In our previous post, we began a discussion of supertasks, or processes with infinitely many steps that are completed in a finite amount of time. Today, we visit the source of the word supertask (or, in its original formulation, super-task), J.F. Thomson’s 1954 paper, “Tasks and Super-tasks”, published in Analysis.
The main thesis of Thomson’s paper is that supertasks are paradoxical, that it is impossible for someone to have completed an infinite sequence of tasks. (He is careful to distinguish between (1) completing an infinite sequence of tasks, which he thinks is impossible and (2) having an infinite number of tasks that one can potentially do. This is essentially the classical distinction between actual and potential infinity that we have discussed here a number of times.)
To illustrate the impossibility of supertasks, Thomson introduces a particular supertask, now known as Thomson’s Lamp. The idea is as follows: Suppose we have a lamp with a button on its base. If the lamp is off, then pressing the button turns it on; if it is on, then pressing the button turns it off. You have probably encountered such a lamp at some point in your life. Now imagine the following sequence of actions. The lamp is initially off. After one second, I press the button to turn it on. After a further 1/2 second, I press the button again to turn it off. After a further 1/4 second, I press the button to turn it on. After a further 1/8 second, I press the button to turn it off. And so on. (As in our last post, we are ignoring the obvious physical impossibilities of this situation, such as the fact that my finger would eventually be moving faster than the speed of light or that the wires in the lamp would become arbitrarily hot.) In the course of 2 seconds, I press the button infinitely many times. Moreover, there is no “last” button press; each press is quickly followed by another. So the question is this: after 2 seconds, is the lamp on or off? Thomson feels that either answer is impossible, and yet one of them must hold:
It cannot be on, because I did not ever turn it on without at once 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 concludes that supertasks are impossible. (I am oversimplifying a little bit here; I encourage interested readers to check out Thomson’s original paper, which is fairly short and well-written.) As anyone passingly familiar with the history of philosophy could guess, though, not everybody was convinced by Thomson’s arguments, and Thomson’s paper initiated a small but lively philosophical discussion on supertasks, some of which we may examine in future posts.
Today, though, I don’t want to focus directly on the possibility of supertasks in general or Thomson’s Lamp in particular, but rather to assume that they are possible and think about what this would mean for a topic of great interest in philosophy: determinism.
Roughly speaking, determinism is the assertion that all events are uniquely determined by what has happened before. Given the history of the universe, its future is fixed in stone. The first modern expression of this type of determinism is typically credited to the French mathematician Pierre-Simon Laplace, who imagined the existence of a demon who knows the precise location and momentum of every object in the universe and is therefore able to calculate all future events.
Determinism is a vast subject, and we don’t have nearly enough time to touch on its many connections with topics such as religion, free will, or quantum mechanics. We do have time, however, to show that Thomson’s Lamp, plus a bit of intuition (always dangerous when dealing with infinity, but let’s allow it for the sake of argument), places serious doubt on the truth of determinism.
To see this, let’s suppose that the supertask of Thomson’s Lamp is in fact coherent, and let’s revisit the question: after 2 seconds, is the lamp on or off? I claim that either answer is possible. To see this, let’s be a little bit more precise. Let’s call our supertask, in which the lamp starts off and we press the button infinitely many times in 2 seconds, Process A. Suppose my claim is wrong and that, at the conclusion of Process A, the lamp is necessarily on (if the lamp is necessarily off, then a symmetric argument will work). Now imagine that Process B is the same as Process A but with the role of “on” and “off” reversed: the lamp starts on and, over the course of 2 seconds, we press the button infinitely many times. One would expect that the final state of Process B would just be the reverse of the final state of Process A, namely that the lamp is necessarily off at the conclusion of Process B. (If this doesn’t seem intuitively obvious to you, try imagining a more symmetric situation in which the switch doesn’t turn the lamp on or off but instead switches the color of light emitted from red to blue.) But now suppose that an observer arrives at Process A exactly 1 second late and thus misses the first press of the button. From their perspective, the remainder of the supertask looks exactly like Process B, just sped up by a factor of two. This speeding up intuitively shouldn’t affect the final outcome, so they should rightly expect the lamp to be off at the end of the process, whereas I should rightly expect the lamp to be on at the end of the process, since I know we are completing Process A. We can’t both be correct, so this is a contradiction.
But if, at the end of the supertask, the lamp could consistently either be on or off, then the state of the lamp after two seconds is not uniquely determined by past events. Laplace’s demon cannot predict the behavior of the lamp after two seconds, even if it knows everything about the universe at all times before two seconds have elapsed. In other words, if we accept the coherence of Thomson’s lamp, then determinism must be false!
This was admittedly a very non-rigorous argument, but I hope it provides some food for thought or at least some amusement. Next time, we’ll return to doing some real mathematics and connect ideas of supertasks with the values of infinite series!
Cover image: “L’empire des lumières” by René Magritte