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.
