Sometimes the mere act of getting out of bed in the morning feels like a supertask, but this is not the kind of supertask I will write about today. Philosophical supertasks are similarly difficult actions but in a vastly different way. They are procedures that delineate an infinite number of tasks that can be completed in a finite amount of time. Questions about supertasks have gripped the minds of philosophers and non-philosophers alike for millennia, probably because they offer no obvious solution. Man’s insatiable thirst for knowledge has led us beyond the confines of our universe’s finite limits and materials to the land of infinite logic. In the 5th-century-BCE, this land of infinity is where our first hero arises: Zeno of Elea.
Zeno is the father to some of the first and most popular supertasks in popular culture: “Achilles and the tortoise” and “the dichotomy.” These thought experiments both involve a race consisting of an infinite series of tasks, each taking half as long and covering half the distance as the last task. In the end, Achilles would need to complete this infinite series of tasks in a finite amount of time to either reach the end of a race or pass a tortoise in a race. Zeno’s conclusion at the time was that motion must be impossible. If you are reading this right now, your eyes are moving, so Zeno must have been wrong, but this perspective largely misses the point of the puzzles. Zeno’s paradoxes weren’t intended to be taken seriously in any sort of logical sense; instead, they were intended to challenge our views of the world: whether it is discrete or continuous, whether time and space can be divided an infinite amount of times, and even the very nature of infinity itself. The discomfort provided by Zeno’s paradoxical claims laid the groundwork for modern ideas of calculus, infinite sums, and convergent series. I believe these modern inventions have adequately explained Zeno’s paradox, but his work has also inspired much more difficult questions surrounding supertasks like Thomson’s lamp.
Thomson’s lamp was J. F. Thomson’s attempt to create a more difficult supertask that arrives at a paradox that cannot be easily disproved by simply looking around the world. The Thomson lamp is a simple lamp with an on and off button, except for its property that it can be turned on and off as quickly as needed. The paradox arises when we turn the lamp on and off zenoianly for two minutes. That is, we turn it on at t=0, turn it off at t=1, turn it on again at t=1.5 and turn it off again at t=1.75. We continue turning the lamp on and off with half as much time between each button press as the last press until t=2. Thomson’s question was then: at t=2, is the lamp on or off? It cannot be on because every time it was turned on, it was turned off immediately, and it cannot be off because every time it was turned off, it was turned on again. An answer to this question would imply the existence of a last step in an infinite series of tasks or even some sort of largest integer. What, then, is the solution to this paradox?
Before we talk about the solution to the paradox, it is essential to first look at the point of the paradox. By designing this lamp, Thomson tries to demonstrate the difference between continuous and discrete tasks. In a sense, it offers a subtle criticism of Zeno’s supertasks, which he might argue aren’t supertasks at all. In the case of Achilles running across the room, there is only one real task (reaching the end of the race), and every other apparent task that Zeno presents is a subtask. These subtasks can be divided and summed up into other tasks which are identical except in time and length. Splitting up the tasks into some arbitrary configuration, even into infinite configurations, does not constitute doing any more or less work than in other configurations. However, there are other supertasks that contain tasks that cannot be divided into homogenous subroutines like this. For example, in the case of the Thomson lamp, the action of turning on the lamp cannot be divided into two or more homogenous tasks. You could say task 1 is pushing in the button, and task 2 is letting go of the button, but these are inherently different tasks and thus, turning on or off the button is some atomic, discrete task. Therefore, pressing the Thomson’s lamp’s button an infinite number of times is actually doing an infinite number of actions and is thus a real supertask.
Philosophers have created versions of Zeno’s supertasks that also seem to blur the line between continuous and discrete tasks. A notable mention is Adolf Grunbaum’s staccato runner which is a modified version of Achilles' race where the runner alternates between being stopped and running twice as fast as Achilles in faster and faster intervals until he has completed the same distance as the original Achilles, but he has started and stopped running an infinite number of times. Prima facie, the discontinuous velocity and instantaneous accelerations of this staccato runner make this seem not only physically impossible but also just like another clear example of a discrete supertask. However, physicist Richard Friedberg derived a mathematical function representing the position of the staccato runner such that the runner maintains a continuous velocity curve that still equals zero an infinite number of times and the runner still completes the race. In this case, is the staccato runner completing discrete or continuous tasks? Well even though Friedberg’s function solves the instantaneous velocity change problem, the staccato runner still completes an infinite number of discrete tasks, unlike Zeno’s Achilles. The task of stopping and starting again is another example of an atomic task, much like those in the Thomson lamp scenario, which cannot be divided or summed up into equivalent types of tasks. A continuous function could similarly be derived for the motion of a finger turning on the Thomson lamp an infinite number of times, but that does not change the fact that there are an infinite number of discrete tasks being accomplished. So the question remains: is a discrete supertask possible.
I have never seen someone do infinite work in a finite amount of time, and you probably have not, either. This is likely due to physical barriers that prevent ever completing a supertask like Thomson’s. For instance, as we press the button faster and faster (assuming that the button moves some nonzero finite distance when it’s pressed), at some point, we would need to be pressing the button faster than the speed of light. Not only is that impossible in our universe, but as we would have approached this point, the amount of energy necessary to accelerate the button as we press it down would approach infinity, and we would end up creating an atomic bomb.
This answer, which denies the physical possibility of the lamp, is not even really an answer at all. What we care about is not whether the Thomson lamp is physically possible; we care if it is logically possible. We care if there is a logically coherent answer to the paradox. To remove the physicality of the situation, it is helpful to create a mathematical analogy representing the situation. Using this analogy, we can determine if the abstract logic of the problem is sound without having to worry about the specifics of the construction of the Thomson Lamp. A useful way we can model this problem as a series of partial sums of Grandi’s series: 1, -1, 1, -1, 1, -1,... This series is a valid analogy of the lamp because it is both infinite in length and switches values after each element much like how the lamp switches states after each task. In particular, after the first task of turning on the lamp, the partial sum is 1, which we will say means on. After the second task, turning off the lamp, the partial sum is 1 - 1 = 0, which we will say means off. In this manner, the partial sum alternates between 1 and 0 after each task, which mirrors the lamp turning on or off. The question of whether the lamp is on or off after 2 minutes can now be rephrased as asking what is the infinite sum of Grandi’s series: 1 -1 + 1 - 1 + 1 - 1 + …. This value would represent the state that the lamp is left in after completing the infinite switches. Unfortunately, there is no singular number you could give which is an answer to this problem. Notice that if we group the terms like this (1 - 1) + (1 - 1) + (1 - 1) + …, then the answer is 0 + 0 + 0 + … = 0, and the lamp should be off at t=2. But if we group the terms like this 1 - (1 - 1) - (1 - 1) - (1 - 1) - …, then the answer is 1 - 0 - 0 - 0 - … = 1, and the lamp should be on at t=2. If that isn’t contradictory enough, we can also show that the sum is equal to ½ if we say S = 1 - 1 + 1 - 1 + … in which case S + S = 1 + (1 - 1) + (1 - 1) + … = 1. Since 2S = 1, then S = ½. Depending on how we rearrange the math, the lamp is either on, off, or somewhere in the middle. Since math is the objective language of logic and it cannot seem to give a coherent answer to this question, maybe the lamp itself is logically impossible.
Another possibility is that the question simply does not offer enough information. Philosopher Paul Benacerraf argues that the reason we cannot determine whether or not the Thomson lamp is on at t=2 is that the premise of the lamp is incomplete. The behavior of the lamp is perfectly defined while t is in the interval [0,2); however, the behavior is not defined when t is greater than or equal to 2. All we are given is a divergent series of states which offers no way to determine the end state of the lamp. Benecerraf thinks that with more information, the answer could be anything and still make logical sense. Are there ways in which we can augment the problem to create a consistent and coherent logical answer? If we allow the lamp to exist in nonbinary states while pressing the button, then the answer can be somewhere in the middle. Alternatively, using the Thomson lamp construction proposed by John Earman and John D. Norton, which uses a bouncing ball to complete the circuit, it is possible in an ideal physical world for there to be an infinite number of bounces yet have the ball come to rest in the end. Depending on the lamp’s construction, this rest state could either be the on or off state. However, the question as it stands, cannot be answered because there is not enough information about the lamp at t=2.
Maybe Benacerraf is right that there is not enough information about the lamp to determine its behavior in infinite situations, but this still avoids answering the original question. The original Thomson lamp–with its binary, togglable switch–has yet to be explained. The question is about accomplishing infinitely many identical discrete actions in a finite amount of time. Is it logically possible for an infinite alternating sequence to have a last step? There cannot be a final state of the lamp. There cannot be a last time that the staccato runner stops. There cannot be a limit to a sequence of partial sums that diverges. The only logical solution is that there is no solution. This leaves us with two logical possibilities: either our current concepts of logic are not consistent, or they are consistent, but there are unanswerable questions.
Our modern conceptions of logic may be incomplete because we are finite beings cursed with never truly being able to know infinity. Maybe infinity deserves special treatment and cannot be accommodated in our current systems. I think it is far more likely that there are simply logical questions we can ask for which there does not exist a correct answer. This is a cool concept, but also scary if we consider all of the unanswered questions humanity faces daily. What if I spend my entire life working on an unanswerable question? Maybe questions act much like languages of a Turing machine in that they are decidable, recognizable, or unrecognizable. The Thomson lamp problem begs the question: are there questions we cannot solve that we can never know we cannot solve? I think philosophers should be the most scared because they are the ones most concerned with problems we do not know how to solve. In every other discipline– engineering, physics, medicine– we can answer questions simply by testing and observing. However, philosophers are obsessed with the unknown and the things that cannot be known. Whether or not a supertask is possible varies depending on the nature of the tasks at hand, but they offer us important insights into what kinds of questions we can answer and which we cannot.