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Zeno's Race Course GREGORY VLASTOS I The first [of Zeno's arguments against motion which "give trouble to those who try to refute them"] says that there is no motion, because the moving [body] must reach the midpoint before the goal... (Aristotle, Phys. 239B 11-13) ? In the same way one should reply to those who pose Zeno's argument, [claiming] that it is always necessary to traverse the half, and these [halves] are infinite (i.e., infinitely numerous), while it is impossible to traverse infinites (Ibid., 263A 4-6). For there are many arguments contrary to common opinion, such as Zeno's that it is impossible to move or to traverse the race course... (Aristotle, Top. 160B 7-9). STARTING AT S, the runner cannot reach the goal, G, except by traversing each of the "halves," i.e., sub-intervals of SG, each of them SG/2 n. Did the argument order these as a progression (n = 1, 2, 3, ..-) or as a regression (n .... , 3, 2, 1)? The latter was the popular version of the argument in antiquity, a and the Aristotelian commentators assumed this had been the form of the Zenonian original. 4 Most modern scholars 5 have thought so too. But the regressive coni Those who may know the account of this paradox and of the Achilles in the chapter on Zeno contributed to Philosophic Classics, ed. by W. Kaufmann (Englewood Cliffs, N. J. : 1961), pp. 27 ft., at 35-40, might note that the present paper is meant to supersede it completely. That chapter was prepared on short notice to meet an urgent pedagogical need and, as I explained at the time (p. 27, n.1), drew on "purely provisional results of work-in-progress." I am now presenting the results of a later study (1963-1964) made possible by a grant from the National Science Foundation, which I hereby acknowledge with thanks. I must also thank those who have read and criticized a draft of this paper. In addition to the specific acknowledgements I shall be making below, I wish to mention my debt to Professor A. Griinbaum whose comments enabled me to remove several false or misleading statements, as well as to Professors J. A. Benardete and R. Sorabji for similar help. 2Aristotle goes on to refer to the next paradox by title ("the one which is called 'the Achilles'"). But he cites no title for the present one, referring to it by a phrase (~ ~LXoro/ze~v:an infinitive, 239B 22) he would scarcely have used if a title (grammatical noun) had been available to him. That this argument should have been called "the Dichotomy" (as frequently in the modern literature) is unlikely: dichotomic sequences occur in several surviving Zenonian arguments (flag. 1; flag. 2; cf. also the Eleatic argument which may be derived from Zeno in Aristotle de Gen. et Cot. 316A 14 ft. and 325 A 8-12 and Porphyry ap. Simplicius, Phys. 140, 1-5). In calling it "the Race Course" (on the strength of Aristotle, Top. 160B 8-9, "Zeno's [argument], that it is impossible to move or to traverse the race course") I am not implying that it was so called in antiquity. References to Zeno's fragments are to Fragments der Vorsohratiker, eds. It. Diels and W. Kranz (6th edition; Berlin: 1952). 8Cf. Sextus, Pyrrh. Hyp. 3, 76 and Adv. Math. 10, 139-141, where it is used by Sextus himself against the Stoics without allusion to its Zenonian authorship. 'See items 20 (Simplicius) and 21 (Philoponus) in H. D. P. Lee, Zeno o] Elea (Cambridge: 1936). But since they do not seem to have known the Zenonian original--their accounts of the paradox seem completely dependent on Aristotle--this has no probative value for the question at issue here. All it tells us is that this is the way they read the Aristotelian accounts (which are ambiguous on this point), and their reading of Aristotle is far from infallible . E.g., E. Zeller, Die Philosophie der Greichen (7th ed.; Leipzig: 1923), I, Pt. 756; J. Burnet, Early G~eek Philosophy (4th ed.; London: 1930), p. 318. (Burnet even imports this into his paraphrase...
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