An inability to sleep, combined with a recent reading of book VI of Republic, led me to the somewhat disordered diversion of creating an Excel spreadsheet to take a look at the so-called "divided line" that Socrates presents. An immediate problem that we should discover in comprehending the line is that it is completely impossible to actually draw it since the precise dimensions are not given in the dialogue (this is, of course, no accident). However, regardless of which numbers are used, so long as we follow the constraint that Socrates gives us (two unequal segments subdivided into two segments that follow the same ratio as the main division), the larger division of the smaller part of the line and the smaller division of the larger part of the line have the same length. This can be shown mathematically as follows:
Line L is divided into segments X and Y, such that Y > X. X is further divided into segments A and B, while Y is divided into segments C and D. Based on the data that Socrates provides, B = X*(Y/(X+Y)), while C = Y*(X/(X+Y)). Both A and B simplify to XY/(X+Y), so A=B.
This is important because it underscores the necessarily circular relationship that exists between B, which Socrates says contains "the animals around us, and everything that grows, and the whole class of artifacts," and C, which contains "a soul, using as images the things that were previously imitated, [which it] is compelled to investigate on the basis of hypotheses and make its way not to a beginning but to an end." Those sorts of knowledge and knowing that correspond to C (Socrates chooses geometry as an example) attempt to use the sorts of things in B ("material" objects) to discover the template that underlies them, the "things themselves." As useful as this enterprise is, it never reaches to the highest level, D, the level of "intellection," because it has began with the things as images and can neither leave this aspect of them behind nor entirely transform it into something higher. Therefore it will be forever "unable to step out above the hypotheses." This knowledge will therefore be drawn back down towards its material origins and seek to reform the material in light of what it has itself created from the material.
Socrates' statement that this sort of thinking is not the highest kind of thinking should be highly interesting to us, since what normally passes for thought is precisely this sort of thinking. That Socrates suggests the existence of an even higher realm, a realm which is neither the astringent ecstasy of the mystic nor the emotional deluge of the romantic, should give us occasion to ask whether we have ever even begun to think in the highest (yet also most original) sense of the word. Despite the tremendous amount of thinking that people all over the "globe" are doing each and every day, it is possible that what is essential in and to thinking remains as an overlooked challenge sheltered in the Socratic dialogues.