In Our Time - Mathematics

Episode Date: May 6, 1999

Melvyn Bragg and guests discuss the way perceptions of the importance of mathematics have fluctuated in the 20th century, the nature of mathematical ability, and what mathematics can show us about how life began, and how it might continue. Galileo wrote “this grand book the universe… is written in the language of mathematics”. It was said before Galileo and has been said since and in the last decades of the 20th century it is being said again, most emphatically. How important is maths in relation to other sciences at the end of the twentieth century - will it ever be made redundant, or is it increasingly crucial to our understanding of the world and ourselves? What insight can it give us into the origins of life, and the functioning of our brains, and what does it mean to say that maths has become more ‘visual’?With Ian Stewart, Professor of Mathematics and Gresham Professor of Geometry, University of Warwick; Brian Butterworth, Professor of Cognitive Neuroscience, University College, London.

This BBC podcast is supported by ads outside the UK. 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 4. I hope you enjoy the program. Hello. Galileo wrote, This grand book, The Universe, is written in the language of mathematics. It was said before Galileo and has been said since,

and in the last decade of the 20th century, it's being said again, most emphatically. So, how important is maths in relation to other sciences at the end of the 20th century? What insight can it give us into the origins of life and the functioning of our brains? And what does it mean to say that mathematics has become more visual? Johnny Meesey and Stuart, who's Professor of Mathematics, Gresham Professor of Geometry at the University of Warwick, and one of the country's most prolific popularisers of mathematics having written or co-authored over 60 books on the subject.

Also an active research mathematician with over 120 published papers. His most recent book was Life's Other Secret, the new mathematics of the living world, published last year. Brian Butterworth is Professor of Cognitive Neuroscience at University College London. His book The Mathematical Brain has just been published. It looks at the way in which the brain deals with numbers and how this is influenced by cultural factors. Currently, he's conducting research in Britain, Italy and China on arithmetical processes in normal and brain-damaged adults and children. Ian Stewart, you quote Carl Friedrich Gail. who you consider to be one of the greatest mathematicians,

who said in the 18th century that maths should be about notions, not notations. Can you enlarge on this? Yeah, I think Gauss was trying to counter the idea that what maths is about is doing calculations with symbols, algebra, and these very formal things. Gauss was saying, no, it's about ideas, and those ideas can be very, very broad. And what we're seeing now is that this point of view is unfolding in a way. that nobody would have anticipated at the time Gauss said it, which is that mathematics in

certain senses is pervading all sorts of areas you wouldn't expect it to turn up in. Can I just stop it over a second? People listening, and myself, I'm listening, think that maths is about adding up, subtracting long division and so on, and people are much more sophisticated in that, but still they get...

So when you say ideas, can you give us an example? Okay, let's take, for example, the question of, when a living creature grows, say a baby growing in the womb. It grows from an embryo, it turns into a fetus, it turns into a child. And there are a whole lot of processes

going on in changes of shape, changes of form, changes of internal chemistry, all of these things. Mathematics has quite a lot to say about that kind of behaviour, that kind of process. And I think in the future it will have a lot more to say about that. So that's the idea,

that is an idea of the way that life develops rather than a simple adding up. You say that it's also becoming more visual. Can you tell us what you mean by that? Mathematicians have spent the last 50 years or so trying to turn visual intuition, informal visual intuition, the feeling we have in our heads for the shapes of things,

into a kind of logically precise way of thinking and things you can write down. And so instead of just using numbers and doing arithmetic, there's a kind of free-flowing mathematics of form, but it is just as mathematical, it is just as logical. The mention of topology, which is rubber-sheet geometry, the idea that a coffee cup is the same as a donut, is what the standard cliché for topology,

because a coffee cup has one hole in it. It's the hole in the handle where you put your finger through. An American-style donut has a hole in it. You can make a plasticine donut and deform it slowly into a coffee cup. And for topologists, those two things are the same. And there's a whole entirely rigorous mathematics based on that which captures the idea of continuous changes in form. Brian Butterworth, what do you have to say about this visual representation?

It's certainly true that one of the key visual areas in the brain, parietal lobe, is one of the areas that. The parietal lobe. That's the bit just behind your ear. Which ear? Both ears. All right. But the key one for mathematics is the bit behind the left ear.

Now, this idea of space is space relative to ourselves, as opposed to space at the final frontier, is dealt with by this part of the brain, and so is mathematics. So it's clear that there are intimate links in the brain between these two notions, and it might well be that somehow has mathematics. mathematicians become more sophisticated.

Some of the ideas of space, which they were previously unable to formalise, have now become better integrated into the symbolic systems that they use. I think the point that I'd want to stress is taking Goussa's idea of a notion, is that at least some of our most basic notions, the notion of number, seems to be built into our parietal lobe when we're born, so that we get a flying start. We're born, if you will, with a kind of start-up kit, which enables us to immediately see that there are three cups on the table

and there are more cups on this table than there are on that table where there are only two. We don't need to learn that. That's already in our brains. I mean, are you comparing it with the language instinct? I am comparing it with the language instinct. And you think it's as innate. You please tell me what you mean by that word.

as the language instinct? Well, I think we actually have better evidence that the number instinct, if you will, is innate. We know which specific bits of the brain there are. Could you tell us, could you give us some of the evidence? Yes. Well, first of all, infants are born able to recognise the number of objects in the display.

At what age? The day they're born. The day they're born? Yes. So the experiments have been done on our... neonates, in the first week of life, you show them an array of, say, two objects, and you show them another array of different two objects, and you keep showing them arrays of two objects, and they get bored with this.

Then you show them three objects, and they start to take an interest again, and then you keep showing them three objects, and they get bored, and you show them two objects again, and they gain interest. Now... What does this prove? It proves that they're sensitive to the number of things in what they see, and at a later age, possibly even when they're born, but we don't know this. I mean, are they more sensitive than the saying, seeing a little puppy by Gids' tail?

Is it a different sort of sensitivity? Is it just a sensitivity to what's impinging on them, or is it a particular numerical sensitivity? It's a particular numerical sensitivity. So you can change the object, so you can show them the same things and then change the object, and they'll be sensitive to that. But they will be more sensitive if you actually change the number of things that they see. So the very young baby is actually more sensitive, it seems, to the number of things around him or her than to what those things actually are.

So we're wired for numbers? So we're wired for numbers. What does that say to you, Inshu? I mean, that's fascinating. And I thought that the evidence that Brian presented in his book for that was quite compelling. This presumably, I don't know, I mean, there's an evolutionary advantage in being able to spot rather early that something around you has changed. there is a new thing has appeared.

You ought to be aware that something new has arrived. Two wolves rather than one, yeah. Yeah, or one extra wolf compared to the two sheep that there were. So, I mean, maybe it comes from that, but actually that's a sort of hairy story, really, isn't it? But, yeah, this does seem to be very strongly wired in. And what's interesting here is it's different from, I mean, my understanding is a lot of our perceptual abilities

are to some extent the potential is wired in but a certain amount of training is needed in the early weeks or months of life. There's a story about kitten. If kittens don't see vertical lines at a relevant time

in the development of the visual part of their brain, then they can't perceive vertical lines very well after that. Now this seems to be much more fundamental than that kind of thing. And that's very interesting. Do you... What I wonder is the extent to which

just how fundamental is that to what I would call mathematics. That's the start-up kit, but then you have a long way to go from there to the kind of thing that I'm assuming goes on inside my head because at some point it tells me about it and I write it down. I'm not saying that every baby is born able to grow up in Ian Stewart.

I'm certainly not saying that. Lucky then. But it does give them, if you like a head start, when it comes to learning the kind of cultural tools that are provided by the culture in which they're born. Now clearly it's very different being born into the 20th century with topology, with non-standard analysis, with all kinds of interesting novel mathematics that was being born, let's say, in the time of Archimedes. So if you give Archimedes, if I wrote down for Archimedes, the greatest mathematician of a, of, antiquity, a simple quadratic equation and asked him to solve it, he wouldn't be able to do it

because he just, from start, wouldn't know the notation. I'm sure he'd figure it out pretty quickly, but it means that that the baby, to become an Ian Stewart, would need to be born into a culture which has those mathematical tools and would have to acquire those mathematical tools, which means a lot of hard work and good education. And, I mean, these tools come in a variety of forms, and some of the most obvious ones you just don't think about. For example, we all have number words. Now, there are some languages in which there are no number.

You mean one, two, three, four. I mean, one, two, three, four. So there are languages in which you don't have number words. For instance. Well, there are languages in Australia, which only have words for one, two, three, and many. And there's Locke talking about the American Indians, isn't there? Yes.

And there's also, there are. But they could count up to 20, and then that was... Well, John Locke, yes, the English philosopher, was surprised that Americans couldn't get above 20. American Indians, we must stress on this. But we have to stress that they were American Indians, and they would use their own fingers to do counting, and fingers of people who are nearby.

And body parts, in Papua New Guinea, one tribe gets up to 33 by counting every body part. We can't describe on this program the last three body parts that the male uses to get... That's right. But actually our own number words are derived from body parts as well. What fascinated me, one of the things fascinating me about your book, is the relationship between a part of the brain and the actual fingers.

Yeah. That the fingers and the mathematical part of the brain are related. Yeah. And now, is it, I mean, that's just terrific. Can you just tell people what's going on there? Okay, well, there's a circuit in the brain, which involves the parietal lobe. And it...

Behind a ear. behind the air, and it links up with a bit in front of the air called the motor strip, which actually does the fine planning of the motor movements and one or two other motor areas in the brain. Control of our thing is actually quite complicated. But the top most decision area seems to be in the parietal lobe. And it may well be, and this is one of the speculations I advance in my book, that what's happened in the course of the development of the individual child and also in the evolution of our culture is that the fingers

gain additional meanings. They gain the meanings of the numbers that are associated with them when a child learns to count on his fingers or to do arithmetic on his fingers. And so you get what's called

activity-dependent brain changes so that the representation of the fingers expands and includes these numerical representations as well. So I ask a very odd question for a neuroscientist. I say why is it that numerical representations are in the parietal lobe, rather than in the temporal lobe, which is just below the ear, where most of our other knowledge is.

So it seems to me that might be one of the answers. And that takes us to send B, the venerable beads finger county. But it also took me on a fantasy about this relationship has always been talked about between mathematics and music. music and that Mozart could have been a wonderful mathematician, so we're told, and wonder whether the fingers which operate the piano are related to the finger. This is, this is we're entering into the sort of, getting way away from what you, you two really know about.

Ian Stewart, you're talking about visual mathematics, and you say that chaos theories emerge from mathematicians using visual geometric maths. Could you tell us how that's happened? I mean, we're all fascinated by chaos theory, and I've been told often enough by people, that we don't understand it well enough. So can you try to make us understand it from that point of view? If you look at where it came from historically, people were trying to solve, let's say, the three-body problem in astronomy.

You've got the Earth, the Moon, and the Sun, and you want to know how they move under gravity. And the answer is it's very complicated. And the way they were trying to solve it was the traditional way, write down the equations, solve them, very much think of it as a symbol manipulation problem, and you've got to calculate lots of numbers and you've got to get some formulas for the answer.

Now it turns out there isn't a formula for the answer. And the reason emerged, the best explanation why there isn't a formula came when people started to think about that problem in a geometric way. It's a bit abstract, but you think about it as the flow of time in a system like that. Things move.

And you kind of map out, not just one path that the planets might move on, but all possible paths. And then what you have is a kind of high-dimensional fluid flowing in a space. And you say, how does that fluid move? And the answer is it's a bit like when you're making bread, you stretch it, fold it, knead it.

And this kneading process, stretch and fold, pull things apart and then fold them back in again, is where chaos comes from. Because if you're stretching things apart, points that start very close together, move further and further, away. If you're folding things back in, they start to become independent of each other. This is the butterfly effect of chaos theory. This is the very small change at some stage can have a

huge effect later on. If you had two little grains of dust in your bread and you keep needing it, after a while, they're in completely different parts of the bread and they're moving around in completely different ways. This is why we need bread, it's to mix it all up. And so it's a geometric picture of the mixing, and that really is the key to chaos. One of the things in your book that interested me is that you're challenging the idea that DNA is the answer to the secret of life, is the secret of life. You give it all due credit and you know far more about it than I'll ever know. But what you're saying is that behind DNA, the deepest structure is mathematical and not biochemical.

Am I right? And if so, can you just give us some... That's the big message. I give DNA. DNA has to be given huge amounts of credit. science of DNA is extremely important. You won't understand life on this planet without it. But the question is, what is it that it plugs into?

What's the background into which it goes? And I think the current view is much too naive. It sees it as some sort of blueprint, some sort of map of the organism. People will talk about finding the bit of DNA that controls the growth of an ear or something like this. And it isn't really like that. What DNA is doing, it's embedded in a much more complex process. the process runs on the basis of the laws of physics and chemistry. If I push an elephant off a cliff, its DNA does not have to specify that it will fall under gravity,

but the law of gravity makes that happen. There's all sorts of processes going on, which biologists are interested in DNA tend to think, oh, these are just background, these are just default, these are going to happen anyway. We don't need to worry about them. But the problem is what the DNA's role is to modify those processes, to kind of drive them through the development. mental landscape. It's like if you're trying to drive

a car, it's no good just having a map. You have to actually know how to turn the steering wheel. You have to know which road's coming up. You have to have a lot of extra things that aren't really in the map. And the way all that stuff works is mathematical. Not in the sense that nature is doing

mathematics, in the sense that human beings have to do mathematics to understand what it is nature is up to. Is this useful? You study Brian Butler. You're studying the brain and your work with stroke victims and accident victims. And as I understand it, please correct me if I'm wrong. The great advantage of that, I mean, unhappy circumstances,

but if somebody suddenly can't count, you can work out which part of the brain has been damaged, which causes them not to count, therefore you can discover which part of the brain controls counting. How is mathematics, what Ian Stewart has been talking about, the mathematics behind DNA, is that relevant to what you're doing? I think what Ian has to say about mathematics is extremely interesting

and about its role in DNA. But also he had a very interesting metaphor in his book, which is the CD metaphor. It's not enough to have a CD, you also have to have a CD player. And that in order to understand how our brains come to be the way they are, it's not enough just to have the CD which contains the, if you like, the programme. You also need to have all the other stuff which turns this program into music, if you will. So we need to understand two things

when we're trying to find out which bit of the brain does mathematics. We need to find out, first of all, which bits we can knock out without affecting mathematics. So if we knock out bits that aren't the parietal lobe, on the whole mathematics, will remain intact. We've got quite a few patients now, including one rather prodigious calculator, whose parietal lobe is intact with everything else seems to be shot to pieces. And also, the other way around, which bits do you knock out,

will affect mathematics. So that tells us where in the adult brain numbers are primarily processed. But we also want to know how it comes about that that bit of brain is the relevant bit. And here, knowing about the DNA is quite important because we now know that there are populations of people with genetic abnormalities,

that is abnormalities of their DNA, who have particular difficulties with numbers. and so we can actually get to, we're getting to look at the CD to find out which bits of it seem to be 40. But then we have to see, of course, how this 40 CD creates music in the CD play and why this music becomes the way it does.

So I think this is a very interesting idea that many people in neuroscience are beginning to get a grip. As a species, both of you really, do you think we could function without numbers? Do you think that we could function with just language but not numbers? Ian Stewart first? I don't, well, I'm sure we could function. I suspect our linguistic abilities would push us in the direction of inventing that. If numbers didn't exist, we would probably invent them.

I mean, there's a massive part of Brian's book, which I pinched on my trail, saying he reads, in the course of reading a newspaper and going through a day, he's about 15,000 numbers you read, it out, something like that. You see, we can't read without a huge amount of visual processing going on the brain which can handle what is actually enormously sophisticated mathematics as the computer engineers who try to get computers to read written symbols. No, there's some very heavy processing goes. Now, a brain that can actually recognise the letter A on the printed page

can probably do arithmetic very easily in comparison. Okay, the connections may be in the wrong place and so forth, but in a sense the sophistication of the programme that's running in the brain to recognise the letter A is much greater than 2 plus 2 equals 4. Is there any way, it's ridiculous, we've only got about seven minutes left, but there's any way that you two can help people, not help, I know what's what I mean to talk about people, help, I hope they're enjoying themselves,

but you've talked to people who both have had strokes and accidents and people in societies which, they're conducting their societies, by only being able to count as 33, as it were. And your institute, that's Brian. You insured are talking about mathematics, which is in a completely different realm from mathematics that normally educated people like myself know about.

Is there any way in which you two are feeding each other? I think so. I mean, in the simple, I read Brian's book, got a lot out of it, and that's changed my thinking on some of these things. In what ways? Well, the idea, he's addressing the inside, story. I'm sitting there thinking, what can mathematics tell us about how the universe outside

works? And he, in a sense, is saying, how can our knowledge of the structure of the material work, the way our brains are actually put together as bits of machinery, tell us about what it's like to be inside a mathematical mind and how does that work? It's kind of the converse problem. And therefore, these points of view, I think, very much they feed off each other. Now, the two of us can't do a huge amount, but science doesn't work that way. But think of scientific communities across the world, putting the inside and the outside together in that way. At the very least, we should get a much better understanding

of how a lot of things work, because I think the problem, until fairly recently, science has tended to be quite compartmentalised. You miss big parts of the picture, and very clearly one of the big changes over the last 10 or 15 years has been that a lot of the boundaries between disciplines are dissolved. Mathematicians can come in a middle with biology and the biologists don't like it,

but every so often they say, oh, that's interesting. Maybe you've got something there. We'll go away and think about it. How important to you, Brian Butterworth, are the conclusions you've drawn from the discovery of discalculia? And could you just explain to people who can, we don't understand it, I can't even pronounce it properly, what we're talking about there? There are some people who are born unable to grasp simple numerical notions. I mean, they're unable to see that there are three cumps on the table. They can, as Ian suggested, learn a language which allows them to conduct a procedure of counting those cups. But unlike me, for example, the vast majority of people,

they can't just see that there are three cumps on the table. These people don't end up... Is it a bit like dyslexia? It's actually more profound than dyslexia in the following sense that for up until the last hundred years there wasn't widespread literacy in the world. There wasn't universal literacy. Literacy wasn't important evolutionarily.

But numbers have always been important evolutionarily. Even lions can use numbers up to a certain extent. Lions meant to be quite stupid. But being able to read and therefore disorders of reading is a result of... not being able to put together bits of brain that were designed for other purposes. But being discalculic

is a bit like being colour-blind. There's a special bit of the brain for seeing the world in a particular way. For colour, it's to seeing... The colour parts of the brain, V4, it's called. It's for seeing the world in colour. We do it automatically, we can't help it. The parietal lobes give us the ability

to see the world in numbers. We can't help it. It's automatic. To come back to my very clumsy question, I do, I do, but the two of you are here together on this programme, and here we are talking. But is it possible between you to have an understanding of mathematical concepts, which is what you're talking about, Ian Seward, brilliantly, I think, and at the same time I have a very limited notion of numbers. There's the question, this is the big, this is back to what we were saying a little bit earlier,

about how do you get from your start-up kid to what I suspect is not as sophisticated as it seems to be. I'm not convinced that when a leading research mathematician sits and does something absolutely amazing in 17-dimensional algebraic topology, that they're as far floating in the heavens above the ordinary mortals as we tend to assume. Because you've got to get there with a fairly ordinary human brain. There's, as Brian says in his book, there's a certain amount of training and other things go in. I think there is at least some sort of... predisposition to be, to respond to training in that sort of direction.

I'm not sure. I don't think you could take anybody and teach them this kind of mathematics. Brian Matam. Well, I think you can teach most people as much mathematics as they would like to learn. Discalculics being an exception. What I think is critical here is how much time they're willing to put into it and how well they're taught. Now, one of the things we don't know,

is whether, and this is a straightforwardly empirical question, is whether people who are born without this number sense are able, for example, to do topology, which doesn't depend so critically on numbers. I mean, I know there's some numerical aspect to topological theorems, but the question is, could they get a sense of geometry of topology, which doesn't depend critically on numbers? Answers we don't know yet.

Do you think you could stimulate parts of the brain to be better at mathematics? Well, you do stimulate parts of the brain to be better at mathematics by training it. I mean, we know that brain areas that are worked heavily tend to become more densely interconnected. They tend to get bigger. So if you give people a lot of maths to do,

then the maths parts of their brain are going to get bigger and better. This is a ludic question to throw at you with a minute to go in, Stuart. But you agree with Galileo, don't you, that the universe is written in the language of mathematics? I'm a little bit ambivalent on it. I guess about 95% of me agrees with Galileo, There's always that other 5% that sits. They're saying, yes, but you're biased.

Not just me as a mathematician, but our whole culture. You're going with quite a good team you've got out there. They could win the league. There's something going on in the universe on a very deep level that's mathematical, but I'm not sure it's the equations the physicists all out and say this is fundamental. I think it's something else. Well, thank you very much. There's lots more to talk about, but maybe we three will meet again.

Thank you, Ian Stewart. Thank you, Brian Butterworth. Thank you for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programmes about history, science and philosophy at BBC.com. com.com.com.com.