Thursday, October 16, 2014

Zeno's Paradoxes


My opinion on Zeno's paradoxes is that they are not real paradoxes, that they are not deep, and that they are in need of dissolution, not resolution. Usually math teachers in high school, and sometimes middle school, present a narrative to their students in which Zeno's paradoxes were long-standing mysteries that differential and integral calculus finally solved. This narrative is echoed in various recesses of pop-math culture. But it isn't true.

The paradox of the arrow, for instance, is claimed to be resolved by differential calculus and the concept of instantaneous velocity. However, it is actually dissolved if we carefully look at what it is that instantaneous velocity means—and what motion at all means for that matter. We say that an object is in motion if it has different locations at different times. It is at rest if it does not. Of course, in defining rest, we must be careful that the object in question may return to a location; it might be better to say that an object is at rest between times A and B if its location remains unchanged over the time interval (A,B). But the point is that saying an object is in motion, or that an object is at rest, requires a consideration of its location over a finite interval of time. It is simply incoherent to say that an object is motionless at a single instant because it's at the same location for that 'entire instant'—yet this is exactly what Zeno does.

Instantaneous velocity, to be sure, actually involves comparing the location of an object at different times, and considering its average velocity over the time interval delimited by these distinct moments. One of the moments is left fixed, and the other varied, so that it is taken arbitrarily close to the fixed time. If the average velocity for this arbitrarily small span of time has a limit, this is called the instantaneous velocity at the fixed moment. However, introducing the concept of instantaneous velocity does not resolve the arrow paradox, but rather carefully avoids making the mistake that Zeno made of comparing an object's location at a single instant of time to itself. But one does not need to correctly conceptualize instantaneous velocity to avoid making Zeno's mistake.

By the way, this dissolution of the paradox is quite ancient; Aristotle offered it in his Physics. We did not need to wait for Newton (as the narrative usually goes) to be confident that things can move.

The other popular paradox has two forms, though they're really variations on the same theme: Achilles and the tortoise, or the dichotomy paradox. Essentially, either can be put as: there is an infinite number of ordered tasks, A, B, C, ..., each (aside from A) a successor to another, and then there is a single event Ω which occurs after all the others. A, B, C, ... are Achilles catching up to where the tortoise previously was, or Achilles being a distance 1 m, 1/2 m, 1/4 m, ... from the finish line of a race; Ω is Achilles passing the tortoise or finishing the race. The claim is that Ω can never happen, since an infinite number of events must occur first.

This has always struck me as a plain non-sequitur. Why the conclusion should follow is never cogently argued, and it's difficult to offer a counter-argument when no real argument is presented. Perhaps it should suffice to point out that the scenario outlined above is mathematically consistent, in case there was any doubt. We can associate the events with numbers: 1, 2, 3, ... aside from Ω, which is just called Ω. The numbers themselves are ordered the usual way, and we have a rule that for any number n, n < Ω.

The ordering relation < (called "happens before") has to satisfy two axioms, which are already satisfied for the numbers. First, it is trichotomous: exactly one of A < B, B < A, or A=B is true. If we consider n and Ω, it is clear that only n < Ω is true, by definition; and if we consider Ω alone then only Ω=Ω is true. Second, the ordering relation is transitive: if A < B and B < C then A < C. This has no implications for a triple with two or more identical elements; and if we consider a triple of distinct quantities (m,n,Ω), with m < n, then the fact that n < Ω requires m < Ω, which is also true by definition. We have consistency.

In fact, the collection of all points on a line segment—such as the last meter of Achilles's race—obeys a strict ordering relation, with absolutely none of the points having an immediate successor or predecessor. Perhaps, for some reason, Zeno had a problem with this. Of course, I can only speculate what he was thinking, but he may well, like Aristotle, have over-anthropomorphized nature, and imagined that it must operate as we humans do in our daily lives: by executing a discrete collection of tasks, one-after-another, each punctuated by the next. But nature does not work like this.

This second paradox is usually claimed to be resolved by integral calculus, but it really has nothing to do with time elapsed or distance traversed. What Zeno had a problem with was one event occurring after infinitely many others, so the relevant mathematics is ordering relations, not integrals. And the 'paradox' turns out to be a transparently bad case of sloppy thinking.

There is another paradox that involves rows of bodies passing each other, but I can't make out what it's trying to say. Nobody really talks about it though, so it probably isn't worth addressing.

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