# Thomson's lamp

Thomson's lamp is a philosophical puzzle that is a variation on Zeno's paradoxes. 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.

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 that there is 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.[1] The sum of this infinite series of time intervals is exactly two minutes.[2]

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

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 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.

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 = 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]

On the Thomson Lamp Paradox, Earman and Norton (1996) write, "The lamp is not paradoxical since any (state of the lamp at the 2-minute mark, ON or OFF) would be compatible with the schedule of switching prior to (that time)." (p.237)

In establishing this schedule of switching, we define Thomson's idealized lamp to be ON only during the following time intervals (in minutes):

${\displaystyle [0,1),[{\frac {3}{2}},{\frac {7}{4}}),[{\frac {15}{8}},{\frac {31}{16}}),\cdots }$

And we define it to be OFF only during the following intervals:

${\displaystyle [1,{\frac {3}{2}}),[{\frac {7}{4}},{\frac {15}{8}}),[{\frac {31}{16}},{\frac {63}{32}}),\cdots }$

We assume that the transitions, switching from one state to another, are instantaneous.

Notice that every element of each of these intervals is less than ${\displaystyle 2}$. Therefore, the state of Thomson's Lamp is undefined for time ${\displaystyle t=2}$ minutes.

We can represent the schedule of switching by a partial function ${\displaystyle f:\mathbb {R} \to \{0,1\}}$ such that:

${\displaystyle f(t)={\begin{cases}1{\text{ (ON) }}&{\text{if }}t\in \cup \{[0,1),[{\frac {3}{2}},{\frac {7}{4}}),[{\frac {15}{8}},{\frac {31}{16}})\cdots \}\\0{\text{ (OFF) }}&{\text{if }}t\in \cup \{[1,{\frac {3}{2}}),[{\frac {7}{4}},{\frac {15}{8}}),[{\frac {31}{16}},{\frac {63}{32}})\cdots \}\end{cases}}}$

where ${\displaystyle t}$ is the elapsed time in minutes.

The function ${\displaystyle f}$ is defined only on the half-open interval ${\displaystyle [0,2)}$. On this schedule of switching, ${\displaystyle f(2)}$ will be undefined. Therefore, if we were to assign a value of either ${\displaystyle 0}$ or ${\displaystyle 1}$ to ${\displaystyle f(2)}$, then that value would indeed be compatible with the above switching schedule on the interval ${\displaystyle [0,2)}$. Contrary to Thomson then, there would be no contradiction in either case.