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.

1. Even if we remove time from the picture and just consider the problem of having an infinite number of conditions to satisfy, the fact that we can specify a process via an infinite list of statements does not mean that it is impossible for such a process to happen. There is an implicit assumption that the only physically feasible processes are those with finite specifications in some particular formal system. Who says that the universe is bound by such rules?

1. " specify a process via an infinite list of statements"

This is very puzzling: why would you think this is about *our specifications*? The Greeks were not worried about *talking about* space as a continuum. They were worried about whether or not it is one.

2. Gene, in your opinion, is there a satisfactory resolution of Zeno's Paradox? Is there a logically tenable argument for why motion can happen in a continuum despite the apparent paradox noted by Greek philosophers?

3. Sure: space is quantized, not a continuum.

4. So the contention is that motion is impossible in a continuum because an infinite number of events would have to happen in order for motion to happen? If I am understanding correctly, the embedded assumption is that it is impossible for an infinite number of events to happen, or to execute an infinite number of moves. That seems to be an assumption in need of justification.

5. I'm going to get back to you... as soon as you count up to infinity. Let me know when you get there, and I will answer this objection.

:-)

6. So what's your point, Gene? If it's simply that we should be careful in reasoning about things that involve infinity then your point is well taken and fully consistent with centuries of mathematical thought.

If your point is not merely cautionary, and you go so far as to claim that Zeno demonstrated the impossibility of motion in continuous space, and that his demonstration has real implications for the physical universe, then you are making a very strong claim about physics. If this claim were true then Zeno would have contributed as much to our understanding of space and time in the physical universe as Einstein. But before we revise the physics books we need to examine the crucial assumption that an infinite number of events cannot happen in finite time.

Cautionary notes are great. Strong claims about physics are something else.

7. I am making no claim about physics whatsoever. Physicists should use whatever models work for doing good physics.

My claim is about the real world, not the abstract world of physics.

2. The excellent In Our Time podcast had an episode about Zeno's Paradoxes a few days ago: http://www.bbc.co.uk/programmes/b07vs3v1

By the way, their guests mentioned that Zeno's teacher Parmenides believed that existence is a single thing and does not move. So, apparently one does not have to be an idiot to believe that things don't really move, at least not if one is an ancient Greek philosopher.

They gave an instructive example of a non-Zeno paradox that was also discussed in ancient Greece. Apparently, beginning with the assumption that the square root of 2 can be written as a fraction, I can prove that even numbers are odd numbers. This paradoxical conclusion implies that my premise is wrong: √2 cannot be expressed as a fraction.

1. "By the way, their guests mentioned that Zeno's teacher Parmenides believed that existence is a single thing and does not move. "

Again, we only have fragments and second-hand reports on Parmenides, so we have to be careful in saying "Parmenides believed..."

I believe he was indicating that the world of physics is not ultimate reality. But we can never really pin this down, unless we find more manuscripts, which seems unlikely.

3. Or was he just pointing out that assuming the continuum/infinite one place requires accepting it everywhere? If we did live in a continuous/infinite universe then infinite limits would be correct response and there would be no paradox or only one in the feebleness of our imagination of handling infinities. An argument more of aesthetics than logic.

1. "If we did live in a continuous/infinite universe..."

Conflating these two things indicates you don't understand this problem. The integers are infinite, but they are not a continuum!