Zeno’s Paradox of the Arrow

Question

Premises And the Conclusion of the Paradox: (1) When the arrow is in a place just its own size, it’s at rest. (2) At every moment of its flight, the arrow is in a place just its own size. (3) Therefore, at every moment of its flight, the arrow is at rest.

If something is at rest, it certainly has $0$ or no velocity. So, in modern terms, what the paradox says is that the velocity of the arrow in "motion" at any instant $t$ (a duration-less duration) of time is '$0$'.

I read a solution to this logical paradox. I do not remember who proposed it, but the solution was something like this:

Let the average velocity of the arrow be the ratio $$\frac{\Delta s}{\Delta t}.$$ Where $\Delta s$ is a 'finite' interval of distance, travelled over a finite duration $\Delta t$ of time. Because an instant is duration-less, and no distance is travelled during the instant, therefore $$\frac{\Delta s}{\Delta t}=\frac{0}{0}$$ or, $$0 \cdot \Delta s=0 \cdot \Delta t.$$ In other words, the velocity at an instant is indeterminate, because the equation above has no unique solution.

This solution denies the concept of a 'definite' instantaneous velocity at some instant $t$ given by the limit of the ratio $\frac{\Delta s}{\Delta t}$ or $$v(t)= \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}.$$

Whatever be the velocity at some instant $t$, how does the above "definition of the instantaneous velocity" or the calculus tell us that the arrow or any other object in motion is moving at an instant? How can something move in a duration-less instant, when it has no time to move? What is the standard modern science solution to understand this logical paradox?

Answer 1

Like all paradoxes, there is no contradiction here, just misuse of logic.

How do you define velocity? If you say

the distance traveled in an extended period of time, divided by that time

well then of course there's no such thing as instantaneous velocity. Asking what something's instantaneous velocity is under this definition is logically equivalent to something like

Let $n$ be the number of apples in a nonempty container of apples. What is $n$ when the container has no apples?

The question doesn't make sense, and simply cannot be answered.

---

Now one can often extend definitions so that terms get defined in new circumstances, consistent with the cases for which they were previously defined. We define velocity as $$ v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}. $$ This is consistent with the old in the sense that if you have a constant velocity $v$ and you travel for an extended period of time, $v$ is given by distance divided by time.

But make no mistake, our new definition goes beyond cases of extended time intervals, and in these cases the old definition still fails, just as it always did. Sure no motion occurs if no time elapses. So what? If no time elapses, the definition of velocity has nothing to do with actual distance traveled over that time.

Some object may have a nonzero velocity because our new definition of velocity says it does, whereas the old definition may have had nothing to say one way or the other. Make no mistake, the old definition does not say the velocity of an object is $0$ if no time elapses. It says the velocity of an object is currently undefined if no time elapses.

Answer 2

How can something move in a duration-less instant, when it has no time to move?

According to elementary calculus, if the position of the arrow is given by the function $x(t)$, then

$$x(t + dt) = x(t) + v(t)dt$$

This defines the instantaneous velocity: it is the ratio of the infinitesimal$^\dagger$ change in position to the infinitesimal change in time

$$v(t) = \frac{dx}{dt} $$

Contrast this with

$$x(t + 0) = x(t) $$

and note that an infinitesimal displacement in space or time is not identical to a zero displacement or "duration-less instant".

$\dagger$ An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity

Answer 3

Since the definition of Velocity is the change in distance traveled divided by the change in time elapsed, and since there is no change in either distance or time ($dx=0$, $dt=0$), Velocity is simply undefined mathematically at a point.

In practical terms this means of course that there never is a instant when the arrow is not moving, so $dx$ is undefined, and so is $dt$ given. The concept of velocity can therefore not be defined at an instant, only over a period of time. There is no paradox, and the question itself is incorrectly phrased.

Momentum is of course defined at any time the object is moving, but is 0 when it is at rest. To argue about "instant velocity" is to argue that momentum is 0 at any/every given point in the object's path of motion. This is of course, ridiculous.

Answer 4

It is a long time since this question is asked. But I see many confusion in the answers. In fact it is simple. The aporia proves that the arrow is at rest but of course in its own system. It omits the relativity of motion. An object is always at rest in its own reference system but not in all reference systems. So in a system of Earth (appr. inertial) it is moving.

Answer 5

The paradox is nonsense. When you see a fast exposure photograph of a moving car it seems to be motionless. So Zeno's paradox effectively asks 'if the car was motionless for the instant that the photo was taken, and if time is just a succession of instants in which the car is motionless, then how does the car move? How does a succession of zero movements ad up to a non-zero movement?'

The answer is that the car is not motionless while the photograph is being taken- it moves by an unnoticeable but non-zero amount. If you half the exposure time, you half the distance the car moves, but it still moves. You can keep decreasing the time interval, and the car still moves by a decreasing amount, but the ratio of the distance moved divided by the time interval remains constant. You can take the time duration as close to zero as you like, and the car still moves some distance during the interval.