Thomson's lamp

Thomson's lamp is a philosophical puzzle that is a variation on Zeno's paradoxes. It was devised in 1954 by British philosopher James F. Thomson, who also coined the term supertask.

Time	State
0.000	On
1.000	Off
1.500	On
1.750	Off
1.875	On
...	...
2.000	?

Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of this infinite series of time intervals is exactly two minutes.

The following questions are then considered:

• Is the lamp switched on or off after exactly two minutes?
• Would the final state be different if the lamp had started out being on, instead of off?

Thomson wasn't interested in actually answering these questions, because he believed these questions had no answers. This is because Thomson used this thought experiment to argue against the possibility of supertasks, which is the completion of an infinite number of tasks. To be specific, Thomson argued that if supertasks are possible, then the scenario of having flicked the lamp on and off infinitely many times should be possible too (at least logically, even if not necessarily physically). But, Thomson reasoned, the possibility of the completion of the supertask of flicking a lamp on and off infinitely many times creates a contradiction. The lamp is either on or off at the 2-minute mark. If the lamp is on, then there must have been some last time, right before the 2-minute mark, at which it was flicked on. But, such an action must have been followed by a flicking off action since, after all, every action of flicking the lamp on before the 2-minute mark is followed by one at which it is flicked off between that time and the 2-minute mark. So, the lamp cannot be on. Analogously, one can also reason that the lamp cannot be off at the 2-minute mark. So, the lamp cannot be either on or off. So, we have a contradiction. By reductio ad absurdum, the assumption that supertasks are possible must therefore be rejected: supertasks are logically impossible.

Discussion

The status of the lamp and the switch is known for all times strictly less than two minutes. However the question does not state how the sequence finishes, and so the status of the switch at exactly two minutes is indeterminate. Though acceptance of this indeterminacy is resolution enough for some, problems do continue to present themselves under the intuitive assumption that one should be able to determine the status of the lamp and the switch at any time, given full knowledge of all previous statuses and actions taken.

Mathematical series analogy

The question is similar to determining the value of Grandi's series, i.e. the limit as n tends to infinity of

$\sum_{i=0}^n{(-1)^i}.$

For even values of n, the above finite series sums to 1, for odd values, it sums to 0. In other words, as n takes the values of each of the non-negative integers 0, 1, 2, 3, ... in turn, the series generates the sequence {1, 0, 1, 0, ...}, representing the changing state of the lamp. The sequence does not converge as n tends to infinity, so neither does the infinite series.

Another way of illustrating this problem is to let the series look like this:

$S = 1 - 1 + 1 - 1 + 1 - 1 + \cdots$

The series can be rearranged as:

$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + \cdots)$

The unending series in the brackets is exactly the same as the original series S. This means S = 1 - S which implies S = ½. In fact, this manipulation can be rigorously justified: there are generalized definitions for the sums of series that do assign Grandi's series the value ½. On the other hand, according to other definitions for the sum of a series this series has no defined sum (the limit does not exist).

One of Thomson's objectives in his original 1954 paper is to differentiate supertasks from their series analogies. He writes of the lamp and Grandi's series,

"Then the question whether the lamp is on or off… is the question: What is the sum of the infinite divergent sequence

+1, −1, +1, …?

"Now mathematicians do say that this sequence has a sum; they say that its sum is ¹⁄₂. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on. I take this to mean that there is no established method for deciding what is done when a super-task is done. … We cannot be expected to pick up this idea, just because we have the idea of a task or tasks having been performed and because we are acquainted with transfinite numbers."

Later, he claims that even the divergence of a series does not provide information about its supertask: "The impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be-associated arithmetical sequence is convergent or divergent."

See also

• List of paradoxes
• Ross–Littlewood paradox
• Supertask
• Zeno's paradoxes
• Zeno machine