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Supertasks, Paradox and Impossibility

24 May 2008 – 3:46 pm by Rich Cochrane

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I was reminded of supertasks the other day and thought they’re interesting enough for a post, particularly as an argument that they’re impossible uses a similar tactic to some arguments against time travel and other speculative activities.

Supertasks

Here’s an example of a supertask. Imagine you had to count up to infinity. How long would it take you? Well, let’s say you count one number every second. Then it will take you an infinitely long time. Clearly this task is impossible, for two reasons: first, at some point you’ll probably die (sorry) and second you can never “finish” the task; even if you had an infinite amount of time to complete it, you’d never actually finally mop your brow, grin and say “phew, that’s it, done”.

But you could consider a different approach. Say “1″. Now wait half a second and say “2″. Now wait a quarter of a second and say “3″. Say “4″ after an eighth of a second, “5″ after a sixteenth of a second and so on. You’ll be finished after 1 second. This is a supertask: a task that accomplishes an infinite number of things in a finite period of time.

Clearly this isn’t physically possible for a human being, or indeed for any machine we could currently construct. But is it logically possible? Might some advanced future society have compters capable of performing supertasks? If so, mathematical mysteries, for example, could be solved by brute force checking of every number.

Thomson’s Lamp

The philosopher James F. Thomson invented a simple tool for thinking about this kind of supertask. Imagine if, instead fo counting, the machine had a lamp on it that simply switched on and off. Thompson’s lamp switches state in the same time intervals as the counting example; that is, it switches on and off an infinite number of times in a second.

So let’s fire up the machine. A second later, is the lamp on or off? Well, if it’s on, it must have turned on at some point during the 1 second period it was running; but if that’s so then at a point shortly after that it would have been turned off. The same argument should convince us it can’t be off either.

Although you need a bit of maths to make it rigorous, you can certainly do that and be quite sure the lamp will be neither on nor off when the machine stops its 1-second run. One way to convince yourself of this it to think of it this way. Say the lamp starts off unlit. Now consider each on-off pair; every time you perform an on-off pair of actions, you leave the lamp unlit again. Even an infinite number of such actions leaves it unlit, so that must be its final state. But now assume the lamp starts unlit, and then the machine starts its run. The first thing it does is turn it on. Then it performs an infinite series of off-on pairs of actions, each of which leaves the lamp on. So the lamp must be on at the end of the process.

This is a contradiction — the lamp must be either on or not on — so that proves the logical impossibility of such a machine ever existing. Whether the light is on or off after one second, we can safely conclude that it carried out only a finite number of state changes, not the infinite number advertised.

There are some problems with this particular argument, and you can have fun finding them for yourself. The thing that interests me here is the form it takes: that since completing a supertask could lead to a paradox, it must be impossible in principal.

Other ingenious examples, like the Ross-Littlewood paradox, make the same sort of argument: supertasks are paradoxical, and paradoxical things can’t happen, so supertasks can’t happen.

Paradox and Impossibility

Time travel paradoxes offer a better-known example of the same argumentative pattern. I could use a time machine to go back in time and murder myself, or the inventor of the time machine, or destroy my own time machine before I’d used it, and so on. This would create a paradox: by performing an action, I made it impossible for me to perform that very action. Such paradoxes show that time travel isn’t possible.

One rebuttal — once popular among science fiction writers, anyway — is that the paradox will get resolved in some way that we don’t expect. I won’t murder my younger self, but the identical twin I never knew I had. I’ll destroy my time machine, but someone will replace it in the night with another one. We know I didn’t succeed in making my action impossible because my action is just whatever I did.

Paradoxes about supertasks are sometimes just as easy to resolve. The specification for Thomson’s lamp defines what it does up to, but not including, the one second mark. At the one-second mark, we need to be told what it will do. Working on the assumption that nature is continuous, we find we actually can’t define this behaviour without a “magical”, “instantaneous” switch of state. So Thomson’s lamp doesn’t work because you can’t specify it without asking for an absurdity, a denial of the laws of physics that’s even worse than the original proposal.

If we can’t find a satisfactory resolution for this paradox, can we reasonably conclude that such a machine definitely, analytically can’t ever exist? I’m not convinced. What if such a machine did exist? Well, we might think there was something wrong with our argument, and that the paradox was not really paradoxical. Or we might say there was something wrong with our logic and that paradoxes are actually possible.

So it seems to me that any paradox that appears to arise from the existence of some imagined object offers us three options:

1. Reject the possibility of the existence of the object;
2. Reject the paradoxicality of the paradox;
3. Accept the paradox and the possibility of the object’s existence.

It might seem obvious which of these we should choose. After all, if a paradox seems convincing, then we have no reason to choose (2). And (3) is unattractive if we feel the logic we’ve been using has done a good job so far. So (1) is the best option.

But that’s all I think it is: the best option. We have no reason not to choose (1), but we do have reasons not to choose (2) and (3). It’s a pragmatic decision. But if we had a reason not to choose (1) then we would, as they say, have ourselves a horse-race. You may want to say that would never happen, because then we’d have a paradox, and paradoxes can’t exist. That would be reassuring, but it begs the question.

We borrowed the image of the geometric series from here. The lamp pic is courtesy of Silenceofnight.