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Ralph Abraham

of cognitive resonance, then mathematics provides the ultimate opportunity for cognitive resonance because the bare bones of cognition itself are represented by these mathematical objects. The strongest resonance of forms takes place in certain special areas, precious little rings of human experience. One is mathematics, another is music, and then of course, mysticism–the three M’s, three crown jewels of beauty. But I wouldn’t know what the experience of beauty really is, and J certainly wouldn’t think a mathematical definition would be appropriate.

DAVID: From chaos theory we know that small errors in calculation can grow exponentially in time, making long-term prediction difficult. With this in mind do you think it’s possible to foresee what life for humanity will be like in the twenty-first century?

RALPH: This idea of the exponential divergence, the so-called sensitive dependence on initial conditions, is very much misunderstood. When a process follows a trajectory on a chaotic attractor, and you start two armchair experiments, two processes, from fairly close initial conditions, then indeed they diverge for a while. But as a matter of fact what is happening is that both of the trajectories go round and round. You can think of yarn being wound on a skein. So they diverge for a while, but pretty soon they reach the edge of the skein, and then they fold into the middle again. They always come back close together again.

They have a certain maximum separation-it might be four inches or something and that’s it. That’s not very scary. They do not diverge indefinitely and go off into infinity. That’s exactly what doesn’t happen with chaotic attractors and that’s why chaotic attractors might be very reassuring to people who would otherwise have anxiety about chaos. Because the chaos in a chaotic attractor is very bounded and the degree to which things go haywire is extremely limited. So that’s the good news, and after you know the process for a while, you know it forever. Chaos is very much the same as the steady state; it’s not scary at all.

Now if our evolutionary track, this species on planet Earth going into the twenty-first century, for example, were modeled by a chaotic attractor, then we can answer the question where will we be in the twenty-first century. Because it would be pretty much the same mess as now. But it’s not modeled very well by a chaotic attractor. A better kind of mathematical object for modeling an evolutionary process is a bifurcation diagram. In this context, a chaotic attractor is changing in time. There may be bifurcations, for example, a catastrophe, a comet or something. Who knows? And it may be that some bifurcations occur under the action of parameters controlled by us, such as how much energy we use, how much waste we make. And that’s why bifurcation diagrams are more interesting than chaotic attractors for modeling our own process. Under this more general kind of model we cannot say where we will be in the twenty-first century. Or if we’ll be.

RMN: Why do you think that the understanding of chaos theory is vital to our future?

RALPH: This fantasy of the importance of mathematics has to do with the idea that we might have a future, that we might have something to do with it, and that conscious interaction with our evolutionary process is possible and desirable. And in this case, things will go better if we understand our process better.

The importance of chaos theory to our future is that it provides us with a better understanding of such processes, the behavior of complex systems such as the one we live in. This is due to the fact that chaotic behavior is characteristic of complex systems. The more complex the system, the more chaotic its behavior. And if we don’t understand chaotic behavior, then we can’t understand the complex system that we live in well enough to give it guidance, make informed decisions, and participate in the creation of our future.

DAVID: Would you tell us about any current research projects that you’re working on?

RALPH: I have an ongoing project with visual music which is just one of a family of related projects having to do with chaotic resonance in cellular dynamical systems. If you had a cellular dynamical system such as a two dimensional spatial array of three-dimensional dynamical systems, and the state of each of the dynamical systems in the two-dimensional array were visible as a color, then you’d see the simultaneous state of this complex system as a colored picture, and the evolution of this system as a movie of colored pictures. This is experimental dynamics and graphic art, all at once.

Complex dynamical systems have very high dimension, they are really hard to see. The conventional methods of scientific visualization, an important field in computer research today, only work for low dimensional systems, for simple systems.

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