Zeno and the denial of plurality

Question

Zeno is well-known as the storyteller of Achilles and the Tortoise and how the tortoise never catches Achilles; which is against our experience; the question of how to square these two notions generally falls to the theory of infinite series; and this is in fact a formalisation of the following physical observation:

That the sequences of displacements that Achilles moves is an infinite series; that we know that the total sum of these displacements must add to a finite sum (since his and the tortoises path do eventually cross); However when we turn to Zenos appearance in Platos Parmenides we find Socrates saying that:

I see, Parmenides, that Zeno would like to be not only one with you in friendship but your second self in his writings too; he puts what you say in another way, and would fain make believe that he is telling us something which is new.

to which he elaborates

For you, in your poems, say The All is one, and of this you adduce excellent proofs; and he on the other hand says There is no many; and on behalf of this he offers overwhelming evidence. You affirm unity, he denies plurality. And so you deceive the world into believing that you are saying different things when really you are saying much the same. This is a strain of art beyond the reach of most of us.

Which hardly appears to be the content of the above argument; for where is plurality denied there (which appears to be the heart of Zenos concerns according to Socrates)?

note:

A possible suggestion is that both motion in terms of displacement and time are measured using the real line; and this concieved as a plurality of points doesn't allow for motion at all. For how can one move from one point to another? For between a point and another is a void.

This sounds un-natural and unintuitive; but consider the real line with whats called the discrete topology where:

the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense

Visually, its as though we took a magnifying glass to the line and saw between the points a void (its of course a magnifying glass with an unaturally high magnification power and not found in any high street store); suggesitively its also called the totally disconnected line. Topology thus ties all the plurality of points together into a unity; it dispells the void; and allows for motion.

Another way of examining the same from a different perspective is to consider sets in set theory: we have a set; its elements are distinct; so we can pick them out one by one; and like taking cards out of a playing pack of cards arrange them on a table distinctly away from each; they are not contiguous; similarly when we conceive the real line as a set of points we can suppose them set away from each other.

additional note:

I'm not by the way disputing the immense mathematical importance of formalising Zenos argument in the mathematical sense; as this is of obvious importance in the theory of infinite series; of which the crux the monotone convergence theorem; and whose importance is re-emphasised in the similar sounding and in a sense equivalent theorem in modern integration theory

Answer 1

Zeno is known not only for Achilles and the Tortoise. He is said to have written a book, but the book hasn't survived (or hasn't been discovered). We know about nine of his paradoxes from Aristotle's Physics. Among them the Dichotomy, the Arrow, the Stadium.

The Arrow argues that when an arrow is shot, at each moment it occupies some place. Therefore, at each moment the arrow is at rest in that place. Therefore, the arrow does not move. For nothing can move and rest at the same time.

The Arrow is, then, an argument against the reality of movement. It pictures every moment as independent of the others.

The Achilles paradox, on the other hand, is not merely against the actuality of movement. Even if we accept that each moment is independent of the others, we still face the Achilles problem: Achilles cannot pass the tortoise, but we see that he does. Therefore what we see is an illusion. And what do we, in general, see? The Achilles seems to be an argument against the reality of time, not merely in the sense of change and movement, but more generally as a static manifold, as a plurality. We are supposed to conclude that time, as we know it, does not exist, not only in the sense of movement and change, but in the sense of a genuine manifold, of a genuine plurality.