Our agenda today is, first of all, a little more historiography, and second, the beginning of the journey. So far -- it's almost the middle of the quarter, and Euclid's knowledge has not yet begun, everything has been background. Not only Euclid's elements, but all of the Greek contribution to the Alexandrian library -- that's Plato, Aristotle, Archimedes, Ptolemy, Appolonius -- had to be packed and shipped out in order to survive before Omar's army arrived in 641 and burned the library. Or maybe the Christians did it in 450 or so. The Sarpeum, the so-called "daughter library," which was even bigger -- it's the New York Public Library [?]. The public library aspect of the [museum?] was burned by a Christian mob in the 5th century. These documents, which have created our history and have been so fundamental to the history of mathematics, would have vanished if they had not been put in safekeeping by an apparently accidental program of conservation. Packages were wrapped up like payloads in a rocket and launched into history, out of Alexandria before the fire consumed them. The meaning of that voyage will be our subject today. I'll call it the Alexandrian Highway. It deals particularly with the cities Antioch, Odessa, and [Misibis?], and the story is very much tied up with the story of early Christianity, the early church, in Alexandria.
The story of the transmission of Greek knowledge to the Arabs and thence to Europe, the crucial stage, the initial stage, is carried by Christianity. Of course the Christians wouldn't have moved except there were all these divisions about mental issues, a few of which we'll discuss today. The minority group was forced out of town, and they hit the road carrying the books to these various destinations. That's the beginning of the journey. This is the strangest and most difficult material we'll come across, I think, although early Islam is a close competitor. I don't think I can succeed in making this clear in a single pass, but that'll be the first pass.
Regarding historiography, I'm trying to suggest, even for people who haven't studied mathematics, that mathematical or geometrical or dynamical models for history have a certain utility for understanding the space-time pattern of history, the meta-pattern of history. They have a particular value in emphasising those crucial moments in history when there is some kind of crisis or, as dynamicists say, bifurcation. Therefore, when we try to use mathematical models to understand history, and we especially want to understand those historical crises, then we have to make use of bifurcations, the mathematical models for crises. That's our overall program, that I call dynamical historiography.
I told you about Flinders Petrie before, who dug up ancient Egypt. This was sometime in the nineteenth century, over a hundred years ago. People who traveled in Egypt noticed that the Egyptian pyramids and other temples like Karnak, Denderra and so on, were oriented toward certain groups of stars that had particular significance in the Egyptian pantheon. Star magic and star myth is part of classical literature, so English scholars of classical literature went to Egypt and speculated about these astrological orientations of the temples. These speculations were published in a popular book by [Piazzi?] Smith that's still read today by pyramidologists. Flinders Petrie was a surveyor in London who was fascinated by this story. As a surveyor, he felt that he could sort of figure out if this was true by doing a careful survey of the orientation of the pyramids. So he headed off to Egypt with his baggage and his measuring instruments, and he got stuck in a loop there that lasted his entire life. He became the first Egyptologist. He dug up the whole of ancient Egypt, and along the way he developed many of the methods of field archaeology which are still used, like when you find pottery you have to dust it with a brush, and then put it in a plastic bag with a label and phylum and keep a computer database and all this stuff
What he dug up was basically buried cities full of toxic waste and refuse and garbage cans and a whole lot of broken pottery. Occasionally he would find a papyrus scroll, but he was able to reconstruct an unbelievably detailed documentary film of a lost civilization mainly from the pottery. He noticed that layers had been deposited in a temporally linear fashion. There was continuous occupation of the place for five thousand years -- the longest-lived civilization in history. With each century, there was another few inches of rubble. As he dug down through this to thirty or forty or fifty feet, he found that every once in a while there was a sudden change of style, just as [Joaquim?] Burkhardt found in walking on Basil. Though the expansion of the city was linear in time, the style changed in a non-linear or even discontinuous fashion as you moved along through the record of time. So, whereas people had conjectured about crises in history, like Burkhardt in 1868, before Flinders Petrie -- Flinders Petrie was the first person who had extensive data upon which to test that theory. He found not only that there were crises in history -- in ancient Egyptian history -- but also that in those crises he was able to analyze a micro-structure of the actual transformation. He believed that there were stages and a universal sequence. He found eight different major bifurcations of history, and by comparing the different ones, and he found a universal pattern.
The pattern that he proposed, that he believed to have documented, was that at first there was a change on the level of metaphor and religion, which then would be followed by a transformation in the mathematical sphere, and then by science and technology -- maybe art is part of it also, styles of decoration of pottery and paintings in temples -- and finally economics. With the aid of the technology based on the science based on the art and math based on the overall paradigm or world view, a greater efficiency at extracting resources from the soil would develop and lead to wealth, and then to downfall. Such as we see in the United States. After discovering this in the field of Egyptian data, he tried to apply it to old Europe as a universal pattern of historical crisis.
By studying Euclid's voyage, i.e., the history of mathematics from 300 B.C. until 1600 A.D., we can see that within that span there are a certain number of historical crises that we might want to identify as bifurcations. To make a more convincing argument, we could just rely upon those crises that have already been identified by Burkhardt, such as the death of Alexander, the beginning of the Hellenistic world, the unification of the Roman empire by Augustus in 30 B.C., or the reign of Constantine in 300 or something A.D. So the Roman empire -- what is the Eastern church -- east of what -- is this just a real centrism or what? -- it's east of Rome, because the Roman empire was the whole world.[PLEASE EXPLAIN] It was difficult to manage an empire that large, especially when there were a lot of barbarians on the northeastern flank, which is very far from Rome. It was necessary to have sort of a secondary capital over in the east. That was Byzantium.
There were two emperors, a major and a minor one, and the minor emperor at this particular time was Constantine. He had a vision in 312 A.D., a very vivid flash that induced him to attack Rome and take over the empire completely. He unified the empire and moved the capital from Rome to Byzantium, which he renamed Constantinople. In about 350 or so all the government officials moved to Constantinople, which they actually referred to as "New Rome," east of Rome. Hence the eastern Roman empire.
These crises are identified as the major crises of history by Burkhardt. Within the span of time we're considering, we could look at each and every one of them with the help of Petrie's model, and I would recommend that in your project, whatever it may be, you use the historiographical concept to try to bring life into what otherwise could be a very dull story. I also suggested that one of the outcomes of Euclid's voyage was that the book, wrapped up for traveling to conserve it from barbarians like the early Christians or early Muslims, as it sped along through one culture and one language group after another, brought with it a kind of a transformation. I kind of suggested that the book itself, or Greek learning altogether, might have actually stimulated a social transformation. As we go along, Ralph's pattern or this suggestion can be tested.
I think that if we look at the historical data we have obtained so far in this series of lectures, we'll find that it's wrong, that it looks like this: some social transformation, namely early Christianity, the historical crisis of Constantine, which means the Christianization of the government of the empire, forced these people out who then carried the book. It's more like Euclid's voyage just happened to be in the baggage car when the forces of history took off -- the train took off for some other reason. Anyway, these are questions of historiographical theory or speculation, which can be compared somehow to the exact sequence of events in the historical data of a particular crisis. I'm suggesting that as a program.
Let me give you just two ideas in historiography, then we'll go on to our story. The second of these two historiographic questions was called the Richardsonian Process of Schizmogenesis by Gregory Bateson. Gregory Bateson was some kind of homogeneous [?] in our recent history. He was part of a scientific revolution of wholistic thought that happened after WWII with the Macy Conferences and the birth of Geo Flight General Systems Theory and Cybernetics. Gregory Bateson was an anthropologist, at one time married to Margaret Meade, who studied wild tribes in (Papowan) and Guinea and Bali and places like that. He observed behavior that couldn't be understood at all from the perspective of our culture and knowledge system. He tried to figure out what these people were doing when they all ran out into the ceremonial hall and took off their clothes and put on the clothes of somebody else of the opposite gender. He realized that it is something like the problem of interpreting the Bible, where you have a text that doesn't make any sense, and when you try to interpret it, you know that you're projecting your own ideas onto it, but you don't know what else to do. So he developed a kind of wholistic theory of interpretation, and one of his main ideas was that things tend to divide in two.
Bateson was inspired by a model by Louis Fry Richardson, who, similar to E.M. Forster, was a Quaker conscientious objector physicist metereologist. In WWI he became an ambulance driver in the European theater and saw so many dead bodies that he decided to devote the rest of his life to the scientific prevention of war. He made a mathematical model for the arms race and submitted it for publication and it was rejected. He tried over and over and over again to have his paper published, but it was never published. He saw WWII before he died in 1968, and he believed that had his paper been published and that kind of research continued, that wars could have been avoided. Gregory Bateson, who knew him, realized that out of an argument between two neighbors, which begins with one of them saying bad words or doing something like tossing a banana peel on a day when the other one has low blood sugar, a very small antipathy tends to grow large, and in growing it splits apart into a more and more unbridgable distance between the two, until the war begins.
The prevention of war would have to go way back before the beginning of the arms race to a technique of dealing with conflict just at the very moment when the schizmogenesis -- the development of a schizm -- happens. A mathematical understanding of the process of schizmogenesis could therefore obviate the need for schizmogenesis to happen among neighbors or people who knew that that process can lead to nuclear winter, and knowing that that outcome was undesirable, to nip it in the bud as it were. That's Richardson's idea.
Gregory Bateson applied this social model to a single person's mind, viewing that as another kind of society in which, when there is a universal process of schizmogenesis, a person becomes schizophrenic, and that led to his theory of the double-bind theory of schizophrenia. It also led to one of the first more-or-less successful methods for treating alcoholism. Gregory Bateson, with a background in anthropology, general systems theory, cybernetics, and mathematical models for the psyche, attempted to influence society by a direct intervention based on mathematical understanding. All of that led him to U.C. Santa Cruz, where he was not only a professor but was appointed by our liberal governor Jerry Brown as a regent of the U.S. system. Bateson was not only a liberal, but also an expert observer of foreign social processes almost impossible for us to understand. He went into the board of regents and delivered as an opening shot his opinion of the U.S. system, which was published as an appendix to his book An Ecology of Mind, called "A Stitch in Time," or something like that. Anyway, he nailed the system. This was around 1972, I think, in the 1970s. He died soon afterwards, so he didn't get to see the working out within our own environment of these kinds of ideas applied to oneself and one's own milieu of evolution.
The schizmogenesis process, as it is a process, is a model for a historical crisis just at the moment when it happens. There were two books published during his lifetime, An Ecology of Mind, and Mind and Nature: An Essential Unity. Fantastic, really. (?) then, specific mathematical model coming from dynamical systems theory, actually called catastrophe theory, for the schizmogenesis process, and it just looks like this. It's one aspect of a mathematical model in which somehow there's a geometric space here, as for example corresponding to the (?) system, the historical data or the social structure measured somehow, in imaginary one-dimensional space, and this way time or some other parameter of the process, and here we see this one thing becomes two, it started with this one thing, which was attractive, so that people wanted to adopt that style of life or something, that one thing continues to exist after this crisis or critical moment is halved, it's there, (?) but it's no longer attractive. It's there but it's even repelling, whereas what's attractive then, there's this solid thing and then there are two different states.[UNEDITED PARAGRAPH--ILLUSTRATION]
You might start out with a group of friends, and in spite of a 100% consensus in community meetings, it still splits into two groups that then migrate to separate towns. There's a whole mathematics that goes with it. This is just a little icon coming from catastrophe theory, or dynamical systems theory, the theory of bifurcations, and it's called the pitchfork bifurcation. It's just one among a family of different models, all of which could apply to a historical crisis. We all have kind of different personalities and shapes, and depending on what the historical data is, you could try to fit this model to it, and if this mathematical model doesn't fit, there are dozens of others that you could try out.
Now, to return to our story, we have all these different lines -- let's shrink them down for a minute. So we'll put the time, we can just start with the largest span of time, 700 B.C. -- zero - - 700 B.A. -- something like that. Then we had Athens -- begins I don't know where, say in the Golden Age, and continues on indefinitely even to this day. Then Alexandria, a little bit later, that (?) also, but we say it has a break maybe like this, continues to exist but not that attractive -- break this line -- Alexandria. Then we're on our way to our next major station, which is Byzantium, that started 300 A.D. at the age of Constantine, and that also ended about the same time because of the arrival of the Muslims. And then we go on to Bagdad, which only began in 700 or 800 and that ended too. So that the A.A., the B.B., and we're trying to construct the track and then watch the progress of the train as it carries the books. There's also, I forgot, the Ancient Worlds -- so that goes right up there -- of Egypt and Babylon. Let's just put Egypt. [AGAIN, UNEDITED]
Right around 500 B.C. with Theles and Pythagorus, a traffic in ancient mathematical ideas to Athens began. Then came a development in the Golden period, and then right at the start of Alexandria, they began using the acquisitions of the (Mutheon) of Athens, which became the library of the paripathetic school upon the death of Theoprastus, the successor of Aristotle. The library was purchased or stolen or whatever by the agents of Ptolome II. Pedadelpos, lover of his sister, married his [WHOSE'S?] sister. They all intermarried. No Ptolome ever married somebody outside the family. Cleopatra for example was -- well, strange family. We ought to really tell the complete story of Athens, from the arrival of mathematics to its departure. While Athens continued as the major philosophical school there was practically no mathematics in Athens. Together with Euclid and his book it all went to Alexandria. But the crisis in history is somehow the most important thing for us to consider.
There was very little development during that period that we have any record of, because the success of Euclid's Elements led to the destruction, or rather the failure to conserve the previous texts. Alexandria was the main mathematical frontier of the world for all this time until the transmission to Byzanitum, and yet I have said practically nothing of what happened in Alexandria between Euclid and the closure of the Pagan schools by Justinian in 529 A.D. But there are some interesting stories.
Our program today is to look at the minor stations on the track which go along here: Antioch, Odessa, Lithidus. I'd like to show this map so you can see where we're going... The Mediterranean, Alexandria, Rome, here's Carthage.It's a place that managed to stay independent of the Alexandrian empire for quite a while, and there's Hannibal of the elephants, and some other good stories. Here is Constantinople, that's Byzantium also known as New Rome. We're far aways from Rome, and when you're trying to maintain an army in this area, in what was formerly Syria, and Turkey, and the Balkans -- it's hard to maintain it with soldiers who are all from Rome. Constantinople was well placed for the Roman garrison. And here is Alexandria, the source of (rouge?) and sesame oil. Ships went back and forth to supply New Rome with groceries. And here is Antioch, and here is Odessa, and here is Lithidus, and here is Ekbatana, Jindishapur -- we'll (?) Astoria eventually -- most of these places were actually on Alexander's route. Anyway, here we are again. Antioch, Odessa, Lithidus. These three places are close together, roughly halfway between Alexandria and New Rome, if going by land. Even if going by sea, you might want to go along the coast in order to sell some of the groceries along the way to this other market. We can find.here Alexander's total world, divided in three[?}: the Ptolomeic and the Salucid Empire. The cities we're talking about are in this Syriac-speaking region as opposed to this Egyptian and Greek speaking region. We're on the road to a different language group.
Today I will put Alexandria in the picture. I previously thought of philosophy and religion as a dichotomy, because the Greeks themselves thought that. If you were a philosopher you wouldn't be religious. All that was considered superstition. Today, however, I like to think more of philosophy and religion as the dichotomy of Aristotle expanded into a continuum, and I can use this continuum, this dimension, to separate out some of the events. I think roughly 300 to 600 [years?] should be enough time. Here's a blowup of that little orange thing. I think I'm going to seriously truncate the data I have here in order to try and make it fit on one board, and here I put in this order A E M, going from left to right on our map...
Our story has to do with the early history of Christianity, the so-called early church and its many bifurcations. (Maybe it's good to have more space.) As we know, the story of Christianity must somehow begin around 33 A.D. when Jesus died. It will help, to begin with, to know that things sort of ended up, after various schizms, with the Eastern church in three parts: the Orthodox church, the Nestorian church, and the Mono(?) church. The Nestorian schizm produced this new branch, and then mono(?)ite or anti-Nestorian -- there were a lot of other things. Now all of this evolution of the church happened in Alexandria, so that would belong on this chart, and then the expelled groups, if they weren't all annihilated, migrated to the northeast and to (Aristotlish?) communities in these towns.
I just want to trace the chronology of this complicated event. First of all, we could put math under philosophy, if we want to Of course Euclid is way up above, and Apollonius and Cuspidcles, Then mathematical developments slowed down, but there was a revival of Platonic philosophy of religion called Neoplatonism, after Platinus. Around the 2nd century we had a revolutionary person, Imonius Sockus by name, who revived, purified and simplified Plato's ideas of two worlds -- the divine world, the celestial sphere which they called perfect, and this human terrestrial sphere, which is a mess. Platinus was the main innovator of ideas after Imonius. He gave lectures in Alexandria, which were recorded by his student Porphiry in a book in nine chapters called the Antiads. Many people in every philosophical tradition still study them. Neoplatonism is still alive today.
The Neo- Platonists were the primary conservators of Euclid's Elements and the other Greek mathematical works. Porphiry, a student of Platinus, wrote down Platinus' lectures in nine books. He also wrote a commentary on Euclid and a commentary on Aristotle's logic, which was very important throughout this track. Iambicus -- these people I'm mentioning are straight text on the history of mathematics -- (?) knew Platonica. Like Porphiry, he was from Syria. He came to Alexandria from Syria bringing not only Babylonian mathematics but a different style of philosophy in which there was a lot of star magic and magical invocations and working of miracles and so on. Proclus, who studied mathematics in Alexandria is a little farther down, 500 [years?]or so, just at the end of the Alexandrian museum. Proclus left Alexandria and went to Athens where he became the last head of Plato's Academy. He also wrote a commentary on the first book of Euclid, which is quite good; it contains all the information we have about mathematics before Euclid in the Platonic Academy, as explained in our text by Caps.
So that's a part of the picture. Now between the religious and the philosophical there were some intermediary groups, particularly we could mention Philo Judeus, Philo the Jew, philosopher of no -- as I said, the two (?) in Alexandria occupied by the Jews comprised the largest Jewish city, and in Jerusalem,????