Talk:Thomson's lamp

message: There can be no answer simply because according to the asymptotal nature of the ever-diminishing periods, the sequence will never actually reach 2.0mins - therefore it is not applicable what the state will be at the point. This is equivalent to trying to solve an equation on a Cartesian plane, where there is an asymptote to e.g. y=0, and trying to find the value for y=-1. Another example: what will be the speed of a car at 200 miles further, while having only 100 miles worth of fuel left.220.244.80.111 (talk) 09:50, 28 October 2010 (UTC)
thread_title: Task is implicitly impossible

message: (inserted for readability Rursus dixit. (mbork3!) 09:19, 12 June 2010 (UTC)
replies: text: The sum of all these progressively smaller times is exactly two minutes
--RickO5
The idea is that 2 minutes will eventually come, the error is in the phrasing. Eventually, the intravles will become infinately tiny, and the bulb will remain in a state of ifinately tiny state changes. The idea behind the argument is that you can not divide by infinity bassically, but 2 minutes will eventually occur anyway. 
The state of the bulb would bassically be whether or not the number of times it is switched is even or odd, but scince the number of times must become infinate, because one is infinately dividing by 2, it is neither odd nor even. one could assume that the bulb is neither on nor off after a certain period of time. (In reality the bulb stops flickering because of the lack of cool down time for the filliment.) 
So, the biggest problem with the argument is the inability of time to be divided by the infinite.
* the biggest problem is that within any physical reality, after a while you bump up against quantum consideration and limits of measurement. My conclusion: Time is not a thing that can be divided, time IS a measurement --JimWae 05:38, 3 January 2006 (UTC)
thread_title: ''(random heading)''

message: The reasoning that S = ½ is a non proof:
"Another way of illustrating this problem is to let the series look like this:
: S = 1 - 1 + 1 - 1 + 1 - 1 + \cdots
The series can be rearranged as:
: S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + \cdots)
The unending series in the brackets is exactly the same as the original series S. This means S = 1 - S which implies S = ½."
Assigning S as it has been assumes that the sequence converges. In fact, it can be shown using the fact that there exist two subsequences of the partial sums (namely partial sums up to an even number and partial sums up to an odd number)--that converge to different limits--that the sum does not converge and so this algebra of S = 1 - S cannot be performed. —Preceding unsigned comment added by 163.1.62.24 (talk • contribs) 00:46, 19 January 2008
:You should read the rest of the paragraph you just quoted: "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)." Do you disagree? Melchoir (talk) 00:56, 19 January 2008 (UTC)
replies: text: One, the connection is presented as "The question is similar to..." in the article, a mild statement which I don't think is in question. Two, the connection is made by Thomson himself in the article that originally introduced the problem. It sounds like you have a problem with Thomson's analysis, which is perfectly reasonable, but Wikipedia should still report on it. Three, yes it is explained: "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 {0, 1, 0, 1, 0, 1, ...}, representing the changing state of the lamp." Melchoir (talk) 00:28, 15 July 2008 (UTC)

text: It is an established method in mathematics, one of several, and when considering a measuring instrument, such as a light detector that integrates all light during say 0.2 sec, a summation can be seen as a physically legitimate operation, although not necessarily unique, nor unambiguous. Rursus dixit. (mbork3!) 09:27, 12 June 2010 (UTC)
thread_title:  Incorrect 

message: Could someone explain this better? The problem appears to be that the person posing the question is demanding to know the state of the lamp at 2 minutes, when according to the statement of the problem at 2 minutes there will be exactly zero time between state changes - so the lamp lacks a discrete on or off state. It is basically stating (in an overly-complicated way) that the lamp is in an undefined state, and them demanding to know what state it is in. In demanding to know the discrete state of the lamp the "paradox" takes advantage of our common-sense assumption that a lamp must be either on or off, but that has already been violated in the statement of the question when it is stated that the lamp will have no time whatsoever between changes at t=2 min. The person posing the paradox is trying to have it both ways, first making us assume a priori that it's possible for the lamp to have an undefined state, then later demanding to know what the state is and declaring it a paradox when the answer is "undefined".
This seems to me like saying that something has no color, and them demanding to know what color it is and declaring it a paradox when no answer an be given that is consistent with my initial contention that it has no color. Am I missing something?  —Preceding unsigned comment added by 128.227.16.53 (talk) 18:46, 14 July 2008 (UTC)
replies: text: It seems to me that you just restated the first paragraph of "Discussion". In any case, the availability of a resolution for a paradox doesn't mean that it isn't a paradox. Your analysis contains an implication that the lamp does not have a state, which is itself a nontrivial argument. Arriving from that conclusion from the stated problem (accelerated flickering) is much more interesting than merely proclaiming that something has no color. Melchoir (talk) 00:25, 15 July 2008 (UTC)

text: Yes, I think you are missing something, and that is that Thompson introduced this thought experiment to question the possibility of supertasks. A supertask is supposed to perform an infinite number of actions, and usually this is accomplished by doing each action in half the time as the previous one, which means that if the first action takes 1 unit of time, then after 2 units of time an infinite number of actions should have been performed. Thompson thought that this was problematic (as in Zeno's paradox, how can an infinite number of actions ever be completed?), and used the Thompson's Lamp to illustrate the problem. To be precise, if you accept the possibility of supertasks, then you have to accept the possibility of Thompson's Lamp, but the problem with that is that the state of the lamp after infinite switches is completely undetermined, which seems strange at the least. Indeed, Thompson found this to be so strange as to declare the Thompson Lamp scenario to be impossible and, by modus tollens, the whole idea of supertasks. So, if you find the Thompson's Lamp scenario to be problematic as well, you may well be on Thompson's side. You may want to take a look at the Balls and vase problem, which is equally, if not even more problematic, and thus can also be seen as a reductio ad absurdum against the possibility of supertasks. Also see the Zeno machine page. —Preceding unsigned comment added by 72.226.66.230 (talk) 14:22, 15 July 2008 (UTC)
thread_title: More explanation?

message: I see this question as being the same as asking what is the last digit of the infinite decimal that equates to 10 divided by 99, i.e. 0.1010101010... Essentially, I interpret the question as asking what happens at the tail end when something occurs 'infinitely many' times.
The real question (and perceived paradox) is that if an infinite decimal can exist in its entirety as a static object, then why is there not a last digit, or if there is one, what is it.  — Preceding unsigned comment added by PenyKarma (talk • contribs) 00:35, 17 February 2015 (UTC)
thread_title: An infinite decimal problem?

message: I have just read this article, so I am no expert.
But it seems to me that we have a definition of a discontinuous function -more and more discontinuous as 2 is approached- for 0<=t<2 and we are asked about the behaviour of the function at 2. If you want me to tell the value at 2 when you only describe it for t<2 i would answer "at 2, this takes whatever value you want, no matter what has happened before". Also note that the function is not clearly defined at times 1,1+1/2,1+1/2+1/4 (is it on or off when it is switching?)
The trick of forcing an answer by saying that all time intervals sum up to 2 seems to need a careful reading. This sum is 1+1/2+1/4+1/8+1/16+1/32+..., not the same type of sum as say 1+1. What is exactly 2 is a limit, the sum of an infinite series, not a sum of any finite number of terms.
A simpler example to illustrate what i mean can be: let the function be 1 at 0, and at times 1, 1+1/2, 1+1/2+1/4,... the function changes to 1 (it was already 1, actually). Now tell me the value at 2. An answer might be: if we assume continuity, it is 1. But without assuming continuity, every value is possible at 2. In the original lamp problem, assuming continuity is impossible.  —Preceding unsigned comment added by 195.235.199.101 (talk) 17:33, 11 November 2009 (UTC)
thread_title: A series or a function?

message: Seeing, or not, an analogy with Grandi’series is a matter of taste. It may, or may not, help think about Thomson’s lamp, but it is a quite different problem.
The state f of Thomson’s lamp is a function of time, perfectly defined for all t such that 0 ≤ t < 2. The definition is given in short in a table at the left of the article. The state does not change except for toggling the switch, so that we not only know that f(0) = On and f(1) = Off, but we also know that f(t) = On for all 0 ≤ t < 1, and  f(t) = Off for all 1 ≤ t < 1.5, et cetera. This a function of a real variable t on the interval [0,2[ (where the last [ means: 2 excluded). It is not a function with a real value: the value of f(t) is in a set with two elements: {On,Off}. You could write it {True,False} or {0,1}, but f is not a function like y = f(t) with some 0 ≤ y ≤ 1 for all 0 ≤ t < 2.
The question is the value of f(2). There is no answer because by definition, by choice, by decision (0 ≤ t < 2), there is no f(2). This is of course no real world lamp (as was justly stressed by others), but in the real world, the lamp would stay in its last state, if nobody comes to put in some other state at t = 2. You could be tempted to say that because a value for f(2) was not provided, one should take f(2) to be the last state On or Off before t = 2. This is in fact the problem put by Thomson.
So the paradox is about there being no last state before t = 2, thus no chance to extend f with this as f(2). It is not about some limit or sum or integral of f, and still less about some limit or sum or integral of another series. The function f has no limit because no such thing has a meaning in the set {On,Off}. Even if we were to treat {0,1} as real numbers in [0,1], f(t) has no limit. Even if there was some alternative concept of a limit with lim f(t) = ½ (t → 2), this would not do because here 0 and 1 mean Off and On. The problem may be clearly stated with the help of f(t) —this is also a matter of taste— but not solved with f(t). -- Dominique Meeùs (talk) 19:56, 25 September 2013 (UTC)
thread_title: Definitely a function

message: A solution is that it is in fact 1/2, because as you get closer to 2 minutes, then the differences between the number of minutes of the time it is on and off rapidly decreases. At the end, it would be blinking, so it is 1/2. 24.1.201.172 (talk) 19:28, 6 June 2010 (UTC)
replies: text: Yes and no. That's one solution, another is that any value between 0 and 1 can be attained, or all at the same time. The paradox is not solved unequivocally. Besides read the article, then source your statements, so that we may use them in the article in the future. This article discussion page is mainly for article discussions, otherwise for technical details that can be used to clarify and cleanup the article. Rursus dixit. (mbork3!) 09:31, 12 June 2010 (UTC)
thread_title: Solution

message: The interesting aspect of Thomson's Lamp is, as stated in the article, the conflict between the indeterminacy of the function at t=2 versus the "intuitive assumption that one should be able to determine the status of the lamp and the switch at any time". This conflict is intended to clinch a reductio ad absurdum argument disproving the concept of supertasks. But it is not enough of a contradiction to do so. Instead, this conflict merely illuminates the incompleteness of the analogy between the mathematical series and the switched lamp. 
The mathematical series is not hard to accept. T=2 is a limit which is approached asymptotically but is never reached over any finite number of terms. There's nothing too challenging about the concept that the number of terms, and therefore the odd/even state of the last term, is undefined. The conflict comes when we try to map this onto the physical lamp and switch. 
The reason that we feel intuitively certain that the lamp's status cannot be indeterminate at t=2 is because the lamp is a physical, tangible object whose state can easily be determined. However, as a physical, tangible object it is not capable to respond to a stimulus of infinitesimally short duration nor is it able to respond within an infinitesimally short time. At some point in the series, the duration of the interval will become less than the minimum response time of the lamp/switch system. Any physical lamp/switch system working within a specified environment will have such a minimum response time, which can be experimentally determined. Even an idealized lamp/switch system will reach a minimum response time beyond which quantum effects distort the outcome. 
Once the minimum response time is reached, the iterations will proceed at a speed determined by the apparatus rather than the mathematical series. Within one to two iterations, T≤2 will be achieved and the status of the lamp will be determinate.
DuardF (talk) 14:20, 25 December 2011 (UTC)
thread_title:  Incomplete Analogy to Mathematical Series 

message: Consider a mathematician who has two minutes to solve a problem regarding a mathematical model which starts out resembling a real-world situation.
At one minute, he believes he has the right answer. Thirty seconds later he has revised his opinion ...
I wonder if the the reply to the theoretical problem is rather "undecidable" than "indeterminate". It seems to be an unanswerable decision problem.
However, since the problem is couched in real-world terms of the 1950s, perhaps a practical answer is called for.
If Thompson did indeed phrase the main question as stated, "Is the lamp switch on or off after exactly two minutes?", the answer is "no".
In the real world, the switch is "broken", no longer a switch, having been employed way outside of its spec. 
In the theoretical model, the dichotomy is also broken, not a number 0 or 1 but some other escaped animal.
Maybe there are alternate realities with different logic. 
If the second question is unambiguous as regards "state", the real answer is again "no"; the theoretical answer is a little more difficult.
It seems that, as to the exact state of the lamp now that two minutes are up, we remain in the dark.
I made a small change to the lede to link to supertask, additional to the bibliographical note in the first sentence.
One is one and one is one (talk) 17:12, 10 September 2012 (UTC)
thread_title:  OR maybe 

message: This edit has three problems. It mentions the limit of a series, which is non-standard terminology. Sequences have limits; series have sums. It links "logically" to Mathematical universe hypothesis and "physically" to Digital physics in a sentence describing Thomson's argument. This is really dubious, since those theories were introduced after Thomson's lamp, so it's reasonable to want a source that makes the connection. Without a source, the links are original research. The link to The Unreasonable Effectiveness of Mathematics in the Natural Sciences is also original research, as it seems to imply a connection which is again unsupported.
I'll edit the prose again, this time using the "sum of series" terminology and preserving the link to Thought experiment. Thanks, Melchoir (talk) 03:41, 13 March 2013 (UTC)
thread_title:  Digital physics 

message: Seeing that the article has been tagged as relying on  sources, the following book may be useful:
* William Poundstone, 1988, Labyrinths of Reason, chapter 8 (pp. 143–159),  (includes bibliography, but not inline citations).
This popular work is a survey (secondary source) of several paradoxes and problems of epistemology discussed in scientific and philosophical literature. The chapter on Thomson's lamp describes various approaches to the problem and compares it to other paradoxes of infinity. (His observation on presenting the problem: "But to answer the riddle of the Thompson lamp would be preposterous. It would be tantamount to saying whether the biggest whole number is even or odd!") The real riddle, of course, is not to find the right answer, but to understand what exactly is wrong with the question. ~ Ningauble (talk) 15:32, 12 January 2014 (UTC)
thread_title:  Secondary sources 

message: Here's how it seems to me. Am I missing something?
At one minute, the lamp is turned off.
If this is instantaneous, then the lamp changes state at one minute. Its status at one minute is undefined. Similarly at 1.5, 1.75, 1.825, ..., though it is defined between these times.
If it is not instantaneous, the problem remains, but with a delay between toggling the switch and the lamp changing state. For example, at one minute it is toggled off; it responds by turning off at 1 minute 1 second, at which time its status is again undefined.
In addition, there will come a point when the delay will be longer than the time between toggles of the switch, and the experiment will crash.  — Preceding unsigned comment added by SputnikIan (talk • contribs) 15:36, 23 February 2016 (UTC)
thread_title:  Is the lamp's status defined at all times prior to @ minutes? 

message: On the Thomson Lamp Paradox, Earman and Norman (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 
Following Earman and Norman's development (p. 236) in establishing this schedule of switching, we define Thomson's idealized lamp to be ON only during the following time intervals (in minutes):
:[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:
:[1, \frac{3}{2}),  [\frac{7}{4},  \frac{15}{8}), [\frac{31}{16}, \frac{63}{32}), \cdots
We assume that the transitions on switching from one state to another are instantaneous. 
Notice that every element of each of these intervals is less than 2. Therefore, the state of Thomson's Lamp is undefined for time t= 2. 
We can represent the schedule of switching by a partial function f:\mathbb{R}\to \{ 0, 1\} such that:
: 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 t is the elapsed time in minutes.
The function f is defined only on the half-open interval [0,2). On this schedule of switching, f(2) will be undefined. Therefore, if we assign a value of 0 or 1 to f(2), then, as Earman and Norman point out, that value would be compatible with the above switching schedule on the interval [0,2). There would be no contradiction in either case.
John Earman and John Norton (1996) "Infinite Pains: The Trouble with Supertasks. In Benacerraf and his Critics," Adam Morton and Stephen P. Stich (Eds.), p.231-261.
:: If you sit at your desk and manually write out the series of the intervals defining f(t), indeed you will never reach the time t=2. This is fine if t is some kind of abstract quantity that never approaches you on its own. The time, however, does not wait for you and it does not allow you to take infinite amount of time to write your series endlessly; it comes upon you whether you like it or not; in other words, t is not always less than 2, and t=2 comes to you unhurriedly but steadily. Before t=2, you were pressing the switch like a demon, at t=2, you have stopped doing it. So there must be a transition from pressing to not-pressing; corresponding to this transition, the lamp must also have had a transition from the ON-OFF changing process to a non-changing state. So whether you like it or not, as time crosses t=2, you are forced to define it at t=2 and thereafter. What is that non-changing state? The paradox remains. --Roland (talk) 02:39, 26 November 2016 (UTC)
replies: text: The answer in this interpretation is "we do not have enough information from the setup to determine what that state is". It could be on, it could be off, it could suddenly turn into a lemon. The question does not give you enough information to tell you what happens. Double sharp (talk) 06:54, 3 February 2021 (UTC)
thread_title:  A Non-Paradoxical Interpretation 

message: The way Grandi’s series is articulated in this article, there is no n. The article then goes on to discuss “when n…” 2601:240:E300:51C0:0:0:0:D619 (talk) 03:02, 30 August 2023 (UTC)
thread_title:  Grandi’s Series