"There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end." The quotation, from Aristotle’s Physics, expresses the principle underlying Zeno’s parable of the race between Achilles and the tortoise. A tortoise challenges Achilles to a race, asking only that he be given a head start. Achilles scoffs at the suggestion but the tortoise explains that the race need not even take place because it would be impossible for Achilles to catch up with him once he had started running from his vantage point some way down the track.
"When you reach the point from which I started, you must admit that I will have run further still, even though it may be a small distance. When you have run that distance, I shall have run further still, and so on ad infinitum. "
At this point, Achilles should have insisted on running the race and proved empirically that the tortoise’s argument was false. This provides a clue to the counterintuitive idea that a fast runner cannot overtake a slow one, which contradicts our quotidian experience. If we stick to the tortoise’s a priori argument we are caught in a logical trap, but if we insist on an empirical refutation the status of logic is downgraded, and we must doubt the applicability of logical argument to physical events.
The tortoise’s argument is one way of defining the race but there are other ways that exclude the paradoxical trap. For example, Achilles could run the agreed distance L (which might have been 1 stadion = 185.4 metres) separately and the tortoise could then have run the shorter distance L/10 say. Timing both events would show that Achilles runs the full distance faster than the tortoise runs the shorter one, the conclusion being that Achilles must have won, without having to overtake the tortoise. The objection to this in those ancient times would be that there were no clocks capable of accurately measuring such short time spans. Zeno might have objected that this was no race at all, because the events took place at different times. If Achilles runs first, he obviously crosses the finishing line first, but if the tortoise goes first he wins if only because he crossed the finishing line before Achilles even starts to run.
So, we are forced to consider the mathematics of Zeno’s argument. We can suppose that the tortoise has a starting advantage of K < L which is sufficiently small to allow Achilles to catch up before the race is over. Careful consideration will show that K can always be set so the tortoise can win but that this is not the case here. The decreasing distances defined by Zeno’s argument can be precisely defined in terms of the relative velocities of Achilles and the tortoise. Let these be Va and Vt respectively. Achilles runs the distance K0 in T1 = K0/ Va. The distance run by the tortoise in time T1 will be K1 = T1. Vt = K0Vt / Va. A series of geometrically decreasing distances can now be defined as (K0 + K1 + K2 + …+Kn) = K0Σn (Vt / Va)n. Since Vt < Va the sum will converge to an infinite limit, which is the point where Achilles catches up with the tortoise being D = K0Va/(Va-Vt).
We can now calculate the time taken by Achilles to cover the distance D which is TD = D/Va. At any time between D/Va and Achilles’ finishing time L/Va, we can claim that the tortoise has been overtaken. This provides an a priori rather than an empirical refutation of Zeno’s argument. However, the objection remains that an infinite summation of the decreasing differences is impossible in the physical world. A way out of this difficulty is to ignore the race track altogether
and consider the relative velocity of Achilles and the tortoise, which is Va – Vt. The time taken by Achilles to close the initial gap K is therefore K0/( Va – Vt) so the distance D = K0Va/( Va – Vt) which is the same formula derived using the infinite summation.
The cleverness or inadequacy of Zeno’s argument is that it is an incomplete model which precludes consideration of the race as a whole, focussing only on the vanishingly small distances between the participants. The argument is a petitio principii in that it assumes in advance that motion is impossible and creates a model that favours this assumption. The existence of other theoretical models which do conform to everyday experience and the intuitions based on it resolve the apparent paradox by rejecting Zeno’s model.
Water clocks existed in Zeno’s time, so the timing of a race by this means should not be ruled out. However, modern circular clocks do provide continual, empirical refutations of Zeno’s argument. The minute hand of a clock does overtake the hour hand several times a day, but the question is: at what time do the hour and minute hands of a clock coincide?
The method of calculation is similar to the one used to find the crossover point in Zeno’s race. The minute hand of the clock completes one revolution in the time it takes the hour hand to move the five minute division between one hour and the next. The relative velocity is therefore 1/12. The first coincidence after 12 O’clock is 12/11 hours = 1h 5m 27.27 sec. Peculiarly there will only be 11 unique coincidences, with no coincidence between 10h 54m 32.72 sec and 12h 0m 0sec. Since the coincidences are independent of the clock face divisions, the latter could be dispensed with. The result would be a clock which divides the diurnal cycle into 22 equal periods.
If a typical clock is observed closely, the minute hand appears to move in one second jerks rather than smoothly. The implication is that it cannot exactly coincide with the hour hand at any time other than 12 O’clock, since there are no other coincidences exactly measured in whole seconds. In this sense, Zeno would be correct in asserting that the minute hand cannot overtake the hour hand at such a coincidental point but would be wrong in concluding that it could not do so after that point. This is because we assume that ‘time’ is a continuous process, rather than more correctly realising it is the nature of the physical process that defines how time is to be interpreted. If physical process involving movement are small but discontinuous, they can be regarded as almost continuous because the smallness of the Planck constant.
The movements of Achilles and the tortoise can be regarded as individual steps, or strides. When Achilles has run the overtaking distance D, he may be in mid stride, between and n and n +1 strides, in which case he will overtake the tortoise on completion of the nth stride. His stride cannot exactly coincide with the point D unless Va is exactly divisible by Va – Vt . If it is, then he must complete n +1 strides before we can claim that he has overtaken the tortoise. In either case, he is bound to win the race unless Paris shoots him in the heel with an arrow first, but Zeno’s argument would also rule out this possibility too. We can conclude from this that Zeno did not believe Homer’s tale or indeed any myths where mythological figures get pierced by arrows. This must have been comforting since on his theory he could never be hit by a missile of any kind.
The idea of a limit that cannot be transcended appears in Einstein’s Theory of Relativity. He argued that, as the velocity of a mass approaches the speed of light, the energy required to accelerate the mass approaches infinity. Since the speed of light is a constant, he concluded that the mass must
increase without limit, which is impossible. Like Zeno’s argument, this mathematical proof seems counterintuitive, but empirically true according to certain scientific observations. Using the earlier Newtonian model, there appears to be no reason to doubt that a mass could reach the speed of light and possibly exceed it. The point is that different theories may lead to different truths. Putting it the other way round, two theories cannot be different and non-contradictory.
Einstein’s formula below, which relates the mass of a moving body to its rest mass, also uses the idea of a limit that cannot be exceeded because of an infinite process. The interpretation is that a moving mass increases without limit as its velocity approaches the finite limit of the speed of light. If we switch the rest mass m0 in the formula with the moving mass m, the result would be a different theory where the moving mass decreases to zero as its velocity increases towards the speed of light.
Zeno’s interpretation was based on his philosophical beliefs. His teacher Parmenides believed that time was an illusion because the universe was a permanent whole that never changed. From this point of view we experience change because our consciousness is incapable of observing the absolute nature of the universe. This approach is similar to Einstein’s four dimensional space time continuum, where events are frozen by regarding time as another spatial dimension. Even the simple Cartesian diagram of parabolic motion illustrates the principle of representing time as a spatial dimension, so Einstein’s formulation was just an extension of this type of description to a fourth dimension, transcending our usual mode of experience.
Zeno’s paradox of Achilles and the tortoise relied upon generating an infinite process which frustrated finite resolution, at least until mathematicians devised a satisfactory theory of limits. When it came to Cantor’s thoroughgoing examination of infinity, a great many paradoxes arose which had to be dealt with. Behind his approach to the theory of infinity lay the kind of absolutism espoused by Parmenides, which was a belief in the existence of absolute infinity rather than Aristotle’s comparatively timid potential infinity. Cantor’s achievement was to show that a hierarchy of infinities had to be defined as a consequence of his set theoretic approach. One of his best known achievements was to prove that the set of irrational numbers could not be counted by the natural numbers. He did this by means of his diagonal argument.
Zeno’s argument was Aristotelian in character, relying on the principle that it was impossible to add up all the increasingly tiny distances that Achilles had to run to catch up with the tortoise. Cantor’s starting point was to assume that the set of natural numbers, like any set, must have a totality, which he called Aleph null. Like Zeno’s argument, Cantor covertly employs the trick of introducing a limit that cannot logically be transcended, to wit the diagonal of digits in a finite square adopted for the purpose of demonstration. This involves the belief that the square can be expanded as far as Omega (the ordinal version of Aleph null) thus preserving the property of excluding certain combinations of digits.
Once the trap is accepted, that the square cannot be greater than the arbitrary number Omega, the result follows that the power set of the natural numbers is non-denumerable. Since the list of irrational numbers has been defined as a permutational power set, the argument degenerates into a
petitio principii. The point being that there is no diagonal associated with a non-square list, so the whole argument is a tautology signifying nothing. While cantor’s diagonal argument is logically sound, like Zeno’s, the model he constructed was erroneous. The consequence is a flawed set theory riddled with paradoxes.
The general issue is the extent to which we should accept counter intuitive arguments on the basis of special definitions which purport to render them logically sound. In the case of Zeno’s imaginary race, this should not be done without careful examination of alternative models. The corollary is that sound logical argument is not a sufficient criterion for truth, since the devil lies in the details hidden in the definitions and covert assumptions lying beneath the surface.