by Melvyn Bragg
Plato wrote a dialogue in which the two of them came to Athens, and Zeno was cast as a defender of Parmenides. That is one way to think of these paradoxes – as Zeno’s attempt to undercut possible objections to Parmenides’ curious thesis. They travelled to the great festivals, such as the Panathenaia, and they would have given demonstrations and public recitations and met people there.
JAMES WARREN: Parmenides is reacting to a tradition of cosmological thinking, of people 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, from water, or air, or something else.
Those people were relying on there being a plurality of things and on there being things that changed and were in motion in order to account for the way the world worked. Parmenides set out to show how that view was grossly mistaken.
Turning to paradoxes, which Zeno would use, Barbara Sattler explained that the word comes from the Greek, being something that is against (para) common expectations or beliefs (doxa).
BARBARA SATTLER: In a philosophical context, by paradox we normally understand that we derive a problematic conclusion from sound premises. It seems we have good starting points and we do right reasoning, and yet we get to a conclusion that is untenable. Why is it untenable? Either because it is inconsistent in itself or it contradicts other beliefs, opinions, that we hold.
One paradox that is quite famous is that of the bald man. We would all agree that, if somebody has no hair, that person is bald. If the person has one hair, we would still call the person bald; two hairs, probably still bald; three hairs, the same, and so on.
BARBARA SATTLER: One hair doesn’t seem to make a difference, but yet, if this person has 10,000 hairs, it seems this person is not bald any longer. So where does that stop? Is it from 100 hairs onwards we say, ‘This person is not bald,’ but at ninety-nine hairs, ‘This person is still bald’? That doesn’t seem to be right.
MELVYN BRAGG: Why not?
BARBARA SATTLER: Because it seems baldness is not a concept or notion where we can give a clear quantitative determination.
We all agree on certain ideas of baldness and we all have a problem with saying when a person stops being bald. What that shows is that there seem to be some notions that can be called ‘vague’. Another example might be: how many grains do you have to take away from a heap of grain before it is no longer a heap? Paradoxes such as these are very fruitful for philosophy.
BARBARA SATTLER: In philosophy, a lot of what we do is actually done conceptually, so our theories are not falsified or verified by the world outside. Paradoxes are very important because they tell us something has gone wrong. You have to go back to your concept and look again.
In Greece, in Zeno’s time, mathematicians were exploring abstract ideas and to these, too, paradoxes would apply. The Egyptians and the Babylonians had been trying to describe the world with the new language of mathematics, but their approach was often geometric and functional.
MARCUS DU SAUTOY: They are measuring areas of land, volumes of pyramids and things like that. But then, in the 100 years before Zeno, we have the Pythagoreans beginning to appear on the scene and they are trying to prove things. They are trying to prove that it is not just a calculation they want to do, they want to produce a proof that something will always work.
Perhaps the idea of analytic thought came from Greeks wanting to do politics, trying to prove that laws would work and would always apply. The idea of paradox was starting to appear at this time in mathematics, perhaps a little after Zeno, as a tool that was a proof, a reductio ad absurdum.
MARCUS DU SAUTOY: Make a hypothesis, for example, that the square root of 2 can be written as a fraction. Then you follow that through, and you end up with a ridiculous conclusion that odd numbers equal even numbers, and then you realise that that’s absurd, it is 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.’
The Pythagoreans discovered that the square root of 2 could be approximated by fractions, more and more; but this idea of teasing out a logical argument, arriving at something absurd, was a very powerful tool for complex problems, such as the one the Greeks had with infinity.
MARCUS DU SAUTOY: Infinity doesn’t seem to exist, I can’t see anything infinite. They have this idea of actual infinity and potential infinity. There is a potential for infinity. For example, Euclid proves that the primes have the potential to go on for ever, but there is a claim that this isn’t an actual infinity, you can’t actually have infinitely many primes, they have the potential to go on for ever.
By using paradoxes, the mathematicians could reveal that their ideas of infinity might be wrong, something that was explored later in this programme.
Another thing to bear in mind, James Warren suggested, was that these paradoxes were playful, a way of Zeno embarrassing an interlocutor just as Socrates embarrassed people. He could take someone and say, ‘Well, you think things move, don’t you?’ ‘Yes, of course I think things move.’ ‘Well, 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.’ ‘Well, surely you would then agree that, to get from A to halfway to B, you would have to get halfway from A to halfway between A to B?’ And so it would continue. This is a dichotomy paradox, the ‘cutting in two’ paradox.
JAMES WARREN: What’s problematic then is that you have got your person to agree that to cross any spatial extension entails an endless series of prior journeys; in order to do something, first I have to do something prior. If that’s an endless series of prior requirements, the killer line will say, ‘But you don’t think you can complete an infinite series of tasks, can you?’
There is obviously a sense in which the impossibility of completing an infinite number of tasks is true, in which case asking someone to cross a room would be asking for something impossible.
The mathematicians and philosophers were grappling with similar problems, if in different ways and with different outcomes. Aristotle thought there was a distinction between potential and actual infinity. Potentially you could think of your journey as including as many subjourneys as you liked, but you do not actually have to do all of those in order to cross the room.
JAMES WARREN: Aristotle is working from the assumption that, of course, Zeno must be wrong, because, of course, things do move, and there are many things. He is 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 carry on writing his book on physics.
Aristotle did not kill the argument over infinite prior steps with this. It continued, it emerged and it re-emerged.
Perhaps the best known of Zeno’s paradoxes is Achilles and the tortoise, which Barbara Sattler noted was a variation of the dichotomy paradox. In this, Achilles, the fastest runner in the ancient world, had a race with a tortoise and gave the tortoise a head start. If we imagine they are racing on a 100m track and the tortoise is starting 10m ahead, Achilles first has to cover those 10m but, during the time that Achilles takes to cover the 10m, the tortoise has moved on. By the time Achilles has reached that point where the tortoise has been, the tortoise will have moved on again.
BARBARA SATTLER: The distance between Achilles and the tortoise will get less and less but will never get to zero. It seems that Achilles, the fastest runner in the ancient world, will never be able to overtake this slow tortoise. That’s the paradox. I don’t think that Zeno wanted to show we will never experience somebody overtaking somebody else. What he is telling us is, ‘Okay, you describe what is going on, and you will get into contradictions.’
In this paradox is the challenge of the infinite, and particularly something called an infinite series, because we are having to add up infinitely many things and are trying to understand whether that is actually physically possible.
MARCUS DU SAUTOY: Let’s
say he does the first step in half a minute; the second step he does in half the time, so a quarter of a minute; the third step in an eighth of a minute; the next step in a sixteenth of a minute. It looks like he is having to do infinitely many tasks, but we understand that he can do infinitely many tasks because it can take him a finite amount of time. This infinite series actually adds up, if you take infinitely many of them, to the answer ‘one’.
Mathematicians had to come up with some way of understanding the adding-up of infinitely many things and whether that had some sort of physical reality, and that advance did not really happen until the seventeenth and eighteenth centuries.
In the physical reality, we do not have a problem with these things, as we see the tortoise being overtaken, but, in mathematics, which we use in order to describe the physical reality, there seems to be a real problem with dealing with infinity.
BARBARA SATTLER: We have, in the nineteenth century, a new way of dealing with actual infinity. Remember Aristotle 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. Then, with Cantor and others, we had this idea: there can be an actual infinity that just needs a different way of dealing with it. That goes against our intuitions, by the way.
Georg Cantor was examining the questions underlying these paradoxes over 2,000 years after they were raised.
There is another famous paradox, which is the arrow in flight. Parmenides argued, and Zeno put it forward, that, if someone shoots an arrow, it never moves but is always at rest. If you think of any point in the arrow’s journey – an instant, a temporal point – at that point, the arrow is occupying a space, exactly arrow-shaped and arrow-sized.
JAMES WARREN: It is not moving within that space, it is stationary at that instant. You can think of it as either too snugly held by space, or there is not enough time for it to do any moving, because we’ve specified that we are talking about a durationless point in time. 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.
The challenge of this arrow, Marcus du Sautoy suggested, was that it was decelerating as it went forwards, so had a different speed at every particular time and, for it to be moving, it had to have a speed.
Bust of Zeno of Elea.
MARCUS DU SAUTOY: If you just take an instant of time, the time interval is zero, the distance that’s gone is zero, but speed is distance divided by time, so it doesn’t have a speed then, does it? Zero distance divided by zero time.
What Newton and Leibniz did was realise that actually this thing did have a speed, but the time interval that you are measuring gets smaller and smaller. Calculus was a way of making sense of this challenge that Zeno had set, as Newton and Leibniz had developed a mathematics to understand a world in flux.
JAMES WARREN: I don’t think Zeno would be impressed by that.
MARCUS DU SAUTOY: Really?
JAMES WARREN: I think that’s mathematically clever, but philosophically not so smart, because you have cheated. You have 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. That’s precisely what is at question. You can’t help yourself to the conclusion that you are trying to get.
In support of Marcus du Sautoy, Barbara Sattler said philosophers did later try to think of motion in that way, with Bertrand Russell and others developing the ‘at-at theory of motion’, where motion was nothing but being at a particular point at a particular time, and the difference between motion and rest was observed from the surroundings. Something in motion would be at a different point in space at the next moment of time and something at rest would still be at the same point.
There is a modern-day effect in quantum physics, Marcus du Sautoy observed, which says that motion does not happen. It is called the ‘quantum Zeno effect’, where an electron can be in two places at the same time but, when you observe one, it has to make up its mind where it is, so it is ‘there’.
MARCUS DU SAUTOY: This is called the quantum Zeno effect because, if I keep on looking at it, I can stop this thing evolving. I have 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, it never has the chance to move because of my observations.
MELVYN BRAGG: That’s like magic, it is marvellous.
MARCUS DU SAUTOY: I know. Anyone who is 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 that don’t move.
The responses to Zeno’s paradoxes have been fruitful; they sparked the natural philosophers, who followed him, to come up with solutions.
BARBARA SATTLER: 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.
The idea of a paradox is still very much used today to tease out and challenge our view of reality. The idea of ‘infinitely many tasks’ was the challenge at the heart of trying to overtake the tortoise, and there have been more recent challenges. Is our universe, as the ancient Pythagoreans thought, finite in its nature and without infinite decimals in its make-up?
MARCUS DU SAUTOY: There are these new challenges called ‘super tasks’. Can you switch on a light, on and off, and on and off, and halve 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?
Is that experiment actually ever physically going to be possible and, if so, what is the end result when you add all of these actions up at the end of the minute – is the light on or off? Such paradoxes, with their roots in Zeno, are still very relevant today when teasing out the nature of reality and our intuitions about it.
CULTURE
We have looked at cultures from around the world. The unexpected consequence of the initial eclecticism is that, over the years, patterns have emerged, ambitions unsuspected in 1998 have been quietly established. For example, we decided to pay full and proper tribute to the great scholars and translators of the early Islamic Renaissance – in the west’s Dark Ages and early Middle Ages. For centuries, their contribution had been ignored or, if recognised, described as a sort of donkey work, as translators heavy-lifting Greek culture into medieval Europe. They did that, too, but, more importantly, their own philosophers, scholars of medicine, and poets greatly enriched and, in some ways, enabled the growth of the western European Renaissance – not as translators but as exemplars.
Similarly, we went to ancient China (though not enough); to ancient India (again by no means enough); to South America (another field to be tilled more sedulously as time goes on). This is not to mine the exotic. It is to chime in with parallels to the western experience; higher civilisations occur at different times in different locations.
The listeners seem to enjoy it when we spread our wings and, because of the ubiquity of the podcast, we receive instant feedback from these programmes – from northern China to Mexico, Hawaii, North America, Australia, Mumbai … It is as though there are those in the world who want to see their own cultures shoulder to shoulder with others and also stay listening to savour the differences. Radio 4 is giving us a unique opportunity.
So where should we begin? Once again, Simon has had the enviable task of selecting a few from the many; ten from about 150. He includes the Epic of Gilgamesh, 4,000 years old, said to be the first great masterpiece of literature, from Mesopotamia in modern-day Iraq. Today we would call it Magical Realism and it would be worth a thesis to see if there is a line from Gilgamesh to Salman Rushdie.
Then, I think, he threw his hat in the air and went for ultimate variety. Frida Kahlo was born in Mexico City in 1907 and, over the decades, despite horrific disabling accidents and perhaps because of her involvement with the left-wing re
volution in Mexico, her work grew, largely under the radar of the oligarch-dealers. Only recently has she been given the highest accolades. Her subject matter emphatically drew on herself as a Mexican, claiming both Mexico and herself as a source of art. And there is so much in Mexico’s past for instance – the Museum of Anthropology in Mexico City is one of the museum wonders of the world. So, if we can wedge in, we can prise open a past that informs both modern Mexico and stands alongside pre-Christian civilisations from the other side of the Atlantic.
When you are given the liberty of a continuing series, you can try things that might not work because you can always pick up the stitch in the following weeks. For instance, we have discussed in detail several paintings on In Our Time. Turner’s The Fighting Temeraire Tugged to her Last Berth, for instance. What most pleased me is that the art historians discussed it as though everyone at the listening end had a print in front of them.
What I most liked was that the contributors on Turner did not stint on detail, the direction of the wind, the relationship of the ship to the sun, what Turner had left out of ‘reality’ and what he had added for his composition.
We did a not dissimilar exercise recently with Beethoven. We/they talked about his music without, in forty-five minutes, hearing a single note – rather like Beethoven himself in his later years – but it was a demand on the listeners’ imagination that they seemed to enjoy.
What brings rapture from the audience can also be a classic novel. Emma brought a surge of Austenites, Middlemarch possessed George Eliot enthusiasts, but the novel that, as it were, scorched the listenership was Jane Eyre.
I don’t do Twitter. Nor do I read it. Save once a week when Simon passes on the tweets about the programme. The contributions generally begin at about 8.30 a.m., when I trail the subject inside the Today programme. On Jane Eyre day, tweeting began at 8.30 a.m. and went on throughout the morning.
