Does Accounting for Time Somehow Resolve Zeno's Paradoxes?
A commenter asks, "Isn't Zeno's mistake thinking that an infinite number of steps cannot be completed in a finite time?"
Upon receiving this question, I realize that I have been making a mistake: I have been presenting Zeno's paradox as it usually, in my experience, is popularly portrayed (e.g., this is what Maor offers): to get to the finish line, a runner has to cover half the distance to the finish line. From there he still has to cover half the remaining distance to the finish line. And from there, he still has to cover… So, 1/2 + 1/4 + 1/8 + 1/16...
But that was not actually how Zeno presented the problem, and discussing this has convinced me that Zeno was correct to present it the way he did. In the common portrayal, the runner can get closer and closer to the finish line, but can never quite get there. However, this leads to the misapprehension that if we just made all of those steps happen in similarly decreasing amounts of time as well, the problem would be solved. And to the misapprehension that Zeno just didn't get limits.
Zeno's presentation is more effective than the pop presentation, because it never gives those misapprehensions any footing to get started. The way Zeno presented the paradox, to finish a race the runner first must get halfway to the finish line. But before he can get halfway there, he first must get a quarter of the way there. And before 1/4, first 1/8, and so on. So Zeno doesn't merely contend that the runner can't finish the race (if space is a continuum). He makes the much stronger contention the runner can't even start it. For any "first move" the runner might make, there is a smaller first move that he will have to make beforehand. If space is a continuum there is no "minimal" first move that will ever get him going!
And positing that the runner can somehow leap past this barrier because all those teeny-tiny first steps will happen really, really fast ignores the fact that Zeno's critique of the continuum applies every bit as much to time as it does to space. If space is a continuum, then time must be one as well. And so in Zeno's model universe, time doesn't keep on ticking, ticking, into the future: no, time, also, can't get going at all, because for time to advance one second, it first has to advance half a second, but before that it must first advance a quarter of a second, and before that… So just as the runner can't get running in the first place, time can't get ticking in the first place either.
Of course, Zeno was not an idiot, and he knew things moved. He is arguing in the context of the Greek discussion of whether φύσις is a single, continuous thing of some sort, a plenum, a continuum, or if it is chunky, atomic. And Zeno is trying to demonstrate that if space/time really were a continuum, neither motion nor time would be possible. Thus, there must be a smallest unit of space (and time), and things move and time advances by multiples of that unit.* (And this, of course, really resolves the paradox, and seems to fit with the findings of quantum physics.) And I am pleased to report that Henri Bergson, unbeknownst to me before yesterday, reached a similar conclusion.
* We only have others' reports about what Zeno actually believed (none of his works survive), so I am being speculative in saying he was not arguing against motion per se, but against motion in a continuum. My interpretation is supported, I think, by Atomism and Its Critics**, and by the fact that it is from an anti-atomist, Aristotle, that we primarily get our report on what Zeno taught. Aristotle would be biased here, and could easily slip into reporting Zeno in a way to make him look as silly as possible, without consciously lying. (Confirmation bias.) But in any case, if what I claim is not what Zeno was arguing, it is what he ought to have been arguing!
** Holy crap, this book is expensive now! I'm sure I did not pay nearly that much for it.