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.)
Nowadays we regard astronomy and harmonics as belonging to the field of "applied" rather than "pure" mathematics, but this was not the case in Plato's time. "Natural science" as we know it was unknown to the ancients; our practice of observation and experimentation to determine knowledge about phenomena had not yet been introduced. Plato and his contemporaries thought calculation to be more important than observation; Plato himself pokes gentle fun at thinkers he considers to be "star-gazers."
For Plato, the same importance of calculation holds true for the study of harmonics, which Pythagoreans had already advanced. Socrates tells Glaucon point-blank that it is not our intent to teach these future Guardians to keep time, or something like three-part harmony or, so to speak, to tap their feet. We are trying to teach these people how to think.
Socrates' allusion to the "shell-toss" preliminary in the children's game may be compared to our "coin-toss" preliminary at the outset of contemporary athletic events.
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Palamedes a hero of post-Homeric stories of the Trojan War.
Daedalus a legendary Athenian inventor, architect, and artist, who according to legend, built the Labyrinth.
Pythagoreans followers of Pythagoras, a philosopher, astronomer, and mathematician of the sixth century B.C.
"rack [the strings] on the pegs of the instrument . . . ." Socrates is referring to music theorists who, in trying to determine precise intervals of pitch, tighten and loosen the strings of a lyre to change the pitch ever so slightly; figuratively, he says they are torturing the strings the way a prisoner might be tortured on a rack, stretching them little by little to make them give up information.
plectrum a thin piece of metal, bone, plastic, etc., used for plucking the strings of a lyre; a pick.