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From: breuel@harvard.ARPA (Thomas M. Breuel)
Newsgroups: net.physics
Subject: Re: Non-linear systems.
Message-ID: <273@harvard.ARPA>
Date: Wed, 9-Jan-85 15:47:14 EST
Article-I.D.: harvard.273
Posted: Wed Jan  9 15:47:14 1985
Date-Received: Sat, 12-Jan-85 07:01:28 EST
References: <209@talcott.UUCP>, <328@rlgvax.UUCP> <384@hou2g.UUCP>
Organization: Harvard University
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> The concept of "predicable, in principle" is usefull if and only if 
> it has some connection with reality.  If 10^70 Cray's calculating for
> 10^10 years cannot predict next year's weather then next year's weather
> is unpredictable.  To argue that the weather is predictable "in priciple"
> is equivalant to saying that "My theory predicts something which is
> IMPOSSIBLE to calculate, but if we could I'm sure it would be correct!"
> 
> It wasn't too long ago that any angle could be trisected, pi could
> be expressed as the ratio of 2 integers, particles positions and momentum
> could be determined simultaneously, ...  In principle, of course.

Sorry, you're wrong. The technical term for "predictable in principle"
is computable. This is a sensible notion of great interest and
significance in modern information science.  What you're aiming at is
the notion of "computational complexity".

But even if a problem is not computable, or if a problem is NP
complete, not all hope is lost. You may not be able to find an
analytical solution for it, or to find such a solution efficiently,
but you may be able to find good heuristics. Indeed, many heuristics
are so good that we don't even notice that they are only
approximations. "In principle", the path of a bullet is not
predictable. Nevertheless, everybody uses Newtonian mechanics to
predict it.

Just a few remarks about your concluding remarks: the trisection of
the angle and the rationality of pi were *unsolved* mathematical
problems. It has now been shown that these problems are unsolvable
(although you can graphically trisect an angle to any desired degree of
accuracy and express pi to any desired degree of accuracy as a
rational number). Whether particle positions and momenta can or cannot
be determined simultaneously is still an unsolved question of modern
physics. The fact that quantum mechanics gives answers that are in
good agreement with experimental data does not imply that quantum
mechanics is by any means "correct". Indeed, it has some highly
questionable points and is in disagreement with other theories that
show equally good argeement with (other) experimental data.

						Thomas.
						breuel@harvard