Chaos and Erodynamics
“Chaos is very much the same as the steady state; it’s not scary at all.”
with Ralph Abraham
Ralph Abraham is renowned for bringing a fresh perspective to mathematical thought. His study of dynamical systems as the building blocks of reality, has led him to extrapolate fundamental mathematical principles into his philosophical outlook . A professor of Mathematics at the University of California, Santa Cruz, he received his Ph.D. at the University of Michigan in 1960. He taught at UC Berkeley, Columbia and Princeton before moving to Santa Cruz in I 96S and has held visiting positions in such various locations as Amsterdam, Paris, Warwick, Barcelona, Basel, Florence and Siena.
He is the author of numerous mathematical books. Linear and Multi-Linear Algebra, Foundations of Mechanics was written with J.E. Marsden and Transversal Mappings and Flows with J.Robbin. He wrote Manifolds,Tensor Analysis and Applications with J.E. Marsden and T. Ratiu, and the highly successful four-volume Dynamics, the Geometry of Behavior with C.D. Shaw. His latest book entitled, Trialogues on the Edge of the West is a group of discussions with Terence McKenna and Rupert Sheldrake on the relationship between science, philosophy and religion.
Traveling through Europe in his twenties, living in a cave in northern India and working as a professional gambler in Las Vegas were all experiences which helped to shape Ralph’s philosophical outlook. He has been active on the research frontier of dynamics in mathematics since 1960, and in applications and experiments, since 1973. In 1975 he founded the Visual Mathematics Project at the University of California, Santa Cruz to explore the use of interactive computer graphics in teaching mathematics. He is the founding editor of Eagle Mathematics and Applied Global Analysis.
We talked with Ralph on March 4th 1989, in the cozy· living room of our dear and mutual friend Nina Graboi, who has often worked as his editor. We found him to be a soft-spoken, intensely thoughtful and down-to-earth character, with the gentle tone of a person who has become philosophically resigned to seeing further than others.
DAVID: Ralph, you’re recognized as one of the leaders in the mathematical study of chaos. Can you tell us what it was that originally inspired your interest in mathematics and the mathematics of vibrations and dynamical systems?
RALPH: Well, I didn’t get interested in dynamics and decide that’s what I was going to study. It was just left foot, right foot, or some series of miracles. It happened like this.
I was an engineer and worked in a physics project, so I became a student of physics. Then one day a physics professor said in class that if you want to understand physics you have to study mathematics. So I changed to mathematics at that point. And I found a mentor, somebody who took care of me and helped me out, a wonderful man, Nate Coburn. I started studying what he was doing because he was my only contact in mathematics. One reason I responded to his program was that it had to do with general relativity. Einstein had been a household word when I was growing up. My father respected Einstein very much. It was said that only eight people in the world could understand Einstein. My teacher apparently could and was writing in that field.
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