After considerations on Zeno's paradox (part 1 and part 2), and the humorous possibilities to Zeno's paradox of motion (part 3), it is time to take beef with the paradox on the nature of what is called a super-task. As stated in the previous post on Zeno, Thomson thought super-tasks to be impossible. It is not, as he points out, that we do not repeat the same tasks indefinitely; but rather that the infinity sequence cannot be dealt with as having a 'mathematical solution'. He gives two examples: splitting a lump of chocolate infinitely into two parts - thus asking us whether we still have chocolate at the end of the sequence; and turning a lamp on and off infinitely - thus asking us whether the light is on or off at the end of the sequence?
Let us take his first example:
| "Suppose (A) that every lump of chocolate can be cut in two, and (B) that the result of cutting a lump of chocolate in two is always that you get two lumps of chocolate" (p. 2) |
The problem with this statement that Thomson observes - in the sense of a Zenoesque infinite sequence - is that we must at some point question when a lump of chocolate stops being chocolate (or a lump for that matter). Thus, this last molecule is either chocolate (in which case premise B is false); or that molecule is no longer chocolate (in which case premise A is false).
This is clear enough - and it has some practical purposes; especially in questions of morality/ethics in politics. Think, for instance, of positive discrimination: when does it stop being 'positive' and ends up simply discriminating (cf. for instance, Brian Barry's Culture and Equality: An Egalitarian Critique of Multiculturalism)?
Thomson's second example is more interesting, and more challenging:
| "There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off . . . Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half-minute, and so on . . . After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off?" (p. 6) |
In this too Thomson follows Zeno's paradox by proposing the same formula: 2 - 1/(2n-1)
At this stage we can give a simple answer to Thomson: the lamp will seem to be ON. Regardless of the continuous switching during the sequence, it will look as if it is ON. Not much of a paradox if you involve physics into mathematics.
But Thomson continues:
| Say that the reading-lamp has either of two light-values, 0 ('off') and 1 ('on'). To switch the lamp on is then to add 1 to its value and to switch if [sic.] off is to subtract 1 from its value . . . What is the sum of the infinite divergent sequence +1, -1, +1, ...? |
A proper mathematical answer to this paradox, and which Thomson agrees with, the sequence has a sum of 1/2 (half). However, as Thomson observes: "this answer does not help us, since we attach no sense here to saying that the lamp is half-on"
So what do we learn from Thomson on this point? What does he tell us with his complicated mathematics? Can we understand this somehow differently, in a non-mathematical jargon? Certainly!
What Thomson tells us can be understood as a questioning of the possibility of our understanding of infinity when it comes to tasks. We may poetically say that there is infinity, in the sense of a journey through life, where each stage is a prelude to another and so on (see this post on Cavafy's Ithaca). However, we cannot meaningfully understand infinity when it comes to action. Every action has an end point. Even when we speak of continuity and repetition, we are still bound to an understanding of our actions as finite accomplishments, which require further actions to either validate the previous ones, or question their results. Indeed, even if we apply Zeno's paradox to Cavafy's Ithaca, we find that there is neither the starting point, nor the end: indeed, that it is the sequence alone that we experience without our understanding of the infinity beyond that point (if there is such at thing at all).
A good example here would be J.S. Mill's view of liberty - liberty is not simply a reckless negative freedom (a lack of restrictions imposed from outside - governments, neighbours, etc.); liberty is a need for a challenge to verify your own opinions and positions, regardless of their offensive nature, or intrusions on your dignity, etc.
Thomson's article can be read on JSTOR, Analysis 1954, Vol. 15 (1), pp. 1-13.
This is a 4-part series on Zeno's paradox of motion.
Part 1: Zeno's paradox of motion in relation to bisection of space (Achilles and the tortoise)
Part 2: Zeno's paradox of motion in relation to bisection of time (Flying arrow is at rest)
Part 3: Further applications of Zeno's paradox: The Ross-Littlewood paradox
Part 4: Further applications of Zeno's paradox: Thomson's lamp
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