Ralph Abraham, Prague Castle, 20 November 1996

Chaos Theory and Space-Time Patterns

A 30 minute lecture and showing of computer graphics on 20 November 1996, at a conference of the Council of Europe entitled:

A New Space for Culture and Society (New Ideas in Science and Art)
19-23 November 1996, Prague Castle

Abstract:

The morphogenesis of the World Wide Web is an exemplar of the new space in social and cultural evolution, and we may try to understand it from the viewpoint of the new paradigm by making use of mathematical tools from coming fractal geometry and chaos theory.

The great epochs

Historians like to divide cultural history into plateaus separated by bifurcations. One of the many people to do this, Sir Flinders Petrie, who was the archaeologist who dug up ancient Egypt, believed that he had discovered a universal sequence of steps which characterizes each major social transformation. The first in this sequence is novelty in mathematics. This is followed by novelty in art, then in science, technology, and finally in economic strength. Then after a plateau there is a collapse. I devoted a good part of my book, Chaos, Gaia, Eros, to try to establish that this universal sequence of Flinders Petrie applies to the present time.

More recently another historian, William Irwin Thompson of Zurich, went further in dividing the cultural history of the world into stages, saying that not only does mathematics precipitate or trigger major social transformations, but also that mathematical novelty characterizes the following stage. He speaks thus of the arithmetic mentality of ancient times, followed by the geometric mentality beginning in Ancient Greece, and the dynamic mentality beginning about 400 years ago. The chaotic mentality is his characterization of the cultural ecology that is rapidly evolving today.

The great bifurcations

There are new branches of mathematics - such as chaos theory and fractal geometry - and there is now in progress a major transformation in mathematical history. This has perhaps been ongoing for a century, but has only very recently come into the public view and the philosophical discussion of our time under the name of the chaos revolution. Of the many different roles that chaos theory can play in the cognitive strategy of our time and in our attempt to understand the pattern and the change of pattern and the metamorphosis of our time, one is the idea of bifurcation, to which Ervin Laszlo has devoted a book, and the idea of the chaotic attractor. These ideas, which are very simple, I would call metaphors of space-time pattern, and when these metaphors are applied to history they give us a new description, or a new understanding, of the process that we experience today. Bifurcation instead of catastrophe.

One of the main goals of chaos theory P not yet achieved P is to provide a taxonomy or encyclopedia of bifurcations, so that we could try to identify our experience of the moment, or the recent past, or possibility of the future, from this list of bifurcation models. But I would like to go on now to another implication of chaos theory for our understanding of our space-time pattern and our evolution into a new space. This aspect of chaos theory is a new branch of mathematics called morphogenesis. This word refers to the application of complex dynamical systems theory to the problem of understanding space-time pattern in its broadest sense.

Biological morphogenesis

The stimulus for this new mathematical area has been biological morphogenesis, a main theme of theoretical biology, since the work of Conrad Waddington in the 1950s. You could say that biological morphogenesis is the Mount Everest of theoretical biology: To understand the mystery of pattern formation in nature. There are whole books devoted to the question of how a leopard gets its spots. Of course the growing embryo somehow gets its spots from the mother leopard. This focuses the main question of morphogenesis, of pattern formation in zoology, on the process of embryogenesis. How does an egg become a chicken?

Probably a majority of biologists, scientists, and people in the street are willing to believe that the egg becomes a chicken or the leopard gets it spots through some mysterious intelligence of DNA alone. Other people hold a another idea, which was boldly described by Ervin Laszlo this morning, namely, that there are fields of intelligence which somehow collaborate with DNA in the guidance of the emergence of form in the embryo. In the process of embryogenesis, as the egg becomes a chicken, there is the development of the nervous system, or neurogenesis, which somehow is an even more mysterious process that could be viewed as a kind of form-guiding system, a radio receiver for instructions from God as it were, for the development of the chicken from the egg.

Mathematical morphogenesis

As a problem in mathematics, people attempted to make mathematical models for morphogenesis. They tried to mimic the process of the egg becoming a chicken, or at least something vaguely resembling a chicken, by using the classical structures of mathematics of the dynamical mentality, ideas that followed from the fantastically original ideas of Sir Isaac Newton. That meant, for example, differential equations, especially partial differential equations. These are the classical mathematical models that, while simply expressed, have enormous implications of complicated space-time structures, such as waves, resonance, interference of waves, and so on. Such models gave rise to, for example, the Brusselator, important work of Prigogine and his group in the 1970s, following the earlier ideas of Rachevsky in the 1930s, and Turing in the 1930s. They could indeed get something like a mathematical chicken to evolve from a mathematical egg.

These mathematical models could not be solved analytically, but could be explored by using the methods of Newton combined with computer simulation. So the new developments in the field of morphogenesis coming from chaos theory require the computer simulation of a mathematical model. To get something like a chicken from something like a mathematical egg requires super-computers to do computations involving thousands and thousands of cells. Each cell demands its own ongoing computation, a big job for a small computer. So altogether, the modeling exercize takes some time on a massive supercomputer, and the resutls may be seen on its graphics screen.

This video

In 1989, when I had access to one of the world's fastest computers for the first time, I made this video. This computer was able to imitate a morphogenetic sequence at a fantastic rate, over a thousand frames per second. Working with John Corliss and John Dorband on the MPP super- computer at the NASA Godard Space Flight Flight Center, we created robot mathematicians who would look at the results of the simulation from time to time and select patterns that were mathematically interesting and then record them on videotape at the rate of ten frames per second so that we could easily view them, and share them with others.

This is just a little background on a kind of new mathematics on the research frontier today, which I am suggesting could somehow inform our cognitive process and help us to understand this very complicated mess we're in today. We certainly can't simulate anything as complex as the biosphere or the global economy, or the connection between these two. Not yet. At the moment we're in an experimental process where we are seeing what kinds of forms, what kind of bifurcations, what kind of special events, in space-time pattern, emerge in these models as favorite forms. The possibility that they can expand our intelligence in understanding the complex system we live in today is very attractive. An application to the World Wide Web I think that everybody can agree that we are in a special moment or hinge of history in which the familiar forms are disappearing and new forms are arising, and we don't actually know what new forms are arising. We may have no power, in fact, to affect this evolution, although the butterfly effect in chaos theory assures us if we just recycle a little bit, that will help a lot. My idea here is:

And this neural system now evolving is, of course, the World Wide Web.

So therefore I'm suggesting to those people who would like to participate in the creation of the future that the observation of the World Wide Web is an important step, a kind of biofeedback, to give us the insight necessary for participation. We have to be able to perceive the evolution of the World Wide Web and the implications for the cultural ecology it is generating, to see these with the aid of some kind of image, a visual image, a mathematical metaphor, poetic and artistic and so on.

Operation Webwatch

Towards this end, Don Foresta and I are engaged in a project called Operation Webwatch, in which we are using chaos theory and fractal geometry to make images out of the World Wide Web viewed as a neural net. A neural net refers to another special area of chaos theory about complex dynamical systems. We're developing a new vocabulary for looking at these things. I'm calling it webometry. Millions of web pages already installed on the Web may be seen as the nodes of a nervous system, a neural net. The connections between them are their hyperlinks. Counting those up in a certain way, we create a simple image, a colored picture, which is a snapshot of the connectivity of the Web as a neural net, engaged in the neurogenesis of our emerging embryonic cultural ecology. It can be seen as a picture, and within this picture are some vague shapes like clouds in the sky, and on successive days, these images change so that, with a little poetic license, we could imagine that we are seeing the nervous system and the structure to guide the development of the bones and the organs and the mind and the personality and so on of this cultural ecology. And through watching this, anyone who tunes into our website would be able to see a movie of the last year or two of the evolution of this neural net and judge for themselves whether or not any interesting structures are emerging with which they want to interact.

Thanks!

Posted 26 May 1997 by Ralph Abraham