Zeno's Paradoxes (Archive Episode)

Summary

Podcast Summary: In Our Time – Zeno's Paradoxes

Podcast: In Our Time (BBC Radio 4)
Episode: Zeno's Paradoxes (Archive Episode)
Date: November 20, 2025
Host: Melvyn Bragg
Guests: Marcus du Sautoy (Mathematician), Barbara Sattler (Philosopher), James Warren (Ancient Philosophy Scholar)

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Overview

This episode explores the enduring intellectual puzzle of Zeno’s paradoxes, devised in the 5th century BC by the Greek philosopher Zeno of Elea. Melvyn Bragg and his panel of experts dive into the historical, philosophical, and mathematical significance of Zeno’s paradoxes, which question fundamental ideas about motion, plurality, time, and space. Through lively discussion, the episode shows both the paradoxes’ historical impact and their continuing relevance in modern mathematics, philosophy, and even quantum physics.

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Key Discussion Points and Insights

Who Was Zeno? (03:33–05:37)

• Historical Context:

  • Zeno lived in the 5th century BC in Elea, a Greek city in southern Italy.
  • He was associated with Parmenides, his tutor/friend, who posited that reality is a single, changeless entity.
  • Zeno’s paradoxes served to defend Parmenides by demonstrating the absurdities that arise if one assumes multiplicity and motion are real.

  Quote (James Warren, 05:11):
  “Plato writes a dialogue in which the two of them come to Athens, and Zeno is cast as a defender of Parmenides. 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 ... clearly things do move.”

What Is a Paradox in Philosophy? (07:28–10:14)

• Definition:

  • From Greek: "para" (against) + "doxa" (belief/expectation): something that challenges common beliefs.

• Role:

  • Paradoxes are fruitful because they point to limitations or vagueness within our concepts and force a re-examination of assumptions.

• Examples:

  • The "bald man" paradox illustrates vagueness: How many hairs must be lost before a man is bald?

  Quote (Barbara Sattler, 08:23):
  “One paradox that's quite famous is the bald man paradox ... we all would agree that if somebody has no hair, then this person is bald ... but is there an exact moment where I can say it's not a heap any longer? Probably not.”

Mathematics, Infinity, and the Greeks (11:37–15:50)

• Historical Development:

  • Greeks developed mathematics from practical geometry to abstract proofs.
  • The Pythagoreans shifted mathematical thought from calculation to logical proof.

• Infinity:

  • The Greeks struggled with whether infinity is real or just a potential abstraction (“potential” vs “actual” infinity).

• Paradox in Proofs:

  • The method of reductio ad absurdum (proof by contradiction) was a major philosophical and mathematical development.

  Quote (Marcus Yasoto, 12:54):
  “The ancient Greeks ... want to produce a proof that something would always work ... 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.”

Zeno’s Paradoxes of Motion (17:43–22:14)

1. Dichotomy Paradox (17:43–18:59)

• Summary:

  • To travel from A to B, you must first reach halfway, then halfway again, ad infinitum.
  • This implies one must complete an infinite series of tasks, which seems impossible.

  Quote (James Warren, 18:17):
  “You’ve got your person to agree that ... in order to cross any spatial extension, that entails an endless series of prior journeys ... it now looks like in order to cross a room, that's asking me to do something impossible.”

2. Achilles and the Tortoise (19:53–22:14)

• Summary:

  • Achilles gives a tortoise a head start. Each time Achilles reaches the point where the tortoise was, the tortoise has moved a bit further. Thus, Achilles can never overtake the tortoise.
  • Demonstrates the problem of summing infinitely many diminishing distances.

  Quote (Barbara Sattler, 21:27):
  "The distance between Achilles and the tortoise will get less and less, but will never get to zero. So it seems that Achilles ... will never be able to overtake the slow tortoise..."

• Common Sense vs Paradox:

  • The panel agrees that while experience refutes the paradox, Zeno seeks an explanation of how overtaking happens without contradiction.

Zeno and the Challenge of Infinity (24:05–27:57)

• Mathematical Resolution:

  • Eventually, mathematicians showed that an infinite series of shrinking intervals can sum to a finite length (e.g., 1/2 + 1/4 + 1/8 ... = 1).
  • The development of calculus and the concept of limits resolved many of Zeno's problems mathematically.

  Quote (Marcus Yasoto, 24:05):
  “We understand this now that he can do infinitely many tasks because it can take him a finite amount of time. This infinite series ... actually adds up ... to the answer one.”

The Arrow Paradox and the Nature of Motion (29:45–35:19)

• The Paradox:

  • An arrow in flight, at any instant, seems motionless because, in a single instant, it does not move.

• Philosophical vs Mathematical Response:

  • Calculus (Newton and Leibniz) allowed defining instantaneous speed, resolving the apparent contradiction for physicists.
  • Philosophers note this assumes motion exists, leaving some underlying conceptual issues unsolved.

• Add Theory & Modern Physics:

  • Bertrand Russell’s “at-at” theory: motion is being at different places at different times.
  • Quantum Zeno Effect: In quantum mechanics, observation can prevent change—a modern echo of Zeno.

  Quote (Marcus Yasoto, 35:19):
  “It’s interesting because actually there's a modern day effect in quantum physics which actually says that motion sort of doesn't happen. It’s called the quantum Zeno effect ... if I keep on observing this, I can actually stop it radiating because it never has a chance to move because of my observations.”

The Legacy and Modern Echoes of Zeno (37:31–42:35)

• Ancient Atomists:
  • Atomism arose as a solution: if matter is ultimately indivisible, there can't be infinite division.
• Philosophical Value:
  • Zeno’s paradoxes don’t offer a new worldview, but instead challenge the very foundations of common sense and philosophy.
• Contemporary Relevance:
  • Paradoxes remain useful in math, science, and philosophy for exposing limits and assumptions.
  • Examples include quantum-level discontinuities, paradoxes in set theory (e.g., Russell’s paradox), and challenges in cosmology.

Is There an Overarching System in Zeno? (39:03–40:15)

• Zeno's method:

  • Not aimed at constructing a positive theory, but at undermining the common sense assumptions behind plurality and motion.

  Quote (James Warren, 39:15):
  “What the paradoxes show is that 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.”

Paradoxes in Modern Thought (Quantum and Cosmology) (40:15–42:35)

• Current Physics:
  • Quantum mechanics and cosmology continually confront paradoxical concepts that echo Zeno’s ancient dilemmas.
  • Issues of continuity, measurement, and the limits of knowledge are not resolved, but recast.

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Notable Quotes & Memorable Moments

• On the Motivation for Paradoxes:

  • “Paradoxes are very important because they tell us, okay, something has gone wrong here. 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.”
    — Barbara Sattler [11:30]

• On Mathematics and Reality:

  • “...there's still the challenge of, with mathematics is really describing reality.”
    — Marcus Yasoto [44:44]

• On Zeno’s Lasting Influence:

  • “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.”
    — Barbara Sattler [38:31]

• On Zeno’s Fruitfulness:

  • “Zeno’s paradoxes are extremely fruitful because they forced the following natural philosophers to come up with some solutions.”
    — Barbara Sattler [38:31]

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Timestamps for Key Segments

• Who was Zeno? 03:33–05:37
• The Nature of Paradox 07:28–10:14
• The Rise of Mathematical Proof and Infinity 11:37–15:50
• The Dichotomy & Achilles Paradoxes 17:43–22:14
• Mathematical Resolutions to Infinity 24:05–27:57
• The Moving Arrow Paradox 29:45–35:19
• Quantum Zeno Effect & Modern Parallels 35:19–38:31
• Overarching System and Philosophical Impact 39:03–40:15
• Relevance to Modern Science & Philosophy 40:15–44:44

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Bonus Material Highlights (45:02–49:10)

• Paradoxes of Plurality:
  • What constitutes a “thing”? How can a line be made from points of zero length?
• Heaps, Vagueness, and Quantum Gaps:
  • The “millet seed” example examines cumulative effects of tiny changes—a precursor to the idea of the quantum threshold.
• Different Types of Paradox:
  • Linguistic (“barber paradox”) vs. conceptual or physical paradoxes.
• The Significance of Paradoxes:
  • They reveal not only puzzles in reasoning but also trigger scientific innovations (calculus, set theory, quantum mechanics).

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Summary

This episode of In Our Time unpacks Zeno’s paradoxes as not just clever puzzles, but profound challenges that have shaped centuries of philosophy, mathematics, and science. It showcases how infinity, motion, and the very nature of reality are still unresolved questions—making Zeno’s ancient thought experiments as relevant now as they were 2,500 years ago. Through accessible examples and engaging debate, the episode underscores the paradoxes’ power to expose the limits and possibilities of human understanding.

Transcript

[James Warren]

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But first, There the last one.

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Enjoy a Coca Cola for a pause that that refreshes.

[Marcus Yasoto]

This is the story of the 1. As head of maintenance at a concert hall, he knows the show must always go on. That's why he works behind the scenes, ensuring every light is working, the H Vac is humming, and his facility shines with Grainger's supplies and solutions for every challenge he faces. Plus 24. 7 customer support. His venue never misses a beat. Call quickgranger.com or or just stop by Granger for the ones who get it done.

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[Marcus Yasoto]

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[Marcus Yasoto]

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[Barbara Sattler]

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[Marcus Yasoto]

Listen on BBC.com or wherever you get your podcasts.

[James Warren]

And now, to mark the end of.

[Marcus Yasoto]

His 27 memorable years presenting in Our.

[James Warren]

Time, we have Melvin Bragg to introduce.

[Marcus Yasoto]

The next in our series of his most cherished episodes.

[Melvin Bragg]

If the title In Our Time works at all is to describe this long period on Earth in in which we humans have tried to make sense of and enjoy the world around us. This is our Time. Who would have thought that Zeno, a Greek philosopher 2 1/2 thousand years ago, even before Socrates, was devising thought experiments that would still be inspiring cutting edge scientists today? That's why we were discussing Paradoxes live at 9am back in 2016, and the audiences loved it. Hello. The ancient Greek thinker Zeno of Elea flourished in the 5th century BC. 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. For example, according to Zeno's paradoxes, motion is not possible. An arrow in flight does not move. The fastest runner in Homer, Achilles could never catch up with the tortoise in a race if he gave it a head start. Philosophers from Aristotle to Bertrand Russell have tried to refute his ideas or explain them with varying success. 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. With me to discuss the paradoxes of Zeno are Marcus Jusotoi, professor of Mathematics and Simonyi, professor for the Public Understanding of Science at the University of Oxford, Barbara Sattler, Lecturer in Philosophy at the University of St. Andrews, and James Warren, Reader in Ancient Philosophy at the University of Cambridge. James Warren, what do we know about Zeno?

[James Warren]

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 he came from Ilia, a town on the west coast of southern Italy. And we know that he traveled a lot in Greece, as people of that sort of. Sort of class did. And he wrote a work, maybe just one work, which included these paradoxes of which we know about. It depends how you count them, perhaps seven, eight, some to do with motion, some to do with plurality. What we can do is put him into some kind of context, intellectual context, that is. So for around.

[Melvin Bragg]

Let's just go into it for a moment or two. Is this a village? Is it a town? Is it known as an intellectual center? What's going on?

[James Warren]

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, it seems to have been quite an intellectual center in particular, because I think one of the most important people in Zeno's intellectual life was again an eleatic with someone from the same city. 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 and that it was changeless and motionless and perfect and so on.

[Melvin Bragg]

When he said there was only one thing, I mean the world was only one thing.

[James Warren]

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.

[Melvin Bragg]

And he was. He was Zeno's tutor.

[James Warren]

Friend, Friend, tutor, something like that. It's not 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. 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.

[Melvin Bragg]

You said quite casually, he wandered over the place of the Greek as people did what did he wander for he go to. How did he look after himself?

[James Warren]

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

[Melvin Bragg]

So we're talking about aristocrats.

[James Warren]

Yes.

[Melvin Bragg]

Well enough off people.

[James Warren]

That's right. And they would travel around and they would often, I think the way in which these ideas were circulated were partly through written books. 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 not even Zeno knows how many books there are. And they traveled to the great festivals like the Athenian Panathenia festival, and they would give demonstrations and public recitations and meet people there.

[Melvin Bragg]

And so he got his teacher, Parmenides. What learning was coming to him through Parmenides. And in the context of the place that time, briefly, what was he reading that influenced him?

[James Warren]

Well, what Parmenides is reacting to is a tradition of cosmological thinking which had been going on for perhaps up to 100 years by now, of people who were attempting to explain the world and how the world worked and functioned, often in terms of identifying some basic principle or element out of which the world was constructed.

[Melvin Bragg]

And like the world was constructed from water.

[James Warren]

From water 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. 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.

[Melvin Bragg]

So natural philosophers of people believe that the world is changing. There are many things in it which are various and unmoving and changing.

[James Warren]

Yes. And that's what Parmenides sets out to show his refute. Grossly mistaken.

[Melvin Bragg]

Yes. Barbara Sutler, what is. What is a paradox in philosophy?

[Barbara Sattler]

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. 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 in a philosophical context. By a paradox, we normally understand that we derive problematic conclusion from sound premises. 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? Well, either because it's inconsistent in itself, it leads to a contradiction, or it contradicts other beliefs, opinions that we hold.

[Melvin Bragg]

Can you Give us a simple paradox. It needn't be one of Zeno's just to get the hang of it.

[Barbara Sattler]

Right? So one paradox that's quite famous is the bald man paradox. So we all would agree that if somebody has no hair, then this person is bald. If this person has one hair, we would still call this person bald. Two hair, probably still bald, three and so on. 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? Is it that from 100 hair onwards we say, oh, this person is not bald, but 99 hair is still bald? That doesn't seem to be right. Right.

[Melvin Bragg]

Why not?

[Barbara Sattler]

Because it seems that with baldness, it's not a concept or notion where we can give a clear quantitative determination. We can't say so. And so many hairs quantify as not being bald. And so. And so many hairs quantify as being bald.

[Melvin Bragg]

But has common sense applies in this?

[Barbara Sattler]

Yeah, this uses common sense that we all agree on certain ideas of boldness and we all have a problem of saying when 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 fussy. We can't really fully determine them. Right? 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 another one, still a heap, 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. Is there an exact moment where I can say it's not a heap any longer? Probably not. 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.

[Melvin Bragg]

I can see it's intriguing and it's a lot of fun. But is there any. Actually, 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 certainly turn out to be running the world, don't they?

[Barbara Sattler]

Okay, so in this case, I think with paradoxes, there's two reasons why they are actually very fruitful for philosophers, right? 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. Either because they show there's something funny about some concepts, like with the bald man, right. They show we are using some concepts that can't be fully determined in the way other concepts can. 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. 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. So our theories and models and concepts very often not just falsified or verified by the world outside. Right. So how do we figure out whether our models are right or not? Well, paradoxes are very important because they tell us, okay, something has gone wrong here. 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.

[Melvin Bragg]

Marcus Yusuto, that time you've written or you've said mathematicians, mathematics as an analytical subject was beginning to emerge. Mathematics were exploring abstract ideas through mathematics. Now, would that, can that be your starting point for talking about paradoxes?

[Marcus Yasoto]

Yes, certainly. I think that before 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. But it's very geometric, it's very functional. They're measuring areas of land, 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, 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 would always work. So, for example, where did that come from? 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. 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 how.

[Melvin Bragg]

Does that root work.

[Marcus Yasoto]

Well, I think that you've got suddenly the Greeks trying to prove that laws will work and that laws will always apply. And so I think it sort of grows out of that sort of change of the city into a political institution. And so I think the ancient Greeks, you see different style of mathematics and what I would really call mathematics, this idea of analytic thinking. But it's interesting that you Know, 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. Make a hypothesis, for example, that the square root of 2 can be written as a fraction, 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. But the paradox is very useful because you can then work backwards and say, okay, something along the way was wrong, 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. So this length cannot be written as a fraction. It can be approximated by fractions more and more. But they realize, using this argument, that there were new numbers here. 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.

[Melvin Bragg]

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?

[Marcus Yasoto]

Well, they did have problems with infinity. And a lot of their mathematics, you can see, is very finite. It's very geometrical. It's about lengths. And this discovery that the square root of two can't be written as to a ratio of two whole numbers. 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. 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. So it's not in the documents, but the volume, the formula that you get, actually, you have to use an idea of infinite divisibility to be able to get that formula as an early form of integral calculus. 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. So they have this idea of absolute actual infinity. And what's the other one?

[Barbara Sattler]

Potential.

[Marcus Yasoto]

Potential infinity. Thank you. There you go. Potential infinity. So there's a potential for infinity. 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. They have the potential to go on forever. So some of these.

[Melvin Bragg]

Let's get back to the paradox. This is where paradox has become very.

[Marcus Yasoto]

Useful because it can tease out.

[Melvin Bragg]

Is it the key key? I mean, is it really that important, the paradox?

[Marcus Yasoto]

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

[Melvin Bragg]

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

[Marcus Yasoto]

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. 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.

[Melvin Bragg]

So that's the idea behind it, is say, let's see what the world is about. Let's see the reality. In one sense, Plato is a dream, but this is mathematical by mathematical analysis, it's a different reality from that which we perceive. 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?

[James Warren]

I think that's right. I think 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. So he takes someone, he says, well, you think things move, don't you? Yes, of course I think things move. Well, wouldn't you, for example, think that you would. You would agree, wouldn't you, that in order to get from A to B, you must get halfway from A to B? Well, yes, of course, I would have to get from halfway from A to B in order to get from A to B. 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. And here you have another example of what we saw in Barbara's bald man case, this repeating premise. 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.

[Melvin Bragg]

This is the dichotomy parent.

[James Warren]

Right, Exactly. So this is what I.

[Melvin Bragg]

Sorry, I split it. Dichotomy, paradox.

[James Warren]

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.

[Melvin Bragg]

But by saying that, what is he saying?

[James Warren]

That's what's problematic.

[Melvin Bragg]

Paradox.

[James Warren]

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. So in order to do something first, I have to do something prior. And if that's an endless series of prior requirements before I even get started, then you can. The killer line will say, but you don't think you can complete an infinite series of tasks, can you? It would be impossible to do an infinite series of journeys. Well, I suppose that's true. 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.

[Melvin Bragg]

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 ways in biff. What did he biff about on this one?

[James Warren]

Well, on this one he thinks, as we've just. It's Aristotle's distinction between potential and actual infinity. He thinks it's misdescribing the job to say that you have to complete an actual series of infinite journeys. Potentially, if you wanted, 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. 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. So he, 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.

[Melvin Bragg]

But the argument goes on. That's the interesting thing, isn't it? Great as Aristotle is, and so on. And so 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, what Zeno says is happening there.

[Barbara Sattler]

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, 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? So let's imagine they are racing on a hundred meter racetrack and the tortoise is starting 10 meters in. 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. 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. 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 again. You know, the same happens. So the distance between Achilles and the tortoise will get less and less, but will never get to zero. So it seems that Achilles, even though he's the fastest runner in ancient world, will never be able to overtake the slow tortoise. Right? So that's the paradox.

[Melvin Bragg]

Did I mean, I said this once before and I'll never say it again after this, did common sense rear its head?

[Barbara Sattler]

So common sense, if you want it, reared 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 can overtake somebody, right? But I don't think that Sino wanted to show we will never experience somebody overtaking somebody else, right? Or somebody covering finite distance. 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? So even though we experience it, we can't give a good explanation of it.

[Melvin Bragg]

I think, I think the paradox is more interesting than common sense actually and obviously leads to more things, don't you think?

[Barbara Sattler]

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. 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 scene for the first time Race.

[Marcus Yasoto]

This is the story of the one. As head of maintenance at a concert hall, he knows the show must always go on. That's why he works behind the scenes, ensuring every light is working, the H Vac is humming, and his facility shines with Grainger's supplies and solutions for every challenge he faces. Plus 24, seven customer support his venue. Never misses a beat. Call quickgranger.com or just stop by Grainger for the ones who get it done.

[Barbara Sattler]

America is changing and so is the world.

[Marcus Yasoto]

But what's happening in America isn't just a cause of global upheaval. It's also a symptom of disruption that's happening everywhere.

[Barbara Sattler]

I'm Asma Khalid in Washington D.C. i'm.

[Marcus Yasoto]

Tristan Redman in London, and this is the Global Story.

[Barbara Sattler]

Every weekday we'll bring you a story from this intersection where the world and America meet.

[Marcus Yasoto]

Listen on BBC.com or wherever you get your podcasts.

[Melvin Bragg]

So can you unpick that more? Can we go into this? Why is this fascinating and why does it continue? I sort of rather brutally said, what about common sense? 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 still is still employed in various ways today. So can you just into that, what's going on?

[Marcus Yasoto]

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. 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? 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. So it looks like he's having to do infinitely many tasks. But we understand this now that he can do infinitely many tasks because it can take him a finite amount of time. This infinite series, a half plus a quarter plus an eighth plus a 16th actually adds up if you do take infinitely many of them to the answer one. And you can sort of see that if you imagine a cake and you cut the cake in half, and then you cut the half in a quarter and then an 8th and then a 16th, 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. So we know that this infinitely many tasks will take a finite amount of time. It's interesting, maybe takes less than a minute. So mathematicians had to come up with some sort of way of, of understanding adding up infinitely many things. And it doesn't mean that adding up anything will always work. 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. Maybe that adds up to something finite. But orig in the 14th century proved that actually, no, that can become as large as you want. So Zeno is already challenging us with how do you understand how to add up infinitely many things in mathematics and that have some sort of physical reality. 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 finite and when they could be infinite.

[Melvin Bragg]

What's fascinating to me, a non mathematician, and I'll go back to you for a moment, Barbara, if I may, is 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. But away they go. What is so fascinating about it?

[Barbara Sattler]

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? We can do this, run killers, can overtake the tortoise, no problem. But yet in mathematics, which we use in order to describe the physical reality, there seem to be a real problem with this, dealing with infinity, right? So our most powerful tool to describe 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. That seemed to get us into contradictions. And if you have a contradiction, then you know you're having trouble with your science. It's not a solid scientific if there's a contradiction at heart, right? So that's why mathematicians were really fighting with that and saying, okay, if we don't want contradiction at the very basis of our science, right? And then in the 17th and 18th century, as Marcus said, there was a new way of dealing with, well, infinite series. Then with Cauchy, we have dealing with limits. We have in the 19th century a new way of dealing with actual infinity. 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. And then with Cantor and others, we had this idea, no, there can be an actual infinity. And that just needs a different way of dealing with it that goes against our intuitions.

[Melvin Bragg]

By the way, James Warren, it seems to me that mathematics is being used in a philosophical way all the way the idea is still, let's go back to Parmenides saying bluntly, the world does not move. Nothing moves. It is one thing. It is not many things that you natural philosophers have been thinking for the last few centuries. It doesn't change all the time. It doesn't move on. It is one thing that is. 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 is only. Is a superficial reading of it because the mathematicians are going for something else, aren't they?

[James Warren]

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. 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. Is worth thinking about. So for example, one of the things Aristotle complains about is that one of these problems that Zeno raises is 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. You can't make a line out of points any more than you can construct a duration out of instance. What, what a mathematical point does is an abstracted. What you're doing is taking an extension that's already there and picking out something out of it. You're not constructing the world mathematically.

[Melvin Bragg]

So let's, let's look at that notion with regard to the arrow in flight or the arrow at rest. Parmenides argued, and Zeno puts it forward, that the arrow never moves. So somebody shoots an arrow and it never moves. What's going on there?

[James Warren]

Right. So the absurd conclusion here is that the moving arrow is always at rest. 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, if you think of any point in the arrow's journey, by point I mean now an instant, a temporal point. So imagine taking a photograph of it that captures 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 at that instant. You can think of it, either it's too snugly held by space or the. There's not enough time for it to do any moving because we've specified that we're talking about an instant or a durationless point in time. But that's the case throughout the Arrow's journey. 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. So it seems to be true throughout the journey that the arrow is not moving.

[Melvin Bragg]

And Aristotle said, well, Aristotle says this.

[James Warren]

Is false because time, a duration, is not made of nows.

[Melvin Bragg]

Yes, duration can have its essentials to it. Can we keep on the arrow because it's such a graphic one, people listening. You have the arrow photograph the arrow, and it doesn't seem to be moving except for that instant. 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, or are we seeing it does at rest? What does at rest mean?

[Marcus Yasoto]

Well, I think it's the challenge of this arrow is changing speed. It's decelerating as it goes towards. So it's got a different speed at every particular time. And it was the real challenge actually, for it to be moving, it has to have a speed. And if you just take an instant of time, the time interval is zero. 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 divided by distance divided by zero time. But you're absolutely right in the way you try to approach that problem, because 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. So if you take the time interval of 1 second before the snapshot you've done, then you've got an average speed of the distance. It's gone over that 1 second divided by the 1 second. 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. So you see though, that the speed is actually tending towards a limit. And Calculus is making sense of this challenge that Zeno has set. Well, what is the speed? It's zero divided by zero. It doesn't have a speed. That's meaningless. It isn't moving. Newton and Leibniz say, no, we have a mathematics now developed by Newton and Leibniz to actually understand a world in flux and be able to say at one instance of time what the speed of the arrow is.

[Melvin Bragg]

James Warren.

[James Warren]

I don't Think Zena would be impressed by that, Really? I think that's mathematically clever, but philosophically not so smart. Because, because you've cheated. You've assumed the arrow is moving 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. You can't help yourself to the conclusion that you're trying to get. And his second point will be, well, surely you would agree then that if, if we can allow ourselves that now is, can be described as an instant, it's true that the arrow isn't moving now. And if it isn't moving now, when, when on earth is it moving?

[Marcus Yasoto]

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. Marlborough, oh my gosh, they're ganging up on me. The map is, oh, this is that racial.

[Barbara Sattler]

I'm ganging up against both of you.

[Melvin Bragg]

Sig at the bridge and the man defending, they're all rushing at him.

[Barbara Sattler]

He defends the bridge in support of Marcus. So philosophers afterwards try to actually think of motion really in that way. So with Russell and others, we have this idea of the add theory of motion, as it's called. 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? 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. So that has been a famous theory, add, add theory. But I would gang up, help James here saying this is a very useful way in mathematics to describe motion and we have come to use it and employ it all the time, doesn't tell us that motion consists of these points. It tells us we can describe motion in this way, in this mathematical way. It's very useful to do that. But it doesn't tell us that we really have understood what's going on with motion in this sense.

[Marcus Yasoto]

Well, it's interesting because actually there's a modern day effect in quantum physics which actually says that motion sort of doesn't happen. It's called the quantum Zeno effect, which is quantum physics says two electrons can be sort of two places at the same time, but when you observe them. So it could be here and there, but when I observe, it has to make up its mind where it is. So it's there. But then, as 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 stuff. So actually this is called the xeno quantum effect, because if I keep on looking at it, actually I can stop this thing evolving. So I've actually brought a pot of uranium into the studio, which is the same effect. If I keep on observing this, I can actually stop it radiating because it can never has a chance to move because of my observations.

[Melvin Bragg]

That's like magic, I mean.

[Marcus Yasoto]

Oh, I know.

[Melvin Bragg]

I mean, I'm fascinated by all this stuff and I'm fascinated by about magic. So there your hands up. So you look at it and it stops moving. Now what's going on? Are you the only one? Can I look at it?

[James Warren]

As long as someone's looking at it.

[Marcus Yasoto]

I mean, this is the challenge of quantum physics and. But it's actually been done in experiments. So Turing was the first to come up with. Alan Turing, the mathematician, to, with, you know, potentially, this is the. The consequences of this. 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, 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 it's the arrow comes towards me.

[Melvin Bragg]

Where's Zeno in all this, Barbara?

[Barbara Sattler]

Well, I mean, Zeno reemerges 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. But I think what that brings up, this 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, you know, the bigger level. So to say, when we move, right, when we move, we think we can talk about continuous motion. And there the question is, well, couldn't it. How do we really explain that getting from one point to the next? Right. Isn't there more to motion? 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.

[Melvin Bragg]

James, you want to come in?

[James Warren]

Yeah. I just wanted to point out that there was an ancient set of, you know, quantum theorists who did indeed react to Zeno in an interesting way. So in, in you can tell a reasonable and plausible story that says ancient atomist theory emerged as a response to Zenonian paradoxes.

[Melvin Bragg]

This is in the 4th century BC.

[James Warren]

Yeah, towards the end of the 5th and 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 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 and you have an indivisible but extended space which you can't cross only a half of. So once you get going, then Zeno's conclusion doesn't follow.

[Barbara Sattler]

I mean, one thing that shows, I think, is that Zeno's paradox is very extremely fruitful because they sparked the following natural philosophers to come up with some solutions. So the atomic solution to say, well, there are indivisible minima and we can't go on dividing infinitely is one. Aristotle is another. 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.

[Melvin Bragg]

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

[James Warren]

Right.

[Melvin Bragg]

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. That was his method of doing it.

[James Warren]

I think the way to put it is that what the paradoxes show is that 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. 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. There are some that seem to be attacking the bare notion of plurality in various ways. And I think it would be hard to think that there was some. Some kind of overarching point to them. And that's, in a way, why they're kind of fruitful, because 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 Aristotle to be bothered by Zeno. 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.

[Melvin Bragg]

Is there any way in which we can start to characterize how these ideas are in play now, Marcus?

[Marcus Yasoto]

Very much so.

[Melvin Bragg]

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 you, you talk in your notes about how these are in play now at the very deep level of.

[Marcus Yasoto]

I think the idea of a paradox is. Is still very much used today to tease out and challenge our view of reality. 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. And I think, you know, Zeno, we talked about quantum physics and the fact that actually the universe may be made out of bits. 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. 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. And there have been sort of more recent challenges. Okay, is that physically possible in our universe? 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. And so there are these new challenges called super tasks. 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, 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 discrete actions? 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 of a minute, 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? So these paradoxes are still very relevant today in teasing out just the nature of reality and our intuition about it.

[Melvin Bragg]

Before we leave this, it might be, I'd like to sort of tip A bow to Parmenides, who, as you were kicking as briefly has his idea of the world not moving got any traction at all.

[Barbara Sattler]

There are some people in philosophy who have now gone back to some form of new monism, right, who say, well, at the deep level of reality, there is just one thing. Some people think it is just a question of dependence. So everything depends on, you know, there being one thing, namely the universe. Other people think, no, it's just in general, there's just one thing. It sounds funny. And monism, I think, is not something that is immediately very attractive to the common sense. But some philosophers in Thomas Schaffer, Michael Della Rocker 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.

[Melvin Bragg]

What does it mean? One thing, one source of energy, one type of energy? Is that what he means? One of the problems we haven't addressed is what's a thing?

[Barbara Sattler]

Okay, that's a very good question. What is a thing? So 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. For him, it was very important that we don't get into contradictions. So the only thing that we can think of is just one thing, something that has no differences, no distinctions, no extension. That doesn't quite sound physical to us. Right. So for him it seemed to be something logical.

[Melvin Bragg]

Do you want to add to that, James?

[James Warren]

No, no, no. I think Barbara's captured that rather well. 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 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.

[Melvin Bragg]

And finally, Marcus.

[Marcus Yasoto]

Well, I think mathematicians think they've sorted these paradoxes out and the invention of infinite series and the calculus can gives us a way to explain them. But I think actually there's still the challenge of, with mathematics is really describing reality.

[Melvin Bragg]

Thank you very much, I enjoyed that. Nice to be back. Barbara Sattler, Marcus Yasoto, James Warren, thank you very much. Next week it's four legs good, two legs bad. Yes, we'll be talking about Animal Farm by George Orwell.

[Barbara Sattler]

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

[Melvin Bragg]

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

[James Warren]

Well, there are those. There are the paradoxes of plurality that we didn't discuss, but they were sort of in the offing when you were asking Barbara what a thing is.

[Barbara Sattler]

Exactly. That's why I thought, should I not jump on? But you wanted to talk about Paminidas.

[James Warren]

And those are very peculiar.

[Barbara Sattler]

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? Well, because there's another thing in between, just air. Okay. But how can I be sure that air is different from that one? And it's. They are in some sense less attractive. But they raised this important question, what makes a thing a thing? Yeah.

[Marcus Yasoto]

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. So if you add something which has zero distance to something which has zero distance, it's still got zero distance. And that was the real challenge of. 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 measure 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 2, PI and something like that. But these numbers have no distance. It's really like the arrow, Zeno's arrow. So how can you have all of these numbers actually make up a line, a ruler?

[James Warren]

That's, I think, very similar to Zeno's millet seed paradox. Right. Which is different again from the how many grains make a heap paradox. But it's, it, it's saying, well, if I drop a single seed, it doesn't make a noise.

[Melvin Bragg]

Well, I think it does. You see, I just. We can't hear it, that's all. And maybe an ant can hear it. There goes a millet seed I'll pop across.

[James Warren]

Oh, but there's a difference between it disturbing the air and it making a noise. But it's clear that a large number of millet seeds do make a noise.

[Melvin Bragg]

Can I rest on mine for a moment? I know. Outgunned completely. If it's still. If it drops, in my view, it'll make a sound. And this sound can be heard by those with ears to hear. Okay, may Be an ant. That may be what he's been waiting for since he woke up. This is breakfast. This is the millet seed.

[James Warren]

So half a millet seed, then, or half of half a millet seed.

[Marcus Yasoto]

The interesting thing is you're getting to quantum physics now.

[Melvin Bragg]

With big ears, baby.

[Marcus Yasoto]

But this is what Einstein won his Nobel Prize for, essentially, is to understand that there are actually thresholds below which you cannot activate things. 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. 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 really still very relevant today.

[Barbara Sattler]

I mean, it's this vagueness paradoxes that we talked about in the beginning about the boldness and the heap of grains. 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. But in with many other concepts, it seems just arbitrary to say, you know, okay, from 5 hair onwards.

[Marcus Yasoto]

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. Something like the barber who only shaves.

[Barbara Sattler]

Those who don't shave themselves.

[Marcus Yasoto]

Which leads. You realize that that's. But I. 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.

[Barbara Sattler]

Right.

[Marcus Yasoto]

But I suppose that that's the point. You're trying to show that there can't be a barber who only shaves those who don't shave themselves. The paradox is a barber.

[Melvin Bragg]

Yeah.

[Marcus Yasoto]

The paradox is resolved by saying this person does not exist. Your hypothesis that there is such a thing. So I think that's, you know, there's a reductio ad absurdum.

[Barbara Sattler]

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

[Melvin Bragg]

The producer comes to make the most announcing. Sorry to interrupt, Simon.

[James Warren]

Who'd like tea?

[Marcus Yasoto]

Or half a cup of tea?

[Barbara Sattler]

Or tea.

[Marcus Yasoto]

Oh, this will run and Run In.

[James Warren]

Our Time with Melvin Bragg is produced.

[Marcus Yasoto]

By me, Simon Tillotson, and it's a BBC Studios production.

[Melvin Bragg]

Hello, it's Ray Winstone. I'm here to tell you about my podcast on BBC Radio 4, History's Toughest Heroes. I've got stories about the pioneers, the rebels, the outcasts who Define tough.

[Marcus Yasoto]

And that was the first time that.

[Melvin Bragg]

Anybody ever ran a car up that.

[Marcus Yasoto]

Fast with no tires on.

[Melvin Bragg]

It almost feels like your eyeballs are gonna come out of your head. Tough enough for you? Subscribe to History's Toughest Heroes on BBC Sound.

[Barbara Sattler]

America is changing, and so is the world.

[Marcus Yasoto]

But what's happening in America isn't just the cause of global upheaval. It's also a symptom of disruption that's happening everywhere.

[Barbara Sattler]

I'm Asma Khalid in Washington, D.C. i'm.

[Marcus Yasoto]

Tristan Redman in London, and this is the Global Story.

[Barbara Sattler]

Every weekday, we'll bring you a story from this intersection where the world and America meet.

[Marcus Yasoto]

Listen on BBC.com or. Or wherever you get your podcasts.