ZENO CANNOT BE CAUGHT ON HIS OWN RACETRACK

Abstract

MR. MooR is quite right in his Note Has Lee Finally Caught Zeno? (MIND, July 1968) that I do not refute Zeno (MIND, October 1965), but he missed the point of my paper: I do not try to refute Zeno. I said Grant Zeno his way of putting the arguments and his conclusions are inescapable. The arguments show that the assumption that continuity can be analyzed in terms of infinite divisibility leads to unacceptable conclusions. Today, continuity is analysed differently, but the result is not to refute Zeno, it is simply to by-pass him. Again, Moor is quite right in thinking the analysis must start with continuities, but he is quite wrong in thinking that the interpretation of the series 1/2, 1/4, 1/8, 1/16 . .. as lengths or distances changes anything in the argument. This series, no matter whether interpreted as points or lengths, is denumerable. The elements of a continuum are neither denumerable nor discrete: they are infinite classes each having no last member. Hence, the above series is not a Dedekind's analysis of continuity is preferred today because it clarifies the relation between rational and irrational numbers (whether they be interpreted algebraically or geometrically as incommensurables) without inadequacy or paradox. I interpret the philosophical lesson of this to be: philosophical analysis should also start with a continuum, not with discrete parts. Mr. Van Valen, in his Note Zeno and Continuity (MIND, same issue) also seems to think that my purpose was to resolve Zeno's antinomies. He invokes mathematical techniques, but I admitted that Mathematical techniques are adequate to deal with the situations. . . They have been since Newton and Leibniz. Van Valen complains that Zeno and much of current philosophical discussion fail to grasp the concept of limit, but the very point at issue is ignored by the concept of limit. What is the justification of substituting the limit for a convergent series because the difference between the two can be made smaller than any pre-assigned quantity? Is this not a ruse to cover saying that if one can make a quantity small enough, he can play that it is not there? (The rules of this game can be stated so as not to lead to logical contradiction.) That two convergent series can be shown to be identical does not answer this question. I do not understand Van Valen's statement that it is not necessary to invoke the continuum. Zeno did not invoke the continuum, he started with it and showed that it cannot be explained as a summation of discrete parts. My paper added verbiage, but the point was essentially the same.