Frozen flight

The more famous of Zeno's two arguments against discontinuity is "The Arrow", which focuses on the instantaneous physical properties of a moving arrow. He notes that if physical objects exist discretely at a sequence of discrete instants of time, and if no motion occurs in an instant, then we must conclude that there is no motion in any given instant. (As Bertrand Russell commented, this is simply "a plain statement of an elementary fact".) But if there is literally no physical difference between a moving and a non-moving arrow in any given discrete instant, then how does the arrow know from one instant to the next if it is moving? In other words, how is causality transmitted forward in time through a sequence of instants, in each of which motion does not exist?

It's been noted that Zeno's "Arrow" argument could also be made in the context of continuous motion, where in any single slice of time there is (presumed to be) no physical difference between a moving and a non-moving arrow. Thus, Zeno suggests that if all time is composed of instants (continuous or discrete), and motion cannot exist in any instant, then motion cannot exist at all.

A naive response to this argument is to point out that although the value of a function f(t) is constant for a given t, the function f(t) may be non-constant at t. But, again, this explanation doesn't really address the phenomenological issue raised by Zeno's argument. A continuous function (as emphasized by Weierstrass) is a static completed entity, so by invoking this model we are essentially agreeing with Parmenides that physical motion does not truly exist, and is just an illusion, i.e., "opinions", arising from our psychological experience of a static unchanging reality.

Of course, to accomplish this we have expanded our concept of "the existing world" to include another dimension. If, instead, we insist on adhering to the view of the entire physical world as a purely spatial expanse, existing in and progressing through a sequence of instants, then we again run into the problem of how a quality that exists only over a range of instants can be causally conveyed through any given instant in which it has no form of existence. Before we blithely dismiss this concern as non-sensical, it's worth noting that modern physics has concluded (along with Zeno) that the classical image of space and time was fundamentally wrong, and in fact motion would not be possible in a universe constructed according to the classical model.

Proto-relativity

The theory of special relativity answers Zeno's concern over the lack of an instantaneous difference between a moving and a non-moving arrow by positing a fundamental re-structuring the basic way in which space and time fit together, such that there really is an instantaneous difference between a moving and a non-moving object, insofar as it makes sense to speak of "an instant" of a physical system with mutually moving elements. Objects in relative motion have different planes of simultaneity, with all the familiar relativistic consequences, so not only does a moving object look different to the world, but the world looks different to a moving object.

This resolution of the paradox of motion presumably never occurred to Zeno, but it's no exaggeration to say that special relativity vindicates Zeno's skepticism and physical intuition about the nature of motion. He was correct that instantaneous velocity in the context of absolute space and absolute time does not correspond to physical reality, and probably doesn't even make sense. From Zeno's point of view, the classical concept of absolute time was not logically sound, and special relativity (or something like it) is a logical necessity, not just an empirical fact. It's even been suggested that if people had taken Zeno's paradoxes more seriously they might have arrived at something like special relativity centuries ago, just on logical grounds. ... Doubtless it's stretching the point to say that Zeno anticipated the theory of special relativity, but it's undeniably true that his misgivings about the logical consistency of motion in it's classical form were substantially justified. The universe does not (and arguably, could not) work the way people thought it did.

In all four of Zeno's arguments on motion, the implicit point is that if space and time are independent, then logical inconsistencies arise regardless of whether the physical world is continuous or discrete. All of those inconsistencies can be traced to the implication that, if any motion is possible, then the range of conceivable relative velocities must be unbounded, corresponding to Minkowski's "unintelligible" G_¥.

What is the alternative? Zeno considers the premise that the range of possible relative velocities is bounded, i.e., there is some maximum achievable (conceivable) relative velocity, and he associates this possibility with the idea that space and time are not infinitely divisible. (It presumably didn't occur to him that another way of achieving this is to assume space and time are not independent.)