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.

after we finish pushing the button — Michael

THERE IS NO AFTER WE FINISH PUSHING THE BUTTON!!

We are supposed to be pushing the button at half of the prior interval. This is infinite. If you end up at 2 minutes, if you finish, you’ve failed the thought experiment or added some new premise.

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?

They claim that it is possible to have completed an infinite succession of tasks in finite time. — Michael

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

I do not claim it is possible to complete this task. “Task” speak is about physical, finite things like lamps and switches, and actually switching the lamp and marking each time it switches as “at one minute” and “at a minute and a half”.

We are not in the physical world. We are using physical world pictures to demonstrate a purely theoretical, mathematical function. To hell with any introduction of actual tasks and actual lamp states at actual times on actual clocks.

If you make a a mark at one distance or one time period and call it “1” and then make a second mark at a further distance or further time and call it “1.5” and then use this pattern to make a further mark at 1.75, you can continue this exercise if you are so inclined to do so infinitely and you will never mark “2”.

I agree actual tasks defined as “finishing” or “completed” that include functions involving infinitely available steps are absurd.

The odd thing is, I think somewhere in here we are seeing the same thing, just not saying what we see so the other sees that we are seeing the same thing.

You need to give a little bit to me to bring me to see your point because nothing I’m saying seems refutable, and isnt being refuted by you.

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

Is this a joke?

Is this a joke? — Michael

I’m just trying to get to one minute with you, step down the first step.

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

Is there any same page or common ground you see in anything I’m saying?

I still don’t know your point if your point is based on refutations of the basic things I’m saying.

Find the common ground so we can walk together or take me to your point. Why do we need to talk about this under the concept of “supertasks” if you think I’m missing something?

Do you mean “after we finish pushing the button” because people get tired and time presses on? Or do you mean after we’ve pressed the button an infinite number of times? Because there can be no such time, and it would certainly not arrive at two minutes.

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.

Ok, so supertasks can’t be completed.

Did you think I was saying anything to the contrary?

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.

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.”

So what am I getting wrong with you? Why are you arguing with me?

Do you think supertasks can be completed?

Do you think there is “finishing” in an infinite task?

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.

I agree they are impossible. So why do you disagree with the other things I’m saying?

Am I saying annything inconsistent with the fact that completing a supertask is impossible?

So why do you disagree with the other things I’m saying? — Fire Ologist

You claimed that the supertask described in Thomson’s lamp does not entail a contradiction. It does, as shown by that previous post.

This contradiction proves that supertasks are impossible.

I’m claiming that because supertasks are impossible, anything posited about some state of affairs after they are completed is irrelevant and non-existent in relation to the task, so it cannot be evaluated for its contradiction or otherwise.

You should look up “refutation by contradiction”.

Look, if in an argument, the premise includes a task that can’t be completed, and the conclusion includes completion of that task, then there is a contradiction, and this contradiction refutes the conclusion. If that is the sticking point then you are right, and I was wrong.

That’s not enough though.

In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. That’s a square circle. We don’t get out of the gate. The infinite is by definition uncountable, so any conclusion based on a premise that includes the “countably infinite” could be said to be irrelevant, because you can’t create a logical connection between the conclusion and a nonsensical, unimaginable thing like the “countably infinite, whether that relation is said to be contradictory or otherwise.

An infinite number of stairs. How is that possible to imagine, to state as a premise, to even picture?

The arguments in these thought experiments don’t get off the ground once you think them through. We have to give the premises some credibility as containing complete thoughts in order to move towards a conclusion. I grant you that if you give the premises credibility (somehow), the conclusions of these arguments may be contradictory, but that is only because of us accepting a picture of infinite stairs, or a race where one of the participants must take infinitely smaller steps.

I’ve addressed all of these premises before. There is no half until after there is a whole. You don’t travel half a distance first then travel the second half and thereby complete the whole. To call a distance “half” you first call another distance “whole” and then cut it in half. The whole always comes first. So when Zeno says Achilles must first travel half, he forgot that Zeno already accounted for the whole so he could claim whatever shorter distance to be some fraction in relation to that whole. So Zeno, like Thompson, and the others, tried to make mathematical relations between numbers correspond to the physical relations between objects. They don’t.

Like you said, you can’t have an irrational number of apples in your fridge, you can’t have a countably infinite number of hotel rooms, or switch a light on and off every half the interval of time prior, or travel any fraction of distance without knowing the denominator (whole distance) first.

In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. — Fire Ologist

See countable sets.

Hotel rooms is nonsense then. There is no such set imaginable.

Just because in mathematics we can distinguish natural numbers from irrational numbers and real numbers, and place certain references to infinite sets in relation to these other concepts, doesn’t mean we can imagine “countably infinite hotel rooms.” You see the hallway there with the room numbers? Or do you see part of the hallway? What floor are we on, if it matters?

If this is a mathematics conversation then why are we ever referring to stairs, lamps, hotels, switches, starting lines at races??

@Michael

I have tracked down but not yet had the time to read Benacerraf's famous paper on the subject,
Tasks, Super-tasks, and the Modern Eleatics (pdf link)

I'll let you know if I find anything of interest. I'm happy to find that my take is officially legitimate.

I’ve addressed all of these premises before. There is no half until after there is a whole. You don’t travel half a distance first then travel the second half and thereby complete the whole. To call a distance “half” you first call another distance “whole” and then cut it in half. The whole always comes first. So when Zeno says Achilles must first travel half, he forgot that Zeno already accounted for the whole so he could claim whatever shorter distance to be some fraction in relation to that whole. — Fire Ologist

It can be rephrased as such:

To run 200m one must first run 100m, then 50m, then 25m, and so on.

As for the paradox:

2-dimensional sensors are placed after 100m, 150m, 175m, and so on. These sensors approach the 200m finish line but importantly no sensor is placed on the 200m finish line.

When each sensor is passed the distance is displayed on a screen, as well as the time that the sensor was passed. This display remains until the next sensor is passed. Nothing else controls the behaviour of the screen.

Say we run at a constant speed. We pass the 100m sensor at 12:00:10, the 150m sensor at 12:00:15, the 175m sensor at 12:00:17.5, and so on.

When we reach the 200m finish line at 12:00:20, what distance and what time is displayed on the screen? There is no sensor on the 200m finish line and so neither 200m nor 12:00:20 will display.

No answer consistent with the premises is possible.

There's also Thomson's paper if you haven't read it yet.

In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. That’s a square circle. We don’t get out of the gate. The infinite is by definition uncountable, — Fire Ologist

When I first saw the phrase "countably infinite", I thought that was absurd, and I still think it is an unfortunately ambiguous description of what it means. I would put it this way - any (finite) part of the infinite set can be counted, even though the whole of the set cannot be counted in one go. But I think that Wikipedia also puts it in a reasonably clear fashion.

Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements — Wikipedia

Though I would have said "even though the counting may never finish due to an infinite number of elements."

If this is a mathematics conversation then why are we ever referring to stairs, lamps, hotels, switches, starting lines at races?? — Fire Ologist

I read somewhere that Hilbert never discussed his hotel after the casual mention of it in a paper, even though it provoked enormous discussion. I'm pretty sure he invented it only to help people realize what infinity means. All these cases play in the border country between the mathematical and the physical; they are entirely imaginary (not in the sense that they are possible, but only in the sense we can imagine impossibilities). Consequently, the normal rules of possibility and impossibility are suspended and people think the fact that they can in some sense imagine them means that they are, in some sense, possible.

when Zeno says Achilles must first travel half, he forgot that Zeno already accounted for the whole so he could claim whatever shorter distance to be some fraction in relation to that whole. — Fire Ologist

I agree with that and for that reason think that to say that the conclusion or limit of the set can be anything at all is misleading. In a convergent series, specifying the limit is essential to defining the series. But that doesn't mean that the function that generates the set can generate it's own limit. In fact, if it could, it wouldn't be an infinite set.
The other problem is that Zeno, and most people since him, lose track of the difference between an analysis and a dissection. If I measure the length of the race-track as being 10 units long, the race track is completely unaffected; my measurements are an analysis. If I then cut the race-track into 1-unit lengths, the race track is affected, and what I have done is to dissect it. One can analyze the race track in terms of a given convergent series. But that does not exclude other analyses, including my analysis of it as 10 units long.
My final issue with Zeno is that he forgets that if Achilles is travelling at a constant speed, he will take less and less time to travel each segment of the series, approaching an infinitesimally small time as the segments become smaller and smaller. And, of course, since he can cover a segment in an infinitesimal amount of time, covering an infinite number of them in a finite time becomes less of a problem.

Sensors are placed after 100m, 150m, 175m, and so on. — Michael

So you never finish placing the sensors. The race never starts. You can’t ask what the screen will display. It’s not a math problem about distance and time.

Or it’s just the same math problem as the other paradoxes, that are only really intriguing because of all the people in the stands watching that tortoise go!

So you never finish placing the sensors. — Fire Ologist

For the sake of the argument the sensors just exist at their locations. We don't have to place them. The thought experiment is only to examine the internal consistency of continuous space and time, not the practicality of carrying out the experiment.

st at their locations — Michael

Locations are in physical space. This isn’t a math problem yet.

Do you not understand what thought experiments are or how they're used?

In the supertasks article, they mention a “hotel with a countably infinite number of rooms”. Right there, at the premise, what does “countably infinite” point to? That’s nonsense. That’s a square circle. We don’t get out of the gate. The infinite is by definition uncountable — Fire Ologist

When I first saw the phrase "countably infinite", I thought that was absurd, and I still think it is an unfortunately ambiguous description of what it means. I would put it this way - any (finite) part of the infinite set can be counted, even though the whole of the set cannot be counted in one go. — Ludwig V

A "countable" set is one that can be placed into a one-to-one correspondence with the counting numbers - integers. Why do mathematicians define "countable" that way? Well, think of how we actually count. You look at a herd of sheep, say, and go "1, 2, 3, ..." The number at which you stop is the number of sheep in the herd. What you have done is you placed the set of sheep into a one-to-one correspondence with a subset of integers. Or you may have used your fingers or beads or scratch marks - either way, counting comes down to placing a set of an unknown size into a one-to-one correspondence with a set of a known size.

Counting infinite sets is just a natural extension of the same idea: you try to establish a one-to-one correspondence between some infinite set and the set of integers. You do not need to actually count every member of the set one by one, you only need to establish a procedure of how you would do it, or even just prove that such a procedure exists. If you can do this, then you have established that your set has the same "size" as the set of integers.

Anyway, this is just a specialist term. It doesn't have to "make sense" to be cogent and useful.

Anyway, this is just a specialist term. It doesn't have to "make sense" to be cogent and useful. — SophistiCat

There isn't a problem with specialist terms. But "cogent and useful" is both cogent and useful as a definition of "make sense". I would rather not have to try to find another definition. "Cogent and useful" can mean different things in different contexts.

you only need to establish a procedure of how you would do it, or even just prove that such a procedure exists. — SophistiCat

I don't disagree. But half the problem, for us ordinary folk, is understanding that procedure, especially if, as in this case, it can't actually be carried out. The difficulty is understanding the difference between "and so on" as laziness, when it could be carried out, but one is too lazy or busy to actually do so, and "and so on" in the context of a mathematical induction, when it can't. In the background, I understand, there are people who have doubts about the validity of mathematical induction.

Locations are in physical space. This isn’t a math problem yet. — Fire Ologist

I think that's too simple. It's about the applied math. The issue is about applying the math to physical space (and time). After all, there is no problem about applying ordinary arithmetic to these situations.

When we reach the 200m finish line, what distance and what time is displayed on the screen? — Michael

Sometimes, I am so slow I cannot believe it. The answer to the question is available, if only you would apply ordinary arithmetic to the problem. All the paradox proves is that an analysis in terms of a convergent series does not apply to the question.

The thought experiment is only to examine the internal consistency of continuous space and time, not the practicality of carrying out the experiment. — Michael

It is as well not to confuse the conclusion you want to draw from the analysis with the point of the thought experiment. After all, Zeno did not draw your conclusion from it. Nor do I.

There isn't a problem with specialist terms. But "cogent and useful" is both cogent and useful as a definition of "make sense". I would rather not have to try to find another definition. "Cogent and useful" can mean different things in different contexts. — Ludwig V

I was referring to how folk who are unfamiliar with specialist terms that are based on words in the ordinary language try to make sense of those terms: they interpret them in light of the more familiar senses of the words. Naturally, this doesn't always work. Misinterpretations happen even in familiar contexts, and they are all the more likely in an unfamiliar domain. And as with neologisms, some just aren't going to like the coining for one reason or another, even when they understand the context. But that alone shouldn't be a barrier to understanding and accepting specialist terms.

In the background, I understand, there are people who have doubts about the validity of mathematical induction — Ludwig V

Yes, ultra-finitists reject mathematical induction as a proof method, but that is a rather extreme position.

The answer to the question is available, if only you would apply ordinary arithmetic to the problem. — Ludwig V

Then what is the answer?