Chapter 8
Achilles and the Tortoise
Sometimes the kids in a philosophy class I conduct will change the topic completely from what I had planned for us to discuss. That happened more than once with a class of third and fourth-graders in an elementary school in Newton, Massachusetts, I met with during the Spring of 1984, and, again, in the following Fall. Those Newton kids came up with some of the most philosophically interesting discussions I have ever taken part in, with anyone. But, more often than not, the kids themselves had shaped the topic we ended up discussing.
One of our most memorable discussions took place on only our third meeting. I had brought along a story beginning about whether time travel is possible. We soon got onto more general issues about the nature of time, and whether time could have a beginning. But then the children changed the subject to whether, not time, but the universe could have a beginning. The discussion that developed, with only minimal input from me, is, I think, the most exciting philosophical discussion among children I have ever witnessed. I quote much of it in my book, The Philosophy of Childhood, with my own comments and observations interspersed. I have put an unedited transcript in Appendix B of this book.
The key idea in that discussion is a notion put forward by Sam, which we could call the “Ground Assumption.” According to the Ground Assumption, nothing can come into existence without there being something for it to appear “on.” Sam’s choice of the preposition, ‘on,’ to state this Ground Assumption is fascinating. It can be compared with Plato’s idea, in his dialogue Timaeus, that there has to be something for things that come into existence to appear in. Plato’s conception of a “receptacle” for things to come to be in has often been understood to be a concept of space. Understood this way, Plato’s idea is that space has to exist first for things to be able to occupy it. Without space, nothing could come into existence.i
Sam’s closely related idea is that there has to be what I’m calling a “ground,” or background, for anything that comes into existence to appear on. And that has to be there first, before anything else can come into existence. “It’s not really anything,” he said; “it’s what other things started on.”
The other main contributor to that memorable discussion was Nick. Sam was ten years old at the time, Nick was nine. Nick began with the Plato-like question: “If there was a big bang or something, what was the big bang in?” But Nick’s main contribution to the discussion was his principle – we might call it the “Birth Principle” – that everything that exists has to have come into existence at some time. As he put this principle, “If something doesn’t start out, that thing isn’t there.”
Sam was quite happy to say that the universe – being what other things appear “on,” is always there – is, as we might say, eternal. But, of course, that conclusion contradicts Nick’s Birth Principle. “If the universe started,” Sam asked Nick, “what would it start on?” Disarmingly, Nick responded, “That’s what I don’t get.”
If you want to think more about this fascinating discussion of what philosophers call “cosmogony” (the beginning of the cosmos, or universe), please check out Appendix B. The discussion there can be taken as a philosophical text worthy of examination and discussion – much like a passage from Plato.
That Newton group had formed itself into what Matthew Lipman calls a “community of inquiry” — and only after three sessions, too! It was quite clear to me that I could discuss with them almost any issue in philosophy. Furthermore, my device of writing story beginnings to convince a class that the issue to be discussed could well be a children’s issue had become unnecessary. Although I did continue to write some story beginnings for our sessions, the kids themselves immediately went directly to the puzzle or perplexity and to the reasoning that brought it out.
This happened, perhaps most clearly, when I decided to try discussing one of Zeno’s famous paradoxes, the Paradox of Achilles and the Tortoise. The Achilles is one of Zeno’s paradoxes of motion, to be distinguished from his paradoxes of plurality. Zeno himself seems to have wanted to show both that the very idea of motion is irrational and also that the very idea of plurality is irrational. We were supposed to conclude from his paradoxes that the world our senses tell us about, a world in which there are, as it seems, many different things, each changing and many of them moving around, is only an illusion. What is, by contrast, real is, Zeno thought, following his teacher, Parmenides, something one and unchanging.
Plato was fascinated with the paradoxes of plurality, which he has his characters discuss at length in his dialogue, Parmenides. He seems not to have been equally interested in the paradoxes of motion. But Plato’s pupil, Aristotle, was quite taken with the paradoxes of motion. He, in fact, is our earliest source for them. So our discussion of Plato for Kids is now moving on to Aristotle, and more specifically, to Zeno. Here is Aristotle’s report of Achilles and the Tortoise:
The second [of Zeno’s paradoxes of motion] is the one called “Achilles.” This is to the effect that the slowest as it runs will never be caught by the quickest. For the pursuer must first reach the point from which the pursued departed, so that the slower must always be some distance in front. (Aristotle, Physics 5.9, 239b18-18, McKirahan, trans.)
This is a puzzle about the infinite divisibility of continuous motion, about which Zeno devised another paradox, called the “Dichotomy.” In one form of the Dichotomy Paradox we are told that we can never get from point A to point B because we have first to get halfway to B, and then halfway along the remaining distance, and halfway along the still remaining distance, and so on. Of course, the series of remaining distances formed by cutting each one in half, then the remainder in half, and so on, is an infinite series. So the core of Zeno’s puzzle is the claim that we can’t get from A to B because we can’t complete an infinite series.
The Achilles Paradox, as Aristotle points out, is both like the Dichotomy and different from it. The Achilles, too, gets us puzzled by trying to convince us that we must complete an infinite series, one step at a time, or otherwise we will never reach our goal. But the Achilles adds an important dramatic flourish. It makes the completion of the required series a “catch-up” game, since the tortoise is given a head start and the runs needed to reach the tortoise are always specified by reference to the location where the tortoise was located. This catch-up game is made even more fun (or, to some people, more annoying) by the recognition that fast runners, in their overconfidence, sometimes do loose real races by underestimating the progress their competition is making. That fact about the psychology of overconfidence, although it makes the Achilles more fun (or more annoying) to think about, actually has, of course, absolutely nothing, logically, to do with the problem about completing an infinite series.
I did try to make up a story about Achilles and the Tortoise for those Newton kids, but they quickly ignored the story line and moved immediately to the reasoning it presented, which was this:
- If Achilles can catch up with the tortoise, Achilles will have to complete an infinite number of catch-up runs.
- Achilles cannot complete an infinite number of runs – nobody can do that. Therefore,
- Achilles cannot catch up with the tortoise.
I had intended that we would focus our discussion on the second premise of this argument and think about why someone, such as Zeno, might think there is no way anyone, even a runner as fast as Achilles, can complete an infinite number of runs. Here is, I think, my favorite way to try to back up premise (2):
A. To be able to complete an infinite number of runs one would need to be able to make some progress toward achieving that goal. But nobody can make any progress toward completing an infinite series. No matter how many runs we complete – so long as the number completed is finite – we still have just as many left to complete, namely, an infinite number of them. So nobody can complete an infinite number of runs, since nobody, not even Achilles, can ever make any progress toward completing that task.
We can also think of other plausible justifications for premise (2). Here is one:
- To be able to complete an infinite number of runs one would have to be able to complete exactly that number, no more and no less. But there is no such thing as completing exactly an infinite number of anything. Infinity minus one is already infinity; and infinity plus one is still infinity.
I thought those Newton kids might come up with (A) or (B), or some other interesting justification for premise (2). But it became clear immediately that they were not interested in the project I had chosen for them. What they were interested in was proving that Achilles can catch the tortoise after all. Paul was first off the mark with this response:
Well I think that, since Achilles can run 100 times faster than the turtle, if Achilles can go 100 yards and the turtle 1 yard, Achilles can still go [another] 100 yards and the turtle goes less.
Paul’s response suggests an argument for the conclusion, ‘Achilles can pass the tortoise.’ It is, what we might call a “counter-argument” to the one Zeno made up. I shall return in a moment to consider the relevance of Paul’s “counter-argument” to the Paradox of Achilles and the Tortoise.
Almost immediately after Paul made his comment Ursula weighed in with her analysis of what might be wrong with the argument:
Well, I think that Achilles is just trying to catch up with the turtle; he’s not trying to get past the turtle. Okay Achilles is right here and the turtle is right there [she gestures], and he runs up and just stays there and the turtle moves further, maybe because he just wants to catch up and he doesn’t want to get past him.
I picked up immediately on Ursula’s suggestion that Achilles might be taking a rest-stop at each time he got to where the tortoise last was. Of course, if he did take rest-stops each time he had caught up to where the tortoise last was, even very brief ones – say stops of only a second or two — it would take him an infinite amount of time to catch up with the tortoise, since infinity times one second is an infinite amount of time.ii I tried to make clear to Ursula and the others that taking a rest at each “catch-up point” was not part of the Achilles Paradox.
Unfortunately, I ignored what was most interesting about Ursula’s suggestion, namely, the problem about whether Achilles can succeed in aiming at just the point at which he will catch up with the tortoise, given their respective rates of travel. I didn’t start thinking about this problem until later. I often discover when I tape-record an exciting discussion like this and only later transcribe it, that I had missed something really interesting and important. I did that in this case.
To simplify the math let’s suppose that Achilles runs only ten times as fast as the tortoise moves. Again, to simplify things, let’s suppose that, at the start of the race, the tortoise is only ten meters ahead of Achilles. A more realistic set of speeds would not change the point I am going to make in elaboration of Ursula’s suggestion.
Given this distance and these rates of travel, when Achilles reaches the point at which the Tortoise had begun the race, T1, the tortoise will be at T2, one meter further along. When Achilles reaches T2, the tortoise will be at T3, one-tenth of a meter (a decimeter) further along. When Achilles reaches T3, the tortoise will be at T4, one-hundredth of a meter (a centimeter) further. Up to this point Achilles has gone 11.11 meters. If we keep adding runs to the point where the tortoise last was, we can come up with a string of ‘1’s as long as you like, for example, this long: 11.111111111111111111111111111111111111111111111111111111111111111111111
But no matter how many 1’s we add, the point specified by our string of 1’s will leave the Tortoise still ahead in the race.
What can we do to specify the catch-up point? Well, if we ever add a ‘2,’ for example, this way:
11.111111111111111111111111111111111111111111111111111111111111111111112
we will have identified a point at which Achilles has already passed the tortoise. But there is no way, using the decimal system, in which we can identify the precise point at which Achilles is dead even with the tortoise.
Thus, in this scenario, and in many others like it, we can identify a very, very late point just before Achilles comes even with the tortoise . Maybe that point will be so close that even our highest-speed cameras will not be able to show that either Achilles is not quite there or has already gone past the tortoise, but we will not be able to specify the magical moment at which Achilles comes exactly even with it. That is a pretty mysterious result.
A practical person will ask, “What difference does it make?” It is true that, for practical purposes, very close approximations are as good as exact specifications. But Zeno can always respond by saying that this “cheating” shows that motion isn’t perfectly rational and so, on his assumption that only what is fully rational is real, there is really no such thing as catching up.
We don’t know that Zeno had exactly Ursula’s worry, as I have tried to fill it out a bit, in mind. But we do know that Greek mathematicians and philosophers of Zeno’s time were enormously troubled by the fact that some numbers, for example, the square root of two, are irrational. They were troubled with the thought that there might be irrationality in the very heart of nature.
I am sorry to this day that I did not follow up Ursula’s suggestion when she made it to bring out the paradox of being unable to specify the “come-even point” for Achilles. I did much better with Paul’s suggestion, which I had myself thought of before.
Here is the conclusion I wrote to the story on the basis of Paul’s suggestion and the discussion that it inspired. Paul’s suggestion, or rather a slight variation on it, is put here in the mouth of the character, Andrea: .
Andrea: Let’s give Achilles an extravagant amount of time to get from his starting point to the starting point of the turtle, say an hour . . . Now all we have to do is to let Achilles run another hour. Then he’ll be so far ahead of that slow-poke turtle that it won’t be funny.
Sarah: But the question wasn’t whether he could get ahead, it was whether he could catch up with the tortoise.
Andrea: Sometimes I think you’ve got only sawdust in your skull, Sarah. There is no way Achilles could come from behind and get ahead without catching up, now is there?
Sarah: You’re just giving us another argument.
Andrea: What do you mean?
Sarah: Your argument is this:
- If Achilles can get ahead of the tortoise, he can catch up with it.
- Achilles can get ahead of the tortoise (by, for example, running at the same speed for another hour). Therefore
- Achilles can catch up with the tortoise.
That’s a nifty argument. But we still don’t know what is wrong with the first one.
Steve (who had been listening and thinking): Wait a minute! Heh, yes, we do know what’s wrong with the first argument. As Sarah said, it’s the second premise – the one about how Achilles can’t complete an infinite number of runs. He can!! That premise is false. To complete an infinite number of runs he just needs to run, say, another hour. Then he’ll be, sometime during that second hour, at some point, B, well beyond the tortoise. And so he will have finished that infinite series of catch-up runs, A[chilles’s starting point] to T1, T1 to T2, T2 to T3, (and so on). After all, you can’t get ahead without finishing that infinite series of catch-up runs.
Sarah: That’s super, Steve.
Steve: That’s me, Super Steve.
Andrea: Ugh!