Authors: David Jay Brown,Rebecca McClen Novick
I had taken very few math courses during that period. I remember two or three very influential courses. One of them was a differential geometry course taught by Raoul Bott who became a very famous mathematician. Some concepts were included in that course that I later found useful in dynamics. So I had some math background, but not the kind of background I would have had if I'd done a Ph.D. under a famous professor of dynamics.
Then I was looking for a job. I had one offer for some place where I didn't want to go and at the last minute, before the school year began, I got a letter from Berkeley offering me a job. In 1960 there wasn't any big mathematical center there, but of course I took it.
After I got to Berkeley I was engaged in rewriting my thesis for publication. One day I discovered that they were having tea in some little room in the back of the building, and I had already been there for two or three months and hadn't met anyone. So I went to the tearoom to meet some people and to find out what was going on. And in this way I discovered a couple of people who later became my best friends in mathematics. They happened to be there in September of 1960, along with a lot of other people that I met. Everybody had just arrived. Overnight, Berkeley had become one of the most important mathematical centers in the world--and I just happened to be there, apparently because of a clerical error.
One of the people I met that day at tea was Steve Smale. I was done rewriting and was looking for something new to do. So I said, "What do you do?" and he said, "Well, stop by the office and I'11 show you." The next day I stopped by his office and we started working together. Later I found out that he was a really famous mathematician. He won the Fields Medal which is the mathematical equivalent of the Nobel Prize for doing the very work that he was showing me.
So I found myself on the research frontier in mathematics, working with some really wonderful people who all thought I was fine, because in this group there was no insecurity. It was just, "This is what we do and if you fit in, fine." So we worked together and had great fun. We had fantastic parties where we played music and danced and got drunk and we did a lot of creative work in what became a new branch of mathematics called "global analysis." And all this happened in just one or two years. Part of this program was "non-linear dynamics" as practiced by mathematicians on the research frontier at that time, using tools called "differential topology." It's a far cry from what people are doing now under the name of chaos, non-linear dynamics, and so on, that you read about in stories like Jim Gleick's book Chaos.
All that I did in those early days was mathematical. It could be explained to a lay person without some very hairy preparation, and I've tried to make that explanation possible in my four picture books called Dynamics: The Geometry of Behavior. The third of these four books is devoted to "tangles." In 1960, Steve Smale and I would take turns at the board drawing these tangles and trying to make some sense out of them and figure out what was going on. Tangles are like the skeleton of a beast. If you go into the Museum of Natural History and there's a skeleton of a dinosaur hanging from the ceiling, you can walk around it and from the skeleton you can imagine the whole thing. But if you saw the whole thing you couldn't see the skeleton inside without an x-ray machine. It's just like a blob. These tangles are the skeletons of chaos. We didn't discover them; they were known to Poincare in 1882 or so.
In 1960 we were just trying to figure out these skeletons and relate them to the eventual behavior of all dynamical systems, which includes practically everything in the world: that's all kinds of processes, including the human process and the process of history itself. All these are dynamical systems, their skeletons are these tangles, and the tangles have aspects known under these words: fractal, chaotic, and so on. But they are much more: they are highly regular, they're dynamic, they're symbolic, they're mythical and they're beautiful. In fact, they're mathematical.
DAVID: Just so that everyone is familiar with the extraordinary work you do, can you briefly explain what chaos theory is about and what role you are playing in this exciting new field of research?
RALPH: Chaos theory is a small branch of dynamics which is a very important region of the intellectual frontier. It overlaps mathematics, the sciences, and computer science, but it's not any of those things. It's not a branch of physics or of mathematics it's dynamics! So we have a really unusual area which is not mathematics and it's not science, it's not a department of the university and there are no dynamicists with titles of "professor of dynamics."
But in spite of the fact that it hasn't been acknowledged, it is a really central human activity and really important to our adventure of understanding the world around us. I would say that its position is mid-way between mathematics and science. Mathematics is not science--science has all these branches, and mathematics is not one of them. Mathematics is completely separate in its philosophical outlook and in the personality of the people who pursue it, who are somehow diametrically opposite to scientists. Scientists are bottom-up in their style of understanding and believing, while mathematicians are sort of top-down. Dynamics is a huge area in between, which comprises the encyclopedia of mechanical models used to understand processes.
Since we have to understand processes in science, dynamics is very important. I do not think that chaos theory is quite so important. The chaos revolution is the biggest thing since the wheel, but I don't think it's fundamentally important. Dynamics is providing us with process models which are much more important than chaos.
The chaos revolution is primarily important because chaos is everywhere. For some reason there was an historical accident, and for six thousand years people repressed chaos to the unconscious. So there has been a totally unnecessary gap where there should have been chaos theory. And the filling of this gap is really a big thing only because the gap was there. But after it's filled, it is perfectly normal to have chaos models, and wheel models, and static models. It was very bizarre that among all these models there was such a huge gap. But now it's filled, now we're back to: "No big deal, aha, fine, so it's chaotic."
But dynamics is offering more. It's offering bifurcation diagrams, catastrophe models. It's offering fantastically good models for processes. And few of these models would actually be there on the shelf for our use in trying to understand the world around us if we denied the existence of chaos--because chaos is ubiquitous in process. You can't model process very well if you're in denial about the existence of chaos. You're certainly not able to model any process which is full of chaos, and that's practically all of them, most especially those involved with life, love and creativity. So we do have something important in dynamics, and chaos has an important role in a sort of double-negative sense. That's what's going on with dynamics.
As far as my work in it is concerned, I think it doesn't matter very much. Some people think I shouldn't waste my time at a computer terminal doing research on specific problems because my role is to go around saying what I just said.
DAVID: What are some of the problems that you see with the present state of American mathematical education and how do you think improvements could be made?
RALPH: Well, I would say a good thing to do with mathematical education in the United States or in the world today would be just to cancel it and start over again from scratch, two or three generations later. The whole thing is in a really dangerous plight. And I've been saying this for years and so have other people, but only recently has the problem risen to a scale of national prominence where even the president and the governor and everybody's saying, "Well hey! our Gross national product might be threatened, because our people are no good at mathematics."
So we have a serious situation. First of all, mathematics is akin to walking as a human experience; it's just really easy. I mean it shouldn't be easy, how can you tell somebody how to walk, you know? But people do find it easy and they naturally learn how to do it. They just watch, and by imitation they can do it. It's the same with mathematics! It's part of our heritage, all of us, to be genius at mathematics. It is a completely human activity. It involves the resonance between prototypical objects in the morphogenetic field and specific examples of similar forms in the field of nature, as they're experienced by human beings through the doors of perception. And as life forms a resonant channel between these two fields, it's just as natural as understanding anything, including walking, playing tennis, and so on. Mathematical knowledge is part of our human heritage.
Furthermore it's essential to evolution. Where there's no mathematical knowledge there can be no evolution, because evolution to a stable life form requires a kind of mathematical, sacred guidance. This can be understood in many different images, the least controversial one being that there would be a harmonious resonance between all of the components, parts, sub-systems and so on involved in the life process. Where there is an disharmonious resonance, or dissonance, there would be some kind of illness whether the organism is a snail, a human, a society, or the all and everything that we know by the name history. So for the harmonious resonance to be maintained during the process of our own growth, or social evolution, evolution requires mathematical understanding. You see the dissonance of the lack of mathematical understanding through the gross national product, or the number of wars, or the spread of AIDS, for example.
Another importance of chaos theory is in correcting a problem in mathematical education that has consisted, in part, of denial. People have been taught the non-existence of some of the essential mathematical forms, namely, chaotic forms. This kind of denial produces an educated adult somewhat less capable than an uneducated adult. So that education which functions in this way is not the same as no education. It's worse, because it destroys intelligence, it destroys functionality, it destroys harmony with the resonance of the all and everything which is necessary for health. Our educational system, in short, is producing sickness and contributing to the global ecological problems on the planet by destroying the native intelligence that children have, the capability they have to understand the world around them in its complexity, in its chaos, in its resonance and harmony and love, destroying it through the inculcation of false concepts and through the production of avoidance mechanisms connected with certain mathematical ideas.
It's a very serious problem. One possible response would be to revise mathematical education so that, within the same system, one would try to provide teachers who are more highly trained. That could only make matters worse, you see, since the teachers are already highly mistrained. Many have already been taught to hate mathematics and so they can only teach hatred for mathematics. They don't really have any idea what mathematics is. For them, it's a knee-jerk response of this dark emotion, so retraining them more wouldn't help. Rather than revision of the schools--which are full of false ideas and bad habits built into the field on a deep level--the most efficacious, practical solution would be the construction of a new educational system outside the usual channels of the school system. This is not too radical, as we have all been brought up to think of our real education as going on outside the school system. In school, for example we do have music classes, yet if parents want their children really to know music, they provide a separate teacher outside the school. We also have religious instruction and dance instruction outside the school--anything that you really want to learn is studied outside of the school. And so also it may be with mathematics.
I think that one practical solution to this challenge to create a school outside of school would be a new breed of learning machines based on computers, educational software, and digital video. Even programs like Hypercard on the Macintosh, for example, could provide alternative education that could be approached by individuals without teachers. So far, however, the creation of educational software has proved to be a very unrewarding activity for authors. And in spite of all different kinds of alternative funding agencies, nobody has seen this as a very important problem although the National Science Foundation, the American Mathematical Society and like organizations have convened conferences to discuss possible solutions of the crisis in mathematical education. The most promising alternative solution at this time has not been funded. And so there are very few existing alternatives for children now. Maybe after another generation or two there will be.
RMN: The principles of chaos theory and other mathematical ideas appear to echo in the myths and philosophies of some ancient cultures; the Greeks had a Goddess of Chaos, for example, and the I Ching is full of references to such ideas. What level of understanding do you think earlier civilizations had of these concepts and how was this expressed?
RALPH: Well, the repression of chaos began with the patriarchal takeover six thousand years ago. So to look at an example of a high culture accepting chaos as part of their mythological pantheon and in their arts and behavior, one has to go back before that takeover. And the most common example of such a culture is Minoan Crete. This culture was excavated by Sir Arthur Evans, and his reconstruction of the temples and religion, etc., have since been seriously questioned by archeologists. In short, there was a controversy as to what were their arts, their social patterns and so on.