Zeno and Uncertainty

Do events of probability 0 ever occur?  This might be called "Zeno's 
Fifth Paradox", since it so closely resembles the four most famous of
Zeno's paradoxes of motion and change.  It's most similar to "The 
Arrow", i.e., before an arrow can reach the target it must reach 
the half-way point...etc, thus motion is impossible. Similarly we 
could imagine Zeno arguing that the probability of the arrow landing 
in a certain region of the target is equal to the ratio of that 
region's area to the total target area.  As the region gets smaller 
the probability gets less, and there is zero probability of the 
arrow's point landing on any particular point of target.  Thus, 
the arrow can't possibly hit the target.

As with Zeno's other paradoxes, this one is easy to resolve from a 
strictly mathematical standpoint, where we are free to *define* 
concepts of limits and measure, but not so easy from the standpoint 
of physics, i.e., does the world really work that way?  For example, 
it's possible to interpret Zeno's "Stadium" paradox as an attempt 
to address the issues of special relativity. ((See the note on 
Zeno's Paradox of Motion.)  Likewise Zeno's paradoxes of
infinite divisibility can be seen as arguments for the fundamental
quantum character of nature.  In a sense, these paradoxes were the 
predecessors of the "ultra-violet catastrophe" of the 19th century, 
i.e., the realization that infinite divisibility (h=0) logically 
implies infinite energy at the high-frequency end of the spectrum 
for black-body radiation.

It's even possible to see in Zeno's arguments some intimation of the
conjugacy between certain pairs of observables, such as position and 
momentum, which underlies the uncertainty principle.  He says nothing
can move in an instant, and since all time is composed of instants,
nothing can ever move.  The mathematical treatment of continuously
varying functions is not really sufficient to answer this from a 
physical standpoint.  The problem involves not just velocity but
momentum, i.e., the persistence of a definite state of motion over
a finite period of time.  How is the existence of this motion (the
information of it) conveyed through a succession of instants, in
each of which the object does not move?  We see here the fundamental
physical fact that if a particle has a definite precise spatial
position, it's momentum is (and must be) completely indeterminate.
No information about motion can be contained in, or conveyed through,
a single point at a single instant.  Conversely, in order for the 
entire inertial motion of an object to be definitely realized in the
present instant, so that it's momentum is fully determinate, it 
cannot have any definite spatial position at all.  Position and
momentum are conjugate observables in this sense, and so they obey
the Heisenberg uncertainty relation, which says the product of the
indeterminacies of position and of momentum cannot be less than a
certain irreducible value. 

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