On Fri, 12 Dec 2003 00:22:21 +0000 (UTC), "Paul P. Budnik Jr."
 wrote:

>
>"Kevin Aylward" 
wrote in message
>news:br7nsr$1nas$1{at}darwin.ediacara.org...

>> But this notion is not relevant to the general question of deterministic
>> systems. We can write down the differential equations for the 3 body
>> system and simply solve them numerically. In principle, from a classical
>> perspective, all of its motion can be absolutly predicted for as long
>> into the future as required. This is what is meant by a deterministic
>> system. Whether or not the bodies will collide will indeed be predicted
>> if the equations are solved for the appropiate time. A non deterministic
>> system by *definition* is *only* when an action can be taken, which in
>> *principle*, can not be predicted.
>
>
>You can only make prediction about this from the differential equations if
>the collision happens. If they do not collide you can never predict that.
>This does have practical importance. When your computer stops responding you
>need to decide if you will reset it and perhaps loose the work you were
>doing or wait and see if it starts responding eventually. Because of
>Goedel's results no one is working on a general solution to detect when a
>compuer program will never again respond, because we know it is impossible
>to solve this problem.
>
>We seem to agree on the facts and I do not want to get into a semantic
>argument about what is meant by prediction as long as we understand each
>other.

Neither of these argments apply to the real world.  Newtonian
mechanics is supposed to be deterministic. However it is well known
(chaos theoy) that tiny uncertainties in initial conditions in
completely deterministic systems can lead to such widely different
solutions that prediction becomes impossible.  You don't even need
quantum uncertainty to generate the non-predictability, only small
inaccuracies in measurements or small outside factors that you failed
to notice -- both factors are present in abudance in any real world
calculation.

The argument about computers stopping has nothing to do with either
Goedel Theorem or the stopping problem (which is related, but another
thing altogether, as has already been pointed out).  When the computer
doesn't respond, you have to figure out whether it is simply crashed
and dead or whether it is in the middle of an enormously long
computation that takes far longer than you are willing to wait.  You
don't need formally undecidable system theory.  The problem of
detecting when a computer is dead is very different.  If the operating
system has indeed crashed, your perfectly good program to tell if the
system is still responding is not liikely to still be running!

Goedel and stopping problems and undecidability and all that depend on
the notion of infinity.  Finite systems simply do not have these
properties.  Turing machines and the stopping problem depend on having
a tape that is indefinitely extendable. No real computer system has
that characteristic.  Even one that stops and says "please insert a
blank formatted disk" isn't because there are only a finite number of
disks possible in our universe.

There is an enormous distinction between making predictions and
decisions in a conceptual, abstract, gedanken world of mathematics,
mathematical logc, and automata theory and making predictions and
decisions to solve practical problems in a reasonable time and with
reasonable certainty given uncertain information in the real, physical
world we inhabit.
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