Zeno's paradoxes

Zeno of Elea's arguments against motion precipitated a crisis in Greek thought. They are presented as four arguments in the form of paradoxes : (1) the Racecourse, or dichotomy paradox, (2) Achilles and the Tortoise, (3) the Arrow, and (4) the Moving Blocks, or Stadium.

1 Suppose a runner needs to travel from a start S to a finish F. To do this he must first travel to the midpoint, M, and thence to F: but if N is the midpoint of SM, he must first travel to N, and so on ad infinitum (Zeno: ‘what has been said once can always be repeated’). But it is impossible to accomplish an infinite number of tasks in a finite time. Therefore the runner cannot complete (or start) his journey.

2 Achilles runs a race with a tortoise, who has a start of n metres. Suppose the tortoise runs one-tenth as fast as Achilles. Then by the time Achilles has reached the tortoise's starting-point, the tortoise is n/10 metres ahead. By the time Achilles has reached that point, the tortoise is n/100 metres ahead, and so on ad infinitum . So Achilles cannot catch the tortoise.

3 An arrow cannot move at a place at which it is not. But neither can it move at a place at which it is. But a flying arrow is always at the place at which it is. That is, at any instant it is at rest. But if at no instant is it moving, then it is always at rest.

4 Suppose three equal blocks, A, B, C, of width l, with A and C moving past B at the same speed in opposite directions. Then A takes one time, t, to traverse the width of B, but half the time, t/2, to traverse the width of C. But these are the same length, l. So A takes both t and t/2 to traverse the distance l. These are the barest forms of the arguments, and different suggestions have been made as to how Zeno might have supported them (for one version, see Bayle's trilemma ). A modern approach might be inclined to dismiss them as superficial, since we are familiar with the mathematical ideas (a) that an infinite series can have a finite sum, which may appear to dispose of (1) and (2), and (b) that there is indeed no such thing as velocity at a point or instant, for velocity is defined only over intervals of time and distance, which may seem to dispose of (3). The fourth paradox seems merely amusing, unless Zeno had in mind that the length l is thought of as a smallest unit of distance (a quantum of space) and that each of A and C are travelling so that they traverse the smallest space in the smallest time. On these assumptions there is a contradiction, for A passes C in half the proposed smallest time. The purely mathematical response only works if we have a satisfactory foundation not only for the arithmetic of infinity but also for the measurement of space and time by its means. The real importance of the paradoxes has lain in the pressure they put on those foundations. For instance, the third paradox suggests that if we are happy to treat a line as made up of extensionless points, and time as made up of instants that occupy no time, then motion is a succession of states of rest. The difficulty with using the fact that an infinite series can have a finite sum as a sufficient solution of the paradoxes has been brought out by considering a lamp set to go on for half a minute, go off for a quarter, on for an eighth…. At the end of the minute, is it on or off ? Neither answer is mathematically acceptable, since there is no last member of the series. So it seems that there can be no such lamp, yet it also seems to be an accurate model of Achilles' completed journey.