To quote from this wikipedia article:
In the arrow paradox... Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in anyone (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, 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.
How is this resolved? It seems a little bit nonsensical but I do get the logic. This has me very confused...

axmls
The explanation I usually hear is because now we know with calculus that an infinite sum can still be finite.

Robert Moore
The argument rests on an ambiguity regarding time itself: does motion beget (entail) time, or does time beget (entail) motion? If the arrow is described as occupying a duration-less "instant" (it would have to have zero duration if it is to be motionless), and if all time is made up of such "instants," we are given to believe that time is always at a stand-still. In other words, time becomes an infinite sum of absolute zeros. So it is like saying that nothing can move because time is just an infinite sequence of zero-time instants. If we say that time never elapses, then naturally nothing can move, but that seems to be entering a circular argument: time cannot elapse because if it does, then motion occurs; motion cannot occur because time never elapses. In other words, Zeno has subtly assumed that time never elapses (hence the arrow never moves).

Calculus helps us see that we can reduce the time interval to shorter and shorter amounts while the rate of motion (change in position divided by change in time) attains a stable value (unaffected by the size of the time interval). Thus, even as the interval approaches zero in the limit, the rate of motion is finite (non-zero). This is far from solving all the mysteries of time and motion, but it is much more satisfactory (and practical) than Zeno's approach.

Isaac0427
Staff Emeritus
Homework Helper
The explanation I usually hear is because now we know with calculus that an infinite sum can still be finite.

That's the other paradox with the arrow.

The argument rests on an ambiguity regarding time itself: does motion beget (entail) time, or does time beget (entail) motion? If the arrow is described as occupying a duration-less "instant" (it would have to have zero duration if it is to be motionless), and if all time is made up of such "instants," we are given to believe that time is always at a stand-still. In other words, time becomes an infinite sum of absolute zeros. So it is like saying that nothing can move because time is just an infinite sequence of zero-time instants. If we say that time never elapses, then naturally nothing can move, but that seems to be entering a circular argument: time cannot elapse because if it does, then motion occurs; motion cannot occur because time never elapses. In other words, Zeno has subtly assumed that time never elapses (hence the arrow never moves).

Calculus helps us see that we can reduce the time interval to shorter and shorter amounts while the rate of motion (change in position divided by change in time) attains a stable value (unaffected by the size of the time interval). Thus, even as the interval approaches zero in the limit, the rate of motion is finite (non-zero). This is far from solving all the mysteries of time and motion, but it is much more satisfactory (and practical) than Zeno's approach.
So is the resolution that velocity can be expressed as dx/dt instead of x/t?

Mentor
How is this resolved?
The average amount of motion over some finite time interval, ##\Delta t## is $$\frac{x(t+\Delta t)-x(t)}{\Delta t}$$. This quantity does not necessarily go to 0 as ##\Delta t## goes to 0.

Robert Moore
Yes, I think that's the resolution from a mathematical and physical point of view.

For historical background, Zeno was trying to prove that change is an illusion. He was apparently a follower of the ancient Greek philosopher Parmenides, who taught that reality is unchanging and, if we seem to perceive change, it is only an illusion. Zeno believed that his proofs of the impossibility of motion and change invalidated our naive observations of the world and that pure reason can overcome this naivete and restore our faith in a static and motionless state of being.

The ancient champion of the opposite view was Heraclitus, who said that the only reality is change itself. He famously said that you can never step into the same river twice (since it is perpetually changing).

Plato is said to have synthesized (harmonized) these two thinkers by saying that they were really referring to two different worlds: the world of change is the one accessible by the senses, the unchanging world is the one accessible through reason. The sensory world is, to Plato, only made up of shadows of the "higher" world, which is filled with his "Ideals," perfect models of reality that cast shadows into our unstable (lower) world.