Zeno's Paradox

Zeno's Paradox refers to the following puzzle. To walk any distance, one must first reach half the distance. After that, to close the remaining distance, one must first reach half of it (3/4 of the total distance). After that, to close the remaining distance, one must first yet again reach half of it (7/8 of the total distance). Et cetera. We conclude that to travel any distance at all, one must travel through infinitely many places. But a finite being cannot travel through an infinite amount of places in a finite time, so this is a paradox.

Standard mathematics' non-solution

It is common for people who know some modern math to mistakenly believe that they how to solve Zeno's paradox. This is because they know identities like the following:

$${\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\cdots =\sum _{n=1}^{\infty }2^{-n}=1.$$

That is, there are situations in standard mathematics where we can sum together an infinite number of terms and get a finite result. Doesn't this prove that Zeno is wrong, and that we can take an infinite number of actions?

No. There are two things being swept under the rug by this argument. The first problem is that it is assuming that the concept of an infinite sum makes perfect sense (which Objective Mathematics takes issue with). The second problem is that it is assuming something about the correspondence

David Deutsch's non-solution

David Deutsch's answer to Zeno is expressed in Chapter 10 of his book in Fabric of Reality. Deutsch, a Platonist, improves upon standard mathematics' non-solution by dealing with its aforementioned "second problem." His answer to Zeno is largely, though not completely, summarized by the following quote:

But what Achilles can do cannot be discovered by pure logic. It depends entirely on what the governing laws of physics say he can do. And if those laws say he will overtake the tortoise, then overtake it he will. According to classical physics, catching up requires an infinite number of steps of the form ‘move to the tortoise’s present location’. In that sense it is a computationally infinite operation. Equivalently, considered as a proof that one abstract quantity becomes larger than another when a given set of operations is applied, it is a proof with an infinite number of steps. But the relevant laws designate it as a physically finite process — and that is all that counts.

In other words, Deutsch is saying that infinite operations exist in the Platonic realm, and the laws of physics determine which infinite operations are physically possible to implement. Walking across a room is an infinite operation, but the laws of physics just so happen to be such that it is physically possible to do it in reality.

The solution

This refutation of Zeno seems obvious in hindsight, but it was first brought to my attention by Harry Binswanger in one of his lectures. The idea is that Zeno's argument is invalid, because the concepts being used by Zeno ("man," "walk," etc.) are not valid at all of the infinitely many scales that he needs to consider.

For example, although it is coherent to talk about a man moving forward by an inch, it is not coherent to talk about a man moving forward by the width of an atom. The reason why is that, at the scale of atoms, it is not clear which atoms are part of the man and which ones are not, nor is it clear what it means for a man---an extended body consisting of many vibrating atoms---to move.

It is possible that those things could be made precise (precisely which atoms get counted as being part of the man?, and precisely what does it mean for the man to move?), but that's besides the point. The point is that our concepts of "man," "walk," etc., were formed by omitted measurements on the perceptual scale, and they are thus not automatically applicable on the atomic scale: lots of work would have to be done to extend those concepts to things that make sense for atoms. Likewise, even if one had those concepts on the atomic scale, one would have to do even more work to extend them to the subatomic scale, then even more work at the scale below that, etc.

But at no point, no matter how much scientific knowledge is acquired, could a concept such as "man" or "walk" be valid at all scales. For Zeno's argument to work, the concepts in his argument would need to be valid at all scales.