Zeno's Paradoxes (Archive Episode)

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Speaker 1 And now, to mark the end of his 27 memorable years presenting in our time, we have Melvin Bragg to introduce the next in our series of his most cherished episodes.

Speaker 7 If the title In Our Time works at all, it's to describe this long period on Earth in which we humans have tried to make sense of and enjoy the world around us. This is our time.

Speaker 7 Who would have thought that Zeno, a Greek philosopher, two and a half thousand years ago, even before Socrates, was devising thought experiments that would still be inspiring cutting-edge scientists today.

Speaker 7 That's why we were discussing paradoxes live at 9 a.m. back in 2016 and the audiences loved it.

Speaker 7 Hello, the ancient Greek thinker Zeno of Ilea flourished in the 5th century BC.

Speaker 7 His great innovation in philosophy was the paradox, a tool to highlight the unexpected consequences of common sense ideas, to question assumptions and provoke new theories.

Speaker 7 For example, according to Zeno's paradoxes, motion is not possible. An arrow arrow in flight does not move.

Speaker 7 The fastest runner in Homer, Achilles, could never catch up with a tortoise in a race if he gave it a head start.

Speaker 7 Philosophers from Aristotle to Bertrand Russell have tried to refute his ideas or explain them, with varying success.

Speaker 7 Innovations in mathematics with Newton and Leibniz went some way to demonstrate flaws in Zeno's arguments, but the questions he raised two and a half thousand years ago about time and space are as relevant as ever and have re-emerged in quantum physics.

Speaker 7 With me to discuss the paradoxes of Zeno are Marcus Yusotoi, Professor of Mathematics and Simonier, Professor for the Public Understanding of Science at the University of Oxford, Barbara Sattler, Lecturer in Philosophy at the University of St.

Speaker 7 Andrews, and James Warren, reader in ancient philosophy at the University of Cambridge. James Warren, what do we know about Zeno?

Speaker 2 Not a huge amount is unfortunately the answer. We know roughly when he was living and working.
He's living, as you said, in the middle of the 5th century BC.

Speaker 2 He came from Ileia, a town on the west coast of southern Italy.

Speaker 2 And we know that he travelled a lot in Greece, as people of that sort of class did, and he wrote a work, maybe just one work, which included these paradoxes, of which we know about,

Speaker 2 it depends how you count them, perhaps seven, eight, some to do with motion, some to do with plurality.

Speaker 2 What we can do is put him into some kind of context, intellectual context, that is.

Speaker 2 So for around,

Speaker 7 is it known as an intellectual centre? What's going on?

Speaker 2 It's a city in the Greek sense of a city. It's a polis, it's an independent city-state that was founded by Greeks from the Greek mainland at some point.

Speaker 2 It seems to have been quite an intellectual centre, in particular because I think one of the most important people in Zeno's intellectual life was again an Eleatic, was someone from the same city.

Speaker 2 This was a character called Parmenides. And Parmenides wrote a very peculiar poem in hexameter verse in the style of Homer.
And Parmenides attempted to set out to prove that there was only one thing

Speaker 2 and that it was changeless and motionless and perfect and so on.

Speaker 7 When he said there was only one thing, you meant the world was only one thing.

Speaker 2 There is just one thing. Yes.
So whatever else you think there is, if it's not this one thing, that isn't actually there.

Speaker 7 And he was he was Zeno's tutor, friend?

Speaker 2 Friend, tutor, something like that.

Speaker 2 It's not a a formal relationship, but Plato writes a dialogue in which the two of them come to Athens, and Zeno is cast as a defender of Parmenides.

Speaker 2 So that's one way to think of these paradoxes as an attempt to undercut possible objections to Parmenides' curious thesis on the basis of common sense assumptions that, well, there are many things, and clearly things do move.

Speaker 7 You said quite casually he wandered over the place of the Greek

Speaker 7 as people did. What did he wander for? Where did he go to? How did he look after himself?

Speaker 2 Well, these aren't people who really have to work for a living.

Speaker 7 So we're talking about aristocrats. Yes.
Well,

Speaker 2 well enough off people. That's right.
And they would travel around, and they would often, I think,

Speaker 2 the way in which these ideas were circulated were partly through written books.

Speaker 2 And Zeno complains that someone's made a pirated copy of his work, so he doesn't know how many of them there are in circulation, which is, I think, a joke on

Speaker 2 not even Zeno knows how many books there are.

Speaker 2 And they travelled to the great festivals like the Athenian Panathenia Festival and they would give demonstrations and public recitations and meet people there.

Speaker 7 And so what was

Speaker 7 got his teacher, Parmenides, what learning was coming to him through Parmenides and in the context of that place, that time, briefly, what was he reading that influenced him?

Speaker 2 Well what Parmenides is reacting to is a tradition of cosmological thinking.

Speaker 2 which had been going on for perhaps up to 100 years by now, of people who were attempting to explain the world and and how the world worked and functioned, often in terms of identifying some basic principle or element out of which the world was constructed.

Speaker 7 And

Speaker 2 from water or air or something else, and describing the various transformations that that element or elements undergo in order to produce the varied and differentiated world that we see around us.

Speaker 2 And they're relying, therefore, on there being a plurality of things and there being things that change and are in motion in order to account for the way the world works.

Speaker 7 So natural philosophers are people who believe that the world is changing, there are many things in it which are various and moving and changing.

Speaker 2 Yes, and that's what Parmenides sets out to show is

Speaker 2 grossly mistaken.

Speaker 7 Barbara Sattler, what is a paradox in philosophy?

Speaker 3 If we just look at the word that comes from the Greek, then that means it's against para common expectations or common beliefs. Doxa, right?

Speaker 3 So that's a paradox against what people normally would assume, what is strange, what is shocking, and therefore what needs explanation. So that's just the meaning of the word.

Speaker 3 In a philosophical context, by a paradox, we normally understand that we derive a problematic conclusion from sound premises.

Speaker 3 So it seems we have good starting points and we do right reasoning, and yet we get to a conclusion that's untenable. And why is it untenable?

Speaker 3 Well, either because it's inconsistent in itself, it leads to a contradiction, or it contradicts other beliefs, opinions that we hold.

Speaker 7 Can you give us a simple paradox? It needn't be one of Xenos, just to get the hang of it.

Speaker 3 Right, so one paradox that's quite famous is the bold man paradox. So we all would agree that if somebody has no hair, then this person is bold.

Speaker 3 If this person is one hair, we would still call this person bold. Two hair, probably still bold, three, and so on.

Speaker 3 But one hair doesn't seem to make a difference, but yet if this person has 10,000 hair, it seems this person is not bald any longer, right? So where does that stop?

Speaker 3 Is it that from 100 hair onwards, we say, oh, this person is not bold, but 99 hair is still bald? That doesn't seem to be right, right?

Speaker 7 Why not?

Speaker 3 Because it seems that with boldness, it's not a concept or notion where we can give a clear quantitative determination.

Speaker 3 We can't say so-and-so many hairs quantify as not being bold, and so-and-so many hairs quantify as being bald.

Speaker 7 But has common sense a place in this?

Speaker 3 Yeah,

Speaker 3 this uses common sense, that we all agree on certain ideas of boldness, and we all have a problem of saying when

Speaker 3 a person stops being bold. And what that shows in this case is that there seem to be some notions and concepts that are what we will call vague.
They are fuzzy.

Speaker 3 We can't really fully determine them, right?

Speaker 3 And there's a sample of them, like for instance, a heap of grain, right? If we have a heap of grain, let's say 10,000 grains, I take away one grain, it's still a heap. I take

Speaker 3 another one, still a heap. A grain doesn't seem to make a difference.
But if I take away so many that I'm only left with one grain, there's no heap any longer.

Speaker 3 Is there an exact moment where I can say it's not a heap any longer? Probably not.

Speaker 3 Philosophers have called this kind of concept vagueness concepts. And there's lots of work done because it's also, in some sense, vague where the vagueness starts, right? So they show.

Speaker 7 I mean, I I can see it's intriguing and it's a lot of fun, but is there any.

Speaker 7 Actually, well, I've discovered on this program for the last goodness knows how many years that the things that seem very odd and eccentric and rather miraculous suddenly turn out to be running the world, don't they?

Speaker 3 Okay, so in this case, I think with paradoxes, there's two reasons why they are actually very fruitful for philosophers, right?

Speaker 3 It sounds ironic because, in some sense, with a paradox, you had a dead end, and you could say, well, okay, now should we not give up? But they are very fruitful for philosophy for two reasons.

Speaker 3 Either because they show there's something funny about some concepts, like Mr. Bold Man, right? They show we are using some concepts that can't be fully determined in the way other concepts can,

Speaker 3 and that tells us perhaps something either about our concepts or perhaps even about the world, that some parts of the world are best described like this, right?

Speaker 3 Then, paradoxes can also be fruitful in one other way, namely that in philosophy, a lot of what we do is actually done conceptually, right?

Speaker 3 So, our our theories and models and concepts very often are not just falsified or verified by the world outside, right? So, how do we figure out whether our models are right or not?

Speaker 3 Well, paradoxes are very important because they tell us, okay, something has gone wrong here.

Speaker 3 You have to go back at your concept and look again whether your assumptions are really as good and true as you thought they are.

Speaker 7 Marcus Yusotoi, at that time, you've written, or you've said, mathematicians, mathematics as an analytical subject was beginning to emerge.

Speaker 7 Mathematics were exploring abstract ideas through mathematics. Now, would that can that be your starting point, Raton, about paradoxes?

Speaker 1 Yes, certainly. I think that

Speaker 1 before

Speaker 1 the ancient Greeks got their teeth into the subject, you've got the Egyptians and Babylonians doing a mathematics, trying to describe the world with this new language.

Speaker 1 But it's very geometric, it's very functional. They're measuring areas of land,

Speaker 1 volumes of pyramids, and things like that. But then the ancient Greeks, and in particular, sort of in the hundred years before Zeno, we have the Pythagoreans beginning to appear on the scene.

Speaker 1 And they're trying to prove things. So they're trying to prove that it's not just a calculation that they want to do, they want to produce a proof that something will always work.
So for example...

Speaker 2 Where did that come from?

Speaker 1 Well, I think it's interesting because I think that the Egyptian and Babylonian mathematics came from the development of the city trying to actually control the land.

Speaker 1 But this idea of analytic thought actually comes from Greeks actually wanting to do politics, and it comes out of the idea of rhetoric and trying to explain.

Speaker 7 How does that route work?

Speaker 1 Well, I think that you've got suddenly the Greeks trying to prove that laws will work and that laws will always apply.

Speaker 1 And so I think it sort of grows out of that sort of change of the city into something, into a political institution.

Speaker 1 And so I think the ancient Greeks, you see a different style of mathematics, and what I would really call mathematics, this idea of analytic thinking. But it's interesting that

Speaker 1 the idea of paradox is starting to appear at this time, perhaps a little bit after Zeno, as a tool, which is this idea of a proof reductio ad absurdum.

Speaker 1 Make a hypothesis, for example, that the square root of 2 can be written as a fraction.

Speaker 1 And then you follow that through, and you end up with a ridiculous conclusion that odd numbers equal even numbers. And then you realize that that's absurd.
It's a kind of paradox.

Speaker 1 But the paradox is very useful because you can then work backwards and say, okay, something along the way was wrong.

Speaker 1 And it was actually the Pythagoreans who discovered, no, this square root of 2, which is a length, it's the length across the diagonal of a square. Each side has unit length.

Speaker 1 So this length cannot be written as a fraction. It can be approximated by fractions more and more, but they realized using this argument that there were new numbers here.

Speaker 1 So the idea of paradox or this idea of teasing out a logical argument which arrives at something absurd is a very powerful tool in actually questioning your assumptions.

Speaker 7 One of the things, a metaphysical thing that the Greeks turned into mathematics was the idea of infinity, which they had problems with. How did they tackle that?

Speaker 1 Well, they did have problems with infinity, and a lot of their mathematics, you can see, is very finite, it's very geometric, it's about lengths.

Speaker 1 And this discovery that the square root of two can't be written as a ratio of two whole numbers.

Speaker 1 If you write it as an infinite, as a decimal, it goes on forever, never repeating itself, was a real challenge to their whole philosophy.

Speaker 1 But in fact, you can look back, even in the ancient Egyptians, in order to calculate the volume of a pyramid, we now know that they must have had some idea of infinitely dividing space.

Speaker 1 So it's not in the documents, but the volume, the formula that you get, actually,

Speaker 1 you have to use an idea of infinite divisibility to be able to get that formula. It's an early form of integral calculus.

Speaker 1 So already these ideas are beginning to sort of bubble up and they're having difficulties with, okay, but you know, infinity doesn't seem to exist. I can't see anything infinite.

Speaker 1 So they have this idea of absolute, actual infinity and

Speaker 1 what's the other one? Potential infinity. Potential infinity.
Thank you. There you go.
Potential infinity. So there's a potential for infinity.

Speaker 1 For example, Euclid proves that the primes have the potential to go on forever. But there's a claim that, well, this isn't an actual infinity.
You can't actually have infinitely many primes.

Speaker 1 They have the potential to go on forever.

Speaker 2 So some of these...

Speaker 7 Let's get back to the paradox. This is where paradoxes become very useful.

Speaker 1 Because it can tease out.

Speaker 7 So is it a key key? I mean, is it really that important, the paradox?

Speaker 1 Well, the paradoxes will be able to reveal that your ideas of infinity might actually be wrong.

Speaker 7 So that's what he's setting out to do.

Speaker 1 Well, I think that he's trying to actually support Parmenides, who isn't bringing a kind of mathematical perspective on the fact that there is no such thing as motion.

Speaker 1 He's trying to actually use this now actually as a mathematical tool to question whether our perceptions of the world are actually correct or not.

Speaker 7 So that's the idea behind it is say let's see what the world is about. Let's see the reality.

Speaker 7 In one sense of Plato it's a dream and but this is mathematical by mathematical analysis it's a different reality from that which we perceive.

Speaker 7 And let's challenge Parmenides who might have been right anyway but by challenging him we might unlock this. Is there something in that, James Warren?

Speaker 2 I think that's right.

Speaker 2 I think that another thing to bear in mind is that these paradoxes are sort of playful and they would have been a way of Zeno embarrassing an interlocutor in the way that you might remember Socrates embarrassing people.

Speaker 2 So he takes someone and he says, well, you think things move, don't you? Yes, of course I think things move.

Speaker 2 Well, wouldn't you, for example, think that you would uh you would agree, wouldn't you, that in order to get from A to B, you must get at halfway from A to B.

Speaker 2 Well, yes, of course I would have to get from halfway from A to B in order to get from A to B.

Speaker 2 Well, surely you would then agree in order to get from A to halfway to B, you would have to get halfway from A to halfway between A to B, and so on and so on.

Speaker 2 And here you have another example of what we saw in Barbara's Baldman case, this repeating premise.

Speaker 2 Once you've granted once that in order to get from one point to another you have to go halfway, that gets repeated and repeated and repeated.

Speaker 7 This is the dichotomy paragraph.

Speaker 2 right exactly so this is what I mean sorry I split a dichotomy paradox right which just means cutting in two dichotomy and that label gets associated with more than one paradox in the sources but Aristotle I think associates it with a paradox of motion in the way that I've been trying to set out so in order to cross a spatial extension you must go halfway but then of course you must go halfway to the halfway and so on and so on.

Speaker 7 But by saying that, what is he saying that's

Speaker 2 going to prove

Speaker 2 paradox. What's problematic then is that you've got your person to agree that in order to cross any spatial extension, in fact that entails an endless series of prior journeys, if you like.

Speaker 2 So in order to do something, first I have to do something prior.

Speaker 2 And if that's an endless series of prior requirements before I even get started, then you can the the the killer line will say but you don't think you can complete an infinite series of tasks can you?

Speaker 2 It would be impossible to do an infinite series of journeys. Well, I suppose that's true,

Speaker 2 and there's obviously a sense in which that is true. In which case, it now looks like in order to cross a room, that's asking me to do something impossible.

Speaker 7 Aristotle rose up against these paradoxes again and again, didn't he? It becomes like a sort of heavyweight championship at one stage. Zeno says this, and Aristotle waves in biff.

Speaker 7 What did he biff about on this one?

Speaker 2 Well, on this one, he thinks, as we've just, it's Aristotle's distinction distinction between potential and actual infinity.

Speaker 2 He thinks it's misdescribing the job to say that you have to complete an actual series of infinite journeys.

Speaker 2 Potentially if you wanted you could you could think of your journey as including however many sub-journeys that you like. But you don't actually have to do all of those in order to cross the room.

Speaker 2 But Aristotle's working from the assumption that of course Zeno must be wrong because of course things do move and there are many things.

Speaker 2 So he's of the opinion that the absurdity of the conclusion licenses you to think there must be something wrong with the argument and he can just move on and carry on writing his book on physics.

Speaker 7 But the argument goes on. That's the interesting thing, isn't it? Great as Aristotle is,

Speaker 7 he doesn't kill it. I mean, it continues, it emerges, it re-emerges.
And Barbara Sattler, probably the best-known paradox is Achilles and the tortoise. Can you tell us what's happening there? Sure.

Speaker 7 Or what Zeno says is happening there.

Speaker 3 Right. So Achilles and the tortoise is basically a variation of the dichotomy paradox that we have just heard from James.
So imagine that Achilles, who is the fastest runner in the ancient world,

Speaker 3 has a race with the slowest runner in the ancient world, a tortoise, as its later tradition calls it. And because Achilles is the fastest runner, he can give the tortoise a head start, right?

Speaker 3 So let's imagine they are racing on a 100-meter racetrack, and the tortoise is starting 10 meters in.

Speaker 3 So what now has to happen is that first Achilles has to cover these 10 meters that the tortoise was given as a head start.

Speaker 3 But during the time that Achilles takes in order to cover this 10 meters, well the tortoise will have moved on. Not very far because it's very slow, but let's say the tortoise moved on for a meter.

Speaker 3 Well, next thing that Achilles has to do is to cover this one meter. During that time, while he's covering this one meter, the tortoise will have moved on yet again, let's say 10 centimeters.

Speaker 3 Again, you know, the same happens. So the distance between Achilles and the tortoise will get less and less, but it will never get to Cero.

Speaker 3 So it seems that Achilles, even though he's the fastest runner in the ancient world, will never be able to overtake the slow tortoise, right?

Speaker 2 So that's the paradox.

Speaker 7 Did, I mean, a guy said this once before, and I'll never say it again after this. Did common sense rear its head?

Speaker 3 So common sense,

Speaker 3 if you wanted, read its head in that some people thought, oh, okay, we can just contradict Sino by getting up and running and showing that, you know, we we can overtake somebody, right?

Speaker 3 But I don't think that Sino wanted to show we will never experience somebody overtaking somebody else, right? Or somebody covering a finite distance.

Speaker 3 Rather, what he's telling us is, okay, and you give me an explanation of how this happens. You give me an account, you describe what is going on, and you will get into contradictions, right?

Speaker 3 So even though we experience it, we can't give a good explanation of it.

Speaker 7 I think the paradox is more interesting than common sense, actually. And obviously it leads to more things, don't you think?

Speaker 3 Well, as you said before, it pops up over and over again. So that shows that people have thought: okay, there's something still going on.

Speaker 3 Something in this paradox shows that if we try to explain motion, change, time, and space, there are still problems that we get into and that get us into these contradictions and that's seen for the first time race.

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Speaker 7 So can you unpick that more? Can we go into this moment? Why is this fascinating and why does it continue to be so? I sort of rather brutally said, what about common sense?

Speaker 7 Of course, I mean, boringly said that. But what is interesting is that the idea goes on.
What is interesting is that the idea is a powerful idea and is still employed in various ways today.

Speaker 7 So, can you just let's go into that? What's going on?

Speaker 1 Well, there's really the challenge of the infinite, and in particular, something called an infinite series, because we're having to add up infinitely many things and understand whether that's actually sort of physically possible.

Speaker 1 So, the way mathematicians eventually resolve this is to say, well, okay, how long does it take Achilles to do this infinite number of tasks?

Speaker 1 So, let's say he does the first step in half a minute, the second step he does in half the time, so quarter of a minute, the third step in an eighth of a minute, the next step in a sixteenth of a minute.

Speaker 1 So it looks like he's having to do infinitely many tasks, but

Speaker 1 we understand this now, that he can do infinitely many tasks because it can take him a finite amount of time.

Speaker 1 This infinite series, a half plus a quarter plus an eighth plus a sixteenth, actually adds up if you do take infinitely many of them to the answer one. And you can sort of see that.

Speaker 1 If you imagine a cake and you cut the cake in half, and then you cut the half in a quarter, and then an eighth, and then a sixteenth, you can see that you'll be cutting each of the smaller pieces in half again, but it won't be any more than one.

Speaker 1 So we know that this infinitely many tasks will take a finite amount of time. And it's interesting, maybe it takes less than a minute.

Speaker 1 So mathematicians had to come up with some sort of way of understanding adding up infinitely many things. And it doesn't mean that adding up anything will always work.

Speaker 1 For example, take the add a half plus a third plus a quarter plus a fifth plus a sixth plus a seventh plus an eighth. You might say, well those are getting very, very small.

Speaker 1 Maybe that adds up to something finite. But Oramay in the 14th century proved that actually no, that can become as large as you want.

Speaker 1 So Zeno is already challenging us with how do you understand how to do add up infinitely many things in mathematics and does that have some sort of physical reality?

Speaker 1 And it really took till 17th, 18th century for mathematicians to come up with some way to understand how to navigate these infinitely many numbers and add them up and understand when they are a finite and when they could be infinite.

Speaker 7 What's fascinating to me, a non-mathematician, and I'll go back to you for a moment, Barbara, if I is

Speaker 7 what grabbed people, mathematicians, about this? Why was this so important to keep studying this? Which was, I think you have used the word, not me this time, in your notes, patently ridiculous.

Speaker 7 But away they go. What is so fascinating about it?

Speaker 3 So, one thing that's so fascinating about it is that it seems in the physical reality we don't have a problem with these things, right?

Speaker 3 We can do this run, our killers can overtake the tortoise, no problem. But yet, in mathematics, which we use in order to describe the physical reality, there seemed to be a real problem with this,

Speaker 3 dealing with infinity, right? So, our most powerful tool to describe the reality and to deal with it, which we use in natural science all the time, right, that seemed to be too weak to deal with that.

Speaker 3 That seemed to get us into contradictions. And if you have a contradiction, then you have a trouble with your science.
It's not a solid scientificion at heart, right?

Speaker 3 So that's why mathematicians were really fighting with that and saying, okay,

Speaker 3 we don't want

Speaker 3 contradiction at the very basis of our science, right?

Speaker 3 And then in the 17th and 18th century, as Marcus said, there was a new way of dealing with

Speaker 3 infinite series.

Speaker 3 Then with Cauchy, we have

Speaker 3 dealing with limits. We have in the 19th century a new way of dealing with actual infinity.

Speaker 3 Remember, with Aristotle, we had this distinction between potential and actual infinity, and there was always this idea there can't be actual infinity, there can only be potential.

Speaker 3 And then with Canto and others, we had this idea, no, there can be an actual infinity, and that just

Speaker 3 needs a different way of dealing with it. That goes against our intuitions, by the way.

Speaker 7 James Warren, it seems to me that mathematics is being used in a philosophical way all the way here.

Speaker 7 The idea is still, let's go back to Parmenides saying bluntly, the world does not move. Nothing moves.
It is one thing.

Speaker 7 It is not many things, as you natural philosophers have been thinking for the last few centuries. It doesn't change all the time, it doesn't move all the time.
It is one thing.

Speaker 7 That's what I proposed. Like a previous person said, it's all water.
He'd done that. And so we're into the fact that it sort of has a comic aspect, as mathematical things,

Speaker 7 is only a superficial reading of it, because the mathematicians are going for something else, aren't they?

Speaker 2 I think one of the things that might emerge from talking about these paradoxes in a mathematical way is the relationship that mathematical analysis has to these kinds of physical cases.

Speaker 2 So the question whether in fact mathematics is an abstracted description of what's going on or that somehow we can construct physical extensions and so on out of mathematical items is worth thinking about.

Speaker 2 So for example,

Speaker 2 one of the things Aristotle complains about is that

Speaker 2 one of these problems that Zeno raises is

Speaker 2 driven by the idea that somehow an extension just is an infinite connection of points. And he says, well, that's just not the case.

Speaker 2 You can't make a line out of points any more than you can construct a duration out of instance.

Speaker 2 What a mathematical point does is an abstracted,

Speaker 2 what you're doing is taking an extension that's already there and picking out something out of it.

Speaker 2 You're not constructing the world mathematically.

Speaker 7 So let's look at that notion with regard to the arrow in flight or the arrow at rest. Parmenides argued, and

Speaker 7 Zeno puts it forward, that the arrow never moves. So somebody shoots an arrow and it never moves.
What's going on there?

Speaker 2 Right, so the absurd conclusion here is that the moving arrow is always at rest.

Speaker 2 And the reconstruction that we get of it from Aristotle goes something like this, that if you imagine an arrow that's being loosed from a bow heading towards a target,

Speaker 2 if you think of any point in the arrow's journey,

Speaker 2 by point I mean now an instant, a temporal point, so imagine taking a photograph of it that captures

Speaker 2 an instant in that flight. At that point, the arrow is occupying a space exactly arrow-shaped and arrow-sized, and it's not moving within that space.
It's stationary

Speaker 2 at that instant. You can think of it that either it's too snugly held by space or there's not enough time for it to do any moving because

Speaker 2 we've specified that we're talking about an instant, so a durationless point in time. But that's the case throughout the arrow's journey.

Speaker 2 You could pick any instant in the arrow's journey, and it would always be the case that at that instant the arrow is stationary.

Speaker 2 So it seems to be true throughout the journey that the arrow is not moving.

Speaker 7 And Aristotle said...

Speaker 2 Well, Aristotle says this is false because time, a duration is not made of nouns.

Speaker 7 Yes, a

Speaker 7 duration can have its sensual state. Can we keep on the arrow? Because it's such a graphic one.

Speaker 7 People listening, you have the arrow, photograph of the arrow, and it doesn't seem to be moving except for that instant.

Speaker 7 But even in that instant, instant, instant, is it not moving fractionally, very fractionally? Are we not seeing it between two so fractional movements that we can't see the movements?

Speaker 7 Or are we seeing it, does that rest, what does that rest mean?

Speaker 1 Well, I think it's the challenge of

Speaker 1 this arrow is changing speed, it's decelerating as it goes towards, so it's got a different speed at every particular time.

Speaker 1 And it was the real challenge of actually, for it to be moving, it has to have a speed.

Speaker 1 And and if you just take an instant of time the time interval is zero and well the distance it's gone is zero but speed is distance divided by time so you're trying to make sense of well it doesn't have a speed then does it zero divide distance divided by zero time but you're absolutely right in the the the way you tried to approach that problem because and you've just invented the calculus Melvin because what Newton Leibniz did is to realize that actually this thing does have a speed but you've got to understand it as the time interval that you're taking gets smaller and smaller and smaller.

Speaker 1 So if you take the time interval of one second before the snapshot you've done, then you've got an average speed, the distance it's gone over that one second divided by the one second.

Speaker 1 Now take the time interval a little bit smaller, and you get another average speed, but it's slightly slower for the half second before then and the quarter second.

Speaker 1 So you see though that the speed is actually tending towards a limit, and calculus is making sense of this challenge that Zeno has said. Well, what is the speed? It's zero divided by zero.

Speaker 1 It doesn't have a speed. That's meaningless.
It isn't moving. Newton and Leibniz say, no, we have a mathematics now developed

Speaker 1 by Newton and Leibniz

Speaker 1 to actually understand a world in flux and be able to say at one instant of time what the speed of the arrow is.

Speaker 2 James Lauren, I don't think Zeno would be impressed by that. Really? I think that's mathematically clever, but philosophically not so smart because you've cheated.

Speaker 2 You've assumed the arrow is moving

Speaker 2 and then have described how it can be moving at a time, at an instant, on the assumption that it is crossing some distance. And that's precisely what's at question.

Speaker 2 You can't help yourself to the conclusion that you're trying to get.

Speaker 2 And his second point would be, well, surely you would agree then that if we can allow ourselves that now is can be described as an instant,

Speaker 2 it's true that the arrow isn't moving now. And if it isn't moving now, when on earth is it moving?

Speaker 1 Well, I would say that moving means it has a speed. And Newton and Leibniz have given you a way to say what the speed of that is.

Speaker 2 Marbury.

Speaker 1 Oh my gosh, are they ganging up on you?

Speaker 2 The man that's all right. No, no, no, no, no.

Speaker 3 I'm ganging up against both of you.

Speaker 1 The singer the bridge and the man defending.

Speaker 2 They're all rushing at him defending the bridge.

Speaker 3 In support of Marcus, so philosophers afterwards tried to actually

Speaker 3 think of motion really in that way. So with Russell and others, we have this idea of the ad-ed theory of motion, as it's called.

Speaker 3 So, that motion is nothing but being at a particular point at a particular time, right? And the difference between motion and rest is just that you look at the surrounding, right?

Speaker 3 And if you look at the surrounding, then something in motion will be at a different point in space at the next moment of time, and something at rest will still be at the same point.

Speaker 3 So, that has been a famous theory, ad-ad theory. But I would gang up, you know, help

Speaker 3 James here saying this is a very useful way in mathematics to describe motion, right? And we have come to use it and employ it all the time.

Speaker 3 It doesn't tell us that motion consists of these points, right? It tells us we can describe motion in this way, in this mathematical way. It's very useful to do that.

Speaker 3 But it doesn't tell us that we really have understood what's going on with motion in this sense.

Speaker 1 Well, it's interesting because actually there's a modern day effect in quantum physics which actually says that

Speaker 1 motion sort of doesn't happen. It's called the quantum Zeno effect, which is

Speaker 1 quantum physics says two electrons can be sort of

Speaker 1 two places at the same time, but when you observe them,

Speaker 1 so it could be here and there, but when I observe it, it has to make up its mind where it is, so it's there.

Speaker 1 But then if I don't look, it starts to evolve again. But if I look very quickly, it's mostly there, and so it collapses back into the there state.

Speaker 1 So actually, this is called the Zeno quantum effect, because if I keep on looking at it, actually, I can stop this thing evolving.

Speaker 1 So I've actually brought a pot of uranium into the studio, which is the same effect.

Speaker 1 If I I keep on observing this, I can actually stop it radiating because it never has a chance to move because of my observation.

Speaker 7 That's like magic.

Speaker 2 I mean, it's magic. Oh, I know.

Speaker 7 I mean, I'm fascinated by all this stuff, and I'm fascinated by magic, so there you go, hands off. So, but you look at it and it stops moving.
Now, what's going on?

Speaker 7 Are you the only one? Can I look at it again?

Speaker 2 As long as someone's looking at it.

Speaker 1 I mean, this is the challenge of quantum physics, but it's actually been done in experiments.

Speaker 1 So Turing was the first to come up with, Alan Turing, the mathematician, with, you potentially this is the consequences of this.

Speaker 1 I mean, actually, anyone who's a Doctor Who fan will know that this is the key to the weeping angels, which provided you keep on looking at them, are these statues which don't move.

Speaker 1 But you look away and then they start moving. So in a way, Zeno is saying, you know, I'm looking at this thing, it's not moving, but if I look away, maybe the arrow comes towards me.

Speaker 7 Where's Zeno in all this, Clambra?

Speaker 3 Well, I mean, Xeno re-emerges with this paradox. So people again have jumped on, you know, the name Zeno because they think there's something similar going on, a similar motivation.

Speaker 3 But I think what that brings up, this

Speaker 3 example that Marcus just gave us, is that we have to ask whether on the quantum level motion works in the very same way as it does on the

Speaker 3 bigger level, so to say, when we move, right? When we move, we think we can talk about continuous motion.

Speaker 3 And there the question is, well, couldn't it, how do we really explain them getting from one point to the next? Isn't there more to motion?

Speaker 3 But on the quantum level, it seems that there is this discontinuous jumps, if you want, right? So motion may work completely different on this level.

Speaker 7 James, you want to come here?

Speaker 2 Yeah, I just wanted to point out that there was an ancient set of quantum theorists who did indeed react to Zeno in an interesting way. So

Speaker 2 you can tell a reasonable and plausible story that says ancient atomist theory emerged as a response to Zenonian paradoxes.

Speaker 7 This is in the fourth century BCE.

Speaker 2 Yeah, towards the end of the fifth and

Speaker 2 going forward. And so, because what they do, when they're faced with the paradoxes, as we've set them out, is they deny the premise that division can carry on endlessly.
That they say,

Speaker 2 there isn't, in fact, an endless series of journeys that I need to make across the room. There's a very large number of them, but eventually you'll get to a point where you can't divide any further.

Speaker 2 And you have an indivisible but extended

Speaker 2 space

Speaker 2 which you can't cross only a half of. So once you get going,

Speaker 2 then Zeno's conclusion doesn't follow.

Speaker 3 I mean, one thing that shows, I think, is that Zeno's paradoxes were extremely fruitful because they sparked the following natural philosophers to come up with some solutions.

Speaker 3 So the atomic solution to say, well, there are indivisible minima, and we can't go on dividing infinitely is one. Aristotle is another.

Speaker 3 All natural philosophers after Zeno, in some way or other, had to find a way to deal with them if they wanted to do natural philosophy.

Speaker 7 Can you see an overarching system, overarching system, in Zeno's paradoxes? I mean, Parmenides, a simple overarching system, nothing moves.

Speaker 7 Let's keep him in mind, because he's the starting point. Zeno, by defending Parmenides, his tutor and his friend, posited the opposite and then tried to destroy the opposite.

Speaker 7 That was his method of doing it.

Speaker 2 I think the way to put it is that what the paradoxes show is that

Speaker 2 the assumption that there are many things and that things move is no less fraught with difficulty and no less absurd than thinking that there's only one thing and it doesn't move.

Speaker 2 So it's difficult to assert a kind of systematic approach to the paradoxes. We have a set of them that seem to deny motion on various counts.

Speaker 2 There are some that seem to be attacking the bare notion of plurality in various ways.

Speaker 2 And I think it would be hard to think that there was some kind of overarching point to them. And that's in a way why they're kind of fruitful, because

Speaker 2 he isn't offering a particular worldview, I think. What he's doing is raising problems for a very, very general set of assumptions.
You don't need to be an Aristotle to be bothered by Zeno.

Speaker 2 You don't need to have a very specific physical outlook to be bothered by Zeno. The premises that he starts with are extremely general and very common.

Speaker 7 Is there any way in which we can start to characterise how these ideas are in play now, Marcus? Very much.

Speaker 7 We've talked about Leibniz and Newton being exercised by them. Bertrand Russell was with his set of sets and so on and so forth.
But

Speaker 7 you talk in your notes about how these are in play now at a very deep level.

Speaker 1 I think the idea of a paradox is still very much used today to tease out and challenge our view of reality.

Speaker 1 Certainly when we're getting down onto the quantum level or the cosmic level, our intuition is generally quite wrong and the idea of paradox is quite important in just saying, look, there's still something to sort out here.

Speaker 1 And I think, you know, Zeno, we've talked about quantum physics and the fact that actually the universe may be made out of bits.

Speaker 1 There may be a shortest distance that you can go, and you can't divide that. Quantum means bitty.
And even time, there is a challenge now that time is quantized and comes in bits.

Speaker 1 And so I think this idea of, I mean, infinitely many tasks, that was what the kind of challenge at the heart of trying to overtake the tortoise is to do infinitely many things.

Speaker 1 And there have been sort of more recent challenges. Okay, is that physically possible in our universe?

Speaker 1 Actually, is our universe, as the ancient Pythagoreans thought, very finite in its nature and made up of, you know, doesn't have sort of infinite decimals in its kind of makeup.

Speaker 1 And so there are these new challenges called supertasks. Can you switch on a light off, on and off, on and off, and half the time between your switching the light on and off?

Speaker 1 And if you do that in one minute, is the light on or off at the end of this? And it doesn't seem to make sense. So sort of challenges, can you do infinitely many sort of

Speaker 1 discrete actions?

Speaker 1 I guess the point about Achilles is that it's a continuous and you can join them up but if you have these discrete things of switching a light on after half a minute, off after a quarter of a minute, on after an eighth of a minute, off after a sixteenth a minute,

Speaker 1 is that actually ever physically going to be possible? And what is the end result at the, when you add all of these up at the minute, whether is the light on or off?

Speaker 1 So these paradoxes are still very relevant today in teasing out just the nature of reality and

Speaker 1 our intuition about it.

Speaker 7 Before we leave this, it might be, I'd like to sort of tip a bow to Parmenides, who, as you were,

Speaker 7 asked briefly, has his idea of the world not moving got any traction at all?

Speaker 3 There are some people in philosophy who have now gone back to some form of monism, right? Who say, well, at the deep level of reality, there is just one thing.

Speaker 3 Some people think it is just a question of dependence. So everything depends on there being one thing, namely the universe.

Speaker 3 Other people think, no, it's just in general there's just one thing.

Speaker 3 It sounds funny and monism I think is not something that is immediately very attractive to the common sense. But some philosophers

Speaker 3 in Johnson Schaefer, Michael Dela Rockefeller and so on, contemporary philosophers have gone back to that and said, okay, metaphysically, it does make sense to say at the ultimate level there is only one thing even though it contains so to say everything.

Speaker 7 What does he mean one thing, one source of energy, one type of energy? Is that what we mean by the terms? Well you think one of the problems we haven't addressed is what's a thing?

Speaker 3 Okay, that's a very good question. What is a thing? Right.

Speaker 3 So it it's also not clear that with Parmenides whether he really thought about the universe or whether he wouldn't think about something that's completely non-physical, right?

Speaker 3 For him it was very important that we don't get into contradictions. So the only thing that we can think of i is just one thing, something that has no no differences, no distinctions, no extension.

Speaker 3 That doesn't quite sound physical to us, right? So for him, it seemed to be something logical.

Speaker 7 Do you want to add to that, James?

Speaker 2 No, no, no.

Speaker 2 I think Barbara's captured that rather well.

Speaker 2 There is an attraction to monism in the sense that it would be nice to be able to find a simple explanation. Simplicity is something that

Speaker 2 natural scientists look for. And what could be more simple than there really only being one thing at all? That sounds like a perfect end result.

Speaker 7 And finally, Marcus.

Speaker 1 Well, I think mathematicians think they've sorted these paradoxes out, and the invention of infinite series and the calculus gives us a way to explain them.

Speaker 1 But I think actually there's still the challenge of what mathematics is really describing reality.

Speaker 7 Thank you very much. I enjoyed that.
Nice to be back. Barbara Satler, Marcus Yisoto, James Warren.
Thank you very much. Next week, it's Four Legs Good, Two Legs Bad.

Speaker 7 Yes, we'll be talking about Animal Farm by George Orwell.

Speaker 9 And the In Our Time podcast gets some extra time now with a few minutes of bonus material from Melvin and his guests.

Speaker 7 What did we not talk about that we should have talked about?

Speaker 2 Well,

Speaker 2 there are the paradoxes of plurality that

Speaker 2 we didn't discuss.

Speaker 2 But they were sort of in the offing when you were asking Barbara what a thing is.

Speaker 2 Exactly.

Speaker 3 That's where I thought, should I not jump on, but you wanted to talk about permenida.

Speaker 2 And those are very

Speaker 3 peculiar. They ask the question basically, what makes one thing one thing, right? And how can we be sure that these two things here, for instance, are two different things?

Speaker 3 Well, because there's another thing in between, just air, okay, but how can we be sure that air is different from that one?

Speaker 3 And they are, in some sense, less attractive, but they've raised this important question, what makes a thing a thing?

Speaker 1 Yeah, I think it also relates to the tension that was raised about how can points, infinitely many points, make a line, because a point has no distance.

Speaker 1 So if you add something which has zero distance to something which has zero distance, it's still got zero distance.

Speaker 1 And that was the real challenge of.

Speaker 1 And you mentioned Cantor, I'm glad you got that in, because Cantor, around that time you're understanding the idea of the continuum there are different sorts of infinity so actually if you take an uncountable number of points it can have measured the idea that infinitely many points with no size can actually be put together to make something with size I mean and that was a real challenge of 19th century mathematics to come up with a way of understanding that ability to measure and make you know a ruler is made up of distances square root of two pi and something like that but these numbers have no distance it's It's really like the arrow, Zeno's arrow.

Speaker 1 So, how can you have all of these numbers actually make up

Speaker 2 a root?

Speaker 2 That's, I think, very similar to

Speaker 2 Zeno's millet seed paradox, right? Which is different again from the how many grains make a heap paradox. But it's saying, well, if I drop a single seed, it doesn't make a noise.

Speaker 7 Well, I think it does, you see.

Speaker 7 I just, we can't hear it, that's all. And maybe an ant can hear hear it.
There goes a millet seed. I'll pop across and see.

Speaker 2 But there's a difference between it

Speaker 2 disturbing the air and it making a noise. But it's clear that a large number of millet seeds can be dropped.

Speaker 7 Can I rest on mine for a moment? I know I'm outgone completely, but still.

Speaker 7 If it drops, in my view, it'll make a sound. And the sound can be heard by those with ears to hear.
Okay. Maybe an ant.
That may be what he's been waiting for since he woke up.

Speaker 2 This is breakfast. This is the millet seed that he wants.
So half a millet seed then.

Speaker 2 Or a half a half a million.

Speaker 1 The interesting thing is you're getting to quantum physics now because

Speaker 2 big ears.

Speaker 1 But this is what Einstein won his Nobel Prize for essentially is to understand that

Speaker 1 there are actually thresholds

Speaker 1 below which you cannot activate things.

Speaker 1 And so this idea of infinitely dividing something was a real challenge. The quantum world says, no, well you can't just keep on lowering the amplitude of a sound wave.

Speaker 1 At some point, it flatlines and there's a gap. And that quantum gap, it's the Planck constant.
So I think that that's why all of these paradoxes are really still very relevant today.

Speaker 3 I mean, it's this vagueness paradoxes that we talked about in the beginning, about the boldness and a heap of grains.

Speaker 3 And in some sense, it also falls in there, saying, can we specify, you know, from three grains onwards, we can hear it or not? And then quantum physics tells us, yeah, we can.

Speaker 3 But with many other concepts, it seems just arbitrary to say, you know, okay, from five hair onwards.

Speaker 1 I was was interested you chose that as a paradox, actually, because I thought you were going to, as soon as you mentioned hair, I thought you were going to go for something like the barber,

Speaker 2 the barber who only

Speaker 1 realised that that's, but I'm glad you didn't choose that because I feel that's just a paradox of language in the sense that this thing cannot exist. But I suppose that that's the point.

Speaker 1 You're trying to show that there can't be a barber who only shaves those who don't shave themselves.

Speaker 2 The paradox is a barber.

Speaker 1 Yeah, the paradox is resolved by saying this person does not exist. Your hypothesis that there is such a thing.

Speaker 2 So I think that's

Speaker 1 a redocto ad absurdum.

Speaker 3 But I think the important thing was there's kind of two different ways in which paradoxes are.

Speaker 7 The producer comes to make the most of the money.

Speaker 2 Sorry to interrupt. Who'd like tea or half a cup of tea?

Speaker 2 Oh, this will run and run.

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