Off to Infinity in Finite Time
In our comparisons of alternative ontologies, some issues
of physics have come up, including the point that Newtonian
mechanics, which is fine for many applications, makes
inaccurate predictions at small distances or high speeds.
There's also a surprising result, which had been conjectured
by Poincare a century ago and more recently proved by Jeff Xia.
Following is a 10-page article that discusses the issue:
http://www.ams.org/notices/199505/saari-2.pdf
Off to Infinity in Finite Time
Xia's proof demonstrates that "noncollision singularities" can
exist in Newtonian systems with 5 or more interacting bodies.
In particular, it's possible for a 5-body system (consisting
of 5 planets or other objects interacting only by gravity)
to eject one of their smaller members at speeds approaching
infinity -- i.e., there's no limit to how fast such a body
could be ejected.
That prediction, which the theory of relativity would rule out,
illustrates that surprising results can still be discovered in
theories that have been very thoroughly studied for centuries.
It shows that even though a theory might have no logical
contradictions, it may make physically impossible predictions
that can go undetected indefinitely. As Dijkstra said, "Testing
can only show the presence of errors, never their absence."
For ontology, the implications are that any theory about
what exists must always be considered at best a useful
approximation whose reliability is limited to those tasks
on which it has been thoroughly tested. Even then, there
may be unpleasant surprises.
As I've said many times before, a major advantage of a
modular system is that it provides "firewalls" between
modules. If a bug shows up in module X, any module
that does not depend on X can be continue to be used
until some revision or correction is made to X.
As an exercise for the reader, following is an experiment
mentioned by the authors of that article, which anyone
can perform at home (or preferably outdoors). It shows
how rapidly a small body (a tennis ball) can be ejected
in a three-body interaction with two larger bodies (a
basketball and the earth).
John Sowa
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To illustrate how energetically a small object can be
expelled from a multibody system:
1. Place a tennis ball on top of a basketball.
2. Drop the pair from chest height onto a hard surface
(such as a concrete pavement).
3. Observe the motion of the tennis ball.
If any list subscribers try this experiment, please submit
a report for the benefit of the armchair philosophers.