More on Plato's Republic

A couple things from my lecture I had to rush through at the end:

First, Plato seems to think that knowledge of the good is necessary if we’re going have knowledge of the other forms.  The good “illuminates” the other forms like the sun illuminates the objects outside of the cave.   Without knowledge of the good, we don’t seem to have knowledge of anything.  The whole foundation collapses.  Furthermore, philosophers need to have knowledge of the good in order to perform their function as rulers of the polis.  This raises a crucial question: how are philosophers supposed to identify the idea of the good?   Plato criticizes mathematical reasoning because it proceeds from unproven axioms or “hypotheses.”  We have to assume the truth of these axioms in order to derive all of the system’s theorems.   As one of my students put it in section, mathematics itself does not have the resources to prove its own axioms. (I wish I’d thought of that!)  By contrast, when philosophers use the method of “dialectic,” they can somehow reach a first principle that is justified or proven rather than assumed.   Some people equate this proven first principle with the idea of the good itself.  And once you’ve attained that, you can derive knowledge of the other ideas including the ideas in mathematics.  But how in the world (or outside of it) are we supposed to arrive at that first principle?   Plato says very little about the method of dialectic in this dialogue, except to suggest that it can lead to knowledge of the good.  Maybe you just have to try the dialectic and--if you’re a philosopher--“you’ll know the good when you see it” (like pornography).  In other words, maybe the idea of the good is something we can intuit once we’ve had the proper training.   We will recognize it when we’re in its presence.  But it seems like you could say the same thing about mathematical axioms, and Plato is dissatisfied with them for that reason.  Intuitiveness or self-evidence doesn’t seem like the right kind of criterion.  So this is a perplexing problem for Plato.

A second issue I raised at the end is how to match up the objects and images in the cave with the lower part of the divided line.   At first glance, it would seem that the shadows on the wall correspond to images of sensible objects and the artifacts held up in front of the fire correspond to the sensible objects themselves.   The problem is that the prisoners are chained in such a way that they can only see the shadows.   But we don’t walk around only seeing our reflections and images, in fact, that’s the exception.  So shouldn’t the prisoner’s be facing the artifacts?  After all, most of the time, we perceive sensible objects and not their reflection. 

Or do we?  Perhaps Plato is saying that our perceptual apparatus does not allow for an objective identification of things like tables and chairs (never mind their ideas or forms).  We see things from different angles and perspectives, and this skews how the objects appear to us.  Furthermore, our eyes play tricks on us—they construct an image for us based on past experience of what works, not how the object really looks.  (See some evidence of these illusions below.)  The only way to properly perceive sensible objects—the second lowest part of the divided line—is to employ measurement and other scientific techniques.  Short of that, we’re stuck looking at images and reflections—even when they aren’t obviously reflections from a mirror or pool or water.   
—Tamler Sommers

Line (a) and line (b) are the same length. Do they appear that way?

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The hearts are moving clockwise. But look at it for a second and they appear to move counter-clockwise.