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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?

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Calculus is the solution. It's not even that modern. Zeno assumes (without saying so) that time is discrete, and observing the world demonstrates that this is pure nonsense. –  dmckee Mar 8 at 16:25
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I don't see a paradox. The arrow is at rest in its own frame, but that doesn't mean that the ground is at rest. In fact, the ground is what is moving according to the arrow. –  Kyle Kanos Mar 8 at 16:48
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I disagree with Kyle Kanos's view of what Zeno intended to show. As far as I know, Zeno's arrow paradox was intended to show that the arrow cannot move in our frame of reference. Zeno wouldn't have referred to "frame of reference" because that concept wasn't isolated until later, but, precisely because the notion of varying the frame of reference wasn't yet under discussion, what Zeno said in "absolute" terms should be understood by us modern folks as being about our frame of reference. In other words, Zeno's paradox is resolved by calculus, not by considering other reference frames. –  Andreas Blass Mar 8 at 17:50

2 Answers 2

up vote 8 down vote accepted

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.

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So, velocity has nothing to do with the question if the object is in motion at some instant or not. Right? –  Samama Fahim Mar 8 at 19:13
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That depends on your definition of "in motion." The sensible definition says velocity is nonzero if and only if the object is in motion. Zeno's definition of "in motion" isn't defined for a single instant. –  Chris White Mar 8 at 19:24

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

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