Lamps, cubes, balls and walls: Zeno problems and solutions

Jeanne Peijnenburg 0 David Atkinson 0 0 J. Peijnenburg D. Atkinson (&) Faculty of Philosophy, University of Groningen , Oude Boteringestraat 52, 9712 GL Groningen, The Netherlands Various arguments have been put forward to show that Zeno-like paradoxes are still with us. A particularly interesting one involves a cube composed of colored slabs that geometrically decrease in thickness. We first point out that this argument has already been nullified by Paul Benacerraf. Then we show that nevertheless a further problem remains, one that withstands Benacerraf's critique. We explain that the new problem is isomorphic to two other Zeno-like predicaments: a problem described by Alper and Bridger in 1998 and a modified version of the problem that Benardete introduced in 1964. Finally, we present a solution to the three isomorphic problems. Our understanding of the conceptual problems raised by Zeno's paradoxes was significantly deepened by mathematical achievements in the nineteenth century (cf. Gru nbaum 1967; Salmon 1970). Yet new Zeno-like problems continue to crop up. A particularly interesting one was formulated by Milos Arsenijevic in 1989, and it has remained unsolved to the present day. In this paper we shall do the following: - We first argue that the problem posed by Arsenijevic can be resolved. For it is equivalent to a puzzle posed by James Thomson in 19541955, which has been solved by Paul Benacerraf in his celebrated paper of 1962. (2) Next we will show that there is another problem, not noticed by Arsenijevic, that is immune to Benacerrafs treatment. (3) We then demonstrate that the new problem is isomorphic to a Zeno-like paradox which has been described by Alper and Bridger (1998). (4) We argue that the new problem is also isomorphic to a modified version of a Zeno-like problem described by Benardete (1964). (5) Finally we explain that the three problems (the new problem introduced in 2, the Alper and Bridger paradox described in 3, and the modified Benardete paradox of 4) can all be solved in the same way. The best approach to the original Benardete paradox, however, is still to be found in Yablo (2000). (1) Consider a cube built up from slabs according to the following rules. Each slab is 1 m wide and 1 m deep, but they are of different heights. The bottom slab is m in height and it is colored red. The next slab, laid down on top of the red one, is only m high and it is green. The following slab, on top of the green one, is 1=8 m high, and is red again. The construction continues ad infinitum with alternating colors, the heights of successive slabs decreasing in the same geometric fashion. Looking down on the completed cube from above, would one see red or green? This problem was first described by Milos Arsenijevic and it is as yet unresolved (Arsenijevic 1989). However, we believe that the cube as described by Arsenijevic is accurately paralleled by the so-called Thomson lamp (Thomson 19541955). Aiming to show that we are still in a Zeno-like grip, James Thomson had presented us with a lamp that is turned on at time 0, off at min, on at min, and so on, ad infinitum. Thomson then asks himself: Is the lamp on or off 1 min after the start? The fact that we seem unable to answer that question was regarded by him as an indication that modern versions of the old Achilles are still with us today. However, Paul Benacerraf famously pointed out the flaw lurking in Thomsons question: it is the incorrect supposition that the rules as laid out by Thomson imply that the lamp will be either on or off after 1 min (Benacerraf 1962). As Benacerraf explains, these rules only tell us that every time the lamp is turned on (off) before 1 min, it is turned off (on) shortly thereafter. Nothing at all is implied concerning the state of the lamp at the 1 min mark. The lamp might be on or it might be off: either possibility is compatible with the rules of Thomsons gamesomething that Thomson himself roundly admitted in his Comments on Professor Benacerrafs Paper: Benacerrafs excellent article puts it beyond doubt that much of [my paper] is mistaken. (Thomson 1970, p. 130). There is a complete parallel between Arsenijevics question Is the cube red or green after completion? and Thomsons question Is the lamp on or off 1 min after the start? Just as the lamps being on or off at the 1 min mark is compatible with the rules of the Thomson game, so the completed cubes being red or green is reconcilable with the rules set by Arsenijevic. Arsenijevics cube and Thomsons lamp are in fact isomorphic, which becomes clear when we look at the assumptions that Thomson and Arsenijevic make. Thomson explicitly assumes that the lamp has to be either on or off 1 min after the start: [The lamp] cannot be on, because I never turn it on without turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction. (Thomson 19541955, p. 95; italics added). In exactly the same manner, Arsenijevic assumes that the cube has to be red or green after completion: It may be said that we dont know what the case may be, but, it seems, it is certain that in any case this must be [red] or [green] (Arsenijevic 1989, p. 37; italics added). We may formalize Thomsons description of his lamp as follows: (A) Every time that the lamp is turned on before the 1 min mark, it is then turned off before the 1 min mark. (B) Every time that the lamp is turned off before the 1 min mark, it is then turned on before the 1 min mark. (C) The lamp is either on or off at the 1 min mark. The parallel formalization of Arsenijevics cube is: (A0) Every time that a red slab is laid down, a green slab is then laid on top of it. (B0) Every time that a green slab is laid down, a red slab is then laid on top of it. (C0) The completed cube is either red or green. The similarity between the Thomson lamp and Arsenijevics cube is further evidenced if we put the first (red) slab in place at time 0, the second slab at min, the third one at min, and so on. If we observe the cube from above during its construction, the color will fluctuate in exactly the same way as does the Thomson lamp. After 1 min the construction will be finished. And just as the state of the Thomson lamp at the 1 min mark is undetermined, so the color of the cube after 1 min is not determined either. The fact that the lamp remains the same whereas the cube only gradually takes shape is not relevant here. What does matter is that in both examples we encounter oscillating, nonconvergent sequences of states. In the one case there is an infinite oscillation between the lamps being on or off, in the other between the cubes looking red or green. In fact, if we were to redesign the lamp so that it switches from red to green rather than from on to off, the color history would be the identical: in either case we would observe a nonconvergent os (...truncated)