Thomson's lamp is a philosophical

puzzle A puzzle is a game, Problem solving, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together (Disentanglement puzzle, or take them apart) in a logical way, in order to arrive at th ...

based on infinites. It was devised in 1954 by British philosopher James F. Thomson, who used it to analyze the possibility of a

supertask In philosophy, a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time. Supertasks are called hypertasks when the number of operations becomes uncountably infinite. A hypertask that inc ...

, which is the completion of an infinite number of tasks. Consider a lamp with a

toggle switch In electrical engineering, a switch is an electrical component that can disconnect or connect the conducting path in an electrical circuit, interrupting the electric current or diverting it from one conductor to another. The most common type of ...

. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being who is 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 In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

of time intervals is exactly two minutes. The following question is then considered: Is the lamp on or off at two minutes? Thomson reasoned that this supertask creates a contradiction:

Mathematical series analogy

The question is related to the behavior of

Grandi's series In mathematics, the infinite series , also written : \sum_^\infty (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a diverge ...

, ''i.e.'' the divergent infinite series * ''S'' = 1 − 1 + 1 − 1 + 1 − 1 + · · · 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

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s 0, 1, 2, 3, ... in turn, the series generates the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

, representing the changing state of the lamp. The sequence does not

converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...

as ''n'' tends to infinity, so neither does the infinite series. Another way of illustrating this problem is to rearrange the series: * ''S'' = 1 − (1 − 1 + 1 − 1 + 1 − 1 + · · ·) The unending series in the brackets is exactly the same as the original series ''S''. This means ''S'' = 1 − ''S'' which implies ''S'' = ¹⁄₂. 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 ¹⁄₂. 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, 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."

Notes

References

* * * * *Earman, John and Norton, John (1996
Infinite Pains: The Trouble with Supertasks. In Benacerraf and his Critics
Adam Morton and Stephen P. Stich (Eds.), p. 231-261. {{DEFAULTSORT:ThomSon'S Lamp Supertasks Paradoxes of infinity Grandi's series

Mathematical series analogy

See also

Notes

References