This sounds straightforward on a casual hearing or reading, but once I sat down to parse its logical structure, I just couldn’t get it. The clearest description of it I’ve found so far is the one in Wesley Salmon’s introduction to this book:
In this paradox, Zeno argues that an arrow in flight is always at rest. At any given instant, he claims, the arrow is where it is, occupying a portion of space equal to itself. During the instant it cannot move, for that would require the instant to have parts, and an instant is by definition a minimal and indivisible element of time. If the arrow did move during the instant it would have to be in one place at one part of the instant, and in a different place at another part of the instant. Moreover, for the arrow to move during the instant would require that during the instant it must occupy a space larger than itself, for otherwise it has no room to move. As Russell says, “It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever”.
It seems that the following is sufficient for the paradox:
P1. An instant is a minimal and indivisible unit of time, i.e. it has no parts.
P2. If an arrow moves in an instant of time, it would have to be at different places at different parts of the instance.
C1. Therefore the arrow cannot move in an instant of time.
Everything after “Moreover” seems superfluous. Moreover, I have no idea how to make sense of that part. What could Zeno mean by an arrow occupying, or not occupying, a space equal to itself? It seems to make a mockery of the concept of occupancy to speak of something not occupying a space equal to itself, as though there is some sort of bubble around its physical shape that it occupies even though its physical shape does not completely fill that bubble. It seems to me that by the definition of occupancy, a body must occupy a space equal to itself, whether it is in motion or at rest. At any rate, how would occupying a space “larger than itself” allow a body room to move? It can hardly move into the space that it occupies, so having it occupy a space larger than itself doesn’t seem to give it any extra room. In any case, does Zeno expect that objects expand when they are in motion so as to give themselves room to move? I suspect this concept of occupancy, as it was originally intended, probably involves some inconceivably strange physics.
Did Zeno really mean that a body must occupy a space equal to itself? Aristotle’s account of the paradox in the Physics is:
Zeno’s reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.
Since I don’t have a copy of the Physics at hand, this was from a free online edition. I wonder if other editions have a clearer phrase than “an equal space”. It’s not clear to me that “an equal space” necessarily means “a space equal to the object itself”. Not that I can think of something else that it could plausibly mean, but as I explained, “a space equal to the object itself” is not plausible either.
Back, then, to what I consider the “main” part of the paradox. My first reaction, or at least my first reaction after reading the paradox closely, is that there is nothing paradoxical about C1. An instant of time has zero length, so of course an arrow does not move in an instant of time. Motion only applies to periods of time, not instants. It would not, for example, make sense to speak of an object that is at rest at all times except t=1, because that physical situation would be indistinguishable from the case when it is at rest at all times including t=1.
I’m cheating. There’s a further component to the argument. Namely:
P3: If the arrow cannot move in any instant of time, then the arrow is at rest at all times.
C2: Therefore the arrow is at rest at all times.
The obvious strategy is to attack P3. One way is to deny that there are such things as instants of time — in short, to deny an atomistic view of time. I can’t imagine how Zeno could salvage his paradox once it is allowed that time is infinitely divisible (of course, this itself he thinks is paradoxical, as he tries to show in his other paradoxes). If instants are denied, then nothing we say about these hypothetical instants can lead us to conclude anything about time in general.
P3 would also be undermined if it were true that motion is a concept that can only be applied to periods of time, per what I argued above. While it is true that we speak of motion at a certain moment, that is arguably a quirk of our language, and need not imply that we mean that an the object moves within the moment in question. What we mean when we say that is that at an arbitrarily small time interval after the moment in question, if we observe the object again, we will find it to be in a different position from where it was at that moment in question. The only way in which we detect bodies to be in motion is by observing the change in their positions at different moments of time, and hence over a non-zero period of time.
It is notable that even in modern calculus, all talk of rates of change is carefully formulated so as not to deal directly with change confined to one point on a function. The standard epsilon proof of a limit (and a derivative at a point is simply a special kind of limit) carefully avoids saying anything about what actually happens at the limit point, instead only referring to what happens as one gets arbitrarily close to the limit. This is why I don’t think calculus offers a solution to Zeno’s other paradoxes concerning the summation of an infinite series — it is just as clueless about what happens at the limit point as Zeno was. But that’s a topic for another post.
