[MUSIC] Timaeus, the world soul. Let's consider Timaeus' story of how the cosmos or the universe comes into being. Now whether this is a story we are supposed to take literally or metaphorically, is much disputed. But the basic conception of the cosmos that Timaeus is working with is pretty much the one that the pre-Socratic naturalists are working with. On this basic picture, which different cosmologists explain differently, the universe is a geocentric system in which all the heavenly bodies, the sun, the moon, the stars, and planets revolve around us here on earth. So, the cosmos is roughly spherical and it has predictable and regular motions. These motions, Timaeus proposes must be due to a soul. Since the soul is the principle of all motion. Now, this is a view we saw a glimpse of in. And we'll see in greater detail in Aristotle. But for the present, the universe can be distinguished into a body, the world's body, and a soul, the world's soul. So, it's actually a kind of animal. Now anima, which is the root for animal, actually comes from the Latin translation of the Greek word for soul. Now, when we are talking about a soul that animates the universe, we are no longer dealing with the tripartite soul of the Republic. Which will turn up again later in the time as. When the creation of human beings is described. The world soul as Plato conceives it is a purely rational soul. In fact, the rational part of our tripartite soul is built along the same principles. Now one of the main points to take away from Plato's picture here is that these principles of rationality are mathematical. In contrast to the world's body which is made of material principles, earth, air, fire, and water. The world's soul is constructed out of an invisible mixture of very rarefied entities, being, same and different. These are identified as the highest kinds in a different dialogue called the Sophist. Now the demiurge mixes these entities together into a coherent mass, no easy feat. And then proceeds to structure the mixture according to mathematical principles. He does so by cutting off a sequence of lengths from the mixture, which he arranges in ascending order of size. You might picture these like the sequence of strings on a harp, or keys on a xylophone, or pipes on an organ. The lengths of the first sections he cuts off constitute a series of double and triple intervals, starting with one. The double intervals are 2,4 and 8. And the triple intervals are 3, 9, and 27. Having cut off and set out in order these series of intervals, the demiurge's next step is to identify within each interval two middle terms or means. Then cut off a strip of soul length that length and then insert it into the progression of strips. Let's look at how this works for the double intervals. Now, one kind of middle term of the two he invokes is the arithmetical mean. The number that is equidistant between the extremes. So, the arithmetical mean between one and two is three over two, and between two and four it's three, and between four and eight it's six and so on. That's the easy one. The other kind of middle term or mean is more complicated to describe and sometimes it gets called the harmonic mean. The harmonic mean between a and b is such that, that for proportion by which it exceeds a, is the same proportion by which b exceeds it. Thus, for the interval between one and two, the harmonic mean is four over three, which is one third larger than one, and one third smaller than two. That is, one third of two is two over three. And if you subtract this from two, you get four over three, the harmonic mean. By the same formula, the harmonic mean between two and four is eight over three and between four and eight is sixteen over three and so on. Next step, Timaeus knows that a unique third interval is defined by the interval between the harmonic mean and the arithmetic mean. If we look at the interval between one and two, where the harmonic mean is four over three, and the arithmetic mean is three over two, we can calculate that the interval between these two is nine over eight. Do the math yourself if you want to check. And lo and behold, that is the same interval between the harmonic and arithmetic means in all the other intervals. And not only that. If we proceed to, as he puts it, fill up the harmonic intervals, that is between for example one and four over three. If we fill them up with successive instances of this nine over eight interval, another regular pattern emerges. Starting this process from our first length, one. We generate the quantities nine over eight then 81 over 64. Remember to multiply by it by nine over eight, not add. And then we get a remainder interval between 81 over 64 and the harmonic mean, four over three. Now, this is less than nine over eight. In fact it's 256 over 243. This is the same remainder we get if we fill in the harmonic intervals between two and four and between four and eight and so on. Now if we continue on filling the remaining spaces at nine over eight intervals the same pattern emerges. There is room for two intervals of nine over eight and then one remainder interval of 256 over 243. We end up with a perfectly regular repeating pattern which musicians and music theorists will recognize as the structure of a Diatonic scale. Where nine over eight is roughly that of the interval of the whole tone, and 256 over 243 is roughly that of the semitone. Well, the harmonic interval, four over three, is that of a musical fourth, and the arithmetic interval, three over two is that of a musical fifth. Now, you might say, well if that's the pattern Plato had in mind, why didn't he just say so? Rather than making us do all this arithmetic to figure out the pattern. But for Plato the mathematics that generates the pattern is what's most important. You can start out with a few primitive operations. A sequence of regular intervals, then two kinds of mean, the arithmetic and the harmonic, and from these you can generate a perfectly regular and repeating pattern. The beauty and regularity of this pattern is Plato's basis for saying that it is good and rational. This is the order that the demiurge's introduces into the sensible world. When he imposes the order of intelligible form on the disorderly stuff in the receptacle. Now to explain how this mathematical structure animates the spherical body of the universe, with all the orbiting planets and stars, Timaeus tells the following story. Now, here's one of the places where it's hard to take the story quite literally. Take this linear progression of intervals, think of it as a huge immaterial keyboard, and cut it lengthwise into two strips each with its ends joined to make a circular band, one band slightly inside the other and crossing it at an angle. We can think of each band as the drive belt of a celestial sphere that defines the orbit of a constellation or a planet. The outer band, which he calls the circle of the same, drives the sphere that contains the so called fixed stars. It rotates east to west. That's the direction in which the constellations appear to move in the night sky to an observer in the northern hemisphere. The inner band, which he calls the circle of the different, rotates in the opposite direction. West to east, the direction the planets appear to move in the night sky to our observer. This inner band now gets subdivided into seven concentric bands, each of them a different size, proportional to the original sequence of doubles and triples. That is one, two, three, four, eight, nine and twenty seven. These seven bands power the orbits of the moon, the sun and the four known planets. Venus, Mars, Mercury, Jupiter and Saturn. At different speeds and different angles. Now as you may know the term planet comes from the Greek for wanderer. A reflection of the fact that from our perspective, the planets occasionally wander off their set course, in the phenomenon that gets called retrograde motion. Now this complex structure of bands and spheres rotating in different directions and at different speeds is what Timaeus proposes as the underlying regularity. That generates the complex observed the motions of the heavenly bodies, as well as accommodates the observation that some of them appear to be more orderly than others. But, what seems less orderly he insist, is actually only more complicated. It maybe very difficult to specify the precise pattern and harder still to observe it, Timaeus says, but the motions of the universe are perfectly regular. He makes it clear that he expects future mathematicians and astronomers to work out the precise details. His main point in the dialogue is that the motion is at bottom, regular and orderly. And as such he claims, it is intelligent. Thus the cosmos is not only alive, having a soul but intelligent. And to the extent that we human beings are intelligent it is because our own individual souls are made of the same ingredient and constructed along the same mathematical principles as the world soul. But enough of mathematical souls for now. Let's turn from the macroscopic picture of the world soul to the microscopic. Because even here, we find that Plato thinks there is order and goodness structuring in the world.