September 9, 1994

Back to Articles on the Public Understanding of Mathematics

Oh I hate math! I can't do this stupid algebra problem.

Hello Katie. I'm the magical math fairy. Having problems balancing your equations?

Yeah, I hate this stuff. It's so boring.

Well, you know Katie that math is more than just solving equations. Math is all about hearing the patterns and symmetry in music.

[synthesizer music]

Cool!

Or calculating the rhythm of motion. Like when a jaguar leaps at you. So you see Katie, mathematics isn't just about number crunching. It's about life.

- Host, Ira Flatow:
- Coming up on
Science Friday: The Beauty in the Beast Called Math.

- This is an edited version of a National Public Radio program which aired on September 9, 1994.
Science Fridayhost, Ira Flatow, interviewed guests Keith Devlin, Dean of Science at St. Mary's College of California in Moraga, and William Dunham, math professor at Muhlenberg College in Allentown, Pennsylvania and author ofThe Mathematical Universe. You are invited to reference Devlin's and Dunham's responses in your public awareness efforts. Their comments may be especially helpful if you are preparing for an newspaper or radio interview.

- Ira Flatow:
- This is Talk of the Nation Science Friday. I'm Ira Flatow. Did grammar school stifle your budding interest in numbers? Did college calculus clobber any nascent love for mathematics? It did for a lot of people, but for some the intrinsic beauty of heaven and earth was unlocked through their study of mathematics. This hour on Science Friday we will try to get past the numbers to talk about the elegance found by the world's greatest mathematicians as they sought to understand nature and we'll explore how artists, philosophers and writers like Escher, Descartes and Lewis Carroll used mathematics in their work. Now let me introduce my guest. Keith Devlin is Dean of Science at St. Mary's College of California in Moraga. He's editor of FOCUS, the news magazine of the Mathematics Association of America, and author of
Mathematics - the Science of Patternspublished by W.H. Freeman and he joins us from KQED in San Francisco. Welcome to the program.

- Keith Devlin:
- Good afternoon Ira. It's good to be here.
- Ira Flatow:
- Which category were you in? Were you in the category where people had such a terrible time with mathematics in grammar school you never wanted to look at it again or did you find the beauty of nature in it?
- Keith Devlin:
- I had a dreadful time with mathematics in high school until I reached calculus and by then I was hooked which is quite the opposite of most people.
- Ira Flatow:
- What hooked you about calculus?
- Keith Devlin:
- I was in school in the '50s and '60s and the sexy careers were aerospace, nuclear physics, telecommunications. That's where all red blooded Yorkshiremen, which is what I'm one of, wanted to be. Moreover, if you wanted to go into those fields, you were pretty well guaranteed a good job, a well paying job, an interesting job. So I actually started out wanting to be a scientist. I wasn't sure whether I was going to be a physicist, an electrical engineer, or something in the science or technology business. I pretty soon learned that I had to know quite a bit of mathematics to do that. And the end result was that I ended up being a mathematician.
- Ira Flatow:
- I've found from my own personal experience that once I could get beyond learning the rote stuff and into some of the formulas that describe the way nature works, you did see sort of the inner beauty, the kind of natural laws of nature that were quite elegant and beautiful to look at. You had to get past that first, number crunching stage.
- Keith Devlin:
- That's a really difficult hurdle. People ask me, because there is a huge debate going on in the nation and the rest of the world, about how do we get education going so that people are hooked on mathematics. It comes downs to motivation. If people are motivated, they will see the beauty because it is there. It is there waiting for anyone to see. But you are right. It is a long haul and it is a painful haul to get there.
- Ira Flatow:
- So what do mathematicians, as they grow up to become mathematicians and observe the world, what do they see? What do they view their role as?
- Keith Devlin:
- There are probably as many different answers in some respects as there are mathematicians. I guess I see myself as just a regular person who looks at the world through a pair of eyes that has been conditioned by years of mathematics. So I actually see mathematical patterns in the world. I see patterns that may not be mathematical but are perhaps amenable to mathematical techniques to study them. But it is just another slant on the world. I look at the world through a certain set of spectacles.
- Ira Flatow:
- Do mathematicians view math as something practical or do they view it as sort of an exercise in mental gymnastics?
- Keith Devlin:
- Oh, almost to a person, mathematicians--professional mathematicians--view it as mental gymnastics. They are almost uniquely not interested in the applications. Many of them don't know the applications, in a sense, don't care. In the back of that we all know that there has been almost no discovery in mathematics in 3000 years of history which hasn't found, not just one, but many applications, many crucial applications. So if people said, why should we pay you to have this great time playing intellectual games, we just say, look at what has happened over the last 3000 years. These great games have led to all the great things in society that we take for granted.
- Ira Flatow:
- Is there anything in nature that math cannot describe? Is it limitless?
- Keith Devlin:
- That is a good question. I am tempted to say that the answer is "no." There are some things that are clearly more or less amenable to mathematics. So the answer is probably "no" although you would have to be very ingenious to see the connections in some respects.
- Ira Flatow:
- Let's talk a bit about cultural changes. You know that a lot of the math that we are exposed to certainly comes from western societies, western cultures. Do Eastern religions look at math, approach it differently than western cultures do?
- Keith Devlin:
- You are on to a area that I know very, very little about. It is the case that a lot of the mathematics that we regard as western actually began in China many thousand of years ago. Way, way back, a lot of the mathematics came from the Indian sub continent. So it is not uniquely western, although nowadays it has a very western slant.
- Ira Flatow:
- I notice from my own study of mathematics and from reading your book that a lot of math that was done many years ago (we are talking hundreds, thousands of years ago) dealt with motion. People wanted to describe, to find a way of using numbers, to describe motion. What is this fascination with numbers and motion?
- Keith Devlin:
- They are two separate things. The fascination with numbers goes way, way back to the very beginning with just counting things. The origin is in commerce and in measuring things. A person needed to know how much land he had or what the boundaries were. Originally there was a necessity for numbers. The fascination really began with the ancient Greeks, where modern learning began, modern western learning. The fascination began with people like Euclid actually looking at numbers and looking at number patterns, looking at properties, primeness and non-primeness. That was an intellectual interest and that was really what symbolized Greek learning.
- Ira Flatow:
- Yes. Especially the thinking of Zeno and Zeno's paradox.
- Keith Devlin:
- Right. And then we get the interest in Zeno's paradox which is very much an interest in motion. How do you use static things to describe motion. The answer to that was not really figured out until the seventeenth century. And really what happened is that mathematicians created the equivalent of the movies -- they created out of a whole moving sequence, a moving sequence of still pictures. And that is really the key to the calculus. Motion is broken down to a sequence of still pictures. But it wasn't easy. It was a couple of thousand years before Newton and Leibniz in the seventeenth century developed what we now call calculus, which was a way of handling motion with the static apparatus of mathematics. And it was another two hundred years before mathematicians really figured out why the calculus worked.
- Ira Flatow:
- If fact, they are still fighting over who discovered it. Right?
- Keith Devlin:
- Well, I am an Englishman, so I am going to say that Newton discovered it--pure and simple. But you are right. The Germans have a claim through Leibniz. And in fact, one of the interesting things about mathematics is that discoveries are almost never made by a single person. Almost always, one or two persons around the world discover something about the same time. Sometimes as many as a dozen people. And that should not really be surprising because mathematicians work in a cultural context and every idea has its time.
- Ira Flatow:
- Well, you know years ago, hundreds, thousands of years ago, there were no science funding organizations to fulfill grants, to pursue science and mathematics. What kinds of people would hire a mathematician? Why would you want a mathematician or why would you find it useful for a mathematician to work for you?
- Keith Devlin:
- Oh, it was really the era of the private donor. I'm actually with a private institution. Therefore I am very aware of the private donor. It was really private donors in the sense of the local duke or the local lord of modern England, titled people, gentried people, landed people, people with money. It really was just the largess of people who felt it was useful to have someone around who was developing mathematics. The other side of the coin is that these people also earned their bread and butter by teaching the sons and daughters of those people--teaching them the basic arithmetic and basic mathematical ideas, which is actually what people like myself do all the time now. We do our own research. We have our own scholastic interests. And society pays us in part for the developments that our work, our research may lead to, but also because we teach their sons and daughters.
- Ira Flatow:
- Let me bring on another guest. William Dunham is a math professor at Muhlenberg College in Allentown, Pennsylvania. He is author of
The Mathematical Universepublished by John Wiley & Sons, and he joins us from WHYY in Philadelphia. Welcome to the program.

- William Dunham:
- Hi!
- Ira Flatow:
- There is a wonderful quote from your book from a female mathematician, and I don't think that a lot of people realize that there are very high-powered female mathematicians.
- Ira Flatow:
- I hope I get her name right. Sonya Kovalesky. And she wrote "many who have never had the occasion to discover more about mathematics, confuse it with arithmetic and consider it a dry and arid science. In reality, however, it is a science which demands the greatest imagination."

- William Dunham:
- A wonderful line.
- Ira Flatow:
- Mathematics is more than just number counting is what she is saying.

- William Dunham:
- It is absolutely. I like this analogy: The mathematics that you learn in elementary school, perhaps the basic algebra where you do drill after drill problem which I am afraid extinguishes a lot of enthusiasm, has been likened to grammar. It is learning grammatical rules, but no one would contend that that is all there is to English. You want to go on and read Shakespeare and
Moby Dickand all those wonderful pieces of literature. Unfortunately, a lot of people stop with the grammar of mathematics, with the drill problems, and don't go on to the literature of mathematics.- Ira Flatow:
- Your book is full of interesting surprises about some of the great mathematicians. One surprise that I came upon was a brief bit about Pythagoras. And it says nothing survives of Pythagoras' work. Nothing! In fact there are some people who doubt that Pythagoras ever even wrote the theorem that carries his name, and other people doubt that he ever existed.

- William Dunham:
- Yes. It is lost in the mist of time, I am afraid. They sometimes describe Pythagoras as "half mythical." We don't have his writing in any formal sense. But there is a lot of tradition in classical mathematics attributing certain things to him.
- Ira Flatow:
- You mean this has been verbally handed down from generation to generation?

- William Dunham:
- I suppose, initially. Now, it did get written down, but it might have been written down 700 years after he worked. And then someone would attribute this great theorem or that great discovery to him. Whether this actually happened as legend suggests, is anybody's guess.
- Ira Flatow:
- What I also find fascinating is your mention of a Chinese proof that is identical.

- William Dunham:
- Yes, there is. It has been dated at roughly the same time as Pythagoras. It is quite old, virtually follows the pathways that people think Pythagoras would have followed and yet was discovered independently.
- Ira Flatow:
- Also fascinating is the whole idea of the hypotenuse. We all know that if you have to walk across the field, it is shorter than going around the sides.

- William Dunham:
- Yes.
- Ira Flatow:
- Why did it take somebody to have to put this down and to have to talk about it so much, and create a proof---whether Chinese or Pythagoras?

- William Dunham:
- Now the idea that the hypothenuse is shorter that adding the two legs, that would have been quite evident to anybody. Sometimes they say that it is evident to an ass because if you put a mule or a donkey at one corner of the triangle, it will walk down the hypothenuse to get its food rather than going down one side and across the other way. It knows the short cut.
The Pythagorean theorem is a little different. It says that the square on the hypothenuse is the sum of the square of the legs. That is far less obvious and is, in fact, quite spectacular that it should work that way. And when the Greeks discovered it, I'm sure when the Chinese discovered it, it was a big deal, really quite remarkable.

- Ira Flatow:
- Why would anybody even think of the squares instead of just the lines? Why would they say, "Hey, this line is one side of a square, and this line is one side of a square, and the hypothenuse is another side of a square?" Why would they think about adding squares up?

- William Dunham:
- It is fascinating, isn't it. Well, when you see the proof, if you look at it, either the Chinese or the Greek one, you can see what is going on. "Oh yes," you say, "I see that." But who had that initial burst of insight? That's the mark of genius. And who knows why somebody, at first, thought to do that.
- Ira Flatow:
- Are there periods in history where suddenly there are burst of mathematical research and then it quiets down? And then there is another burst, a renaissance of it. And if there is, is there a reason for it

- William Dunham:
- I think that there are such periods. It is hard to tell the reason. Certainly in classical Greece, there was this wonderful period stretching from Euclid to Archimedes with all sorts of amazing discoveries. Professor Devlin mentioned the discovery of calculus in the late seventeenth century. That was a wonderful period. I might put in a plug for Professor Devlin's book called
Mathematics: the New Golden Age. There he's writing about the discoveries today going on in mathematics which is perhaps another period which will be remembered for its mathematical innovation.- Ira Flatow:
- We're in a golden age today?
- Keith Devlin:
- Maybe I should come in now since Bill is so kind to plug my book.
- Ira Flatow:
- That's your cue. Why would you consider today to be a golden age in mathematics?
- Keith Devlin:
- Just the amount of mathematics that is being studied, the amount that is being developed. A lot of current mathematics started with the computer. The computer actually came out of mathematical research. Its existence was predicted by mathematicians in the 1930s, in fact, before then. The modern computer has thrown up a whole series of problems that are very definitely mathematical and a lot of current research. So we've got a whole range of different new mathematics thrown up by a new computer.
- Ira Flatow:
- There is a good match between musicians and the ability to program music. Have you ever heard that?
- Keith Devlin:
- A lot of people have pointed out the analogy between mathematics and music. And you can see one analogy when you look at a sheet of music or a sheet of algebra. Unless you have been trained to read it, you just see garbage. Musicians who have been trained can look at a sheet of music and see the tune in their heads. They can perhaps hear it performed by a large orchestra. To a mathematician who has been trained in mathematics, you can look at a page of algebra and see what is behind it. You can play this mathematical symphony in your head.
- Ira Flatow:
- Do you see it as objects, or do you see it as numbers?
- Keith Devlin:
- I see it as objects, very abstract objects that don't have any color, that don't have specific shape.
- Ira Flatow:
- I think that is the real difference between somebody who loves mathematics, who see mathematics and somebody who just sits down and see a page of boring numbers. I was reading your books last night and my wife looked at me and she said, "Oh! Gosh! Look at the numbers in that book. You're reading this?" And I've to admit that I don't see the objects there either. I still see the numbers. At what point do you get to see the objects behind it?
- Keith Devlin:
- In my case, I was about seventeen or eighteen when I suddenly saw the light and mathematics was never the same again. And from then on, I didn't want to do anything else in the world.
- Ira Flatow:
- Now let me bring up a sore point that we have been talking about over the weeks and months. It has to do with physics. A lot of physics is mathematics. If they are talking about physics in 24 dimensions, you know string theory, super string theory, you know it's got 24 dimensions, it's got 10 dimensions, whatever, how do you see that object
- Keith Devlin:
- You see it in an abstract way. It is not the same as picturing a three dimensional object. What you do is you see aspects of it. So you look at little bits and you sort of synthesize it into a whole. It is very analogous to looking at a photograph. If you look at a two dimensional photograph, or a TV screen, you see it as a three dimensional object. That is what is going on with higher dimensions. You see certain aspects of them and you somehow try to synthesize them in your mind
- Ira Flatow:
- I think some of the most familiar, sort of mathematically artistically abstract painting might be Escher paintings or drawings. What other artists knew mathematics and said, "This will help me become a better painter, or express my feelings better?
- Keith Devlin:
- There was certainly Albrecht Durer and da Vinci. The renaissance period certainly had ideas of projected geometry and so forth. Max Weber with his work on the fourth dimension. The cubist program is very, very mathematical if you look at it. It's looking at things and almost digitizing them and pulling them apart into mathematically precise patterns. Many, many artists, perhaps not even explicitly, did not even realize that they were looking at things mathematically. But when you look at what they produced, it was mathematical
- Ira Flatow:
- Dr. Dunham, is Isaac Newton considered the number one mathematician?

- William Dunham:
- He is way up there, near the top, I guess, in the AP rankings. Usually they cite Archimedes as the great Greek mathematician. He did things that were centuries ahead of his time. Quite an awesome intellect. Newton is usually put up there. There is a German mathematician name Carl Friedrich Gauss, who may not be so familiar to the general listener, but Gauss is usually ranked near the top
- Ira Flatow:
- We have to stop with Gauss, and tell everybody the test that the teacher gave him when he was four years old. Do you know that story? Add the numbers from one to one hundred.

- William Dunham:
- Yes. Supposedly, who knows if this was accurate, the teacher wanted all the children in the class to add up the first 100 numbers. Little Carl Friedrich came running up almost instantly with the answer correct He had done it by very cleverly listing the numbers from one to a hundred in ascending order; beneath it listing them in descending order, adding column-wise and each column will add up to be one-hundred-and-one and there are one hundred of them. Take half of that and you've got the answer. He was quite insightful, even at that young age. And that was just the first sign of a career that would stretch for decades and produce all sorts of glorious mathematics.
One personal favorite is Leonhard Euler who was a Swiss mathematician from the eighteenth century who did a volume of work that is almost unbelievable. They have been publishing his collected works all this century and they are not done yet. They are up to volume 77 or something. And there will be many, many more volumes that will run into the twenty-first century. He filled bookshelf after bookshelf with mathematics producing something like 800 pages a year, every year of his life. Some people point out that most people cannot even type that fast, let alone do mathematics that fast. He also did not just a great quantity of work, but enormous quality -- incredible discoveries that answered questions that had been lurking around for a long time, or that started whole new branches of mathematics. But what I think is the particularly interesting angle on this is that for much of his life Euler was partially blind. And for a good portion of it at the end, he was totally blind. And yet, that did not in any way reduce his output. He would dictate mathematical formulas to scribes who would write them down. It was an awesome and really quite inspirational performance by this amazing person.

- Caller John:
- My question is about topology. The four color-map problem was solved by massive computer power. Was it an elegant solution or was it a brute force solution?
- Keith Devlin:
- It was both. The problem goes back a hundred-fifty years or so, may longer. It is a simple problem. You've got a map of the country. What is the minimum numbers of colors that you are going to need if you want to color the different counties or the different states. And when you are coloring, the idea is to distinguish them. So you want to have different colors for adjacent states or adjacent counties. What's the minimum color it would take for any map. If you pick a map off the shelf in a library or a bookstore, you will be able to do it with three or four colors almost certainly. Certainly you can do it with four colors because it was found to be true in 1974 by two American mathematicians, Apple and Hawkins. Actually one was German by birth. They finally proved conclusively that whatever map you draw with billions, zillions, trillions of whatever countries or states you can always color it with at most four colors. You don't need to buy more than four pots of paint if you want to color a map no matter how complicated the map is. The final solution to that problem in 1974 was a mixture of very, very clever mathematical analyses using ideas that people had tried when the problem was first raised by Guthrie in London, together with raw, brute computing power. It was a modern age proof in the sense that you could not have done it without the computer. But you could not have done it without the people as well. It was a symbiosis of human, mental, intellectual ability and machine. It was both brute force and mental exercise. There have been at least a couple more proofs like that since. And I think that we are just going to have to live with proofs like that.
- Ira Flatow:
- Yes. It is true, isn't it Keith, that when it came out, it was rather controversial among mathematicians.
- Keith Devlin:
- And to some extent, I think that it is still a little controversial. This questions whether the proof is true, whether it is correct. But as many other people point out, it is true even if a computer is not used. Modern proofs of results can stretch out over hundreds of pages. You know that you could write a whole book about a single proof. And then it is anyone's guess as to whether that proof is right or not. It is very easy to find a mistake in a proof and say that it is wrong, but you can never be sure that a complicated proof is right whether or not you are using a computer. And people tend to make mistakes more than computers, providing that you write the right programs.

- Caller Jim:
- I wanted to dispute the general idea that in school children should take a lot of mathematics. I don't see any advantage to taking, say, algebra I and maybe a little trigonometry. This business of learning calculus and higher mathematics would be useful only to people who tend to exploit it as an interest in life or as a means to pursue a professional life. I don't see how learning, for instance, calculus is going to do a person any good who is going to work in an office.
- Ira Flatow:
- Jim, you don't think that you might have a nicer or a better appreciation of nature or how things work by learning a little calculus?

- Caller Jim:
- Well, I suppose so but I think that in today's environment, they would spend their effort more profitably learning more chemistry, or nutrition, or child raising, or something like that. How many people in their life time as adults will have to solve a problem in trigonometry?
- Keith Devlin:
- Essentially, Jim is right. But you could say the same thing: Why teach them to appreciate drama? Why teach them to read good literature? Why teach them to play an musical instrument? You know, we can live without any of these things, but the more we know, the more we understand, the more ways we have looking at our world, the richer we all are. And when you cut out something, you are closing a door. In my own case, if I had not studied calculus, I would never have seen this wonderful world that I have explored.

- William Dunham:
- If you do not know mathematics, you cut out a lot of options in your future. At Muhlenberg, where I teach, many majors require some sort of mathematics aptitude and knowledge even if you don't quite expect it. You come along and want to be a psychologist. Well, you'd better know some statistics. You want to do economics or business, you'd better know calculus for that. There are all sorts of ways that mathematics creeps into the various disciplines and subjects and if you have not seen it, you are going to have a devil of a time.
- Ira Flatow:
- Let's talk a little bit about women in mathematics. I think that if you polled a hundred people on the street that they might name one woman Nobel prize winner -- maybe Madam Curie. But if you asked someone to name a woman mathematician, I think they would be hard pressed to come up with even one name.

- William Dunham:
- Possibly so. I devote a chapter in my book to this issue because women are certainly an under-represented group in mathematics which is not to say that they have not contributed. There have been women down through the centuries. But my thesis is that women have had to face obstacles and try to overcome hurdles that men can not even imagine in terms of their mathematics. The act of discouragement of women in mathematics has been really quite horrible and one that has people like Immanuel Kant, the great philosopher, saying that women might as well have beards as trouble their pretty heads about geometry. I believe that is a direct quotation. That is not very encouraging. Other women were not allowed to go to college. It is hard to believe these days, but there were barriers. Women had to eavesdrop at the door to learn the mathematics. And women were not allowed to have university positions and the sort of support that you get when you are at a college or university which is necessary to do mathematics. So the obstacles were everywhere and I think that that certainly accounts for the low numbers of women in the history of mathematics. I think that things are changing. I certainly hope so. I cite a statistic, that nowadays in the U.S. about half the math majors at U.S. colleges and universities are women. And I think that's a statistic that would have been unthinkable about a century ago.
- Ira Flatow:
- That's amazing. In fact we've heard stories on this program about the nineteenth century patent office refusing patents from women without their husbands co-signing them.

- William Dunham:
- The stories are terrible, There was a twentieth century algebraist named Emmy Noether. She wanted a position at a university in Germany and the male faculty were frumping about this, "Oh No! Here comes a woman wanting a job." And, supposedly, a very great German mathematician named David Hilbert rose to speak and said something like this: "Gentlemen, Gentlemen. The sex of the candidate should have no bearing on this. We are a university and not a bathing establishment."
- Ira Flatow:
- Of course, to be any mathematician years ago, you had to have a lot of leisure time because it basically involved sitting around and thinking a lot. You had to be independently wealthy or have somebody supporting you.

- William Dunham:
- Absolutely. And one finds some of the women who were prominent in mathematics back through history had that luxury of being well-to-do, had tutors, or had brothers with tutors. Being affluent certainly helped. Archimedes, supposedly, was a royal person.
- Ira Flatow:
- Is there an age when you know that you are going to have it or not? I mean when, as you mentioned that breakthrough that you made where you actually saw the numbers in dimensional shapes, you can actually look at a page and see it.
- Keith Devlin:
- No. I don't think that there is an age. You come across people who quite late in life, in their forties, fifties, sixties, even seventies, suddenly decide, for whatever reason, that they want to look at it and they manage to break through and see the spark. I guess, the older you are the more difficult it gets. But I don't think that there is a time barrier.
- Ira Flatow:
- There are a lot of cases in history where scientists were not great at mathematics and I think that probably the most famous person was Einstein. When he was talking about relativity, he had to get his friend to give the math that he needed.
- Keith Devlin:
- I tell the Einstein story to both my children when they complain about mathematics. It seems to work. Einstein is a good hero figure.
- Ira Flatow:
- Is there an age when mathematicians are finished. I mean, when you think of physicists as you look at the history of physics, their golden age is in their twenties, early thirties, and then they are sort of quiet the rest of their lives.

- William Dunham:
- There is that sense about mathematicians--that you had better achieve it while you are young because it disappears later in life. I'm not sure that that is altogether true. There are certainly some very good mathematicians that are quite senior, quite up in years, but there also are these prodigies as we were alluding to earlier where someone at a very young age will do something very impressive. A genius can emerge quite early, like Gauss, for instance.

- Caller Janet:
- Hello. I am referring to Kurt Godel, the mathematician in Germany, in the thirties. What I'm curious about really is Godel's influence on mathematics and the structure of it or the organization of the various branches.
- Keith Devlin:
- It has been enormous. In fact, Godel had a whole bunch of results in the area known as mathematical logic which is the study of patterns of language, and reasoning, and deduction; looking at the deductive process itself. And subsequent to Godel the whole branch of mathematics that was not solely due to him, known as modern mathematical logic, just blossomed. It was that branch of mathematics that really led to the modern computer age. So the influence of Godel in many, many ways has been enormous.
- Ira Flatow:
- There has been talk recently about whatever happened to fractal geometry. I mean that it is a nice mathematical way of making pretty pictures but it has not developed the kind of practicality that people thought it would. Is that true?
- Keith Devlin:
- I guess the answer is true and false. The reason for the huge media attention was the pretty pictures. I mean they were glorious pictures. They were new mathematical objects. The mathematics going on behind them, the whole mathematics of dynamical systems, is still going on. It's a lot of work. It's just that mathematics is doing what it always does. There are still people working on use of fractals to do data compression, to compress data and store it in more efficient ways. It is connected to work in differential equations which has to do with aeronautic design, and so on, and so on. It is just no longer of interest to the media.
- Ira Flatow:
- Whenever you talk to mathematicians or whatever, they always show you pictures of knots. I mean they are making knots and saying this is a mathematical something or other. This knot has the same number of lobes as the other. They are identical although they look totally different. Why are knots so fascinating?
- Keith Devlin:
- Knots and fractals and one or two other subjects come up frequently because they are the kind of things that you can actually draw a picture of either on a computer screen or on a book or so forth. When I was writing my book, Freeman's
Mathematic - Science of Patternsthat was meant to be sort of a coffee table book of pictures, I was very careful to weave a theme that allowed me to mention parts of mathematics that were visual. If you try to illustrate a book, you either use the photograph of the inventor, the discoverer, the mathematician, or you pick a part of mathematics that you can illustrate. Knots, fractuals, surfaces, various kinds of topological things can be illustrated visually so they tend to get a lot of attention when you're trying to communicate mathematical ideas to non-mathematicians.

- William Dunham:
- And the donut hole is very popular--the donut not having a hole.
- Keith Devlin:
- Well, the story about the donut, by the way, we are talking about a regular ring donut. When we say the donut doesn't have a hole, what we mean is the mathematician talking about the donut is talking just of the surface of the donut--just the actual brown, white, silvery, sticky stuff that is on the outside, the stuff that you do not want to get on the lounge sofa. That's the surface. That's what the mathematician talks about when he talks about the donut or she talks about the donut, the surface does not have a hole in it. The hole is, as we say, in the space that the donut is sitting in.
- Ira Flatow:
- Well, there you have it. Great conversation for the cocktail party tonight: donuts and how they don't have holes. Thank you very much for joining me this hour. William Dunham is professor of mathematics at Muhlenberg College in Allentown, PA and author of
The Mathematical Universefrom Wiley. And Keith Devlin is Dean of Science at St. Mary's in Moraga, CA. and is author ofMathematics - the Science of Patternsfrom W. H. Freeman. Thank you both for joining me.

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