Justice, Geometry, Big, Small and Good

At the beginning of Plato’s Republic there is some fairly straightforward discussion on whether it’s better to be just or unjust.  So far, so good.  Thrasymachus argues that it’s much more advantageous to be unjust than just and that the power that can be derived from unjust behaviour proves this.  Then we get into the whole construction of a hypothetical state by Socrates so he can show how things in macrocosm reflect the internal operations of the average individual.

At this point the discussion really starts to make my head hurt.  Socrates introduces some concepts that  even those he is engaged with have difficulty grasping.  Specifically things like absolute beauty, which  appears to be a contradiction… at first.  Beauty seems like a pretty subjective thing.  Along the way he tries to help clarify what he is saying by constructing images to illustrate his ideas.  The cave and the subdivision diagram in particular.  I don’t know what else to call the second.  It is this second image that I have been thinking a lot about.

You have to imagine then, that there are two ruling powers, and that one of them is set over the intellectual world, and the other over the visible.

Now take a line which has been cut into two unequal parts, and divide each of them again in the same proportion, and suppose the two main divisions to answer,  one to the visible and the other to the intelligible,  and compare the subdivisions in respect to their clearness and want of clearness.,  and you will find the first section in the sphere of the visible consists of images.  And by the images, I mean in the first place, shadows, and in the second place reflections in the water and in solid, smoothed and polished bodies and the like …

Imagine now the other section, of which, this is only the resemblence, to includes the animals which we see and everything which grows or is made.

Next proceed to consider the manner in which the sphere of the intellectual is to be divided.

Thus:  There are two subdivisions, in the lower of which the soul uses the figures given by the former division as images; the enquiry can only be hypothetical, and instead of going upwards to a principal descends to the other end; in the higher of the two, the soul passes out of hypothesis, making no use of images as in the former case, but proceeding only in and through the ideas themselves.

OK, so I took a stab at drawing this out.

What I got out of this is that just like the sun provides light that reflects off of objects in the physical world and causes them to block light casting a shadow or perhaps a reflection, the Good reflects of knowledge and causes reflections of an another kind in the realm of the intellect.

After thinking about it more I realize that even that is too simplistic by far.  It’s hard for me to read this and get it.

I do not quite understand your meaning, he (Thrasymychus?) said. 

But revisiting it, I see I glossed over something:

… You are aware that students of geometry, arithmetic, and the kindred sciences assume the odd and the even and the figures and three kinds of angles and the like in their several branches of science; these are their hypotheses, which they and every body are supposed to know; and therefore they do not deign to give any account of them either to themselves or others; but they begin with them, and go on until they arrive at last, and in a consistent manner at their conclusion?

… Although they make use of the visible forms and reason about them, they are thinking not about these but about the ideals which they resemble.

Ok I’m going to make a big jump to my conclusions.  To get to it though, I have to recap an argument I had with someone about the nature of justice and the Good.  Basically the crux of it was that my opponent said jusitice was relative, where as I said it was absolute.  My opponent stated that if someone was hungry and stole from you, it might be just to forgive them.

I said that kindness or mercy plays no role in justice.  Justice is always represented in terms of a scale, tit for tat, the blind lady with the scales, fair is fair, and all that.  I might be justified in retrieiving the stolen items, but I might choose to forgive and forget.  However my potential mercy would have nothing to do with what was just.  There is something that is above justice and that is the good.

I think I won the argument.  Justice doesn’t equate with Niceness.  I think Socrates was saying the same thing when he suggested justice in society was equated with each person doing what they were most suited for and nothing else.   Then he compares the role of the sun and the Good in realms of the visible and intelligible.

Now lets go back to the beginning of this and think about the concept of absolute bigness and smallness.  Since something can’t be it’s opposite (or it wouldn’t exist) it seems diffieult to say something is both big and small at the same time.  My ring finger is bigger than my little finger but smaller than my middle finger.  It is both big and small at the same time!  Hence we realize that the concept of big or small is absolute, but at the same time only a mental abstraction.  Similarly, a geomentric point has no size, or as Euclid defined it: “A point is that which has no part.”  It is something that does not really exist in the physical world, but rather only an intellectual construct.

It seem to me that what Socrates is saying is that by assuming and accepting basic hypotheses without questioning them, we are doomed to end up with the self same conclusions each and every time.

How does quantum theory jive with the axiom: “The shortest distance between two points is a straight line.”

 I think one of the things that Socrates was trying to convey in all of this was that by focusing in on Justice, these guys were missing what was important.  In reality justice is just a mathematical equation, unless tempered by the good, it’s of little value.