Thomson's lamp

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Thomson's lamp is a philosophical puzzle based on infinites. It was devised in 1954 by British philosopher James F. Thomson, who used it to analyze the possibility of a supertask, which is the completion of an infinite number of tasks.

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 in a logical way, in order to arrive at the correct or fun solution of the puzzle. There are different genres of puzzles, such as crossword puzzles, word-search puzzles, number puzzles, relational puzzles, or logic puzzles.

James F. Thomson (1921–1984) was a British philosopher who devised the puzzle of Thomson's lamp, to argue against the possibility of supertasks.

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 includes one operation for each ordinal number is called an "ultratask". The term supertask was coined by the philosopher James F. Thomson, who devised Thomson's lamp. The term hypertask derives from Clark and Read in their paper of that name.

Contents

TimeState
0.000On
1.000Off
1.500On
1.750Off
1.875On
......
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 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. [1] The sum of this infinite series of time intervals is exactly two minutes. [2]

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, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

The following question is then considered: Is the lamp on or off at two minutes? [1] Thomson reasoned that this supertask creates a contradiction:

It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction. [1]

Mathematical series analogy

The question is related to the behavior of Grandi's series, i.e. the divergent infinite series

In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written

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. [3] The sequence does not converge as n tends to infinity, so neither does the infinite series.

Integer Number in {..., –2, –1, 0, 1, 2, ...}

An integer is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5 1/2, and 2 are not.

Sequence ordered list of elements; function with natural numbers as domain

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order matters. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers or the set of the first n natural numbers.

Limit of a sequence value that the terms of a sequence "tend to"

In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to". If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.

Another way of illustrating this problem is to rearrange the series:

The unending series in the brackets is exactly the same as the original series S. This means S = 1 − S which implies S = 12. 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 12.

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 12. 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. [4]

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." [5]

See also

Notes

  1. 1 2 3 Thomson 1954, p. 5.
  2. Thomson 1954, p. 9.
  3. Thomson 1954, p. 6.
  4. Thomson p.6. For the mathematics and its history he cites Hardy and Waismann's books, for which see History of Grandi's series .
  5. Thomson 1954, p. 7.

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