Thomson's lamp
Encyclopedia
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:
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.
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
s 0, 1, 2, 3, ... in turn, the series generates the sequence
{1, -1, 1, -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:
The series can be rearranged as:
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,
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."
Puzzle
A puzzle is a problem or enigma that tests the ingenuity of the solver. In a basic puzzle, one is intended to put together pieces in a logical way in order to come up with the desired solution...
that is a variation on Zeno's paradoxes. It was devised by philosopher James F. Thomson
James F. Thomson (philosopher)
James F. Thomson was a British philosopher who devised the puzzle of Thomson's lamp , to argue against the possibility of supertasks -Academic career:...
, who also coined the term supertask
Supertask
In philosophy, a supertask is a quantifiably infinite number of operations that occur sequentially within a finite interval of time. Supertasks are called "hypertasks" when the number of operations becomes innumerably infinite. The term supertask was coined by the philosopher James F...
.
| 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 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
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of all these progressively smaller times is exactly two minutes.
The following questions are then considered:
- Is the lamp switch 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
Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
, 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
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
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s 0, 1, 2, 3, ... in turn, the series generates the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
{1, -1, 1, -1, ...}, representing the changing state of the lamp. The sequence does not converge
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
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:
The series can be rearranged as:
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
Cesàro summation
In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...
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
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
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
- SupertaskSupertaskIn philosophy, a supertask is a quantifiably infinite number of operations that occur sequentially within a finite interval of time. Supertasks are called "hypertasks" when the number of operations becomes innumerably infinite. The term supertask was coined by the philosopher James F...
- Ross-Littlewood paradox
- Zeno's paradoxesZeno's paradoxesZeno's paradoxes are a set of problems generally thought to have been devised by Greek philosopher Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is...
- Zeno machineZeno machineIn mathematics and computer science, Zeno machines are a hypothetical computational model related to Turing machines that allows a countably infinite number of algorithmic steps to be performed in finite time...