These brief notes indicate a fuller account that can be drawn out by reading the central books of Plato’s Republic. [I have typed out a Comprehensive exam answer on the Sun, Line and Cave, “On the Philosophic ascent in Plato’s Republic,” in the WordPress menue hovering in the philosophy section.]
The account may be said to begin from a point made by Seth Benardete, that there seems to be no place for political philosophy in the three images of Sun, Line and Cave. We argue that it is there, but veiled, as might be demonstrated by following suggestions in the works of Leo Strauss and Alan Bloom.
The cave is “by nature.” Here: Where is the place of political philosophy in the Sun, line and cave images? Bloom: “Only by constant reference back to the divided line can one understand the allegory of the cave.” The key to the demonstration is in Book 10. What fun! Plug in the city.
Bloom: The poetic images are to be used as the geometers use the figures they draw… Strauss: The human things are the key to understanding all things. There are no artificial things outside the cave. Man is the highest image. City and soul have the same form, in Books III-V.
3/3 Draw the three images, sun, line and cave, together on the same paper. Where do the images of the poets go? And images as the the three part city and soul? Now watch the soul, as the female element is added. And at 501, the imago Dei: the cause of the laws, Genesis 9:6; 1:26.
The particulars of the image are of course very important, and the account recurs at 517 and 535). There are shadows and phantoms seen in water, divine appearances in water, and such, and the things themselves, including humans. Note that the entire pattern of the line- a segmented line divided again in the same proportions- is repeated outside the cave. The account makes up what reminds of Jacob’s ladder, a pattern helpful for ascending and descending in metaphysics. To say the least, there is a bit more to Plato than linguistic universals and the originals of math.
There is of course the one and the two, and within these, the philosopher and king, one king of the visible, while the good is king of the intelligible. Third from a king and virtue” is a key phrase, from Book X.