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. — Michael

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.

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 — Michael

P2 is false. As shown by P1. But look, Michael. We have not said anything new to each other in about a week. May we not just agree to disagree? I already pointed out that you claim P2 is true essentially by claiming it's true. Circular.

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 — Michael

How in this great vast wonderful world of ours, does P3 justify P2? They're not even related. You are just confused about what it means to iterate an infinite sequence in math. It doesn't stop, because there's no last element. But under the successive halving hypothesis, all numbers are recited. One day you are going to wake up and this is going to be as clear to you as it is to me. Till then, I can't really repeat it any more times than I already have.

Again:


(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.


When you understand that both these are true, you will be enlightened. I implore the math gods to bring you this insight in a dream. Else my fingers are going to fall off repeatedly explaining it to you.

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

Ok thanks for that. So something is metaphysically impossible if it's necessarily false.

But I do believe that under that definition, I have shown that a Zeno-like supertask is NOT metaphysically impossible, because it is NOT necessarily false. A Zeno-like supertask is indeed POSSIBLE under the assumption that time is like the real numbers, and that an interval of 1 second is equal to an interval of 1/2 second plus 1/4 second, dot dot dot, exactly as it works for the mathematical real numbers.

I would be the first to admit that such a think is highly unlikely. But it is not inconceivable, and therefore is is not necessarily false, and therefore it is not metaphysically impossible.

That is my rejoinder to your claim that supertasks are metaphysically impossible. I can conceive of a circumstance in which they're possible.

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.[/quote]

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

Thompson's lamp shows absolutely no such thing. And of all the paradoxes you could think of, Thompson's lamp is the least relevant thought experiment possible. It just shows that there's no natural way to define the limit of a sequence of alternating 0's and 1's. A trivial observation. Nothing to do with the metaphysical impossibility of supertasks.

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

I'm going to skip engaging with the rest of this. I have articulated my objections to your claims as clearly as I humanly can; and many times over already. I haven't said anything new to you in my last half dozen posts. You just prefer not to engage with my arguments. I wish you would.

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.

There is a fundamental problem with identifying supertasks with series limits, namely the fact that literally infinite summations are not expressible in calculus, given that they cannot be written down.

A formalist is free to use the name "1/2 + 1/4 + ..." to denote 1, but the formalist cannot interpret "1/2 + 1/4 + ..." as an expression implicitly representing part of an infinite summation, because the formalist considers expressions to have no meanings other than being finite states of a syntactical parser when proving a theory in a finite number of steps.


Frege fell into a similar trap as the supertaskers in the Grundgesetze when he proposed his law V. He wanted there to be a one-to-one correspondence between every function and it's representation as a table of values, even in the case of functions with infinite domans. So he proposed Basic Law Vb with disastrous consequences:

{x∣Φx} = {x∣Ψx} → ∀x(Φx ↔ Ψx).

To a finitist or potentialist, Law Vb can be interpreted as introducing fallacies of induction into Set Theory, since they will likely interpret the sets-as-extensions on the left hand side as denoting a finite amount of observable information, and they will likely interpret the function on the right-hand side as denoting an unbounded amount of implicit information, meaning that they cannot regard Law Vb to be a reliable rule of induction. Furthermore, according to their reasoning Law Vb cannot be regarded as constituting a definition of the right hand side, unless one gives up the idea of functions having infinite domains).

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.

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). — fishfry

I'm entirely in favour of the project, but, to be honest, I don't think it is worth dying for.

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

I think that's the first time I've encountered anyone on these sites who understands the difference between "discrete" and "discreet". Not patronizing, just saying.

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. — fishfry

Now you have me a bit puzzled. In my book, that means that the equation is about the complete series, which seems at odds with the idea that it can't be completed. What does "complete" mean? Or does it mean the sense in which it is "always already" complete? (see below)

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

It might be easier to understand if you thought of them as regarding all possible worlds as equally possible. I could understand that. I hope they don't mean that all possible worlds are equally actual....
But some people tend to think only of one kind of possibility - logical possibility. But there many other sorts - physically possible, legally possible, practically possible, etc. etc. I say that possible means different things in different contexts, but it may be that I should be saying there is cloud of possible worlds for each kind. Or maybe physically possible worlds are a subset of logically possible worlds. It's all very confusing. But I shouldn't get too snooty. There is, apparently, a need to this concept in modal logic, but I don't understand what it is.

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. — fishfry

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. — fishfry

Well, in that case, you are also traversing the infinitely many possible points along the way, as well as the convergent series based on "<divide by> 3" and all the other series based on all the other numbers, plus all the regular divisions by feet or metres. Or maybe you could decide that all these ways of dividing up your journey are in your head, not in the world. Think of them as possible segments rather than chunks of matter or space.

But in math, 1/2 + 1/4 + ... is added together all at once. And the sum is exactly 1, right now, right this moment. — fishfry

Yes. Thanks for clarifying that for me. That's what I was trying to express when I started babbling on about "always already" in that post that you couldn't get your head around. The comparison with Loop program captures what I've been wrestling with trying to clarify. All that business about getting (or not getting) to the end... It's important though that it's a physical process which takes time. You can switch it off at the end of 60 seconds, and see how far it got, but it won't have completed anything, will it?

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. — fishfry

There's no clear criterion for what is conceivable and what is not, in spite of generations of logicians. It seems pretty clear that some people have a much more generous concept of that than I do. There are famous philosophical issues around that many people seem able to conceive of, but I can't. I don't know what's wrong with me.

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 ... :-) — fishfry

Make sure you get one of the ones that you can't finish drinking. You would not be popular if you passed round one of the ones that you can't start drinking.

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. — fishfry

You won't have bothered with this exchange - his comment, my reply:-

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. — Ludwig V

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.

This is an important difference and is why so many "solutions" to Thomson's lamp (and other supertasks) miss the point entirely.

If the lamp is turned on after 30 seconds then, unless turned off again, it will remain on for all time. This is why if you claim that supertasks are possible then you must be able to give a consistent answer as to whether or not the lamp is on or off after 60 seconds. If you cannot, because no consistent answer is possible, then this is proof that the supertask is metaphysically impossible.

It is necessary that the lamp is either on or off after 60 seconds, and for it to be either on or off after 60 seconds it is necessary that the button can only been pressed a finite number of times before then. — Michael

My impression of Benacerraf is that he is defining Thomson's Lamp as a boolean valued function

$l : \overline {\mathbb {N}} \rightarrow \mathbb {B}$

on the domain of the extended natural numbers $\overline {\mathbb {N}}$ which introduces an additional point $\omega$ at "infinity", and then arguing that the value at $\omega$ can be chosen arbitrarily and independently of the function's limiting value, if any. But if this the case, then he isn't engaging with Thomson's argument and has merely shifted the goal posts to declare victory in an incomparable axiomatisation.

But the point about Frege's Law Vb also applies to the extended natural numbers; Thompson's lamp when defined as the function $l$ has a domain consisting of two definite and maximally separated points 0 and $\omega$ and a number of points between 0 and $\omega$ that is intensionally described as being countably infinite. However, if Frege's Law Vb is rejected for reasons mentioned previously, then although $l$ still has the aforementioned intensional properties, it does not possess an extensionally well-defined number of points, in which case it cannot be considered to represent the metaphysical notion of a supertask.

Essentially, mathematical analysis will fail to persuade unless one is already a true believer of supertasks.

Essentially, mathematical analysis will fail to persuade unless one is already a true believer of supertasks. — sime

That about sums it up.

It is necessary that the lamp is either on or off after 60 seconds, and for it to be either on or off after 60 seconds it is necessary that the button can only been pressed a finite number of times before then. — Michael

That looks to me like a valid argument to the conclusion that the Thompson lamp is a physical impossibility, because switches and electrical currents are not infinitely fast. What's wrong with that?
My problem is that I don't understand what metaphysics is, if it is not logic.
I may be wrong, but it seems to me that those who believe in supertasks think that by stipulating that the lamp can be switched infinitely many times in a limited time, they can make it so. But they can't. Supertasks play on the difference between the physically possible and the logically possible to create an illusion.

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.

So the fact that the status of the lamp at t1 is "undefined" given A is the very proof that the supertask described in A is metaphysically impossible. — Michael

This is a contradiction, therefore Thomson's lamp shows that it is logically impossible to have pushed a button an infinite number of times. — Michael

That's right. But there's nothing special about the lamp. It is impossible to complete any action an infinite number of times.
I wasn't accurate enough in my last post. The representation of the series is misleading. (1/2,1/4,1/8...) invites one to think of the dots as an abbreviation for a longer sequence, which could be written out in full. But there is no possibility that it could be written out in full. The notation does not define an end,

In the message you linked to, you concluded:-

So the fact that the status of the lamp at t1 is "undefined" given A is the very proof that the supertask described in A is metaphysically impossible. — Michael

I think that this is what @fishfry was saying. (Substituting "logically impossible" for "metaphysically impossible".)

So what is the argument about?

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

Your post seems to add nothing new, and does not appear to engage with any of the points I've made. I have nothing to add till I see a need to write something I haven't already said.

I've enjoyed the chat but progress has not been made in some time.

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).
— fishfry
I'm entirely in favour of the project, but, to be honest, I don't think it is worth dying for. — Ludwig V

I love to spread the gospel of Cauchy and Weierstrass.

I think that's the first time I've encountered anyone on these sites who understands the difference between "discrete" and "discreet". Not patronizing, just saying. — Ludwig V

I'm sure I'm not the only one. I've always been a finicky speller.

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.
— fishfry
Now you have me a bit puzzled. In my book, that means that the equation is about the complete series, which seems at odds with the idea that it can't be completed. — Ludwig V

Well "completed" is a loaded term, but it has no mathematical meaning. Mathematicians don't use the word. Given a sequence, it may or may not have a limit. The word completed is not a technical term.

On the other hand, the real numbers are the completion of the rationals, meaning that the reals consist of all the limits of all the Cauchy sequences of rationals. But that's a technical definition, it doesn't have the overtones you're giving to it.

What does "complete" mean? Or does it mean the sense in which it is "always already" complete? (see below) — Ludwig V

"Complete" is not an applicable mathematical term. Unless you want to say that sqrt(2) completes the sequence 1, 1.4, 1.41, 1.414, ... That's an acceptable usage. But it doesn't mean there is any kind of magic jump at the end. It just means the terms of the sequence are arbitrarily close to the limit.

It might be easier to understand if you thought of them as regarding all possible worlds as equally possible. — Ludwig V

I even disagree with that. But some of the possible worlds people (David Lewis, I believe) claim that the possible worlds are real.

I could understand that. I hope they don't mean that all possible worlds are equally actual.... — Ludwig V

Most comprehensively in On the Plurality of Worlds, Lewis defended modal realism: the view that possible worlds exist as concrete entities in logical space, and that our world is one among many equally real possible ones. -- David Lewis

The day I read that is the day I gave up on caring about possible worlds.

But some people tend to think only of one kind of possibility - logical possibility. But there many other sorts - physically possible, legally possible, practically possible, etc. etc. I say that possible means different things in different contexts, but it may be that I should be saying there is cloud of possible worlds for each kind. Or maybe physically possible worlds are a subset of logically possible worlds. It's all very confusing. But I shouldn't get too snooty. There is, apparently, a need to this concept in modal logic, but I don't understand what it is. — Ludwig V

Well it all went over my head when I took a MOOC on the subject.

Well, in that case, you are also traversing the infinitely many possible points along the way, as well as the convergent series based on "<divide by> 3" and all the other series based on all the other numbers, plus all the regular divisions by feet or metres. Or maybe you could decide that all these ways of dividing up your journey are in your head, not in the world. Think of them as possible segments rather than chunks of matter or space. — Ludwig V

Yes. If time is like the real numbers, then Zeno supertasks are a everyday occurrence. You execute one when you roll over in your sleep. This is the point I made to @Michael.

But in math, 1/2 + 1/4 + ... is added together all at once. And the sum is exactly 1, right now, right this moment.
— fishfry
Yes. Thanks for clarifying that for me. That's what I was trying to express when I started babbling on about "always already" in that post that you couldn't get your head around. The comparison with Loop program captures what I've been wrestling with trying to clarify. All that business about getting (or not getting) to the end... It's important though that it's a physical process which takes time. You can switch it off at the end of 60 seconds, and see how far it got, but it won't have completed anything, will it? — Ludwig V

The mathematical series sums immediately. The loop takes time and never achieves its limit, since computing resources are bounded. Under the thought process experiment of "adding the next term" at successively halved time intervals, I'd say it completes in finite time. But that confuses people because we're conflating math and physics.

There's no clear criterion for what is conceivable and what is not, in spite of generations of logicians. It seems pretty clear that some people have a much more generous concept of that than I do. There are famous philosophical issues around that many people seem able to conceive of, but I can't. I don't know what's wrong with me. — Ludwig V

I agree. @Michael keeps saying supertasks are metaphysically impossible, and I think they're metaphysically possible.

In the message you linked to, you concluded:-
So the fact that the status of the lamp at t1 is "undefined" given A is the very proof that the supertask described in A is metaphysically impossible.
— Michael
I think that this is what fishfry was saying. (Substituting "logically impossible" for "metaphysically impossible".) — Ludwig V

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, and not at the limit; and it's not a real lamp, so turning into spaghetti is no more unrealistic than cycling at arbitrarily small time intervals.

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.

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.

I agree. Michael keeps saying supertasks are metaphysically impossible, and I think they're metaphysically possible. — fishfry

Now I'm confused. I thought you didn't know what "metaphysics" means - or what metaphysics is.

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. — fishfry

I'm puzzled now about "natural". If the terminal state of the lamp is not defined, there is no way to define it - natural or otherwise. Or, possibly better, any arbitrary state will do. Hence the plate of spaghetti.

the lamp problem, which only defines the state of the lamp at the points of the sequence, and not at the limit; — fishfry

Yes, of course - and since it is not defined, Michael can derive a contradiction - two equally possible or impossible states.

"Complete" is not an applicable mathematical term. Unless you want to say that sqrt(2) completes the sequence 1, 1.4, 1.41, 1.414, ... That's an acceptable usage. But it doesn't mean there is any kind of magic jump at the end. It just means the terms of the sequence are arbitrarily close to the limit. — fishfry

H'm. I would be quite happy with that acceptable usage. But it suggests that 1,1.4,1.41, 1.414... is incomplete, and we are back with the temptation to think that series can somehow be completed. It's probably better to stick with "not applicable".

Under the thought process experiment of "adding the next term" at successively halved time intervals, I'd say it completes in finite time. But that confuses people because we're conflating math and physics. — fishfry

I think that's the heart of the problem. My only hesitation is that the lamp is imaginary, so it sits on an ill-defined boundary between the two. I'm very suspicious of the idea that anything anyone can imagine is (logically) possible. Twin Earth is a good example. But there's a raft of others.

possible worlds exist as concrete entities in logical space, — fishfry

I don't know what to say. Ryle would go on about category mistakes. In poetry (or politics) people sometimes talk of a "tin ear". That's exactly what this is - a rhetorical gesture that confuses "concrete" with "well defined" and with - well - concrete. It's protesting too much. There must be some repressed doubt going on there.

Well it all went over my head when I took a MOOC on the subject. — fishfry

You are lucky. It will spare you a world of grief and confusion. Modal logic can look after itself.

The nested interval construction can be explicitly written down. I perhaps am not sharing your vision here. — fishfry

The system is telling me that you mentioned me in the context of this comment in the thread on the Fall of Man paradox, but I can't find any mention of me. But the system is doing some weird things anyway, so I'm not going to worry about it. I do regret not having been aware of the thread sooner. I thought it had something to do with theology.

I can't contribute to the discussion you are involved in there, but this quotation:

I concur that this narrative couldn't unfold in our physical reality, but your argument doesn’t address the core of the paradox. The inclusion of God and the Garden of Eden in the story was specifically to lift us beyond our finite limitations. — keystone

does make me think that the same problem, of the interface between mathematics and empirical reality, is at the heart of this paradox as well.

Supertasks play on the difference between the physically possible and the logically possible to create an illusion. — Ludwig V

What I've explained though, is that infinite divisibility is really incoherent due to self-contradiction. So the supertask is not even logically possible. It just appears to be, when not subjected to critical analysis.

So the matter of doing something an infinite number of times never comes about, because the conclusion of an infinite number of times is only produced from the premise of infinite divisibility. If we remove that premise, and just start talking about doing something an infinite number of times, it's obvious that there is no end to such a task. It's only the contradictory notion, that a finite thing can be divided an infinite number of times, which produces the paradox.

From past experience I understand that @fishfry is very slow to accept the reality that some principles employed by mathematicians are incoherent.

What I've explained though, is that infinite divisibility is really incoherent due to self-contradiction. So the supertask is not even logically possible. It just appears to be, when not subjected to critical analysis. — Metaphysician Undercover

.... and when one analyses it, it is a confused mixture of physical possibility and logical possibility, each of which are coherent on their own.

It's only the contradictory notion, that a finite thing can be divided an infinite number of times, which produces the paradox. — Metaphysician Undercover

A finite thing certainly cannot be divided an infinite number of times, if by "divided" you mean "physically divided", subject to clarification of what you mean by a finite thing.
At the same time, it is possible to divide it into halves, quarters, etc. (how many fractions are there?) and into feet, inches, etc. and into metres, etc, and according to an indefinite number of other units of measurement. To physically divide in one of those ways excludes dividing it in any other way, so you can't divide it by all those things at the same time. But those possibilities do all exist, all at the same time.
On the other hand, the number 2 is finite, in one sense. Yet it cannot be physically divided at all (because it is an abstract thing), yet it can be divided by a familiar mathematical operation, and that operation can be applied to it an infinite number of times. (No, I'm not talking about space or time.)

From past experience I understand that fishfry is very slow to accept the reality that some principles employed by mathematicians are incoherent. — Metaphysician Undercover

I wonder if that's because the principles that you are applying to mathematics do not apply to mathematics? For example, numbers are abstract objects; they do not exist in space and time. Geometry is not about physical objects, but about ideal objects, which do not exist in space and time. Abstract entities that do not exist in space or time are not subject to the restrictions you wish to impose on space and time - obviously. You may or may not regard such entities as not true (or not real) objects, but that's neither here nor there.

and when one analyses it, it is a confused mixture of physical possibility and logical possibility, each of which are coherent on their own. — Ludwig V

I don't think infinite divisibility is a logical possibility, that's the point I'm making. Infinite division is logically impossible.

A finite thing certainly cannot be divided an infinite number of times, if by "divided" you mean "physically divided", subject to clarification of what you mean by a finite thing. — Ludwig V

I mean to "divide" in any sense of the word. The qualification of "physical" is irrelevant. Division is an action, an operation, and we are talking about the possibility of dividing something an infinite number of times. Do you think that this is logically possible? It's like counting, which is another activity. Do you think it's possible to count infinite numbers?

t the same time, it is possible to divide it into halves, quarters, etc. (how many fractions are there?) and into feet, inches, etc. and into metres, etc, and according to an indefinite number of other units of measurement. — Ludwig V

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.

Yet it cannot be physically divided at all (because it is an abstract thing), yet it can be divided by a familiar mathematical operation, and that operation can be applied to it an infinite number of times. (No, I'm not talking about space or time.) — Ludwig V

No, the mathematical operation of division cannot be applied to an infinite number of times, for the reason explained above. If the action is completed it is not infinite, and if it continues indefinitely, at any stage in its progression, it is not infinite either. Quite simply, there will always be more dividing to do before an infinite number of times is achieved, and if you stop it is not achieved either. Therefore it is very clear that such an activity (infinite division) is logically impossible. In general, an infinite activity, or operation, is logically impossible.

My problem is that I don't understand what metaphysics is, — Ludwig V

I can answer this. Metaphysics is about what physically is, and physics is about what physically is measured. That's a crude definition, but what it comes down to is that the phrases 'physically possible' and 'metaphysically possible' mean the same thing. You can't have one without the other. Metaphysically possible means that there exists a metaphysical interpretation where the thing in question is physically possible.
So for instance, physical determinism is a metaphysical issue. There are some valid interpretations that are deterministic, and some valid interpretations that are not. Therefore determinism is metaphysically possible, and therefore determinism is physically possible.

Everybody posting seems to treat 'metaphysically possible' as some weird sort of realm between physically and logically possible, resulting in confusion when nobody can come up with an example of something distinct.
So any argument against the lamp, or recitation of numbers, becomes a logical argument because these arguments have no application to physics. They are all metaphysically impossible for the very reason that they are physically impossible.

Supertasks play on the difference between the physically possible and the logically possible to create an illusion.

This seems to say it. It is a logical issue, but with applications to the physical when the scenario in question doesn't involve physical impossibilities.

After completing the supertask the lamp must be either on or off — Michael

I am willing to accept this statement, but you are not willing to engage with any of the faults identified with your logic. Hence I can only presume you have no counters to them, resorting only to changing the subject every time a fallacy is pointed out. I for the most part have dropped out due to this lack of engagement.

It is impossible to complete any action an infinite number of times. — Ludwig V

This is Zeno's strategy. Just beg your conclusion.

The notation does not define an end, — Ludwig V

There it is. Not possible due to the asserted necessity of a bound of something which by definition has no bound. All the arguments against seem to take this form. Even Zeno avoided this fallacy, and his argument was made before the mathematics of infinite sets was formalized.

Your post seems to add nothing new, and does not appear to engage with any of the points I've made. I have nothing to add till I see a need to write something I haven't already said. — fishfry

Myself as well. I have dropped out some time ago, and not surprisingly, nothing new has been posted. But I did chime in to define 'metaphysically possible' since the term seemed to be used in a way in which it was somehow meaning something different than physically possible, which it cannot be.

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

I respectfully disagree. You've addressed none of my points.

Now I'm confused. I thought you didn't know what "metaphysics" means - or what metaphysics is. — Ludwig V

I don't know what @Micheal means by metaphysically impossible. I know what metaphysics means. More or less. Not an expert.

I'm puzzled now about "natural". If the terminal state of the lamp is not defined, there is no way to define it - natural or otherwise. Or, possibly better, any arbitrary state will do. Hence the plate of spaghetti. — Ludwig V

Contrast with the staircase. The walker is present at each step, and the terminal state is undefined. If we define the terminal state as "walker is present," that is natural, ie continuous. If the walker is defined to be absent at the terminal state, that's discontinuous: 1, 1, 1, 1, ... with terminal state 0. That's unnatural.

Likewise Cinderella's coach. Coach at midnight minus 1/2 second, coach at mid minus 1/4, etc. Pumpkin makes it discontinuous. The natural continuation would be for it to remain a pumpkin.

I'm defining natural as continuous.

Yes, of course - and since it is not defined, Michael can derive a contradiction - two equally possible or impossible states. — Ludwig V

There is no natural continuation. No terminal state that makes 0, 1, 0, 1, ... continuous.

H'm. I would be quite happy with that acceptable usage. But it suggests that 1,1.4,1.41, 1.414... is incomplete, and we are back with the temptation to think that series can somehow be completed. It's probably better to stick with "not applicable". — Ludwig V

Yes, the completion in that case is sqrt(2). That's how we define the irrationals as particular sequences of rationals.

I think that's the heart of the problem. My only hesitation is that the lamp is imaginary, so it sits on an ill-defined boundary between the two. I'm very suspicious of the idea that anything anyone can imagine is (logically) possible. Twin Earth is a good example. But there's a raft of others. — Ludwig V

My point is that the lamp is fictitious and violates the laws of physics. So its terminal state need not be on or off. It could be a plate of spaghetti. That is no more fictitious than the lamp itself.

I don't know what to say. Ryle would go on about category mistakes. In poetry (or politics) people sometimes talk of a "tin ear". That's exactly what this is - a rhetorical gesture that confuses "concrete" with "well defined" and with - well - concrete. It's protesting too much. There must be some repressed doubt going on there. — Ludwig V

I don't just oppose the modal realists. I think they're insane. Or trolling.

You are lucky. It will spare you a world of grief and confusion. Modal logic can look after itself. — Ludwig V

My feelings exactly.

The system is telling me that you mentioned me in the context of this comment in the thread on the Fall of Man paradox, but I can't find any mention of me. But the system is doing some weird things anyway, so I'm not going to worry about it. I do regret not having been aware of the thread sooner. I thought it had something to do with theology. — Ludwig V

I no longer know what any of these threads are about. Perhaps I never did.

From past experience I understand that fishfry is very slow to accept the reality that some principles employed by mathematicians are incoherent. — Metaphysician Undercover

That's funny, coming from someone who can't understand the axiom of extensionality because you don't understand material implication.

Logical equivalence does not imply "the same as". I have no problem with the axiom of extensionality. I have a problem with people who conflate the axiom of extensionality with the law of identity, to interpret that axiom as saying two equal things are the same thing.

Logical equivalence does not imply "the same as". I have no problem with the axiom of extensionality. — Metaphysician Undercover

Our conversations have proven and confirmed to me that you do have a problem with the axiom of extensionality. You are unable to engage with the mathematical formalism, and therefore you do not understand what it says, and how it is to be used. And the reason you can't engage with the formalism is that you don't seem to understand material implication.

That's what I determined. You've said nothing to convince me otherwise recently. If I have misconstrued your position, I'd be grateful for any correction. But I don't think I have, because before that you refused to even acknowledge my proof that 2 + 2 = 4 from the Peano axioms.

That just shows you won't/can't engage with the symbolism. Either way, if you are not willing to do that, then there's little else I can say.

[Note: These next two paras inserted a bit later]

For what it's worth: The axiom of extensionality is the definition of a symbol. Nobody is saying it means anything at all. If you denied that sets exist, I would agree with you. They're mathematical fictions.

But they are useful, because we can base almost all modern math on them; and two, they're interesting in their own right. You are the only one trying to give the axiom of extensionality metaphysical implications that are not really there.

I have a problem with people who conflate the axiom of extensionality with the law of identity, to interpret that axiom as saying two equal things are the same thing. — Metaphysician Undercover

Are these people in the room with us right now?

But I don't think I have, because before that you refused to even acknowledge my proof that 2 + 2 = 4 from the Peano axioms. — fishfry

I have no problem acknowledging that 2+2=4. I have a problem with people who claim that "2+2" symbolizes the same thing that "4" does. And so, I refused to accept your claim to have proven that "2+2" signifies the very same thing as "4" does. Simply put, if the right side of an equation does not signify something distinct from the left side, mathematics would be completely useless.

You can say that I have a problem with formalism, because I do. Like claiming that accepting certain axioms qualifies as having counted infinite numbers, formalism claims to do the impossible. That is, to remove all content from a logical application, to have a logical system which is purely formal. If such a thing was possible we'd have a logical system which is absolutely useless, applicable to nothing whatsoever. Attempts at formalism end up disguising content as form, producing a smoke and mirrors system of sophistry, which is riddled with errors, due to the inherent unintelligibility of the content, which then permeates through the entire system, undetected because its existence is denied.

You can say that I have a problem with formalism, because I do. Like claiming that accepting certain axioms qualifies as having counted infinite numbers, formalism claims to do the impossible. — Metaphysician Undercover

Formalism as a philosophy considers mathematics to be reducible to a finite single-player sign game of perfect information in which proofs refer to deterministic winning strategies, and hence Formalism does not support the Platonic interpretation of abstract mathematics as denoting actually infinite objects, whatever the formal system concerned.

So I think your problem is actually with Platonic myths that have become psychologically wedded to innocent formal definitions, and in particular the formal definitions of limits and total functions that are ubiquitously misinterpreted in both popular and scientific culture as denoting a non-finite amount of information, E.g as when the physicist Lawrence Krauss misleads the public with nonsense about the physical implications of Hilbert Hotels.

sign game of perfect information — sime

See that phrase, "perfect information"? That's why I say formalism attempts to do the impossible. In other words, it assumes an ideal which cannot be obtained, therefore it's assumption is necessarily false.

So I think your problem is actually with Platonic myths that have become psychologically wedded to innocent formal definitions, and in particular the formal definitions of limits and total functions that are ubiquitously misinterpreted in both popular and scientific culture as denoting a non-finite amount of information, E.g as when the physicist Lawrence Krauss misleads the public with nonsense about the physical implications of Hilbert Hotels. — sime

I view formalism as a form of Platonism. It's a Platonist game in which the participants deny their true character, that of being Platonist. Notice "perfect information" is the foundational feature of Platonist idealism. That perfection is the only thing which supports the eternality of Platonic ideals. So formalism and Platonism are really just the same thing, even though the formalists will claim otherwise.

See that phrase, "perfect information"? That's why I say formalism attempts to do the impossible. In other words, it assumes an ideal which cannot be obtained, therefore it's assumption is necessarily false. — Metaphysician Undercover

Perfect information isn't an assumption of formal reasoning, rather it is regarded to be a necessary condition of the meaning of "formal" reasoning in that it is by definition finitely deducible and does not require appealing to unformalized intuitions about infinite and ideal objects. Most importantly, the condition of perfect information ensures that formal reasoning cannot interpret an expression such as {1,2,3,...} as representing an abbreviation of some ideal object; the former expression must either be formally treated as a finite object of some type, else the expression must be considered illegal.

It is actually by sticking to formal reasoning that the illusion of the ideal is never obtained. The opposite impression is due to Platonists disguising themselves as formalists, which might be said to even include Hilbert himself.

Formalism makes the reasonable demand that whatever informal intuitions originally motivated the construction of an axiomatic system, and whatever informal interpretations one might subsequently give to the signs of that system, the methodology of theorem-proving should be purely algorithmic and make no appeal to such intuitions, whether such intuitions be rooted in platonism or in Kantian intuition.

I view formalism as a form of Platonism. It's a Platonist game in which the participants deny their true character, that of being Platonist. Notice "perfect information" is the foundational feature of Platonist idealism. That perfection is the only thing which supports the eternality of Platonic ideals. So formalism and Platonism are really just the same thing, even though the formalists will claim otherwise. — Metaphysician Undercover

The irony of Hilbert, is that his formalism ultimately led to the rebuttal of his own informal intuitions about infinity, namely his presumption that a closed axiomatic system must possess a finite representation of it's own consistency. Had Hilbert better understood the implications his formalism, and especially the finite formal meaning of The Law of Excluded Middle which he apparently accepted for instrumental purposes, then Godels incompleteness theorem might not have come as a shock to him. It is evident that Hilbert was a methodological formalist who didn't mean to insinuate that mathematics was a meaningless game void of semantics, but only that the terms used to denote sets, formula and constants shouldn't require interpretation for the purposes of theorem proving. Unfortunately, his intuitions misled him regards to the outcome of his formal program.

If we inspect the finite activity of theorem proving in a formal system, we see that every term that is informally interpreted as denoting an "infinite object" only possesses finite conditions under which the term is introduced into a theorem and under which the term is eliminated from a theorem.

Different formal systems can be regarded as differing only in regards to their ability to distinguish types of finite object. E.g Intuitionism that formalizes choice-sequences can distinguish uncompleted finite sets from ordinary finite sets, whereas ZFC as a theory of first-order logic can only distinguish finitely defined functions from finite sets - so whilst ZFC might be informally said to be a theory about "infinite sets", this isn't the proof-theoretic formal meaning of ZFC, and so a formalist is free to reject the platonic myths that surround ZFC.

No, the mathematical operation of division cannot be applied to an infinite number of times, for the reason explained above. — Metaphysician Undercover

Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. — Wikipeida

I know it is only Wikipedia, but I'm sure that more authoritative references could be found.

What is wrong with that?