On Wed, Oct 03, 2007 at 12:20:10PM -0400, Richard Loosemore wrote:
> Second, You mention the 3-body problem in Newtonian mechanics.  Although 
> I did not use it as such in the paper, this is my poster child of a 
> partial complex system.  I often cite the case of planetary system 
> dynamics as an example of a real physical system that is PARTIALLY 
> complex, because it is mostly governed by regular dynamics (which lets 
> us predict solar eclipse precisely), but also has various minor aspects 
> that are complex, such as Pluto's orbit, braiding effects in planetary 
> rings, and so on.

Richard, we had this conversation in private, but we can have it again
in public. J Storrs Hall is right. You can't actually say that the
3-body problem has "various minor aspects that are complex, such as
Pluto's orbit". That's just plain wrong. 

The phonomenon you are describing is known as the "small divisors
problem", and has been studied for several hundred years, with a
particularly thick corpus developed about 150 years ago, if I remember
rightly. The initial hopes of astronomers were that planetary motion
would be exactly as you describe it: that its mostly regular dynamics, 
with just some minor aspects, some minor corrections.  

This hope was dashed. The minor corrections, or perturbations, have 
a denominator, in which appear ratios of periods of orbits. Some of
these denominators can get arbitrarily small, implying that the "small
correction" is in fact unboundedly large. This was discovered, I dunno,
several hundred years ago, and elucidated in the 19th century. Both
Poincare and Einstein made notable contributions. Modern research 
into chaos theory has shed new insight into "what's really going on"; 
it has *not*, however, made planetary motion only a "partially
complicated system".  It is quite fully wild and wooly.  

In a very deep sense, planetary motion is wildly and insanely 
unpredicatable.  Just becaouse we can work out numerical simulations 
for the next million years does not mean that the system is complex 
in only minor ways; this is a fallacious deduction.

Note the probabilites of pluto going bonkers are not comparable
to the sun tunneling into bloomingdale's but are in fact much, much
higher. Pluto could fly off tommorrow, and the probability is big
enough that you have to actually account for it.

The problem with this whole email thread tends to be that many people 
are willing to agree with your conclusions, but dislike the manner in
which they are arrived at. Brushing off planetary motion, or the 
Turing-completeness of Conway's life, just basically points to
a lack of understanding of the basic principles to which you appeal.

> This is the reason why your original remarks deserved to be called 
> 'bullshit':  this kind of confusion would be forgivable in an 
> undergraduate essay, and would have been forgivable in our debate here, 
> except that it was used as a weapon in a contemptuous, sweeping 
> dismissal of my argument.

Actually, his original remarks were spot-on and quite correct.
I think that you are the one who is confused, and I also think
that this kind of name-calling and vulgarism was quite uncalled-for.

-- linas

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