see Grünbaum 1967, articles in Salmon 2001, Huggett 1999, and the entry on    Zeno’s Paradoxes) hold that  the arguments are reductios of the theses that space and time are continuous (the Achilles and the Dichotomy) or discrete (the Moving Blocks and the Arrow). Consider the Dichotomy: a runner can never complete a run from point A to point B. First, the runner must move from A to a point halfway between A and B (call it C). But between A and C there is yet another halfway point (D), and the runner must first reach D. But between A and D there is yet another halfway point … and so on, ad infinitum. So the runner, starting at A, can never reach B. The argument assumes that it is impossible to pass an infinite number of points in a finite time. Similarly, Zeno produced paradoxes showing that plurality is impossible: if things are many, contradictions follow (Plato’s Parmenides 127e1ff.; Zeno in DK29B1/LM20D5, D6; DK29B2/LM20D7, R6; and DK29B3/LM20D11); there were also purported proofs that place is impossible (DK29A24/LM20D13a, R22, R23) and that things cannot have parts (the Millet Seed, DK29A29/LM20D12a, D12b, R16).