Sorry about that. I typically select the entire post and hit Quote, and it seems to lose a lot of the attribution. — fishfry

Yes, I find that as well. I work round it by selecting only the quoted text, not including the link that gives the attribution. Then, you can hit "quote" and the system does pick up the attribution. Then, if you separately select the response, it is copied and attributed in the normal way.

The limit is not part of the sequence. so that doesn't run the sequence backward. I am not sure what point you are making about the sequence. The dots merely indicate that the sequence progresses indefinitely. — fishfry

Neither am I, on reflection. I was trying to articulate the point that one can count forward, but not backward, so I don't think anything is at stake.

"If you know what you're doing you're not learning anything." Think I read that somewhere. — fishfry

Yes, I like that. I'm a bit of a contrarian, so I'm tempted to reply that I don't need my surgeon to learn anything while he's cutting me open. Indeed, I would be rather concerned if I thought he was. It applies better to artistic, experimental, open-ended activities - like philosophy and maybe mathematics, at least sometimes.

I'm out of my depth on that. Don't understand what's meant by realism or anti-realism. Simply don't believe that 2 + 2 = 4 has a truth value before some intelligent entity shows up to pass judgment. — fishfry

If you don't understand what realism vs anti-realism means, you have understood correctly - as I see it. Some people would argue that the proposition that "2+2 = 4" does indeed only have a truth-value only when someone passes judgement on it but that 2+2 = 4 independently of anyone doing that i.e. is objectively true. There's a temptation to think that mathematical truth is eternal, i.e. always has been true, always will be true, whatever happens. But that's a mistake. It makes no sense to assign a place in the time series to 2+2 = 4; there is no meaningful way of doing that. (Grammarians recognize a tense that is called the timeless present which is exemplified in propositions like this.)

Ok. Don't think I disagreed with anything you said. — fishfry

I'm glad it made sense.

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.

This is so sensible to me, I understand completely.

Though I would disagree with 'or we never stop'.

Since it should be P4, and it introduces the proposition that we could complete the supertask. In this case, there must be infinite time. In infinite time, it's no longer a supertask, but a measure of a greater and lesser infinity. If we imply an infinity and then imply a greater infinity, there is the middle of the two that we can call X, and for X, we can assume all natural numbers were recited, just like we assume natural numbers go on forever.

I see no contradiction in Thompson's lamp, only a failure to define the terminal state.
— fishfry
See here. — Michael

I followed your link and found this quotation from Benacerraf's Tasks, Super-Tasks, and the Modern Eleatics. I've put the passages of interest in bold and italicized the passage quoted from Thompson for clarity.

Thomson's first argument, concerning the lamp, is short, imaginative, and compelling. It appears to demonstrate that "completing a super-task" is a self-contradictory concept. Let me reproduce it here:

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. ... 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 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.

Pause here. I think Thompson means that

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.

contradicts

But the lamp must be either on or off.

That seems to be true.

But the passage continues (Benacerraf's words): -

Rarely are we presented with an argument so neat and convincing. This one has only one flaw. It is invalid. Let us see why. Consider the following two descriptions:
A. Aladdin starts at t0 and performs the super-task in question just as Thomson does. Let t1 be the first instant after he has completed the whole infinite sequence of jabs – the instant about which Thomson asks "Is the lamp on or off?" – and let the lamp be on at t1.
B. Bernard starts at t0 and performs the super-task in question (on another lamp) just as Aladdin does, and let Bernard's lamp be off at t1.
I submit that neither description is self-contradictory, or, more cautiously, that Thomson's argument shows neither description to be self-contradictory (although possibly some other argument might).

That also seems to be true. The three sentences in bold in the first passage are not individually self-contradictory, but the conjunction of the three (the concept of a supertask) could be described as self-contradictory. Nor are Benacerraf's A or B self-contradictory. They could be both true, if a third state that is neither on nor off were possible. Perhaps Benacerraf was assuming that there isn't.

But @fishfry says that the final state is not defined. That would indeed be a third state which is neither on nor off. The idea that this is the case, is supported by the fact that both Thompson and Benacerraf feel the need to consider both alternatives. The point is simple enough - the definition of the infinite set is such that there can be no last step, in virtue of the definition, every step has a successor. So the last step is not defined. You may be thinking that there must be a last step in a convergent series in the series that we've been considering here, it is 1, or 0. But those are limits, not last steps. The series itself by definition cannot reach that limit, so 1 (or 0) cannot be a step in the series. One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series.

I think I'll leave it there.

PS Since I started writing this, the link to the post that I copied this quotation from seems to have become non-functional. Very odd.

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

That's true, but seems to be a purely physical limitation. It raises the question whether that means it is really on or off, or a some sort of in-between state. Fluorescent lights flicker on and off all the time (at least if they are running on AC, and we just say they are on. And it is true that for practical purposes there is no relevant difference between that light and sunlight or candle-light. — Ludwig V

Jgill talked about how the lamp would "appear", and this implies a sense observation, and empirical judgement. The point I made is that the description describes something far beyond our capacity to sense, so it is incoherent to talk about how this described thing would "appear".

Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent.

OK. You and @fishfry both believe that the supertask is impossible. But you believe that is because it is contradictory and fishfry believes that it is because the last step is not defined. Am I right about that?

Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent. — Metaphysician Undercover

There is no doubt that it is easy to do that. But it seems that people disagree about whether the scenario makes sense or is incoherent and even if they do agree, they still disagree about why.

Jgill talked about how the lamp would "appear", and this implies a sense observation, and empirical judgement. The point I made is that the description describes something far beyond our capacity to sense, so it is incoherent to talk about how this described thing would "appear". — Metaphysician Undercover

I agree that this isn't really about anything empirical, but it sort of seems to be.

Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent — Metaphysician Undercover

In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. Flashing 0 and 1 cards would seem to be a zero with a one through it. These are models of the supertasks in a rough sense.

Admittedly not the real deals which are metaphysical fantasies.

There is no doubt that it is easy to do that. But it seems that people disagree about whether the scenario makes sense or is incoherent and even if they do agree, they still disagree about why. — Ludwig V

It's easy to see why it's incoherent. Start out with the concept of infinite. We can easily see why it is beneficial to allow for numbers to be infinite. This allows that there is no limit to our capacity to count any quantity, or to measure any size of thing, because we can always have a large number. In the case of division though, we may assume that infinite divisibility would allow us to divide anything anyway, but this is really incoherent. That is because division implies, or requires logically, that there is something, an object of some sort, to be divided, and its divisibility will always be dependent on the sort of thing that it is. An object, or thing is a unity of some type, and as such there is always limits to its divisibility, whatever unifies also determines divisibility.

To propose a thing which is infinitely divisible is an incoherent proposition, because as a thing, it is already necessarily limited in its divisibility. This is the issue of the proposition of a finite object being infinitely divisible. That is incoherent because the finiteness of the object limits its divisibility. In whatever way it is finite, its divisibility is limited accordingly.

I agree that this isn't really about anything empirical, but it sort of seems to be. — Ludwig V

This is the trick of the whole thing. It really is about empirical things. These empirical things are space and time, each of these is known through experience. Then we take these empirical things and pretend that they are absolutely abstract, purely ideal, and stipulate ideal principles like infinite divisibility. Then, someone creates a scenario, like the lamp or the op, which utilizes this purely ideal feature of infinite divisibility. Now we do not properly separate the purely ideal from the empirical, in our minds, so that "empirical time" interferes, and we say that 60 seconds must pass, it has to because experience tells us that it will. But that is allowing "time" to be an empirical thing.

In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. — jgill

The problem though, is that in the prescribed scenario there is no such thing as "a certain rate". The rate is not constant, but rapidly increasing. The only constant is the rate of increase. That rate of increase is what I say is incomprehensible and incoherent.

In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light. — jgill

It's a bit more complicated than that. Bulbs like fluorescent ones flicker, but the light really is constant. It's like what is called "motion illusion" or the φ phenomenon. Film and television both rely on it. In a sense, the motion is an illusion, but in another sense, it isn't. The illusion of constant light, paradoxically, is real.

The problem though, is that in the prescribed scenario there is no such thing as "a certain rate". The rate is not constant, but rapidly increasing. The only constant is the rate of increase. That rate of increase is what I say is incomprehensible and incoherent. — Metaphysician Undercover

Yes, but it is not difficult to abandon the (pseudo-physical) lamp for a purely abstract version, which does not have the same problems.

One is to align the infinite sequence "0,1,0,1,0,1...." with the steps in the regress and to ask whether 0 or 1 can be assigned to the last step. An even simpler version is to align the natural numbers with the steps in the regress and ask whether the final step is an odd or an even number.

Those are clearly abstract sequences, and don't have the complications of trying to align the steps of a physical sequence to the steps of the regress. I agree that trying to do that is very problematic and almost certainly incoherent.

This is the trick of the whole thing. It really is about empirical things. These empirical things are space and time, each of these is known through experience. Then we take these empirical things and pretend that they are absolutely abstract, purely ideal, and stipulate ideal principles like infinite divisibility. — Metaphysician Undercover

I don't have a problem with ideal principles. They are very useful. We need infinite divisibility for the same sort of reason that we need infinite numbers. The infinite numbers guarantee that we can count anything. Infinite divisibility guarantees that we can measure anything (that is measurable at all). Limitations on either are physical.

In the case of division though, we may assume that infinite divisibility would allow us to divide anything anyway, but this is really incoherent. That is because division implies, or requires logically, that there is something, an object of some sort, to be divided, and its divisibility will always be dependent on the sort of thing that it is. An object, or thing is a unity of some type, and as such there is always limits to its divisibility, whatever unifies also determines divisibility. — Metaphysician Undercover

I think you are being misled by the temptation to take the divisibility of "medium-sized dry goods" as the paradigm of divisibility. But even that depends on the level of description you are applying, or, if you prefer, the level of analysis you are using.
The weight of a medium sized dry good is infinitely divisible. We couldn't measure its weight or calculate its centre of gravity it if it wasn't.
The colour of something isn't divisible at all. (That's why colour is disregarded in physics and treated as a mental phenomenon, whatever that is.)
You may say that these are not "objects", but that reinforces my impression that you are unduly focused on just one phenomenon in the physical world.

Then, someone creates a scenario, like the lamp or the op, which utilizes this purely ideal feature of infinite divisibility. Now we do not properly separate the purely ideal from the empirical, in our minds, so that "empirical time" interferes, and we say that 60 seconds must pass, it has to because experience tells us that it will. But that is allowing "time" to be an empirical thing. — Metaphysician Undercover

You may be right. I'm afraid that I'm like Augustine. I don't know what time is, though I do know, of course, what time it is right now and what time I woke up.
The great trap with infinity is the temptation to try to get round the endlessness of the sequence by hacking through each step in it.

Yes, but it is not difficult to abandon the (pseudo-physical) lamp for a purely abstract version, which does not have the same problems. — Ludwig V

The point I've been arguing since the beginning of the thread, is that if we abandon the empirical, and adhere strictly to the prescribed, purely abstract version, then nothing indicates that 60 seconds will pass. You see in both the op, and the lamp example, 'there will be a condition at 60 seconds' is an unwarranted conclusion simply added on, and not derived from the initial premises. The initial premises, being the described activity do not allow that 60 seconds will pass. This conclusion, 'there will be a condition at 60 seconds' is pulled from empirical evidence, and is completely inconsistent with the prescribed purely abstract version.

We need infinite divisibility for the same sort of reason that we need infinite numbers. The infinite numbers guarantee that we can count anything. Infinite divisibility guarantees that we can measure anything (that is measurable at all). Limitations on either are physical. — Ludwig V

No we don't need infinite divisibility, for the same sort of reason that we need infinite numbers, for the reasons I described. Any thing which is to be divided has its divisibility determined by the sort of thing that it is. And each type of thing is divisible in different ways. And, the way that the thing is divisible must be determined prior to division, or else we attempt to do the impossible. There is nothing that is divisible infinitely, therefore this ideal needs to be excluded as necessarily an attempt to do the impossible.

This is not the same as infinite numbers. The countability of a multitude cannot be determined beforehand, as the divisibility of a thing must be determined beforehand. So, we need infinite numbers because we do not know how large the multitude will be until we count, but we do not need infinite divisibility because it is impossible to divide something prior to knowing its divisibility, and the possibility of infinite divisibility is already excluded as actually impossible.

That is a fundamental feature of the difference between a unity and a multitude. If a unity is composed of a number of parts, then the number of parts is necessarily finite. To have infinite parts would violate the finitude of the boundaries implied by the concept "unity". But the concept of "multitude" has no such implied boundaries, principles of limitations. Therefore a "unity" is always limited in its divisibility, limited by the principles which make it a unity rather than just a multitude (I cannot say "multitude of parts" because "part" implies a whole), and a multitude is not necessarily a whole, so its countability is not necessarily limited.

I think you are being misled by the temptation to take the divisibility of "medium-sized dry goods" as the paradigm of divisibility. — Ludwig V

No, I am talking about the divisibility of anything, in an absolute sense. There is no such thing as a unity which has absolutely zero limitations on its divisibility. That is a fundamental feature of what it mean to be a unity.

The colour of something isn't divisible at all. — Ludwig V

Colour is very much divisible. It is a collection of distinct wavelengths, and I believe it is divided by the harmonic principles of the Fourier transform.

There is nothing that is divisible infinitely, therefore this ideal needs to be excluded as necessarily an attempt to do the impossible. — Metaphysician Undercover

.... apart from a geometrical straight or curved line. I grant you that that is a concept of an abstract, ideal object. I grant you also that such division does not necessarily affect the unity of the object in any way.

It is a collection of distinct wavelengths, and I believe it is divided by the harmonic principles of the Fourier transform. — Metaphysician Undercover

That is the best representation of colour that physics can manage. But most people do not know about wave-lengths or Fourier transforms. So when I choose a red coat to wear to-day, how do I manage that? The colour that I am aware of is divisible in the sense that there are many colours and shades of colours. These correspond only roughly to the wavelengths of light.

No we don't need infinite divisibility, for the same sort of reason that we need infinite numbers, for the reasons I described. — Metaphysician Undercover

So how can we be sure that anything can be measured in terms of metres, if metres cannot be divided so that they exactly measure the length we are measuring?

I need to think about how to tackle your metaphysics.

An afterthought. Do I understand rightly that your analysis of wholes and parts applies to physical objects, and not to mathematical ones?

Yes, I find that as well. I work round it by selecting only the quoted text, not including the link that gives the attribution. Then, you can hit "quote" and the system does pick up the attribution. Then, if you separately select the response, it is copied and attributed in the normal way. — Ludwig V

Thanks, guess I can no longer just copy/quote the whole response any more. Have to do it one paragraph at a time.

Neither am I, on reflection. I was trying to articulate the point that one can count forward, but not backward, so I don't think anything is at stake. — Ludwig V

Nothing is at stake for me or you, but our friend @Michael seems to regard it as important. It's obvilus that you can't count the natural numbers backwards.

"If you know what you're doing you're not learning anything." Think I read that somewhere.
— fishfry
Yes, I like that. I'm a bit of a contrarian, so I'm tempted to reply that I don't need my surgeon to learn anything while he's cutting me open. Indeed, I would be rather concerned if I thought he was. It applies better to artistic, experimental, open-ended activities - like philosophy and maybe mathematics, at least sometimes. — Ludwig V

Agreed! That's why job applicants should not say they're looking for a challenging position. Employers want someone who can do the job!

If you don't understand what realism vs anti-realism means, you have understood correctly - as I see it. — Ludwig V

LOL. I feel better then.

Some people would argue that the proposition that "2+2 = 4" does indeed only have a truth-value only when someone passes judgement on it but that 2+2 = 4 independently of anyone doing that i.e. is objectively true. There's a temptation to think that mathematical truth is eternal, i.e. always has been true, always will be true, whatever happens. But that's a mistake. It makes no sense to assign a place in the time series to 2+2 = 4; there is no meaningful way of doing that. (Grammarians recognize a tense that is called the timeless present which is exemplified in propositions like this.) — Ludwig V

We must be in agreement on that then.

Ok. Don't think I disagreed with anything you said.
— fishfry
I'm glad it made sense. — Ludwig V

Ok!!

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 — Michael

I believe you're equivocating in the sense that you don't distinguish between the successively halved time intervals as:

a) An abstract, hypothetical, mathematical exercise; and

b) A physical situation that is incompatible with known physics and that may or may not be incompatible with the true nature of the world.

Can you clarify which sense you mean?

If you mean (a), then of course we CAN recite every natural number, so P2 is wrong; if (b), we can't currently and likely will never be able to.

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.

I see no contradiction in Thompson's lamp, only a failure to define the terminal state.

PS Since I started writing this, the link to the post that I copied this quotation from seems to have become non-functional. Very odd. — Ludwig V

Was this from you to me? That post of @Michael disappeared for me as well.

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.

Was this from you to me? That post of Michael disappeared for me as well. — fishfry

I still can't find it. I copied the quoted passage into my message, but not the commentary. Which is a pity.

c) it is metaphysically possible to recite the natural numbers at successively halved intervals of time — Michael

There's another strictly philosophical issue. I know that metaphysics overlaps with logic. I'm still trying to work out whether it is identical with logic.

Supertasks cannot be performed in any possible world. — Michael

suggests to me that it is a question of logic.
Subject to that, I do agree that c is false. I think that those who disagree with you think that they can stipulate by definition some sense in which it is true.

↪Michael
OK. You and fishfry both believe that the supertask is impossible. But you believe that is because it is contradictory and fishfry believes that it is because the last step is not defined. Am I right about that? — Ludwig V

I have not said that. I have said that I have no strong opinion about supertasks and am entirely comfortable arguing either side.

.... apart from a geometrical straight or curved line. I grant you that that is a concept of an abstract, ideal object. I grant you also that such division does not necessarily affect the unity of the object in any way. — Ludwig V

Assuming lines to be infinitely divisible is problematic, just like assuming time or space to be infinitely divisible is problematic. As much as we like to claim that such a divisibility is "logically possible", it's really not. Such an assumption produces an unintelligibility similar to infinite regress, which is an incoherency, therefore illogical.

So when I choose a red coat to wear to-day, how do I manage that? The colour that I am aware of is divisible in the sense that there are many colours and shades of colours. These correspond only roughly to the wavelengths of light. — Ludwig V

I don't follow you. Why would you need to know something about divisibility to choose a colour?

So how can we be sure that anything can be measured in terms of metres, if metres cannot be divided so that they exactly measure the length we are measuring? — Ludwig V

Again, I do not follow. Metres can be divided. We have centimetres and millimetres. But when we measure, at some point an approximation is made, a rounding off.

An afterthought. Do I understand rightly that your analysis of wholes and parts applies to physical objects, and not to mathematical ones? — Ludwig V

I don't think that this is relevant. I believe the analysis applies to all objects. But there is a problem with supposed "mathematical objects", and this is that we assume them to be infinitely divisible. And this assumption creates incoherency. This incoherency renders the supposed objects as not true objects.

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. — Michael

Ok. Possible worlds. I actually took a class where we talked about that, but I have a hard time understanding the concept. There are people who think possible worlds are real. I'm not one of them. And the whole metaphor is lost on me.

But there's a problem. You are assuming that which you wish to prove. You say, "Supertasks cannot be performed in any possible world." But that's the thing you're trying to convince me of. Nevermind that I don't necessarily believe in supertasks myself, but regardless, you are trying to convince me that there are no supertasks. So you can't just state that there are no supertasks. Right? Just a basic point of logic.

And even then I reject the claim on its own merits. I could argue (not that I do, but that I could -- hope that's clear) that if time is modeled by the real numbers (agreed, that is a dubious assumption) then I perform a supertask every time I get up to go to the kitchen for a snack. I named my refrigerator Zeno.

If I can make that argument, then there is some possible world in which it's true. Namely, in any possible world in which time is accurately modeled by the standard real numbers, supertasks are commonplace, every day occurrences.

I argue -- in fact I believe I've made this argument several times already -- that because supertasks are abstractly conceivable, they are NOT metaphysically impossible.

A



In short, (1) You assumed what you're trying to prove; and (2) Your assumption is wrong. Supertasks are conceivable and as far as anyone knows, even physically possible. It's for the physics of the future to know.

Also, it's conceivable that future physics will incorporate physically instantiable infinities, and that supertasks will be possible. I just don't see how you can ignore that possibility.

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. — Michael

(a) and (b) are mathematical truths we all agree to, I hope. But "metaphysically impossible," I don't even know what that means.

If I make the assumption of successive halving, I can recite all the numbers in finite time. I agree this is a purely fictitious, hypothetical game.

We agree that it is physically impossible, relative to currently known physics.

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

By analogy with previous scientific revolutions, we must allow for the possibility that actualized infinities could become part of physics; and that physical supertasks might be witnessed in the Superconducting Supertask Collider of the future. This is a grant proposal waiting to happen.

I just don't see how you can call this metaphysically impossible. So was heavier-than-air controlled flight, once.

Your argument rests on the assumption that (c) follows from (a) and (b), but it doesn't. — Michael

I don't even know what you are talking about. No argument I make assumes a and b and concludes c. (a) and (b) are trivialities. (a) is freshman calculus and (b) is the very definition of an infinite sequence. Together they don't imply anything. You're imagining some argument I didn't make and wouldn't make.

(c) is proven false by P3, as well as arguments like Thomson's lamp. — Michael

I wish we were having the same conversation. "My argument" assumes (a) and (b) and concludes (c)? I can't even relate that to anything I can think of. What do (a) and (b) have to do with what's metaphysically possible? And what does metaphysically possible even mean? I should have asked that earlier. I don't know what the phrase means to you. And it's not one I use in my own mind, myself. I think just about anything is metaphysically possible, if I were pressed to give an opinion on the subject. If not pressed, the thought of metaphysical possibility never enters my mind. So probably I have no idea what you mean.

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. — Michael

I have surely presented my reasoning, which I will repeat here.

P1) It is metaphysically possible that time is accurately modeled by the standard real numbers;

P2) 1/2 + 1/4 + 1/8 + ... = 1 in the real numbers;

C1) A Zeno supertask is possible, by moving in any direction for one second, and thereby traversing a countably infinite sequence of finite intervals of time [0, 1/2], [1/2, 3/4], [3/4, 7/8], ...

C2) Therefore supertasks are metaphysically conceivable.

Secondly:

P3) It is conceivable that future physics will allow for supertask; and therefore

C2) Supertasks are metaphysically conceivable.

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

And please reframe your argument about my claiming (a) and (b) imply (c), when in fact I make no such argument.

I still can't find it. I copied the quoted passage into my message, but not the commentary. Which is a pity. — Ludwig V

Lost in the ether, forever.

Lost in the ether, forever. — fishfry

I expect we'll survive.

I have not said that. I have said that I have no strong opinion about supertasks and am entirely comfortable arguing either side. — fishfry

Thanks for clarifying that. I find it quite hard to remember what everyone's position actually is. It gets lost in all the detail.

One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series. — Ludwig V

I would be very grateful if you could help me clarify this. When you say:-

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

That's not quite as simple as it looks. The left-hand side will never equal the right-hand side as long as I try to make them equal by adding further steps in accordance with the same rule (...1/16, 1/32...). That's what it means to say that 1 is the limit, not the last step. But if I add 1/8 again, the two sides will be equal. Does that count as completing the sequence?

Ok. Possible worlds. I actually took a class where we talked about that, but I have a hard time understanding the concept. There are people who think possible worlds are real. I'm not one of them. And the whole metaphor is lost on me. — fishfry

Whether possible worlds count as real depends entirely on what you mean by "real". For some people, "real" comes down to true. If it is possible that it will rain tomorrow then possible worlds are real because it is true that it will rain tomorrow. For others, a possibility is not actual, so cannot be real.

And even then I reject the claim on its own merits. I could argue (not that I do, but that I could -- hope that's clear) that if time is modeled by the real numbers (agreed, that is a dubious assumption) then I perform a supertask every time I get up to go to the kitchen for a snack. I named my refrigerator Zeno. — fishfry

Quite so. But I think there is a confusion going on here. If you'll allow a temporary and artificial distinction... Roughly, it's the difference between an analysis, which doesn't change or affect its object, and a division or separation which does. That's the difference between measuring a plank of wood as 10 cm long and cutting it into 1cm lengths. The first is an analysis, the second is a division.
There are infinite ways in which I can mark out the plank, and they are all true at the same time and the physical object that is the plank is unaffected by any of them. True, the marks will be physical objects, so there will be limits to what I can do. But the system allows me infinite possibilities, including a convergent series. None of these makes the slightest difference to the plank. So when you visit Zeno for a beer, the fact that there are infinitely many analyses of your journey does not make the slightest difference. It's all in your head.

(Here's a thought. When you drink your beer, you have to drink 1/2 of it and then 1/4 of it and then... Your beer will never be finished. :smile: But then, a similar argument would show that you can't even start drinking it. :sad: )

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

It simply isn't clear. "Metaphysics" is a word looking for a meaning. There is some connection with logic, but what differentiates the two is a mystery.

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.

Again, I do not follow. Metres can be divided. We have centimetres and millimetres. But when we measure, at some point an approximation is made, a rounding off. — Metaphysician Undercover

Quite so. And we know that it is an approximation because we know what more and less accurate or precise measurement is. The exact measure, in the physical world, is the limit that empirical measurements can approach and never reach. That's mathematics and logic.

I don't think that this is relevant. I believe the analysis applies to all objects. But there is a problem with supposed "mathematical objects", and this is that we assume them to be infinitely divisible. And this assumption creates incoherency. This incoherency renders the supposed objects as not true objects. — Metaphysician Undercover

I'll take that. I wouldn't put it the same way, but it's near enough. I think, by the way, that you would have a tough job to convince mathematicians that there is an incoherency in the concept of the infinite. But that's not my problem.
Irrational numbers are built in to the system, so are recurring numbers, and so is infinity. You can't have one without the other. Sad, but true.

Your beer will never be finished. — Ludwig V

Where do i get one of these metaphysical beers?

I think, by the way, that you would have a tough job to convince mathematicians that there is an incoherency in the concept of the infinite. — Ludwig V

I clearly explained though, it isn't "infinite" which is incoherent, it is "infinite divisibility" which is. "Infinite divisibility" is a specific application of the term "infinite" which is incoherent. It is incoherent because the concept of "infinite" is incompatible with, inconsistent with, or contradicts, what is implied by the concept "divisible". Therefore the two together as "infinite divisibility" is self-contradicting.

Mathematicians have made "infinite" into a new term, which really has very little resemblance to its metaphysical roots. This is very evident in set theory. There is no consistency between "infinite" in mathematics and "infinite" in metaphysics. Because of this, mathematicians are incapable of understanding metaphysical "infinite". There is no word for it, as that word is used for something else in mathematics, so the concept escapes their grasp. This leaves mathematics, and mathematicians in general, as fundamentally incapable of dealing with the metaphysical problems involved with the concept "infinite".

Where do i get one of these metaphysical beers? — Metaphysician Undercover

At your local metaphysical beer shop, of course. I'm sure Google knows its address and will give you directions. (Shops never stock both metaphysical and mathematical beers at the same time. They fight, you know - very messy!)

I clearly explained though, it isn't "infinite" which is incoherent, it is "infinite divisibility" which is. "Infinite divisibility" is a specific application of the term "infinite" which is incoherent. It is incoherent because the concept of "infinite" is incompatible with, inconsistent with, or contradicts, what is implied by the concept "divisible". Therefore the two together as "infinite divisibility" is self-contradicting. — Metaphysician Undercover

Of course you did. I'm sorry. But in any case you've just accepted that mathematical objects aren't true objects. So what's the problem?

Mathematicians have made "infinite" into a new term, which really has very little resemblance to its metaphysical roots. — Metaphysician Undercover

I would be happy to accept that there are two concepts of infinity here. I think that may be because their concept has its roots in mathematics, whereas the metaphysical concept has roots elsewhere..

This leaves mathematics, and mathematicians in general, as fundamentally incapable of dealing with the metaphysical problems involved with the concept "infinite". — Metaphysician Undercover

So we just have a case of Domains of Magisterial Authority, and no need to fight about it.
Our only remaining issue is whether the problem of Achilles and the tortoise and Thompson's lamp is a mathematical problem or a metaphysical problem. Or maybe it's just a question of understanding two solutions to the same problem. They clearly won't be incompatible.

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. — Michael

I think it would be better to stick with the strong claim. At least it is more comprehensible.

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). — Michael

Yes Kripke does claim that. But he waters down the meaning of "necessarily". For him, it no longer means "in all possible worlds", but "in all possible worlds in which certain conditions hold". But contingent means, or used to mean, "true or false depending on certain conditions". So, on this account "necessarily" means what "contingent" used to mean. Talk about having your cake and eating it!

Of course you did. I'm sorry. But in any case you've just accepted that mathematical objects aren't true objects. So what's the problem? — Ludwig V

The problem is the incoherency. Maybe, if there was no incoherency inherent within mathematical objects, they could be true objects.

So we just have a case of Domains of Magisterial Authority, and no need to fight about it. — Ludwig V

There's still the issue of incoherency, and the work of the metaphysician is to seek and destroy that sort of evil. Therefore, the fight is on.

Our only remaining issue is whether the problem of Achilles and the tortoise and Thompson's lamp is a mathematical problem or a metaphysical problem. — Ludwig V

It's both, and that's the reason for the fight.

Or maybe it's just a question of understanding two solutions to the same problem. They clearly won't be incompatible. — Ludwig V

When one uses incoherent principles, and the other does not allow incoherency, then they will not be compatible.

Thanks for clarifying that. I find it quite hard to remember what everyone's position actually is. It gets lost in all the detail. — Ludwig V

Yes, I am experiencing that myself. No idea what's being argued in some of these threads or what claims anyone is defending.

I try to take a useful role around here clarifying some of the vague and imprecise mathematical ideas people have. Misunderstandings around mathematical convergence and limiting processes abound in supertask and Zeno discussions. My main goal is to try to rigorize and clarify the mathematical discussions, without necessarily taking strong positions on the paradoxes themselves. I don't care if there are supertasks or not, but I am driven to straighten out the bad thinking around limits (or die trying, is more like it).

One might say that one cannot complete such a series. I'm not sure of my ground here, but I think you will find that everything depends on what is meant by "complete" and it won't mean completing a recitation of all the steps in the series. — Ludwig V

Don't remember what that was in reference to, but based on the next para perhaps we're talking about mathematical infinite sums?

I would be very grateful if you could help me clarify this. When you say:-
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
That's not quite as simple as it looks. The left-hand side will never equal the right-hand side as long as I try to make them equal by adding further steps in accordance with the same rule (...1/16, 1/32...). That's what it means to say that 1 is the limit, not the last step. But if I add 1/8 again, the two sides will be equal. Does that count as completing the sequence? — Ludwig V

The two sides are equal. Unlike a For loop in programming, the left side is not a process or a sequence of steps. The left side happens "all at once" in exactly the same way that 1 + 1 = 2. It's true right now and it's true for all time. 1 + 1 and 2 are different text string expressions for the same abstract object, the number we call 2.

Likewise 1/2 + 1/4 + 1/8 + ... and 1 are two text string expressions for the same abstract object, namely the number we call 1.

In programming with think of loops as taking place in time. And they do. A loop in a programming language is a notation for a physical process that inputs energy and outputs heat and takes a certain amount of time. And of course the computer process never finishes, it just hits the computational limits of the computer and doesn't get any larger after a certain point.

But in math, 1/2 + 1/4 + ... is added together all at once. And the sum is exactly 1, right now, right this moment. And the reason the sum is 1 is because we cleverly define it to be that way. We define the sum of that series to be the limit of the sequence of partial sums: 1/2, 3/4, 7/8, ... And that sequence has the limit 1, because we have carefully set up the definition of a limit in such a way that it's true.

Am I addressing or at least understanding your concern? It's a common one when it comes to convergence.

In math, the notation 1/2 + 1/4 + 1/8 + ... does NOT denote a process or a sequence of discrete steps.

Rather, it's the sum of infinitely many numbers, taken all at once. And it evaluates exactly to the same abstract number that the notation "1" evaluates to. It is no difference than 1 + 1. It's 2 right now, not later one when the process completes. It's not a process. It's a sum. A binary operation. Two numbers go in, one comes out. Or infinitely many numbers go in, and their sum comes out. Immediately. No resources such as time or energy are consumed. The sum just is.

Whether possible worlds count as real depends entirely on what you mean by "real". For some people, "real" comes down to true. If it is possible that it will rain tomorrow then possible worlds are real because it is true that it will rain tomorrow. For others, a possibility is not actual, so cannot be real. — Ludwig V

Some people regard all possible worlds as equally true. That viewpoint doesn't resonate with me.

Quite so. But I think there is a confusion going on here. If you'll allow a temporary and artificial distinction... Roughly, it's the difference between an analysis, which doesn't change or affect its object, and a division or separation which does. That's the difference between measuring a plank of wood as 10 cm long and cutting it into 1cm lengths. The first is an analysis, the second is a division.
There are infinite ways in which I can mark out the plank, and they are all true at the same time and the physical object that is the plank is unaffected by any of them. True, the marks will be physical objects, so there will be limits to what I can do. But the system allows me infinite possibilities, including a convergent series. None of these makes the slightest difference to the plank. So when you visit Zeno for a beer, the fact that there are infinitely many analyses of your journey does not make the slightest difference. It's all in your head. — Ludwig V

You lost me here. I believe I was arguing to @Michael that it's at least conceivable that we execute a Zeno walk on the way to the kitchen for a snack; and that therefore, the idea is at least metaphysically possible. That's all I'm saying.

Oh maybe I understand ... you're saying that just because the path can be infinitely subdivided, does not mean that I'm actually executing that sequence. I think I disagree. I have to traverse each of the segments to get to the kitchen.

But all I'm saying is that it's at least conceivable; in which case it's not metaphysically impossible. I don't have to argue strongly that it's true; only that it's at least barely conceivable.

(Here's a thought. When you drink your beer, you have to drink 1/2 of it and then 1/4 of it and then... Your beer will never be finished. :smile: But then, a similar argument would show that you can't even start drinking it. :sad: ) — Ludwig V

Right. Aleph-null bottles of beer on the wall, aleph-null bottles of beer. You take one down, pass it around, aleph-null bottles of beer on the wall ... :-)

You could probably help me out by clearly defining metaphysically impossible.
— fishfry
It simply isn't clear. "Metaphysics" is a word looking for a meaning. There is some connection with logic, but what differentiates the two is a mystery. — Ludwig V

I think this was @Michael I was asking. Did my quoting get messed up? Michael keeps saying supertasks are metaphysically impossible, and I want to make sure I understand what he means by that.