Infinitely divisible

The first two arguments are usually interpreted as critiques of the idea of continuous motion in infinitely divisible space and time. They differ only in that the first is expressed in terms of absolute motion, whereas the second shows that the same argument applies to relative motion. Regarding these first two arguments, there's a tradition among some high school calculus teachers to present them as "Zeno's Paradox", and then "resolve the paradox" by pointing out that an infinite series can have a finite sum. This may be a useful pedagogical device for beginning calculus students, but it misses an interesting and important philosophical point implied by Zeno's arguments. To see this, we can re-formulate the essence of these two arguments in more modern terms, and show that, far from being vitiated by the convergence of infinite series, they actually depend on the convergence of the geometric series.

Consider a ray of light bouncing between an infinite sequence of mirrors as illustrated below:

On the assumption that matter, space, and time are continuous and infinitely divisible, we can conceive of a point-like particle (say, a photon) travelling at constant speed through a sequence of mirrors whose sizes and separations decrease geometrically (e.g., by a factor of two) on each step. The envelope around these mirrors is clearly a wedge shape that converges to a point, and the total length of the zigzag path is obviously finite (because the geometric series 1 + 1/2 + 1/4 + ... converges), so the particle must reach "the end" in finite time. The essence of Zeno's position against continuity and infinite divisibility is that there is no logical way for the photon to emerge from the sequence of mirrors. The direction in which the photon would be travelling when it emerged would depend on the last mirror it hit, but there is no "last" mirror.

Similarly we could construct "Zeno's maze" by having a beam of light directed around a spiral as shown below:

Again the total path is finite, but has no end, i.e., no final direction, and a ray propagating along this path can neither continue nor escape. Of course, modern readers may feel entitled to disregard this line of reasoning, knowing that matter consists of atoms which are not infinitely divisible, so we could never construct an infinite sequence of geometrically decreasing mirrors. Also, every photon has some finite scattering wavelength and thus cannot be treated as a "point particle". However, these arguments merely confirms Zeno's position that the physical world is not scale-invariant or infinitely divisible. Thus, we haven't debunked Zeno, we've merely conceded his point. Of course, this point is not, in itself, paradoxical. It simply indicates that at some level the physical world must be regarded as consisting of finite indivisible entities. We arrive at Zeno's paradox only when these arguments against infinite divisibility are combined with the complementary set of arguments (The Arrow and The Stadium) which show that a world consisting of finite indivisible entities is also logically impossible, thereby presenting us with the conclusion that physical reality can be neither continuous nor discontinuous.