That's our agenda. First a short lecture on Athens. Then another tour of the locker. And then there's a documentary film about Alexander the Great. That's our agenda today.
Now the overall picture, our master plan, I made a map last time. They're the ancient sources, the predecessors of the Greeks, which you have now, everybody has read about in chapter 1, not covering that in the lectures. Then Athens corresponds to chapters 2 and 3. My official agenda is Euclid's voyage. So all of which precedes Euclid of course, but the idea here is to trace the origin of the material, the content of Euclid's book, Storcia.
Next, Alexanderia corresponds to chaters 4 and 5, then Byzantium, we have no chapter on that I don't think. And Bagdad, chapter 7, Cordova and Polermo, two stops, two alternate routes after Baghdad, chapter 8, and then Renaissance Europe, chapters 9 and 10. So this is just a rough correspondence between the many subjects of the lectures, what I call the main stations, the main stopping places on the route of Euclid's voyage, and the readings in the text, rough correspondents. So we'll be going at different rates and there won't be an exact correspondence between the lectures and the text.
Then, I had made a figure previously that showed kind of a bar charts of Athens, Alexandria and so on, and we can put here the ancient world, especially Sumer, Babylon and Egypt. And we have a kind of a space-time picture then, a historiogram if you like, showing the diffusion of the mathematical concepts into Athens and then from Athens to Alexandria, then at a certain point we get our book and following that to Byzantium, to Bagdad, and so on. So this is the overall tour guide, and our agenda today is just to expand part of the history of Athens, particularly the part from the beginning of mathematical history in Athens up until the migration of the mathematical text from Athens to Alexandria. It's a fairly short period.
Oh yeah, I have a main idea in mind, which I mentioned last time, a bifrication, and this is more of an implosion explosion, so this part of the picture right here, the input and output at this node, we can also draw this way, is the critical part of Greek history as far as mathematics is concerned, the mathemtical golden age, and we have a lot of different sources: Babylonian, Egyptian, Chinese, Indian and so on. And then out, there are various different subjects. For a brief period there's a kind of a unification of everything and most especially religion, philosophy, and mathematics. These subjects all are learned in the Platonic Academy, and this (?)fication has probably a lot to do with the earlier history of mathematics where we're not going into detail, but following this period, in fact after Aristotle, we have separately religion, philosophy, mathematics, and all the sciences one by one. So this implosion or explosion -- sometimes it's call the Greek miracle -- I'll call it this bifrication - - that explicitly is my subject today.
So my plan is to draw this in such a way that we can try to see something like this, a space-time pattern blowing up successfully different zooms of this Greek period. So if you read in the text and then there are all these events, dates I mentioned, like Plato for example, you wonder what exactly is the birthday of Plato, then you can look in the index of the book and you have the dates, then it says when Plato started the Academy and then Plato died and so on. Extracting the basic data and making a list, you have what's called a chronology. That's just a list of dates, basically. And this kind of list is a very important and useful abstract of the information in the text. But better is what I call a chronograph where we try to arrange those events not only linear order but also linear geometry, so if there's a lot of events in a short time you can see this acceleration just by a glance at the chronograph, and if there are long gaps in the timespan where there are no events, then you see there's a decceleration, essentially, in the events recorded in the history. So all of these historiograms are examples, I guess, but I'm particularly going to recommend here this chronograph method as an aid when you're reading and taking notes and so on, because even in a first reading, even skimming a chapter in the text like our chapter for last week, 35 pages, it's a fair amount of reading considering the amount of detail there. What I want you to do is essentially skim and extract the data that's important for your own personal view of the story that's told. There's much too much information in these chapters in the text to try to learn or even read in detail. So this would be one method. You skim the text and you create your own chronograph.
So my first one here -- a chronograph of Athens -- goes on for essentially the whole time. Of course Athens still exists, but ancient Athens begins around 1700 B.C. and ends in 5 or 6 or 700 A.D. So this is a large enough timespan to record all the names and dates mentioned in chapter 2 you're reading for this week on Greek mathematics. And in here I'm imagining a linear scale of time, and then using this as a ruler to right alongside the main dates. I'll just give a few of these will be enough for now. So first there's the time of Homer and Hessius, the great epic poets who began the tradition of Greek literature, had nothing to do with mathematics, and this is so far back that their dates are not very exactly known, so I won't write any dates. Next, Theles however, the mathematical pioneer mentioned briefly in the text and his dates, 624 to 547, he is said to have traveled to Egypt and brought back mathematical information, so his name actually belongs on one of these incoming lines. And then Pythagorus, likewise mentioned in the text, born in 527 and died in 497, and he is said to have traveled to not only Egypt, but also Babylonia. So these are the main lines, at least that we know about, responsible for the initial input to the miracle. And they were also great innovators. Theles is credited with the whole idea of Greek science, investigation, logical proof, and theorem definitions and proofs and so on, and Pythagorus was -- Everything that's written about Theles and Pythagorus is myth and it's not even sure that people of this name lived, so they're either, they're gods or their mythical figures or they're actual people. And so these dates are not very firm. But people wrote biographies of Theles and Pythagorus many years later, centuries later, way down almost at the end of this period we have historical records of these people, but after such a long span of time after their lifetime that they can't be regarded as definitive. Anyway, Theles was a sort of a lonely teacher type but Pythagorus started and entire religious movement very similar to hippie communes in the 1960s where he had his students living communally in a house. They were vegetarians, they dressed all in white, and they regarded mathematics as their primary religious discipline.
I might write Socrates, though I'm running out of space here, to put these together in strict chronograph, but we could do like this [writing]...Socrates...Now we're getting to an important part of our story. As told last time by Paul Lee, Socrates is not mentioned in our text and is not regarded as the chief figure or any figure as a matter of fact in the history of mathematics. Nevertheless he was one of two main inspirations for Plato who is the main figure in this Greek miracle, especially regarding the unification of religion, philosophy, and mathematics.
So who's next? So Pythagorus is very early in our story, and yet Socrates, who's a main figure, was born just shortly after Pythagorus died. So Socrates could be essentially the grandson of Pythagorus, and Plato the son of Socrates, and so on, so we're getting now into a denser scale of generations as it were. And the next generation after Plato, Aristotle. And then, of course, leading up to Euclid, which is one of these outlines. Here is Euclid carrying all of Greek mathematics from Athens to Alexandria, and here he is on our list. I'll stop here. Eventually all of this data gets to the class locker, it takes me about a week after each lecture. Two lines I'd like to add here are for the two schools, the Academy -- Academy means playing field and Plato took this name or this name was given to his school because of its location and this began in 385, and this bifrication actually occurs between Plato and Aristotle and as we take some small piece of this chronograph and blow it up to -- zoom into a larger view, we'll see more details on this generation between Plato and Aristotle in order to dissect, as it were, the cultural- historical background of this great bifrication. So this other school started in 335, so 50 years later, this one by Plato, and this one by Aristotle, after he split, called the Leucium. And this also named after the location of the school which was in a temple of Apollo Leuceus(?). The Lyceum in English. So all of these words and names and so on I'll pronounce various different ways depending on how I'm thinking or doing it right or making an error and so on. I don't insist on any authentic pronounciation of these names, because it's just too difficult. The names started in Attick Greek, then they're translated into Byzantine Greek which is essentially like modern Greek, and then into Latin when you get these letters attached, and transformed a little bit to facilitate Latin pronounciation, and then from Latin maybe to Arabic and so on. So with each change, the pronounciation, the spelling and everything is changed beyond recognition, and now we have English. We have the dictionary. If you look in your encyclopedia how to pronounce this name it will say Pythagorus, but in Ancient Greek that would be `Pitagorus.'
[in response to question] That must be a mistake. This is the dates of his writing, though I don't actually know the dates of his birth and death.
These ended in 529 when Justinian closed all pagan schools and these among them. But the high time of the Academy was just the part during Plato's lifetime. Plato died in 347, so that's just shortly after it started. That's the high period. And Aristotle died in 322, so just in each case about a dozen years we have the high time. And of course the Platonic Academy was the original university. It had students and a curriculum. And the curriculum is described in detail in Plato's Republic. And there used to be four branches of so-called mathematics were taught, and one of them is astronomy and one is music and the other two are algebra, what we would call algebra they called arithmetic and geometry.
Before putting this away, let me put 300 B.C. here. Around -- this is the approximately the time when Euclid carried the book, as it were, from the Platonic Academy to Alexandria.
Next I want to blow up this period into another chronograph. So here we have Plato born and Socrates was something like 40 years old when Plato was born, so a little while later he died. So sometime in the latter part of this span must be the time when Socrates was the teacher of Plato. So just shortly after Socrates' death Plato was traveling in Sicily, and there he met Arcutus. Arcutus is one of the more important mathematicians in the span between Pythagorus and Plato, and somehow the combination, the chemistry between the death of Socrates and the meeting with Arcutus led to Plato's decision to create the Academy. Because only two years follow, and that's a reasonable -- and of course that's approximately the time of Aristotle's birth, so as far as mathematics is concerned, the golden years in the Academy precede any involvement by Aristotle. He is really another generation. But eventually Aristotle grows up, goes to the Academy, is a main student, is highly praised by Plato, and then Plato dies. And on his death his nephew Spectapus succeeds as head of the school, though maybe Aristotle had ambitions for that post. Anyway, he was disappointed, and he left Athens. So the mathematical golden years begin in 375 and end with Plato's death.
That period I'll blow that up further in a moment. Just now we'll finish this chronograph number two. What happened shortly after this event, the end of the golden years, is Aristotle goes, he's moving around and eventually he goes to the court of Phillip the Second of Macedonia, and becomes the tutor of Alexander the Great, then 13 years old, and after the murder of Phillip in which possibly Alexander is implicated -- sounds like modern times -- Alexander becomes king and at that point Aristotle's job as tutor ends and he returns to Athens and creates his own school, Leuceum or Lyceum. And then this too ends here -- I don't know how these are related. Alexander dies in 323 and Aristotle dies in 322. So when Aristotle dies then his school is taken over by the (Theoprostus?) and that event is also part of our bifrication picture, because this bifrication is really complete when Aristotle dies. Alexander died essentially of grief after his lover died of the flu, his lover the main general and second in command of Alexander's army. We'll come to that later, in the video.
Okay then, just blowing up these golden years, one important step is Plato writes The Republic, which is interesting for a lot of reasons, just one is the description of the curriculum in the Academy. Another, at the end we have the Myth of Er that Paul talked about last time. And there's also the idea how to properly run a government. Later on in his life Plato got a chance to try this out. He was invited by the king of some island and went there to put his ideas in practice on the ideal society, and within two years they'd put him in prison. Anyway, shortly after that, Theatetus arrives at the Academy around 370 with Leotomus and he wrote about the Platonic (science?) and now we're coming to the mathematical development in the Academy in the golden years, which became Euclid's Elements. Shortly afterwards Leon, he wrote Elements. Elements was -- Strykia is, just means that the mathematical inventions, the creations of the various students and scholars working in the Academy were put together in logical order according to the fantasy of what mathematics ought to be developed by Theles. But elements of Theatetus, Leotomus, Leon, and so on, did not survive, because after -- down here, when Euclid wrote his Elements, they were so complete and so well done according to the view of the time, the Aristotelean view, that they superceded the earlier texts, and they were no longer copied, and therefore they vanished. Probably they were still in existence in 500 A.D., but when the Arabs burned the Alexandria library, then they were lost. So after Medieval times, Euclid's Elements become the oldest extant text. But it was simply a synthesis of these earlier Elements.
350 we have Eudocus and Tridius -- he also wrote Elements. These are mentioned in the text as you'll be reading this week. And then 347 Plato's death. Aristotle leaves, Spectapus takes over, and so on. So this is the background and here is the development of the content of Euclid's Elements, which will be our main story then as this we can regard as a packaging. Greek knowledge was packaged, fit to travel in a substantial box. Euclid's Elements, off it went, and brought the Greek miracle to one culture after another, until it arrived here at the end of our story. Or it arrived in London, translated into English in 1587 by John Dee.
Well this is the map? What is the content? I'm leaving it to you to figure that out by reading. I'll recommend two different sources. One is our text that as you'll see is exceptionally mathematical. There is a brief mention of parallel developments in philosophy and science and that's it, so if you want to read further there are many interesting works in the Sci Lab and some of them I have put on reserve. The main idea of Pythagorus is maintained by Plato, so from Pythagorus to Plato we have more or less a continuous tradition. Although the religion of Pythagorus faded, and then it was revived so that Plato was called a neo-Pythagorean. So one thing you might like to know is what was the religion of Pythagorus, and there are many books, especially recently there are very good new studies of Pythagorus and his religion and the beliefs of his people, who are called Pythagoreans. They were eventually stamped out by a backlash from the conservative community that lived around them. The main idea is one in which -- as described in detail by Plato -- that mathematical objects enjoy a category of existence between the divine and ordinary realities. So that somewhere halfway between the realm of the gods and the realm of people, there is this very real but not substantial universe of mathematical objects. That's the main idea of Pythagorus. And they thought of them not only geometrically as halfway between heaven and earth, but also in status, that mathematical objects were essentially divine. They were one step below the gods. So the number 1 was something like a god and was worshipped and understood in terms of spiritual significance about the unity of nature. And then, after Plato, this disintegration between Plato and Aristotle, which we see here. Plato is still a Pythagorean, Aristotle a complete athiest as it were, I mean without -- he was a religious person, but the religion was totally separated from science, so nature was to be understood only from the bottom up, only through the actual perceptions through the senses. You try to making sense of nature through your own experience, that there would be no divine pattern in nature. That was the idea of Aristotle, which is then manifest in the later philosophies, the main ones are the Epicureans after Epicurus and the Stoics. So immediately following Aristotle, or a contemporary of Aristotle, we have these philosophical schools that are somehow the antithesis of the Pythagoreans. So to understand the content here, the real meaning of this bifrication you have to study, read Plato, and also read the Epicureans and the Stoics, and try to figure out the difference between their views. And the Epicureans and the Stoics are essentially modern, I mean they make sense to people today, and on the other hand Pythagorus and Plato say things that it's very hard for us to understand.