7.6: Further Mathematical Studies

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See 526c-531c. After numbers, Socrates recommends studying “geometry” (shapes, both in two and three dimensions), “astronomy” (“the motion of things having depth,” adding the fourth dimension of time to the three dimensions of geometry), and “harmonics” (the proportions that generate musical harmony). The aim of these further mathematical studies is to turn the attention of students away from the transient particulars and toward the eternal forms. This is why Socrates recommends studying problems of a sort that do not require for their solution information gathered through the externally directed senses of the body. There is no need to develop skill at observing the shapes of crystals, the motions of the planets, or the subtleties of audible sounds – indeed, interest in these things could even be counterproductive – when the point is to have the rational part of the soul “purified” of concern for particulars and “rekindled” in preparation for philosophical inquiry.

Most people appreciate the usefulness of mathematics, but some people love it for its own sake. What is it about mathematics that these peoplelove?
People skilled in mathematics are often said to be good at abstract thinking. What is abstract thinking?
The Greeks were aware, on the basis of experiments with altering the length of strings equal in tension and tubes equal in diameter, that the proportion of one to one half (one string or tube being twice the length of the other) generates the musical interval of the octave, that the proportion of one to two thirds generates the fifth, and that one to three fourths generates the fourth. Socrates criticizes his contemporaries for taking the numbers at work in these “audible concordances” seriously, but failing to investigate, apart from sense perceptions, “which numbers are in concord and which are not.” What could he mean by this? Is there a kind of inaudible, purely mathematical, harmony that can be investigated through the study of ratios?