Chaos and Analysis

Currently reading:
Prose Edda of Snorri Sturlson
   trans. by Jean I. Young

I've been thinking about chaos theory recently, and I was wondering how it fits with my recent statements on logic. Paleolithnick had also suggested the relevance to my thoughts a while back on free will, a topic which I may well cover when I return to the "Defense of LFW" series (it will come back eventually, really, I need to get the paper in readable form by October 5th. And I'll return to the statements about causation and whether it affects logical analysis at some point as well; maybe in a year or two).

I'm going to make some rough-hewn suggestions which may send my inner mathematician into seizures, but he can work them out more later if it bothers him that badly. I'm going to be pulling at too characteristics of chaotic systems. The first has to do with the lack of predictability. This is due to two things: first, sensitivity to initial conditions. This means that no matter how close you get to the starting point of the system, if you're not exact, you'll get arbitrarily far away as time goes on. This in itself isn't the only issue; a simple exponential or even quadratic function does the same thing. However, one can at least get a feel for the function and try again in those cases, and determine how far off one started by how the ending looks. A chaotic system twists and turns on itself so much that one can get completely lost in it, without any idea of how on- or off-target one is. Further, while if you come at the system in some directions it remains stable, if you come at it in other directions it is unstable. An example of this could be the climate; some changes to it will be canceled out, others will balloon, and if pushed passed certain boundaries, it'll completely change. If you come at it from an unstable direction, things'll go haywire and you're prediction will be nothing like the real thing.

However, despite this inability at being able to predict the system at an analytical level, the system displays macro-level regularities. These are called "strange attractors." More or less, if you look at the graph of a chaotic system, you can see a pretty picture rather than a tangled ball of yarn. An example would again be found in weather patterns: we can't predict the weather with that much accuracy, and while we can improve we'll never be able to look too far in advance, but at the same time we have seasons. I don't know what the weather will be like tomorrow, but we are entering fall.

So, in sum, low-level analytical details, if they don't start at the right place, can (and most likely will) create an overall picture of the world that is wrong in all of what it has to so. Further, getting closer to the right answer doesn't necessarily mitigate this, and the starting point is also the most controversial point in a well-formed argument. However, even if we can't get at the details very well, by looking at the different trajectories of different theories, we can see broad patterns arising. These patterns then exist, even if they themselves are not analyzable. All of this assuming that we are in a chaotic system, whatever that might mean, or that our epistemological position is analogous in appropriate ways to chaotic systems.