How is this resolved? It seems a little bit nonsensical but I do get the logic. This has me very confused...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.
axmls said:The explanation I usually hear is because now we know with calculus that an infinite sum can still be finite.
So is the resolution that velocity can be expressed as dx/dt instead of x/t?Robert Moore said: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.
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.Isaac0427 said:How is this resolved?
Zeno's Arrow Paradox is a philosophical thought experiment that questions the possibility of motion. It was proposed by the ancient Greek philosopher Zeno of Elea in the 5th century BC.
The paradox states that in order for an arrow to reach its target, it must first reach half the distance, then half of the remaining distance, and so on infinitely. Therefore, it can never actually reach its target since there will always be a smaller distance to cover.
There are several proposed solutions to the paradox, including the concept of infinite divisibility and the use of calculus. However, the most widely accepted solution is the concept of "actual infinity," which states that an infinite number of smaller distances can be traversed in a finite amount of time.
Zeno's Arrow Paradox has been used to challenge and explore the concepts of time, space, and motion in modern physics. It has also influenced the development of calculus and the concept of infinity.
Yes, Zeno's Arrow Paradox is still discussed and debated by philosophers and scientists today. It continues to challenge our understanding of motion and the nature of reality.