Confusing, infuriating, and frustrating – these words are often used to describe the paradoxes of this ancient Greek philosopher. To this day, people continue to be baffled by Zeno of Elea's contradictory statements.
In this episode, we'll explore three of the most famous of Zeno's paradoxes, and their legacy on the world of philosophy and mathematics.
[00:00:00] Hello, hello hello, and welcome to English Learning for Curious Minds, by Leonardo English.
[00:00:11] The show where you can listen to fascinating stories, and learn weird and wonderful things about the world at the same time as improving your English.
[00:00:20] I'm Alastair Budge, and today we are going to be talking about someone called Zeno.
[00:00:26] Or rather, we aren’t going to talk about the man, so much, but his paradoxes, his contradictory statements.
[00:00:35] These are statements and riddles that have puzzled philosophers and thinkers for going on three thousand years.
[00:00:42] They puzzle me, they will probably puzzle you, and no doubt they will continue to puzzle people for thousands of years to come.
[00:00:50] So, let’s not waste a minute, and talk about the paradoxes of Zeno.
[00:00:57] Let me start with a simple statement.
[00:01:01] If you meet a person on the street and their only words to you are “I am lying, I am not telling the truth”, what do you think?
[00:01:13] Well, you probably think it’s a slightly odd thing for a stranger to say, but when you dig a little deeper and you really think about it, you realise that it is a problematic statement.
[00:01:27] If someone says “I am not telling the truth”, then their statement is both true and false, they are both telling the truth and they are lying.
[00:01:38] If their statement is true, that they are not telling the truth, then they are also telling the truth.
[00:01:46] Similarly, if their statement is false, and they are lying about not telling the truth, then they are telling the truth, which contradicts their statement.
[00:01:57] Confusing, right?
[00:01:59] This is perhaps the best-known paradox, and it’s called the “Liar’s Paradox”.
[00:02:06] It is infuriating and frustrating on many levels, because it seems ridiculous, illogical, it doesn’t make sense.
[00:02:15] But on another level it makes perfect sense, it simply outlines how logic can be used to defy common sense.
[00:02:24] And this is the crux, the most important part, of a paradox. You can use logic and theory to prove that something is the case, even if we know that it is not.
[00:02:37] The grandfather of the paradox, even though he wasn't behind The Liar's Paradox, was a man called Zeno, Zeno of Elea.
[00:02:47] Now, we don’t know a huge amount about the life of Zeno, and what is known, or believed to be known about the man is far less interesting than his paradoxes.
[00:02:59] He was a Greek philosopher, and lived in the 5th century BC, in the ancient city of Elea.
[00:03:06] Confusingly, perhaps, Elea was in what we call Ancient Greece, but it's now in Italy, in the province of Campania, just south of Naples.
[00:03:16] This isn’t a paradox. It’s easy to understand, as back in the 5th century BC, this area was populated and controlled by the Greeks.
[00:03:26] Anyway, this was where Zeno was born, grew up, and created the paradoxes he would go down in history for.
[00:03:35] So, to the paradoxes themselves.
[00:03:38] Historians believe that there may have been up to 40 different paradoxes, but only 9 survive to this day. They are fascinating and infuriating in equal measure. We don't have time to talk about them all today, but we'll talk about three of his most famous ones, and if you find this topic interesting, well there is a lot of material all over the Internet on the subject.
[00:04:05] So, the first of Zeno's paradoxes that I'd like to talk about today is one I remember being told about by a Maths teacher when I was about 14, and being completely struck by it.
[00:04:18] This paradox is called Achilles and The Tortoise.
[00:04:23] Achilles, if you remember your Greek history, is a Greek warrior who can, amongst other things, run very fast.
[00:04:32] And a tortoise is the animal with a shell on its back that is known for walking very slowly.
[00:04:39] So, picture this situation. There is a race between Achilles and a tortoise, something very fast and something very slow.
[00:04:49] Because the tortoise is very slow, it is given a head start, an advantage. Achilles allows it to start 100 metres ahead of him.
[00:05:00] The race is to a tree that is 100 metres in front of the tortoise.
[00:05:06] So, you have Achilles at point A, the tortoise at point B, which is 100 metres ahead of Achilles, and you have the finish line 100 metres ahead of the tortoise and 200 metres ahead of Achilles.
[00:05:22] The tortoise only needs to travel 100 metres to the finish line whereas Achilles needs to travel 200 metres - 100 metres to catch up to where the tortoise started, then the remaining 100 metres to the finish line.
[00:05:38] There is a countdown, 3, 2, 1, bang goes the firing pistol and both Achilles and the tortoise set off at the same time, running as fast as they can to reach the finish line.
[00:05:53] Who would win the race?
[00:05:55] Well, Achilles would, of course. You don't need to be a great sports commentator or to know the top speed of a tortoise to know that.
[00:06:04] But, at least according to Zeno, you can logically prove that Achilles will never be able to overtake the tortoise, therefore it will be impossible for him to win the race.
[00:06:17] How, you might be thinking?
[00:06:19] Well, remember that the tortoise started 100 metres before Achilles. Assuming that the tortoise travels at a constant speed, and does not stop, by the time Achilles has reached the point where the tortoise started, the tortoise will have moved on, and will be at a different place.
[00:06:41] Let’s say, for sake of ease, it takes Achilles 10 seconds to run 100 metres, he travels at 10 metres per second.
[00:06:50] And the tortoise is much slower. Let’s say it takes him 10 seconds to go 1 metres.
[00:06:58] After 10 seconds, where is Achilles and where is the Tortoise?
[00:07:04] Achilles after 10 seconds is at the position the tortoise started, and the tortoise is 1 metre ahead of him.
[00:07:12] You could say, well this is ridiculous, in another 10 seconds Achilles will have reached the tree and the tortoise will only have moved 2 metres from where it started.
[00:07:23] But here comes the paradox.
[00:07:26] You can use the same logic as we used for Achilles to catch up to where the tortoise started to logically prove that Achilles will never be able to overtake the tortoise, he’ll never be able to go past it.
[00:07:41] Achilles will continue to gain ground on the tortoise, he will continue to close the distance between him and the tortoise, but in the time it takes for Achilles to reach the point where the tortoise was, the tortoise will have moved on.
[00:07:58] If this is a strange thing to get your head around, to understand, just imagine that the distances are much larger.
[00:08:06] Let’s say there is a 10km race, and Achilles gives the tortoise a 9km head start, so Achilles needs to travel 10km and the tortoise needs to travel 1km. It might take 60 minutes for Achilles to run 9km, by which time the tortoise might have moved 100 metres, let’s say. Then it might take Achilles 30 seconds to run 100 metres, and in that 30 seconds the tortoise might have moved 1 metre.
[00:08:36] Then it might take Achilles 0.1 seconds to run 1 metre, and in that 0.1 seconds the tortoise will have moved forward 0.1 centimetres.
[00:08:48] Then in that 0.1 seconds the tortoise will have moved forward 0.1 millimetres.
[00:08:54] So the distances keep on getting smaller and smaller, but logically Achilles can never overtake the tortoise, because in the period of time that he has run to catch up with the tortoise, the tortoise has continued.
[00:09:07] Obviously, this seems ridiculous, because we know that in the real world Achilles would be able to overtake the tortoise, our real world experience proves that.
[00:09:20] And of course Zeno knew this too, he simply wanted to show how logic and mathematics sometimes give results that contradict what we know to be true.
[00:09:32] Let me tell you another one, which is very similar to the first, in that they both rely on distances getting smaller and smaller to an infinitesimal size.
[00:09:43] This one is called the Dichotomy Paradox.
[00:09:47] To introduce it, let me start with what again might seem like a ridiculous question.
[00:09:54] Let's say you are standing 100 metres away from your friend and you walk towards them in a straight line, you'd get there, right?
[00:10:03] Of course you would.
[00:10:05] Zeno, however, has a paradox that logically and mathematically proves that this is impossible, and that all motion and movement is an illusion.
[00:10:18] Let me explain.
[00:10:20] You are standing on the ground and your friend is 100 metres away.
[00:10:25] To reach your friend you first need to get halfway there.
[00:10:30] Then, you need to travel half of the remaining distance, so the remaining ¼ of the way.
[00:10:36] Then you need to travel half of this remaining ¼, so ⅛.
[00:10:40] You are currently ⅞ of the way there, so you’re almost there. But first you need to travel half of the remaining 1/8, so 1/16.
[00:10:51] And you keep on going, to the smallest imaginable distance.
[00:10:56] And what was that smallest possible distance? For Zeno there was no answer to this, because numbers and distances could be divided infinitely. There is no “final” distance, because every distance every number can be divided again.
[00:11:15] So, to go back to the problem of you walking 100 metres, or any distance for that matter, in a straight line to meet your friend.
[00:11:24] First you need to travel half the distance, then you need to travel half the remaining distance, so a quarter, then you need to travel half of the remaining distance, so an eighth. And then you need to travel half of that, so one sixteenth.
[00:11:39] And because there is no end to how small these numbers can be divided, there is an infinite amount of distances to be travelled. And it is impossible to travel an infinite distance, therefore all movement and motion is impossible.
[00:11:56] There is even another, flipped, version of this paradox, which has the same idea of someone needing to travel a fixed distance, but before they get to the halfway point between them and their destination, they need to travel half of that distance, so a quarter, and before they get to a quarter they need to get to the halfway point of a quarter, so an eighth, and so on.
[00:12:20] Because there are an infinite number of distances to travel to even start the journey, the logical conclusion is that it is impossible to even start, and that all movement must be an illusion.
[00:12:34] Now, our eyes and our reality tell us that this cannot be true; movement is not an illusion.
[00:12:42] And in fact, this paradox has baffled and confused mathematicians and philosophers for millennia. There are now some solutions to it, namely that it is possible to add up an infinite number of things and get a finite answer, thereby proving that logic does match our real world expectations.
[00:13:03] One way of solving this, and by proving that it’s possible to add up an infinite number of things and get a finite answer, is by imagining an object. In some examples a square is used, in others it's a cake.
[00:13:20] First, let’s take the example of a square.
[00:13:24] Let’s say this square is 1 metre by 1 metre, so it has an area of 1 metre squared. If you divide it in two, then two again, then two again, then two again, and keep on dividing it in two to infinity, Zeno’s dichotomy paradox would suggest that the number is infinity, the square has an area of infinity.
[00:13:48] But we know that the total area of all of the divided squares, all the space in the square, equals 1 metre squared, which proves that you can add up an infinite number of measurements and still receive a finite answer.
[00:14:05] And if you prefer to think about the example of a cake, well imagine cutting half a cake, then the remaining quarter, then an eight, a sixteenth, and so on. You know that there is a finite amount of cake, no matter how much you keep on cutting.
[00:14:21] Now, perhaps even more mind-bending and unusual is The Arrow Paradox.
[00:14:29] For this, Zeno asks us to imagine an arrow flying through the air. If you don’t like the idea of an arrow, you can imagine anything - a stone, an aeroplane, a cat being catapulted through the sky. I’ll continue to use the example of the arrow, because that’s what Zeno used.
[00:14:49] Now, think of the idea of time. Time is only a series of moments, fixed periods in time, of “nows”.
[00:15:00] In these moments, the arrow is fixed in its position, it doesn’t move. And as time consists only of an infinite number of moments, of nows, in which an arrow doesn’t move, logically an arrow cannot move. Movement is impossible.
[00:15:20] If this is a bit difficult to get your head around, let me give you an example that I think is helpful. Zeno lived 2,500 years ago, so he didn’t have this example, but imagine a photograph of an arrow in the air.
[00:15:36] At that exact moment in time, if that moment in time is zero seconds, the arrow is still, it isn’t moving. The photograph captures it at exactly where it was at that point in time, not moving.
[00:15:53] So, to continue the logical argument, if time consists only of these moments in time where the arrow isn’t moving, logically the arrow cannot move through the air.
[00:16:06] Again, this defies common sense, it goes against what we know to be true in the real world.
[00:16:14] For philosophers and mathematicians, this was clearly frustrating. If logic and maths are to be a way of explaining what we know to be true, how is it that they can also be used to prove that something we know to be true is in fact false?
[00:16:32] As such, mathematicians and philosophers have spent literally thousands of years attempting to disprove the paradoxes of Zeno.
[00:16:42] Aristotle interrogated the paradoxes several hundred years after Zeno created them. The Italian scholar Thomas Acquinas tried in the 13th century.
[00:16:51] The French philosopher Jean-Paul Sartre and the English philosopher and mathematician Bertrand Russel gave it their best attempt in the 20th century.
[00:17:00] And people are still trying.
[00:17:04] And this is the legacy of Zeno and his paradoxes.
[00:17:08] He was the first Western philosopher to really grapple with the concept of mathematical infinity, a concept that is incredibly important in modern mathematics.
[00:17:19] When he created these paradoxes, he most likely thought of them as a fun thought exercise, a fun and interesting way of testing logic and reason against what we know to be true in the real world.
[00:17:33] In all probability he had little idea that these paradoxes would continue to baffle and confuse mathematicians and philosophers, to this very day.
[00:17:42] And if the past two and a half thousand years are anything to go by, people will continue to be baffled by them for many thousands of years to come Ok then, that is it for this little exploration of the Paradoxes of Zeno, one of the first Western philosophers to have grappled with the concept of infinity, and get people asking all sorts of questions about what is real and what is not.
[00:18:10] As always, I would love to know what you thought about this episode.
[00:18:14] Did you know about Zeno before?
[00:18:16] Have you ever studied any of these paradoxes?
[00:18:19] What other kinds of paradoxes do you know about?
[00:18:22] I would love to know, so let’s get this discussion started.
[00:18:26] You can head right into our community forum, which is at community.leonardoenglish.com and get chatting away to other curious minds.
[00:18:35] You've been listening to English Learning for Curious Minds, by Leonardo English.
[00:18:40] I'm Alastair Budge, you stay safe, and I'll catch you in the next episode.
[END OF EPISODE]
[00:00:00] Hello, hello hello, and welcome to English Learning for Curious Minds, by Leonardo English.
[00:00:11] The show where you can listen to fascinating stories, and learn weird and wonderful things about the world at the same time as improving your English.
[00:00:20] I'm Alastair Budge, and today we are going to be talking about someone called Zeno.
[00:00:26] Or rather, we aren’t going to talk about the man, so much, but his paradoxes, his contradictory statements.
[00:00:35] These are statements and riddles that have puzzled philosophers and thinkers for going on three thousand years.
[00:00:42] They puzzle me, they will probably puzzle you, and no doubt they will continue to puzzle people for thousands of years to come.
[00:00:50] So, let’s not waste a minute, and talk about the paradoxes of Zeno.
[00:00:57] Let me start with a simple statement.
[00:01:01] If you meet a person on the street and their only words to you are “I am lying, I am not telling the truth”, what do you think?
[00:01:13] Well, you probably think it’s a slightly odd thing for a stranger to say, but when you dig a little deeper and you really think about it, you realise that it is a problematic statement.
[00:01:27] If someone says “I am not telling the truth”, then their statement is both true and false, they are both telling the truth and they are lying.
[00:01:38] If their statement is true, that they are not telling the truth, then they are also telling the truth.
[00:01:46] Similarly, if their statement is false, and they are lying about not telling the truth, then they are telling the truth, which contradicts their statement.
[00:01:57] Confusing, right?
[00:01:59] This is perhaps the best-known paradox, and it’s called the “Liar’s Paradox”.
[00:02:06] It is infuriating and frustrating on many levels, because it seems ridiculous, illogical, it doesn’t make sense.
[00:02:15] But on another level it makes perfect sense, it simply outlines how logic can be used to defy common sense.
[00:02:24] And this is the crux, the most important part, of a paradox. You can use logic and theory to prove that something is the case, even if we know that it is not.
[00:02:37] The grandfather of the paradox, even though he wasn't behind The Liar's Paradox, was a man called Zeno, Zeno of Elea.
[00:02:47] Now, we don’t know a huge amount about the life of Zeno, and what is known, or believed to be known about the man is far less interesting than his paradoxes.
[00:02:59] He was a Greek philosopher, and lived in the 5th century BC, in the ancient city of Elea.
[00:03:06] Confusingly, perhaps, Elea was in what we call Ancient Greece, but it's now in Italy, in the province of Campania, just south of Naples.
[00:03:16] This isn’t a paradox. It’s easy to understand, as back in the 5th century BC, this area was populated and controlled by the Greeks.
[00:03:26] Anyway, this was where Zeno was born, grew up, and created the paradoxes he would go down in history for.
[00:03:35] So, to the paradoxes themselves.
[00:03:38] Historians believe that there may have been up to 40 different paradoxes, but only 9 survive to this day. They are fascinating and infuriating in equal measure. We don't have time to talk about them all today, but we'll talk about three of his most famous ones, and if you find this topic interesting, well there is a lot of material all over the Internet on the subject.
[00:04:05] So, the first of Zeno's paradoxes that I'd like to talk about today is one I remember being told about by a Maths teacher when I was about 14, and being completely struck by it.
[00:04:18] This paradox is called Achilles and The Tortoise.
[00:04:23] Achilles, if you remember your Greek history, is a Greek warrior who can, amongst other things, run very fast.
[00:04:32] And a tortoise is the animal with a shell on its back that is known for walking very slowly.
[00:04:39] So, picture this situation. There is a race between Achilles and a tortoise, something very fast and something very slow.
[00:04:49] Because the tortoise is very slow, it is given a head start, an advantage. Achilles allows it to start 100 metres ahead of him.
[00:05:00] The race is to a tree that is 100 metres in front of the tortoise.
[00:05:06] So, you have Achilles at point A, the tortoise at point B, which is 100 metres ahead of Achilles, and you have the finish line 100 metres ahead of the tortoise and 200 metres ahead of Achilles.
[00:05:22] The tortoise only needs to travel 100 metres to the finish line whereas Achilles needs to travel 200 metres - 100 metres to catch up to where the tortoise started, then the remaining 100 metres to the finish line.
[00:05:38] There is a countdown, 3, 2, 1, bang goes the firing pistol and both Achilles and the tortoise set off at the same time, running as fast as they can to reach the finish line.
[00:05:53] Who would win the race?
[00:05:55] Well, Achilles would, of course. You don't need to be a great sports commentator or to know the top speed of a tortoise to know that.
[00:06:04] But, at least according to Zeno, you can logically prove that Achilles will never be able to overtake the tortoise, therefore it will be impossible for him to win the race.
[00:06:17] How, you might be thinking?
[00:06:19] Well, remember that the tortoise started 100 metres before Achilles. Assuming that the tortoise travels at a constant speed, and does not stop, by the time Achilles has reached the point where the tortoise started, the tortoise will have moved on, and will be at a different place.
[00:06:41] Let’s say, for sake of ease, it takes Achilles 10 seconds to run 100 metres, he travels at 10 metres per second.
[00:06:50] And the tortoise is much slower. Let’s say it takes him 10 seconds to go 1 metres.
[00:06:58] After 10 seconds, where is Achilles and where is the Tortoise?
[00:07:04] Achilles after 10 seconds is at the position the tortoise started, and the tortoise is 1 metre ahead of him.
[00:07:12] You could say, well this is ridiculous, in another 10 seconds Achilles will have reached the tree and the tortoise will only have moved 2 metres from where it started.
[00:07:23] But here comes the paradox.
[00:07:26] You can use the same logic as we used for Achilles to catch up to where the tortoise started to logically prove that Achilles will never be able to overtake the tortoise, he’ll never be able to go past it.
[00:07:41] Achilles will continue to gain ground on the tortoise, he will continue to close the distance between him and the tortoise, but in the time it takes for Achilles to reach the point where the tortoise was, the tortoise will have moved on.
[00:07:58] If this is a strange thing to get your head around, to understand, just imagine that the distances are much larger.
[00:08:06] Let’s say there is a 10km race, and Achilles gives the tortoise a 9km head start, so Achilles needs to travel 10km and the tortoise needs to travel 1km. It might take 60 minutes for Achilles to run 9km, by which time the tortoise might have moved 100 metres, let’s say. Then it might take Achilles 30 seconds to run 100 metres, and in that 30 seconds the tortoise might have moved 1 metre.
[00:08:36] Then it might take Achilles 0.1 seconds to run 1 metre, and in that 0.1 seconds the tortoise will have moved forward 0.1 centimetres.
[00:08:48] Then in that 0.1 seconds the tortoise will have moved forward 0.1 millimetres.
[00:08:54] So the distances keep on getting smaller and smaller, but logically Achilles can never overtake the tortoise, because in the period of time that he has run to catch up with the tortoise, the tortoise has continued.
[00:09:07] Obviously, this seems ridiculous, because we know that in the real world Achilles would be able to overtake the tortoise, our real world experience proves that.
[00:09:20] And of course Zeno knew this too, he simply wanted to show how logic and mathematics sometimes give results that contradict what we know to be true.
[00:09:32] Let me tell you another one, which is very similar to the first, in that they both rely on distances getting smaller and smaller to an infinitesimal size.
[00:09:43] This one is called the Dichotomy Paradox.
[00:09:47] To introduce it, let me start with what again might seem like a ridiculous question.
[00:09:54] Let's say you are standing 100 metres away from your friend and you walk towards them in a straight line, you'd get there, right?
[00:10:03] Of course you would.
[00:10:05] Zeno, however, has a paradox that logically and mathematically proves that this is impossible, and that all motion and movement is an illusion.
[00:10:18] Let me explain.
[00:10:20] You are standing on the ground and your friend is 100 metres away.
[00:10:25] To reach your friend you first need to get halfway there.
[00:10:30] Then, you need to travel half of the remaining distance, so the remaining ¼ of the way.
[00:10:36] Then you need to travel half of this remaining ¼, so ⅛.
[00:10:40] You are currently ⅞ of the way there, so you’re almost there. But first you need to travel half of the remaining 1/8, so 1/16.
[00:10:51] And you keep on going, to the smallest imaginable distance.
[00:10:56] And what was that smallest possible distance? For Zeno there was no answer to this, because numbers and distances could be divided infinitely. There is no “final” distance, because every distance every number can be divided again.
[00:11:15] So, to go back to the problem of you walking 100 metres, or any distance for that matter, in a straight line to meet your friend.
[00:11:24] First you need to travel half the distance, then you need to travel half the remaining distance, so a quarter, then you need to travel half of the remaining distance, so an eighth. And then you need to travel half of that, so one sixteenth.
[00:11:39] And because there is no end to how small these numbers can be divided, there is an infinite amount of distances to be travelled. And it is impossible to travel an infinite distance, therefore all movement and motion is impossible.
[00:11:56] There is even another, flipped, version of this paradox, which has the same idea of someone needing to travel a fixed distance, but before they get to the halfway point between them and their destination, they need to travel half of that distance, so a quarter, and before they get to a quarter they need to get to the halfway point of a quarter, so an eighth, and so on.
[00:12:20] Because there are an infinite number of distances to travel to even start the journey, the logical conclusion is that it is impossible to even start, and that all movement must be an illusion.
[00:12:34] Now, our eyes and our reality tell us that this cannot be true; movement is not an illusion.
[00:12:42] And in fact, this paradox has baffled and confused mathematicians and philosophers for millennia. There are now some solutions to it, namely that it is possible to add up an infinite number of things and get a finite answer, thereby proving that logic does match our real world expectations.
[00:13:03] One way of solving this, and by proving that it’s possible to add up an infinite number of things and get a finite answer, is by imagining an object. In some examples a square is used, in others it's a cake.
[00:13:20] First, let’s take the example of a square.
[00:13:24] Let’s say this square is 1 metre by 1 metre, so it has an area of 1 metre squared. If you divide it in two, then two again, then two again, then two again, and keep on dividing it in two to infinity, Zeno’s dichotomy paradox would suggest that the number is infinity, the square has an area of infinity.
[00:13:48] But we know that the total area of all of the divided squares, all the space in the square, equals 1 metre squared, which proves that you can add up an infinite number of measurements and still receive a finite answer.
[00:14:05] And if you prefer to think about the example of a cake, well imagine cutting half a cake, then the remaining quarter, then an eight, a sixteenth, and so on. You know that there is a finite amount of cake, no matter how much you keep on cutting.
[00:14:21] Now, perhaps even more mind-bending and unusual is The Arrow Paradox.
[00:14:29] For this, Zeno asks us to imagine an arrow flying through the air. If you don’t like the idea of an arrow, you can imagine anything - a stone, an aeroplane, a cat being catapulted through the sky. I’ll continue to use the example of the arrow, because that’s what Zeno used.
[00:14:49] Now, think of the idea of time. Time is only a series of moments, fixed periods in time, of “nows”.
[00:15:00] In these moments, the arrow is fixed in its position, it doesn’t move. And as time consists only of an infinite number of moments, of nows, in which an arrow doesn’t move, logically an arrow cannot move. Movement is impossible.
[00:15:20] If this is a bit difficult to get your head around, let me give you an example that I think is helpful. Zeno lived 2,500 years ago, so he didn’t have this example, but imagine a photograph of an arrow in the air.
[00:15:36] At that exact moment in time, if that moment in time is zero seconds, the arrow is still, it isn’t moving. The photograph captures it at exactly where it was at that point in time, not moving.
[00:15:53] So, to continue the logical argument, if time consists only of these moments in time where the arrow isn’t moving, logically the arrow cannot move through the air.
[00:16:06] Again, this defies common sense, it goes against what we know to be true in the real world.
[00:16:14] For philosophers and mathematicians, this was clearly frustrating. If logic and maths are to be a way of explaining what we know to be true, how is it that they can also be used to prove that something we know to be true is in fact false?
[00:16:32] As such, mathematicians and philosophers have spent literally thousands of years attempting to disprove the paradoxes of Zeno.
[00:16:42] Aristotle interrogated the paradoxes several hundred years after Zeno created them. The Italian scholar Thomas Acquinas tried in the 13th century.
[00:16:51] The French philosopher Jean-Paul Sartre and the English philosopher and mathematician Bertrand Russel gave it their best attempt in the 20th century.
[00:17:00] And people are still trying.
[00:17:04] And this is the legacy of Zeno and his paradoxes.
[00:17:08] He was the first Western philosopher to really grapple with the concept of mathematical infinity, a concept that is incredibly important in modern mathematics.
[00:17:19] When he created these paradoxes, he most likely thought of them as a fun thought exercise, a fun and interesting way of testing logic and reason against what we know to be true in the real world.
[00:17:33] In all probability he had little idea that these paradoxes would continue to baffle and confuse mathematicians and philosophers, to this very day.
[00:17:42] And if the past two and a half thousand years are anything to go by, people will continue to be baffled by them for many thousands of years to come Ok then, that is it for this little exploration of the Paradoxes of Zeno, one of the first Western philosophers to have grappled with the concept of infinity, and get people asking all sorts of questions about what is real and what is not.
[00:18:10] As always, I would love to know what you thought about this episode.
[00:18:14] Did you know about Zeno before?
[00:18:16] Have you ever studied any of these paradoxes?
[00:18:19] What other kinds of paradoxes do you know about?
[00:18:22] I would love to know, so let’s get this discussion started.
[00:18:26] You can head right into our community forum, which is at community.leonardoenglish.com and get chatting away to other curious minds.
[00:18:35] You've been listening to English Learning for Curious Minds, by Leonardo English.
[00:18:40] I'm Alastair Budge, you stay safe, and I'll catch you in the next episode.
[END OF EPISODE]
[00:00:00] Hello, hello hello, and welcome to English Learning for Curious Minds, by Leonardo English.
[00:00:11] The show where you can listen to fascinating stories, and learn weird and wonderful things about the world at the same time as improving your English.
[00:00:20] I'm Alastair Budge, and today we are going to be talking about someone called Zeno.
[00:00:26] Or rather, we aren’t going to talk about the man, so much, but his paradoxes, his contradictory statements.
[00:00:35] These are statements and riddles that have puzzled philosophers and thinkers for going on three thousand years.
[00:00:42] They puzzle me, they will probably puzzle you, and no doubt they will continue to puzzle people for thousands of years to come.
[00:00:50] So, let’s not waste a minute, and talk about the paradoxes of Zeno.
[00:00:57] Let me start with a simple statement.
[00:01:01] If you meet a person on the street and their only words to you are “I am lying, I am not telling the truth”, what do you think?
[00:01:13] Well, you probably think it’s a slightly odd thing for a stranger to say, but when you dig a little deeper and you really think about it, you realise that it is a problematic statement.
[00:01:27] If someone says “I am not telling the truth”, then their statement is both true and false, they are both telling the truth and they are lying.
[00:01:38] If their statement is true, that they are not telling the truth, then they are also telling the truth.
[00:01:46] Similarly, if their statement is false, and they are lying about not telling the truth, then they are telling the truth, which contradicts their statement.
[00:01:57] Confusing, right?
[00:01:59] This is perhaps the best-known paradox, and it’s called the “Liar’s Paradox”.
[00:02:06] It is infuriating and frustrating on many levels, because it seems ridiculous, illogical, it doesn’t make sense.
[00:02:15] But on another level it makes perfect sense, it simply outlines how logic can be used to defy common sense.
[00:02:24] And this is the crux, the most important part, of a paradox. You can use logic and theory to prove that something is the case, even if we know that it is not.
[00:02:37] The grandfather of the paradox, even though he wasn't behind The Liar's Paradox, was a man called Zeno, Zeno of Elea.
[00:02:47] Now, we don’t know a huge amount about the life of Zeno, and what is known, or believed to be known about the man is far less interesting than his paradoxes.
[00:02:59] He was a Greek philosopher, and lived in the 5th century BC, in the ancient city of Elea.
[00:03:06] Confusingly, perhaps, Elea was in what we call Ancient Greece, but it's now in Italy, in the province of Campania, just south of Naples.
[00:03:16] This isn’t a paradox. It’s easy to understand, as back in the 5th century BC, this area was populated and controlled by the Greeks.
[00:03:26] Anyway, this was where Zeno was born, grew up, and created the paradoxes he would go down in history for.
[00:03:35] So, to the paradoxes themselves.
[00:03:38] Historians believe that there may have been up to 40 different paradoxes, but only 9 survive to this day. They are fascinating and infuriating in equal measure. We don't have time to talk about them all today, but we'll talk about three of his most famous ones, and if you find this topic interesting, well there is a lot of material all over the Internet on the subject.
[00:04:05] So, the first of Zeno's paradoxes that I'd like to talk about today is one I remember being told about by a Maths teacher when I was about 14, and being completely struck by it.
[00:04:18] This paradox is called Achilles and The Tortoise.
[00:04:23] Achilles, if you remember your Greek history, is a Greek warrior who can, amongst other things, run very fast.
[00:04:32] And a tortoise is the animal with a shell on its back that is known for walking very slowly.
[00:04:39] So, picture this situation. There is a race between Achilles and a tortoise, something very fast and something very slow.
[00:04:49] Because the tortoise is very slow, it is given a head start, an advantage. Achilles allows it to start 100 metres ahead of him.
[00:05:00] The race is to a tree that is 100 metres in front of the tortoise.
[00:05:06] So, you have Achilles at point A, the tortoise at point B, which is 100 metres ahead of Achilles, and you have the finish line 100 metres ahead of the tortoise and 200 metres ahead of Achilles.
[00:05:22] The tortoise only needs to travel 100 metres to the finish line whereas Achilles needs to travel 200 metres - 100 metres to catch up to where the tortoise started, then the remaining 100 metres to the finish line.
[00:05:38] There is a countdown, 3, 2, 1, bang goes the firing pistol and both Achilles and the tortoise set off at the same time, running as fast as they can to reach the finish line.
[00:05:53] Who would win the race?
[00:05:55] Well, Achilles would, of course. You don't need to be a great sports commentator or to know the top speed of a tortoise to know that.
[00:06:04] But, at least according to Zeno, you can logically prove that Achilles will never be able to overtake the tortoise, therefore it will be impossible for him to win the race.
[00:06:17] How, you might be thinking?
[00:06:19] Well, remember that the tortoise started 100 metres before Achilles. Assuming that the tortoise travels at a constant speed, and does not stop, by the time Achilles has reached the point where the tortoise started, the tortoise will have moved on, and will be at a different place.
[00:06:41] Let’s say, for sake of ease, it takes Achilles 10 seconds to run 100 metres, he travels at 10 metres per second.
[00:06:50] And the tortoise is much slower. Let’s say it takes him 10 seconds to go 1 metres.
[00:06:58] After 10 seconds, where is Achilles and where is the Tortoise?
[00:07:04] Achilles after 10 seconds is at the position the tortoise started, and the tortoise is 1 metre ahead of him.
[00:07:12] You could say, well this is ridiculous, in another 10 seconds Achilles will have reached the tree and the tortoise will only have moved 2 metres from where it started.
[00:07:23] But here comes the paradox.
[00:07:26] You can use the same logic as we used for Achilles to catch up to where the tortoise started to logically prove that Achilles will never be able to overtake the tortoise, he’ll never be able to go past it.
[00:07:41] Achilles will continue to gain ground on the tortoise, he will continue to close the distance between him and the tortoise, but in the time it takes for Achilles to reach the point where the tortoise was, the tortoise will have moved on.
[00:07:58] If this is a strange thing to get your head around, to understand, just imagine that the distances are much larger.
[00:08:06] Let’s say there is a 10km race, and Achilles gives the tortoise a 9km head start, so Achilles needs to travel 10km and the tortoise needs to travel 1km. It might take 60 minutes for Achilles to run 9km, by which time the tortoise might have moved 100 metres, let’s say. Then it might take Achilles 30 seconds to run 100 metres, and in that 30 seconds the tortoise might have moved 1 metre.
[00:08:36] Then it might take Achilles 0.1 seconds to run 1 metre, and in that 0.1 seconds the tortoise will have moved forward 0.1 centimetres.
[00:08:48] Then in that 0.1 seconds the tortoise will have moved forward 0.1 millimetres.
[00:08:54] So the distances keep on getting smaller and smaller, but logically Achilles can never overtake the tortoise, because in the period of time that he has run to catch up with the tortoise, the tortoise has continued.
[00:09:07] Obviously, this seems ridiculous, because we know that in the real world Achilles would be able to overtake the tortoise, our real world experience proves that.
[00:09:20] And of course Zeno knew this too, he simply wanted to show how logic and mathematics sometimes give results that contradict what we know to be true.
[00:09:32] Let me tell you another one, which is very similar to the first, in that they both rely on distances getting smaller and smaller to an infinitesimal size.
[00:09:43] This one is called the Dichotomy Paradox.
[00:09:47] To introduce it, let me start with what again might seem like a ridiculous question.
[00:09:54] Let's say you are standing 100 metres away from your friend and you walk towards them in a straight line, you'd get there, right?
[00:10:03] Of course you would.
[00:10:05] Zeno, however, has a paradox that logically and mathematically proves that this is impossible, and that all motion and movement is an illusion.
[00:10:18] Let me explain.
[00:10:20] You are standing on the ground and your friend is 100 metres away.
[00:10:25] To reach your friend you first need to get halfway there.
[00:10:30] Then, you need to travel half of the remaining distance, so the remaining ¼ of the way.
[00:10:36] Then you need to travel half of this remaining ¼, so ⅛.
[00:10:40] You are currently ⅞ of the way there, so you’re almost there. But first you need to travel half of the remaining 1/8, so 1/16.
[00:10:51] And you keep on going, to the smallest imaginable distance.
[00:10:56] And what was that smallest possible distance? For Zeno there was no answer to this, because numbers and distances could be divided infinitely. There is no “final” distance, because every distance every number can be divided again.
[00:11:15] So, to go back to the problem of you walking 100 metres, or any distance for that matter, in a straight line to meet your friend.
[00:11:24] First you need to travel half the distance, then you need to travel half the remaining distance, so a quarter, then you need to travel half of the remaining distance, so an eighth. And then you need to travel half of that, so one sixteenth.
[00:11:39] And because there is no end to how small these numbers can be divided, there is an infinite amount of distances to be travelled. And it is impossible to travel an infinite distance, therefore all movement and motion is impossible.
[00:11:56] There is even another, flipped, version of this paradox, which has the same idea of someone needing to travel a fixed distance, but before they get to the halfway point between them and their destination, they need to travel half of that distance, so a quarter, and before they get to a quarter they need to get to the halfway point of a quarter, so an eighth, and so on.
[00:12:20] Because there are an infinite number of distances to travel to even start the journey, the logical conclusion is that it is impossible to even start, and that all movement must be an illusion.
[00:12:34] Now, our eyes and our reality tell us that this cannot be true; movement is not an illusion.
[00:12:42] And in fact, this paradox has baffled and confused mathematicians and philosophers for millennia. There are now some solutions to it, namely that it is possible to add up an infinite number of things and get a finite answer, thereby proving that logic does match our real world expectations.
[00:13:03] One way of solving this, and by proving that it’s possible to add up an infinite number of things and get a finite answer, is by imagining an object. In some examples a square is used, in others it's a cake.
[00:13:20] First, let’s take the example of a square.
[00:13:24] Let’s say this square is 1 metre by 1 metre, so it has an area of 1 metre squared. If you divide it in two, then two again, then two again, then two again, and keep on dividing it in two to infinity, Zeno’s dichotomy paradox would suggest that the number is infinity, the square has an area of infinity.
[00:13:48] But we know that the total area of all of the divided squares, all the space in the square, equals 1 metre squared, which proves that you can add up an infinite number of measurements and still receive a finite answer.
[00:14:05] And if you prefer to think about the example of a cake, well imagine cutting half a cake, then the remaining quarter, then an eight, a sixteenth, and so on. You know that there is a finite amount of cake, no matter how much you keep on cutting.
[00:14:21] Now, perhaps even more mind-bending and unusual is The Arrow Paradox.
[00:14:29] For this, Zeno asks us to imagine an arrow flying through the air. If you don’t like the idea of an arrow, you can imagine anything - a stone, an aeroplane, a cat being catapulted through the sky. I’ll continue to use the example of the arrow, because that’s what Zeno used.
[00:14:49] Now, think of the idea of time. Time is only a series of moments, fixed periods in time, of “nows”.
[00:15:00] In these moments, the arrow is fixed in its position, it doesn’t move. And as time consists only of an infinite number of moments, of nows, in which an arrow doesn’t move, logically an arrow cannot move. Movement is impossible.
[00:15:20] If this is a bit difficult to get your head around, let me give you an example that I think is helpful. Zeno lived 2,500 years ago, so he didn’t have this example, but imagine a photograph of an arrow in the air.
[00:15:36] At that exact moment in time, if that moment in time is zero seconds, the arrow is still, it isn’t moving. The photograph captures it at exactly where it was at that point in time, not moving.
[00:15:53] So, to continue the logical argument, if time consists only of these moments in time where the arrow isn’t moving, logically the arrow cannot move through the air.
[00:16:06] Again, this defies common sense, it goes against what we know to be true in the real world.
[00:16:14] For philosophers and mathematicians, this was clearly frustrating. If logic and maths are to be a way of explaining what we know to be true, how is it that they can also be used to prove that something we know to be true is in fact false?
[00:16:32] As such, mathematicians and philosophers have spent literally thousands of years attempting to disprove the paradoxes of Zeno.
[00:16:42] Aristotle interrogated the paradoxes several hundred years after Zeno created them. The Italian scholar Thomas Acquinas tried in the 13th century.
[00:16:51] The French philosopher Jean-Paul Sartre and the English philosopher and mathematician Bertrand Russel gave it their best attempt in the 20th century.
[00:17:00] And people are still trying.
[00:17:04] And this is the legacy of Zeno and his paradoxes.
[00:17:08] He was the first Western philosopher to really grapple with the concept of mathematical infinity, a concept that is incredibly important in modern mathematics.
[00:17:19] When he created these paradoxes, he most likely thought of them as a fun thought exercise, a fun and interesting way of testing logic and reason against what we know to be true in the real world.
[00:17:33] In all probability he had little idea that these paradoxes would continue to baffle and confuse mathematicians and philosophers, to this very day.
[00:17:42] And if the past two and a half thousand years are anything to go by, people will continue to be baffled by them for many thousands of years to come Ok then, that is it for this little exploration of the Paradoxes of Zeno, one of the first Western philosophers to have grappled with the concept of infinity, and get people asking all sorts of questions about what is real and what is not.
[00:18:10] As always, I would love to know what you thought about this episode.
[00:18:14] Did you know about Zeno before?
[00:18:16] Have you ever studied any of these paradoxes?
[00:18:19] What other kinds of paradoxes do you know about?
[00:18:22] I would love to know, so let’s get this discussion started.
[00:18:26] You can head right into our community forum, which is at community.leonardoenglish.com and get chatting away to other curious minds.
[00:18:35] You've been listening to English Learning for Curious Minds, by Leonardo English.
[00:18:40] I'm Alastair Budge, you stay safe, and I'll catch you in the next episode.
[END OF EPISODE]
