Bumping up against limits

This brings us to the last of Zeno's four main arguments on motion, "The Stadium", which has always been the most controversial, partly because the literal translation of its statement is somewhat uncertain. In this argument Zeno appears to be attacking the only remaining alternative to the unintelligible G_¥, namely, the possibility of a finite upper bound on conceivable velocity. It's fascinating that he argues in much the same way that modern students do when they're first introduced to the concept of an invariant speed in the theory of special relativity. He says, in effect, that if someone is running towards me from the west at the maximum possible speed, and someone else is approaching me from the east at the maximum possible speed, then they are approaching each other at twice the maximum possible speed...which is a contradiction.

To illustrate the relevance of Zeno's arguments to a discussion of the consequences of special relativity, compare the discussion of time dilation in Section 2.13 of Rindler's "Essential Relativity" with Heath's review of Zeno's Stade paradox in Chapter VIII of "A History of Greek Mathematics". The resemblance is so striking that it's tempting to imagine that either Rindler consciously patterned his discussion on some recollection of Zeno's argument, or it's an example of Jung's collective unconscious. Here is a reproduction of Rindler's Figure 2.4, showing three "snapshots of two sequences of clocks A, B, C,... and A', B', C', ... fixed at certain equal intervals along the x axes of two frames S and S':

These three snapshots are taken at equal intervals by an observer in a third frame S", relative to which S and S' have equal and opposite velocities. Rindler describes the values that must appear on each clock in order to explain the seemingly paradoxical result that each observer considers the clocks of the others to be running slow, in accord with Einsteinian relativity. Compare this with the figure on page 277 of Heath:

where again we have three snapshots of a sequence of clocks (i.e., observers/athletes), this time showing the reference frame S" as well as the two frames S and S' that are moving with equal and opposite velocities relative to S". As Aristotle commented, this scenario evidently led Zeno to the paradoxical conclusion that "half the time is equal to its double", precisely as the freshman physics student suspects when he first considers the implications of relativity.

Surely we can forgive Zeno for not seeing that his arguments can only be satisfactorily answered - from the standpoint of physics - by assuming Lorentzian invariance and the relativity of space and time. According to this view, with it's rejection of absolute simultaneity, we're inevitably led from a dynamical model in which a single slice of space progresses "evenly and equably" through time, to a purely static representation in which the entire history of each worldline already exists as a completed entity in the plenum of spacetime. This static representation, according to which our perceptions of change and motion are simply the product of or our advancing awareness, is strikingly harmonious with the teachings of Parmenides, whose intelligibility Zeno's arguments were designed to defend.

Have we now finally resolved Zeno's "youthful effort"? Given the history of "final resolutions", from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of "Rorschach image" onto which people can project their most fundamental phenomenological concerns (if they have any).

--- This article is copyright protected. All rights reserved. This article is for personal use only. Other use, especially reproduction, storage in data bases, publication and transmission to third parties – also in parts or in edited form – without BCG´s prior written permission is not permitted. ---