A variant of Thomson's Lamp

In the classic Thomson’s Lamp paradox, the lamp has a switch such that each time you press it, it toggles between on and off. The lamp starts turned off, say, before 10:00, and then the switch is pressed at 10:00, 10:30, 10:45, 10:52.5, 10:56.25, and so on ad infinitum. And the puzzle is: Is it on or off at 11? It’s a puzzle, but not obviously a paradox.

But here’s an interesting variant. Instead of a switch that toggles on or off each time you press, you have a standard slider switch, with an off position and an on position. Before 10:00, the lamp is off. At 10:00, 10:45, 10:56.25, and so on, the switch is pushed forcefully all the way to the on side. At 10:30, 10:52.5, and so on, the switch is pushed forcefully all the way to the off side.

The difference between the slider and toggle versions is this. Intuitively, in the toggle version, each switch press is relevant to the outcome—intuitively, it reverses what the outcome would be. In the slider variant, however, each slider movement becomes irrelevant as soon as the next time happens. At 10:45, the switch is pushed to the on side, and at 10:52.5, it is pushed to the off side. But if you skipped the 10:45 push, it doesn’t matter—the 10:52.5 push ensures that the switch is off, regardless of what happened at 10:45 or earlier.

Thus, on the slider version, each of the switch slides is causally irrelevant to the outcome at 11. But now we have a plausible principle:

1. If between t₀ and t₁ a sequence of actions each of which is causally irrelevant to the state at t₁ takes place, and nothing else relevant to the state takes place, the state does not change between t₀ and t₁.

Letting t₀ be 9:59 and t₁ be 11:00, it follows from (1) that the lamp is off at 11:00 since it’s off at 10:00, since in between the lamp is subjected to a sequence of caually irrelevant actions.

Letting t₀ be 10:01 and t₁ still be 11:00, it follows from (1) that the lamp is on at 11:00, since it’s on at 10:01 and is subjected to a sequence of causally irrelevant actions.

So it’s on and off at 11:00. Now that’s a paradox!

Thomson's core memory paradox

This is a minor twist on the previous post.

Magnetic core memory (long obsolete!) stored bits in the magnetization of tiny little rings. It was easy to write data to core memory: there were coils around the ring that let you magnetize it in one of two directions, and one direction corresponded to 0 and the other to 1. But reading was harder. To read a memory bit, you wrote a bit to a location and sensed an electromagnetic fluctation. If there was a fluctuation, then it follows that the bit you wrote changed the data in that location, and hence the data in that location was different from the bit you wrote to it; if there was no fluctuation, the bit you wrote was the same as the bit that was already there.

The problem is that half the time reading the data destroys the original bit of data. In those cases—or one might just do it all the time—you need to write back the original bit after reading.

Now, imagine an idealized core not subject to the usual physics limitations of how long it takes to read and write it. My particular system reads data by writing a 1 to the core, checking for a fluctuation to determine what the original datum was, and writing back that original datum.

Let’s also suppose that the initial read process has a 30 second delay between the initial write of the 1 to the core and the writing back of the original bit. But the reading system gets better at what it’s doing (maybe the reading and writing is done by a superpigeon that gets faster and faster as it practices), and so each time it runs, it’s four times as fast.

Very well. Now suppose that before 10:00:00, the core has a 0 encoded in it. And read processes are triggered at 10:00:00, 10:00:45, 10:00:56.25, and so on. Thus, the nth read process is triggered 60/4ⁿ seconds before 10:01:00. This process involves the writing of a 1 to the core at the beginning of the process and a writing back of the original value—which will always be a 0—at the end.

Intuitively:

1. As long as the memory is idealized to avoid wear and tear, any possible number—finite or infinite—of read processes leaves the memory unaffected.

By (1), we conclude:

2. After 10:01:00, the core encodes a 0.

But here’s how this looks from the point of view of the core. Prior to 10:00:00, a 0 is encoded in the core. Then at 10:00:00, a 1 is written to it. Then at 10:00:30, a 0 is written back. Then at 10:00:45, a 1 is written to it. Then at 10:00:52.5, a 0 is written back. And so on. In other words, from the point of view of the core, we have a Thomson’s Lamp.

This is already a problem. For we have an argument as to what the outcome of a Thomson’s Lamp process is, and we shouldn’t have one, since either outcome should be as likely.

But let’s make the problem worse. There is a second piece of core memory. This piece of core has a reading system that involves writing a 0 to the core, checking for a fluctuation, and then writing back the original value. Once again, the reading system gets better with practice. And the second piece of core memory is initialized with a 1. So, it starts with 1, then 0 is written, then 1 is written back, and so on. Again, by premise (1):

3. After the end of the reading processes, we have a 1 in the core.

But now we can synchronize the reading processes for the second core in such a way that the first reading occurs at 9:59:30, and space out and time the readings in such a way that prior to 9:59:30, a 1 is encoded in the core. At 9:59:30, a 0 is written to the core. At 10:00:00, a 1 is written back to the core, thereby completing the first read process. At 10:00:30, a 0 is written to the core. At 10:00:45, a 1 is written back, thereby completing a second read process. And so on.

Notice that from around 10:00:01 until, but not including, 10:01:00, the two cores are always in the same state, and the same things are done to it: zeroes and ones are written to the cores at exactly the same time. But when, then, do the two cores end up in different final states? Does the first core somehow know that when, say, at 10:00:30, the zero is written into it, that zero is a restoration of the value that should be there, so that at the end of the whole process the core is supposed to have a zero in it?

Another way to turn Thomson's Lamp into a real paradox

In Thomson’s Lamp, a lamp is (say) off at 10:00, and the switch is toggled at 10:30, 10:45, 10:52.5, and so on, and we are asked whether the lamp is on or off at 11:00, neither option being satisfactory.

As it stands, Thomson’s Lamp is a puzzle rather than a paradox. There does not seem to be any absurdity in the answer being “on” or the answer being “off”.

In Infinity, Causation and Paradox I tried to generate a paradox from Thomson’s Lamp. But here is perhaps a better way. Start with this premise:

1. Removing any number of interactions with a system none of which changes a system will not affect the system.

Now, consider these complex interactions with the lamp system:

• Toggling the switch at 10:30 and at 10:45

• Toggling the switch at 10:52.5 and at 10:56.25

• …

Two successive togglings do nothing, so each of these is an interaction that does nothing. By 1, removing them all makes no difference. Now, we know that if we remove them all, the lamp will be off at 11:00, since its switch will not have been toggled even once since 10:00. So, we have established:

2. The lamp will be off at 11:00.

But now consider these complex interactions:

• Toggling the switch at 10:45 and at 10:52.5

• Toggling the switch at 10:56.25 and at 10:58.125

• …

Again, each of these is an interaction that makes no difference. So if we remove them all, by 1 that won’t change anything. But if we remove all these interactions, we have a lamp that is on at 10:31 (since we still have the 10:30 toggling) and then never has its switch toggled. Thus, we have shown:

3. The lamp will be on at 11:00.

So, indeed, we now have a paradox.

Supertasks and empirical verification of non-measurability

I have this obsession with probability and non-measurable events—events to which a probability cannot be attached. A Bayesian might think that this obsession is silly, because non-measurable events are just too wild and crazy to come up in practice in any reasonably imaginable situation.

Of course, a lot depends on what “reasonably imaginable” means. But here is something I can imagine, though only by denying one of my favorite philosophical doctrines, causal finitism. I have a Thomson’s Lamp, i.e., a lamp with a toggle switch that can survive infinitely many togglings. I have access to it every day at the following times: 10:30, 10:45, 10:52.5, and so on. Each day, at 10:00 the lamp is off, and nobody else has access to the machine. At each time when I have access to the lamp, I can either toggle or not toggle its switch.

I now experiment with the lamp by trying out various supertasks (perhaps by programming a supertask machine), during which various combinations of toggling and not toggling happen. For instance, I observe that if I don’t ever toggle the switch, the lamps stays off. If I toggle it a finite number of times, it’s on when that number is odd and off when that number is even. I also notice the following regularities about cases where an infinite number of togglings happens:

1. The same sequence (e.g., toggle at 10:30, don’t toggle at 10:45, toggle at 10:52.5, etc.) always produces the same result.

2. Reversing a finite number of decisions in a sequence produces the same outcome when an even number of decisions is reversed, and the opposite outcome when an odd number of decisions is reversed.

(Of course, 1 is a special case of 2.) How fun! I conclude that 1 and 2 are always going to be true.

Now I set up a supertask machine. It will toss a fair coin just prior to each of my lamp access times, and it will toggle the switch if the coin is heads and not toggle it if it is tails.

Question: What is the probability that the lamp will be on at 11?

“Answer:” Given 1 and 2, the event that the lamp will be on at 11 is not measurable with respect to the standard (completed) product measure on a countable infinity of coin tosses. (See note 1 here.)

So, given supertasks (and hence the falsity of causal finitism), we could find ourselves in a position where we would have to deal with a non-measurable set.

Thomson's Lamp and change

Start with two Thomson's Lamps. They each have toggle switches and are on at 10 a.m. The switches are toggled at 10:30, 10:45, 10:52.5 and so on. Now suppose, as is surely possible, that regardless of what, if any, state the lamps would have had at 11 a.m., aliens come and instantaneously force the first lamp to be on at 11, and force the second to be off (say, by breaking the bulb!).

Now some but not all causal interactions are changes. Which lamp's on/off state did the aliens change? There are four possible answers:

1. The first but not the second
2. The second but not the first
3. Both
4. Neither.

Symmetry considerations rule (1) and (2) out of court. Can we say that in both cases, the aliens changed the on/off state of the lamp? Surely not. For if something can have only two states, it can't be that each of the two possible state inductions counts as a change. Moreover, if the lamp were to have changed state, what state did it change from? For inducing an on state only counts as a change if the induction starts with the lamp in an off state, and vice versa. But the induction didn't start with the lamp on, nor did it start with the lamp off. That leaves only last option: Neither.

But it seems that if an object has been persisting, and a causal interaction induced a state in that objection, that causal interaction either was a state-change or a state-maintenance. So if in neither case did the aliens change the state of the lamp, then it seems that in both cases they maintained the state. But we get analogues of (1)-(4), and analogues of the above arguments also lead to the conclusion that in neither case did the aliens maintain the state.

So the Thomson's Lamp story forces us to reject the dichotomy between state-change and state-maintenance.

Here's another curious thing. It seems that the following is true:

5. If an object has state A at t₁ and non-A at t₂, then the object's having state non-A at t₂ is the result of a change.

But applying (5) shows that both lamps' final states are the results of a change. But that change must have thus been from the opposite state. And yet the final state doesn't follow right after the opposite state.

One could use this as an argument against the possibility of infinitely subdivided time. Alternately, one could use this as an argument against principles like (5) and the idea that the concepts of change and maintenance are as widely applicable as we thought them to be.

Thomson's lamp and two counterfactauls

Thomson's lamp toggles each time you press the button and nothing else affects its state. The lamp is on at noon, and then a supertask consisting of infinitely many button presses that completes by 1 pm, and the question is whether the light is on or off at 1 pm. There is no contradiction yet. But now add these two claims:

1. The state of the lamp at 1 pm would not be affected by shifting the times at which the button presses happen, if (a) all the button presses happen between noon and 1 pm, and (b) we ensure that no two button presses happen simultaneously.
2. If we removed one button press from the sequence of button presses between noon and 1 pm, the state of the lamp at 1 pm would not change.

Given this intuition, we do have a problem. Suppose that our sequence of supertask button presses occurs at 12:30, 12:45, 12:52.5, and so on. Then shift this sequence of button presses forward in time, so that now the sequence is at 12:45, 12:52.5, 12:45.25,and so on. By (1) this wouldn't affect the outcome, but by (2) it would as we will have gotten rid of the first button press. That's a contradiction.

So if we think Thomson's lamp is possible--which I do not--we need to deny at least one of the two counterfactuals. I think the best move would be simply to deny both (1) and (2), on the grounds that the connection between the state of the lamp at 1 pm and the button presses must be indeterministic.

Thomson's lamp and the Axiom of Choice

Consider Thomson's lamp: a lamp with a pushbutton switch that toggles it on and off. The lamp starts in the off position, and then in the next half minute the button is pressed, and in the next quarter it is pressed again, and then in the neight eighth again, and so on. Then at the end of the supertask, the lamp is either on or off.

Now keep the lamp but change the story. During each of the ever shorter intervals, a coin is flipped and the switch is pressed if it lands heads, and not pressed if it lands tails. Moreover, the final state of the lamp depends on the results of the coin flips in the following ways:

1. The results of the coin flips determine the final state of the lamp.
2. For any sequence of coin flip results, if any one (and only one) coin flip had a different, the lamp's final state would have been different, too.

Surprisingly, the existence of a lamp that would work in this way implies a version of the Axiom of Choice. To see this, notice that if the coin flips are independent and fair, then the subset of the probability space where the lamp's final state is on is nonmeasurable.[note 1]

But of course, on some technical assumptions, the existence of a nonmeasurable set requires a version of the Axiom of Choice. So if we read the Thomson's lamp story in such a way that the final outcome is determined by which presses are made and which aren't, in such a way that changing a single press changes the final outcome, that story seems to commit us to a version of the Axiom of Choice.

Conversely, it is easy to use the Axiom of Choice for pairs to prove the existence of a function such as would be implemented by the lamp.[note 2]

Thomson's lamp

Thomson's lamp has an on-off switch. It begins in the "off" position. At noon the switch is toggled, and the lamp comes on. Half a minute later, the switch is toggled, and the light goes off. A quarter of a minute later, the switch is toggled again, and the light comes on. And so on. There are no other switch flippings than these, and the switch survives at least until 12:01 pm. At 12:01 pm, is the switch on or off?

As paradoxes go, this one seems really flimsy. As best I can see, the argument to a paradox is something like this:

1. Time is actually infinitely subdivided.
2. If time is actually infinitely subdivided, the story of Thomson's lamp is possible.
3. Necessarily, if the story is true, then the switch is either on or off at 12:01.
4. Necessarily, if the story is true, then the switch is not on at 12:01.
5. Necessarily, if the story is true, then the switch is not off at 12:01.

The argument for (4) is, presumably, that after every time the switch is on, there is a next time when it is off, and the argument for (5) is similar.

There are a couple of ways of showing what's wrong with the argument. Here is one. In order to argue for (4) and (5), it needs to be a part of the story that

6. At each time t after noon, the position of the switch is the result of the last switch-flipping event prior to t.

For suppose that we deny this. Then we can allow that the switch is on at 12:01, but not due to any switch-flipping event. Or we can allow that the switch is off at 12:01, but not due to any switch-flipping event. After all, perhaps, the switch, instead of being flipped, just undergoes a quantum leap from one position to another.

Fine, then, says the paradoxer: Add (6) to the story.

However, now (2) becomes false. The defender of actually infinitely subdivided time can simply deny (2), since the story is plainly inconsistent: the position of the switch at 12:01 is determined by the last flip before 12:01, but there is no last flip before 12:01. It is a story as plainly inconsistent as this one: "Whether there is an obligatory side of the street to drive on is determined by the content of the will of the king of France. And France is a monarchy." The question of the position of the switch is rather like asking: "If atheism were true, would God want us to be atheists?"

Perhaps the paradoxer will say that that was her whole point, but nonetheless the defender of actually infinitely subdivided time has to affirm that this inconsistent story is possible. But why? It is an easy game to construct inconsistent stories by including a stipulation that something is sufficient to determine something, and then adding to the story something that denies the existence of the determiner. In addition to my present king of France story, consider this one:"A lamp with an on-off switch that can only have two positions, on and off, is produced ex nihilo by God at t₀. The position of the switch at any time is fully determined by how it has last been flipped." Then ask: What is the position of the switch at t₀? Obviously, we have an inconsistency in the story—if the lamp came into existence ex nihilo at t₀ it came into existence with the switch in a particular position, but that position was not determined by a flipping.

But does not the defender of actually infinitely subdivided time think that a lamp switch's being flipped in the supertask way is possible? Certainly. But she has to hold that this is only possible in those worlds in which either something other than the last flip determines the position of the switch at 12:01 or the Principle of Sufficient Reason is violated (I don't think there are any such) or both.