I wanted to start with the Archaic Smile, because this is really the period of Homer as the source of the Western tradition. If you want the equivalent of this smile in terms of Greek literature, it's the Homeric poems. The question is, what are they smiling about? That is extremely difficult to answer, because the smile is so enigmatic and mysterious. The smile begins to disappear from the sculptings of Greece as the culture develops and as sculpting itself develops. You move from the period of the Archaic Smile, which is on all of the archaic figures, to more and more fully formed and expressive figures. And what's instructive about following the history of Greek sculpting is that it mirrors in effect the development of the emergence of rational self-consciousness. So just as the Greek human being comes into a fully formed, individualized ego-centered rationality, so do the sculptings.
What I want to characterize for you is the transition that Greek culture went through from this Archaic period to the Classical period that is the subject of this course, or the period to today, the highpoint of Athenian culture. The best way to do this is to go from the oral culture to what's called the literature culture. This is Homer and this is Plato. Plato in effect signaled the emergence of literacy. Socrates wrote nothing, so he is in a way is the major figure immediately prior to the development of literacy, which occurs in Plato, probably the greatest writer who ever lived. I don't think anyone can match the Platonic dialogues. What happened there, I think, has universal significance. I didn't understand it, even though I was groping for formulations that would characterize it, until I started to think about Euclid as a result of my collaboration with Ralph. All of a sudden I realized that you can talk about Socrates the way you can talk about triangles, so that the Euclidian definition of a triangle is in some sense equivalent—I don't want to say identical, but I'm not sure if the word equivalent is just right—but somehow, the universal significance of Euclidian Geometry is one way of describing the universal significance of the appearance of Socrates as the eruption of rational self-consciousness.
Another way that I've tried to put this is to compare Socrates to Jesus the Christ, so that what appeared in the Jewish culture in terms of the appearance of the Christ appeared in Greek culture in terms of the appearance of Socrates. It's remarkable that Socrates was referred to as the Savior by three or four subsequent philosophical schools. What Jesus was to Israel culture and then in a sense to subsequent culture, Socrates was to Greek culture -- as the bearer of the new being that appeared in Socrates as the manifestation of rational self-consciousness. It had some kind of saving power, just as did the messianic appearance of Jesus the Christ. In the case of Socrates, it was because he overcame the anxiety of having to die, which was partly why he was regarded as the savior. So you have Jesus the Christ on the one hand as a figure of universal significance, the bearer of salvation, and you have Euclid on the other hand as the manifestation of geometry, two for everyone everywhere, for all time, I don't care what any of you think. It's just that I like to think about Socrates in that way. He is the one in whom rational self-consciousness was made manifest. It's just the case.
Socrates represents the transition to rational self-consciousness. Now I'll try to characterize for you what happened as a consequence, because it's a complete cultural shift, what Ralph would call a bifurcation. In a way it's a little chaotic, because one cultural mode supplants and supercedes another one. Today a lot of you are struggling with this supercession, because another way of characterizing this is a transition from a native culture or a native society, to a civilized one and eventually, an industrial one. In some sense this is the beginning of what many of us now deplore, namely the burden of rational self-consciousness that in some respects has completely broken down in our time, so that some of us think that science, now the bearer of rational self-consciousness, is evil. I get a slight rush, a slight thrill, to be able to say those words in a classroom at UCSC.
To understand the meaning of rational self-consciousness, you have to understand what was superceded and what was departed from. In the Homeric period you've got to try to think of what it's like to be in a culture without books, a culture without literacy, a culture where no one in the general public writes or has the ability to write. There is a written language, Linear A, Linear B, that is reserved almost exclusively for the work of scribes, but there is no literacy in the common public. Everything is transmitted by word of mouth. The entire cultural tradition is transmitted in effect in the Homeric Poems. You could call the Homeric Poems a tribal encyclopedia, which gives you the models for action, so that almost everything you do or think or say has reference to the Homeric Poems. What do you do now? Well, you do what Agamemnon did in a similar situation. What do you think this or that is? In fact, the word think doesn't really work. Nobody thought. What you have is protothought.
Some kind of rudimentary thought processes do not become what we call thinking until Socrates. That's what's so difficult to understand. Maybe the best analogy is to try to get you to imagine what it was like before you learned how to read. Now most of you probably can't remember yourself at that age, because we learn how to read so early, maybe at age 5 or 6. Let's say the period from birth to 5 is the Archaic Period of Ancient Greece. When you learned to read, you really went through a kind of trauma, because you have a kind of amnesia regarding the character of your consciousness before you learned how to read. That's how radical the moment is when you become literate. It displaces you from the kind of open fullness of your imagination and your rapport with things prior to literacy.
One of the most remarkable things about Homer is that he has this seemingly completely intact participation in and empathy with all natural forces, and the Homeric Similes are a linguistic expression of that. The army swarming down from the hillside is like a swarm of bees, which Homer characterizes in painstaking ways in order to establish the similarity in the comparisons. In a way all of the Homeric poems, especially the Iliad, are constructed around similes that unite natural forces by virtue of comparison one with the other, and everything is comparable. You get these parallel columns that explode in terms of their associate meaning, where it's all intact. Everyone is at one with these powers.
What happens when you make the transition to rational self-consciousness and literacy and writing, is that you have the subject-object split, the separation of the knower from the known. Rational self-consciousness is predicated upon the subject-object split and the separation of the knower from the known. Which means that you have to have a fairly fully formed ego in order to be the bearer of a rational self-consciousness that puts you up over and against everything that you want to think about.
What happened with the Homeric figures is they didn't know how to think. They were almost completely at the mercy of the play of images in their minds. Here's how Homeric language indicates that: the word psyche, which is our word for consciousness, comes to full expression in Socrates—he's really one of the first figures to talk about the psyche. The word psyche in Homer means breath. Ideally, life breath. And the only time the word psyche in the Homeric poems is used is when you die, that's to say when you gasp your last breath and your psyche leaves. It's only referred to in Homer as the last gasp, the life breath that goes out of the person as it departs from the body, which in Homer is pretty much just a bag of blood. One way to get rid of the life breath or to allow it to depart is to stick it with a spear, and there it goes. You could say that the whole development of rational self-consciousness from Homer to Socrates is to bring this last gasp into a fully-formed rational self-consciousness. In a way, that's why Socrates defined philsophy as having one foot in the grave.
Another example would be muse. That's the Greek word for mind. What does it mean in Homer? To smell. (Tulos) is my favorite word of all the words I know. It means vitality or vigor or courage. In Homer it's the rattling of the gland in the chest, the thymus gland. What's amazing about these Greek words in Homer is that they're quasi-bodily functions or organs. Quasi. They're just beginning to lift off. The psyche is just beginning to become a little more than the breath, but not yet. So by the time you get to Socrates you have a fully formed, conscious psyche called Socrates. And he knows it. In Homer you were a kind of battlefield for conflicting forces, there was no personalized rational center, and therefore no one who was able to think. A Homeric hero could not easily abstract from his situation in order to be able to think about it or talk about it. In a way Odyseus is uniquely the one who can do that. In Odyseus you have a figure who can seemingly think about things in a kind of rudimentary way. He talks to himself. He talks to his (tumos). And he has a kind of rudimentary way of reasoning things out. "Should I take this course or should I take that course?" he wonders. For which he's known as the man of many guises, the wily strategist.
Odyseus is rather unique in terms of doing that. Most of the Homeric heroes are at the mercy of the play of forces that they're subjected to, not least of all the gods. You could say that the gods in the Homeric poems function as the objectified powers of the mind that eventually become subjectively integrated. Whenever something significant or important happens, the Homeric hero says, "It must have been a god who prompted me to do that". He can't really fathom the springs of his own action because he doesn't have a centered self. He has gods who form the sources for action and become the references which one has recourse to describe why something has happened.
By the time you get to Socrates all of these forces in effect have been integrated into a centered self in correlation with a structured world. This happens thanks to the emergence of literacy and writing. I'd like you to have at least a kind of correlation between the emergence of rational self-consciousness and the emergence of literacy. They go hand in hand. It is followed by the culture of the book, of which we are prsently witnessing the end.
Now the best book—actually there are two books worth pursuing, because it's not only interesting in terms of the culture, it's interesting for your own experience. You all go through this, because the culture went through it. The childhood of Western culture, in the move from oral to literate, is your own childhood. Your own effort to become rationally self-conscious, for better or worse, and literate, makes you experience the same transition that the culture went through. So in a way they're mirror each other. It's an example of ontogeny recapitulating phylogeny. What the individual now undergoes, the race once underwent. It's marvelous to think about how this has affected you in the same way it's affected the race.
Consider the example of the move from native culture all the way up to industrial society in terms of the extent to which rational self-consciousness has worked its way through Western history and now is a worldwide, almost tyrannical, imperial imposition. One mourns the passing of native cultures because of the success, the triumph of Western rational self-consciousness which eventuated an industrial society. That rings a bell, doesn't it? We see these struggles played out now in terms of all kinds of issues that you face every day.
So this is the story of that transition, and in a way Euclid is the signal expression of it in terms of why this has become such a universal enterprise. Because if geometry, according to Euclid, is true for everyone everywhere, well then it's possible to impose it as a universal enterprise. That's why the geometrician becomes a person of infinite tasks and opens up an infinite horizon. But now it's in effect breaking down, it's had its great day, and that's why there is this kind of jaundiced attitude toward the success and triumph and ultimate self-destruction of the history of rational self-consciousness culminating in industrial society. We departed from native consciousness, oral consciousness, figurative consciousness, and made a wholesale effort to in effect wipe it out. The effort to promote literacy -- I mean why would an American Indian want to learn how to read? Why would he want to forsake his oral consciousness in favor of the Western trick? Because he wants to become an engineer? There are all kinds of themes that show up in terms of this transition. It's good to get a sense of how you characterize oral culture in order to catch the evolution from oral culture to literate culture, and this, at least, is one convenient way to do it.
One of the great breakthrough books I was going to mention is by a German philologist named Bruno Schnell. It's called The Discovery of the Mind. The first chapter is on Homeric anthropology and the Homeric figures. These figures are at the mercy of a play of forces. They don't really know how to think in our sense because they have no concepts per se. There are kinds of proto concepts—like the word psyche. It still means life breath, but eventually becomes the word for consciousness, the concept of consciousness. Another way to put this is to go from image to concept, from the comparison of images and similes to logic--from a mass of flighty images, almost hallucinations, as the field of one's consciousness, to cognition.
The other great book is by Eric Kavlar, called Preface to Plato. This is probably the best single volume on this issue that I know of. Poetry is Preserved Information, The Homeric Encyclopedia, Epic as Record vs. Epic as Narrative, The Oral Sources of the Helenic Intelligence, The Homeric State of Mind, The Psychology of the Poetic Performance—that's one of the best chapters in the book. It shows that because you don't really know how to think as yet, you're not a fully formed, centered self. When the minstrel sings, everybody goes into a trance; it's as if you hallucinate the scenes of the epic--they become the vivid play of forces in your trance state.
You might say that it's Socrates' central effort to wake you up from that trance state in order for you to think for yourself, maybe for the first time. In all the early dialogues, nobody knows what that means. They reach an impasse, and Socrates says, "What is truth?" So they say something that they know from Homer, like, it's when so-and-so did so-and-so. Socrates says, "No, I don't want an example, I don't want an instance of it. I want to know what it is in itself per se." What Socrates wants is a definition. He wants to know what truth is as a concept. Define it. Conceptualize it. Don't give me an image of it. Everybody is annoyed because they don't know what he's talking about. They don't know how to think. They can't bring themselves out of imagery into conceptual rationality. But it's the beginning. Once you bump against it in discussion with Socrates, it's as if something has become. That's why he calls himself a midwife. Something can now be born out of you, namely, rational self-consciousness.
There's also a chapter in that book on the separation of the knower from the known, and what it means for Socrates to talk about the psyche as the concept for consciousness, and what it means to become rational in terms of thinking about things rather than being at the mercy of them through a play of images.
Okay, now why should we bother with Socrates? Well, because if you take Plato, who is something like the Gospel writers, you have the gospel according to Plato. Something appears in Socrates that comes to fulfillment, much like Jesus in reference to the Israeli prophets. Just as the way was prepared for Jesus in terms of the prophetic movement, so that he comes as a culmination of what was promised, and is the fulfillment of what was promised, so does Socrates. Every figure prior to Socrates in the succession of Greek thinkers is known as pre-Socratic. Because he caps it. And then Plato, like the Gospel writers, writes in the dialogue what he has received in his transmission. Some revelatory charge went from Socrates to Plato just as it went from Jesus to the Gospel writers. The dialogues are texts that bear witness to this event.
If Socrates is the embodiment and the manifestation of rational self-consciousness full of grace and truth, and if the Platonic Dialogues are the Gospels According to Plato of this event, bearing witness to it, then this is an event of universal significance, and the only comparison that confirms for me what I mean by that is Euclid. Therefore Socrates is a triangle which you can actually express in the form of a classic syllogism that in a way symbolizes this moment of rational self-consciousness. All men are mortal, Socrates is a man, therefore Socrates is mortal. That's the triangle. And that's what it means to know how to think in terms of a syllogism, which is the structure of thought in its logical coherence.
So we got Jesus, we got Socrates, we got Euclid, as a stone soup. These are turning points in history, Colli Aspers called them the axial age. I like Ralph's notion of bifurcation, because things aren't the same, everything is in effect transformed accordingly, and everybody has to somehow cope with it in terms of how they respond to it. One of the great features of the Platonic dialogue is when you come into the presence of Socrates—hey hey hey—this is an existential moment the likes of which nobody has ever known. Socrates, in a way, is the bearer of the mysteries. Plato likes to play off the notion of the Socratic masks, claiming that Socrates' ugliness masks his inner beauty. If you can see the inner beauty of Socrates, you can see into the mysteries of the true and the good and the beautiful.
Everyone thinks that Plato developed the notion known as idealism. He said that there's a heavenly world of essences somewhere that smells like perfume, and you could only find them in the sky by and by if you take some kind of a journey. This heavenly world of ideas is very remote from the life and the cares we have to cope with every day. What is often missed in terms of reading the Platonic dialogue is that Socrates is the embodiment of these principles. He's the existential manifestation of them. You can't really get it unless you pay attention to the Platonic myth. Our favorite is the one known as The Myth of Er at the end of The Republic.
The Republic is the dialogue that talks about the transition from the oral to the literate, and the conflict between the Socratic culture, or the Asymmian culture, and the Homeric culture. This creates a big problem in terms of how to educate the youth. We can't be singing the Homeric songs to them anymore, you know. They sound like they're totally stoned, you might as well just give them some kind of psychedelic drug as give them Homer, and we all know what psychedelic drugs do to a person's rational self-consciousness. That's why the argument in The Republic is, "Let's get rid of the poets, let's banish them from the Republic in order to finally get people to think and to begin to think for themselves! So that's the issue of the Republic.
A new stage of culture now occurred, thanks to Socrates and to the emergence of rational self-consciousness. It means the old culture has to be replaced. This whole phantamagoria of gods and scenes and the play of forces that one is at the mercy of has to go. Let's just take control of ourselves and our lives and begin to develop our minds so that we can think about things and not be at their mercy. Let's bring our will into coordination or into a coordinate with our mental powers as they can now be developed. That's the theme of The Republic. We've got to devise a new curriculum in order to produce new human beings. A new stage of human culture has been achieved, which The Republic celebrates. That's why a revolution has to occur in the educational curriculum and the structure of how one teaches and how one develops consciousness formation.
At the end of The Republic there are three great moments. One is very subtle, one is very obvious and one is really complicated. The Republic begins by using the word (cotabin?), "I went down..." I don't know if it's Socrates who starts talking, but somebody says, "I went down to the pareis," which is the harbor of Athens, "and there I watched a festival for the Asian goddess Bendis whom some sailors had brought into Athens and now they celebrate it every year." It was quite a spectacle. People rode around on horses and carried torches, it went on at night and it was just incredible. Bendis is one of the goddesses of the underworld, and the person who speaks went down to the pareis to watch this festival to the goddess of the underworld.
That's how The Republic opens. In the middle of the book there is the descent into the caves that contains probably one of the greatest parables that anyone has ever written, The Parable of the Caves. That, in a way, is Plato's or Socrates' attempt to describe the transition from the Homeric to the Socratic culture. Homeric figures are sitting in a cave. They are chained, and they have to look at a wall on which shadows of images are being cast. They are at three removes from reality, but they think that that's what they see. This is the kind of phantasmagoria in the Homeric poems that's like some kind of hallucination removed from reality. At last somebody finally breaks free from their chains, Socrates goes out into the light of day, sees reality for what it is. He goes back down to tell the people, but they're annoyed and they kill the guy. Which is the fate of Socrates. This forms the center of the dialogue.
The Myth of Er is at the end. This is a myth of the transmigration of souls, it's a myth of the afterlife. It's one of the great expressions of what you could call symbols of eternal life in terms of the myth of reincarnation, and it's extremely subtle and fantastic. Once I got it, I said ah-ha. This is Plato throwing a net. This myth is like a net that you want to throw back over the whole dialogue in order to express the dynamics of what it means to encounter Socrates. The only way Plato could express this was to project it into a myth of the the afterlife in order to open up the meaning of the moment in which one encounters Socrates. Something of eternal significance occurs and you better get it, or else. Because those are the stakes. It's played off against the myth of descending into the underworld by Odyseus in The Odyssey. When Odyseus tells the story of how he descended into the underworld at the (cord) of the (pheakians) to King Alsinowis(?) he tells just so much of his descent, and then he breaks it off. There's a little interlude, and then he continues telling the story.
The Myth of Er, in terms of that structure, is identical. There is a break-off, Socrates goes "Hey now, hey now, maybe I've now done the same as what Homer has done to you, you've gone on this mythical journey of Er, gone into the afterlife, and there you saw what happens to souls after they die, now come on, come on, wake up. I mean let's think about it." It sounds almost like a little sermon. "Now shouldn't we really strain every nerve to understand the significance of these symbols and think about them in terms of their meaning for our life?" Then he goes back into telling the myth again. Somebody hass actually counted the lines, so that the lines to the break are identical in Homer and The Myth of Er. It's deliberately patterned after the descent of Odyseus into the underworld.
At the end of the myth, when Er goes to the center of the universe to see what happens—his people are about to be born again into their new life—an unknown prophet comes out. Now at the center of the universe, the goddess of necessity, Ananke, fate herself, is sitting on the throne, and before her are her three daughters, past, present and future—Lakysus, Atropos and Closo. They turn the spindles of the universe that are centered on the lap of their mother, the great goddess Ananke, fate herself. And unto this altar at the center of the universe where the spindles of the universe turn on the lap of Necessity, steps forth an unnamed prophet who addresses the assembled souls as they are about to select their new lives, as they are about to be reborn, and he says to them, "Creatures of (Adais), ephemoroy. Now is another cycle of mortal generation where birth is the beacon of death. No one will choose for you. Each one will choose their own destiny. God is blameless." Then the prophet throws out numbered lots, and according to your number, you get in line to select from the patterns of lives what you will become upon your rebirth. It would take a couple hours for us to explore the symbolism of this myth, part of which is mathematical, part of which is Pythagorean. There's a number symbolism imbedded in the myth. But suffice it to say that the myth is an effort to illuminate what it means to encounter Socrates, who is the unnamed prophet before whom you decide who you are.
That's an example of Plato's effort to express and illuminate what appeared in this figure which you can only call an event, and the event is Socrates. This event of rational self-consciousness then becomes of universal import, like Euclidian Geometry. God knows I tried to get into the subject matter of Ralph's course, and God knows I spent hours...I started last summer, not particularly in anticipation of this particular hour but of Ralph's project regarding Euclid. He wants to do a CD-ROM as a consequence of having taught this class, and I'm happy to help him with it—so I got this book. It is Husserl's The Origins of Geometry, with an extremely long introduction by Jacques Derrida, who is just the hottest thing going, folks. You must have heard of him. There are guys up here who would just drool over my holding up his book. I mean he is so hot, it's just—I went to the Black Oak Bookstore in Berkeley. I've done this now for maybe five years, and there're like 10 to 20 new books on Derrida everytime I go there.
This is the first thing Derrida wrote. He's famous for what's called deconstruction. In a way this is an effort to take apart the whole show, from Socrates to now, and seeing how it's all kind of coming unhinged. It's in a state of disarray. Deconstruction ain't all that hard. In effect it's deconstructing itself, and Derrida is the witness at the funeral to give the eulogy. The first thing he wrote dealt with Husserl's effort to talk about the Origins of Geometry in just the way I've done, namely by examining what happened historically to make this subject matter come through a guy named Euclid who opened up a transcendental realm that entailed infinite tasks forever after for everyone who wanted to be a geometer. One question I wanted to ask is, is there progress in geometry? Are there problems still being formulated and solved in geometry as an ongoing historical task? You'd think that after 2500 years they'd have the field pretty well mapped. Is that the case?
Ralph: After 25 centuries.
After 25 centuries. It's the same thing, isn't it? Anyhow, this is what I tried to crack: In The Origins of Geometry there is the most intensive discussion about what it means when a historical moment becomes an infinite horizon of infinite tasks, namely the subject matter of geometry, which is an example of the idealization of any kind of knowledge. True for everyone everywhere at every time, no matter what, and therefore it becomes an example of how to inquire into the ideal, into ideal formations, and into knowledge traditions, par excellence. It's probably the best example of the formation of an ideal tradition that then opens an infinite horizon for anyone to enter at any point. It's an amazing discussion. What interested me most of all is Husserl's notion that geometry is ethical. Now c'mon, where did he get that from? Geometry is ethical? What has prompted Ralph and me to go back to these early sources of the tradition is that we're interested in the fact that they haven't broken off into independent, even completely separated disciplines.
No one in the philosophy department here had any interest whatsoever in mathematics. If we had a really good logician, maybe the logician would be interested in mathematics to some extent through symbolic logic. But we didn't have a good logician, and so there was no interaction whatsoever beween the philsophy department and the mathematics department. No philosopher, in my experience of teaching for 20 years, ever once mentioned Euclid or any theme that had any bearing on mathematics. And yet there's a strong mathematical component in Plato coming from Pythagorus, and there's a strong mathematical component in Aristotle. What interests Ralph and me is that this kind of unity of disciplines in the ancient world even includes theology. There is a sacred geometry which gives you a sense of how geometry is part of the whole cosmic harmony, and you get a sense of the meaning of geometry for spirituality. The whole neoplatonic tradition was an effort to in effect score the geometry of the soul. You can't find anybody but Ralph, as far as I know, in a mathematics department in the United States, who has any interest in that, because the whole thing has become so flat. So it was a little bit of a surprise to find Husserl talking about geometry as ethical, simply on the basis that you should take responsibility for what you think, even if you're a geometer. It's part of the universal meaning of what you're doing that you take responsibility for. Go tell that to the Bulgarians! You can even go from the ethical implication in geometry to the theological one.
I once read a short paper by somebody, Frega maybe, that said there is no such thing as non-Euclidian geometry. It's a contradiction. I mostly go along with that. I mean I have a hard enough time with Euclidian geometry, let alone a non-Euclidian geometry. There's this wonderful legend that in a way characterizes the scene. This is called the Delian [Delphian?]Prophecy or the Delian Oracle. The speaker is Simeus, one of the Theban conspirators. He is reporting on a recent visit to Egypt that he made in the company of Plato. "As we traveled from Egypt, certain Delians came upon us near Karia and begged Plato, as a Geometer, to solve for them an unusual oracle proposed by the god. The oracle was that there would be a respite from present ills, like the plague, for the Delians and the other Greeks when they had doubled the altar in Delos. But as they were having a ludicrously bad time with the construction of the altar on account of their unfamiliarity with the proportion which provides the double in length, they failed to notice that when each of the four sides was doubled, they were effecting an eight-fold increase in the solid space. They summoned Plato for assistance in the puzzle, and he, recalling the Egyptians, said that the god was making sport with the Greeks for their neglected education, taunting them for their ignorance, and demanding that they engage in Geometry, and not just as a past-time." Get it? They didn't know how to do it and therefore they weren't going to overcome the plague. "For surely it is the test not for an intellect which is inferior and perceives dully, but rather for one trained to the limit in the constructions of lines, to take two means in proportion by which alone the figure of a cubic body is doubled by being increased in the same way by every dimension. This, he said, Eudocsys of Nidus of Helicon of Zyzecus would work out for them, but they shouldn't think that the god wished this, but rather than he enjoined all the Greeks to abandon war and its ills and to consort with the muses and by soothing the passions by a reasoning in mathematics, to live together profitably and without harm."
That's as Greek as you can get in terms of this Delian oracle and the significance of geometry for doubling an altar that's a cube. Now one can go from there to the tradition that wants to make geometry practical, let's say, in terms of warfare, and the whole Galileo thing about how you dump a missile on somebody, and the Platonic tradition resists that. They wanted geometry to be an ideal pursuit. You did geometry in terms of its significance as a mental undertaking. It's transcendental and doesn't have to have any practical application. You work in the realm of the mind with geometry and don't worry about any kind of inventions or applications from it. That becomes a major conflict in terms of mathematical tradition. Some people are interested in the mechanics of geometry and in applying it mechanically, which is opposed by the Platonic tradition that wants to keep it as a mental subject matter in order to maintain it as ideal.
I've got a prize essay contest I'd like to propose to this class. We can figure out what the prize is, but the theme is, "Why did Pythagoras have a golden thigh?" If anyone wants to write a short essay answering that question, Ralph and I will determine who writes the best one and—let's say it'll at least be a hundred bucks. We have a hundred-dollar prize essay contest on the question, "Why did Pythagoras have a golden thigh?" You can think of this, at least ostensibly, as a study in the history of mathematics and philosophy. I'll just give you one clue in terms of the golden thigh: was it a tattoo of an isosceles triangle? Obviously in gold. Was it a golden tattoo of an isosceles triangle? So, anybody have any questions?