In the conversation earlier, it was decided that the future Guardians are to be trained in gymnastic and the arts early in their education and nurture. As they mature, they are to be introduced to various levels of mathematics and thoroughly schooled in them in order to train them intellectually so that they may become adept at abstract thought. Socrates suggests that the studies should move from the simplest to the most complex in this order: arithmetic, plane geometry, solid geometry, astronomy, and harmonics. We bear in mind that the future Guardians are to be schooled thus in order that they may one day understand the Forms and Goodness itself.
Following their rigorous study in mathematics, the future Guardians are to be trained in Dialectic, which field of study has been discussed earlier in the conversation. (Here we should review that summary and analysis having to do with the four levels of intellect, the Analogy of the Line, and the Allegory of the Cave.)
