Thomson's lamp

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

Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose 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 all these progressively smaller times is exactly two minutes.

The following questions are then considered:
*Is the lamp on or off after exactly two minutes?
*Is the lamp switch on or off after exactly two minutes?
*Would it make any difference if the lamp had started out being on, instead of off?

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.

One response is that one must consider how much time is spent moving the switch. Questions of lamp physics aside, one can simplify the problem to flipping a single bit of information to either a 0 or 1 state. If the flip takes any constant positive amount of time, then an infinite number of flips would take forever. So the only way this paradox will reach the 2 minute mark, under the assumption of constant flip time, is if the flip is not a delaying factor — essentially, if the flip takes zero amount of time. Yet if one can change the state of a bit instantly, then what does the question of a bit's state at a certain time mean? One could turn it off and on again without any time passing. One could even turn it off and on an infinite number of times. This response, however, does not deal with the case where successive flips take less and less time, so that the entire supertask can be performed in the given two minutes.

One possible solution to this problem, at least in the physical world, is provided by special relativity, i.e. the existence of a speed limit. That is, no one and nothing would be able to flick the switch infinitely fast, as would be required at the end of the sequence. There is a limit (the speed of light) to how quickly we can flip the switch.

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 {0, 1, 0, 1, 0, 1, ...}, 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." [Thomson p.6. For the mathematics and its history he cites Hardy and Waismann's books, for which see "History of Grandi's series".] 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." [Thomson p.7]

References

*cite journal |last=Thomson |first=James F. |authorlink=James F. Thomson (philosopher) |title=Tasks and Super-Tasks |journal=Analysis |volume=15 |issue=1 |year=1954 |month=October |pages=1–13 |url=http://links.jstor.org/sici?sici=0003-2638%28195410%2915%3A1%3C1%3ATAS%3E2.0.CO%3B2-Y |doi=10.2307/3326643

See also

* Supertask
* Balls and vase problem
* Zeno's paradoxes
* Zeno machine