A New Approach to Katharsis
We have always had a literary or possibly medical view of katharsis in the Poetics, one way or another. I would like to try another direction:
1) I am convinced that the Socratic elenchus is based on a mathematical model. First something is proved to be equal to something else, then another thing is proved to be equal to the same "something else". Sincethe first and second things are not the same (by definition) the argument cancels itself, and we are left with aporia. This has always been taken as a philosophical statement, but since mathematics are in Greek terms philsophical in nature, I read the elenchus as a direct anticipation of what Euclid was later to codify as "Things equal to the same thing are equal to each other" and if not equal to each other, the process is wrong. Used for self- clarification, the mathematical base is apparent, to be found in the axiom to Bk I of Euclid. [If a = b, and b = c, then a = c, but if in the real world it does not, we goofed in our thinking. Socrates is exploring philosophical goofs, a good thing to do for a philosopher!.
2) Katharsis can be seen as a mathamatical process of cancellation, if it operates on two factors which are inherently inverses of each other.If eleos and deos can be seen as inverse emotional states, which is hard to do since we do not have an authentic Greek psychological discipline, then the "processing" of these two together, at the same time while watching the same dramatic sequence, should reveal a "null". In short, if I am computing (+x + -x) I get O. Now the problem would be this: Are pity and fear inverses? I believe they are, eleos being an outgoing emotion from the drama-watcher TOWARD a person of action, it has a positive (yang) quality, emptying the beholder's "heart" (Aristotle's wrong theory...). But deos is quite the opposite, it is a shrinking, in pulling emotion, which fills the "heart" with apprehension, sense of personal danger, risk. It sucks inward, yin-wise.If these two can be combined at a single time, the beholder may go out of the theater with a "null" or empty feeling.
I think this is often the feeling one gets after watching good modern Cinema, this null-ness, often almost numbness, which opens the mind to calmness, re-consideration, thoughtfulness.These are the qualities many of use feel after reading a Greek drama right- through.Mathematical patterns are often used by Plato for other than mathametical purposes.
A prime example of reaching a philosophical null, and then trying a new direction, is the puzzle in the middle of the Meno about triangles, which defies every mathematical solution, and leaves us with a "method of approximation", clearly intentionally. The rest of the Meno goes on to non-mathematical, mystical views seen through priestly eyes. Here is the beginning of a "farewell to mathematics", the turning point.And when we reach the Myth of Er, it will be shapes (topology) not mathematically calculable geometry, with which Plato has replaced his earlier faith in current mathematical developments to lead him to the higher reaches of pure thought.