Talk:Thomson's lamp

Task is implicitly impossible

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)gabe76

Benacerraf makes that point as well. See the Supertask article. —Preceding unsigned comment added by 128.113.89.57 (talk) 01:26, 29 October 2010 (UTC)

(random heading)

(inserted for readability Rursus dixit. (^mbork³!) 09:19, 12 June 2010 (UTC))

How can this quote be true? "The sum of all these progressively smaller times is exactly two minutes." The sum of an infinite progression of intervals that are halved would never reach 2 minutes. But maybe that's the crux of the paradox and I'm just not getting it. :-) If so, seems more like a koan to me.

Just like there's a formula to quickly find the sum of all integers to 10,000 there's one for an infinite series that converges. Click on the word sum in the article (or below) --JimWae 2005 July 3 23:25 (UTC)

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)

• Good point, time increments for change can be devided but not time itself. I think thats a problem left unresolved on some of the other paradox pages. --RickO5

--

Quantum physics aside, there are two mathematical aspects to this problem:

• At 2 minutes the lamp will be switching on and off an an infinite rate

• The exact 2 minute mark occurs instantly - i.e. in an infinitely small length of time

The solution would seem to lie in the division of the infinite by the infintesimal, which I'm not sure has a defined meaning in mathematics.

It is obviously physically impossible, but it is not conceptually impossiible, as we've been conceptualizing it here. This would lead me to assume that the answer is a CONCEPT rather than a physical QUANTITY. Trying to derrive a physical quantity from such a question is like asking "What is Truth divided by Beauty?" —Preceding unsigned comment added by 84.92.193.137 (talk) 17:50, 28 August 2008 (UTC)

Mathematical proof...

I'm marking this stuff as original research. I doubt that the mathematical theory of convergent series is really capable of "proving" anything about the ill-posed metaphysical question at hand. And I find it hard to believe that a reputable source has made this assertion. Melchoir 04:50, 2 January 2007 (UTC)

I've finally skimmed Thomson's original paper. He says the opposite of this article: that the divergence of the series has nothing to do with the impossibility of the supertask. I'll be rewriting the section soon. Melchoir 11:00, 13 January 2007 (UTC)

Done! Melchoir 19:46, 13 January 2007 (UTC)

I think this question is the same as asking: what is the limit of sin(1/x) when x goes to 0? The answer is: it doesn't exist. See this link: http://www.math.washington.edu/~conroy/general/sin1overx/ —Preceding unsigned comment added by 201.53.83.199 (talk • contribs) 19:33, 12 September 2007

It's certainly related, but not the same. Again, the metaphysical problem does not so easily reduce to mathematics. Melchoir 20:23, 12 September 2007 (UTC)

Incorrect

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)

I disagree with the use of this series as a "solution" to the problem. Why sum the results? This is not explained, and makes no sense. I think it's a discredit to mathematics to have this "mathematical" solution to the problem on this page. —Preceding unsigned comment added by 70.150.87.29 (talk) 21:18, 16 April 2008 (UTC)

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)

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)

More explanation?

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)

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)

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)

An infinite decimal problem?

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)

A series or a function?

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)

The third question asks "would it make any difference if the lamp had started out being on, instead of off?" With regards to the state on/off of the lamp at t=2 minutes, it does not make any difference, but the total time the lamp has been on is (2/3)*2 minutes if it starts on and (1/3)*2 minutes if it starts off. This is what the description of the procedure determines, the integral of the function from t=0 to t=2, not the value of the function at t=2. --84.127.78.170 (talk) 15:57, 12 November 2009 (UTC)

Definitely a function

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)

Solution

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)

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)

Speed Limit

Assuming the switch is made of mass, which it would be if it were a lamp, then the speed of light is indeed the speed limit but the switch cannot be flicked at that speed. In actuality, it would be up to a near infinitely small number less than the speed of light itself. If the switch is not made of mass, then the speed of light can be achieved. I don't know if this affects the calculations in any way or not, I just thought this would be good to add. 69.153.116.124 (talk) 03:31, 3 November 2010 (UTC)

When you flip the switch, nothing happens at the light immediately. There is a delay because the voltage at the bulb does not change instantly in response to the change at the switch. You have to calculate how long it takes the signal to travel.

71.109.149.173 (talk) 18:26, 11 January 2014 (UTC)

Please observe WP:NOTFORUM and WP:OR. Thanks. Paradoctor (talk) 00:48, 12 January 2014 (UTC)

Off either way

The fuse would blow if one tried to turn on and off that fast. 220.255.1.82 (talk) 12:45, 28 August 2011 (UTC)

More than than the friction from flicking it on and off an infinite amount of times in such a short span of time would vaporize the switch completely, effectively disconnecting the circuit, the lamp would at the end be off. — Preceding unsigned comment added by 161.49.249.254 (talk) 00:19, 9 December 2011 (UTC)

No, it's on either way, unless it's an LED light. Most others lights remain on for a small interval after the flow of electricity stops. For example, an incandescent light remains on until the filament cools.

Keep in mind that the flow of electricity in an a.c. current starts and stops, and reverses direction, repeatedly, but no one considers these states as being "off". A light is not off until the power is off long enough for it to stop glowing.

The state of the light (glowing or not glowing) is whatever it is when it is receiving electricity 1/2 the time and no electricity 1/2 the time.  : When the intervals get small enough, it no longer matters whether it is, or is not, receiving electricity at that moment.

71.109.149.173 (talk) 18:24, 11 January 2014 (UTC)

Please observe WP:NOTFORUM and WP:OR. Thanks. Paradoctor (talk) 00:46, 12 January 2014 (UTC)

Incomplete Analogy to Mathematical Series

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)

OR maybe

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)

Digital physics

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)

Secondary sources

Seeing that the article has been tagged as relying on {{primary}} sources, the following book may be useful:

• William Poundstone, 1988, Labyrinths of Reason, chapter 8 (pp. 143–159), ISBN 0385242611 (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)

Is the lamp's status defined at all times prior to @ minutes?

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)

The state of Thomson's Lamp is simply unspecified for $t\geq 2$ minutes

Thomson's Lamp can be defined to be ON only in the following time intervals (in minutes):

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

Whereas, it can be defined to be OFF only in the following time intervals:

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

Notice that every element of each of these intervals is less than 2. Therefore, the state of Thomson's Lamp is unspecified or undefined for time $t\geq 2$. --Danchristensen (talk) 18:20, 4 March 2016 (UTC)

That's a reasonable interpretation of the situation, although as a logical argument, it seems to beg the question. Is it attributable to a published source? If so, it should be added to the article. (If not, it's original research.) Melchoir (talk) 21:08, 4 March 2016 (UTC)

Original research? It's only a single paragraph, four sentences long stating the obvious. It seems very unlikely that anyone would challenge these observations.

--Danchristensen (talk) 04:10, 5 March 2016 (UTC)

The claim that the lamp is on during [0, 1) and off during [1, 3/2) is dubious. Is the lamp's state defined at t=1? Just above this topic on the talk page, SputnikIan claims that it isn't. Now, I am not interested in their answer to this question, or to yours. I am only interested in treatments within the published literature. Is there a published work that defines the lamp's state using these half-open intervals?

Another problem is the word "Therefore". You say that the lamp's state is defined only in these time intervals, and therefore, the state is undefined for t>=2. That's begging the question. Why is the lamp's state defined only in these time intervals? To answer that question, we already have to explain why the state is undefined for t>=2. Well, once we've done that, there's no point in messing about with the intervals. They don't add any insight; they're just filler. Melchoir (talk) 23:31, 7 March 2016 (UTC)

Remember that the whole point of the article is to demonstrate a supposed contradiction arising from infinitely many changes of states occurring in a finite interval:

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.

So, this was not meant to be an exercise in electrical engineering. The idealized lamp and switch are part of a thought experiment to demonstrate a mathematical principle. There was no mention of any transition times between the ON and OFF states, so I assume there are none. It just makes sense in this context.

--Danchristensen (talk) 19:41, 8 March 2016 (UTC)

The whole point of the Wikipedia article is to summarize the published literature. It is not to resolve the issue one way or the other. I've replaced that paragraph with Thomson's original language, which is more concise anyway. Melchoir (talk) 21:54, 8 March 2016 (UTC)

There is a reference to a paper by Benacerraf on this aspect of the "paradox" here:

"Are there other consistent ways to describe the final state of Thomson’s lamp in spite of the missing limit?

"Benacerraf (1962) pointed out a sense in which the answer is yes. The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. It may still be possible to “complete” the description of Thomson’s lamp in a way that leads it to be either on after 2 minutes or off after 2 minutes. The price is that the final state will not be reached from the previous states by a convergent sequence. But this by itself does not amount to a logical inconsistency." Stanford Encyclopedia of Philosophy

You also have a reference to the original paper in your References section

--Danchristensen (talk) 17:44, 7 June 2018 (UTC)

Re: Mathematical analogy proof based on falsehood

The summation does not converge. It is then a falsehood to say that it is equal to some real number S. And all things follow from a falsehood. --Danchristensen (talk) 19:56, 4 March 2016 (UTC)

I think Thomson's article and the Wikipedia article both adequately label the series as divergent, not convergent. It is admittedly misleading when Thomson says "its sum is 1/2", since the unqualified word "sum" is almost always reserved for convergent series, at least in the writing of contemporary mathematicians. But Thomson's meaning is clear enough, since he cites Hardy's Divergent Series for this claim. It is true that most summation methods for divergent series assign this series a regularized sum of 1/2; for instance, its Cesaro sum is 1/2. The main editorial question for us is how to explain that distinction. Of course, we need to avoid falsehoods! I think the current wording: "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." is good enough. Melchoir (talk) 21:04, 4 March 2016 (UTC)

From the last sentence in the article, "Later, he [Thomson] 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.'"

Here, Thomson himself seems to be dismissing as irrelevant these generalized definitions for sums of series as a red herring. As such, there should really be some mention of my simple observations in the article.

--Danchristensen (talk) 13:29, 7 March 2016 (UTC)

Thomson is dismissing as irrelevant the distinction between convergence and divergence. This is the opposite position from insisting that the sums of convergent series are more relevant than the sums of divergent series.

Anyway, what observations do you want mentioned? Melchoir (talk) 23:37, 7 March 2016 (UTC)

No contradiction proven

From the article:

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.

We are talking about a partial function $f:\mathbb {R} \to \{0,1\}$ such that:

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

Left unspecified or undefined is the value of $f(t)$ for $t\geq 2$ or $t<0$.

Even if we add the assumption that $f(2)=1$, we would not be able to prove that "there must have been some last time, right before the 2-minute mark, at which it was flicked on." By the above definition, there is no last switching on or off before the 2-minute mark. Likewise if we were to assume $f(2)=0$. Since we cannot prove it, there is no contradiction.

--Danchristensen (talk) 13:40, 8 March 2016 (UTC)

I've replaced that paragraph with Thomson's original argument, which doesn't rest on knowing the "last" switch. In fact, Thomson explicitly stated otherwise: "But in any case it should be clear that no assumption about a last task is made in the lamp-argument. If the button has been jabbed an infinite number of times in the way described then there was no last jab and we cannot ask whether the last jab was a switching on or a switching-off. But we did not ask about a last jab ; we asked about the net or total result of the whole infinite sequence of jabs, and this would seem to be a fair question." Melchoir (talk) 21:58, 8 March 2016 (UTC)

The argument now reads:

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.

Seems a bit "hand-wavy" to say the least. The partial function $f$ is well defined on the interval $[0,2)$ and it is undefined at the 2-minute mark. So, no contradiction can be obtained by simply making the additional assumption that $f(2)$ is either $1$ or $0$.

--Danchristensen (talk) 23:03, 8 March 2016 (UTC)

Look, if you find a published source that makes this argument, you're welcome to add it with a citation. To be honest, I doubt that you'll find your argument in the literature, for reasons that I've expressed above. But it would be interesting to be proven wrong! Melchoir (talk) 23:50, 8 March 2016 (UTC)

You might have a look at:

Berresford

Earman and Norton p.236

--Danchristensen (talk) 02:03, 9 March 2016 (UTC)

That's progress! Feel free to integrate them into the article. Melchoir (talk) 05:50, 9 March 2016 (UTC)

Would it suffice to quote Earman and Norman: "The lamp is not paradoxical since any [state of the lamp at the 2-minute mark] would be compatible with the schedule of switching prior to [that time]." p.237

--Danchristensen (talk) 19:53, 9 March 2016 (UTC)

Ideally we would take a paragraph or two to describe how Benacenaf and Earman & Norman arrive at that conclusion. But for now, sure, any mention of their argument is better than nothing! Melchoir (talk) 20:36, 9 March 2016 (UTC)

A Non-Paradoxical Interpretation

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)

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)

--Danchristensen (talk) 21:59, 9 March 2016 (UTC)

Grandi’s Series

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)