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Ralph Abraham

might not fit someone’s preconception based on an intuitive understanding of these chaos concepts. So I don’t know if social change is going to be characterized by chaos or not. I guess it might, according to some measures and observations, and might not, according to others.

DAVID: Do you see the process of evolution as following a chaotic attractor, and if so does that mean there is a hidden order, so to speak, to evolution? May what has appeared to evolutionary biologists as chance and randomness actually be a higher form or order?

RALPH: No. I think that the understanding of dynamical systems theory presented in popular books is extremely limited and a lot of physicists for example have studied attractors exclusively while as I said the mathematicians have been studying the separatrices. Attractors are very important in modeling physical processes in some circumstances, and that is very fine, but when you’re speaking about evolution, if you want to make models for an evolutionary process, then probably the best modeling paraphernalia that mathematics has to offer you are the response diagrams of bifurcation theory. Bifurcations have to do with the ways in which attractors appear out of the blue, or disappear, and the way in which one kind of attractor or size of attractor changes into another.

These transformations appear in scientific data and in mathematical models in a much smaller variety of transformation types than you would suspect. And dynamical systems theory, at the moment, is trying to accumulate a complete encyclopedia of these transformation types called bifurcation events. Bifurcation events assembled in some kind of diagram would provide a dynamical model for an evolutionary process. Therefore, the actual attractors involved are almost of no interest. From the bifurcation point of view it doesn’t matter if the process is static, periodic or chaotic. What’s important is whether the attractor appears or disappears. And here there is plenty of room for chance and randomness.

And so as bifurcation theory becomes better known, I think the style of making models of process will undergo a radical and very exciting revision. The main point of my books, Dynamics: The Geometry of Behaviour, is to present the beginning of the bifurcation encyclopedia as far as it is known to date. There are about twenty-two different events there.

DAVID: Do you think it’s possible to form, or have you already formed, a mathematical theory to explain the phenomenon of how consciousness interacts with the material world?

RALPH: No. There are models, specific mathematical models, for different perceptual functions of human mammalian physiology which represent the frontier of neurobiology today. One example is Walter Freeman’s model of the olfactory bulb. These models are mathematical objects known as cellular dynamical systems, which include neural-nets and excitable media as special cases. These mathematical models for perception pertain to the question of how consciousness interacts with the natural world. And they comprise a conceptual frontier today. In that context, what would an idea be?

In the context of the olfactory bulb, what is a smell? So it turns out that from the perspective of reductionist science, along with its mathematical models, a smell is a certain space-time pattern on the olfactory cortex, a pattern of excitation. The cortex consists of a sheet of oscillators side by side vibrating. A certain pattern in their frequency, phase relationship, and amplitude, is a smell. There is a certain picture, where inside a region there is a larger oscillation, and outside, a smaller one. This picture is recognized as a smell.

This kind of modeling does provide the possibility of making a simple model for the natural world, a simple model for consciousness, and a simple model for the interaction between the two. The interaction model, in this cellular dynamics context, is based on resonance. A lot of my work has to do with vibration and resonance phenomena in this context and has provided a specific mechanism for the transfer of a space-time pattern from one such medium to another. However, these mechanical models may be too simple to provide intuition as to such things as how your mythology, your perceptual filter, function so as to limit your perception of the natural world to a certain paradigm in your consciousness? Such models, which I think is the essence of your question, would have to do with a more linguistic or symbolic approach rather than at the mechanical model level.

DAVID: Could you define beauty in a mathematical way?

RALPH: People do say mathematics is beautiful, and some mathematical objects are certainly beautiful. Whatever beauty is, if you could define it in some way, it would include mathematics within it somehow. If you define it, for example, in terms

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