If you switched on the light this morning and walked to the bathroom, you've put yourself in the middle of a philosophical conundrum. In one case, you have accomplished what's known as a supertask. In the other case, you've done something that shows supertasks are impossible.
What are Supertasks?
Supertasks are an infinite number of things done in a finite amount of time. Ever since they have been described, people have been arguing about whether they are possible or not. Admittedly, at first it looks like common sense to simply say "no," and move on with your day. But before you make up your mind, consider the fact that you accomplished your first supertask as soon as you were born.
Supertasks were first proposed by Zeno of Elea, a philosopher who was considered a smartass even by ancient philosopher standards. He was fond of paradoxes, and came up with one which proved that motion is impossible. For a person to get from one place to another, they have to pass by a point half way between those two places. For them to get to the half-way point, they have to get somewhere half way between that. Since there will always be more half-way points, a person will have to pass through an infinite number of points in a finite amount of time. That's not possible, argued Zeno, and so all motion is impossible.
That was the first supertask, then, and it is obviously doable. In fact, that's the tack philosophers took to refute it. No talk about the fundamental nature of reality, and no arguments about continuous versus discrete space — because we can move, Zeno's argument was clearly wrong. Either there is a certain scale at which there is no longer half-way point between to places, or supertasks are possible. So you accomplished a supertask by walking anywhere, taking a single step, or getting from one end of the birth canal to another. (I guess all you c-section kids were late bloomers or something. Don't worry. You caught up. Although whether anyone can catch up to someone with a head start is yet another debatable supertask.) What could be simpler than a supertask?
On the other hand, supertasks are completely impossible. That comes from no less an authority than James F. Thomson, the man who coined the word "supertask." He came up with his own version of a supertask, which he believed showed that they could not be done. Following Zeno's lead, he went with the idea of halves, but instead of halving an amount of space, he chose to halve an amount of time.
Thomson's Lamp is meant to be switched on and off very quickly. This is a good thing, because Thomson wants you to switch it on and off an infinite number of time in two seconds. The lamp starts out in the on position. At the one-second mark, you turn it off. At the 1.5-second mark, you turn it back on. At the 1.75-second point you turn it off again, and so on and so on, until you reach two seconds. The clock hits the two-second mark. You reach out your hand. Is the lamp on or off?
Thomson argues that, if supertasks were possible, there should be an answer to that question. But is there an answer? Let's say the lamp is on at the two-second mark. If the lamp is on as you stretch your hand out at the two-second mark, it had to have flipped on just before the two-second mark. But if it was flipped on at that time, then halfway between that time and the two-second mark, it had to have been flipped off.
That means the lamp is off as you stretch out your hand, but if it is off, it had to have been off at some time before the two-second mark. Which means that halfway between that time and the two-second point, you had to have switched it on. That means the lamp is on. Which means it had to have been turned off before, making it off again. The point is, supertasks are impossible.
Or are they? Obviously, there are practical limits to any investigation you try to conduct on supertasks (although, if they actually are possible, there theoretically shouldn't be). Do you think it's possible to accomplish a supertask? If so, what supertask would you propose?
[Via Infinite Pains, Stanford Encyclopedia of Philosophy, MathLair, Mathematics]