# Thompson's Lamp **URL:** \ **Category:** Great Debates\ **Created:** [December 3, 2021, 11:17pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473 "2021-12-03T23:17:30Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Max\_S](https://avatars.discourse-cdn.com/v4/letter/m/46a35a/32.png) [@Max\_S](https://boards.straightdope.com/u/Max_S)\ **Post date:** [December 3, 2021, 11:17pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/1 "2021-12-03T23:17:30Z")
Continuing the discussion from [Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/91): Here is a dedicated topic for this tangent. > [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/91): > > Consider [Thomson’s lamp](https://en.wikipedia.org/wiki/Thomson%27s_lamp): it’s switched on after a second, off after half a second, on again after a quarter, and so on. What is its state after two seconds? Clearly, it can’t be on: for every time that it has been switched on, it also has been switched off. And for the same reason, it can’t be off! But it must be in either of these states (those being the only states available to it). So whatever state it is in can’t be a function of its history. ~originally posted by **Half\_Man\_Half\_Wit** ---
**Author:** ![Max\_S](https://avatars.discourse-cdn.com/v4/letter/m/46a35a/32.png) [@Max\_S](https://boards.straightdope.com/u/Max_S)\ **Post date:** [December 3, 2021, 11:17pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/2 "2021-12-03T23:17:51Z")
> [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/96): > > This is only a paradox because it assumes time is continuous, while also assuming the state of the lamp is discrete. If time were discrete the infinite series becomes a finite series. If the state of the lamp were continuous, the mathematical answer ‘ _S_ converges at 0.5’ is applicable. It would mean perhaps the switch is half way between being on and off, or the lighting element is still warming up. ~Max. ---
**Author:** ![Max\_S](https://avatars.discourse-cdn.com/v4/letter/m/46a35a/32.png) [@Max\_S](https://boards.straightdope.com/u/Max_S)\ **Post date:** [December 3, 2021, 11:18pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/3 "2021-12-03T23:18:18Z")
> [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/99): > > It doesn’t assume time is continuous, and ‘converges to 0,5’ isn’t the mathematical answer. Each point in time at which a switch happens is after a rational time interval \frac{1}{2^n} is rational for all n ), and the rational numbers aren’t a continuum (there’s always an irrational number between any two rational numbers). That’s nitpicking of course—you presumably meant that the difference between two successive switches shrinks without lower bound. That’s true, but it’s true for time in all of our best physical theories (in fact, time is a continuum in all of our best physical theories)—the fundamental nature of the Planck time being only conjectural at this point. More to the point, the series that determines the state of the lamp after two seconds is 1 - 1 + 1 - 1 + … The limit of this isn’t 0,5—it’s divergent, so it simply has no well-defined limit. ~originally posted by **Half\_Man\_Half\_Wit** ---
**Author:** ![Max\_S](https://avatars.discourse-cdn.com/v4/letter/m/46a35a/32.png) [@Max\_S](https://boards.straightdope.com/u/Max_S)\ **Post date:** [December 3, 2021, 11:19pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/4 "2021-12-03T23:19:27Z")
> [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/100): > > > [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/99): > > > > More to the point, the series that determines the state of the lamp after two seconds is 1 - 1 + 1 - 1 + … The limit of this isn’t 0,5—it’s divergent, so it simply has no well-defined limit. > > Although I suppose that’s only true if the state change is instantaneous, which might be what you meant. If it’s not, then at some point, we’d reach a situation where state changes could no longer be completed before the next switch happens. ‘0,5’ might then be a sensible answer if the ramp-up and ramp-down are symmetrical, although they probably wouldn’t be. But really, this is unrelated to the point I was making, which was [ed. note: continued in [Lack of Freewill](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/100) topic] ~originally posted by **Half\_Man\_Half\_Wit** ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [December 4, 2021, 1:09am UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/5 "2021-12-04T01:09:10Z")
A few problems here. Firstly, it’s a bit weird phrasing the problem as being that each “on” is followed by an “off” and vice versa. The problem is really that there are infinite steps before 2. If I flip the switch every 1 second then it would also be true that every “on” is followed by an “off” and vice versa, yet it would be trivial to say that the lamp is switched off at precisely 2.0 Secondly. we can ask the question of whether infinite sets like this can be manifest in physical reality – and bear with me here because I am not going to make a claim either way. e.g. does an electron really have an infinite number of possible paths that the universe somehow “computes”? If the answer is “yes” then the universe must be capable of resolving paradoxes like the lamp: it must be (somehow) capable of executing infinite steps to arrive at a simple discrete result. If the answer is “no” then the paradox is irrelevant to empirical claims like the existence of free will. Either way it doesn’t seem to support the desired conclusion. Finally I just want to reiterate in this thread my position that the concept of free will is nonsense (note: this is a very different thing from saying you don’t make decisions or act according to your own preferences). You think, you plan, you change your mind, you dream…and all this is causally connected to the universe because no other concept of making a decision makes any sense… ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [December 4, 2021, 7:39am UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/6 "2021-12-04T07:39:25Z")
> [@Mijin](#): > > Firstly, it’s a bit weird phrasing the problem as being that each “on” is followed by an “off” and vice versa. The problem is really that there are infinite steps before 2. > If I flip the switch every 1 second then it would also be true that every “on” is followed by an “off” and vice versa, yet it would be trivial to say that the lamp is switched off at precisely 2.0 Because it wouldn’t be switched on again for the last time it was switched off, and hence, not every time it was switched off was followed by it being switched on again. The phrasing directly implies the infinite number of steps (such a thing is only possible in an infinite set), but the infinity isn’t itself the problem—think of a lamp that is switched on at every rational time point, and switched off at every irrational time. Then, between each two ‘offs’ there’s an ‘on’ and vice versa, but it’s not switched on for every time it was switched off (there being vastly more irrational numbers than rational ones), and there’s no problem with its state after two seconds, which will be ‘on’. > [@Mijin](#): > > Secondly. we can ask the question of whether infinite sets like this can be manifest in physical reality – and bear with me here because I am not going to make a claim either way. e.g. does an electron really have an infinite number of possible paths that the universe somehow “computes”? That the universe ‘computes’ anything, of course, already limits the possible options—clearly, such a process as that of Thomson’s lamp is non-computable, so you’re kind off saying, well, then the universe must be somehow capable of computing the uncomputable, and hence, you’re smuggling in the underlying assumption that Thomson’s lamp is intended to challenge (namely, that there’s a mechanistic way to account for every process in the universe). As argued in the free will-thread, if the universe has the resources of resolving Thomson’s lamp-like situations, which our best current physical theories tell us it does, then it has the resources of completing the infinite regress encountered in the notion of free will (and it will also not be computable). > [@Mijin](#): > > If the answer is “no” then the paradox is irrelevant to empirical claims like the existence of free will. The existence of free will isn’t an empirical claim. At most, our experience of it could be called ‘empirical’, but as that data is subjective and hence, not publicly available, I don’t think many would agree with such a notion of ‘empirical’. > [@Mijin](#): > > You think, you plan, you change your mind, you dream…and all this is causally connected to the universe because no other concept of making a decision makes any sense… This is what I was talking about in the other thread: you’re here appealing to the notion of causal connection as if it was completely unproblematic—but it’s subject to difficulties just as great as (and IMO, equivalent to those of) the notion of free will. ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [December 4, 2021, 12:45pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/7 "2021-12-04T12:45:00Z")
> [@Half\_Man\_Half\_Wit](#): > > Because it wouldn’t be switched on again for the last time it was switched off, and hence, not every time it was switched off was followed by it being switched on again I don’t think you properly read my objection, because no, in my version there are still infinite steps, just a finite number before 2. It’s an illustration that “Every on is followed by off, and vice versa” is not actually what creates the problem, it’s the infinite steps before 2. > [@Half\_Man\_Half\_Wit](#): > > then it has the resources of completing the infinite regress encountered in the notion of free will (and it will also not be computable). There is no reason that free will needs to be an infinite regress – that would just add _one more_ problem to the concept. Because one could explain where the “will” comes from hypothetically in one step. So what this comes down to is that **if** we posit one extra problem regarding the concept of free will, and **if** the universe can deal with infinite regress, then **that particular** problem has been solved. I think this level of argument says all we need to about how defensible the free will concept actually is. > [@Half\_Man\_Half\_Wit](#): > > The existence of free will isn’t an empirical claim. At most, our experience of it could be called ‘empirical’, but as that data is subjective and hence, not publicly available, I don’t think many would agree with such a notion of ‘empirical’. Right, so it’s a philosophical claim, and we’re back to square one. We have no reason in the first place to posit the existence of this thing. Prior to neuroscience maybe the notion of decisions being causally disconnected made a kind of intuitive sense – that’s the beginning, middle and end of the evidence for the concept. > [@Half\_Man\_Half\_Wit](#): > > This is what I was talking about in the other thread: you’re here appealing to the notion of causal connection as if it was completely unproblematic—but it’s subject to difficulties just as great as (and IMO, equivalent to those of) the notion of free will. Please elaborate. ---
**Author:** ![Bonum\_Legatum](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bonum_legatum/32/6183_2.png) [@Bonum\_Legatum](https://boards.straightdope.com/u/Bonum_Legatum)\ **Post date:** [December 4, 2021, 1:17pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/8 "2021-12-04T13:17:25Z")
The way that the problem is stated reminds me of a problem in computer programming that uses recursion. For these, the program must have an explicit stated point at which the recursion ends. Otherwise the program just keeps recursing and crashes. So consider the original statement: It only describes the condition of the lamp - the position of the switch - at any point in time up to two seconds. ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [December 4, 2021, 2:04pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/9 "2021-12-04T14:04:10Z")
> [@Mijin](#): > > I don’t think you properly read my objection, because no, in my version there are still infinite steps, just a finite number before 2. I don’t think I follow. At what point in time, in your model, would there have been an ‘off’ for every ‘on’, and vice versa? (In an effort to limit thread cross-posting, I’ve replied to the rest of your post [over in the other thread](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/105).) ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [December 4, 2021, 2:41pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/10 "2021-12-04T14:41:02Z")
> [@Half\_Man\_Half\_Wit](#): > > I don’t think I follow. At what point in time, in your model, would there have been an ‘off’ for every ‘on’, and vice versa? No point in time. I never made such a claim. I simply pointed out that the “every X is followed by the Y” is not the issue here, it’s the infinite steps prior to 2. You could have infinite presses of a _single_ red button and there would still be an issue of if the button gets pressed at 2.0 given that the recursion never gets there. It doesn’t actually matter anyway, I’m just saying it’s an unusual way to present this problem. ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [December 4, 2021, 3:08pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/11 "2021-12-04T15:08:24Z")
> [@Mijin](#): > > I simply pointed out that the “every X is followed by the Y” is not the issue here, it’s the infinite steps prior to 2. It’s exactly the issue. The problem arises because every time it has been turned on, it’s also been turned off again, and every time it’s been turned off, it’s also been turned on again—hence, it can neither be on (because if it’s on, it would’ve been turned off), nor off (same). The fact that this leads to an infinite construction is secondary, because such a state of affairs is only possible for an infinite set of state transitions. In your construction, it’s simply never the case that it’s been turned off for every time it’s been turned on, and vice versa—at every point in time, it’s been turned on or off one more time than the other. Hence, the problem doesn’t arise. Now, you could take some system like the one you’re describing, chuck it into a rotating black hole, and jump in after it in such a manner as to encounter it after an infinite proper time has passed (according to the Malament-Hogarth construction), and then you’d have the same issue again; but again because, then, for every time it’s been turned on, it’s been turned off, and vice versa. ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [December 4, 2021, 4:58pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/12 "2021-12-04T16:58:23Z")
> [@Half\_Man\_Half\_Wit](#): > > In your construction, it’s simply never the case that it’s been turned off for every time it’s been turned on, and vice versa Why do you keep saying that? The number of times my lamp gets turned on or off is the same as the number of integers. Are you saying that there’s a last integer? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 4, 2021, 5:25pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/13 "2021-12-04T17:25:03Z")
The lamp’s state is not specified at 2 seconds, nor at any later time. We are accustomed, when something’s state is not specified at some point, to assume that the state at that point is the limit of known states which approach that point (formally, in mathematics, this would be called “removing a removable discontinuity”). However, in this case, we cannot make such an assumption, because the limit does not exist. Incidentally, this is also the answer (or lack of answer) to another, related puzzle, where you have a “bag of balls”, each with a number on it, and at every step, you remove the lowest-numbered ball and add the next two higher-numbered balls. So for instance, you start with ball 1 in the bag, then remove ball 1 and add balls 2 and 3, then remove ball 2 and add balls 4 and 5, then remove ball 3 and add balls 6 and 7, and so on. The question then asks how many balls will be in the bag after all of the steps are performed. The usual answer is that the bag will be empty, because every ball gets removed at some step or another, but this is incorrect, because this is also a situation where the limit does not exist, and therefore the state of the bag at the end is not specified. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [December 4, 2021, 6:15pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/14 "2021-12-04T18:15:01Z")
> [@Chronos](#): > > this is also a situation where the limit does not exist The limit with respect to which topology? 😄 ---
**Author:** ![FlikTheBlue](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/fliktheblue/32/10972_2.png) [@FlikTheBlue](https://boards.straightdope.com/u/FlikTheBlue)\ **Post date:** [December 5, 2021, 12:35am UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/15 "2021-12-05T00:35:25Z")
I just read the Wikipedia article. It seems like a modern version of Zeno’s paradoxes. Those seem to all fail in the real world as paradoxes due to quantum physics, and this seems like another example. In reality, wouldn’t it take at least one Planck time to flip the switch, and thus there wouldn’t be a real world limit at two seconds? ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [December 5, 2021, 8:40am UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/16 "2021-12-05T08:40:32Z")
> [@Mijin](#): > > Why do you keep saying that? The number of times my lamp gets turned on or off is the same as the number of integers. The number of times your lamp gets turned on or off, after t seconds, is t. Hence, for all t, there is one switch that hasn’t been followed by its opposite. So at no point in time t—i.e. never—is it the case that it’s been turned off for every time it’s been turned on, or vice versa. ---
**Author:** ![Mijin](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mijin/32/9369_2.png) [@Mijin](https://boards.straightdope.com/u/Mijin)\ **Post date:** [December 5, 2021, 12:37pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/17 "2021-12-05T12:37:02Z")
Exactly. The necessary part is steps before t, not the endless chain of On Off. ---
**Author:** ![epolo](https://avatars.discourse-cdn.com/v4/letter/e/b487fb/32.png) [@epolo](https://boards.straightdope.com/u/epolo)\ **Post date:** [December 5, 2021, 9:26pm UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/18 "2021-12-05T21:26:49Z")
> [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/99): > > > [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/99): > > > > It doesn’t assume time is continuous, and ‘converges to 0,5’ isn’t the mathematical answer. Each point in time at which a switch happens is after a rational time interval \frac{1}{2^n} is rational for all n ), and the rational numbers aren’t a continuum (there’s always an irrational number between any two rational numbers). That’s nitpicking of course—you presumably meant that the difference between two successive switches shrinks without lower bound. That’s true, but it’s true for time in all of our best physical theories (in fact, time is a continuum in all of our best physical theories)—the fundamental nature of the Planck time being only conjectural at this point. More to the point, the series that determines the state of the lamp after two seconds is 1 - 1 + 1 - 1 + … The limit of this isn’t 0,5—it’s divergent, so it simply has no well-defined limit. > > ~originally posted by **Half\_Man\_Half\_Wit** Even without the limit of the Planck time, at some point the switch - no matter how small it is - would need to be moving faster than the speed of light. Thompson’s Lamp can’t exist in the real world. Not that that removes its usefulness as a thought experiment. ---
**Author:** ![Max\_S](https://avatars.discourse-cdn.com/v4/letter/m/46a35a/32.png) [@Max\_S](https://boards.straightdope.com/u/Max_S)\ **Post date:** [December 7, 2021, 12:03am UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/19 "2021-12-07T00:03:50Z")
> [@Lack of Freewill doesn't mean lack of choice](https://boards.straightdope.com/t/lack-of-freewill-doesnt-mean-lack-of-choice/954128/100): > > Although I suppose that’s only true if the state change is instantaneous, which might be what you meant. If it’s not, then at some point, we’d reach a situation where state changes could no longer be completed before the next switch happens. ‘0,5’ might then be a sensible answer if the ramp-up and ramp-down are symmetrical, although they probably wouldn’t be. > [@epolo](#): > > Even without the limit of the Planck time, at some point the switch - no matter how small it is - would need to be moving faster than the speed of light. Thompson’s Lamp can’t exist in the real world. Exactly. The demon’s hand is presumably limited to the speed of light, as is the position or state of the switch’s physical lever, &etc. After a certain point the lever will not be able to reach its fully “on” and “off” position before it must change direction. Eventually it must stop moving entirely. The stopping point is necessarily _before_ 2 minutes have elapsed. Assuming the demon is forward thinking enough to stop before reaching full-on so he has enough time to get back below the threshold, the stopping point will be when the _lever_ is exactly halfway between on and off. The resultant state of the _switch_ would depend on the specific switch, I think simpler switches would technically be “on”, due to arcing that could cause melting or even a fire. More expensive switches would have a dead zone in the middle which is handled by springs based on the previous state (eg: if off, you must lift it past 100 degrees to switch on & if on, you must push it down under 80 degrees before it switches off). Can’t remember the word for that. ~Max ---
**Author:** ![epolo](https://avatars.discourse-cdn.com/v4/letter/e/b487fb/32.png) [@epolo](https://boards.straightdope.com/u/epolo)\ **Post date:** [December 7, 2021, 3:06am UTC](https://boards.straightdope.com/t/thompsons-lamp/955473/20 "2021-12-07T03:06:25Z")
> [@Max\_S](#): > > Exactly. The demon’s hand is presumably limited to the speed of light, as is the position or state of the switch’s physical lever, &etc. After a certain point the lever will not be able to reach its fully “on” and “off” position before it must change direction. Eventually it must stop moving entirely. The stopping point is necessarily _before_ 2 minutes have elapsed. > > Assuming the demon is forward thinking enough to stop before reaching full-on so he has enough time to get back below the threshold, the stopping point will be when the _lever_ is exactly halfway between on and off. The resultant state of the _switch_ would depend on the specific switch, I think simpler switches would technically be “on”, due to arcing that could cause melting or even a fire. More expensive switches would have a dead zone in the middle which is handled by springs based on the previous state (eg: if off, you must lift it past 100 degrees to switch on & if on, you must push it down under 80 degrees before it switches off). Can’t remember the word for that. > > ~Max Actually, if you’re trying to figure out what would happen in a real world attempt at Thompson’s Lamp we don’t even need to go as far as relativistic effects. Long before then the speed of switching will be pumping so much energy into the system that the mechanism will fail due to metal fatigue, melting, or whatever. The switch and probably the lamp will be neither on nor off; they will be destroyed. The universe doesn’t seem to allow “naked” infinities. [Next page](https://boards.straightdope.com/t/thompsons-lamp/955473.md?page=2)