## Thursday, June 18, 2020

### 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:

1. 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:

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

So, indeed, we now have a paradox.

Majesty of Reason said...

This is really interesting. I have three comments:

(A) I guess one of my questions is: in what sense are these things truly 'interactions' above and beyond the individual switch togglings? The difference between the two scenarios you described seems to be purely verbal: considered in terms of the actual switch togglings, the two series are wholly and utterly identical. As such, surely we should conclude that the states of the lamps at noon are the same. It's hard to see why we should give any significance whatsoever to the use of 'interactions'.

(B) Also, might we simply take this scenario as a reductio of (1)?

(C) Does this version of the paradox (the one you outlined in the post) fall prey to the usual objection to Thomson's Lamp in the literature (i.e. that the state of the lamp at 11:00 is simply not deducible from the setup of the situation, since the domain of times is open in the later-than direction; no truths restricted solely to a description of the lamp's state *before* 11:00 entails anything about its state *at* 11:00)?

Thanks!!!

Majesty of Reason said...

It's also not immediately clear how causal finitism could solve this paradox. It doesn't seem that this involves any infinite causal chains -- the fist scenario involves no switch togglings. The second scenario involves a single switch toggling at 10:30. Does 'removing' an infinite number of interactions require an infinite causal chain? Couldn't 'removing' these interactions simply obtain in virtue of 'not performing' them? And if that's the case, then it seems adopting causal finitism isn't doing any work in resolving the paradox. But perhaps I'm missing something! :)

Majesty of Reason said...

Okay -- come to think of it, here's a way to get causal finitism to play a role.

Simply adapt (1) to the following:

(1*) Including/adding any number of interactions with a system none of which changes a system will not affect the system.

With (1*) in hand, we don't need to remove or eliminate any interactions; instead, we can get the contradictory lamp states at 11:00 by means of defining the first and second scenarios as you did originally.

Majesty of Reason said...

Quick note: In my first comment in (A), I meant that there is no difference between the two scenarios *before doing the removing*. :)

Michael said...

Alex,

I don't understand the paradox. These states are not equivalent. The first has the switch not hit at 10:30. The second as you note does not have the switch hit at 10:30. This is no different than saying if we start with a lamp off and do paired toggles we end up with an off lamp, and also if we do no toggles. And that if we start with a lamp on we end up with an on lamp if we do paired or no toggles.

IanS said...

A nit-pick. Removing any toggle changes the state of the system at some time. Maybe the idea of (1) is that each ‘complex interaction’ has a start time and an end time such that removing that interaction would leave the system unchanged outside the interval [start time, end time]. Modified (1) would then say that if these intervals are disjoint, then removing all the interactions would leave the system unchanged outside the union of the intervals.

Dominik Kowalski said...

Alex has a good idea so I will reconstruct the paradox:

Premise 1 is not only plausible, but, in the case of Thomsons Lamp pretty much undeniable. Two successive toggles cancel each other out and hence make no difference.

Suppose now that the lamp is on on 10 p.m. and we apply Premise 1. It follows that the toggles make no difference and such can be removed. If that were to go for all toggles, the lamp remains on on 11.p.m.

If we however spare the toggle at 10:30 p.m. the lamp is off and it remains off if we apply Premise 1 to all the toggles afterward, so that the lamp is off on 11 p.m.. Thus we get contradictory results. If you think about it, scenario 1 indicates that the amount of toggles between 10 and 11 is even, scenario 2 says its uneven.

I'm sure Alex has something to correct or contribute, but I think this reconstruction is at least helpful.

Dominik Kowalski said...

Mor,

1) my post above answers this.

2) don't see how. It's not like it would make a difference to the answer of the question whether Thomsons lamp is on or off at 11 if we remove two successive toggles in the midst of it.

3) the counterexample requires a teethshattering bullet to bite. I suppose you could go down that extreme humean route, but the absurdities that follow from that, especially empirical, are obvious. The causal principle applied here is way weaker than a standard PSR and its acceptance shouldn't be that controversial, especially if it means that the alternative is to throw the intelligibility of causality out of the window.

4) Causal finitism solves this paradox, because on it time is discrete and the scenario impossible. (Don't know whether it can be applied to “flowing“ time as well, maybe it can, but Aristotelians accept discrere time anyway, since causality is prior to it). The absurdity only arises because it is supposed that time is infinitely divisible.

“Infinity, Causation and Paradox“ pp. 40-46+ ch. 8 for help.

Majesty of Reason said...

Dominik Kowalski,

I'll briefly touch on what you said, especially point (4) of yours.

You state: "Causal finitism solves this paradox, because on it time is discrete and the scenario impossible."

There are a number of problems with this.

(1) As Alex argues convincingly in his book, Causal finitism doesn't entail that time is discrete, and hence it's false that "on it time is discrete". Alex argues that causal finitism provides *some evidence* for the discrete nature of time and space, but he is emphatic that causal finitism doesn't *imply* them.

(2) Also, it doesn't follow that the scenario is ruled out as impossible under the supposition of the discreteness of time. Consider again what is *actually* happening here. In the first scenario, all that actually happens is: a lamp is off at 10:00. Nothing toggles its switch from 10:00 to 11:00. So, the lamp is off at 11:00. In the second scenario, all that actually happens is: a lamp is off at 10:00. The switch is toggled a single time at 10:30. Nothing toggles its switch from 10:30 to 11:00. So, the lamp is on at 11.

Crucially, this does *not* involve infinite divisibility of time, or the continuity of time, or anything. All we did was *not perform* the original supertask that does require the continuity of time. But *not performing* a supertask doesn't require continuity of time. (Since even under the discreteness of time, it's still that case that we do *not perform* a supertask).

This goes back to what I wrote here: Does 'removing' an infinite number of interactions require an infinite causal chain? Couldn't 'removing' these interactions simply obtain in virtue of 'not performing' them? And if that's the case, then it seems adopting causal finitism isn't doing any work in resolving the paradox.

Dominik Kowalski said...

The problem with your proposal is that it is ignoring the paradox hinted at rather than confronting it head on. It ignores the puzzle presented and Pruss would do the same if the “removing“ and “not performing“ were explanatorily equivalent in this context. I don't see though why we should take it that way. If I'm right in my first post then the contradiction generated is to be found in the fact that only one of the scenarios above is possible. Both scenarios deviate from the same origin, but make different assumptions on whether the amount of toggles is even or uneven. This is also why “not performing“ in this scenario must be different from “removing“. I take it that the latter in this context is a tool for bringing the contradiction to light. If those steps were instead not performed, then I don't see how the paradox got resolved. It didn't answer whether the light would be on or off in a Thomsons lamp scenario.

The kicker is that for the opponent of causal finitism neither scenario 1 or 2 is satisfactory since both vindicate it. But they are exhaustive.

I was wrong about discreteness, but it doesn't seem to affect the argument.

I also don't understand what you mean when you say that the question is whether removing an infinite number of interactions requires an infinite causal chain, but if I derived it correctly from the context, then I answered it

Alexander R Pruss said...

Perhaps relevant to Ian's comment, here's a potential counterexample to my principle 1. Suppose there is a piece of paper in a cool oven that is turned off. I take the piece of paper out for a day. During that day my son bakes a pie in the oven. Next day I come back and put the piece of paper back in once the oven has cooled. My interaction (taking the paper out and putting it in) in one sense doesn't change the system: the system is in the same state before the interaction (a piece of paper in a cool oven) as after. But if the interaction were removed, the piece of paper would have been in the oven while my son was baking (unless he noticed it) and it would have burned up. This looks like a counterexample.

What I want is a concept of a "null interaction", one that doesn't affect a system overall. But it's pretty hard to define. However one defines it, it's pretty intuitive that a double flip of a switch (in an idealized system--in a real system, each flip puts wear and tear on the switch, makes an annoying sound, etc.) is such an interaction. As per Ian's suggestion, we definitely want to restrict 1 to interactions on disjoint intervals.

But the fact that it's a lot harder to define a null interaction than I thought when I wrote the post makes me worried about this whole approach.

Alexander R Pruss said...

MoR:

The final state of the bulb has infinitely many togglings in its causal history, which violates causal finitism.

Dominik:

It's not quite right to say that causal finitism implies time is discrete. See Section 4 of Chapter 8 for a discussion of how one can have continuous time and causal finitism.

Majesty of Reason said...

Alex:

Thank you for your feedback. I think there's an ambiguity surrounding the word 'remove'. If by 'remove' we mean 'take out of the process', i.e. eliminate the steps and refrain from performing such steps, then removing the infinitely meany switch togglings means we only perform finitely many switch togglings (say, the one at 10:30 and no more, since we removed and hence did not perform the ones after it).

A different sense of 'remove' could be a kind of subjunctive sense, as in: We *actually perform* infinitely many switch togglings, but *were* we to remove (refrain from performing) the specified sequence of interactions, this *would* not have made a difference to the end state/condition of the system.

I believe you meant this latter meaning, and it is this that facilitates a paradox, since we get:

(A) Were we to remove the togglings from 10:30 onward, this would not have made a difference to the state of the lamp (in which case we can conclude that the lamp is *off*); and
(B) Were we to remove the togglings from 10:45 onward, this would not have made a difference to the state of the lamp (in which case we can conclude that the lamp is *off*); so
(C) The lamp is actually both on and off.

And that, certainly, is a paradox. I hope I've accurately untangled what you originally meant in your post. I think there may have been some slight unclarity in the original post on this distinction between the different senses of 'remove'. Though, of course, I could be mistaken :)

wrf3 said...

The Thompson Lamp paradox is no different from Zeno's paradox. It asks for a measurement that cannot be made and therefore the only correct answer is "can't be done as instructed." The only way to actually get to a measurement is to quantize time, space, or energy. And if you do that, with one metric the answer can be determined by classical calculation; by another, the answer is "we can't possibly know until we measure it."

Andrew Dabrowski said...

Anyone want to comment on the parallel to

\sum_{n=0}^\infty (-1)^n

? Mathematicians aren't bothered by this - the sum simply isn't defined. Why should philosophers be bothered by Thomson's lamp? The paradox only obtains if one assumes the state at 11 is defined, but that assumption is unwarranted.

Another problem with infinity vis-a-vis causality would be infinite causal chains which take up only a finite amount of time. But none of these problems arg.e for causal finitism, just that the laws of the world must be carefully defined.

Alexander R Pruss said...

Well, if it is possible to press the button infinitely many times, *something* would happen at 11. There may not be a fact as to what specifically would happen, but *something* would, and that might itself be enough to generate a paradox (e.g., because there might be no possible explanation as to why it happens).

Andrew Dabrowski said...

Since it is not possible to press the button infinitely often in finite time, there is no need to define the state of the light in that eventuality. The universe is just a set of laws - maybe not even that.