Dynamical Systems & Altered States
W.M.T. '95

I'm very honored to be here.  It's probably a challenge for all of us just to have a mathematician addressing this group. We owe it to Steve Brooks, who has dragged me here and got us all into this. I don't know exactly what happened, because I've been busy. I've really been around the block recently. I got on the airplane and looked at this file that I snatched at the last moment.  It says Telluride, 1995, and I looked in there trying to remember what was supposed to happen here, and then I saw the title for this evening's talk, "Altered States and Dynamical Systems". I realized that what we have here is essentially an impossible task, and I nearly aborted.  So   rather than fail, I thought that I would just switch the topic -- I must have been in an altered state when I thought up that title. But then I kind of got a second wind, and I recognized that this topic, if I could try to address it,  would be kind of the deepest sharing that I could do, which I've never done, and in this audience, there would be the possibility of--well, not of success, but of something happening. So for primarily selfish reasons, I want to try to go ahead as scheduled.

There are three reasons why this is almost impossible, quite apart from the fact that it's embarrassing and dangerous to be honest.  One is that we don't know what altered states are, so it's hard to talk about them, and the other is that we don't really know what dynamical systems are, so it's hard to talk about that. And the idea of putting them together or in any way accepting the possibility that there is a relationship between them other than that they have been my life, is...All I can do now is to ramble on chaotically without any expectation of linearity, and we'll see what happens. I thought that if I were to do this properly, it would take two weeks, but if I did it the only possible way I can in these circumstances, it would be really short.  O.K., so I'd like to make a short try, and then with your questions I think we'll be able to go on from there.

I'd like to start with the question, "Why are we interested in altered states?"  A lot of people are interested in altered states, of course. One reason I heard on the radio driving here from Grand Junction this afternoon.  It was in a song that said,  "Our love will disappear if we don't take it someplace new. It's no good to be always in the same place. It just won't do." That's one reason to seek altered states, it's like adventure sports.  But for me there's more. There's this feeling that I abhor that our civilization is on a death track. We like to be optimistic. We don't like to be pessimistic, so we don't actually like to give voice to these dark fears, especially in Telluride. Nevertheless, the nagging thing is there and the most optimistic view we can have is that something we may do could somehow be a little help. Chaos theory tells us that small causes can have huge effects, like the so-called butterfly effect.  We are encouraged, like Andrew, to start to school.  The doctors--what is the trouble with these doctors?  They're in  some kind of a rut.  They're not stupid, they're nice people, they're trying to be helpful, and I don't know if it is the medical school or the whole school from kindergarten on or the whole culture including the billboards, the television, that creates this rut--what Rupert Sheldrake calls a runnel which is a really big track that mice make by going up and down for so long that the walls are so deep that you just can't get out. ???

One image of what we're doing here now is Doris Lessing's image in Shikasta, where she said, "This planet has been attached by a dark force from a dark planet."    The energy is being sapped. Everything is dying. We have chosen to incarnate, and she describes in detail how that's like jumping out of an airplane, I don't want to have one more incarnation, to get into this body, the cell, embriogenesis.??? We try to enjoy the process of incarnation, try not to get too caught up in the physical and do what we came to this planet for. It's not very optimistic. It's a hospice operation. ** 

I'm not that negative.  Most of us are not that negative, but I think everybody is pretty much agreed that we need a paradigm shift. We need the biggest shift of the century, or maybe the biggest shift in the millenium, or maybe the biggest shift in 6000 years, according to which book you read.  In my book, Chaos, Gaia, Eros, I try to convince the readers, if any, that we are right now in the biggest shift in 6000 years. Well, that's supposed to be kind of an optimistic view.  Look at how many evils can be wiped out if we can do this shift, and it is shifting. The old structures are in a major meltdown. And if we see that as an opportunity for change, we can be optimistic, even though it's quite difficult living through a meltdown. 

I'm going to talk about altered states, and why we are interested in  them. We want a paradigm shift, but we don't know exactly how to trigger it. Something's happening, but what will the outcome be? Can we influence the outcome in any way by thinking something, doing our exercises in the morning or whatever? And here is where mathematics is helpful. If we had the best understanding of the whole dynamic of a (phase? faith?) transformation, the best targeted strategy for understanding this kind of thing that we could possibly have, then we would have a 1% of a 1% chance of intervening correctly and doing something that wouldn't just bring on further disaster. 

My life has been 35 years of teaching mathematics and chaos theory in universities here and there, and I like to think that this was potentially useful, because that's where young people get their education.  If I could swerve them just the tiniest bit with whatever experience and mathematical understanding I have achieved, they would soon come out with a different paradigm, because Chaos Theory is a major paradigm shift, at least for the sciences, and the sciences we know are a large part of our problem. As a branch of philosophy they have gone wrong, but they have their good side and their bad. The good side is that they can do surgery on your knee, and the bad is the reductionism, the specialization, so that one person doesn't know what's happening in the next discipline. By working at the university I felt I could be the little butterfly that could have a large effect. Eventually, however, I realized that it's too late for university students. Their idea of mathematics has already been destroyed in K through 12.  If we could change the curriculum totally in the elementary school, we would have a better chance of precipitating a major social transformation and thereby creating a reasonable future, or at least of getting this train off the death track it's on now and buying us another 50 or 100 years, so other people can try to figure it out. 

I promise I'll talk about altered states. Just be patient one more minute. You think I'll never get to it, but please have faith.
 
Now, I've actually been promoted from professor at the University of California at Santa Cruz to Curriculum Consultant at an elementary school K-12. [applause] And I've really been excited about the possibility that this could make a difference. We've designed a new curriculum based on a completely different idea. It mainly deals with content rather than method, and the content is based on what I called the Sheldrake Principle, which means that world cultural history is mapped onto the grades. So in kindergarten you learn about paleolithic, aboriginal life. In the first grade you have the agricultural revolution, and in the second grade the first cities. You end up in the sixth grade in ancient Greece, then come the Middle Ages, and finally there's the computer revolution in grade eleven.  In grade twelve you talk about the future. This is just an idea. It's different, and it's one that students and teachers alike are really excited about. I just came from a symposium--I'm getting close to the altered states--where the mentors, that is to say the cultural historians, the experts on partnership and gender issues and other so-called mentors were preparing the teachers to go into the classroom when we realized that these teachers had been trained in the system, and there was no way that they could do it differently. There was just no way. So it's a complete strike-out, right? Wrong! 

This is why--you'll be disappointed, it's so simple--why we're interested in altered states.  O.K.,altered states. I can confess to a period of six years of altered states in the very distant past.  It was followed several years later by my discovery of computer graphics as a way of doing research in mathematics along with the discovery of the intimate relationship between altered states and mathematics, which has sort of become my theme.

Altered states are a way out of the rut. I'm not saying that everybody in a society has to experience altered states, but there need to be a few. That is what shamanism is all about. As soon as you're out of your culture and understand how relative it is in the context of another culture, your view in the altered state somehow puts everything in a different perspective. It's a change which can't really be undone, although some people claim that they have healed up and become the same as they were before. 

My personal view about altered states goes something like this: In the new curriculum for the elementary grades K through 12, we have taken special care to map the major social transformations of cultural history, for example agriculture and the wheel, onto the major psychological transformations of children in their development, in the sense of Piaget for example. There are certain special moments in history and in a child's development. I believe that adults also have the potential of catastrophic bifurcation, a quantum leap into another level of consciousness. That sequence begins at age 4. There's a big change in the connectivity. At age 7 or 8 it melts down and new connections grow--those are the special moments. And there are possibilities after adulthood.  With the aid of our guides in altered states--botanical ones and spirit ones, I don't know how you want to think about this--what happens is the initiation of a new stage of development, so that our consciousness grows and we're led out of--We see that our culture was always false. It does seem to be in kind of a stuffy rut. There they are in Beirut and Bagdad still doping it out and nobody agrees on anything.

Like all traditional cultures, we have a deep and abiding interest in altered states  because of the belief that our further evolution will be impossible without them. I still didn't say what I thought they were, and I want to approach that mathematically. So now I'll talk about dynamical systems and then come back to altered states from that point of view. 

Dynamical Systems Theory is a branch of mathematics. It's the newest branch of mathematics, only 400 years old, and it's the home base of Chaos Theory and the Theory of Bifurcation. What it's all about is the study of space-time patterns. A space-time pattern is a pattern that moves. A pattern that doesn't move is a special case of space-time pattern. Circles and triangles are space-time patterns in a trivial sense. But if you have a triangle inside a circle and the triangle rotates, that's a space-time pattern. Our cognitive apparatus includes a pattern recognition capability especially for space-time patterns, which allows us to recognize that certain cloud formations are associated with oncoming rain, and that certain patterns of waves in the ocean indicate that sailors should beware. What I'm telling you is, that's mathematics. 

It's hard for me to talk about mathematics because everybody has an idea about what it is, and these ideas are totally different from the idea that I have, which is shared by professional mathematicians including official statements of the American Mathematical Society and so on. It's not just me. But it's not taught in school, so that most people think of mathematics as something that died a long time ago, is not very interesting, and machines do it. I used to think it was impossible to catch on to mathematics without doing any mathematics, I'd put people down for trying. But by now it's been proved to my satisfaction that the non-mathematical idea of dynamical systems, space-time patterns and so on, the non-mathematical ideas that I'm going to talk about in a minute are the most important ideas, and one way to grasp them is through  mathematics--the 800 pages of my book that Steve mentioned. I'm not even sure there's any other way. But when you acquire these concepts and can talk with these words and think with them, you have attained what is now called dynamical literacy. It's an extension of language. It's a cognitive strategy. It's a way of talking and thinking in which space-time patterns have a language that goes with them. Helping people to attain dynamical literacy is probably the most important mission of mathematics right now because we're engaged in these transformations, we are engaged in dynamical processes. The largest scale dynamical process that we're engaged in is the whole thing, and in order to understand that we need dynamical literacy. We need the help of everything we've got that enhances intelligence about space-time patterns in order to recognize the patterns that we're in.

Here are just a couple of ideas of dynacal literacy, and I'll try to apply them or illustrate them. Native speaker speaks dynamical literacy about altered states--we'll see what happens. This is the impossible part. OK, a dynamical system has a space, usually a geometrical space, an imaginary space. I say imaginary because it's a very high dimension and therefore you can't exactly see it. Mathematicians have developed tricks to see it, and when you have visual hallucinations, what you see is like what I'm talking about.  This kind of space-time pattern is a sequence of states in a space of a very * dimension. So for example if it does this and then repeats it over and over again, that means a trajectory that goes around in a circle. Each point on the circle represents a picture that you perceive as a picture. You perceive it as a picture because it takes all your senses to grok what is happening in a space of such high dimension. 

In a dynamical system, the whole idea is that this is what mathematics is. There is a very compact secret rule which is responsible for all the manifestations. So when the trajectory goes around in a loop, that's because at each instance it's obeying a rule, a law, that says, "Well if you're there you go in that direction at a certain speed." I'm not saying this is how reality is. I'm describing a certain kind of mathematical object called a dynamical system. And in these systems what happens as the action continues is that it settles down to a certain kind of thing which might be very very irregular in the case of a chaotic attracter. What you could recognize by watching the space-time pattern is called its attracter. It's part of the information. And that's the part that the mathematics of dynamical systems theory is about. These attractors represent states--dynamical states, not static equilibrium. How you feel on a certain day might be a whole progression with a ninety-minute cycle and a six-hour cycle and so on, and that whole cycle is a state, a dynamical state. In mathematics, most especially with the aid of computer graphics, you learn to recognize these states. 

When you watch different trajectories they go to different attracters. That's one of the amazing things about dynamical systems. There are things like for example good and evil, or different nations, different economies, that have different attracters which are in a way competing and trying to attract all the action going on. A map of this imaginary space, which is colored in such a way as to show where all the attracters are and all the points that end up at each attracter, is called the basin portrait. That's the most important object in dynamical literacy. This is a way of thinking. Here is a dynamical system. If we could know all about it through extensive experiments with super computers or something, we would end up with this map, which, in the case of roulette that Steve mentioned in The Eudomonic Pie, was a scam where little computers were installed in shoes to play roulette successfully. With extensive computation you create a map in advance that shows how all these initial conditions end up at that number, aand all those at this number. Knowing this map you know where to place your bets so that you end up a winner. And that's what we want to do with our own cultural history. We want to place a bet which is our intention, our action, our lives, and place it on a winning number. 

So back to altered states. If we were to use mathematics to make models for it, we would probably accept that the brain and the mind are different things. Later on, if they turned out to be the same, if those physicalists were correct, we could always identify them. It seems to me, however, that the efforts of neurophysiology to make a model for the mind are doomed. I won't say that I believe the mind is out there. I 'll just say that I don't believe the mind is in here. If you get the difference. Belief is not one of my virtues. 

So we would try to create a mathematical model for the neurophysiology of the brain, and another one for the mind. The one for the mind, the datum that describes that point in this probably infinite dimensional space would be ideal. And feelings and smells and all that. ???  In the neurophysiological model it's the physical state with all the neurotransmitters, the neurons, the actual plasic ?? transport, the cytoskeleton and all that stuff.  The task of making a mathematical model for the neurophysiological brain is well advanced in neuroscience. I've been working on it myself since the mid-1970s with one of the most advanced models of Walter Pribram, professor of Physiology at U.C. Berkeley. It uses what we call the Complex Dynamical Systems Theory to make a model in which all the states are chaotic, and yet they think pretty much the way a brain thinks. I don't imagine for a moment that such a model is a model for a mind. But it's certainly converging on the possibility of a very elaborate model for a brain. 

The brain-mind problem is something like the body-mind problem. Maybe the brain is part of the body and part of the brain, it's sort of the interface between the mind and the body. It's certainly worthwhile knowing how the brain works. Anyway, this kind of mathematics, Dynamical Systems Theory, is capable of making very elaborate and interesting and useful models. They're not good models. They're only good for understanding, but not good for predicting. Because Chaos Theory says prediction is impossible, but understanding is not. So when we have a model for the brain and for the mind, and they are similar in their structure, the brain is maybe a lower dimensional one where you have a lower dimensional space...in an altered state you can simultaneously see the mind and the brain, and the way that they go together is kind of a resonant, sympathetic vibration, where something here produces a like something there, and thanks to this knowledge yogis are able to slow down their heartrate or go out of the body here and come back in a different place so that functionality is restored.

Well that gives you an idea about Dynamical Literacy and its possible application to making models for the brain and mind. If all of that were history or science fiction or whatever you'd call it, then we could try to make a model for altered states, particularly  for that aspect of altered states which feels like a breakthrough to another kind of consciousness that doesn't in any way contradict the old consciousness but simply builds upon it, like a house upon the foundation, like all those other Piagetian stages of psychological development.

Suppose that there is a dynamical system which is the model for somebody's brain and mind, and that this somebody is about to eat an amenita muscaria or something. This plainly physical thing--an herb, a botanical, chemical thing--obviously makes things happen in the brain which neurophysiologists are beginning to understand quite well, and then it has this effect on the mind. I'm going to end with an idea of how the mathematical model can suggest the process of a breakthrough which isn't undone when the mushroom is digested. That would be a change in the model for the brain and mind in which the attractors exist in different places, and there are a lot of them. These are the things that you see when you see. That's why one experience after another reveals familiar patterns. Different people have seen familiar patterns too, and then there are books with drawings. 

Through the experience the map is changed. Imagine that we're playing roulette and you have to put the bet on the right number. It's important to know where the line is that separates the losing bets from the winning bets. If we think of ordinary reliaty and alternate reality as different regions in this geometrical model for the mind, if that line between those states is a clean line and the attractor of ordinary reality is far from that line, then it's quite likely that you would never get over that line. But when you ingest something and you get over it, I suggest that the math has been changed in the following way: the boundary between those two states is thickened and complexified so that it becomes a fractal. We have many examples of this in experimental models studied by computers. Through the fractalization of the boundary between one state and another, the ease of crossing the boundary is obtained, because then you can get close to it and it needs just the slightest breeze and you're out of one loop and into another. Imagine a pot of white paint and put one drop of black paint in the middle. You have the black and the white. If you're deep in the white zone, it's really hard to find that black drop. So then you put the paddle in and stir, and you end up with this marbled thing. Let's say that the white is ordinary reality--and wherever you are in the white there is a bit of black that's real close, so it's very easy to get to it again. 

Well, that's how it's possible to make mathematical models for a lot of psychological things. This is not the same as the models for consciousness made by Penrose and Hemeroff and D.H. Walker and people that used quantuum mechanics. This is a mathematical model. It says nothing about the actual physical reality. The model is just a convenience for the sake of thought. More elaborate models in which you feel at home make it possible to think better about these experiences, and therefore to manage them better. Anyway, that's the promise of mathematics. It's not the same as living better through chemistry.

There you have it.