I wrote about Zeno’s Achilles paradox – along with the Dichotomy (a.k.a. Racetrack) and the lesser know Plurality – here. You should read that before reading this. In that post, I didn’t mention Zeno’s other very famous paradox: the Arrow. Though a huge majority of philosophers think Zeno’s other paradoxes were cleared up by calculus and set theory, many consider the Arrow still unsolved. I’ll be arguing that the Arrow, though it appears to deal with separate issues, is really the same Achilles paradox in a different form.

Consider an arrow traveling down a trajectory. Take a snapshot at one static instant. In this one duration-less instant, the arrow is at rest. It is not moving, insofar as motion requires movement from one point to another, and this arrow is, in this one instant, in only one spot. So the arrow is at rest in one instant, at rest again in the next, at rest again in the next. If the arrow is at rest at every instant along its journey, how can we say that the arrow is in motion? When exactly does this arrow travel?

One aspect of the Arrow is simple enough to dissolve: being at rest, defined as “not being in motion,” only makes sense over some duration of time. Motion is *defined* as distance traveled over some period of time, so it makes no sense to even inquire as to whether motion is happening or not happening when we’re limited to a time range of 0. It is therefore not correct to say that the arrow is at rest in one duration-less instant. It is more accurate to say that it is in only one place. This answers one of the questions posed by the paradox: we would *not* say that the arrow is “in motion” during the single duration-less instant, though we would also not say that it is at rest. Without time, motion is simply not a meaningful question for it.

But this doesn’t resolve the paradox altogether. The question remains: if time is composed of duration-less instants, when exactly does the arrow travel? Look at it this way: at some instant in time T_{1}, the arrow is at some distinct point in space D_{1}. At a later instant in time T_{2}, the arrow is at point in space D_{2}. T_{1} and T_{2} are both duration-less instants, and all of the instants in between them are also duration-less instants. Time consists only of an infinite series of such duration-less instants. If all these instants are duration-less, when and how does the traveling actually happen?

Without questioning continuous space/time, I can imagine only one possible solution: accepting that motion is nothing more than what we perceive when an infinite series of static instances manifests one after the other. This leaves us with a stop motion booklet picture of movement. One static instant drops on top of another on top of another, each instant featuring objects in slightly different positions. In this image, what we normally consider to be the actual motion itself – the continuous or fluid animation gliding from instant to instant – is an illusion.

At first glance, this picture of motion does not seem all that tough to swallow, especially when we consider an important difference between the Arrow’s motion and the stop motion booklet: the booklet has gaps of time in between every image. During these gaps, there is no image. For Zeno’s arrow, however, though every instant is static, every instant is also *instantaneously* followed by the next instant, without intermittent gaps. This difference makes it intuitively easier to accept motion as nothing more than the succession of static instants.

Yet a problem remains. What do we mean by saying that every instant is *instantaneously* followed by the next instant? Does “instantaneously” mean that no time passes between an instant and the next? If that is the case, then time doesn’t move forward at all. The two adjacent instants are actually the same instant. So that can’t be our answer.

Does that mean there has to be some time in between adjacent instants? But then the instants are not adjacent after all. Some instant must be wedged in between. We could posit some kind of “in-between time” that isn’t composed of instants as being what lies in between adjacent instants. But what would be its metaphysical status? The possibility of a “time in between adjacent instants” seems too absurd to be worth pursuing.

Let’s look at the same problem from a slightly different angle. I said earlier that every static instant is followed instantaneously by *the next*. This cannot be strictly correct, since the series of static instants is – like the series of Achilles’s runs – infinitely divisible. No static instant has a distinct next instant. This is what causes the problem with analyzing the meaning of a moment being instantaneously followed by the next – there *is* no next.

To summarize thus far, Zeno’s Arrow suggests a picture of motion as being nothing more than *the instantaneous succession of distinct instants, each featuring objects at different spatial points, and in which each instant in the succession has no distinct next instant*.

Is this coherent? Isn’t it built into the definition of “succession” that each item in the succession is followed by a distinct next?

One possible answer is that it doesn’t have to be. The meaning of “succession” can be agnostic with regard to whether every item has to be followed by a distinct next. The rational numbers, for example, form a succession in which no member has a distinct next – there’s always an infinite number of members in between any two members of the succession. Even so, the rational numbers can still be considered a succession because, for any two items in the series, we can always say which comes before or after the other. Mathematicians call this a *densely ordered set*.

Here is the heart of the problem. Let’s accept that 1) time is composed of distinct instants; 2) time progresses forward; and 3) instants form a densely ordered set. From this it follows that time moves forward by *infinitesimal* increments – that is, increments which are infinitely small but not zero. But the infinitesimal exists only as an idea. When it comes to manifest reality, measurements are always either some finite number or zero. There is no third.

Applying the idea of a densely ordered set to time is squeezing progression and stasis into the same concept. It does not solve the problem. It merely gives it a name.

This should sound familiar. Analyzing the Achilles paradox led us to identify its source in the idea of densely ordered space: how can an infinite number of dimensionless points constitute a finite length? The analogous question can be asked here: how can an infinite number of duration-less instants constitute a finite duration?

This is because Achilles and the Arrow are really the same problem. Achilles comes down to: how can Achilles actually achieve an infinite number of distinct tasks? “Achievable” and “infinite” are mutually exclusive. We can frame the Arrow in the same way: how can the universe manifest an infinite number of distinct instants (within a finite duration)? “Manifest” and “infinite” are mutually exclusive.

It’s important to stress that the problem is not with infinity as a mathematical idea. Mathematical objects exist as such only in the mind. There is no problem with applying densely ordered sets, or other kinds of infinite sets, to purely mathematical objects (such as the rational numbers). It is when we apply them to the *physically manifest* that we get problems. It makes no sense to say of a physical event that it is both endless and completed. Using densely ordered space/time to solve Zeno’s paradoxes does just that.

Zeno’s Arrow is, therefore, unsolved for the same reasons that Achilles is unsolved. Like Achilles, it *would* be solved by discrete space/time, if that weren’t so counterintuitive. Both paradoxes are really variations on the fundamental problem of Zeno’s Plurality – that is, the problem of using densely ordered sets as our model for physical motion.

Other posts about Zeno’s paradoxes: