Jameson York
Starting, Ending, Passing: The Impossibility of Motion
Zeno proposed a number of paradoxes, most famous of which is
“The Arrow.” Other paradoxes of motion are “Achilles and the Tortoise”
and the “Dichotomy,” each of which is essentially the same, as they
each show a different portion of motion being impossible using
variations on the logic. These paradoxes are obviously false, (as you
can surely attest, being in this classroom, rather than still trying to
leave your bed, or pass a slower car on the road, or trying to reach
your seat), but proving it intellectually is difficult.
The arrow paradox is that it will be impossible for an arrow to
strike a target, because it would have half of the remaining distance to
travel forever. The Dichotomy paradox states that it is impossible for
anyone to move anywhere, because before they can travel half the
distance to a target, they have to travel half of that distance, and so
on (the opposite of the arrow paradox). The final variation originally
was that Achilles, giving a tortoise a hundred foot head start could
never beat the tortoise in a race, because in the time it takes Achilles
to reach the point where the tortoise started, it would have moved
forward a much smaller distance, but still moved forward. Then when
Achilles reaches that point, the tortoise has moved again, and so on
forever.
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There are a number of ways to defeat this logical paradox. A
common method is to refuse the idea that distance is infinitely
divisible. This means that it an arrow reaches and hits its target
because although the distance is halved many times, it reaches “
”
(Planck Length) which is the smallest distance, or 1.61624x10-35m. So
a 5 mph car hits a wall by traveling at lP /( lP *3600/8046.72m), which
is math talk for the speed of lP per 7.23085x10-36 seconds (such a
small portion of a second, we have no device able to correctly time it –
the most accurate atomic clocks are synchronized to 10-7 seconds, for
some perspective). We can similarly do the math based on the Planck
Time, but that number is such a small fraction of even the time we
have here (10-44), we would end up with a smaller distance than even
the Planck Length, which is theoretically impossible. This shows that
completing an infinite number of actions (portions of movement) to
get to the target is no longer required, as there is a finite number of
units of range to cross.
For the starting paradox, the lP method is just as valid, and the
math is the same, supposing the car starts at full speed. The passing
is slightly more complex, but still thwarted. The problem with the
passing paradox is that the target point moves as well. If a man is
running at 10lP/sec and a tortoise is moving at 1lP/sec with a 20lP head
start, then 1 second later the man would be 11lP away from the
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tortoise. One second more gets him 2lP away from the tortoise. So the
man moves the 2lP to reach the tortoise, which takes only .2 seconds,
and insufficient time for the tortoise to have moved another 1lP away,
and since it is only possible to move in a complete lP, in another .1
second the man has moved 1lP past the tortoise, and when the tortoise
finally makes its 1lP, the man will be 7lP beyond that new point,
disproving the paradox. Note, these objects are moving at a
ridiculously slow speed in order to simplify the math and prevent
yoctoseconds from coming into play.
I personally have a problem with this refutation. I don’t see any
good reason not to have a ½lP, if not just in a purely intellectual sense.
It may not be possible to see or measure or in any way gauge
anything smaller than Planck’s units, but that doesn’t mean that they
don’t exist all the same. Just as travel above 299,792,458 m/s is
considered impossible, there is nothing to prevent us from imagining
something traveling at 300,000,000 m/s.
Easier for me to accept is the fact that it is possible to complete
a limited infinite set of tasks within a finite period of time. My “limited
infinite” is to mean an infinite set that is conceivably ended, such as
traveling every portion of ½ the remaining distance of a 100 meter
travel – the infinite set approaches zero. An “unlimited infinite” would
be something such as running north forever, with no stated limit other
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than the end of forever – the infinite set approaches infinity. That is to
say, it is possible to move two feet forward in a time less than forever,
while still preserving every (infinite) degree of space traveled before
reaching the end point. The reason I can so blithely accept something
that seems at the face to be irrational is Speed = Distance/Time.
When a car is traveling toward a wall, the speed it travels at is
constant (just as the arrow traveling in the paradox), so Distance/Time
will always reduce to the same value (2 = 18/9 and 2 = 4/2). If we
assume the car’s speed is 5 m/s (11.18 mph), and the target is 100
meters away, then the time to reach the target is 5 = 100/t, or 20
seconds. Following the paradox, half the distance is 5 = 50/t, so it
takes 10 seconds to reach the half way mark. Half again is 5 = 25/t, or
5 seconds. Essentially, for each halving of the distance, it halves the
time it takes to complete that task.
If we continue this trend for long enough, we reach a point
where the distance traveled to half the distance remaining is
indistinguishable from zero, say 10-100 meters, the time that it will take
to travel that space is 2*10-101 seconds. This continues until it is an
infinitely small distance to travel in an infinitely small amount of time,
where infinitely small becomes zero rather than simply approximating
it.
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Perhaps the simplest and most elegant refutation is to simply
dispute the definition of infinite. It makes sense to split infinite sets
into Limited and Unlimited, because they describe vastly different
number sets. An Unlimited infinite would retain the standard definition
of infinity, wherein it is an unbounded and indefinably large number or
amount. A Limited infinite is not unbounded, and is, in fact by
definition, bounded on both ends – a car traveling at a wall with
infinite degrees between has both a start and ending bound of the set,
starting at 100 meters and ending at 0 meters. Strictly speaking,
anything moving to a target has bounds, and so isn’t infinite by the
classic definition. Something that is not infinite is finite, so by strict
definition, it is possible to reach a target simply because although it
seems infinite, it is not because it does not meet the requirements of
an infinite set so because it has an end, it is possible to reach it.
The argument that it is impossible to reach a target because you
have to do an infinite number of tasks is false, assuming you have
more time to complete the actions than the actions take. And when
moving the last second to hit a target, although it takes an infinite set
of actions to complete, the time it each takes is infinitely small. The
only way to prevent an infinite set of actions that takes a finite amount
of time to complete from completing is to prevent the time from
progressing. For an infinite action to be impossible to complete, it
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must take an infinite (non-bounded) amount of time. That said, it is
impossible, assuming an infinitely divisible unit of length to move half
the distance, then move half the distance, then again, and in the end
reach the target – in this case the action is not finite in duration, so
cannot be completed. An arrow hitting a target, however, does not
change it’s speed (for ease of math – it does change speed, but not in
the same way that halving the distance, then again and again does,
for the speed change is insignificant for the purposes here unlike
stopping and starting.) This lack of change means that the constant
speed and set time means that the distance is traveled in that time is
mathematically set (and never in dispute). The improbability of
traveling through a (limited) infinite set of actions is overcome when
shown to have a limited time in which those actions are performed.
The end result is that in order to keep the arrow from hitting the
target, time must stop 0.00..(∞)..01 seconds before it hits or sooner.
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