Do you think supertasks can be completed? — Fire Ologist

No. I’m trying to prove that they are impossible, as clearly shown in this post.

And supertasks didn’t come up until later in the post and really another way of incorrectly claiming there is anything relevant to the lamp problem at two minutes. — Fire Ologist

Thomson’s lamp is a thought experiment designed by Thomson to prove that supertasks are impossible, with Thomson being the person who coined the term “supertask.”

The concept of the “super task” is not essential to anything I’m saying. — Fire Ologist

Supertasks are the topic of this discussion. They are what the rest of us have been arguing about for 26 pages.

You could perhaps start with the SEP article.

I am not going to add confusion and complexity to this by starting to discuss “super tasks”. — Fire Ologist

Is this a joke?

THERE IS NO AFTER WE FINISH PUSHING THE BUTTON!! — Fire Ologist

There is according to those who claim that supertasks are possible. They claim that within two minutes I can finish pushing a button an infinite number of times, allowing me to then carry on my day.

Do you not understand the discussion we're having?

I'm not redefining the premises.

Before we even consider when the button is pushed, it is implicit in the thought experiment that the lamp both existed (as a lamp) before we start pushing the button and continues to exist (as a lamp) after we finish pushing the button, and that for its entire existence nothing other than pushing the button controls the lamp.

The button pushing is just one aspect of the wider environment. You and fishfry seem to ignore this wider environment, treating the supertask as nothing other than some infinite sequence of values.

I think you mean between 22:00 and 23:00, assuming the two minute mark is 0:00. — Fire Ologist

No I don't. I mean exactly what I said.

That doesn’t contradict the premises, because the premises never touch the lamp at two minutes. — Fire Ologist

Yes they do. These are our background premises:

P1. Between 00:00 and 23:59, nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at 00:00
P5. The lamp is either on or off at 12:00

Only then do we consider the manner in which the button is pushed.

If the button is only ever pushed at 11:00 then the lamp is on at 12:00.

If the button is only ever pushed at 11:00, 11:30, 11:45, and so on, then the lamp can neither be on nor off at 12:00. This is a contradiction.

Note especially P1 which is where fishfry and Benacerraf go wrong.

And because lamps are either on or off at all times, but you can’t deduce the state from the premises, we don’t have a contradiction. — Fire Ologist

Not only is no state deducible from the premises, no state is consistent with the premises.

The lamp is turned on if and only if the button is pushed when the lamp is off.
The lamp is turned off if and only if the button is pushed when the lamp is on.

Neither of these are possible at t₁ if a supertask has been performed between t₀ and t₁. But the lamp must be either on or off at t₁. This is the contradiction.

As a side note, you’ve proven that reasoning with you about how infinity works is a supertask. — Fire Ologist

I understand how infinity works. The problem is that fishfry doesn't appear to understand how a lamp works.

But I've already discovered that Benacerraf and others have viewed this problem exactly as I do. So my position is valid, or you should take your objections up with them. The problem does not defined the lamp state at midnight and it can be anything you like. — fishfry

I addressed his paper a month ago. See also here for a more formal argument.

As for the screen, it can only ever display the time that the button was most recently pushed. If the button is only ever pushed at 23:00 then at midnight the screen displays 23:00. Notice that the rule about how the screen works still applies at midnight, long after the button has been pushed. This rule also still applies at midnight if the button is only ever pushed at 23:00, 23:30, 23:45, and so on.

And because the button is not pushed at midnight the screen cannot display 00:00 at midnight, even though 00:00 is the limit of the sequence.

There is no tiny little interval before midnight where the sequence has ended, leaving the lamp in a particular state. — fishfry

This is why the supertask is impossible. A lamp can only be on if something causes it to change from off to on. This is as true at midnight as it is at any time before and after midnight. It’s a fundamental fact about lamps. Therefore a lamp cannot be modelled by the sequence off, on, off, …, on.

the limit of a sequence has no immediate predecessor in the sequence, and that's just a mathematical fact — fishfry

I have always accepted this; it's the reason that the supertask is proven impossible.

A lamp being off must always precede it being on, and so the sequence off, on, off, ..., on cannot model a lamp.

Midnight. ... Of course the clock says midnight at midnight. — fishfry

It's not a clock. It's a screen that displays the time that the button was most recently pushed.

If the button is pushed at 23:00 then the screen will display 23:00 until the button is pushed again at 23:30, and then the screen will display 23:30 until the button is pushed again at 23:45.

The button is not pushed at midnight and so the screen will not display 00:00 at midnight, even though 00:00 is the limit of the sequence.

Thompson's lamp is a rather poor example of a supertask, because its underlying sequence can not be made to converge to a limit. — fishfry

A supertask is any infinite succession of tasks performed in finite time. Having that task be to push a button is as good a task as any, regardless of what pushing the button actually does.

I'll concede your point that the lamp is impossible. That does not necessarily entail that supertasks in general are impossible. — fishfry

The lamp has two buttons. One button turns the lamp on and off, the other does nothing. The same mechanism is responsible for pushing both buttons. If it is possible to have pushed one of these buttons an infinite number of times then it is possible to have pushed the other button an infinite number of times.

If Thomson's lamp proves that it is impossible to have pushed the button that turns the lamp on and off an infinite number of times then it proves that it is impossible to have pushed the button that does nothing an infinite number of times.

And we can replace pushing a button with literally any other task and the same conclusion follows. Therefore, Thomson's lamp proves that all supertasks are impossible.

At midnight the Supreme Button Pusher flips a coin and turns the lamp on or off, accordingly. I don't see any problem. — fishfry

If the Supreme Button Pusher turns the lamp on at midnight then it must have been off before He turned it on, because that's what "turning on" means. If He turned it off at midnight then it must have been on before He turned it off, because that's what "turning off" means.

For the entire continuous time that the lamp exists – not just the time during which the button is pushed – it being off must precede it being on, regardless of magic or divine intervention. As such the sequence off, on, off, ..., on makes no sense, much like having an imaginary number of apples in the fridge makes no sense.

We must accept that the button can only be pushed $n \in \mathbb{N}_0$ times between t₀ and t₁.

But also note P1 in the argument above, implicit in Thomson's argument. You don’t get to just introduce God to deflect from the incoherent causal consequence of having pushed a button an infinite number of times.

Thompson's lamp is not a good example of a supertask, because the sequence doesn't have a limit or any natural termination point. — fishfry

A supertask is just performing an infinite succession of tasks in finite time. Pushing a button, which just so happens to turn a lamp on and off, is as good a supertask as any.

But let’s assume that pushing the button displays the time that the button is pushed – persisting until the button is pushed again – and is pushed (only) at successively halved intervals of time starting two minutes to midnight. What time is displayed at midnight?

The sequence approaches midnight but because the button is never pushed at midnight the display can never show midnight, but must show something because it’s never turned off. It always only displays the time that the button was last pushed, but in this scenario there is no last button push, entailing a contradiction.

As Thomson says, "the impossibility of a super-task does not depend at all on whether some ... sequence is convergent or divergent."

It's not like either number is somehow more physical than the other one — fdrake

I’m not claiming otherwise. I’m only claiming that I cannot have an imaginary number of apples in my fridge.

That some number is sensible isn’t that it’s sensible to use that number in any and every context.

Similarly, the sequence 0, 1, 0, …, 1 might make sense in some contexts, but the sequence off, on, off, …, on doesn’t make sense in the context of a lamp being turned on and off by pushing a button.

Tell that to an electrical engineer or quantum physicist, both of whom use imaginary and complex numbers as essential tools of their trades. — fishfry

I didn't say that imaginary numbers don't have a use. I said that I cannot have an imaginary number of apples in my fridge.

You really reject the complex numbers, negative numbers, fractions, and irrationals? — fishfry

No.

It's a premise of the thought experiment that there is a sequence of steps at successively halved intervals of time. — fishfry

Yes, and this premise is proven false. See refutation by contradiction and modus tollens.

I already showed you how to model the process using the inverse powers of 2. You ignored that, since it refutes your argument. — fishfry

It doesn't refute my argument because it doesn't address my argument.

My argument is that the lamp cannot be on unless it is turned on (whether by pushing a button or by magic), as argued above.

Therefore, off, on, off, ..., on makes no sense in context. The lamp being off must precede the lamp being on. Given that this is inconsistent with the premise that the button is pushed at successively halved intervals of time it is proven impossible in principle to push the button at successively halved intervals of time.

Your argument doesn't prove that. — Ludwig V

Others do, like Zeno's and Bernadete's.

This leads us to think that there is some sort of miracle involved in arriving at the fridge to get a beer. — Ludwig V

There's no miracle. Motion isn't continuous; it's discrete.

$0, 1, 0, ..., 1\\0, 1, 0, ..., 0$

Such sequences may make sense in the context of abstract mathematics but they do not make sense in the context of a lamp being turned on and off.

As a comparison, even though imaginary numbers have a use in mathematics it is more than just physically impossible for me to have $\sqrt{\text{-}1}$ apples in my fridge; it is metaphysically impossible.

No pretend physics can allow for me to have an imaginary number of apples in my fridge and no pretend physics can allow for the above two mathematical sequences to model the state of a lamp over time.

With Thomson's lamp, these are our premises:

P1. Nothing happens to the lamp except what is caused to happen to it by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t₀
P5. The button is pushed at least once between t₀ and t₁
P6. The lamp is either on or off at t₁

And these are our conclusions:

C1. If the lamp is on at t₁ then the button was pushed to turn it on, prior to which it was off
C2. If the lamp is off at t₁ then the button was pushed to turn it off, prior to which it was on
C3. The button is pushed $n \in \mathbb{N}_1$ times between t₀ and t₁

These conclusions prove that a supertask is not performed.

The basic confusion is not understanding that an infinite sequence has no end. — fishfry

I understand that it has no end. That is why I am arguing that it is metaphysically impossible for an infinite succession of button pushes to end after two minutes.

And that’s precisely why the question of whether the lamp will be on or off at two minutes will never present itself. — Fire Ologist

It does present itself because the lamp must be either on or off after two minutes as per the law of excluded middle.

We want to know what would happen to the lamp if we were to push the button to turn it on and off at successively halved intervals of time within two minutes. Nothing else happens to the lamp except what we cause to happen to it by pushing the button.

Is it left on or off after we stop pushing the button? If you can’t answer this then you must accept that supertasks are metaphysically impossible.

How is that? How is it on or off at or after two minutes? — Fire Ologist

Because it's a lamp. If it exists at 12:02 then it's either on or off, and it exists at 12:02.

It cannot be a function of a switch that operates by switching every half of the prior interval. — Fire Ologist

And that's precisely why supertasks are impossible.

We are imagining, for the sake of argument, that we are in some alternate universe with physical laws that allow us push a button to turn the lamp on and off at successively halved intervals of time. We want to know what happens to the lamp if we do this (and only this).

As you say, we don't get to "just [assume] something exterior to the premise about time and lamps."

So the only premises we are allowed to work with are:

P1. If the lamp is off and the button is pushed then the lamp is turned on
P2. If the lamp is on and the button is pushed then the lamp is turned off
P3. The lamp is off at t₀
P4. The lamp is either on or off at t₁
P5. The button is pushed at successively halved intervals of time between t₀ and t₁

So we ask: is the lamp on or off at t₁?

If you cannot provide a coherent answer then you must accept that the premises are inconsistent. Given that there's nothing problematic about P1 - P4, we must accept that the problem lies with P5. It is necessarily false. Supertasks cannot be performed.

Because the switch is not designed to ever present the question. — Fire Ologist

I don't know what you mean by this.

Given that the lamp must be either on or off after two minutes we must ask the question. If you cannot provide a coherent answer then you must accept that your premise – that the lamp has turned on and off an infinite number of times – is necessarily false, and so that supertasks are metaphysically impossible.

Or more precisely, not designed to function at or after two minutes. — Fire Ologist

That's also true about the first two scenarios – neither switches after 1 minute and 30 seconds – and yet we can still answer the question about the lamp after 2 minutes, and even after 2 years.

The lamp is off. It turns on and off only as described:

Scenario 1
The lamp turns on after 1 minute.

Is the lamp on or off after 2 minutes? It's on.

Scenario 2
The lamp turns on after 1 minute and off after a further 30 seconds.

Is the lamp on or off after 2 minutes? It's off.

Scenario 3
The lamp turns on after 1 minute, off after a further 30 seconds, and so on ad infinitum.

Is the lamp on or off after 2 minutes?

In all three scenarios the switch is "designed to function within two minutes."

The lamp is off. After one minute the lamp turns on. Is the lamp on or off after two minutes? It's on because it was turned on after one minute and then never turned off again.

If you can't apply the same reasoning to the lamp having turned on and off an infinite number of times before the end of two minutes then you must accept that it makes no sense for the lamp to have turned on and off an infinite number of times before the end of two minutes.

I don’t understand. How do you ever arrive at the two minute mark? — Fire Ologist

I don't know what you mean by "arrive" at the two minute mark. Two minutes just pass. That's how the world works.

Imagine I am facing a clock with my back to a lamp. The experiment starts at 12:00. Through some automated process the lamp turns on and off at successively halved intervals of time. When the clock shows 12:02 I turn around. Is the lamp on or off?

That means that any answer whatever is equally valid — Ludwig V

That's not true, as I explained here, and as I alluded to above. It is not just the case that whether the lamp is on or off after two minutes is undefined but that the lamp cannot be either on or off after two minutes.

As Thomson says in his paper, "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."

Now if I can just get Michael to agree! — fishfry

I have always agreed that the sequence "0, 1, 0, 1, ..." does not converge.

I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes. See my previous post and my initial defence of Thomson on page 13.

But which is not defined. — Ludwig V

It's more than that; the lamp can't be on and can't be off, even though it must be one or the other. This is a contradiction, and so therefore the supertask is proven impossible in principle.

I'm not at all clear how the ordinary logic of cause and effect would apply in the context of hypothetical physical laws. But we are clearly not dealing with the ordinary physical world, and that leaves us free to imagine anything at all. — Ludwig V

If it's on at t₁ then either it was left on before t₁ or it was left off before t₁ and then turned on at t₁.

This is a straightforward logical point that does not depend on what the physical laws are.

To make it very simple, Thomson's lamp proves that these premises are logically inconsistent:

P1. The lamp is turned on and off only by pushing the button
P2. If the lamp is off and the button is pushed then the lamp is turned on
P3. If the lamp is on and the button is pushed then the lamp is turned off
P4. The lamp is off at t₀
P5. The button is pushed at successively halved intervals of time between t₀ and t₁
P6. The lamp is either on or off at t₁

A supertask is not simply an infinite sequence of numbers.

In our hypothetical scenario with hypothetical physical laws we are still dealing with the ordinary logic of cause and effect.

It is implicit in the thought experiment that it is only by pushing the button that the lamp is caused to turn on and off, but strictly speaking this premise isn't necessary as the logic applies regardless of the cause – even if it's magic.

If the lamp is on then something caused it to turn on, prior to which it was off. If it is turned on then it stays on until something causes it to turn off.

Given this, if the lamp is on at t₁ then either:

a) it was turned and left on prior to t₁, or
b) it was turned and left off prior to t₁ and then turned on at t₁

But as Thomson says, "I did not ever turn it on without at once turning it off ... [and] I never turned it off without at once turning it on", and so both (a) and (b) are false. Therefore the lamp is not on at t₁. Similar reasoning shows that the lamp is not off at t₁ either.

But it's not my only solution. I've said (several times) that "Lamp is on" and "Lamp is off" are also valid solutions. — fishfry

The lamp is on only if the button was pushed to turn it on, prior to which the lamp was off. Even if you want to introduce magic it is on only if magic turned it on, prior to which the lamp was off.

So if you want to say that the lamp is on after two minutes then you must accept that at some final time prior to two minutes the lamp was off. This is a simple logic fact.

Except the supertask doesn't allow for this. So, as Thomson argues, the lamp cannot be on after two minutes. And for the same reasoning cannot be off after two minutes. This is a contradiction and so the supertask is proven impossible in principle.

My point is that once we've entered the realm of speculative fantasy, where do we stop? — fishfry

We stop at the single issue being discussed: performing some action at arbitrarily small intervals of time. So taking the code here we assume that each line is run in an instant with the exception of the wait i *= 0.5 line which waits for the specified time in seconds. The logic of the code still behaves exactly as we would expect. Thomson is asking us what is output when echo isLampOn runs.

If your only solution is to insert the line isLampOn = 'a plate of spaghetti' after while (true) { ... } and before echo isLampOn then you are not answering the question as posed.

If you cannot make sense of the echo isLampOn line without inserting some arbitrary code before it then you must accept that it doesn't make sense for while (true) { ... } to complete. The arbitrary code you are trying to insert is a smokescreen to disguise this impossibility, exactly like your magic turning the lamp into a plate of spaghetti.

you haven't demonstrated any contradictions in TL — SophistiCat

Thomson does that himself in his paper. I am defending his paper and explaining why Benacerraf's response to it fails. See here where I first brought it up.

Perhaps you could explain which part of my (or Thomson's) reasoning you reject? You're a coder so perhaps you could even address the code here. Simply saying "it's wrong" is hardly a meaningful criticism. If that's all you have to say then I will simply reply with "it's right".

nor linked it to continuous motion. — SophistiCat

I did so in the post yesterday. Just as if we push a button an infinite number of times within two minutes the lamp can neither be on nor off after two minutes, if we run through an infinite succession of sensors when running a mile the lamp can neither be on nor off after finishing the run.

Given that the lamp must be either on or off, this is a contradiction, and so therefore it is proved that one cannot have run through an infinite succession of sensors.

You are just restating - reimagining - Thompson's Lamp thought experiment, which has nothing to do with continuous motion as such — SophistiCat

I’m using Thomson’s lamp to show that continuous motion entails contradictions.

and repeating once more your baseless conclusion — SophistiCat

It’s not baseless. I’ve explained it quite clearly here and here and in many other comments.

The problem is that if motion is continuous and if the sensors are set up as stated then the lamp can neither be on nor off after the run is completed, which is a contradiction.

One or more of the premises is necessarily false. So either motion is not continuous or we cannot set up sensors at an infinite succession of halfway points. The latter would seem to suggest that there aren’t an infinite succession of halfway points and so would entail the former anyway.

Unlike Zeno's thought experiments, which deal with examples of ordinary motion — SophistiCat

Continuous motion suffers from the same problem. We can imagine sensors at each successive half way point that when passed turn a lamp on or off. Is the lamp on or off when we finish our run?

The simple solution is to say that motion isn’t continuous. Discrete motion at some scale is a metaphysical necessity.

Quantum gravity and quantum spacetime are viable theories so it’s not a hard pill to swallow.

So there is no "logical" way to connect the sequence, with its arbitrary terminal state, which you can define as on or off. — fishfry

A supertask is not simply an infinite sequence.

With a supertask we are given some activity to perform and we assume that it is physically possible to perform this activity at successively halved intervals of time. We are then asked about the causal consequence of having done so.

We do not get to introduce additional (and nonsensical) premises such as "and then the lamp magically turns into a plate of spaghetti, prior to which the lamp was neither on nor off."

The lamp must be either on or off after two minutes. If the lamp is on after two minutes then it is on only because the button was pushed to turn it on, prior to which the lamp was off. If the lamp is off after two minutes then it is off only because the button was pushed to turn it off, prior to which the lamp was on. The supertask doesn’t allow for either of these scenarios and so is proven impossible in principle.

I think you'll find that's because it makes no sense to answer the question.

In other words, it also makes no sense to answer the question with "on" or "off". — Ludwig V

The lamp is either on or off at t₁. The fact that it makes no sense for it to be on and no sense for it to be off if the button has been pushed an infinite number of times before that is proof that it makes no sense for the button to have been pushed an infinite number of times.

Manhattan voted 85% for Joe Biden, and registered Democrats outnumber Republicans eight to one in New York. The Biden/Harris campaign and a whole host of anti-Trump Democrats pay the judge's daughter an obscene amount of money to work for them. A simple change of venue would have been an appropriate fix. — NOS4A2

It's a New York crime so was always going to be tried in New York.

If it makes you feel better, his Florida trial will be in an area that heavily favours Republicans, so you can be grateful of a biased jury in his favour, and with a biased judge he appointed.

I suppose prosecutors would have had to prove that Trump first new about this law, and then intended to violate it. — NOS4A2

Ignorantia juris non excusat.