In Our Time - Mathematics and Platonism

Episode Date: January 11, 2001

Melvyn Bragg looks at the deep claims made for mathematics, the discipline some believe to be the soul and true key to the understanding of all life, from the petals on the sunflower to the pulse in our wrists. The notion that mathematics is akin to theology might take some taking in at first. But from the first, in the West, they were. To Pythagoras, numbers were mystical and “prove” God. To Plato, who, it is claimed, has driven mathematics for over two thousand years, the ideals beyond the reality of our lives are to be found in mathematical perfections, immutable truth, God again in numbers. Are mathematics there in the universe, waiting to be discovered as the great ocean lying before Newton - or are they constructs applied by us to the universe and imposed rather than uncovered? It’s a long way from chalky sums on the blackboard and the first careless swing of the compass. Galilei Galileo wrote, “The Universe cannot be read until we have learnt the language and become familiar with the characters in which it was written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word”. But is he right that mathematics is the script in which the universe was written, or is it really just one of many possible systems that humankind has invented to interpret our world? Is mathematics is a process of invention or a voyage of discovery?With Ian Stewart, Professor of Mathematics and Gresham Professor of Geometry, University of Warwick; Margaret Wertheim, science writer, journalist and author of Pythagoras’ Trousers; John D Barrow, Professor of Applied Mathematics and Theoretical Physics, University of Cambridge.

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or wherever you get your pods. Thanks for downloading the In Our Time podcast. For more details about In Our Time and for our terms of use, please go to BBC.co.com.uk forward slash radio four. I hope you enjoy the program. Hello, Galileo wrote, The universe cannot be read until we've learned the language and become familiar with the characters in which it was written.

It's written, he wrote, in mathematical language, and the letters are triangles, circles, and other geometrical figures without which means it's humanly impossible to comprehend a single word. But is he right that mathematics is the script in which the universe was written, or is it really just one of many possible systems that humankind has invented to interpret our world? With me to discuss whether mathematics is a process of invention

or a voyage of discovery. Is Ian Stewart, Professor of Mathematics and Gresham Professor of Geometry at the University of Warwick, author of many books, including Nature's numbers? Also whether this is a science writer and journalist, Margaret Wirtheim, author of Pythagoras's trousers, and John D. Barrow, Professor of Applied Mathematics and Theoretical Physics at the University of Cambridge, and the author of The Universe that Discovered itself.

Ian Stewart, if people remember anything from maths at school, it's Pythagoras' theorem, and Pythagoras in 6th century BC is credited with being the leading force in the origins of mathematics in the West. What was his philosophy of mathematics, and what would you say he is supposed to have given to it? The interesting thing about Pythagoras is his names come down to us, but it's really the cult that he was heavily associated with that we know most about. We know amazingly little about Pythagoras himself. In fact, we don't know if certainly he existed.

The cult was an interesting mixture of number mysticism and what would now be considered serious mathematics. So the number two is the male principle, the number three is the female principle, five is two plus three, so that represents marriage, all this kind of thing. This is one of the things the Pythagorean did, and that in a sense is not really very interesting mathematically. But they also discovered lots of interesting patterns in numbers, general patterns, things like triangular numbers, 1 plus 2, plus 3, plus 4 makes 10, and so on, that kind of thing. And they found patterns

in triangular numbers. And they discovered the five regular solids, the cube and it's more exotic relatives and Pythagoras' theorem, which is the one, everyone remembers the name and a friend of mine went to us, was talking to someone and said, you know, Pythagoras's theorem, what's that? And they said, oh yeah, A square plus B squared equals C squared. He said, what's A, B and C?

Can't remember. So, whereas, of course, Flanders and Swan fans will all know that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the two adjacent sides. They may have trouble remembering what a hypotenuse is, but it's the long side of the triangle. Anyway, so the Pythagorean's and their mystical founder discovered this amazing fact about right-angled triangles, which is if you know what two of the sides are, then the third

one can be calculated. And a huge amount of maths comes from that one fact. Would you like to add to that, John Barrow, would you like to talk a bit about the place that Pythagoras' idea of numbers has in his version of the universe. Yes, one of those important differences between Pythagoras' sect and what they thought about numbers and what mathematicians today might think, you could sum up

by saying it's the difference between numerology and mathematics. You see, Pythagoras in many situations thought that there was some intrinsic meaning to the numbers themselves. So if he found that there was seven of one thing somewhere in

nature, it should be related to everything. else that had a sevenness about it. But nowadays, mathematicians don't think there's anything magical or mystical about the numbers themselves that we're interested in the relations between numbers. So mathematics is really the study of all possible patterns, as the way I see it. This is probably something that Pythagoras would have liked. He liked to see analogies and relationships between things. So when he could strike pendulum and gongs, and make sounds, he noticed how the tones would change with the length of the gong that he was striking.

How good while time, do you think that Pythagoras, whether he existed or not, had gone to be ensured, do you think that that time in 6th century BC, in the West could be considered the beginning of, the beginning of the driving force of mathematics? Yes, I think that's true. Pythagoras did something that is really incredibly important for the world, and that is that he's not the first person who started to,

explore mathematics seriously, and in fact, it was the Babylonians who discovered the Pythagorean theorem we know. Pythagoras picked it up from them. But what he did was he gave the world something else. Pythagoras is the first person that we know of who came up with the idea that mathematics could be the language of the physical world. Other cultures had developed mathematics, but no other culture beforehand had had the idea that mathematics would be the ultimate language to describe physical reality. And it really is to Pythagoras that we owe this. So not only was he the huge inspiration for getting serious mathematics going in the West, but more importantly, I think he's the philosophical inspiration for the idea that we could find mathematical patterns in the

physical world. And it's interesting that for most of the ancient Western world, the Greeks, the Romans, and for the first 1,500 years of Christianity, that idea was actually rejected because the alternative scientific tradition in the ancient world was the Aristotelian one, and Aristotle very specifically rejected the idea that mathematics could describe physical reality. And that in many ways, I think, is one of the things you can, the ways you can interpret the scientific revolution, that the scientific revolution is the time in Western history and indeed in world history when that transition was made, when people, when our culture as a whole,

effectively took on the Pythagorean idea that mathematics is the language through which we could study the physical world. You threw away the idea that mathematics wasn't really, that Pythagoras got his theorem from the Babylonians. You mean the theory about the Hippuccaneus? That's right, yes, yes. It's well and truly known that the Babylonians

had this long before Pythagoras and believed that Pythagoras went to Babylon. I mean, as Ian says, his life is clouded in shadow. He lived at that time, you know, before there was sort of concrete historical records, but it's well and truly believed that he got it from the Babylonians. And incidentally, the Chinese discovered the theorem

independently, but the Babylonians knew it long before Pythagoras, and it believes that that's where he ultimately got it. But there's no question that he himself made mathematical discoveries, as Ian said, particularly the patterns in the numbers like the triangular numbers and square numbers. Sorry, do you want to say some very interesting Indian parallels at the same time?

literally at the same time, six century BC. Just about the same time. And it's the Chinese about the same time, sixth century BC. I'm not actually sure when the Chinese discovered it. And the Babylon is what? About 8th century BC. Are we getting? Fine, sorry, John. But what the Indians wanted to do for religious reasons, they like to have family and village altars that might be just as large as this table in front of us. This table is a normal size table, all right, fine.

But for religious reasons, the altars were made out of, made in shapes of strange creatures like falcons. and birds which have an unusual angular shape and would be made of lots of small bricks of triangle and rectangular and square shape. Now if things went badly for you, you had to appease the gods in some way and you did this by doubling the area of your altar.

Now this is a non-trivial problem if your altar is not just the simple square or even if it is a simple square. And so there was a great book of recipes of geometry as to how you change the area of your altar. called the Sulba Sutra, the Book of the Chord. And this really contains lots of geometry of the Pythagorean sword. Interesting that geometry is linked with divinity in a way.

But also with altars. There was a great tradition that Pythagoras sacrificed an ox to the gods when he found his theorem. This couldn't possibly be true because he didn't eat animals. Not something that his sect would have done. But the suspicion was that this came with the tradition, from the Indian culture, that Pythagoras and triangles and theorems of this sort

were about altars and sacrifice. But the link, one of the interesting things is that the link between mathematics and religion is very strong in many cultures. And as Ian rightly points out, the Pythagorean cult was as much a religious community as it was a proto-scientific community and indeed in the Pythagorean community to be inducted into the mystery of numbers

and to learn mathematical theorems, one had to go through rituals of purification because they believed that what they were doing, that Pythagoras actually associated the numbers with the gods. In fact, he literally associated the numbers 1 through 10 with the major gods of the Greek pantheon. For him, Apollo was the primary god, and he associated that with number 1,

which in Pythagoras's mind, one was almost like, you know, the ultimate God. And so Pythagoras believed that studying the relationships between numbers was actually understanding, the properties of divine beings. Can you move on to Plato? Ian, Plato believed that mathematics was the finest training for the mind. Above his academy, he wrote,

let no one unversed in geometry enter here. And that academy lasted for about 900 years. So it was perhaps the most influential mathematical university in Western history. But can you explain, why did he think it was the finest training for the mind, first of all? And secondly, what would you say was his theory of mathematical forms? Well, the Greeks developed mathematics and more and more they turned it into a very logically rigorous way of thinking.

And so you would start from very simple statements that nobody could reasonably disagree with, or at least that everyone knew what those statements were. And then you could build upon those and you would prove Pythagoras's theorem, not just discover it experimentally or have hand-waving arguments that said it ought to be true. and Plato is kind of looking at this tradition and formalising in his own philosophical way, what's going on. And what does he come up with thy, John Barrow? I mean, what's his notion of mathematical forms, Plato's notion? Well, he's interested in the unchanging aspects of the universe. These are the things that he thinks are most fundamental,

most important, most worthy of study. So although the Greeks move to the same, study of mathematics for its own sake, it appears. So they're not just counting and using their fingers, but they had the idea, as Ian said, of laying down axioms, you know, rules for the game, and then exploring what sorts of games could follow from that. But Plato was persuaded that there existed behind reality, perfect blueprints of everything that we see around us. And the things that we see like this cup here are just poor versions, poor copies of the eternal blueprints.

Can I just bring it slightly more today by going to the second century idea with Ptolemy who discovered, made a breakthrough, discovered the predicted the cycle of the planets. Now, it was used for navigation successfully for many years, indeed for many centuries, this great discoveries of Ptolemy, and yet it was based fundamentally on wrong mathematics

because it's based on the idea that the sun and the planets revolved around the earth. What does that tell you about mathematics? It doesn't have to be true to be effective, or what would you draw from that, John Barrow? If you think of mathematics as really just being the catalogue of all possible patterns that there could be, it's then no mystery why the world is mathematical. If we or anything is to exist in a coherent way, there have to be patterns of some sort. And some of the entries in that catalogue of possible patterns will be. be instantiated and will appear in nature and others will not.

The mathematics is the only thing related to science that we do, which is really infinite, there's no limit to the number of mathematical structures and patterns which could be invented. And there's no reason to expect that all of them are actually manifested in nature. Some are. And this is really one of the messages of how modern mathematics is very different from the mathematics of Plato and Pythagoras. You take something like geometry. The Greeks discovered the rules and regulations of what we now call Euclidean geometry, so geometry on flat, plain surfaces, where the interior angles of triangles add up to 180 degrees.

Now, right until the beginning of the 19th century, it was believed that this was the one and only possible geometry, and that this is how the world actually was. And so the discovery of Euclidean geometry in its rules had a deep, philosophical and religious significance for many people, because it showed that human thinking could get at part of the ultimate truth of reality. And so there was a major shock to the system when the early 19th century mathematicians discovered that you could invent other geometries,

which were logically self-consistent, but they were not Euclidean. So they were geometries which applied on curved surfaces. So if you looked at a Euclidean triangle in a curved mirror, you would see a rather bent triangle. And so all of a sudden, mathematics was about the free invention of rules and regulations, which just had to be logically consistent. And they didn't actually have to have examples in the real world that you could point at. And people started writing peculiar books and pamphlets with titles like non-Euclidean forms of government

or non-Euclidean economics, that non-Euclidean became a sort of byword for something that was relativist and anti-establishment and so forth. So I think this is one of the messages of modern mathematics is that it works jolly well as a description of nature. It doesn't have to, but you can invent any structures you want. Mathematical existence just means it's logically self-consistent. Margaret Boisheim, can I come to the point of the nature of mathematics, whether it's a process of discovery, whether mathematics are there,

whether calculus is there, whether Newton and Leibniz discovered it or not, or whether mathematics is there, or whether it's an invention. They don't exist without being observed. Could you give us your views on that? Yes, the question is generally put in that terms, is it discovered or invented? And I think that that is actually a little bit of a wrong way of characterising the problem.

And I think I agree with what John has just said, that I have always believed that mathematics is the language of pattern. It is the formal language in which we can discuss formally the properties of patterns and relationships between various patterns. And as John said, because the world is structured, because it's not complete chaos, there must be patterns in it. So it's not surprising that we find mathematical patterns in nature. But I don't think that necessarily means that those patterns are invented. However, what I do reject is the notion that those patterns have some archetypal transcendent, existence without conscious human beings there to find them.

So if there were on other planets, there are not mathematics on other planets. Other planets without human beings do not have mathematics? Well, it doesn't have to be human beings, but I think there has some conscious, yes, conscious intelligent life. Absolutely. I mean, I would argue that in the absence of conscious intelligent beings, there is no such thing as numbers. The numbers don't have a platonic or Pythagorean transcendent existence.

in and of themselves. Sorry, but how can you believe both that the university's universe is ordered, as you said a few sentences ago, and that it doesn't exist without an observer? The universe has an existence. Then the question is... Well, isn't it its existence to a great extent dependent on its mathematics?

As three of you have been argued. Yes, I don't think the existence of the universe is dependent on mathematics at all. I think that mathematics is a language, that we have found, which is an extremely powerful language because it's a language that can discuss the formal properties of patterns. But don't you think the part of the order is due to the working of mathematics? I ask Ian Stewart to come in here, and I'll come back to you. Yes.

I think I'm going to sit on the fence a little bit here, but there are two possibilities here. One is that human mathematics is our understanding of something that is genuinely there as a basic ingredient. of the universe, that the gob was a mathematician and that we are picking up bits and pieces of it. The other extreme is to say, no, no, no, mathematics is like it is because that's the way our minds work, that's the way we've evolved to structure the universe around us. I think there's a middle ground which says, look, a lot of these patterns must be real in some sense, but we haven't got the ultimate understanding of what they are. The whole history of science shows that what we thought 100 years ago was the ultimate pattern of gravitation or something turns out to be pretty

good, but it's really something else that seems to be better. And I think that process just keep going. It's a workable version of the way the universe works, like Ptolemy was and like Newton was, and like always time we've seen it and so on. Now, I agree with Margaret, and I think most mathematicians, if you pin them into a corner, will agree that what we call mathematics on this planet, and it's the only one we know, is really a kind of shared experience of human minds. It's a mental construct, but it's not one that we've just made up for its own sake.

We've made it up because we see a lot of things in the best of. That's why I think the problem is this dichotomy of is it discovered or invented. I think those words are problematic particularly. Would aliens have... They're difficult to answer. I think I'm sorry, I'm going to maintain the integrity of the question. I don't think it's problematic. I think that you're finding it quite difficult to answer

because it seems to me that you've all been saying, a me taking from Pythagoras and Plato, a fundamental place of mathematics in the formation and ordering of the universe, and I'm saying, well, is that in the universe in itself, or is that in the universe because we see it? I think it's an interesting question. If you don't think it's an interesting question, we're going to lose the next five minutes. It's a fascinating question.

John Barrow wants to come in. There's a couple of things to say. I mean, first I think it's a mistake to lump all mathematics into one sort of bag and say, you know, is mathematics discovered or is it invented? I mean, suppose that Plato was correct and mathematics lived out there, a great body of it, and we sort of discovered it by, some extrasensory perception. Well, having discovered, no, but imagine we did, but having discovered it,

we would be free to use what we'd discovered to invent other generalisations and things that flowed from it. That's what happened in the last two-th century-hundred years. So I think it's clear that we do invent some bits of mathematics for purposes of generalisation, but it's as well to remember that our minds which, you know, do this generalisation, which do this studying, are the result of a process of natural selection in which selection acts in a real environment and selects against something that really

exists. So the fact that we have ears that have all sorts of interesting properties as acoustic detectors which evolved before we were conscious witnesses to the fact that there really is something called sound that has very definite properties that an evolutionary process can select or optimise reception for. And so

when we end up up with minds which can process information in logical ways and have intuitions about geometry and counting. I think this is a reflection of the fact that embedded in the world, the world around us, there are aspects of what we would call it a mathematical sort, which are true, they're part of reality. Margaret. Yes, I mean, I think that's a very important point,

that the mathematical patterns we find in nature are absolutely a part. part of reality. And they're very powerful parts of reality because they allow us to make very concrete predictions. For instance, the mathematics of quantum physics has given us microchips. Nature does exhibit these very complex mathematical structures that we can then turn into practical applications.

But the question, I think, of his mathematics discovered or invented, I think, can be in one way looked at from a question that's been asked by the mathematician and philosopher Brian Rotman, which is, where do the number of the number, numbers come from. Now the Platonist answer to this is the numbers have, or Pythagorean answers, that they have some

transcendent existence above and beyond the physical world. There will always be numbers. If there was no physical world, or in some sense prior to the existence of physical world, numbers had an existence out there, as John said, in some sort of disembodied realm. But Brian Rotman

suggests this idea. He says, look, numbers come from the process of counting, so if I can go one cup, two, cup, three, cup, four cups. There are four cups there, and in some sense four has an existence. But if there weren't any conscious beings who could count, do numbers have an existence?

And his answer is no. And I think that's a very valid point. If there are no beings who can do the counting, in what sense can you say the numbers exist? But that implies that in the end nothing can exist without an observer. I think that's an invalid point. You think Margaret's an invalid point.

Brian Rotman's point, I think is an invalid point. Can you explain why? Well, in astronomy you have a nice way of getting at this because when you look out into space in enormous distances, you are also looking back in time. So when you receive light from distant quasars, that light left with all its bar codings telling you what atoms produced the light

billions of years ago, billions of years before there was any life on Earth, And so we can tell that there really was an ordered structure. There were things like counting, there were interrelationships between frequencies of a sort we understand today. Long before there was anything like people, long before there were anything like planets and stars. So I think we, in a curious way, have direct observational evidence that there were patterns of a numerical sort, certainly, before there were people to do the counting. But again, we're just getting at this point that maths is about patterns and the universe is about patterns. The mystery, I think, which is now shifted somewhere else,

is the fact that the patterns are often extraordinarily simple, that the universe is much easier to understand in many respects than we might have suspected. Ian, sure, is Plato still alive in mathematics? I mean, we're coming to the end of this conversation. Are mathematicians still looking for what's transcendent? I mean, I'd like to throw a little spoke in the work. It seems to me, it does seem to me, and this is very personal

and probably completely relevant, but that music's the best evidence for the perfection of mathematics. And it's interesting that Pythagoras talked about the harmony of the spheres, and Mozart was a great mathematician as a boy. This is a long-running undercurrent, and there is something to it. If you talk to professional mathematicians, on the one hand, they know that what they do is a collective human activity, and on the other hand, they're all closet platonists.

I am. What is a closet? Let's have me out. Okay. Confess your closet platinism. Okay. While you're trying to do creative mathematics, while you're trying to, is it invent, is it discover, I don't know, but I'm trying to do it, something new. You're trying to solve a problem that's not been solved before.

You have the very strong feeling that the answer, there is only one answer, and it's sort of sitting there, and your job is to find what it is. you can't make it up as you go along. You can make up your exploratory route towards finding it, and we all do. But you have this mental image of moving through some sort of landscape or some sort of world looking for things. And in order to carry out that process, it makes it much more possible to do it. If while you're doing it, you have this illusion that these things exist. How can I look for something if it isn't there? I think what's really going on here, it's not so much music, it's a narrative.

We're telling ourselves a story. A mathematical proof is a story. We're acting out a narrative in our minds. And this is the way mathematicians discover or discover or invent, discover, or invent, discover, disvent new things. But if you then grab them and say, now, okay, that's what it feels like when you're doing it, but is it really that. Some of them will say, yes, it is really that.

They get very aggressive about it. But most will say, well, no, this is just a convenient way of thinking about it. and it's a very effective way of thinking about it. So Platonism is alive and well in the operative working philosophy of mathematicians, but they wouldn't formalise it into something and say, in the sense that a real hardcore philosopher would formalise it and say, this is what's really going on. John Barrow, what's your comment on that?

Well, say first, if you were to ask mathematicians what their subject was, you would get a strangely ambiguous answer. if you went around universities asking historians what their subject was or chemists, they would probably be able to tell you without any difficulty, but most mathematicians probably couldn't tell you. Why is that? Well, because they have this dichotomy about the invention and discovery, whereas Monday to Friday, as Ian said,

they might sort of act as though they were Platonists. If you caught them at the weekend in the armchair and ask them, they will be rather more philosophical and circumspect about it. So it's a curious ambiguity and puzzle that you can have different, different sectors of this huge subject, having quite different views about what its nature is. So for some people, it's just a game. It's like chess, you know, that you lay down rules and regulations, and you simply explore the finite steps and consequences of that. You catalogue all the possible moves, all the possible positions that all the possible games can have.

And other people, notably Roger Penrose, regard mathematics as very much existing in the platonic realm. and we pick out these counterintuitive ideas from that realm. Margaret Watton, finally. Yes, I mean, I think this is, I think Platonism is still alive and well among mathematicians and perhaps even more so among physicists that I've spoken to. But I think it raised a very interesting question. If the mathematics is in the world, then what is doing the calculating? And apparently the physicist Richard Feynman became a bit obsessed with this idea.

Are the atoms out there actually calculating these very complex. mathematics structures, or are the structures somehow there, and we are the ones who somehow reflect the calculating onto them? We'll have to end there. Everything but mathematics must come to an end, said Paul Erdos. Perhaps he's right. He's right this time. Thanks for listening.

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