BEGINS

I use the word "math" in the title of this morning's talk, something which I have learned never to do, so I expected, and  must actually be looking at, only the true devotees of mathematics.  I had announced the title as "Mathematics, the Environment and Social Transformation", and I guess what I had in mind were two topics: mathematics and the environment, and mathematics and social transformation.  I'm going to speak about those two things, but not in that order.  Also, time permitting, under the category of "loose ends" I may try to say something more about education, mathematics education.  I've already talked about that a little bit, and the reason why I didn't put "education" in the title of this talk is that at the time of submitting the titles I hadn't begun my current obsession with elementary school education, which started only about two months ago.

So let's consider the question of mathematics and social transformation first.  Why should we be interested in social transformation?  Well obviously, the reason is that we're in one.  I think opinion is divided as to whether we are in a major social transformation, or experiencing the end of the world.  Being an optimist, my friend Terence McKenna, who's been talking about the end of the world now for the past ten years, has begun to hedge and say, "Well, maybe what we're looking at is a major social transformation."  It's never too late to become optimistic once you get this idea of a major social transformation.  One way to understand it is to look at our world's cultural history and try to spot other similar events such as the Italian Renaissance, for example.  Maybe there was a little transformation in the 1800's called the Enlightenment, and then there was the Renaissance.  In art history, European art is classified into different periods like Renaissance, Baroque, Classical etc.  

There is a perceptible change of style between these different periods, which is kind of an indicator, or graphic evidence, for a quantum leap or some kind of catastrophic shift in culture, even in popular culture.  People have studied popular song lyrics and tried to identify through the frequency of the occurrence of certain key words when there is a shift in popular interest that is significant enough, for example, to base a bet on, or an investment in the stock market.  There's a whole theory of cycles based on popular song lyrics.  So whether it's popular music, popular art, or other historical events, we can look at history and see when in the smooth stream of cultural change there is a sudden radical shift, Almost any year in history contains such a shift;   there are bigger ones and smaller ones.  Our idea of what's happening today is that we're in a big one.  If we want to understand it and put it in historical perspective, we might look in history for the largest, quickest, most radical social transformations ever to take place on the planet and study those as models.

Now one reason why I'm doing this is that unlike the present one, all of the past social transformations had an afterwards. This sort of gives me confidence that we will have an afterwards, and this is soothing to the spirit when we see all structures disintegrating around us -- the world economy, for example, or the Western medical system.  We can feel optimism instead of pessimism when we believe in a process of metamorphosis like when the caterpillar turns into the butterfly.  That particular metamorphosis is in fact one of the most spectacular ones observable in nature.  Birth is pretty spectacular, and  so is death, but particularly appropriate for social transformations as a model is the metamorphosis of the caterpillar into the butterfly. The butterfly doesn't live very long and soon dies, but still, there is a transformation which is somehow guided by a mechanism or guidance system that is beyond the understanding of science at the moment.  We know that the chicken comes from a single cell, the egg, by a process which conventional scientists believe to be guided by DNA.  But in the creation of the butterfly from the caterpillar, it's not at all clear how DNA could have any effect, because in the morphogenetic processes that are taking place there's a complete meltdown of the caterpillar into a soup that then reassembles itself into a butterfly.

In my book Chaos, Gaia, Eros I've tried to make a case for three particular social transformations as the biggest ones ever to come down the pike.  I even identify them with the words "chaos," "gaia" and "eros."  That's a long story, but obviously I consider it of deep and enduring value, otherwise I wouldn't have written a book about it.  I'm going to skip that story and the reason for associating these words with these three transformations for the time being  and just say what they are.

The one that I associated with the word "gaia" is the agricultural revolution.  A huge change in the style of life, a huge change in population density and numerical strength of the human species that took place [unintelligible] It is possible to name a year for these things, because the occurrence of a major social transformation, like the metamorphosis of the caterpillar to the butterfly, is a space-time process that, even though catastrophic, still occupies a certain period of time.  In the case of the agricultural revolution, what happened, according to recent reconstructions, is that there were two or three nearby centers where the idea got started, and then it proceeded outward in circles, ever growing like ripples in a pond when you drop a pebble.  It took some two or three thousand years to sweep over the entire planet.  We can see  that this was a catastrophic change that seemed like an earthquake to people living in a certain spot.  When this change swept over them, life changed completely, yet the motion of this change was glacial in its scale. It moved very very slowly -- word of mouth being carried by people who traveled, the messengers bringing a new theory to the stay-at-homes.  

This particular transformation has been studied in minute detail by cultural anthropologists.  The last tribe to join the agricultural transformation, the Kalahari !Kung, accomplished this particular transformation under the gaze of dozens of graduate students in archaeology and anthropology from Harvard in 1974.  So just exactly what happened is extensively documented.  Many features seen in these recent examples, when extrapolated into the past give some idea why weapons and warfare became more popular after the agricultural revolution.

Anyway, it was a big transformation, and I'm claiming that it's on the same scale as the one we're experiencing today.  The next one on this scale, the one that I associated with the word "eros," was the arrival of patriarchy.  It coincided with the discovery of the wheel and with the advent of writing, around 4,000 BC.  There are roughly six thousand years between these two major events.  There is a theory -- in fact there are many theories that I've described under the heading "Dynamical Historiography".  They are mathematical or pseudo-scientific theories about  the mechanism of a major social transformation.  
  
Our main scholar and instigator of all that followed is Sir Flynders Petrie. He was a curious    character. He was a professional surveyor in London at  the turn of the century. He read some recently published books about the orientation of Cheops' pyramid and other monuments in Egypt.  These claims of high scientific knowledge on the part of ancient civilizations was as controversial then as it is now. He thought as a professional surveyor he was uniquely qualified to go to Egypt and measure those alignments and see if, as a matter of fact, they corresponded to stellar events. So he took off for Egypt with his large cases of surveying instruments and stayed there for the rest of his life. He dug up the whole of ancient Egypt and created most of the techniques practiced today in field archaeology: dusting with a brush, labeling, putting stuff in bags and so on. He was the first person to dig down through the detritus of eight thousand years of civilization, and in doing so he discovered art-historical changes in the context of sculptures, paintings, pottery decorations, literature and so on. Digging down he uncovered another few centuries, another transformation every few feet. He discovered eight of these major social transformations in the history of ancient Egypt, the longest lived civilization on record, and he believed that he observed in them a universal sequence or mechanism where the transformation on the mathematical level came first, to be followed by the scientific, the technological, the economic, painting, sculpture--all of these happening in sequence. Since the sequence was the same in each of the eight major social transformations, he thought that there was some kind of law of nature at work, and that this sequence, which I call the Petrie Sequence, should apply to major social transformations in other cultures. He tried to analyze Old Europe from this perspective, with mixed results. This is the subject of his book, Cultural Revolution, published in 1911.

What interested me particularly from my unique and arcane focus of mathematics is the primacy of mathematics in the Petrie sequence. We do seem to see something like that today.  Chaos Theory began a century ago, and the Chaos Revolution in the sciences began only recently, in the past 20 years. In terms of our experience, and most especially my own experience, I have  intimate knowledge of the paradigm shift in the various sciences occasioned by Chaos Theory, because they took place in my career. I wonder if the Petrie Theory, or his idea of the primacy of mathematics, could be an especially advanced indicator, so that if you knew of a paradigm shift in mathematics you could place your bets in the stock market and pick up your profits later. This could somehow be tested by looking back at those major social transformations as advertised. Let's just take one for example, the one I associated with Eros.

In 3850 B.C. plus or minus, the Indo-Aryan people, the Indo-Europeans were migrating from the north in Cossackstan down into Persia, India, Sumer, Babylon and so on, bringing with them the new social structure which is now ours--the patriarchal, hierarchal, war-like, violent society in which we live, and along with it the first alphabet, the beginning of writing, of written records, the beginning of history. There's a book called History Begins in Sumer by a Samuel Noah Kramer. It's a wonderful book, detailing step by step the way in which writing actually began and historical records of everything were kept--decisions made by the king, complaints by the people, ebb and flow of financial reserves in the kingdom and all the stuff  that goes with the patriarchy, writing, and the wheel.

I think that the wheel is somehow the most obvious social transformer of this whole sequence. Still, it is the concommitance of these different revolutionary developments happening at more or less the same time. According to Flynders Petrie's theory, they would then be concommitants of some less visible trigger, which would be a mathematical trigger. The question I'm proposing is, could that be the case?  

In the case of the wheel, the revolution was a space-time pattern in which the first wheel, which was a pottery wheel, has been traced archaeologically in the following way: the pottery wheel leaves traces in pottery. Pottery shards are the most enduring remnants of any culture. So when teams of field archaeologists go out and dig, the first thing they come upon are pottery shards. If the pottery was made on a wheel, there are round marks on the pottery that prove that it was made on a wheel.  They've essentially dug down all over the world looking for pottery shards which are datable with radio carbon techniques and dated all the pottery shards made on wheels. These dates form a pattern of concentric circles with a center in the Middle East, which is known exactly. Somebody got the idea of the pottery wheel, and it expanded in these circles very slowly. It took from 4000 B.C. to 500 B.C. to reach England. Maybe the English Channel  slows down communication by voyagers, but of course there was always trade all over the world from 100,000 B.C. onwards. 

It was a major transformation and it traveled slowly, like a glacier.  A major social transformation is a space-time pattern. You have to recognize the movie, not any particularly event in the movie. Now the wheel, the first wheel, the pottery wheel, the first material tool wheel soon turned into a cartwheel, and the cartwheel made it possible to locate the fields farther away from the house. That empowered the idea of cities, and so cities began. As soon as you had the agricultural revolution you had settlements, in the field essentially, with the animals living on the first floor and the people upstairs. You kind of get used to the smell and the flies. I've lived this way in India. With the advent of the cartwheel, the produce could be brought from the farm to the city, and that's when giant cities like Shumer and Babylon and Egypt and Canaan and Jericho and so on spring up. Joshua Fought the Battle of Jericho--that was in 7000 B.C.--so that's the timespan. In that way the wheel can in fact be seen as the precipitator for major social transformation via cities. It's a plague that has grown more and more, until the reverse movement back to the field began just recently.

And now I want to make an argument for the primacy of mathematics in two of these concommitant major social transformations or developments: the wheel and writing. The history of the alphabet is like archaeology: you start digging on the surface and go backwards.  You get the story in backwards order and then later on you tell it in frontwards order as if that's the way you learned it. The history of the alphabet, of all these different alphabets that we have today--the cyrilic, the sominican and so on--goes back to a common root. But when it comes to the question of who actually used a symbol for what when it became an alphabet, there is kind of a vacuum in the theory, because that transcends cultural history in a way. Our best theory about this--David Moran and others--is that the alphabet was created as an astrological sign system,  and that the letters of the alphabet correspond to the so-called mansions of the moon in Chinese astrology, the constellations of the Northern Pole divided into 27 polar zodiacal constellations, and the signs for those constellations became the alphabet, ABC and so on. Well, theories like this can't be proven. It has no competitor, by the way. If we understand the study of the motion of the planets and the stars and their rising and setting and their correlation with the seasons as the main mathematical enterprise of early peoples, then we can say that mathematics led to the alphabet, according to this scenario. 

Now let's talk about the wheel. How do we get the wheel from mathematics? Isn't it a pottery wheel? Well, and how do you get a pottery wheel?  Here's the story. It is generated by the observation that the Mayan civilization did not have a wheel. They did not have carts. They did not have cartwheels. They didn't have pottery wheels. They didn't have pottery made out of wheels. This civilization has been very extensively studied because of its proximity to great universities and its tremendous mystery. Even now they're still decoding that culture's literary records which escaped the flames of Bishop Delonda. The wheels that the Mayan peoples had were toy wheels, and it's suggested that the toy wheels actually preceded the larger wheels that were used as tools in the process of cultural discovery that brought about the wheel. It is  conjectured that the origin of the toy wheel was as a model for the heavens. The Mayan calendar is the most advanced in the world. We observe time by watching stars rise and set on a horizon. Sir Norman Lockier, the early pioneer astronomer-royal of England and early pioneer of archaeo-astronomy, said that the horizon was the telescope of ancient peoples. That's why Stonehenge and Carnack and monuments like that are the main evidence for archaeo-astronomy.  These monuments are telescopes in the sense that they allowed you to  observe the rising and setting of certain stars on the horizon as they moved back and forth along a short arc, and that's a clock and you find the calendar.

Observing the position of the sun against the constellations, it soon became clear that it moves back and forth in a belt, which we now call the zodiacal belt. I've tried to figure this out myself by using a hot tub as my observatory and studying the stars without having read and understood all the books on archaeo-astrology.  I tried to figure it out for myself, how does it work? Where does the sun rise, where does it set? What is the phase of the moon, what is it doing?  I just kind of figured it out from scratch in order to recreate the step-by-step process by which shepherds, who watched the sky instead of television, originally figured this out. It's almost impossible to figure it out without making a little three-dimensional model. You end up with this object called a Hemeris Mill from the book by the same name, the great testament of archaeo-astronomy. It's a millstone, a thick wheel, on the cylindrical surface of which you could paint the zodiacal constellations and map the path of the sun as it winds snake-like up and down past the horizon around the zodiacal wheel, like a spinning top on an axle that you could twist with your fingers in order to imitate the motion. So if you tried to figure out the sky, early mathematics, with the aid of a model, you'd end up with a toy wheel, which could then become--

Maybe I'm carrying on a little too long with this, but it's one layer of fantasy upon another, trying to pursue at least the possible plausibility of the Petrie Sequence in the context of one of the largest social transformations ever to come down the pike. I would never do this if I weren't me, I guess.  Unlike most people, thanks to the accident of missing high school, I ended up loving mathematics.  I just think it's the most wonderful thing, beautiful in itself and providing all that's necessary to glorify the soul, without needing to be useful. However, it has been useful. It has been utilized throughout time in gaining understanding about the environment, the world we live in, the relationships between peoples, the time, the heavens empowering agriculture, the wheel, the major social transformations up to now, so why not now? Hence I'm putting Chaos forward as the main element in the whole spectrum of the concommitant elements that comprise the major social transformation we're in today--the feminist revolution, concern for the environment, population explosion, chemical-industrial dependence on electricity, warfare, peace, nuclear contamination--who would put math first? Well, I am. That's the story of math and major social transformation, and let me repeat that the reason we're interested in this is we are in one now. The better we understand it, the more comfortable we'll feel that something we do seeking to influence the outcome can actually affect the outcome in a desired way, so that we end up with a better world--a butterfly world after the disintegration of this caterpillar world that we see behind us. 

The Chaos Theory contains a theory called Theory of Bifurcation which is even more technically helpful for making specific models for major social transformations. What are the steps that can be expected? What are changes which are very unlikely from the point of view of mathematical models? This is somehow very empowering for optimism and for effective social action. That's my conviction anyway.

That naturally leads to the environment. My next topic, Math and the Environment, is a smaller scale version of the same thing. We live in a really complicated system, too complex to grok with our futile human minds. There is the stone with the molten core, the motion of the drifting continents; then there is the water with its very complicated currents, and then the biosphere  where the water is full of life forms from the microbial to giant whales. The land is covered with green stuff and also with this microbial mat, ** The soil is the largest living thing on the planet, and the atmosphere with its hurricanes and its climate variations, and climate has certainly been the main motor for cultural evolution. It's well understood that an Ice Age is coming soon, and that means that it would be intelligent to invest in real estate in Portugal and sell your land here on the coast.

All of these systems are so complicated that we can't understand them, and they are all  interacting as one, the whole enchilada. The wholistic thing is too big to grok even though we have satellite photos--no chance of grokking the whole thing. Even an ecosystem--when we look at Bridal Veil Falls or Bear Creek or some other ecosystem, we see that one small ecosystem has all the climate variations, the weather variations, the temperature, the carbon dioxide, etc., etc. The complexities are difficult to understand. Some people have a special intelligence for complex systems, a kind of  gestalt intelligence--The Howard Gardener Theory of Multiple Intelligences or something--and there are people who become naturalists and always know where they are in the forest. They are sensitive to influences and have a special talent for that. 

Mathematics comprises the special talent for understanding complex systems. Under its pseudonym, Complexity Theory,  Chaos Theory in particular provides a way for modeling complex systems, but the models that mathematics makes for these systems admittedly are no good. They're not an exact model of life. Their predictions are valueless. Nevertheless they do help us to understand a little bit. It's like card counting when you play Blackjack. The advantage of a fraction of a percent is enough to turn you from a loser into a winner. It feels a lot better to be a winner, as the casino is an institution devoted to losers. So the mathematical modeling of complex systems--impossible, unthinkable before the chaos revolution--now gives us a way to see a little further into the distance, into the haze of complex systems. We could say that we can understand what we can understand. I mean we can understand simple systems and we can understand systems that are a little bit complex. 

If we think of a space like the universe, in which the more complex systems are farther away, then each person has a natural horizon of complexity. We can see all the complexity out to so far, and beyond that we can't see at all.  Some people's horizons are a little bit bigger than others', and the effect of Chaos Theory is to provide a complexity telescope, which I call the Chaoscope, so that we can see substantially farther past the normal horizon of complexity. It doesn't give us the power to understand the whole thing, but it increases enormously our capability to understand ecosystems on the scale that we live in. Mathematical modeling of ecosystems is a connection between math and the environment which can enhance our understanding of the environment, particularly in the language of nature which is chaos, something we could not do before the advent of chaos theory, never mind the computer revolution. We needed chaos theory, and we needed the computer revolution in order to  interact with all those complex systems that are beyond our complexity horizon. All of this is something very very new.

One of the amazing developments was a commercial computer game called Simmer. Many of you know it. A very small company, consisting maybe of one or two people, Maxus Software, made a computer game for the Macintosh called Sin City. It was very popular and gave them the financial resources to do a second game. What they chose to do is the history of the biosphere as a video game with which you could interact, as it were, scientifically. For example, what would happen if we had ten percent more oxygen at the beginning of that big bang in four billion years B.C.?  What if there were more of this or less of that? You can set the computer running in the evening, and when you come in in the morning you see what it would be like in 1995 in terms of the main elements of the biosphere--the amount of oxygen, the temperature and so on. This commercial effort leapfrogged the efforts of  large national laboratories like NCAR by working on the same thing and doing it in a way that was popular with children and commercially successful. That game is the epitomy of what I see as the educational value of mathematics and chaos theory in the context of the environment. This kind of game can be developed and put on the World Wide Web so that anybody can play it, comprising educational environments or resources for people who are interested and want to learn these things. 

There are some excellent games on a more useful scale than Simmer on the World Wide Web right now. For example there's one called "Grouse", some of you may be familiar with it. It has to do with the effect of a military base on the environment, especially the Grouse population in Eastern Washington State. There's another one called "Cyberfarm in Illinois", where almost all the farms of a certain county have been wired up to censors connected to computers connected to the Internet. You can cruise those farms from Japan or anywhere else and observe the basic parameters like pH and moisture in the soil and find out if those farms would be good locations for growing a crop of brown rice or something like that. That provides for interaction with real farms on a countywide basis for people both in the county and outside. It's not exactly like a computer game, but the whole World Wide Web is essentially a computer game, and these are its more interesting installations as far as education and the environment is concerned. All of these installations have mathematical bases coming from Chaos Theory and created by Chaos Theorists.

So, once again, the role of Mathematics in the environment that I'm advertising is an increase in the complexity horizon and an improvement in our sensitivity to nature by enabling us to decode to a slight degree her language of chaos and fractal geometry. These forms are the basic alphabet for communication in nature, and practicing these new forms of mathematics that are computer based can enhance our sensitivity to nature.

Now I've talked about education in a general way, and I have to admit that I'm amazed that I'm so late in coming to this realization. I've been complaining for a long time about the state of mathematics in our culture. As long ago as 1975  I began a computer-based project at the University of California in Santa Cruz to revolutionize the teaching of mathematics within the university by using computers and changing the curriculum and so on. I was trying to work against the math anxiety and math avoidance reflex which abounds in our society, as well as trying to work against the very unfortunate gender bias in the world of mathematics, where there are as few women mathematicians now as there were since ancient Greece--in fact, since historical records have been kept. It hadn't really occurred to me, although the evidence was full in my face, that university students are already destroyed mathematically by elementary school. It was too late to change the curriculum and to add computer graphic method visualization techniques--it had to be done in the elementary school. 

I've described my interest in the elementary school curriculum as a whole previously. Based on Sheldrake's principle that things are learned more easily in historical order, I had revolutionized my own teaching in the university by changing it from mathematical order to its opposite, historical order, in the presentation of even the most technical classes. It took me until just the last week or two to really begin to envision a historically based math curriculum for the elementary grades, K-12. Basically, to teach mathematics in historical order, you have to use Euclid as a textbook. Euclid's Elements is the second most popular book of all time, the most important mathematical textbook of all time. There are thirteen books, each just 30 or 40 pages long. It has been the one and only mathematical textbook from Plato's Academy up until recently. I studied it myself in junior high school. At that time it was still used as a textbook. By now I think it's been abandoned everywhere, although in high school gemoetry classes you do find material that comes from Euclid's Elements, but it's taught in a completely different way--the wrong way, opposite to the way that Euclid taught mathematics himself.

Euclid's Elements was written in 300 B.C. It collected all the mathematics from previous civilizations and put them in historical order. For example in Book 5 and in Book 7, the same subject is treated. How come Euclid devoted Book 7 to the subject which had already been treated in detail in Book 5? historians ask. The answer was, because he was honoring history, and Book 5 was an earlier treatment of the same subject.  Sheldrake's Principle, applied to math education in the elementary school, says you have to somehow use Euclid as a textbook. However, this is a very very difficult textbook. It's hard for professional mathematicians to understand. I'm trying to mitigate that now by creating a new edition of Euclid's Elements in which all of the proofs and statements are illustrated with little drawings that progress step by step, making it understandable at least to teachers. But there's so much material here that I myself spend about an hour a day in what I call mathematical meditation when I get up in the morning, translating Euclid. I think it's about as fast as I could go, and by Christmas this year,  I'll have done six books. So at the pace of an elementary school or even of university education,  it will take several years to go through this book. It will be necessary to extract a little material here and there in Euclid in order to make a curriculum. A lot of elements in Euclid are constructions, as they're called, not theorems with proofs. Toconstruct an equilateral triangle with a given base the way the ancients did is by stretching a rope, and this is something that children can do in teams. The constructions extracted from Euclid comprise a curriculum which is doable in terms of physical movement, like Montessori rods applied to geometry. 

Still, there are too many constructions in Euclid for the elementary grades. So another idea would be to select a subset of Euclidian constructions that center around a theme. If you could find the right theme you would end up with just the right number of constructions that could be done in the curriculum K-12. I found such a theme by accident. It is called the Golden Section. Many people have heard of it because it's extensively used in the arts, especially the arts of the Renaissance, in architecture, in sculpture, and in paintings. So far this exists only as an idea, but I hope within a couple of months to have a website describing what I call The Golden Thread as a possible curriculum for math education in schools. 

The ideal way of doing this would be to obtain mathematics as an empowerment for the purpose of understanding instead of misunderstanding what I and other people are saying about the role of mathematics in social transformation and its possible effect in giving us optimism and informing our political actions. The role of mathematics in the environment is to make us more sensitive to the language of nature. These possibilities can only become realities when we have an educational system which is either mathematically empowering or at least mathematically neutral so that people's inherent mathematical talent is not destroyed in anti-educational institutions like elementary schools. 

So there you have it. Thanks.