Can someone give me a layman's explanation of the importance of the "cave sections" in Plato's Republic?
I'm also interested in the "myth of the metals" in the same book.
As almost every word written by Plato (and Aristotle) the Cave Allegory is interpreted today in various ways:
As Plato is one of the most read writers of the Ocidental culture, almost every branch of knowledge along the history tried to interpret his allegories by its vision. But if you want a suggestion of a simple tutorial, I liked this one: http://faculty.washington.edu/smcohen/320/cave.htm
The "Allegory of the Cave" is a section in Plato's Republic featuring a group of prisoners in an underground cave who see shadows on a wall, and believe those shadows to be all of reality, as contrasted with an escaped prisoner who journeys upwards to the sunlit world above.
Plato's own explanation is that the cave is the ordinary world as we experience it, and what we typically think of as "reality" is the shadows on the wall. Escaping from the cave is entering the world of the mind, and the sunlit realm is the world of abstract concepts and ideals.
(The myth of the metals is entirely separate, and is a comparison of different types of people to different precious metals -gold for the rulers, bronze for the craftsmen, etc.. In a later section of the Republic, Plato makes the controversial suggestion that teaching people that their souls are composed of different kinds metals will make them more likely to accept a stratified social hierarchy.)