The lamp starts switched on, and the timer starts. Another flick will turn the lamp off. —Preceding unsigned comment added by 128.227.16.53 (talk) 18:46, 14 July 2008 (UTC). 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. 2 I doubt that the mathematical theory of convergent series is really capable of "proving" anything about the ill-posed metaphysical question at hand. If it is not instantaneous, the problem remains, but with a delay between toggling the switch and the lamp changing state. [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. —Preceding unsigned comment added by 70.150.87.29 (talk) 21:18, 16 April 2008 (UTC). The trick of forcing an answer by saying that all time intervals sum up to 2 seems to need a careful reading. { 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. is defined only on the half-open interval The scientist continues toggling the lamp, waiting one-half of the previously waited time between toggles. or . First Known Solution to Thomson’s Lamp Attached is a draft academic paper detailing what I believe to be the first known solution to the paradox posed by Thomson’s Lamp. — Preceding unsigned comment added by SputnikIan (talk • contribs) 15:36, 23 February 2016 (UTC). This sum is 1+1/2+1/4+1/8+1/16+1/32+..., not the same type of sum as say 1+1. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on. Therefore, if we assign a value of 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 chapter on Thomson's lamp describes various approaches to the problem and compares it to other paradoxes of infinity. {\displaystyle f(2)=1} One minute later Thompson turns the lamp off, thirty seconds later turns it on, fifteen seconds later it's off again, etc. It cannot be on, because I did not ever turn it on without at once turning it off. It seems to be an unanswerable decision problem. But let's first see why the solution can not be mathematical: After a total sum of two minutes of toggling, what is the state of the lamp (on or off)? I wonder if the the reply to the theoretical problem is rather "undecidable" than "indeterminate". 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. For even values of n, the above finite series sums to 1; for odd values, it sums to 0. f . 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. This is equivalent to trying to solve an equation on a Cartesian plane, where there is an asymptote to e.g. This is not explained, and makes no sense. It is not a function with a real value: the value of f(t) is in a set with two elements: {On,Off}. The state of Thomson's Lamp is simply unspecified for, Re: Mathematical analogy proof based on falsehood, http://www.math.washington.edu/~conroy/general/sin1overx/, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, generalized definitions for the sums of series, https://en.wikipedia.org/w/index.php?title=Talk:Thomson%27s_lamp&oldid=844881204, Creative Commons Attribution-ShareAlike License, the biggest problem is that within any physical reality, after a while you bump up against quantum consideration and limits of measurement. This domain has expired. —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?" If Thompson did indeed phrase the main question as stated, "Is the lamp switch on or off after exactly two minutes? Therefore, the state of Thomson's Lamp is unspecified or undefined for time {\displaystyle f(2)} In the original lamp problem, assuming continuity is impossible. 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 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. --84.127.78.170 (talk) 15:57, 12 November 2009 (UTC). {\displaystyle [0,2)} 2 ) Maybe there are alternate realities with different logic. 2 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. service provider for further assistance. {\displaystyle f(2)=0} At 2 minutes the lamp will be switching on and off an an infinite rate; The exact 2 minute mark occurs instantly - i.e. It may, or may not, help think about Thomson’s lamp, but it is a quite different problem. I'll edit the prose again, this time using the "sum of series" terminology and preserving the link to Thought experiment. = Thomson's Lamp can be defined to be ON only in the following time intervals (in minutes): Whereas, it can be defined to be OFF only in the following time intervals: Notice that every element of each of these intervals is less than 2. Seeing that the article has been tagged as relying on {{primary}} sources, the following book may be useful: This popular work is a survey (secondary source) of several paradoxes and problems of epistemology discussed in scientific and philosophical literature.

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