Thomson's lamp is a philosophical
puzzle
A puzzle is a game, problem, or toy that tests a person's ingenuity or knowledge. In a puzzle, the solver is expected to put pieces together ( or take them apart) in a logical way, in order to find the solution of the puzzle. There are differe ...
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 in ...
, 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 o ...
. 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, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
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 diverg ...
, ''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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 cal ...
, 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)
In mathematics, a limit is the value that a function (or sequence) app ...
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 parentheses 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."
See also
*
List of paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their ...
*
Ross–Littlewood paradox
*
Zeno's paradoxes
Zeno's paradoxes are a series of philosophical arguments presented by the ancient Greek philosopher Zeno of Elea (c. 490–430 BC), primarily known through the works of Plato, Aristotle, and later commentators like Simplicius of Cilicia. Zeno de ...
*
Zeno machine
In mathematics and computer science, Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that are capable of carrying out computations involving a co ...
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