Zeno also has another paradox, that of the two arrows:
Consider a moving arrow as it would appear at an instant in time, and consider an arrow that is standing still as it would appear at that instant. The two ‘snapshots’ are identical. Thus (so Zeno argues) there is no difference between an arrow that is moving and one that is standing still, and the whole idea of motion is an illusion.
When the paradox is resolved, it is not necessarily proved that motion exists, only that Zeno’s reasoning does not prove that motion does not exist.
The description of the arrows that is given by Zeno does not distinguish between the still and the moving arrow. Thus, the alternative to admitting that he is right in saying that there is no difference between the two arrows, is to say that there is something wrong with his description, i.e. that giving the position of the arrow at a single time (a ‘snapshot’) is not a complete description of the arrow.
Indeed, there are two ways in which the description might be incomplete:
(1) An appropriate description may be one containing information about more than one time (e.g. a collection of snapshots, rather than just one).
(2) The description of an arrow at one instant of time may contain more than just information about position (i.e. you ignore relevant information if you just look at a snapshot).
Consider the intuitive notions of past, present and future. The present ‘feels’ special: the future seems indefinite, the past seems to live only through traces in the present. But what is it that distinguishes the present from the other two? There seem to be two possibilities:
(a) The present is indeed real and past and future are not. But that means presumably that the past was real, the future will be real, and the present is real now. But there is a problem with what is meant by "real now"
(b) Past, present and future are equally real. There is no absolute distinction between them, only a relative distinction between events that are earlier, simultaneous or later than each other. But is that enough to explain why the present feels special (or why time seems to flow)?
Pursuing option (1) above seems to commit one to option (b), in the sense that if the appropriate description of an arrow - one that makes motion real rather than illusory - includes information about the arrow at different times, then it would seem that these different times must all be equally real as well.
Something needs to be added to the instantaneous description of the arrow. This something will be the instantaneous velocity of the arrow. Intuitively, one might rather think of velocity as defined between two points (the distance between the two points divided by the time needed to travel between them). Indeed, it is also possible to talk about velocity as a property defined at one single instant. This is made possible by use of the calculus. And in fact, Newton, who was one of the inventors of the calculus, developed it precisely for the purpose of talking about instantaneous velocities (and instantaneous accelerations).
Imagine an object which is at point P at time t, and moves to Q by time s (say, moving in a straight line). Its (average) velocity v between the two points P and Q is
v = |P-Q| : |t-s|
where |P-Q| stands for the distance between the two points and |t-s| for the (positive) difference between the two times. Velocity in this sense is clearly a property of the object pertaining to more than one point or instant.
There is one special case, however, when velocity thus defined can be said to be a property of the object at any single time; namely, when the average velocity of the object is the same between any two points along its trajectory (uniform motion).
Assume that the object goes through the points P and Q, P’ and Q’, P’’ and Q’’ etc. (not necessarily in that order), at the times t and s, t’ and s’, etc., respectively. If for any pair of such points
v = |P-Q| : |t-s| = v’ = |P’-Q’| : |t’-s’| = v’’ = |P’’-Q’’| : |t’’-s’’| = ...,
it can be justified in being said that not only that the object has the same velocity between any two points, but that it has a velocity at all points and that this velocity is always the same.
The more difficult case is when these average velocities are not all equal, and this is where one needs the notion of limit defined above.
To define the instantaneous velocity at a time t (when the object is at point P), let us consider any infinite sequence of times t1, t2, t3, ..., converging to our given time t, and consider the corresponding points P1, P2, P3, ..., where the object is at those times. The average velocities can be calculated between P and P1, between P and P2, between P and P3 etc., thus obtaining a sequence
v1, v2, v3, ...
If this sequence has a limit v, this limit can be taken to be the velocity of the object at the time t. In the terminology of the calculus, taking such a limit of rates of change is called taking a derivative.
But now, there is a property of the arrow at a single time, namely the instantaneous velocity, which can distinguish between a moving arrow and an arrow that is standing still.
So we can resolve Zeno’s paradox by saying that Zeno is not giving us the complete description of the arrows at a single time; indeed, the information given on a snapshot of the arrows does not include their velocities, and is thus an insufficient basis for a distinction. Zeno is trying to convince us that there is no difference, but he has not given us all the information we are entitled to.