In Our Time - Maths and Storytelling

Episode Date: September 30, 1999

Melvyn Bragg and guests discuss the relationship between maths and storytelling. Is there a hidden mathematical logic in stories? The American mathematician John Allen Paulos thinks so. It’s an intriguing thought. Patterns, measurement, the logic of jokes, numerology from Leviticus to Alice in Wonderland, but does it really go to the square root of fiction? According to anthropologists, both have similar origins - in our prehistoric ancestors’ need to measure and assess the world around them. Both mathematics and stories need a shape and structure to make any sense. But does it go further than that? Is it possible to apply mathematical logic to literature or to reduce a joke to an algebraic equation? Or are literary imagination and scientific substance irreconcilable?With John Allen Paulos, Presidential Scholar of Mathematics, Temple University, Philadelphia and author of Once Upon a Number - The hidden mathematical logic of stories; Marina Warner, novelist, historian, critic, former Reith Lecturer and Visiting Professor at Birkbeck 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 programme. Hello, I'm joined today by John Alan Paulos and Marina Warner to examine the links between mathematics and storytelling. According to anthropologists, they have similar origins

in our prehistoric ancestors need to measure and assess the world around them. mathematics and stories need a shape and structure to make any sense. But does it go any further than that? Is it possible to apply mathematical logic to literature or to reduce a joke to an Albreik equation or other literary imagination and scientific substance irreconcilable? Jolin Pallas is presidential scholar of mathematics at Temple University of Philadelphia. He's the author of innumeracy, mathematical illiteracy and its consequences, which became an instant bestseller and was subsequently translated into 11 languages. Described as America's favorite mathematician, His latest book is called Once Upon a Number,

the hidden mathematical logic of stories, which looks at the relationships which tie the realm of mathematics and the world of stories together. The novelist, historian and critic Marina Warner is less sure of the links and similarities between mathematics and storytelling, a former wreath lecturer and currently a visiting professor at Birkbeck College London.

Her latest book, No Go the Bogueman, was a study of fear, and she has a special interest in legends, symbolism, and storytelling. John Alan Paulus, could you give us a gist of it? What do you think that maths and storytelling have in common and what do they contribute to each other? Well, the relationship's complicated one.

One point that can be made is that stories provide a context for mathematical ideas and applications. Almost any mathematical notion that's introduced can be placed into a story, a little vignette that clarifies it. Such a... random walk, if you're talking about random processes, the drunkard's random walk or exponential growth, he put a grain of sand on the first square of the checkerboard,

two on the second, four on the third, and so on, and so on. And it's two to the 64th on the last one. So also these stories of vignettes kind of exposed the limitations of mathematical notions. I mean, one in one is two, but if you take a couple, of popcorn and a cup of water and add them together, you only get about a cup and a half of soggy popcorn. So it's not that the mathematics is fine, this particular application isn't. More generally, I mean, stories, the narrative development of mathematical ideas and theorems

is important. Too often in mathematics education, there's a theorem, there's Koshis theorem, girdles theorem, central limit theorem, whatever, and they just kind of come out of the woodwork and Nobody knows the story, the development, the relevance of it to other notions in mathematics and physics. And going in the other direction, mathematical notions help clarify the structure of stories, probabilistic notions, notions from information theory, recursive function theory, help us get a grip on stories. And moreover, if you look at news stories, often the – statistics or whatever provide some muscle for this story? Now there's a lot there.

You're going to have to sort of try to decode that. Are you saying that I can see that theorems like enough for this conversation stories, but are you saying that inside mathematics itself, starting with 1, 3, 4, 5, 6, 7, 8, 9, 10, there is a strict logical similarity to the telling of a story once upon a time. You call it once upon a number. So once in a time, a little girl or a little boy lived in a little of a dark wood. Now, is that similar to, what is that similar to in mathematics? How is the logic, which is the word you use?

How is the logic similar? Well, I think the logic is quite different in a way. I mean, mathematical logic, for example, is extensional, which means it allows substitutions. If you have a three in a mathematical computation, you can replace it with the square and nothing changes. But that's substitutability isn't true in narrative logic,

intentional logic. It's kind of a nebulous... Yeah, but I'm interested at the moment in the similarities. Okay. So one more crack, and then I'll turn to and bring it. You say their origins are similar. Can you just give us some idea of the similar? We can talk about differences in the moment. What about the similarity of there?

Well, I mean, I think the development of stories and of mathematics were were kind of prosaic. I mean, mathematical counting led to formal arithmetic, measuring fields, and so on, led to Greek geometry,

and in something like the same way, I would imagine, stories developed out of more primitive, practical sorts of communications, like, hey, there's a behemoth over there, or come here, or how'd you get that fire started? And then gradually, these things were strung together into stories and then into literature.

in the same way that Greek formal mathematics was a development, refinement of these everyday rules of thumb that preceded them. Marina Warner, do you see the similarity in origination between stories and mathematics? Are you convinced that, or have you any evans, is there any evidence, that the one preceded the other?

Well, there's certainly a relationship in language because you get, in English, for example, you get the relationship between teller, like the bank teller, which is counting, which is like tax. and the idea, of course, telling a tale. And also there's a very deep symbolic tradition of taking and weighing and measuring the life of a person,

which actually goes through, of course, to St. Michael, the Archangel holding the scales at the last judgment, but goes back to ancient Egypt where your soul was weighed in the pan at death and seen if you were light or heavy. And you had to be light. You had to weigh against the feather of justice. and that was the sum, I mean a mathematical idea, the sum of your deeds and your character.

So very deep down, there is this, as it were, metaphorical idea of accumulation and of counting and of taking the tally of someone. You might interrupt me, in addition to tell her, I mean, words like a count. And the count is a story as well as a count. But I think that what is more difficult is the point, actually, that you were making more earlier. And that is the idea that there is some kind of, resemblance in structure.

And I think the difficulty there is that while we can talk about lots of things like doubles and the use of doubles, which are very, very strong in fairy tales, you can use, you can talk about repetition, you know, the idea of three wishes, the idea of nine ravens, the idea of 12 dancing princesses, all the ways that, I think those have got a different relationship, they're not structural. That is to do with the whole tradition of oral storytelling before print. I think there's a great need for pattern As you point out in your book

John that in the Bible people will seek patterns to do with stories from taking certain words and patternizing Is the need for pattern something which they have in common I mean in a sense counting your sheep is putting a pattern on your sheep isn't it And telling a story Well one of the ways it's important to count

And to make relationships in storytelling is, of course, to do with kinship systems. And that's very, very strong in the Bible and very strong in early myths all over the world. You think of Leviticus in the Bible? Yes, and, of course, in Leviticus, it's also dietary prohibitions, which are also very important

because they position you with regard to nature and, of course, in society, and they mock you out. But those are stories that are, again, about telling, making a network, making a pattern, but it doesn't mean that the story itself is structured. What worries me about the idea that there's embedded structure in stories

that is mathematical, is that then it sends you back to some slightly fixed position, which is the sort of formalist position, or even perhaps sometimes a union position, that you can't actually make new stories, that we're somehow hardwired to be caught up in this kind of, you know, this world of solid. Well, actually, you don't think, but John Alan Powell thinks that the mathematicians can be more creative, and more visionary than novelists and playwrights. That's what you're saying in your book, one of the things you say in your book. And one presumes that there's a common imagination.

You're not saying, Marie, that the imagination is sort of sliced up into little compartments and one can deal with fiction but not having anything to do with mathematics. I should imagine, I'm talking to, you know, I'm sure, I shouldn't imagine that mathematics have to be mathematicians have to be as imaginative as anyone else who's trying to invent something new. I couldn't agree more. I mean, I think the attempt to divide the mind up into little parts, the literary part, the mathematical part, is wrongheaded for a high quote better term.

And you're talking about patterns. I mean, searching for patterns is what basically what art and science is about. I mean, we're little islands of order in this kind of roiling thermodynamic sea of swirling static. And we have to search for whatever patterns are out there to stay alive. And whether we do it through art or through science, matters to some extent in another sense it doesn't

we need patterns to survive I'm just trying to hold to the similarities before we move on because Galileo famous as I said the language the book of life I discovered the secret of the book of life it is written

the book of life is written in the language of mathematics and he obviously used that analogy with great deliberation so what do you make of that I think that the world in some sense

without subscribing to full-blown Platonism I mean is mathematical in the Galileo sense. In the Galilean sense. And so, I mean, without, as I say, I mean, being a platonist, there's a book written recently, the number sense by Stanislaw Dahan, which has lots of nice items in it about the mathematical brain,

Brian Butterworth wrote another book recently with the same kind of theme. But one of the contentions that Dahan makes is, maybe mispronouncing his name is that numbers are social constructs, that somehow they're in our heads, and that seems clearly not to be the case, that they have some sort of independent reality, and not just numbers, but numerical relations or relations in general. So I'm not quite sure where I'm going with that, but...

Do you think that the number... Sorry, Marian, you say what you want, then I'll ask a question. No, I was just going to say that the difficulty is that... I'm not going to comment on whether numbers are inherent or constructed, but the difficulty is that any kind of... kind of use or application of them, especially outside the mathematical field, I mean, outside an equation, the minute you start applying them to stories,

or even to anecdotes, or to the kinds of stories you tell in your book, which are not, you know, which are to do with news items or fears, panics that take root, that then you actually don't have the platonic purity of the number. The number then does become a figure in this storytelling, and I think that you can't use numbers to kind of, as it were, rectify the mistakes of stories. It seems to me that they are figures in the stories.

Well, I mean, but by numbers, I mean, I don't mean only numbers. I mean, I think you don't want to have too narrow a conception of what mathematics is. I think part of the problem is that the mathematics of stories, narrative logic, intentional logic, or we want to characterize this nebulously defined field, isn't well developed yet. I mean, people don't understand intentional logic very, very clearly. But I think people sense that there's something there, that we can't formalize the notion of situations,

we can formalize the notion of structures, and that it just hasn't been done. I mean, and there's a desire to do it, and people often, I mean, bring in mathematical metaphors in a kind of wrong-headed way. I mean, there's a hoax in the United States last year or a couple years ago. As physicist Alan Sokol at NYU, I mean, sent this article into, as you know, this journal is full of scientific buzzwords and it was accepted and it was total nonsense. So, I mean, there's a hunger for mathematical metaphor, scientific insight.

But, I mean, there's more of a will to employ them than there is a knowledge about them, which is still informed. Can I just, I'm just stick to this, make the last point. I'm going to move on the next second. John has said that he sees mathematics can be just as visionary. In fact, he would claim sometimes more visionary and creative than novelists and dramatists. Now, would you concede that that's a possibility? And if so, what does that say about the two apparently different disciplines?

Well, I think actually he modifies that a bit in his book. You don't quite say that because you do agree that one of the differences is that math is, as it were, not subjective. You can't have a theorem coming out in two different ways, but you can have a story coming out in two different ways, as indeed many great writers have done. You come at the same problem from different viewpoints, and you get a different moral dilemma emerging from that.

And of course, any kind of drama going back to the Greeks, but certainly Shakespeare will also go around that. So that the mathematical equation... It doesn't modify it much. I found the quotation. Mathematics and scientific ideas frequently are more creative and visionary than novels or flies. That's pretty strong.

But frequently allows the fact that... can only that than the law of probability. What is the probability? I know, right. But I think that one of the, I mean, there is obviously a certain sort of beauty and wonder in these mathematical formula

which I don't understand. But I do think that the point of wonder and of the, and a fantasy, is something that is very hard to factor in if you're trying to see what is mathematical about stories. I've heard mathematicians talking about their work and talking about what's going in their head while they're

thinking about, which is almost precisely the same as descriptions I've read of writers and artists what's going inside their heads. I think the problem is that the way mathematicians think about the subject when they're doing mathematics which is full of little stories,

pictures, arrows, kind of half starts and so on, is very different than what's presented. I mean, once they've got the theorem they scrupulously cleanse it of any evidence that it was thought of by a human being and it had a history or

whatever. I mean, it's often in a very rigorous way. I mean, this is a definition, this is a lemma, here's a theory, here's a corollary, and it's done elegantly. Rather like religious texts did in a way, which were also a sort of story behind. Can I ask you, Marina Warner Warner? Early in this century, as you know, Vladimir Prop in 1922 wrote about, tried to reduce, thought he had reduced folk stories to almost mathematical structures. Yes. The princess in a folk story had a function. And he did, no, what do you make it out? Well, there were 31 functions, and seven spheres of action. And this was in the 20s. And his book, The Morphology of the Folk Tale,

was extremely influential after it was translated in the 60s, or the 50s, late 50s into English. So it had a kind of time lag of influence. But there were many other attempts. And what's interesting about these formalizing attempts is, of course, that everybody came up with different numbers of functions and different numbers of spheres of actions and different numbers of factors and things.

And you do see, I mean, the limits of that theory it seemed to me to be that you lose a sense of the context of response, not just the creator of the stories. You were talking about visions in the mind of the mathematician and the writer being similar. It's to do with the context of response. And for instance, Vladimir Prop himself says that the princess and the father can never be divorced.

They function always together. Well, that's a clear effect of social convention. in the society where, you know, these stories developed, where the princess is given in marriage by her father and therefore the functions. Now you will never say that a daughter and her father were the same function as any of us who have any children. No, this is no longer at all how society works. So there you have a loss, a leaching out of something that I've always wanted to put back into the looking at myth or fairy tale or indeed other forms of fantastic writing, which is that one must see what function it's performing for the actual receivers and those receivers change over time.

An example is something like the wicked stepmother who has been seen for a long time as an archetype, a figure of constantly recurring in Chinese. Children are 100 times more liable to murder their stepmother than their mother, yeah. Recent statistics, sorry. Really, yes. Well, I mean, that may have a great basis in experience. I mean, one doesn't know,

Well, we're coming on to statistics. We're with the solemn statistician here. We've got to be tough. But it seems to me that one must forget that one of the reasons there were so many stepmothers, wicked or not, is that women... But women were dying in childbirth. Until the 19th century, this was the main cause of female death. So the stepmother was a very, very common figure in people's lives.

I don't think it's just a psychoanalytical hostility towards, you know, the bad mother that is expressed in the fairy tales. That's stick to numbers. Nevertheless, these formalists said something. And maybe Joseph Campbell's hero of a thousand faces took that up as well. And certainly when Umberto Echo did a breakdown of the Fleming novels. He said the Bond novels in Fleming. He said these seven things happened. Enter Bond, enter villain, Bond and villain meat, enter girl.

Now, what do you make of that attempt to relate? I mean, because your book also says stories and mathematics diverge in certain ways. Do you think that is useful or not? I think it's useful, but I think, again, the tools aren't in place. I mean, to some extent, it's a useful typology, as long as you don't put too much weight on it. I mean, seven, eight categories, 31, 33. I mean, it's just an artifact. And if people recognize it as such, it's useful as far as it goes.

I mean, I think tools from recursive function theory, artificial intelligence, where you analyze so-called scripts, like what do you do when you go into a restaurant, what you do when you go to a movie and so on. I mean, is another way to get at the structure of stories, the structure of intentional logic. The fact is, I mean, that there are kind of fairly easily recognizable structures to stories, to interchanges. But formalizing this logic hasn't been done yet. There's something called situation logic where you try to formalize situations. And I think that that told some promise. again it's

a laudable attempt but it doesn't go very far and it isn't all that impressive One of the things that I was puzzled by and didn't know how you would express it or what you'd say about it is what seems to me to be the defining characteristic of a lot of the fairy tale material

and mythological material and that is transformation metamorphosis that something changes into something else the beast turns into a beautiful young man the poor, you know, peasant girl in the rags turns into the princess. Now, that's very, very deep in...

Now, how do you express that in... Because a number, if changed, is changed. It's part of what I was saying about the extensionality. I mean, in mathematics, three, square of nine, cube or 27 means the same thing, but in, you know, in everyday life, that's not the case. I mean, Superman equals Carr Kent, but nevertheless, or, you know,

there's a story about... But the kid calls home, and he asks for the cat, and his brother says, the cat died. And he said, oh, can't you break that to me more gently? And he says, couldn't you have told me, you know, the cat's on the roof, and then you couldn't get it down. I call it next time, and gradually, in this way, say the cat died. And the brother apologized for being so brusk and telling him about his cat's death. Anyway, the guy says, well, anyway, how's mom? He says, well, she's on the roof.

Okay, so, I mean, in 30 seconds, the meaning of on the roof changes, and there's a no way present to capture that but in court it is in the wider mathematics I mean we mustn't think of mathematics as multiplication tables no right things are changing a lot all the time as well aren't they but changing I mean according to rules

I mean according to functions and theories change but a perception change as a result of what we know I mean Einstein bent Newton and so on yeah but still he bent Newton but Newton is, I mean, for all practical purposes, I mean, unless you're really...

So there's no relationship between mathematics and what Marina was... Not in that sense, not right. You talk about a relationship between mathematics and humor, and a structure of a joke, and we know from what we've read and what people have said, particularly comedians, writing comedians, particularly. I actually do talk very specific,

almost sometimes in engineering terms, about how they pace, play, work out, and deliver a joke or a line of comedy on it. And we've got the great example of we can bring Lewis Carole in here for all sorts of reasons, mathematician. So you say they exist on the same continuum. What do you mean by that? Well, you can think of both mathematics and humours being forms of intellectual play,

different ends of the continuum. In the mathematics, more intellectual, play, humor more playful, but in the middle are brain teasers, puzzles, paradoxes. But, I mean, many jokes have a sort of quasi-mathematical structure. A lot of the same operations and structures that are common in mathematics, re-reversal, juxtaposition, self-reference. Why do philosophers ask so many questions? Why shin philosophers ask so many questions?

Redukti-a-a-absurdum is used all the time in mathematics for the sake of disproving some assumption that whatever in humor is used for the pleasure of the reductio. self-reference these various syntactical operations that Ulippo, the French group, is involved in substituting the seventh word following a noun in some holy book and you get something funny or whatever. And then there's self-reference, non-standard models, is a notion in logic.

I mean, you go to a computer dating service, ask for somebody who's short, gregarious, likes formal clothes, cold weather and the computer sends you a penguin. It's a non-standard model of these axioms. In some sense, the structure is the same as non-Euclidean geometry. What satisfies this, this, and that axiom of Euclidean geometry without the parallel postulate, and you get this saddle-shaped surface. So the logic of many jokes is mathematical, quasi-mathematical.

But the effect of many jokes is also because of the community listening, and their values being overturned. Right, I'm talking about the formal aspects. The trouble is that if you take that kind of meaning out of it, that sort of social and ethical meaning, you actually lose so much of the psychological and moral context. But I'm not talking about that. Of course, yeah.

That's the problem of actually saying that because people attach themselves to these notions of structure and then say, well, jokes all sorts of be like that. And then you get the target joke, the joke which makes the butt of someone. And very often the Prattful joke is to do with actually demeaning someone else. All right. And we have to think about that. The psychology, the emotional climate is much different than the structure. We've just got a few minutes, but let me dive right.

Louis Carroll was a fine mathematician, and he wrote Alice in Wonderlandland. Great work of imagination. Now, does that, as it were, prove your case or prove your case? First with you. Well, I think, I mean, I like, one thing about Carol that's always struck me is that he seemed bothered and preoccupied, obsessed with the same kinds of misunderstandings of language and language games as Wittgenstein did.

I mean, you know, you talk about having a pain in your foot, your foot's in your shoe, you don't talk about having a pain in your shoe because Ian is not a transivulation. And there's so many instances in Carroll of these kind of misunderstandings

of language games. But you see that Alice in Orlando is formed deeply by his mathematics. That's what I'm getting at. Oh, yeah. I mean, she puts your hand on top of her head. Am I growing? That's not a way to find out. Jam today, jam yesterday, jam tomorrow, but not jam today. I mean, that misunderstands dates, or would you be so kind as to stop a minute? I'd be kind enough, but I'm not strong enough. And there are all these sorts of misunderstandings of the language.

You see, these are mathematically based, they're rooted. Yeah, I think one problem is that people have a very narrow conception of mathematics, It's computation or multiplication of division, or, as Carol would say, uglification and derision. And, I mean, if, and it's understandable. If all everybody ever did in English class is diagram sentences for 12 years, it wouldn't be real surprising if when I got to college, they didn't have a keen appreciation for literature.

But given suitable allowance for hyperbole, that's what most people's mathematics education is like. It is aglification and derision. And they have this very narrow conception where, you know, which doesn't involve stories, which they think of mathematics is very hierarchical. And I think these misconceptions are part of the reason people find it such a stultifying distasteful subject.

Well, not everybody, I think. Not everybody, but so many. Too many. Marina. Well, I think that Carol is very, very aware of how the adult world pressed on children, and he took the child's part with Alice. And he actually gives the Red Queen, who's one of his tyrants, but he gives her the point that I would make about it, He says, what you suppose is the use of a child without any meaning.

Even a joke should have some meaning, and a child's more important than a joke, I hope. So he was interested in the child as a person and was worried that these logical structures which he played with so brilliantly, which were to do with jokes and puzzles and games, did actually somehow lose the sense of the person in the child. Oh, and I think that there's a joke that's not very funny, but it's relevant. Two hard-headed scientists are talking, and they're saying, well, why do we do away with these narrative notion stories? I hate that stuff.

That's just stick to facts and numbers. And the other one says, I couldn't agree more. It just resolve mathematical facts, scientific facts, and so on. And the joke, such as it is, is that they're using these notions of hate and clarity. And so mathematics presupposes and grows out of these murky notions of storytelling, narrative, and so on. Thank you very much. That was a gallop, wasn't it? But I enjoyed it. I hope you did. That's John Olin-Powlus and Marina Warner. And thank you for listening. We hope you've enjoyed this Radio 4 podcast. You can find hundreds of other programs about history, science and philosophy at BBC.com.com.com.com.