Beyond Infinity? Russell and Zeno

The paradoxes of motion created by Zeno of Elea (ca. 490-ca. 430 B.C)(1)  have posed some of philosophy’s most intriguing and durable problems. Thinkers from Aristotle onwards have sought to answer Zeno’s riddles, which according to Bertrand Russell have “…afforded grounds for almost all theories of space and time which have been constructed from his day to ours.”(2)  Russell, for his part, believed the paradoxes to have been adequately resolved by nineteenth century mathematics, outlining what he saw to be their solution in his 1914 work Our Knowledge of the External World.(3)  This essay will discuss Russell’s analysis of the paradoxes, and will contend that despite his conviction that mathematicians had at last discovered their “true solution,”(4)  Russell’s argument is not ultimately the definitive refutation of Zeno’s paradoxes he believed it to be.

To begin, it is important to understand exactly what the paradoxes were intended to prove. It is generally believed that Zeno created the paradoxes in order to defend his teacher Parmenides, who was often derided by his contemporaries for teaching that all reality is a single, unchanging whole, and that all our perceptions of motion, space and time are illusory.(5)  Zeno wished to defend his master by showing that it was in fact his opponent’s views of reality which led to the greater absurdity. The paradoxes were therefore intended to vindicate Parmenides by proving that motion is logically impossible, and therefore illusory.(6)  They are known respectively as The Dichotomy, The Achilles, The Arrow and The Stadium.

Importantly, not all of the paradoxes are thought to argue against the same views of motion. It has been suggested that while The Dichotomy and The Achilles argue against an infinitely divisible, continuous model of space and time, The Arrow and The Stadium(7) argue against the opposing view that space and time are discrete.(8) Russell for the most part agreed with this, but also indicated that he in fact considered all four paradoxes to be “perfectly valid”(9)  if pitted against a discrete conception of motion. It is therefore only Russell’s analysis of The Dichotomy and The Achilles that I will focus on here, as he clearly wished to argue for a continuous, infinitely divisible model of space and time,(10)  and agreed that it is only these two paradoxes which argue against this model.(11)

The Dichotomy and The Achilles are essentially two forms of the same paradox. The first states that there is no motion, because that which moves must first travel half the remaining distance before it arrives at it’s goal, and so on ad infinitum. The second states that the quicker will never overtake the slower, because that which is pursuing must first reach the point from which that which is fleeing started, in which time the slower will have moved further ahead, and so on ad infinitum.(12)  The problem is essentially the same in both: that if space is continuous and infinitely divisible, one must traverse an infinite number of points one by one in order to reach a goal, which cannot be done.

There is much in these arguments that Russell accepted. He believed, as I have mentioned, in the infinite divisibility of space. He was also willing to accept that for the runner to reach his goal, he would first have to reach an infinite number of middle points.(13)  He differed from Zeno of course, in thinking that this could be done, yet I will contend that his arguments towards proving this were largely based on a misinterpretation and underestimation of the paradoxes.

In discussing The Racetrack and The Achilles, Russell interprets Zeno to be arguing that because the runner must traverse an infinite number of finite distances, and each traversal will take a finite amount of time, it will therefore take the runner an infinite amount of time to reach his goal, because “…the sum of an infinite number of finite times must be infinite, and therefore the process will never be completed.“(14)  In this, he purports to find error in Zeno’s reasoning, because, as he claims, an infinite number of finite times do not necessarily make up an infinitely long time. Russell proves this by showing that the infinite number of finite fractions which make up the series ½, ¾, 7/8, 15/16… fits entirely between 0 and 1,(15)  and therefore an infinite number of time periods, each greater than zero, do not necessarily sum to an infinitely large total. This means that while it does take the runner an infinite number of finite times to reach the end, it does not take him an infinitely long time. He concludes that “…the view that an infinite number of instants make up an infinitely long time is not true, and therefore the conclusion that Achilles will never overtake the tortoise does not follow.“(16)

Yet here Zeno’s argument is couched in terms which do not convey its true strength. Russell refutes the view that an infinite number of finite time periods must make up an infinite length of time, yet the paradoxes are not necessarily dependent upon such a view. They hinge rather on the much simpler and more powerful argument that motion and time, if imagined as a succession through an infinitely divisible continuum, both require never-ending processes to be completed, and therefore both imply contradictions. The series ½, ¾, 7/8, 15/16… can never reach 1, because every possible bisection, no matter how minute, still leaves a finite fraction yet to be traversed. This applies equally to time and motion, for just as the runner must traverse infinite points one by one in order to reach his goal, the time must traverse infinite moments one by one to reach the final moment. Thus, the true core of the paradox is not that it will take the runner forever to reach his goal, but simply that if the runner does reach his goal he will have progressed one by one through infinite points and moments, and will therefore have completed a never-ending process, which is logically impossible.(17)

Russell’s oversight seems to be that he does not realize or acknowledge that continuous time is equally as vulnerable to the paradoxes as continuous motion. In fact he seems to simply assume that continuous, infinitely divisible time is self-evidently possible, defining the position of a moving body as “a continuous function of time.”(18)  He writes: “If half the course takes half a minute, and the next quarter takes a quarter of a minute, and so on, the whole course will take a minute.”(19)  What he argues here is essentially that because both space and time are infinitely divisible, there will be a corresponding moment for every one of the infinite points the runner moves through. The runner will therefore reach his goal simply when the time reaches the appropriate corresponding moment. Carefully considered, it is easy to see why this response is inadequate.

As we have seen, the crux of the paradoxes is that the concept of continuous motion and time presupposes the completion of never-ending processes. The paradoxes demand an explanation as to how such never-ending processes could be completed. Russell sidesteps this however, arguing that infinite processes simply must be able to be completed, because according to his definition of motion the runner will reach the end when the time reaches 1 minute, and thus will have traversed infinite points. Yet this does not address the key concern, as Russell’s definition of motion is precisely what is being questioned by the paradoxes, because it presupposes the completion of such never-ending processes.

On this deeper question of whether or not an infinite series, such as ½, ¾, 7/8, 15/16… can ever be completed, Russell is ambiguous. He first writes that “…the points are reached one by one, and, though they are infinite in number, they are in fact all reached in a finite time.”(20)  Yet ten pages later he admits “If you set to work to count the terms of an infinite collection, you will never have completed your task.”(21)  There must then, exist an important difference between ‘reaching’ an infinite number of points and ‘counting’ to an infinite number of points, yet what this difference may be is not made clear.

By way of explanation, Russell offers that “…it is not essential to the existence of a collection, or even to knowledge and reasoning concerning it, that we should be able to pass its terms in review one by one.”(22)  Yet after admitting ten pages earlier that his conception of motion does in fact require infinite points to be reached one by one, Russell’s argument is now soundly unconvincing. Zeno’s paradoxes do not question the ‘existence’ of the collection ½, ¾, 7/8, 15/16…, nor do they question whether we can have ‘knowledge and reasoning concerning it’. What they question is whether one can reach all of its terms one by one, and it is ultimately here, at the heart of the paradox, where Russell is found without a valid response.

As we have seen, the crux of the paradoxes is that if the runner reaches 1, he will have travelled beyond the end of a never-ending series, which is logically impossible. Several modern writers have derived thought experiments in order to show the difficulty of postulating the completion of an unending series. One of these was James Thompson, who imagined a lamp which is turned on after ½ a minute, then off again after another ¼ of a minute, then on again after 1/8 of a minute and so on, ad infinitum.(23)  Thompson then asked after 1 minute has passed, will the lamp be on or off? It cannot be on, because for every time it was turned on it was subsequently turned off before the minute had elapsed. Yet it likewise cannot be off, because every time it was turned off, it was turned on again before the minute had elapsed as well. Thompson’s point is that discussing the completion of such infinite processes leads to an absurdity.(24)

Wesley Salmon responded to Thompson in his Space, Time and Motion. He countered that the lamp might be either on or off after 1 minute has passed. We have no way of knowing, because Thompson has only provided information about whether it is on or off at any point prior to 1, and not at 1 itself.(25)  However, while this argument seems sound, it appears to simultaneously undermine the notion that 1 can be reached through such infinite bisections. The reason Thompson’s instructions do not provide information about what happens at 1 is precisely because the series ½, ¾, 7/8, 15/16… never reaches 1. That is exactly what Zeno’s paradoxes have argued for two and a half millennia, and was doubtlessly the point Thompson wished to make.

Salmon’s reply to this is that while Thompson’s ‘switching function’ provides no information about the lamp’s position at the 1 minute mark, the runners ‘motion function’, (i.e. his position at various points in time), “…provides a suitably appealing answer to the question about the location of the runner at the conclusion of his sequence of runs.“(26)  Salmon, like Russell before him, is effectively saying that according to the continuous model of space and time, which presupposes infinite processes being completed, the runner will reach 1 when the time reaches 1, thus after the ‘conclusion’ of an infinite series of runs. Consequently, infinite processes can be completed, and the continuous model of space and time is free from contradiction. Salmon employs the same intellectual sleight of hand as Russell here, effectively using the continuous model of space and time to prove itself, and ignoring the core logical problem of the paradoxes. Such an argument is not altogether much more sophisticated than that offered by Diogenes the Cynic, whose refutation of Zeno is said to have consisted of silently standing and walking across the room.(27)

Thus, notwithstanding Russell’s bold proclamations that the “true solution”(28)   had at long last been found by mathematics, it seems his attempts to rescue continuous, infinitely divisible space and time from the clutches of The Dichotomy and The Achilles were ultimately unsuccessful. Yet perhaps we should not be surprised. As Alfred North Whitehead remarked, “To be refuted in every century after you have written is the acme of triumph…No one ever touched Zeno without refuting him, and every century thinks it worthwhile to refute him.”(29) In fact, the dawning century looks like it may well refute Zeno all over again in an entirely different manner. It has been suggested that quantum physics will ultimately prove that space and time are not infinitely divisible after all, but are ultimately discrete at the smallest scale.(31)   This of course was the view attacked by Zeno with his latter two paradoxes of motion, The Arrow and The Stadium. Exactly where ‘Quantum Physics’ vs. The Arrow and The Stadium will lead us is uncertain, yet one thing seems assured. It will be an intriguing duel.

(1) Roy Sorenson, A Brief History of the Paradox, Oxford University Press, New York, 2005, p.45
(2)Bertrand Russell, Our Knowledge of The External World, Allen & Unwin, London, 1961, p.183
(3)Ibid, pp.159-188
(4)Ibid, p.164
(5)Ibid, p.174
(6)Wesley Salmon, Space, Time and Motion, Dickenson Publishing Co., California, 1975, p.31
(7)Although The Arrow and The Stadium are more difficult to grasp than The Dichotomy and The Achilles, they are well explained at http://www.mathpages.com/rr/s3-07/3-07.htm (last viewed at 19/06/07 1am) which summarizes them as follows: The Arrow: If everything is either at rest or moving when it occupies a space equal to itself, while the object moved is always in the instant, a moving arrow is unmoved. The Stadium: Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time
(8)Salmon, Space, Time, and Motion, Op cit, p.35
(9)Russell, Our Knowledge, Op cit, p.174
(10)Ibid, pp.135-158
(11)Ibid, p.174
(12)http://www.mathpages.com/rr/s3-07/3-07.htm at 19/06/07 1.09am
(13)Russell, Our Knowledge, Op cit, p.177
(14)Ibid.
(15)Ibid, 178
(16)Ibid.
(17) http://en.wikipedia.org/wiki/Zeno’s_paradoxes at 20/06/07, 11.54pm
(18)Russell, Our Knowledge, Op cit, p.142
(19)Ibid, p.177
(20)Russell, Op cit, p.177
(21)Ibid, p.187
(22)Ibid.
(23)Sorenson, A Brief History, Op cit, p.55
(24)Salmon, Space, Time, and Motion, Op cit, p.44
(25)Ibid, p.46
(26)Ibid, p.46
(27)Simplicius, On Aristotle’s Physics, 102.22, http://plato.stanford.edu/entries/paradox-zeno/ at 19/06/07, 2.00am
(28)Russell, Our Knowledge, Op cit, p.164
(29)Alfred North Whitehead, Essays in Science and Philosophy, (New York: Philosophical Library, 1947), p.114
http://en.wikipedia.org/wiki/Spacetime at 20/06/07, 12.14pm