MDMD(4) p.123 small re-write

[Brian D. McCary]

Interesting, if tangential thread.  

David C. writes, in part:
"The other idea is "non-linearity," which is the phenomenon of huge 
effects that are out of all proportion to their tiny causes.  The classic 
image is that a butterfly flaps its wings in Manchuria, and this is 
ultimately the cause of a blizzard in North America."

As I recall, the importance of "non-linearity" was as a limiting case
for modeling.  (This does, in a way, relate to Pynchon).  I'm gonna get
windy here, since it's probably the only way to really explain the 
relevance of the issue.

All math models of natural phenomena are approximations.  The assumption one
makes when creating a model is that if you make small changes to the input,
you will end up with smallish changes in the output.  The classic example is
weather:  The assumption, for a long time, was that weather was basically 
deterministic, and that if you could adequately model the temperature, wind,
humidity, (ect) distribution, at a given instant, you could generate accurate
long term weather predictions.  Errors in the outcome of the predictions were
going to arise because your model had small errors in the input conditions, but
by making the input condition approximations increasingly accurate, you would
make the predictions increasingly accurate.  What modelers started to find was that
even for simple models, big changes in the long term predictions would result
from small changes in the inputs.  This implied a certain indeterminancy in
systems previously considered basically determinant.

This doesn't so much mean that the butterfly changes or causes the weather.  
Rather, it means that if you run one model, starting in July, with no butterfly
wings in Siberia, and a second one, including a local increase there the
butterfly is flapping, one model may predict a blizzard in Akron in six months,
and the other one might not.  It is a limitation on the usefulness of the model.
This same effect (which limits model efectiveness) shows up in most long term
modeling of any complex system, including most economic or political ones.

One line of inquiry was "How simple does a system have to be to guarentee
good long term predictive power?"  Several examples were found, including the
Mandelbrot set, where seemingly simple single equations of one variable may
be shown to defy prediction.  It's been a while since I looked at anything about
the Mandelbrot set, but it is, as I recall, a plot indicating convergence 
characteristics of solutions to a simple equation, using complex numbers.  The
"self similar" nature is an indication that for some regions in the complex
plane, the convergence characteristic for one point may not be predicted from
the convergence characteristic for a neighboring point, no matter how close
you pick the neighboring point.  If there are simple systems for which this
is true, there is little difficulty believing that it may be true for more
complex systems.

ob Pynchon:  Several times in GR (I don't recall where, right now) Pynchon spins
beautiful poetry about Calculus, epsilon and delta, dv/dt kinds of things.  
The enourmous power of calculus lead many to hope that it could ultimately lead
to good long term predictions about the world.  It does so, but only in a general
sense.  The chaos, or complexity, of the natural world prevents these predictions
from getting very accurate.  For instance, one would think, launching rockets from
Northern Europe to London, that there is one exact launch vector which will result
in a direct hit on a particular target building, or even a target person.  Chaos 
suggests that this may not be so, and that even if it is, that you haven't a prayer
of measureing enough variables to determine that vector.  The rocket *can't* be 
aimed at you by people.
	On the other hand, all those variables provide the necesary conditions to
invoke the Mean Value Theorem, which means you can use Gaussian statistics to
predict rocket fall patterns, (using the Poisson distribution) which is why 
Roger Mexico's map looked like it did.  Which returns us to the question of why
Leutenant Slothrop's map looked like *it* did...


Brian McCary