Solving a Paradox. What does it Mean?
By Giuseppe Gori
In
the arrow paradox, Zeno uses the example of an arrow in flight. He
states that for motion to occur, the arrow must change the position
which it occupies, but for every such position we see no motion, thus
motion does not exist.
Modern
formulation:
(From:
https://bit.ly/2v30Q8X) “At
every instant of time there is no motion occurring. If
everything is motionless at every instant, and time is entirely
composed of instants, then motion is impossible.”
As
reported by Aristotle: "If everything when it
occupies an equal space is at rest, and if that which is in
locomotion is always occupying such a space at any moment, the flying
arrow is therefore motionless."
This
is a paradox, as we know that motion exists.
So,
what’s wrong with Zeno’s syllogism?
Explanation:
The
concept of infinity
helps
us with
mathematical
functions
“at the limit”,
but it
is
possibly
confusing when,
in
the physical world, we need to evaluate
motion
(from
point A to point B)
or speed (an interval of space divided an interval of time).
The
term infinite means “not
finite”.
A
finite
interval (of space, or time), can be thought of as the
sum of a
finite
number of very small intervals, as small as we want (infinitesimal),
but each with its own dimension.
Conversely,
an infinite
number, of dimensionless points or instants cannot add up to a finite
interval(1).
In
the first case, we can calculate the intervals of space and time,
thus we
can evaluate motion
and speed. In the second case, we
can’t.
Zeno,
in Aristotle’s words, asks us to use the concept of a
dimensionless(2)
instant
to imagine the arrow “at rest”.
Today,
with our familiarity with photography, we
can easily
imagine
such
a
snapshot
of
the arrow,
but
we
cannot
evaluate its
state
of motion without
some other information.
In
other words, by
considering
only
that dimensionless instant, without
measurements of time and space, we
cannot evaluate if, or in which direction, the arrow may be moving.
However,
Zeno
suggests
that we could
evaluate the arrow’s locomotion
at any such
moment
and
declare it “motionless”.
Thus,
Zeno’s
syllogism
breaks down.
We
could
explained
this contradiction to Zeno a
bit more figuratively:
We
distinguish between a
snapshot of the arrow, taken
at one instant in time,
and the arrow it depicts.
Zeno
transfers
the motionlessness we observe
on
the snapshot of the arrow, to the arrow that is the
subject of the snapshot. This
is not a valid logic operation, at any time.
Now,
Zeno may respond that he knows nothing about photography, and that
he
is talking about the real arrow, as we imagine it in a dimensionless
instant in time.
We
can then
respond
by saying that, if
we are given
information
about an arrow in a dimensionless instant, and
no other space/time
information,
we cannot say that
the
arrow
is
motionless,
even if we
can imagine it
motionless.
NOTES:
(1) An
infinite series can never complete,
and form a finite interval. It
can converge, or get arbitrarily close to, but not actually equal a
finite sum.
(2) If
that
instant
had
measurable
dimensions,
we could measure the corresponding
intervals
of space and time, and calculate the speed of the arrow, invalidating
the premise.
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