Russell: The Journal of Bertrand Russell Archives

Russell and Zeno's arrow paradox by Paul Hager ON RUSSELL'S ACCOUNTS of Zeno's Arrow Paradox, Gregory Vlastos comments that there "seem to be almost as many Zenos in Russell as there are Russells.'" Zeno of Elea is, in fact, a philosopher whom Russell often discusses,2 and Vlastos' remark appears to be amply justified when we note that, for example, in his 1903 Principles ofMathematics Russell was asserting that Weierstrass had vindicated Zeno and established that "we live in an unchanging world and ... the arrow, at every moment of its flight, is truly at rest",3 yet by his 1914 Our Knowledge ofthe External World Russell had adopted the opposite view ofthe arrow that at "a given instant, it is where it is ... but we cannot say that it is at rest at the instant."4 Reversals such as this are, of course, the basis I G. Vlastos, "A Note on Zeno's Arrow", in R.E. Allen and D.]. Furley, eds., Studies in Presocratic Philosophy (New York: Humanities Press, 1975), II: 184-200 (at 199). 2 The principal sources are "Recent Work in the Philosophy ofMathematics", The International Monthly, 4 (July 1901), reprinted under the title "Mathematics and the Metaphysicians " in Mysticism and Logic (London: Longmans, Green, 1918); The Principles ofMathematics (Cambridge, 1903); "The Philosophy of Bergson", Tile Monist, 22 (July 1912), republished with a reply by H. Wildon Carr and a rejoinder from Russell as The Philosophy of Bergson (London, Glasgow and Cambridge: Bowes and Bowes, 1914); and Our Krwwledge of the External World (London and Chicago: Open Court, 1914), Lectures v and VI. The Bergson lecture of 1912 was later included in the chapter on Bergson in Russell's A History ofWestern Philosophy (New York: Simon and Schuster , 1945; London: Allen and Unwin, 1946). This chapter was severely cut in the British second edition of 1961, the material on Zeno being part of the omissions. 3 P. 347. See also "Mathematics and the Metaphysicians", p. 63, for the same claim. 4 P. 142. (Page references to Our Krwwledge are to the rev. 1926 Allen and Unwin ed.) 3 4 Russell summer 1987 of the famous C.D. Broad remark that "Mr. Russell produces a different system of philosophy every few years...."5 Vlastos himself highlights Russell's apparent changes of mind about Zeno's arrow by suggesting that different Russellian accounts of the paradox ascribe different assumptions to Zeno. Thus Vlastos views the Our Knowledge account as imputing to Zeno the central assumption "that there are consecutive instants", yet much later, in the History of western Philosophy, Russell had, according to Vlastos, produced another interpretation which centres on the different Zenonian assumption "that there can be no motion unless there are instantaneous states of motion."6 Since none of Zeno's writings have survived, our knowledge of the paradoxes of motion derives from secondary sources. This scantiness of direct evidence has led to a proliferation of interpretations and reconstructions of the arguments, so Russell would perhaps not be alone if he had, indeed, changed his interpretation of the Arrow Paradox as frequently as Vlastos suggests. Nonetheless, despite the evidence of vacillation outlined above, I will argue that Russell consistently maintained a single interpretation of the Arrow Paradox. The apparent differences and changes noted above will be seen to be differences of emphasis stemming from developments in Russell's doctrines concerning space and time, developments which can, in fact, be shown to underlie all of the major changes in Russell's philosophy.7 Accordingly, I will present my reconstruction of what Russell took Zeno's argument to be and then show how the differences emphasized by Vlastos are more apparent than real. Of course, Russell himself never set out the complete argument explicitly. However, the subsequent discussion will show the textual fidelity of my reconstruction. RECONSTRUCTION OF RUSSELL'S ACCOUNT OF ZENO'S ARROW PARADOX (I) Finite intervals of spaces and times consist of series of points and instants. (2) The series of points and instants are either finite or infinite. SIn J.H. Muirhead, ed., Contemporary British Philosophy, First Series (London: Allen and Unwin, 1924), p. 79. 6 Vlastos, p. 199. Vlastos is apparently...