Zeno’s Paradoxes

An immediate concern is why Zeno is justified in assuming that the arrow is at rest during any instant. It follows immediately if one assumes that an instant lasts 0s: whatever speed the arrow has, it will get nowhere if it has no time at all. But what if one held that the smallest parts of time are finite—if tiny—so that a moving arrow might actually move some distance during an instant? One way of supporting the assumption—which requires reading quite a lot into the text—starts by assuming that instants are indivisible. Then suppose that an arrow actually moved during an instant. It would be at different locations at the start and end of the instant, which implies that the instant has a ‘start’ and an ‘end’, which in turn implies that it has at least two parts, and so is divisible, contrary to our assumption. (Note that this argument only establishes that nothing can move during an instant, not that instants cannot be finite.) So then, nothing moves during any instant, but time is entirely composed of instants, so nothing ever moves. A first response is to point out that determining the velocity of the arrow means dividing the distance traveled in some time by the length of that time. But—assuming from now on that instants have zero duration—this formula makes no sense in the case of an instant: the arrow travels 0m in the 0s the instant lasts, but 0/0 m/s is not any number at all. Thus it is fallacious to conclude from the fact that the arrow doesn’t travel any distance in an instant that it is at rest; whether it is in motion at an instant or not depends on whether it travels any distance in a finite interval that includes the instant in question. The answer is correct, but it carries the counter-intuitive implication that motion is not something that happens at any instant, but rather only over finite periods of time. Think about it this way: time, as we said, is composed only of instants. No distanc...

...e is traveled during any (View Highlight)

instant. So when does the arrow actually move? How does it get from one place to another at a later moment? There’s only one answer: the arrow gets from point at time 1 to point at time 2 simply in virtue of being at successive intermediate points at successive intermediate times—the arrow never changes its position during an instant but only over intervals composed of instants, by the occupation of different positions at different times. In Bergson’s memorable words—which he thought expressed an absurdity—‘movement is composed of immobilities’ (1911, 308): getting from to is a matter of occupying exactly one place in between at each instant (in the right order of course). For further discussion of this ‘at-at’ conception of time see Arntzenius (2000) and Salmon (2001, 23-4) (View Highlight)