Athens, Alexandria, Byzantium, Bagdad, etc., these are the major stations on the trip. It's hard to remember, but try thinking of it as AA BB -- Athens, Alexandria, Byzantium, Bagdad. Today's agenda is Byzantium. We should arrive at (?) station #3 and take a tour there. It can't be done in a day or a week. If we had enough money we'd put up at a good hotel and stay for an entire year in Byzayntim.
I want to correct a mistake I made last time about the birth and death dates of Pilo Judeus. He was born in 20 B.C. and died around 50 A.D. I had him a little earlier on the chart. Pilo remembers one of the mergers between the agnostics and the philosophical and religious traditions in Alexandria. What emerges is this figure: philosophy/religion bridge in say minus 100, minus 300, minus 400, plus 400, or whatever. Here we have Plato, and of course Aristotle, and most importantly, we have Stoics and Neoplatonists. Here we have the Orphic religion of Ancient Greece about which I didn't speak too much, though it was the most important religion of Ancient Greece. It came into Alexandria in disguise as the merry deity Dionysus, who was renamed Serapus. Later, as Alexandria developed, came the Jews and the Christians, and in between these brides [?] of Pilo and Anostics. That summarizes almost everything we've talked about as a foundation and background for understanding the generation of Greek mathematics.
I thought I'd like to share with you just a couple of passages from Gregory Bateson's book Mind and Nature: A Necessary Unity, which I mentioned last time. First, let me explain schizmogenesis. This word comes up in many of his lectures in the earlier book, Steps to an Ecology of Mind. He uses it in the context of a paper on the culture of the (Yatmo?) people in New Guinea. He was puzzled by their behavior and couldn't understand the "typing of the sexes" as it was called then. The initial ideas were based on Patterns of Culture, a book by Ruth Benedict that I mentioned before. Bateson says:
"(?) this typology [this is about two psychological types] as an abstract map onto which I dissected my descriptions of Yatmo men and women. This dissection, and especially the fact of differentiating the typing of the sexes, which would have been foreign to the ideas of patterns of culture, went away from typology and into questions of process [that is for us dynamics, as in dynamical historiography, chaos theory, fractal geometry, postmodern mathematics and son on]. It became natural to look at the Yatmo data as exemplifying various interactions between men and women which would create in the men and the women that differentiation of ethos which was the base of my typology of persons. I looked to see how the behavior of the men may promote and determine that of the women and vice versa. In other words, I proceded from a classification or typology to a setter of the processes that generated the differences summarized in the typology. But the next step was from process to a typology of process. I labeled the processes with the term 'schizmogenesis' and had them put a label on the processes. I went on to a classification of them. It became clear that a fundamental dichotomy was possible. The processes of interaction that showed the general potentiality of promoting schizmogenesis, that is first determining character within the individuals and beyond that creating intolerable stress. The processes were in fact classifiable into two great genera -- the symmetric and the complementary. I applied the term 'symmetric' to all those forms of interaction that could be described in terms of competition, rivaly, mutual emulation, and so on. That is, those in which A's action of a given kind would stimulate B to action of the same kind, which in turn would stimulate A to further similar actions, and so on. If A engaged in boasting, this would stimulate B to further boasting and vice versa. In contrast, I applied the word "complementary" to interactional 'complementary' to interactional sequences of which the actions of A and B were different but mutually fitted each other. For example, dominance-submission, exhibition- spectatorship, dependence-nurturance. I noted that these paired relationships could likewise be schizmogenesis. For example, that dependency might promote nurturance, and vice versa. At this point I had a classification or a typology, not of persons but of processes, and it was natural to swing from this classification to ask about what might be generated by interaction among the named processes. What would happen when symmetrical rivalry, which by itself would generate symmetrical schizmogenesis of excessive competition -- what would happen when this symmetrical rivalry were mixed with complementary dependence-nurturance? Sure enough, there were fascinating interactions between the named processes. It turned out that the symmetrical and complementary themes of interaction are mutually negating. That is, as mutually opposite effects on relationship. So that when complementary schizmogenesis, for example dominant submission, has gone uncomfortably far, a little competition will relieve the strain. Conversely, when competition has gone too far, a little dependency will be a comfort."
So here, in just two pages of this book, we see the characteristic working of the mind of Gregory Bateson. He starts with a typology of personality into two different types of personality, thinks of the processes that generated them, tries to classify these, thinks of the process of those, tries to classify that, and then permutes them in a Pythagorean scheme to try to account for all these different personality processes that extend from the individual psyche to the group, from the group psyche to the whole of history, and evolve as an idea, essentially a mathematical idea for the evolution of all and everything -- what we generally call morphogenesis. He applies this one model everywhere. It is essentially a mathematical scheme that tries to understand everything in perception, history, literature and so on. He applied this nurturance-dependency theme to alcoholism and developed one of the first successful therapies -- well I don't know how successful it was, so let's say, relatively successful therapies for alcohol dependency.
When he came to UCSC in the 1970s -- I think I met him first at the old Catalyst around 1973 -- he looked at the university system and at the students in his classes. His classes were very popular, but I think attendance was limited to 10 or 15 people. You'd come in, everybody would sit down, and he wouldn't say a word. If anything happened whatsoever, it was up to you. He developed (sams). Experiencing the university, he got an idea based on the theme of metapattern, or what we would call mathematical structure, for education itself. Upon being appointed a regent of the UC system by the liberal governor Jerry Brown in 1978, he presented a paper for the first meeting which is reproduced at the end of this book. I 'd like to read you a little bit of a memorandum circulated to the Regents of the University of California in August 1978, called "Time is Out of Joint":
"At the meeting of the Committee on Educational Policy, July 20, 1978, I remarked that current education processes are a rip-off from the point of view of the student. The present note is to explain this view. It is a matter of obsolescence. While much that universities teach today is new and up to date, the presupposition of a premise as a thought upon which all our teaching is based are ancient and are, I assert, obsolete. I refer to such notions as A, B, and C. (A) The Cartesian dealer when separating mind and matter, (B) The strange physical rhythm of the metaphors which we use to describe and explain mental phenomena, that is power, tension, energy, social force, etc., (C) Our anti-aesthetic assumption, borrowed from the emphasis which (?), Locke and Newton long ago gave to the physical sciences, namely that all phenomena, including the mental, can and shall be studied and evaluated in quantitative terms. The view of the world, the latent and partly unconscious epistemology which such ideas together generate is out of date in three different ways: A, B, C. (A) Pragmatically, it is clear that these premises and their corrolaries lead to greed, monstrous overgrowth, (??) and pollution. In this sense, our premises are daily demonstrated faults, and the students are half aware of this. (B) Intellectually, the premises are obsolete in that systems theory, cybernetics, wholistic medicine, ecology and Gestalt psychology offer demonstrably better ways of understanding the world of biology and behavior. (C) As a base for religion, such premises as I have mentioned become clearly intolerable and therefore obsolete about 100 years ago. In the aftermath of Darwinian evolution, this was stated rather clearly by such thinkers as Samuel Butler and Prince Quipratkin. Already in the 18th century, William Blake saw that the philosophy of Locke and Newton could only generate dark, satanic mills."
He ends by stating, "So we come back to the place from which we started, seeing that place in a wider perspective. The place is the university, and we, this board of regents. The wider perspective is about perspectives, and the question posed is this: Do we as a board foster whatever will promote in students, in faculty and around the boardroom table, those wider perspectives which will bring our system back into an appropriate synchrony or harmony between (rumor? river?) and imagination. In short, as teachers, are we wise?"
He said this in 1978. I'd like to put another correction in here, about the minor stations. Where we give a list of three -- Antioch, Odessa, and Nissibus -- and their beginning dates in order, I think they were 330, 363, and 490, I want to mention a fourth: Jimdi Shapur, the camp of King Shapur, begins in 531, and therefore is a little out of the time frame we're discussing now. That's why I didn't put it on the map previously, but here we have a map, our main picture, in which we see Athens, beginning with the golden years, 375 to 350 B.C., then Alexandria, beginning with the golden years of Euclid, 285 B.C., and then I put these three little ones down here, and we'll add a fourth. And then Byzantium -- these are sort of going in parallel and there's interactions in between, and quite a while later comes Bagdad, our agenda for next week. Jimdi Shapur, as we see on a map, if we ever do see it on a map, is close to Bagdad. These minor stations can be thought of as a sort of trunkline leading to Bagdad, along with (?) near to Byzantium. There's a sort of trade route in the traffic of mathematical ideas from Alexandria to Byzantium to Bagdad, in which these minor stations figure several times repeatedly in sequence, so it's kind of hard to put them in linear order.
It may be good to keep the distances in mind. Athens, Alexandria, Byzantium -- over land this is about 1000 miles, like from here to the Oregon border. And over land this is about 300 miles. I'll come back to Jimdi Shapur later when discussing Bagdad next week. Another useful map idea about which I haven't spoken too much so far is called histomap. There are a couple of famous examples of a histomap of the entire world history published in a book which is sort of a very long banner that unfolds. It looks like a book, but when you unfold it you get this very long banner in which we have time going to the right, and space going to the left. The Greek world is shown as resembling the trunk of a tree, and the bifurcations I'll show are more or less like branches growing on a tree. Branches can arrive like roots and depart as branches. So here we have the Roman Empire and then we have the Greek world, and the Roman world gets bigger and bigger while the Greek world gets smaller and smaller. That's how you get the Roman Empire. Then the Roman Empire splits in two, into east and west, and then they get together again in one of the major bifurcations of history.
That (?) called crisis of history with Constantine let's say in 330 when the capital was moved from old Rome to new Rome, from East Rome to West Rome. There were some other kinds of bifurcation, when East Rome, Byzantium, came to dominate. So East Rome was called Byzantium, and old Rome became essential Eastern Byzantium, and that corresponds to an enlargement of the empire on the eastern end, under Justinian, in 529. Here are some important dates in Byzantine history: portrayed in this way, a style called a histomap, it makes the bifurcations particularly evident. This old world bifurcation sort of suggests furcation forking into two, that's the history of the bifurcation concept. (?) Casiole was the translation in French of the German Abzweigung, which means dividing into two It was introduced by Jacoby in 1838. What we'll now call the pitchfork bifurcation is just one example. This is a normal pitchfork bifurcation, and here is the reverse bifurcation. Here's one where something simply gets larger, as for example in the crisis of history represented by Alexander. That's next. Here are some where many different tiny Greek states were unified into a huge empire. This one, like this one, is what we call a stilted bifurcation. This kind of graphical representation can be useful too in your reading and in your projects.
So now we're ready for Byzantium, our major station #3. In a way it might be the most difficult, because, to begin with, we know so little about it. Hardly anybody has read a word about the Byzantine empire, it's just considered too arcane, too difficult. Indeed, it is plenty weird. And Alexandria is pretty weird too, if you read the story in detail. It's a story that's unbelievable, amazing, beyond science fiction. It's wonderful to read and yet horrible in its bloodiness and its ripping people apart, hacking them up with shells and blinding closest relatives and killing them in their sleep -- Alexandria is incredible. Byzantium is very similar. It, too, has a culture that lived a thousand years. Very clever, politically, very powerful militarily, and very wealthy with fantastic and novel developments in art and literature and so on. Alexandria and Byzantium are kind of a parallel, and very close together.
There was an important prophet who went back and forth over this land, passing these minor stations repeatedly during the overlap of those cultures for about 300 years. There's a lot of data here that is difficult to access. Most everyone overlooks Byzantium. For example, in our text, History of Mathematics: An Introducation by Mr. Katz, there's no mention whatsoever of Byzantium. The theme of this series is Euclid's voyage, and Euclid's voyage has this important stop where in fact books 14 and 15 were written. If it weren't for the mathematicians of Byzantium, whose names nobody in Santa Cruz has ever heard, we wouldn't even have Euclid.
Byzantium means city in Latin, Byzantiun in Greek. It was just a tiny little village until Constantine came along, and then he named it Constantinople in Latin, or Constantapolis in Greek, and it became the capital of the Roman Empire. After Justinian the Roman Empire ceased to be called the Roman Empire and became the Byzantine Empire because the culture was completely different. The population came largely from Asia Minor -- a lot of Armenians or Syrians and Jews who also spoke Greek and identified with Alexandria and Athens. Even though they lived in the Roman Empire they were not speaking Latin, they rejected Latin. In church maybe Latin could be used, but Greek was used in daily life and eventually it even became the main language of the Orthodox church. It was a completely different culture from Rome. In this sort of bifurcation there's a swing, and then there's some connection to Rome, it's stretched and stretched and finally breaks, and then it snaps over to the completely different culture. That empire also uses the name Empire, first the Roman Empire and then the Byzantine Empire, and the Byzantine Empire is also called Byzantium. This causes a certain confusion. For us the emphasis would be on the city, where the universities or academies and mathematicians were primarily working.
I want to show you some maps. It would be best if I could give all of this in order, but I suppose that's impossible.We miight start with this Penguin Atlas of Ancient History that I've been relying on, because, as I explained, all the maps are of the same area on exactly the same scale, so you can (?) it in a flip book. Here's the last page, which shows the trade routes. Here's Alexandria, Constantinople, Athens is right there, kind of small, those are close and these are far, Alexandria and Constantinople. Here you had to go through the water through Antioch, and that way. And here we actually can see Jimdi Shapur. This blue one is the Mediterranean, this is the Black Sea, and there's a narrow neck between the Black Sea and the Mediterranean Sea. This little bit is the Adriatic, which is full of islands, some of which are important in our story; that's part of Greece. And this is Turkey, the Armenians are up here at the end of the Black Sea. The former Persian empire was taken over by the Salutids, and here isArabia with the wild tribes, which will figure importantly in our story next week. Here's Odessa and Nissidus. I showed this last time. Here's another (?) date, 362 A.D., that's 30 years after the foundation of Constantinople as the capital of the Roman Empire. Here's the Roman Empire, there again the Mediterranean and the Black Sea, so here in this little neck and a very narrow strait going through it, you can practically throw a stone across. Constantinople is just on the left or western side of that isthmus, and that little mouth-shaped lake in between here is the Sea of Marmara. Here's the Nile, Alexandria is here, Constantinople there, Rome, formerly the center of the Roman Empire, is here. The center shifts, it doesn't look like that big a deal, because most of the action -- trying to fight off the Persian Empire and Osterbrof -- happened on this side.
Melville does a long search over 1000 entries for the title word 'Byzantium'. Let me show you the main resource, the Cambridge Medieval History, Vol. 4, the Byzantine Empire, in two parts, Part 1, Byzantium and its neighbors, is a fairly thick work, and Part 2 is just a little bit thinner. These two together comprise by far the best resource. Nothing comes close for the history of Byzantium. Now let me show you some maps. Here's the Byzantine Empire. There's a zillion labels here, so the states, the cities, the counties, everything is identified. The most interesting feature of the map might be the boundary. This says the word 'extant' around the year 1025. It includes the whole (group), the eastern half of the boot of Italy, including (Bali?), which is where you get the ferryboat to go to Greece nowadays. Then there's Crete, one of the main sources of Greek culture in the beginning, and here's Macedon and Pela, where Alexander came from, Constanople, Sea of Marmara, and here, (?), so Armenians are living in this area. Trebazon, the famous city in classical history. Over here the Euphrates and the Tigris river are going down to the Persian Gulf, and somewhere around here is the location of present- day Bagdad, just off the edge there. And here of course is Odessa and Antioch. Just below Odessa is Halon, another city that will eventually be of great importance in our story. These are a few maps of Byzantium. There are a lot of big and very detailed maps in this book.
We'll focus on the city. Here's the city of Constantiple, the Sea of Marmara, I showed you that on the larger map, the Bosporus, which is this narrow strait. There's another little thing here, the Golden Horn, and I've seen this promentary in a description in the literature of a trip up the Sea of Marmara in a boat approaching this city, and it was extraordinarily beautiful as seen from the sea. The palace area is down here, near the sea, and there are monasteries everywhere. Here is the Hippodrome where they had chariot races that were like the Super Bowl or something. Everyone attended every chariot race. There were teams called the Blues and the Greens, and people following the two teams became very important political factions and had riots after the game and tried to take over the entire empire from the king.
One more map, I want to show you the minor stations. Here's an interesting book, proceedings of a conference held in 1980 in Los Angeles. There's a phenomenal amount of detail coming from very recent scholarship about Syria and Armenia in the formative period. This map shows -- this is blue, the Mediterranean, and here is Antioch, and here Odessa, that we showed you in smaller scale on the earlier map, and here is Haron, and here is Nissibus. So here are the minor stations -- Antioch, Odessa, Nissibus, shows on this map. Now these (?) are like train tracks, these are trade routes, "Ancient trade routes of Ancient Mesopotamia after Charlesworth" it says here. And in the neighborhood of Nissibus, this little rectangle is blown up larger here and it shows a whole bunch of monasteries that are important in the early history of Christianity, especially regarding the Nescarian heretics. Anyway, this is a region of interest in our story. These X's mark the boundary of the Byzantine Empire in this area. And you'll see indeed that Nissibus was right on the border between the Byzantine Empire and the Syrian enemies, with their capitol in Damascus, and eventually captured and this border then, in history, is moving back and forth. So it's sort of like the beach. It's a fractal boundary between two different cultures, and fractal boundaries facilitate diffusion, and now diffusion is one of the key ideas in our story, so the moving back and forth of this boundary where here it's shown in the year 363, so that's, again, just a mere 30 years after the foundation of Byzantine Empire, and before the time of Justinian. This boundary moving back and forth, of course, when it's down here, then we had these cities accessible to the Christian diaspora, the heretics, the medocubites and Nestorians come in this way, suddenly this area given back to Persia, the boundary moves up, then these places are incorporated in the Persian kingdom and they have meanwhile have been briskly translating the books from Greek into Syriac, which is a dialect of Aramaic, which is the language of the later Old Testament, after the pentatuc(?) in Hebrew. And semitic language related to Hebrew. That's the maps. [GOT TIRED OF EDITING THE MAPS, SO I SKIPPED THIS PARAGRAPH, OKAY?]
I'd like to give a more detailed idea what it was like in Byzantium, so maybe I will carry Byzantium over into next week. Maybe we could see a slide show with folk music of the time, or if I can find a video, I'll show some of that and read some stories of everyday life in Byzantium; it's pretty amazing. Today, let me continue with a chronology in which we put mathematics on the map. I've neglected this with our previous major stations because you have a text to read that tells the story in pretty complete detail. However, our text doesn't mention Byzantium, so I'm going to give a chronograph of Byzantium's philsophy and mathematics. It's a pretty long story. We'll be primarily interested in what happens when the Greek literature is received around 330 or so from Alexandria.
There was constant traffic back and forth between mathematicians going from Byzantium to Alexandria and vice versa, beginning somewhere up here, and ending around 800 when our train moves on to Bagdad. First let's identify our main periods. By the way, the only story I could find with any detail at all about Byzantine mathematics is in this book, Part 2 of the Volume 4 on Byzantium. There's an article called "Byzantine Science" that's pretty long, phenomenally detailed, and with an outstanding bibliography. The author's name is Vogel. This, I think, is 1959. There should be some more material, but I've looked hard and haven't found it.
Vogel divides our times as follows, in three parts, from 527, the beginning of the reign of Justinian I. In the back of Part I you have the list of rulers, and of course there's an awful lot of them, beginning with Constantine. There's Constantine I in 324, Constantius II, Julian, Jovian, (?), Odesseus I, Icabius, Odesseus II, Markin, Leo I, Leo II, Vino (?), Vino again, and Astasius I, Justin I, Justinian I, 526 to 565. So there were more than a dozen rulers between Constantine and Justinian. Anyway, this kind of data is enebrus [?]when you're reading it because, since you never heard of these names before, you have no background in the reading. If you did have a chapter in Katz on Byzantine mathematics, you wouldn't understand a word. Who's Andronicus through Peralogus? What's the Armenian dynasty? and so on.
There are the geneological tables with this list of rulers that lasted(?)only for one year or less. This huge list is divided into families, showing the descent from the parent to the child, which is eventually broken by a murder from a rival. The list of rulers is broken into groups called dynasties and there are 10 dynasties: the dynasty of Heracleus, the Assyrian Dynasty, the Armalian Dynasty, the Macedonian Dynasty, the Dynasty of Ducus and so on. Each of those dynasties has a geneological table, and these are given in full. You can see that all of them married their sisters and their brothers, until there was a sudden break. These are the resources that allow us to understand this story.
This period goes down to 829, the beginning of the rule of Theophylus. That's the first of three periods treated separately by Vogel; the next one goes down to the fourth crusade in 1204, when the crusaders captured Byzantium briefly, but the Byzantines got it back after a few years, and that's B. The Byzantine Empire ends with the Turkish conquest in 1453. Any remaining Greek books were to be banned, so the scholars were very anxious to get them out of there and carried them to Italy, and that's when the Renaissance happened. We don't have room over there, ao we'll put them at the beginning of Bagdad B2, and we'll be primarily interested for our story in part A and the first little bit of part B.
Here are the names of some mathematicians in this period. First we have Antemeus. He wrote on (comics?), and he was also the architect of the church of Hagia Sophia, St. Sophia, the most magnificent church in Christiandom at that time in this century. When he died in 534, the church was not finished. Construction was taken over by his student Isadore, and Isadore wrote and extended the results of Archimedes. These people were complete scholars. Constantine started a university that didn't get very far, and Theodopheus, another ruler, revived it in 425. Historians always call it the university. Some people say, well, it shouldn't really be called a university. I don't know many students were there, but usually, as in the case of the Alexandrian New Salon, there were the scholars studying in the library. Maybe a very small number of students who were just so obsessed with the subject that they had to hang on there, could be kept away. In other words, the school aspect of the academy was very small, like our graduate school. It became a center of Greek learning in 425.
We have to assume that they obtained every basic book of importance from Alexandria. Copies were made. Byzantium was very wealthy, as was Alexandria. People were sent with gold coins to arrange the "xeroxing" of a manuscript, which would take 4 to 6 months full- time salary for a person or two, and then bring them back. The library was growing. Some very competent students of Plato, Aristotle, the Stoics, and the Neoplatonists and so on came to the university, so it became a center for the study of the Quadrivian -- it wasn't called the Quadrivian until a little bit later. Quadrivian comprises the four subject of arithmetic, geometry, astronomy and music. The trivian, from which we get trivial pursuits, the trivian is logic, rhetoric and grammar. These subjects were probably combined with certain topics of Christology, because the university was a state university and the state was a religious state. Later on, the so-called secular school was formed that was independent of Christianity. Anyway, this university was the home for some but not all of these mathematicians.
Hupsicles wrote books 14 and 15 of Euclid, of the Elements. You can see by looking at these books that they demonstrate a fantastic technical capability in mathematics. These people had mastered everything in the elements from Book 1 to Book 13 and were able to extend the methods to write further works of the same quality.
In 612, Steven, primarily a philosopher(?). In 726, the university was closed. Actually it was opened and closed repeatedly, so maybe you shouldn't take that too seriously. What's important for us is that in 863, the new emperor Bargas opened a new university, a secular university, and he chose Leo as director of the university. He was the founding father of the (?) University and was called Leo the Mathematician. He also wrote on philosophy, but he was primarily a mathematician. 800, the year of his birth, is about the year that Bagdad started its university under the Milab Kalif Al'Mamun. There's a huge and fantastic history of Islam before Al'Mamun. But the (?), that oscillating border between the Byzantine Empire and the Arabic Empire, after the rise of Islam, is kind of a plug between the Arabian lands and Europe. You have to go through this narrow neck between the Sea of Marmara and the Black Sea, and there's this fortified city or empire with fantastically brilliant generals whose techniques are regarded as the most brilliant military inventions in history, and they are still doing it today, confounding the United Nations and NATO and so on. They held back the forces of Islam, which were also militarily very competent, and the oscillation created the diffusion of classical knowledge between Europe and Asia.
In Bagdad they had full information about the existence of universities on the other side of the line, and Al'Mamun decided they needed them. He invited Leo, who was an unemployed scholar teaching in the street, as it were. Leo went to Bagdad to have a look, and was simply (?). When Fidus heard that Leo had an offer from the competing culture, he would not be outdone and created this university, invited Leo to head it, and essentially prevented Al'Mamun from getting hold of Greek mathematics, but Al'Mamun got them anyway by buying the books later on, and you'll hear that story next week.
Leo's our main man in this list. Hupticles is also important for his contribution to our baggage train, and that's the story up to this critical time of the 9th century. Again the university fell into decline and was revived in 1045. This time, Michaelselis was put in charge, a name that's important in the Renaissance. We'll be hearing more about him later. So that's the story in brief. As you can see, there's a lot more to it. The music, the architecture, the art, the churches, the architecture of the church, the employment of art in the church, the church as the pinnacle of harmony of mathematics, science, art, music, liturgy, poetry, criticism and so on -- all of this phenomenal city[?] [culture?] was pretty much hidden by the political exploits of various kings and their neurotic families.
There's one aspect from the art history angle that I find particularly interesting. All churches have icons, and this eventually became an important controversy in the history of Byzantium. There was a ruling family and a school of philosophers who didn't think there should be any icons, and this almost split the empire. It's hard to understand why this should matter. We have pictures of things in churches or not, is this like graven images? To understand it requires having the mindset of Euclid, from the time of Euclid, which is not so easy. Well, Euclid wrote several books that were all studied here in Byzantium and in Alexandria, not only the Elements. There's also the the Data, the Porisms, the Optics, and so on. The Optics in particular describes a theory of vision similar to Rupert Sheldrake's, which states that the mind extrudes an aspect of itself from the eye. This is not necessarily to be confused with physical rays of light. The mind is extruded through the eye. It reaches out and engulfs the things you're looking at and creates the contact with that thing, especially if it's looking back. Stars that emit light have a divine aspect. They are looking out and seeing us and we're looking back, and therefore the stars can influence life because the celestial information contained within is being extruded in a stellar, celestial version of mind which is related, contained in God's mind, but it's not the whole of God's mind. This is something like the Sanskrit theory of the Okosh.
The mind is extruded through the eye. Different minds interact and mingle, and the connection with God is the illuminationist or animationist philosophy of Athens, Alexandria, Bagdad and India and so on, and they all essentially evolved from Euclid's Optics. Right up to the late Renaissance, or the time of John Dee and Robert Flood, this concept was taken very seriously. It's only recently, maybe after Newton or something, that it was debugged, and now there is a kind of renaissance of Euclid's optical theory. Well, that's background. Now here we are in Byzantium and they are fighting viciously over the question whether there should or should not be icons in the church. The icons in the church were regarded as windows. In looking at the icon, your look, the extension of your mind, could go through the icon into what it represented, thereby making contact with the divine. They had only icons with very divine things with which you wanted to make a complete and concrete spiritual contact when you went to church. That was the idea.
Iconography is sort of about the icons, but our historians got more and more fooled by the representation, or were interested in the representation as opposed to the represented, due to the fact that in their own worldview, in their paradigm, there was no such thing as divine, so therefore how could you look at it? That's how the whole meaning of icons, iconography and the spiritual connection was lost from the European tradition, but it's definitely part of the mindset, the paradigm of Byzantium from beginning to end as well as the paradigm of Alexandria from beginning to end. It all depends on the idea that there is a divine realm, and early Christianity is just one form of worship. The main Greek religion of Orphism deals with the trinity of Chaos, Gaia and Eros, or with Dionysus, Zeus and Athena. These gods and goddesses in the Pantheon of Babylon, Egypt and Greece, arose in a process of schizmogenesis from the one-and-only unique mind. There was no conflict between monotheism and polytheism, because monotheism says there's only one mind, and polytheism, that on the next level down it has a bifurcation into parts which can be identified in different ways. They don't have to be taken totally seriously, because like the stars in the sky that we see in different constellations, these would be different ways to project the images of various myths and concrete images from ordinary reality upon the myriad of stars. The gods and goddesses weren't taken that seriously, you could change their names because so many different gods would be representations of the same principle. The same group of stars appear in different myths under different names. Those people knew they were all the same, but we look in the Encyclopedia of Greek Mythology by Robert Grays and say, oh this one's Zeno, this is Euripides, and so on.
This aspect of paradigms, without which you can understand nothing in history, has its own morphogenesis or the schizmogenesis of the paradigm itself, which is called Cosmogenesis or something. This kind of picture is not on the plane of political action, economic boundaries, king's power, armies and so on, but on another plane, without the fundamental myths, the characters, the aspects, the perceived, of the divine realm. That has an evolution in time also, with occasionally gradual changes and sometimes catastrophic changes or cosmogenical crises. And this happens, I suppose, while on the grossest level we think that we have a continuum from A1 to A2, the minor stations, and B1, even though Christianity intervenes. You can decide whether you consider Christianity a crisis in history or not. We have Orphism here, we have Christianity here, and there, with the advent of B2, we have Islam. It is possible that in mathematics there could also have been a schizmogenesis, a bifurcation in the paradigm, in the very interpretation of things such as number and distance, angle and dynamical process and so on. Looking back at it from where we are now, and having available to us an outline of the paradigm of the early Christians, the early Islam, the Alexandrian period, or the Helenistic one, we can read all these (?) and (?) and so on, but we just can't really understand them.
Here's just one example of what it would mean in terms of the view of history, to have available to us the Euclidian paradigm . That paradigm was carried along with the book Optics. Next week we'll see it go from the Eastern Orthodox church into Islam, whereby Islam gains, for the first time, Euclidian Optics, an emanationist or illuminisionist philosophy and religious paradigm. Underneath the exterior of styles of dress and hats and hair, there's actually a continuity from the Greek tradition that eventually covered the whole globe, and then finally is somehow extinguished through the enlightenment or the rise of scientific materialism or whatever it is which has us in its grip now. Let's say you're a practicing Christian, maybe an Episcopalean or Anglican. You go to prayers in the morning and at night and hear the rituals in medieval or ancient latin with the tones sounding the right pitch and rhythm in the right meter, understanding all that you understand, yet without any relationship whatsoever between what you understand and what was understood in the Byzantine Church or in the early Christianity of Alexandria. How can we overcome the barrier of these filters in our own worldview? One possibility is mathematics, mathematical history. It could be useful, and it might be the easiest realm in which to make these paradigm jumps. Or maybe not. Anyway, that's what we're up against in trying to understand Byzantine history. I wish we had more time for this.