As far as I can see Timaeus provides a coherent model of the universe that is based on observation. (Plato describes this in his Timaeus 1.) Unfortunately, the observations are poor, although easy to make, and the model wrong.
This recounting does not follow Timaeus exactly -- I am bringing in others memories and ideas, but I am pretty sure the thesis is as follows:
Everything on this earth rots or changes. That can be observed by looking at rotting meat or the erosion of mountains.
On the other hand, certain things are eternal and unchangeable. Numbers are one of them. The number three always is the number three. It never changes to two or four. Similarly, right triangles, the kind you think about and about which you devise theorems, such as Pythagorean, remain always the same also.
Triangles provide a hint of how the universe must be constructed:
Triangles you make from wood, for example, or from a length of rope, are always imperfect. However, the triangles you imagine, about which you repeat theorems, are and always remain perfect.
Each person can observe himself or herself imagining a triangle, but no one else can observe that person doing the imagining. This is quite unlike an imperfect triangle made from a length of rope, which two or more people can see together.
In any event, as individuals, we can observe two kinds of thing:
The observations do not depend on someone else's report, or anything like that; we make them ourselves by ourselves, so they are convincing.
The next question is what, if anything, can we observe in this universe that is perfect, or, if not perfect, rots or changes more slowly than old bread?
The answer lies in the heavens. Except for those that wander and temporary stars move suddenly across the sky, stars keep their positions with respect to one another, although the whole sky does move around the world each day, which just a slight delay that is related to the position of the sun. Moreover, the stars' positions repeat pretty exactly from one year to the next (but because of precession and proper motion, not from one millennium to the next). (The wanderers are the sun, moon, and planets; they move within the band of the Zodiac. Shooting stars are more properly called `meteors', which is a word that comes from Aristotle, who figured they must have more to do with meteorology than with the further heavens.)
So the universe as we observe it is closer to perfection than anything else we can jointly observe; but it is further way than the kind of perfection anyone can observe in his or her mind by imagining a triangle.
Hence, it is worth considering what the universe as a perfect object must be like. Timaeus reckons it is a sphere. In that sphere, there are parts that are the same and parts that are not. This is a simple consequence of thinking about drinking; either the wine is in the cup or it is not. (Well, obviously, one of those annoying people could point out the drop on the edge and ask where it is; but if you exclude that possibility -- if you imagine wine either in a cup or drunk -- then you can construct a logic that succeeds. So the general premise that there are things that are the same and things other makes sense.)
We look at the sky, and observe that there are two aspects that appear to match this division: on the one hand, the part that repeats itself, the part that is the `same' each time. That is the celestial equator. Then there is a part on which the wandering stars move. They stick within a band, but that band is not on the celestial equator, although it crosses it. That band, the ecliptic, must the the `other' part.
Another observation, first reported by Pythagoras, that when a string is twice as long as another, but of the same material and pulled as tightly, when strummed, the longer string sounds an octave lower than the shorter. A string one and a half times as long as another sounds a fifth lower.
In fact, you can make up a musical scale by adjusting string lengths in ratios of 3 to 2 and the like. The only trouble is that the final note lacks a nice simple ratio. This is not unlike the problem with the square root of two, which also lacks a nice simple ratio. Likewise, the circumference of a circle is not exactly three times its diameter; indeed the circumference is not 22 divided by 7 times the diameter, although that ratio is closer than a ratio of 3 to 1.
These factors are clearly observable: but then again, this world in which we live is imperfect. We must live with exceptions.
When we think about geometry, anyone can create in his or her mind imagined, perfect, three dimensional objects: but only five convex ones; no more.
If me make an assumption that does not involve observation, we can say that the most basic kinds of object match the objects.
What are these basic objects? These we can observe. But to separate them from everything else about us, we must think.
First, we stand on something solid, the earth. That is a necessity. So earth is one of the basic objects. Since cubes do not move much, earth matches cube.
Second, water washes away mountains. Some say that the world we live in was once a mixture of water and earth, a mud, but the earth was allowed to settle out, and some of the water evaporated, so we have earth on which to stand. Water is important. Moreover, when you fill a goblet with water, you can feel its weigh, and if you move your hand in a pool, it pushes back. Water is peculiar, universal, and clearly a different element from earth.
Both earth and water exist in air. Water sometimes evaporates into air, although it condenses out, so it is not really air. Air does not push when you wave your hand in it, but you can feel a push in a strong wind. So air is more ethereal than water.
Fire is like air in that it is ethereal, but not like air in that we are not surrounded by fire. Indeed, fire can kill us; and it hurts when we stick our finger for too long into a flame.
Thus, we have four elements, earth, water, air, and fire. (There are five Platonic solids; Aristotle adapted the fifth solid to a fifth element, that of the heavenly sphere, which is not of the world in which we live. He called it the `fifth essence' or `quintessence'.)
Earth matches cubes, as we said. Water flows; it matches the icosahedron, which rolls on its 20 sides pretty well. Air and fire match the octahedron and tetrahedron respectively. The fifth element matches the dodecahedron. Interestingly, the band of the `other' in the sky, the Zodiac, has twelve parts and a dodecahedron had twelve sides.
All this makes sense, and it is partly based on the kinds of observation, either of the external world or of internal imaginings, that anyone can make.
In addition, you can go on to consider human features, such as hunger and disease, and create a model that fits.
Timaeus' whole description fits together. Moreover, it derives in a large part from observation, and only in a small part from cultural presumption. So we can see why it had such a long run, outlasting the fall of two civilizations, the Classical and the Medieval.
Troublesome astronomical observations were explained by Ptolemy. He showed that the locations of heavenly wanderers could be predicted by presuming that they, the planets, the moon, and the sun, moved upon circles upon circles or upon eccentric circles.
Although someone could have followed the method before Timaeus, it was not until the 1640s or there abouts, more than a millennium after Ptolemy, and two after Plato, that some astronomers began to use cathedrals as large pin hole cameras. They projected an image of the noon day sun on the floor or wall and marked it. The location of the image rose and fell with the seasons. After taking measurements for a generation or so, they found that Ptolemy's and Copernicus' predictions of the locations of the image of the sun were less accurate than those of Kepler.
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