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Ralph Abraham

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

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