Is is not the case that "logically impossible" implies "metaphysically impossible"? — Ludwig V

So what are you arguing about? — Ludwig V

So we can agree that the consequent is false. — Ludwig V

Clearly not by you. Could've easily included it. So why didn't you? — Outlander

You missed out "The lamp is either on or off at all times." — Ludwig V

That seems to be true, so Benacerraf is right. — Ludwig V

I don't know if a button is pushed or not at the terminal time. Who says it's not? — fishfry

The terminal state is arbitrary. — fishfry

Why don't you just run the code and see? — Ludwig V

But since you've put the argument in a list, I'd make explicit all the premises. — TonesInDeepFreeze

var isLampOn = false

function pushButton()
{
  isLampOn = !isLampOn
}

var i = 120

while (true) {

  wait i *= 0.5 // seconds
  
  pushButton()

}

echo isLampOn

Lamps that switch state in arbitrarily small intervals of time? — fishfry

If "nothing other than pushing the button turns the lamp on or off," then at midnight, the button pusher pushes the button and turns the lamp on or off, per your premise. — fishfry

With the lamp, there is no possible way to assign a terminating value that makes any particular sense. Instead, absolutely any answer will do. On, Off, or as I facetiously said earlier, a plate of spaghetti; to emphasize the arbitrariness of the choice. — fishfry

var isLampOn = false

function pushButton()
{
  isLampOn = !isLampOn
}

var i = 120

while (true) {

  wait i *= 0.5
  
  pushButton()

}

echo isLampOn

Not just that it was off and then turned on, but rather that it was off at time t1 and on at time t2. That is, that it's not just a matter of the lamp having been off previously but rather that there is an off state that is an immediate predecessor of the on state and that that extends to 12:00 too so that for the lamp to be on at 12:00 there must be an immediate predecessor state in which the lamp was off, mutatis mutandis for the lamp being off at 12:00. Thomson mentions this. It's a premise that needs to be stated. — TonesInDeepFreeze

(1) The first task is impossible to be performed. The second task is impossible to be performed. The third task is impossible to be performed ...

Quantified:

For all tasks, there is not a performance of any of them.

I think that is not what you mean.

(2) It is not possible for there to be a single performance of all the tasks.

Quantified:

There is not a performance that performs all the tasks.

I surmise that is what you mean.

I wouldn't write "it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on" because it can be understood in sense (1).

It is not possible for the first dancer to do a flip today, for the second dancer to do a flip tomorrow, and so on.

I would take that to mean that none of the dancers can do a flip on their appointed day. — TonesInDeepFreeze

As I mentioned, that is a premise that you don't include in your own argument. As I mentioned:

"his argument includes the premise that there is a state at 12:00 and that that state must be determined by an immediate predecessor state but that there is no immediate predecessor state."

I can't imagine anyone denying that there is no immediate predecessor state, but some partisans who don't accept the argument deny that the state at 12:00 must be determined by an immediate predecessor state. So you must include the premise that the state at 12:00 must be determined by an immediate predecessor state — TonesInDeepFreeze

It's not a matter of continuousness but rather of density. — TonesInDeepFreeze

If you don't mean "Therefore, it is impossible for the first task to be performed at 11:00, the second at 11:30, the third at 11:45, and so on" then it should be considered scratched. — TonesInDeepFreeze

But I wouldn't take P2 as a given without justification. — TonesInDeepFreeze

As I mentioned, C1, as you wrote it, is a non sequitur. That it is impossible for infinitely many tasks to be performed in finite time does not entail that there is a finite upper bound to how many tasks may be performed in finite time, let alone that each of the tasks is impossible to be performed. But maybe you didn't mean C1 as you wrote it. — TonesInDeepFreeze

I'm not firmly opining as to whether A implies B — TonesInDeepFreeze

nor as to whether B is possible. — TonesInDeepFreeze

I'm not sure whether the argument is modally valid — TonesInDeepFreeze

First, though, what sense of 'possible' is meant? Thomson discusses physical possibility and logical possibility. If I'm not mistaken, he doesn't mention metaphysical possibility. Of course, discussion doen't have to be limited to Thomson's context, but 'metaphysical possibility' requires even more explication. — TonesInDeepFreeze

Logical possibility is usually considered the broadest sort of possibility; a proposition is said to be logically possible if there is no logical contradiction involved in its being true. "Dick Cheney is a bachelor" is logically possible, though in fact false; most philosophers have thought that statements like "If I flap my arms very hard, I will fly" are logically possible, although they are nomologically impossible. "Dick Cheney is a married bachelor," on the other hand, is logically impossible; anyone who is a bachelor is therefore not married, so this proposition is logically self-contradictory (though the sentence isn't, because it is logically possible for "bachelor" to mean "married man").

Metaphysical possibility is either equivalent to logical possibility or narrower than it (what a philosopher thinks the relationship between the two is depends, in part, on the philosopher's view of logic). Some philosophers have held that discovered identities such as Kripke's "Water is H₂O" are metaphysically necessary but not logically necessary (they would claim that there is no formal contradiction involved in "Water is not H₂O" even though it turns out to be metaphysically impossible).

Nomological possibility is possibility under the actual laws of nature. Most philosophers since David Hume have held that the laws of nature are metaphysically contingent—that there could have been different natural laws than the ones that actually obtain. If so, then it would not be logically or metaphysically impossible, for example, for you to travel to Alpha Centauri in one day; it would just have to be the case that you could travel faster than the speed of light. But of course there is an important sense in which this is not possible; given that the laws of nature are what they are, there is no way that you could do it. (Some philosophers, such as Sydney Shoemaker, have argued that the laws of nature are in fact necessary, not contingent; if so, then nomological possibility is equivalent to metaphysical possibility.)

(1) We may question P2. — TonesInDeepFreeze

(2) C1 doesn't follow from P1 and P2. — TonesInDeepFreeze

It is incorrect to infer that infinitely many tasks may be completed in finite time from the premise that there is no finite upper bound to how many task may be completed in finite time. I would put it this way: For for any finite number of tasks, there may be a completion of all the tasks. But that does not imply that there may be a completion of all of infinitely many tasks. — TonesInDeepFreeze

Where in the paper does Thompson say that? — TonesInDeepFreeze

And one may take it as a premise or as an established fact that there is a shortest distance and a shortest duration. But perhaps one may also logically take a hypothetical premise that that is not the case. — TonesInDeepFreeze

Did Thomson make that argument? Was that part of his answer to the paradox? — TonesInDeepFreeze

If something is infinitely divisible, and you are to say into how many parts it shall be divided, you have $\mathbb{N_0}$ alternatives from which to choose. This is not to say that $\mathbb{N_0}$ is one of them. And if something is infinitely divisible, then the operation of halving it or halving some part of it can be performed infinitely often. This is not to say that the operation can have been performed infinitely often.

Mathematics doesn't say there is no limit to the ways objects may be divided. — TonesInDeepFreeze

That's the first I've heard of any use of transfinite numbers in this thread. I don't think they are relevant - more, I very much hope they are not relevant. — Ludwig V

But space or time being infinitely divisible does not entail that supertasks are possible. — Ludwig V

Yes, that's what I thought. I think the concept of a valid paradox is a bit confusing. — Ludwig V

Thompson says that there's a false premise, which is that infinitely many tasks cannot be completed in finite time. He says that there's no finite upper limit to the number of tasks that can be completed in finite time, but that not infinitely many can be completed in finite time. — TonesInDeepFreeze

I'm puzzled. I thought you thought that Thompson's paradox was flawed and therefore invalid - as Thompson did, didn't he? — Ludwig V

What does "paradox is valid" mean? Does it mean that the premises indeed entail a contradiction. — TonesInDeepFreeze

The analogy with imaginary numbers and apples is amiss in this regard: Yes, apples are counted by integers, not imaginary numbers, so indeed imaginary numbers are not the correct kind of number to count with. But distances and durations are measured by real numbers, so smaller and smaller real numbers are not a difference in the kind of number. — TonesInDeepFreeze

My question was about mathematics not physics. — TonesInDeepFreeze

If there is a maximum number of divisions, then what is that maximum number? — TonesInDeepFreeze

Einstein’s General Theory of Relativity describes the properties of gravity and assumes that space is a smooth, continuous fabric. Yet quantum theory suggests that space should be grainy at the smallest scales, like sand on a beach.

One of the great concerns of modern physics is to marry these two concepts into a single theory of quantum gravity.

Now, Integral has placed stringent new limits on the size of these quantum ‘grains’ in space, showing them to be much smaller than some quantum gravity ideas would suggest.

...

Some theories suggest that the quantum nature of space should manifest itself at the ‘Planck scale’: the minuscule 10^-35 of a metre, where a millimetre is 10^-3 m.

However, Integral’s observations are about 10,000 times more accurate than any previous and show that any quantum graininess must be at a level of 10^-48 m or smaller.

Come on Michael. Fire Ologist explained the problem with "placing", and you said, we could assume that they are already placed. Now I show you the problem with "already placed", and you say we can assume placing. What's the point in switching back and forth, when both are shown to be problematic? — Metaphysician Undercover

What these "supertasks" show us is that there is a disconnect between the conceptual structures of mathematics and the concepts of the empirical, natural philosophy, (science). — Metaphysician Undercover

To avoid the problem , you just assume the impossible. There is a limit to the number of sensors which can exist in that space, depending on the size of the sensors, Because a sensor takes up space. Or, are you assuming that an infinite number of sensors can fit in a finite space? — Metaphysician Undercover

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

So you never finish placing the sensors. — Fire Ologist

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

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

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