Something that is both common to these two paradoxes and essential to
each is that space be divided into a series of points. In the Dichotomy, the
distance from the moving object to the goal must be divided into a series of
mid-points, and in the Achilles, the paths of Achilles and of the tortoise must
each be divided into a series of points ("the point where the tortoise
used to be," "the point where the tortoise has moved on to").
So it would seem that one way of dissolving the paradoxes is to keep them
from even getting off the ground by denying that space is composed of points.
If space is a fluid continuum, and not a jolting series of discrete points,
then either Zeno would have to find a radically different way of describing his
paradoxes, or he would have to admit that they cannot be described, and so are
not genuine paradoxes. And, after all, does it not seem true that space
is a continuum? Raising my arm does not seem to require momentary startings and stoppings, as if my flesh jumps imperceptibly
from one point to the next.
I think, however, that noble as it may be, this strategy does not succeed
in ridding us of the paradoxes. For even if metaphysically space is continuous,
we can still ask where the tortoise moves on to when Achilles has reached where
the tortoise used to be. In other words, even if space is not really composed
of points, we can still divide it into points in thought, though not in
reality. We cannot physically separate the convexity of a basketball (if viewed
from the outside) from its concavity (if viewed from the inside), yet we can separate
them in thought and consider the one without considering the other. Likewise,
although physically there are no spheres tangential to a plane, we can say all
sorts of things about such shapes by imagination alone. A way to put this for
spatial points is that even if any magnitude of distance is a continuum, we can
still freeze motion along that distance (either in thought or by taking a quick
photograph of the object), and ask various questions about where the object is
at that frozen moment, e.g., about where it is "at that
point," how fast it is moving "at that point."
One observation about the paradoxes is that they seem to reduce to this
common denominator: Zeno is claiming (apparently truly) that an infinity of
tasks, each of which takes a finite amount of time, cannot be completed in a
finite amount of time. If this description does correctly distill the essence
of the paradoxes, then we see that the paradoxes are really about time, not
primarily, as they appear to be, about space. If so, then we can see that our
solution attempted above was doomed to failure, for we are apparently not going
to solve a temporal paradox by messing with our concept of space. Let us
instead investigate time.
Here we might note that Zeno gets away with the paradoxes only because he
treats time differently from the way he treats space. That is, we will run out
of time if we have a finite number of time points in which to perform an
infinity of tasks (in these cases spatial tasks), but if we can divide the
spatial magnitude infinitely, why can we not do the same with the temporal
magnitude? So a distance of one foot can be traversed in a time of one
minute, for although the foot contains an infinity of space points, the minute
contains an infinity of time points. Zeno would have us think that we run out
of time points, for he says that any allotted time span will expire by the time
the moving object reaches its goal (Dichotomy) or by the time Achilles catches
up to the tortoise (Achilles). But if there are an infinity of space points and
an infinity of time points, then we can have a one-to-one correlation between
each space point and time point (i.e., for each space point there will be a
time point), and we will never run out of time points.
At this point (!) it seems that we have beaten back the ghost of Zeno.
But what we have actually done, I fear, is to have unwittingly given him more
ammunition with which to make his case. For now Zeno can turn to us and say,
"By Zeus, not only is it
impossible to cross any quantity of space (because it is
infinitely divisible, and each little interval needs to be crossed), but you
have just shown me that it is impossible to cross any quantity of time.
That is, it is impossible for an interval of one minute to pass, because first
the mid-point must pass, and before that the mid-point of the mid-point must
pass, et cetera, and each little interval takes some time to pass, and
with an infinity of intervals, we would then need an infinity of time in order
to live through one minute. Since this is absurd, time must indeed not move,
which is to say that time does not exist. And this thrills me for it greatly
aids my Parmenidean cause; if there is no time, then of course there can be no
change or motion, and then we need not put up with any of that Heraclitean (or, more accurately, Cratylean)
claptrap."
What we took to be a clever insistence upon a parallel treatment of space
and time comes to be used against us as showing that matters are simply twice
as bad as we had thought. Hoisted with our own petard into the Parmenidean One.
What to do? Short of a full solution of the paradoxes, let me make a few
remarks.
I still want to treat the Dichotomy and the Achilles in the same general
way that I treat the Arrow and the Stadium. That is, I do not dismiss the Arrow
and the Stadium as mere verbal tomfoolery or linguistic hucksterism. I take the
paradoxes seriously, yet I resist the view that taking them seriously commits
me to agreeing with Zeno's conclusions about them. The conclusion that I do
draw from the Arrow and the Stadium is not that motion does not happen, but
that it happens over moments and not at a moment (Arrow), and
that distance traveled and velocity attained must be described relative to some
reference point, and this relativization cannot be omitted (Stadium). So what
lesson about motion am I to conclude that we should learn from the Dichotomy
and the Achilles? Something about motion, or time, or change?
I can't rightly say as of now. My suspicion, though, is that the solution
to these two most difficult of Zeno's paradoxes lies in the clarification of
infinite divisibility. In other words, to get a refined sense of what it is to
be infinitely divisible is to solve these paradoxes.
One observation is that a magnitude is not shown to be infinitely long
just because it is infinitely divisible. There are, then, two ways for a
magnitude to contain an infinity of points: to be infinitely long (that would
certainly do it) or to be finitely long but infinitely divisible. This does not
solve the paradoxes, for Zeno can agree with all this.
A theme common to the paradoxes is that there are tasks that can be
subdivided infinitely into sub-tasks, with each sub-task requiring a small but
finite amount of time. But what seems conceptually muddled about this is that
for something to be infinitely divisible is not for there to be a point
at which we have an infinite number of finite intervals. At any point in the
process of division we can have only a finite number of finite intervals (e.g.,
within one foot for there to be 1 million intervals each of 1 one-millionth of
a foot, or 1 million intervals with 999,999 having a combined magnitude of
S=1/2, 1/4, 1/8,..., and the remaining interval picking up whatever the
difference is between S and 1 foot).
After all, dividing infinitely cannot give us an infinity of segments,
for then we would still have some dividing to do, viz., dividing those
segments. In fact, we would have an infinity of dividing to do; infinity
- 1 gadzillian = infinity. One of the quirks of
infinity is that it is not true that there are twice as many positive
integers as there are odd positive integers; there are no more members of the
series 1,2,3,4,... than there are in the series 1,3,5,7,..., for we will never
run out of odd positive integers before we run out of positive integers.
Although this may make infinity seem either beastly or just plain silly, it is
actually a perfectly sane version of limitlessness; bewilderment comes from
thinking of infinity (wrongly) as a number ("just really, really big; you
know, with lots of zeros and commas"), and so as limited. Zeno, then, had
better be able to generate his paradoxes without talking of "an infinite
number of intervals."
"Infinite divisibility" must then be understood not as
"being divisible into an infinite number of parts," or even as the
equivalent (although sans the phrase "infinite number")
"being divisible into an infinity of parts," but rather as
"being forever divisible" or "being divisible without end."
In other words, what is infinite about infinite divisibility is not the number
of parts you get, but the process of dividing. It is the process, not the
result, that is infinite.
Impossible it is for us get a infinity of parts, segments, or intervals.
And so how Zeno dupes us is that he sets up a division impossible to complete
and then asks us what the outcome of completing it is. We are, of course, left
speechless. At which point Zeno leaps up saying, "See? No moving object
can get to its goal, and Achilles cannot catch up to the tortoise!" What
we must do somehow is resist becoming conceptually hogtied by trying to answer
Zeno's fundamental question, "What is the end product of a process of
division that can have no end?"
What I have not done is to offer a solution to the paradoxes. Even if
everything I have said here is true and relevant (mirabile dictu), it is
still open for Zeno to say, "Well and good, but isn't it still true that
in order for Achilles to catch the tortoise he will first have to get to where
the tortoise just was, and by that time the tortoise will have moved on to a
new point....?" And of course Zeno is right that I have not directly
addressed the paradoxes. So he wins ("for now," I somewhat
resentfully add), and these ramblings amount to mere grumblings.
Any thoughts?
Appendix on Zeno’s Paradox of the Arrow
When I discuss Zeno’s paradox of the
arrow (which Zeno offers as a different paradox than the Achilles or
Dichotomy), I have often represented the key to dissolving the paradox, and so
the lesson to be learned from the paradox, to be something like “Motion doesn’t
happen at a time, at a moment, at an instant, but rather –through
an interval of time, over a span of moments, over instants.”
I have now come to suspect, however,
that such a dissolution of the Arrow is too simple. Or at least that it is importantly
incomplete. Here is why.
Physicists, not a lot given to
sloppy thinking, find fully meaningful such questions as “What is the
instantaneous velocity of projectile P
at time t?” If this is meaningful, then I fear that the
way I had proposed for dissolving the Arrow paradox is illegitimate (or, again,
at least incomplete).
After all, to talk of an
instantaneous velocity is to talk of a velocity at a time, and to talk
of velocity at a time is to talk of traversing a distance at a
time. Yet how can distance be traversed at
an instant?
As McLaughlin indicates (“Resolving
Zeno’s Paradoxes,” Scientific American,
Nov. 1994, p. 88, sidebar), differential calculus and its use of limits are
needed in order to calculate any instantaneous velocity. So perhaps calculus is needed in order even
to understand instantaneous
velocity.
Here’s a try: one makes sense of talk of instantaneous
velocity by actually taking velocity over an interval of time (so, by
taking distance traversed over an interval of time—so far, so good), and then
making the interval thinner and thinner, infinitesimally, with the limit of
zero. So the thickness of each interval
would be 1/n as n→∞. Mighty
slim.
The rub, however, is that it’s this
last part that gets to me, for it resurrects the conceptual troubles with
making sense of infinitesimals. I know
those intervals are small, but how
small? If they’re infinitely small, then
taking a lot of them—even an infinity of them—wouldn’t be enough to get you a
definite (that is, non-zero) quantity.
Adding up a series of infinitely thin time slices wouldn’t be enough to
get you a loaf of time—not even a meager, crusty loaf.
And if the infinitesimal intervals
are not infinitely small, and so have
some real—even if teeny weeny—width to them, then an infinity of them would be infinitely large. Hence, even a ravenous Achilles could not
finish any loaf of bread, since each loaf would be infinitely large.
This indicates to me, however
unhappily, two things. First, although I
am quite happy to nod my head in agreement with Them That Know, insofar as I’ll
grant that the calculus is a wonderful calculation
device, I am prevented from thinking that calculus sheds any conceptual light on what’s going on
beneath the calculation. And it is this
conceptual cellar that Zeno occupies.
Second, I am now resigned to
thinking that the only satisfactory dissolution to the Arrow paradox would have
to be a satisfactory dissolution to the Achilles and Dichotomy paradoxes
too. The Arrow proves more difficult to
defend against than I had first thought.
Send me comments at mstaber at smcm
dot edu.
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