 Hi, I'm Peter Adamson, and you're listening to the History of Philosophy podcast, brought to you with the support of King's College London and the Leverhulme Trust. Today's episode, You Can't Get There From Here! The Eliatic Philosophy of Zeno and Melissaus. Imagine you're standing at the baseline of a tennis court. That's the line at the back, where you serve from. It's time to change sides, so you're about to walk straight across to the other end of the court. You set off undertaking a journey of almost 24 metres, the length of a regulation court. It doesn't seem particularly daunting. When you get halfway, of course, there will be a net blocking your path, but you figure you'll cope with that problem when you get to it. Maybe you can leap over it jauntily, the way tennis players used to do at the end of the match. But, before you get to the net, which looms some 12 metres in the distance, you'll be arriving at the line drawn halfway across your side of the court. As those of you who are tennis experts know, this is called the service line. At that point, you will have completed a quarter of your trip, having walked about 6 metres. But of course to do that, you first got to get halfway to the service line. You need to walk 3 metres forward. And of course to do that, you've got to walk half of those 3 metres. It's starting to look a bit more daunting now, isn't it? In fact, now that you think about it, there is quite literally an infinite number of things you need to do to walk to the far side of the court. To get to any point, you will need to travel halfway to that point, and halfway to that halfway point, and so on. There's no end to the halfway points you'll need to visit. You better bring a packed lunch. This thought experiment, obviously without the tennis court, was invented by a man named Zeno of Elea, who was born in the early 5th century BC. He was the associate and student of Parmenides, who we looked at last week. Along with Melissa of Samos, Zeno was the most important follower of Parmenides. Zeno is renowned for the paradoxes he invented in support of Parmenides' theory that being cannot change or be more than one. The word paradox is another one of these English words that comes from ancient Greek. Para means, in this case, against, and doxa means belief, so if something is para doxon, it is contrary to our beliefs. Zeno's paradox of the halfway points, usually called the stadium or dichotomy paradox, is paradoxical in this sense. It shows you that it's impossible to move, and yet, of course, you still believe you can move. In fact, you move all the time. It's at this point that you need to do some philosophy. Paradoxes are a great way to introduce people to philosophy, because they force us to think hard about things we normally take for granted. There are many kinds of paradoxes in philosophy, for instance, moral paradoxes. There are two burning buildings in front of you, and one is your mother, and the other, several innocent children. You can only save the occupants of one building, what should you do? Then there are paradoxes about time and space. For instance, you hop in a time machine, go back, and stop yourself from being born. But then, how did you exist in order to go back in time in the first place? This of course is the basis of that philosophical classic, the film Back to the Future, the only case I know of where the philosophy of time comes together with a DeLorean. There are also logical paradoxes, for instance, the barber's paradox. Imagine a barber who shaves every man in the village who doesn't shave himself. Does the barber shave himself or not? Or the liar's paradox. If I say, what I'm now saying is a lie, is that a lie or not? And so on. Obviously, different paradoxes call for different kinds of solutions. You might say that burning building and other such moral puzzles are not really paradoxes, it's just there's no right answer. The world sometimes lands us with tragic situations that cannot be resolved morally. In the case of Back to the Future, we might simply say that such paradoxes show time travel to be impossible. Similarly, we might say that the barber in the logical paradox just can't exist. In each case, the paradox forces us to stop and think about the subject at hand, whether it be walking across tennis courts, the nature of time, or moral dilemmas. But we have to assume that the paradox is meant to be resolved in one way or another. Hardly anyone wants to say that it both is and is not possible to walk across the tennis court, or that the sentence, this is a lie, is both true and false. When people think about the dichotomy paradox, they naturally enough assume that the task is to spot the error in Zeno's reasoning. This approach assumes that the philosopher's job is to defend common sense against the paradox. Since we clearly do walk across tennis courts, there must be a mistake somewhere. Of course, this is a perfectly reasonable response, but it isn't what Zeno was hoping for. What he wanted to do, as a follower of Parmenides, was to show that motion really is impossible. Parmenides, as we saw last week, had argued that all motion and change are impossible, because they would involve non-being in some way. Parmenides argued positively, starting out from first principles, in particular that one can speak and think only about being, but never non-being. From this principle, he establishes that being is unchanging and eternal, a perfectly balanced sphere. Zeno takes a different approach, which is more destructive. With his dichotomy paradox, he tries to show us that the concept of motion is itself beset by contradiction. That this was Zeno's method is confirmed by Plato, who makes Zeno a character in a dialogue starring, and named after, Parmenides. In Plato's Parmenides, he shows Socrates discussing Zeno's book with Zeno himself. Socrates asks whether he has understood the goal of Zeno's book rightly, to show that belief in change and multiplicity leads inevitably to self-contradiction. Exactly right, says Zeno. There is some uncertainty about how Zeno's book was structured, and whether all the paradoxes ascribed to him were contained in this one book. But we know that he did produce a whole series of paradoxical arguments, some of which were apparently paired together. For instance, he used one argument to show that if things are many, then things are finite, and then he had another argument to show that if things are many, then things are infinite. Taking the two arguments together, we can conclude that if things are many, they must be both finite and infinite, which is a contradiction. Thus things are not many, instead being is one, just as Parmenides taught. We have information about a number of Zeno's paradoxes. Some are rather complicated, so I'll discuss only a few of them, especially since I'm already almost halfway through this podcast. Zeno, I refute you thus. First let's return to the dichotomy paradox. Here it is again, without the tennis court. Whenever you move from A to B, you have to move to C, the point halfway between A and B. To do that, you have to move to D, the point halfway between A and C, and so on. There will be an infinite number of such points, meaning that to move from A to B, you have to perform an infinite number of tasks, which is impossible. Or to put it as Aristotle does when he relates the paradox, you have to come into contact with infinitely many things, namely the halfway points. But why is this impossible? It's either because you only have a finite time to visit all these points, or simply because no one can do an infinite number of things. Another of Zeno's paradoxes makes pretty much the same point. We imagine Achilles racing a tortoise, with the tortoise getting a head start. The much faster Achilles tries to catch up, but every time he reaches the point where the tortoise just was, the tortoise has moved on at least a little bit further. And when Achilles reaches that new position, the tortoise has moved on again. Achilles cannot overtake the tortoise, no matter how fast he runs. Can we refute Zeno, other than by getting up and walking away, and thus proving that motion is possible after all? It's often thought that somehow modern mathematics, and especially our modern notions of infinity, have solved the dichotomy paradox. The idea would be, I guess, that we now have no problem in accepting that the series 1 half, 1 quarter, 1 eighth, 1 sixteenth, and so on, just adds up to 1. In fact, we might say that the number represented by that series just is 1. So that's how the distance is covered. But this is really no solution. After all, Zeno's paradox relies precisely on this fact, that the whole distance is equal to the sum of the infinite series. That's not to say that Zeno's understanding of infinity is the same as ours, but this may not be the decisive issue. The question is not only about mathematics, but about time and space. Certainly, we can now mathematically model the idea of dividing up a space or a motion into infinitely small parts. But Zeno's paradox isn't only about a mathematical model. It's about time, space, and motion. His question to us will be, okay, you've got your mathematical model, but how does it relate to what is really going on? Aristotle offers what I think is a more relevant response to Zeno. He says that one can divide the time needed to move from A to B right along with the distance from A to B. For instance, if it takes you 20 seconds to get from A to B, it will take you 10 seconds to get halfway there, 5 seconds to get a quarter of the way there, and so on. It's simply a mistake on Zeno's part to think that one needs an infinite amount of time to visit this infinite number of points, because the divisions will apply to both the time and the distance. This response, though, assumes that Zeno was worried about the motion taking an infinite amount of time, which isn't obvious. As I said, his point might be that it is impossible to perform an infinite number of partial motions, in the finite amount of time it takes to perform the whole motion. In that case, Aristotle's response does not really solve the paradox. This mention of time brings us to another brilliant argument of Zeno's, the paradox of the arrow. Here we consider an arrow in mid-flight on its way from archer to target. Now, consider how things are with this arrow at any one moment during the flight. Imagine it captured by a stop-motion camera. In this freeze-frame moment, it isn't moving at all, because at least a little time has to pass for anything to move. As Zeno says, it is at rest, because it is, as he says, against something equal. What he means by this is I suppose something like this. At this instant, the arrow is exactly aligned with the bit of space it is occupying, so it is at rest with respect to that space. Of course, this will be true for any instant during the flight of the arrow. This then is the paradox. At any instant during the arrow's flight, it seems to be hovering motionless in the air, yet over the whole time of its flight, the arrow apparently moves from bow to target. Again, we naturally assume that there's something wrong with the way Zeno has described the situation. This is how Aristotle reacted. He complains that time is not made up of instants, which he calls nows, that have no duration. Rather, if you divide time, you have to divide it into periods of time, maybe very short periods like a millionth of a second, but even a millionth of a second takes some time to elapse, and the arrow will move a little while that little time passes. Of course Zeno would just disagree. He would say that if motion cannot happen now, it cannot happen at all. And this is the intended conclusion. As a follower of Parmenides, he simply doesn't believe that anything can move. Not all of Zeno's paradoxes though concern motion. Some have to do with our even more basic assumption that there is more than one thing in the world. Remember, Parmenides claimed that all being is unchanging, eternal, continuous, and one. This would mean, for instance, that you and I are the same thing, unless neither of us exists at all, which is pretty hard to believe either way. But Zeno's paradoxes try to persuade us that if we assume that more than one thing exists the results are just as bad. For instance, he tried to show that between any two distinct things, there must be an infinity of other things. As with the dichotomy and the arrow, we need to speculate about exactly how this argument should work, but I think the idea was something like this. Imagine you've got two objects, call them A and B. Well, they must be separated from one another, because they aren't just one continuous object, so there must be some third object C, which is separating A and B. But now, why is C distinct from A on the one hand and B on the other hand? There must be some fourth thing separating A and C, and a fifth thing separating C and B. As with the dichotomy, we can repeat this argument over and over. The only way to escape is to give some explanation of how two things can be touching each other directly without being a single object because they form a continuous body. Of course, one could try to give such an explanation. Again, Aristotle is the first to try. He devotes a whole discussion to what it means for two distinct things to be in contact. In a way, Aristotle's response is a tribute to the fruitfulness of Zeno's paradoxes. Even if they do not really tempt us to believe that all being is one, they do force us to ponder the nature of such things as motion, time, and physical contact. This is why philosophers still find his paradoxes useful. Like Aristotle, they can formulate their positive ideas about motion and so on by explaining how Zeno's paradoxes should be resolved. We find a rather different approach to Parmenides' legacy in the other great eliatic of the 5th century BC, Melissa of Samos. Sharp-eared listeners will remember that Samos is also the island that gave us Pythagoras, in the eastern Mediterranean off the coast of Ionia. So when we call Melissa an eliatic, we mean that he followed the philosophy of Parmenides of Elia, not that he was from Elia. Unlike Pythagoras, Melissa's played a major role in the history of Samos. The historian and philosopher Plutarch tells us that Melissa led his people in a naval battle against the Athenians, and won handsomely. This battle was fought around the middle of the 5th century BC, which probably means that Melissa was, like Zeno, a generation younger than Parmenides. He may have been younger still than Zeno. In any case, his philosophy is clearly an attempt to develop and defend Parmenides, but not without departing from the master on some points. Melissa follows the method of Parmenides, rather than Zeno. That is, he argues positively that all being is unchanging and one, instead of inventing paradoxes to undermine motion and multiplicity. Like Parmenides, he starts from the idea that being cannot have started. To do that, it would have to come from non-being, which is absurd. He now applies this point to space as well as time. Whereas Parmenides had said, apparently quite seriously, that being is spherical in shape, Melissa denies that being has any limits at all. After all, if it had limits, there would have to be non-being beyond those limits, and there is no such thing as non-being. Thus he calls being unlimited or infinite. It's the revenge of Anaximander's principle, the aperon. These developments of Parmenides' ideas show that the Eliatic philosophy wasn't just a static, received doctrine. Rather, the theory of unchanging being itself changed, and was taken in different directions by Zeno and Melissa's, even at the price of contradicting the teachings of Father Parmenides. Another one of Melissa's ideas, though, would no doubt have delighted Parmenides. This is his argument against the possibility of motion. Again, he starts by ruling out non-being. In this case, the sort of non-being he discusses is emptiness. There cannot be a place with nothing in it, again because there is no such thing as nothing. To put it another way, void is impossible. Now Melissa's points out that, if there is no void, then that will make motion impossible. After all, there will be no empty place for anything to move into. The whole of being will be like a train compartment, packed so tightly that no one can budge, a familiar experience to those of you who, like me, live in London. In the wake of this argument of Melissa's, anyone who wants to defend the common sense idea that motion does exist has two possible responses. One would be to agree that there is no void, but insist that there is motion anyway. Whenever one thing moves, something else is displaced. Imagine the people on the train compartment shuffling along, perhaps with difficulty, each moving into the space of their neighbour in front while they give up their place to the person behind. This was Aristotle's view. The other response would be to say that, except in the London Underground at rush hour, no place is totally full. There is indeed void. These move around in this emptiness, banging into each other. That is more or less what we think today. Of course, we now know that outer space is empty, or mostly empty. But also, every physical body down here on earth turns out to consist of more empty space than full space, or at least that's what I was told in high school. Atoms and molecules exist in the void. What I wasn't told in high school is that this conception is a very old one. It was first conceived in response to the Eliatics, and perhaps especially in reaction to Melissa's. It was developed by the ancient atomists, Leucippus and Democritus. They are the pre-Socratic thinkers who seem to come closest to the scientific doctrines we actually accept nowadays. But unsurprisingly, the atomic theory of these ancient atomists was very different from modern atomic theory. Ancient atomism, like the Eliatic philosophy of Parmenides, Zeno, and Melissa's, was founded more on conceptual analysis than empirical investigation. But it's no less interesting for that. So join me next time for the atomists on the history of philosophy, with void, but without any gaps.