Well, in the way philosophy pictures them yes. I moved the discussion here because the article above provides some history of the parallel picture that neuroscience labors under. Philosophy has never liked being wrong so the fact that we can be (and that we are responsible for that) leads it to create the conclusion that we must not have direct access to the world (or we are ensured it), that we only see the “appearance” of something, or that our individual perspective is somehow partial or lacking or individual (my “sensation” or “perception”). — Antony Nickles

There are plenty of good reasons, supported by science, to believe indirect realism over direct realism, as I discussed at length here.

But I don't understand how we got to this point. You were saying something about us wanting to help each other if we're in pain, and somehow conclude from this that indirect realism is false? Your reasoning is confusing.

Michael, This post may be of interest to you. — fishfry

I'm afraid it's not, because it doesn't address the issue of supertasks.

For supertasks, we have this:

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

Q. If the button is pushed an infinite number of times between t₀ and t₁ then is the lamp on or off at t₁?

It makes no sense to answer this question with "a plate of spaghetti" or "$1\over2$".

Have you even read Thomson's paper? This is the most relevant part:

There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off. So if the lamp was originally off, and you pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half-minute, and so on, according to Russell's recipe. After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off? It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. 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. But the lamp must be either on or off. This is a contradiction.

The answer to his question isn't "a plate of spaghetti", it isn't "0", it isn't "1", and it isn't "1/2".

The only coherent answers are "on" and "off" – except as he argues, it can't be either, and so therefore it is impossible, even in principle, to have succeeded in pressing the button an infinite number of times.

I think the judge and jury were partial — NOS4A2

Perhaps. Trump is a divisive figure. Most people either hate him or worship him. But if there's evidence of crimes then he still needs to be prosecuted. How would you go about finding an impartial jury, and what makes you think that this wasn't already done in this case?

the crime was made up — NOS4A2

I don’t really know what this means. He was prosecuted under 175.10 - Falsifying Business Records In the First Degree, with the intent to violate 17-152 - Conspiracy to Promote or Prevent Election.

the conviction was bought and sold — NOS4A2

I don't really know what this means either. Are you suggesting that the jury were paid to find him guilty?

Biden’s Banana Republic prevails. Trump is now a Mandela, and they destroyed the justice system to rig another election. — NOS4A2

You think the jury was planted?

The code here is effectively the same as a recursive function.

My point is that I think that the disagreement between you and fishfry is about different ways to make the same point. — Ludwig V

I'm arguing that supertasks are metaphysically impossible. He's arguing that supertasks are metaphysically possible.

The ideas of consciousness, sensation, appearance, reality, are all manufactured by philosophy, partly to feel like we are necessarily special, as I discussed above. — Antony Nickles

What do you mean by saying that they're manufactured? Are you saying that they're a fiction? You need to prove that, not simply assert it – and so we're doing philosophy.

The contradiction is the result of the fact that there is no criterion set for the final step in your process - i.e., the end state is undefined. — Ludwig V

That's precisely why supertasks are impossible.

The lamp is off at t₀. The lamp must be either on or off at t₁. But if the button has been pushed an infinite number of times between t₀ and t₁ then the lamp can neither be on nor off at t₁.

It's strange that some are taking the very thing that proves that supertasks are impossible as proof that the proof isn't a proof.

Witt would be showing how this “problem” and ontology are manufactured by our human desires. — Antony Nickles

So because we only care about aspirin when we have a headache then it follows that first person private sensations don't exist, or that if they do exist then they are the same for all people?

That's obviously a non sequitur.

Maybe the way to put this is that equating our pains is not how pain is important to us. If this situation actually did happen, what would matter to us about comparing pains would be attending to one or other of us. Philosophy abstracts this discussion to a place of equating pains, and then creates “sensation” as a kind of object, rather than just me expressing how I feel (which is too vague), so that knowledge might stand in the place of our having to react to someone in pain. What it wants is to be sure of the other person (and what to do), and not have to make the leap of faith of treating them as a person in pain. — Antony Nickles

When we're discussing something like the hard problem of consciousness and the ontology of sensations then it very much matters to us if our pains are the same or not.

All you seem to be arguing is that when we're hungover after a night of heavy drinking then we should care more about whether or not there is some aspirin. I doubt anyone disagrees. But I fail to see the relevance of this on a philosophy forum.

Right, but this might be because one is feigning agreement because they are pitying the other, or being stoic, and maybe not some way for our pain to be “truly” the same, which philosophy perhaps simple creates in order to impose the requirement we wanted all along. — Antony Nickles

Or it's because the sensation I have when I stab myself in the arm is unlike the sensation you have when you stab yourself in the arm, and so our pains are not the same and we don't know one another's pain.

As a matter of connection and to identify with the other person, we say our pain is the same, that we know the other’s pain. — Antony Nickles

That we say it isn't that it's true.

Again, Witt’s point is not to be right — Antony Nickles

Then I will simply say that Wittgenstein is wrong and so we shouldn't listen to what he has to say.

By contrast, Benecerraf et al argue along more classical lines, by defining an abstract completion of the sequence that doesn't contradict Thompson's premises — sime

I think it does. We need to examine the process in reverse, and remember that the lamp is on iff the lamp was off and the button was pushed to turn it on. We're discussing a supertask after all, not simply the infinite sequence {off, on, off, ...}.

If the lamp is on at t₁ then either:

a) the button was pushed to turn the lamp on before t₁ and then it was left on until t₁, or
b) the button was pushed to turn the lamp off before t₁ and then it was left off until t₁ when the button is pushed to turn it on

Neither (a) nor (b) are possible given the defined supertask – the lamp is never left either on or off – therefore the lamp cannot be on at t₁. And then the same reasoning shows that it cannot be off at t₁ either. Yet it must be either on or off. This is a contradiction.

This fact has nothing to do with one's interpretation of mathematics (and nothing to do with the limit of some proposed infinite sequence of numbers).

Are you arguing that Thompson's sequence is finishable hypothetically, but without possessing a definite end value? — sime

No, I'm saying that it isn't completable, even hypothetically. The fact that there is no definite end value is just one way to demonstrate that it isn't completable; the lamp must be either on or off after two minutes, but if the button has been pushed an infinite number of times before then then it cannot be either.

In which case your argument would be closer to constructive mathematics based on intuitionistic logic, rather than to intuitionism. — sime

I don't think it has anything to do with mathematics. This is perhaps clearer if we don't consider the button to turn the lamp on and off but instead consider it to alternate between two or more colours.

What number would you assign to the colour red, and why that? What number would you assign to the colour blue, and why that? Shall we use $e$ and $i$, because why not?

The logic of the lamp just has nothing to do with some sequence of additions and subtractions. The code here properly demonstrates the logic.

Excellent observation. What Witt would do is create a situation and give examples of what we’d say. “I’m in pain” “Me too” “But I have a headache.” “Me too!” “Mine’s a shooting zing behind my ear” “Right! Boy, I know your pain.” Thus why he will conclude that, as a matter of identity, to the extent we agree, we have the same pain (PI # 235). — Antony Nickles

The fact that we use the same word "pain" to refer to your sensations and to my sensations isn't that your sensations are the same as my sensations.

I don't think it has anything to do with formalism or intuitionism or anything like that.

Our starting premises 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

Thomson asks the following question:

Q1. If the button is pushed an infinite number of times between t₀ and t₁ then is the lamp on or off at t₁?

Compare with:

Q2. If the button is pushed an odd number of times between t₀ and t₁ then is the lamp on or off at t₁?
Q3. If the button is pushed an even number of times between t₀ and t₁ then is the lamp on or off at t₁?

Benacerraf claims that we can simply stipulate that the lamp is on at t₁.

There are two problems with this.

The first is that we cannot simply stipulate the answer. I cannot stipulate that the answer to Q2 is "off" and I cannot stipulate that the answer to Q3 is "on". The answers must be deducible from the premises.

The second problem is that the following is deducible from the first three premises:

C1. If the button is pushed an infinite number of times between t₀ and t₁ then the lamp is neither on nor off at t₁

This is because if the button is pushed at least once then for the lamp to be on at t₁ the button must have been pushed to turn and leave it on for t₁, and for the lamp to be off at t₁ the button must have been pushed to turn and leave it off for t₁, neither of which are possible if the button is pushed an infinite number of times between t₀ and t₁.

C1 contradicts P4.

Given that P4 is necessarily true, it follows that the antecedent of C1 is necessarily false. So it is metaphysically impossible to have pushed the button an infinite number of times.

You can define the terminal state to be on, off, or a plate of spaghetti and be consistent with the rules of the game. — fishfry

No you can't. I addressed this in my initial defence of Thomson here, and even more clearly below.

You're confusing yourself on this point.

You're claiming that "a plate of spaghetti" is a coherent answer to the question "is the lamp on or off after two minutes?"

So I think the confusion is yours.

That's funny, because according to SEP, he used the asymptotic density of the sequence 0, 1, 0, 1, ... to argue that the final state must be 1/2. He made that exact argument using that exact sequence. So you are mistaken, because Thompson has used the exact same reasoning I did. — fishfry

He discusses the sequence and its sum, but only to show its irrelevancy, hence the earlier quote.

From his paper:

What is the sum of the infinite divergent sequence +1, -1, +1, ...? Now mathematicians do say that this sequence has a sum; they say that its sum is 1/2. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on.

If there is a sense of "know" that means "acknowledging, recognizing", then you are saying that we do know another's pain (at least, sometimes). I agree, but this is contrary to your earlier statements that we do not know another's pain. — Luke

Is there a difference between knowing someone's pain and knowing that someone is in pain?

Because there's certainly a difference between knowing someone's mother and knowing that someone has a mother.

BUT!!!!!! You have not defined the terminal state. So why do you think there should be a sensible answer for what it is?

I don't see how your expressing the problem in pseudocode adds anything. We all have agreed to it long ago, even before you wrote it down. That's the premise of the problem. But the question is about the terminal state, which is not defined. — fishfry

The terminal state isn't just undefined; any proposed terminal state is inconsistent. The lamp cannot be either on or off after two minutes even though it must be either on or off after two minutes. This is a contradiction, therefore it is impossible to have pushed the button an infinite number of times.

After all, there is no number that can serve as the limit of the sequence 0, 1, 0, 1, ... — fishfry

We're discussing the consequence of having pushed a button an infinite number of times, not the limit of some infinite sequence of numbers. These are two different things.

As Thomson says, "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."

I don’t understand your reasoning at all.

Here's an even simpler demonstration:


In this case there is no wait time between operations; the next occurs "immediately".

Does this script end in an instant, or does it continue forever? If it ends in an instant, does echo isLampOn output true or false?

But given the meaning of while (true), it must continue forever; echo isLampOn never runs.

If you think that this changes by introducing a wait time within the loop then you've obviously made a mistake.

division by 2 ... results in even number of total divisions. — SpaceDweller

I don't know what you mean by this.

The wait(i *= 0.5) simply means that pushButton is called at successively halved intervals of time, i.e. it is called for the first time after 60 seconds, for the second time after a further 30 seconds, for the third time after a further 15 seconds, and so on.

Under assumption that those 2 minutes must pass the lamp will therefore be off with console.log(isLampOn) — SpaceDweller

Given what while (true) { ... } means, it is logically impossible for console.log(isLampOn) to ever run.

2 minutes must pass, and while (true) { ... } must still be running after those 2 minutes.

@fishfry

You wanted something new so see the above.

Here's some pseudocode to demonstrate Thomson's lamp:


According to those who claim that supertasks are possible, after two minutes the echo isLampOn command will run. This is the first contradiction; the while (true) command ensures that this cannot happen.

But even assuming that echo isLampOn does run, it cannot output true and cannot output false, but also cannot output anything other than true or false. This is the second contradiction.

fishfry's solution is to inject some additional code after while (true) { ... }, assigning some arbitrary value to isLampOn, but in doing so he is no longer addressing the problem as posed.

See also this.

What is wrong with that? — Ludwig V

There's nothing wrong with defining, or performing, a recursive function. There is a problem with claiming that it is possible to have completed a recursive function.

You may say that it is possible to divided indefinitely, but that does not mean that infinite divisibility is possible. Take pi for example. You can get a computer to produce the decimal extension for pi, "indefinitely", but you never succeed in reaching an infinite extension. Divisibility is the very same principle. Some mathematical principles allow one to divide indefinitely, but you never reach infinite division. That is because infinite division, therefore infinite divisibility, is logically impossible. — Metaphysician Undercover

Thomson makes a similar point:

People have, I think, confused saying (1) it is conceivable that each of an infinity of tasks be possible (practically possible) of performance, with saying (2) that is conceivable that all of an infinite number of tasks should have been performed. They have supposed that (1) entails (2). And my reason for thinking that people have supposed this is as follows. To suppose that (1) entails (2) is of course to suppose that anyone who denies thinking (2) is committed to denying (1). Now to deny (1) is to be committed to holding, what is quite absurd, (3) that for any given kind of task there is a positive integer k such that it is conceivable that k tasks of the given kind have been performed, but inconceivable, logically absurd, that k + 1 of them should have been performed. But no-one would hold (3) to be true unless he had confused logical possibility with physical possibility. And we do find that those who wish to assert (2) are constantly accusing their opponents of just this confusion. They seem to think that all they have to do to render (2) plausible is to clear away any confusions that prevent people from accepting (1).

But there's nothing special about the lamp. It is impossible to complete any action an infinite number of times. — Ludwig V

Yes, as I further explained in this comment.

No. I'm saying that there's no natural way to define the terminal state. There are lots of ways to defined it. I define it as a plate of spaghetti. That's entirely consistent with the rules of the lamp problem, which only defines the state of the lamp at the points of the sequence — fishfry

No it's not, as explained here.

and does not appear to engage with any of the points I've made — fishfry

As far as I can see I've addressed everything you've said.

After completing the supertask the lamp must be either on or off, but as I explain here, Thomson's lamp shows that if we have pushed the button an infinite number of times then it is logically impossible for the lamp to be either on or off after the supertask is completed. This is a contradiction, therefore Thomson's lamp shows that it is logically impossible to have pushed a button an infinite number of times.

There is a fundamental problem with identifying supertasks with series limits — sime

This is the kind of mistake that Benacerraf makes in his response to Thomson, as explained here.

The lamp is not defined as being on or off at particular times; it is turned on or off at particular times by pushing a button.

It is an important distinction that some are failing to acknowledge.

P2 is false. As shown by P1. — fishfry

The argument form I am using is called modus tollens and is valid:

If P then Q. Not Q. Therefore, not P.

P = we can recite the natural numbers at successively halved intervals of time

Q = we can recite every natural number in finite time

“If P then Q” does not prove “P” and does not prove “Q”. So you are wrong to claim that P1 shows that P2 is false.

How in this great vast wonderful world of ours, does P3 justify P2? They're not even related. — fishfry

If we stop on some finite number then we don't recite every natural number. If we never stop then at no time have we recited every natural number. Therefore P3 entails P2.

(1) The sequence 1, 2, 3, 4, ... never stops. It has no last element. You can always find the next one.

(2) Under the successive halving hypothesis, all numbers are counted. Because as can be plainly seen, there is no number that isn't. — fishfry

The "successive halving hypothesis" leads to contradictions – namely Thomson's lamp and reciting every natural number in finite time – and so it is necessarily false.

But it is not inconceivable, and therefore is is not necessarily false, and therefore it is not metaphysically impossible. — fishfry

It entails contradictions. Therefore either it isn't conceivable or contradictions are conceivable. If the latter then being conceivable does not entail metaphysical possibility. If the former then you are simply mistaken in claiming it conceivable; you are failing to fully understand what it means to perform a supertask.

Thompson's lamp does not lead to a contradiction showing that supertasks are impossible. That's your interpretation, which you are failing to explain or defend to my satisfaction. — fishfry

I explained it here. I think it very clearly shows that having pushed a button an infinite number of times leads to a contradiction. And here I explain that this reasoning extends to all supertasks.

I simply do not understand why you jump to saying that means it's metaphysically impossible. — fishfry

Because it leads to contradictions as shown by Thomson's lamp, defended here and expanded on here.

Also because it's the conclusion of this sound argument:

P1. If we can recite the natural numbers at successively halved intervals of time then we can recite every natural number in finite time
P2. It is metaphysically impossible to recite every natural number in finite time
C1. Therefore, it is metaphysically impossible to recite the natural numbers at successively halved intervals of time

I justify P2 with this tautology:

P3. If we start reciting the natural numbers then either we stop on some finite number or we never stop

You could probably help me out by clearly defining metaphysically impossible. — fishfry

Metaphysical impossibilities are things which are necessarily false; e.g. see Kripke's Naming and Necessity in which he argues that "water is H₂O" is necessarily true even though not a priori (i.e. logically necessary).

But I would even go so far as to say that supertasks are logically impossible (as shown by the above argument and Thomson's lamp). I simply went for the phrase "metaphysical impossibility" because it's the weaker claim and so easier to defend.

Can you clarify which sense you mean? — fishfry

Metaphysical impossibility. Supertasks cannot be performed in any possible world. P3 is a tautology, P2 follows from P3, and so C1 is necessarily true.

When a mathematician says that 1/2 + 1/4 + 1/8 + ... = 1, they don't mean that you can perform this calculation with pencil and paper before lunchtime. They mean that the two expressions on either side of the equal sign denote the same real number. — fishfry

Here are three distinct propositions:

a) 1/2 + 1/4 + 1/8 + ... = 1
b) there is a bijection between this geometric series and the natural numbers
c) it is metaphysically possible to recite the natural numbers at successively halved intervals of time

(a) and (b) are true and (c) is false. Your argument rests on the assumption that (c) follows from (a) and (b), but it doesn't. (c) is proven false by P3, as well as arguments like Thomson's lamp.

You can continually assert that (a) and (b) are true, and I will continually agree, but until you can present actual evidence or reasoning to support (c), I will always reject it as per the above.

You should read beyond the quote to where I respond to explain the error in Benacerraf‘s reasoning.

I’ll keep it simple @fishfry. This is my argument:

P1. If we can recite the natural numbers at successively halved intervals of time then we can recite every natural number in finite time
P2. We cannot recite every natural number in finite time
C1. Therefore, we cannot recite the natural numbers at successively halved intervals of time

I justify P2 with this tautology:

P3. If we start reciting the natural numbers then either we stop on some finite number or we never stop

See also here for my defence of Thomson's lamp and here where I explain that this reasoning applies to all supertasks, further justifying C1.

What is "evidence" in a metaphysical realm? — jgill

That’s for those who assert the truth of (2) to answer. As it stands it’s just an assertion, and contradicts the tautology given in (1), so I have every reason to reject it.

How do you make this conclusion? — Metaphysician Undercover

It’s the conclusion of those who use the finite sum of a geometric series as proof that a supertask can be completed. If I recite the first number after 30 seconds, the second after 15 seconds, and so on, then I have recited them all and so stopped after 60 seconds, even though there is no largest number for me to stop on.

I think it’s nonsense as it contradicts the tautology given in (1), and so I reject (2).

To make this very simple, we have two competing claims:

1. If we start reciting the natural numbers then either we stop on some finite number or we never stop

2. It is metaphysically possible to recite the natural numbers at successively halved intervals of time

If (2) is true then we can stop without stopping on some finite number.

Some take this as proof that (1) is false. I take this as proof that (2) is false.

I think that (1) is a tautology whereas no evidence has been offered in support of (2).

But that's YOUR hypothesis, not mine. — fishfry

It's not mine. It's the hypothesis of those who claim that supertasks are possible. They try to use such things as the finite sum of a geometric series to resolve Zeno's paradox. They claim that because time is infinitely divisible it's possible for us to perform a succession of operations that correspond to a geometric series, and so it's possible to complete an infinite succession of operations in finite time.

I have been arguing firstly that it hasn't been proven that time is infinitely divisible and secondly that if we assume such a possibility then contradictions such as Thomson's lamp follow.

I was very clear on this in my reply to you on page 4, 22 days ago:

We can determine whether or not something entails a contradiction. If time is infinitely divisible then supertasks are possible. Supertasks entail a contradiction. Therefore, time being infinitely divisible entails a contradiction. — Michael

Most of the last few pages has been me trying to re-explain this to you, e.g. 10 days ago:

These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed. — Michael

---

You have repeatedly asked me what happens if we go backwards, saying "1" at 60 seconds, "2" at 30 seconds, and so forth. That also is a purely hypothetical thought experiment. Why on earth are you proposing hypothetical non-physical thought experiments, then saying, "Oh that's impossible!" when I attempt to engage? — fishfry

It was brought up for two reasons. The first was to address the flaw in your reasoning. That same post 10 days ago was very clear on this:

Argument 1
Premise: I said "0", 30 seconds after that I said "1", 15 seconds after that I said "2", 7.5 seconds after that I said "3", and so on ad infinitum.

What natural number did I not recite?

...

Argument 2
Premise: I said "0", 30 seconds before that I said "1", 15 seconds before that I said "2", 7.5 seconds before that I said "3", and so on ad infinitum.

What natural number did I not recite?

...

These arguments only show that if I recite the natural numbers as described then I have recited all the natural numbers, but this does nothing to prove that the antecedent is possible, and it is the possibility of the antecedent that is being discussed. — Michael

If argument 1 is proof that it is possible to have recited the natural numbers in ascending order then argument 2 is proof that it is possible to have recited the natural numbers in descending order.

It is impossible to have recited the natural numbers in descending order.

Therefore, argument 2 is not proof that it is possible to have recited the natural numbers in descending order.

Therefore, argument 1 is not proof that it is possible to have recited the natural numbers in ascending order.

The second reason I brought it up was a proof that it is impossible to have recited the natural numbers in ascending order.

If it is possible to have recited the natural numbers in ascending order then it is possible to have recorded this and then replay it in reverse. Replaying it in reverse is the same as reciting the natural numbers in descending order. Reciting the natural numbers in descending order is impossible. Therefore, it is impossible to have recited the natural numbers in ascending order.

Or if you don't like the specific example of a recording, then the metaphysical possibility of T-symmetry might suffice.

Either way, the point is that it's special pleading to argue that it's possible to have recited the natural numbers in ascending order but not possible to have recited them in descending order. It's either both or neither, and it can't be both, therefore it's neither.

You yourself proved P2 true — fishfry

No I didn't.

Your argument is analogous to this:

If I am immortal then when will I die of old age? I won't. Therefore, I have proved that I am immortal.

Agreeing with what follows if we can recite the natural numbers at successively halved intervals of time doesn't prove that we can recite the natural numbers at successively halved intervals of time.