That seems to me a good response, though not quite the knock-out blow one would hope for. — Ludwig V

If there was an easy knock-out blow to it, it wouldn't be a topic on philosophy/mathematical discussions.

This logical consequence can be shown when the experiment is explained more clearly:

A1. At t0 the lamp is off
A2. The button is pressed only as described by this sequence of operations: at t1/2 I press the button, at t3/4 I press the button, at t7/8 I press the button, and so on ad infinitum

Compare with:

B1. At t0 the lamp is off
B2. The button is pressed only as described by this sequence of operations: at t1/2 I press the button

The status of the lamp at t1 must be a logical consequence of the status of the lamp at t0 and the button-pressing procedure that occurs between t0 and t1 because nothing else controls the behaviour of the lamp.

If no consistent conclusion can be deduced about the lamp at t1 then there’s something wrong with your button-pressing procedure. — Michael

I see that you have an opinion, and that you are attempting to rationalize this opinion. But you leave some pretty low hanging fruit in this post, and rather than have me point them out and you denying whatever it is I post, I invite you to step into my shoes and critique the above. If your opinion was the opposite, what portions of the above argument would you put in bold and say is wrong?

I want to see if you are aware of the issues against which you are arguing.
Your response to me never seems to be along such lines. Instead of pointing out faults in my assertions or whatever, you simply ignore the argument and post yet another rewording of a counterargument, making the same mistakes (from my point of view). So show me that you at least know where I think those mistakes are being made.

The important part is in bold. If there is a problem with the button-pressing procedure, which there is in the case of A2, then this problem remains even if the button is broken and doesn't actually turn the lamp on — Michael

OK, the bold line is telling. There is something wrong with the procedure. I've pointed it out in several posts. The lamp isn't broken. That violates the mathematical definition of how the thing works. There is no physical lamp since physics cannot do what is described.

A clock ticks 1 time per second.
You start with a cake.
Every second the clock ticks, cut the cake in half.
Make the clock variable, it ticks n times a second.
The limit clock as n tends to infinity applies an infinity of divisions to the cake in 1 second. There is no final operation. — fdrake

This is not a supertask, not even as the tick rate increases arbitrarily high, because the cake (if it is continuous, which a physical one isn't) is going to take forever to consume at any clock rate.

This is not a supertask, not even as the tick rate increases arbitrarily high, because the cake (if it is continuous, which a physical one isn't) is going to take forever to consume at any clock rate. — noAxioms

Why? The ticks per second is also going to infinity.

I don't really think it matters whether this is a supertask or not, though. It was an attempt to give an example that hits @Michael's argument.

The lamp starts off. Every time the clock ticks a lamp turns from off to on or from on to off as applicable. Thomson's lamp shows that this leads to a logical inconsistency. — Michael

By providing a standard mathematical object which is infinite, has no final element, tends to an end state, and has an infinite number of occurrences ("steps"), but occurs in finite time.

By providing a standard mathematical object which is infinite, has no final element, tends to an end state, and has an infinite number of occurrences ("steps"), but occurs in finite time. — fdrake

I’ll repeat something I said above:

The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:

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

The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible: — Michael

The lamp starts off. Every time the clock ticks a lamp turns from off to on or from on to off as applicable. Thomson's lamp shows that this leads to a logical inconsistency. — Michael

Might show it's logically possible tho.

Might show it's logically possible tho. — fdrake

You think that the super task I described might be logically possible? How would you start it?

The fact that there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 does not prove that the following supertask is metaphysically possible:

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

How does one start such a supertask? — Michael

I have explained this repeatedly. If you have the sequence 1, 1/2, 1/4, 1/8, ... on the number line, the points go right to left. If you start at 0, the limit of the sequence, and move to the right by any nonzero amount, no matter how small, you necessarily jump over all but finitely members of the sequence. That's by virtue of the fact that 0 is the limit.

It's certainly true, and I've agreed to this many times, that an infinite sequence has a beginning but no end; so that you can not iterate through it in reverse. How you get from this utterly trivial fact to some kind of cosmic conclusion, I can not fathom.

From Tasks, Super-Tasks, and the Modern Eleatics:

What conclusions are we to draw from this rather heady mixture of genies, machines, lamps, and fair and foul numbers? In particular, has it been shown that super-tasks are really possible – that, in Russell's words, they are at most medically and not logically impossible? Of course not. In a part of his paper that I did not discuss, Thomson does a nice job of destroying the arguments of those who claim to prove that super-tasks are logically possible; had there been time I should have examined them. In the preceding section I tried to do the same for Thomson's own neo-Eleatic arguments. I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his (supposing that I did destroy them). — Michael

Ok. I asked for a reference. Now I have no idea what I'm supposed to conclude from this. That one person thinks supertasks are impossible and another does.

Can you answer a specific question that I've asked you?

Do you understand that mathematically, if you take a step, no matter how small, from 0, you necessarily pass over all but finitely many elements of the sequence 1, 1/2, 1/4, 1/8, ...?

It's really important me to know if you at least understand this mathematical fact.

So there is a common understanding of what the issue is. Your disagreement is about different ways of responding to it. Don't you think? — Ludwig V

You tagged three people before that quote so I'm not sure if this is for me. But I did say I wasn't sure I understood the staircase problem so if I've got that wrong, so be it.

Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said. — Ludwig V

I'm for that. The mathematics could not be more clear. Once you start talking about mythical lamps and staircases, the examples have all the moral force of Cinderella's coach. Fairy tales. Thanks for the quote.

But this problem makes me think that they were wrong. One issue that comes to mind is the issue of making a 2-dimensional map of a 3-dimensional sphere. Euclid doesn't work (accurately). But the problem is resolved by developing a different geometry, which breaks some of Euclid's rules. (I realize I'm oversimplifying here, but I hope I'm not hopelessly mistaken.) — Ludwig V

Yes, Gauss and Riemann et. al. Not sure how that helps us with the lamp, the staircase, or Cindarella's coach.

One point to take into account here. This is a thought experiment, so, while the mathematics is real, the horse is not physical, but imaginary, and the difficulty is to work out what rules apply to that in-between context. — Ludwig V

The horse. Uh oh did I miss a story about a horse? Is that a horse of a different color?

Not sure what you mean by potential cardinality.
— fishfry
Pick a number, say 27. I believe it has been shown that there exists a set the cardinality of which is 27, if that's valid terminology. — noAxioms

Yes that's true. There's a set of cardinality 27. One such is the set {0, 1, 2, 3, ..., 26}. There are others, of course.

One could also reference aleph-26, — noAxioms

$\aleph_{26}$ is vastly larger than 27. It's infinite, for one thing, whereas 27 is finite. Not sure where you're going with this.

but I'm not sure that one can prove that no sets exist with cardinalities between the ones labeled 1 through 27. — noAxioms

Of course many such sets exist, as shown by the von Neumann encoding of the natural numbers.

I am not understanding your point. Of course there are sets of all finite cardinalities. And since cardinal numbers are themselves defined as particular sets, there are sets of all cardinalities.

Point being that you get no increase in computational power from parallelization.

I beg to differ. A 16 processor machine can sustain a far greater work load than a single-processor machine. The Cray machines were highly parallelized (SIMD architecture) in which thousands of floating point operations were performed by every instruction. These machines were great for stuff like weather simulation. — noAxioms

No. I'm talking about computability theory. A Cray supercomputer has no more computational power than I do using pencil and an arbitrarily large sheet of paper, which which I can implement a Turing machine.

You are thinking of complexity theory, in which the time and space resources of computations are important.

But in computability theory, a function is either computable or not. If it is, it can be computed by pencil and paper (taking a very long time, of course). If it's not, no supercomputer will help.

As an example, consider the Euclidean algorithm to compute the least common divisor of two integers. It's a simple algorithm that can be executed using pencil and paper. If I had two trillion-digit numbers, I could not feasibly do the computation with pencil and paper; but I could still do it in principle.

Complexity theory is about what can be done feasibly. Computability theory (Turing machines etc.) is about what can be done by algorithms, whether the computation is feasible or not.

A Cray supercomputer can not compute anything that I can't compute with pencil and paper. But it can do so much more feasibly once the inputs become large.

I found a SEP article on the subject.

https://plato.stanford.edu/entries/computability/

No function is computable by a parallel process that's not already computable by a linear process.
With that I agree. But that same function can also be done by paper & pencil. You said 'powerful', a reference to how fast the work is completed, and more processors helps with that. — noAxioms

Computational power. With the complexity/computability distinction, I believe we're in agreement.

Coloring the steps reduces to the lamp.
I notice that any scenario with a contradiction involves invoking magic. Suppose this physically impossible thing (infinite gods, stairs requiring faster-than-light speed, lamp switches that operate without delay. No magical measurement of something nonexistent. Zeno doesn't do that. No magic invoked, and the first premise thus produces no paradox. — noAxioms

Ok. Not sure where we're going with this. I'm happy to do a Zeno supertask by walking across the room. Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.

My Quora feed gives me a lot of cute cat pics lately. Makes me happy. Quora certainly used to be a lot better.
Oh it serves its purpose, but correct answers are not promoted above the others, and apparently a great deal of their posters don't know what they're talking about when it comes to stuff like this. — noAxioms

Lot of troll accounts on the site and lots of people who don't know what they're talking about. Believe it's something to do with the Quora owners trying to make some money. Sadly I'm a bit addicted to the site.

Zeno's horse is quite real. Almost none of the others are. — noAxioms

The Zeno Wiki page doesn't mention a horse. Did I miss something? @Ludwig V mentioned a horse too.

Exactly so. I have correct my post. I meant valid and wrote 'sound' in haste. A simple application of modus ponens shows the lack of soundness of Zeno's conclusion iff empirical knowledge is given any weight.

The conflicting premise which would be used to disprove this, the limitations of divisibility
The conflicting premise seemed to be a denial of the completability of a supertask. He never suggests a limit to divisibility. — noAxioms

In Zeno's Achilles and the tortoise, empirical knowledge shows that Achilles will pass the tortoise. But empirical knowledge has problems like what Hume showed with the problem of induction. Because of this, empirical knowledge does not prove the supertask to be impossible.

That the supertask is not completable is not denied, that it is not completable is what actually leads to the problem. In Zeno' paradox Achilles never catches the tortoise because the supertask is never completed. By the premises of the op, Icarus cannot reach the bottom of the staircase, just like Achilles cannot reach the tortoise. So "the supertask" by the nature of what it means to be a supertask, cannot be completed.

The problem is that empirical evidence shows us that tasks will be completed, Achilles will pass the tortoise, and in the op 60 seconds will pass. This shows that the supertask as a fiction. However, due to the problem of induction, empirical evidence does not provide a proper proof. That is why I suggested we look at the divisibility of time as the means for providing a better proof.

Why? The ticks per second is also going to infinity. — fdrake

The sum of an infinite set of identical finite numbers is not finite, no matter how small the number being summed. It needs to complete in finite time to be a supertask.

I don't really think it matters whether this is a supertask or not, though. It was an attempt to give an example that hits Michael's argument.

does not prove that the following supertask is metaphysically possible: — Michael

I was wondering about what is actually meant by 'metaphysically possible' or 'logically possible'. The latter is probably the same as 'mathematically possible', but I'm wondering how the former is distinct.

I notice you ignored my prior post. That itself indicates to me that you do not intent to actually consider points made against your stance. I was hoping for more confidence in it.

No. I'm talking about computability theory. — fishfry

Gotcha. No argument then. As I already pointed out, you had referenced power instead of computability: "there's no difference in computational power between parallel and serial processing." and I took it as a statement of work over time.

I brought this up in my simulation-theory topic. A simulation of Earth to a precision sufficient for consciousness can be done by pencil and paper, or by dominos falling, The latter is really interesting: set up dominos so that you get the function of a Turning machine. Not easy, but it seem that it can be done.

Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.

Plank length is not a physical limit, only a limit of significance. If I have it right, any pair of points separated by a distance smaller than that is not meaningfully/measurably distinguishable from just the two being the same point. It doesn't mean that the two points are necessarily the same point.
But I gave some QM examples that suggest a non-continuous model of reality.

The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too.

Yes. Search for 'horse' in the last 20 posts or so.

But empirical knowledge has problems like what Hume showed with the problem of induction. Because of this, empirical knowledge does not prove the supertask to be impossible. — Metaphysician Undercover

Because of this, empirical knowledge doesn't prove pretty much anything to be possible or impossible. That's why science theories are supported by evidence and not by proofs. They'd be theorems, not theories, if they were provable.

That the supertask is not completable is not denied, that it is not completable is what actually leads to the problem. In Zeno' paradox Achilles never catches the tortoise because the supertask is never completed.

I beg to differ. That simply does not follow from the description. Zeno describes a physical completable supertask, which is only as possible as the soundness of his first premise.

Achilles will pass the tortoise, and in the op 60 seconds will pass. This shows that the supertask as a fiction.

Again I differ. The supertask (if that premise is true) is not fiction. I mean, my opinion is that there isn't a physical supertask, but opinion isn't evidence, and I have no evidence (let alone proof) that it isn't a supertask.

Declaring something to be impossible is a strong claim and requires strong evidence.

No. I'm talking about computability theory.
— fishfry
Gotcha. No argument then. As I already pointed out, you had referenced power instead of computability: "there's no difference in computational power between parallel and serial processing." and I took it as a statement of work over time. — noAxioms

Ok fair enough. I think of power as computability but perhaps I could have been more clear.No. I'm

I brought this up in my simulation-theory topic. A simulation of Earth to a precision sufficient for consciousness can be done by pencil and paper, or by dominos falling, — noAxioms

I doubt that consciousness is computable, nor is the universe, and I utterly reject the notion that consciousness can be simulated by any computational device. Period. Consciousness is not a computable phenomenon.

We are not computers no matter how many TED talkers declare it so. Just as we aren't Newtonian machines, as we thought we were in the Newtonian machine age; nor are we flowing fluids as the Romans thought, in the age of their great waterworks. After all if we're computations, what are the odds we'd figure that out right when we're in the age of computation? It's historical relativism or whatever the phrase is.

But I do agree that if consciousness were computable, then the computation could be carried out by pencil and paper. Glad you made that point. Because if so, then where is the conscious mind? In the pencil? In the paper? In the air? In a neural network? I reject the idea.

Do people really think their web browsers or word processors are having subjective experiences? I know they think their neural networks are. The idea's absurd. Don't get me started :-) I would characterize myself as a mysterian.

The latter is really interesting: set up dominos so that you get the function of a Turning machine. Not easy, but it seem that it can be done. — noAxioms

Yes, I saw a domino logic gate on Youtube a while back. Any physical substrate will do. For computation, not consciousness. For consciousness you need some secret sauce not yet understood. It will turn out to be something other than a digital computation.

Perhaps it's some kind of analog computation, but that's not the same thing.

Whether someone regards that as a supertask or tells me I forgot about the Planck limit and so forth are different issues.
Plank length is not a physical limit, only a limit of significance. If I have it right, any pair of points separated by a distance smaller than that is not meaningfully/measurably distinguishable from just the two being the same point. It doesn't mean that the two points are necessarily the same point.
But I gave some QM examples that suggest a non-continuous model of reality. — noAxioms

If reality is not continuous that goes a long way to solving Zeno. If it is continuous then walking across the room is a supertask.

The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too.
Yes. Search for 'horse' in the last 20 posts or so. — noAxioms

I have the bad habit of only responding to my mentions, but I'll take a look. Thanks.

ps -- I checked out the Simulation thread and from there, saw your initial post in the "What is the Simulation Hypothesis" thread, and I agree with everything you said. I especially appreciated the distinction between simulation and VR, which is something a lot of the simulation discussions miss.

The sum of an infinite set of identical finite numbers is not finite, no matter how small the number being summed. It needs to complete in finite time to be a supertask. — noAxioms

Aye that is true. I wrote wrong. I was imagining a clock that speeds up in its ticking to ape a convergent geometric series.

Maybe I'm not being clear, so I'll try one more time.

Here are two proposed supertasks:

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

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

Here is our premise:

P1. In both (a) and (b) there is a bijection between the series of time intervals and the series of natural numbers and the sum of the series of time intervals is 60.

However, the second supertask is metaphysically impossible. It cannot start because there is no largest natural number to start with. Therefore, P1 being true does not entail that the second supertask is metaphysically possible.

Therefore, P1 being true does not entail that the first supertask is metaphysically possible.

If you want to argue that the first supertask can end despite there being no largest natural number to end with, and so is metaphysically possible, then you need something other than P1 to prove it.

Ok. I asked for a reference. Now I have no idea what I'm supposed to conclude from this. — fishfry

I said this:

Can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum? It's not something that we can just assume unless proven otherwise. Even Benacerraf in his criticism of Thomson accepted this. — Michael

You responded with this:

Feel free to give a reference, else I can't respond. — fishfry

I gave you this reference:

I think it should be clear that, just as Thomson did not establish the impossibility of super-tasks by destroying the arguments of their defenders, I did not establish their possibility by destroying his.

So I ask again: can you prove that it's metaphysically possible for me to halve the time between each subsequent recitation ad infinitum?

I beg to differ. That simply does not follow from the description. Zeno describes a physical completable supertask, which is only as possible as the soundness of his first premise. — noAxioms

I think you misunderstand Zeno's paradoxes. Zeno concluded that Achilles cannot overtake the tortoise. That is explicit. And therefore, it indicates that he is arguing that the supertask is not completed. The "supertask" of passing an infinite number of spatial divisions is never finished, therefore the faster runner never surpasses the slower.

The paradox is that physical evidence indicates that the faster runner always does overtake the slower, in reality, even though the logic proceeding from fundamental axioms proves that the faster overtaking the slower is a supertask which cannot be completed.

Due to the strength of the empirical evidence, we are led toward the conclusion that the fundamental axioms concerning the continuity of space and time, and the infinite divisibility of those continuums, must be faulty. Those axioms are the "unsound premises".

Declaring something to be impossible is a strong claim and requires strong evidence. — noAxioms

This is why we cannot simply accept the empirical evidence, and conclude that the supertask is a descriptive impossibility derived from faulty axioms. Empirical evidence is known to be unreliable. So, we need stronger principles to demonstrate the actual faults in the axioms.

I doubt that consciousness is computable — fishfry

what, because consciousness is not a physical process, or that physical processes cannot be simulated? You seem to be in the former camp. If that's the case, then no, it probably isn't computable.

After all if we're computations, what are the odds we'd figure that out right when we're in the age of computation?

Pretty much 1-1 odds. That's when the terminology became part of our language. You describe yourself in terms of the things you know.

We are water. The vast majority of mass would be lost (as would consciousness) if the water was taken away. Lots of pipes going here and there. It's a pretty good description for the Roman days.

Because if so, then where is the conscious mind? In the pencil? In the paper? In the air? In a neural network?

In the process.

Yes, I saw a domino logic gate on Youtube a while back.

Gawd, I spelled it 'Turning' machine. More typos.
Anyway, yes, the discussion was inspired by that. Any moron can create a domino or gate, but creating a nor gate gets tricky. Any gate can only be used once, so it's impossible to create say a flip flop, normally a trivial thing created with a pair of nor gates.

I've not seen the video, but mention of it inspired me to design a Turing machine with the technology. Can dominos be used to run a physical simulation? I think it's possible since I found not obvious roadblocks. I'm tempted to start a topic on it, but not here since it isn't a philosophy topic at all.

Perhaps it's some kind of analog computation, but that's not the same thing.

I've also programmed analog computers in school, never on the job. It's a different sort of thing, I tell ya.

ps -- I checked out the Simulation thread and from there, saw your initial post in the "What is the Simulation Hypothesis" thread, and I agree with everything you said. I especially appreciated the distinction between simulation and VR, which is something a lot of the simulation discussions miss.

Your view of consciousness is modelled by a VR. One big distinction is that a VR cannot be implemented with paper and pencil (or dominos).

I was imagining a clock that speeds up in its ticking to ape a convergent geometric series. — fdrake

OK, that would be pretty much what has been the topic of discussion this whole thread. If it completes in finite time, it's a supertask. Don't forget the inverse case where the clock starts fast and slows down to its final tick.

I think you misunderstand Zeno's paradoxes. Zeno concluded that Achilles cannot overtake the tortoise. That is explicit. — Metaphysician Undercover

Correct, but a second unstated premise must be assumed in order to draw this conclusion, since without it, one can only say that the tortoise cannot be overtaken at any particular step. That second premise might well be that supertasks cannot be completed. That premise is indeed in contradiction with the first premise and empirical observation. At least one of the three is wrong.

even though the logic proceeding from fundamental axioms proves

Fundamental axioms? None of the premises are that. They're both easily doubted.

Due to the strength of the empirical evidence, we are led toward the conclusion that the fundamental axioms concerning the continuity of space and time, and the infinite divisibility of those continuums, must be faulty.

Or the premise of supertasks being uncompletable is wrong, or that empirical evidence isn't as strong as is asserted.

Asserting that your premise of choice must be the faulty one is a mistake.

OK, that would be pretty much what has been the topic of discussion this whole thread. — noAxioms

I imagined you lot were talking about metaphysical rather than logical possibility. @Michael made a comment to the effect that such a construction was logically impossible. Which would be odd, seeing as such an object has a model in set theory. Even if it's not physically or metaphysically possible. All I wanted to add.

Which would be odd, seeing as such an object has a model in set theory — fdrake

Does it? Consider these two supertasks:

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

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

The first is reciting every natural number in ascending order and the second is reciting every natural number in descending order.

Does the second have a model in set theory? Is the second logically possible?

That there is a bijection between the series of time intervals and the series of natural numbers and that the sum of the series of time intervals is 60 says nothing about the possibility of (b) and so says nothing about the possibility of (a) either.

The sequence {1/2n} for n=1 to infinity has a finite sum, 1. That's numbers and an infinite set. So yes, geometric sequences have a model in set theory. They also have a finite sum, no last operation, and contain an infinity of operations. You can think of the partial sum, up to the nth term, as the total time elapsed on clock whose ticks last 1/2n. The limit construction exists without contradiction. The reason being that the sequence gets arbitrarily close to its greatest lower bound (0). And there is no smallest element, since the greatest lower bound isn't in the set.

How can a sequence of operations in which each operation is performed only after the previous operation is performed complete without there being a final operation? — Michael

So there is no final operation, there is an infinite sequence of operations, and it completes in a finite time. Since the completion time is just the limit of the partial sums.

There's your model. Logically possible. I leave now.

So you’re claiming that it’s logically possible to have recited the natural numbers in descending order. That’s evidently absurd.

↪fdrake So you’re claiming that it’s logically possible to have recited the natural numbers in descending order. That’s evidently absurd. — Michael

Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility!

Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility! — fdrake

It is logically impossible to have recited every natural number in descending order because it is logically impossible to even start such a task.

The Zeno Wiki page doesn't mention a horse. Did I miss something? Ludwig V mentioned a horse too. — fishfry

I am so sorry. I started a hare by mistake. The horse first appeared in this comment

Ryle might have called it a category mistake and talked of putting a physical harness on a mathematical horse or (better, perhaps) putting a mathematical harness on a physical horse, He and many others thought that nothing further needed to be said. — Ludwig V

So a horse here is shorthand for whatever physical object one is trying to put into mathematical harness. Zeno's horse is the tortoise, or Achilles, or both.

I was wondering about what is actually meant by 'metaphysically possible' or 'logically possible'. The latter is probably the same as 'mathematically possible', but I'm wondering how the former is distinct. — noAxioms

I asked about this earlier in this thread. You can find what I got from it here. I'm not at all clear what people who use the term metaphysics mean by it. For the time being I'm treating the "metaphysics" and "logic" as co-terminous, if not synonymous.

Plank length is not a physical limit, only a limit of significance. If I have it right, any pair of points separated by a distance smaller than that is not meaningfully/measurably distinguishable from just the two being the same point. It doesn't mean that the two points are necessarily the same point. But I gave some QM examples that suggest a non-continuous model of reality. — noAxioms

I have been wondering about exactly that point, and trying to work up the courage to articulate in this context. Thanks. If physics requires a non-continuous model of reality, then so be it, but then it would be empirical (physical) and wouldn't affect the geometrical concepts, would it? If what happened to the question whether matter was continuous or not is anything to go by, I think that a third alternative (not yet available) is most likely.

Plank length is not a physical limit, only a limit of significance. If I have it right, any pair of points separated by a distance smaller than that is not meaningfully/measurably distinguishable from just the two being the same point. It doesn't mean that the two points are necessarily the same point. But I gave some QM examples that suggest a non-continuous model of reality.
— noAxioms
I have been wondering about exactly that point, and trying to work up the courage to articulate in this context. Thanks. — Ludwig V

Well my quote above is not given from authority. Planck units are just a standard of natural units. A Plank length is a small distance, but the fact that they know that distance down to at least 7 significant digits means that far smaller space units are meaningful. Still, wiki says "Since the 1950s, it has been conjectured that quantum fluctuations of the spacetime metric might make the familiar notion of distance inapplicable below the Planck length", which is similar to what I was trying to convey.

There's lots of useful stuff on the wiki Planck units page that's better expressed than me trying to paraphrase it here.

Interestingly, a unit of Planck energy is said to be the equivalent of the chemical energy contained in the fuel tank of a fairly large car. Certainly not the minimum meaningful energy. One unit of Planck force is an even larger silly number. Such is the way with natural units.

If physics requires a non-continuous model of reality, then so be it, but then it would be empirical (physical) and wouldn't affect the geometrical concepts, would it?

A more complex model for the universe does not effect a simple geometric model at all, no. The simple model simply isn't fully applicable to the reality it is supposed to describe, just like Newtonian physics isn't fully applicable to the same reality, despite the fact that they'll continue to teach it in schools.

If what happened to the question whether matter was continuous or not is anything to go by, I think that a third alternative is most likely.

Somebody still suggests that matter is continuous? I mean, that sort of went out the window a couple centuries ago.

I imagined you lot were talking about metaphysical rather than logical possibility. — fdrake

Actually, I've been asking about the distinction between those two. Nobody has really answered. A nice example (not a supertask example if possible) of something that is one but not the other would be nice.

So you’re claiming that it’s logically possible to have recited the natural numbers in descending order. That’s evidently absurd. — Michael

It may grind against your intuitions, but no logical argument against it has been presented. That you personally find it 'evidently absurd' carries no weight.
How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down.

Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility! — fdrake

It is logically impossible to have recited every natural number in descending order because it is logically impossible to even start such a task. — Michael

I think that's actually a very difficult issue to resolve. It's basically the same question as whether an infinite regress is logically possible. I believe the infinite regress actually is logically possible, and it requires a contradictory premise to make it impossible.

Correct, but a second unstated premise must be assumed in order to draw this conclusion, since without it, one can only say that the tortoise cannot be overtaken at any particular step. — noAxioms

If no particular step can overtake the tortoise, then the tortoise, by the described motion cannot be overtaken. Where's the need for another premise?

That second premise might well be that supertasks cannot be completed. — noAxioms

Following from the described premises, the supertask cannot be completed. It is logically implied that there is always further distance for Achilles to cover before overtaking the tortoise. Therefore the described task can never be completed. There is no further premise required, it is a logical conclusion, the described "supertask" cannot be completed.

That premise is indeed in contradiction with the first premise and empirical observation. At least one of the three is wrong. — noAxioms

The conclusion that Achilles cannot overtake the tortoise does contradict empirical evidence, that's the reason it's called a paradox. But I do not see how it contradicts "the first premise". Which premise would that be? The argument is valid, so how could it contradict a premise?

How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down. — noAxioms

This is not the issue. It clearly does not have a start. The question is whether it is logically necessary for such a task to have a start. This is argued in "first cause" arguments.

@Michael @fdrakeThe problem is that what is described is an activity, and the way that we understand activities is that they have a beginning. Activities are all physical. But if you remove that requirement of "physical", then the activity might be proceeding without a beginning, just like it could proceed without an end. It's the way that we look at the difference between past and future, which makes it difficult. If there is no such difference, then the past supertask must be logically possible, just like the future one.

Actually, I've been asking about the distinction between those two. Nobody has really answered. A nice example (not a supertask example if possible) of something that is one but not the other would be nice. — noAxioms

I don't have a particularly clear cut example to distinguish the two. And I don't know if the concept of logical possibility survives the existence of nonclassical logics. But what I envision for logical possibilities are things which don't by themselves, and with no other contextual information, entail a contradiction. Any possible fact is logically possible.

As for the merely logically possible - as in logically but not metaphysically possible - , I imagine procedures like Banach Tarski. Turning a sphere into two spheres using only the material in the first sphere. But that's just because I can't imagine a concept of space used in metaphysics (like extension) that makes central use of non-measurable sets (things with ill defined extension in principle).

Physically possible? That's getting hard. A universe that contains violations of the second law of thermodynamics is metaphysically possible. Like Lord of the Rings, Harry Potter. In the sense that there's a self consistent narrative going through those works of fiction whose behaviour is impossible to translate to our universe, those universes would be metaphysically but not physically possible.

So when I hear @Michael talking about the impossibility of a geometric series "completing" (so to speak) due to being unable to recite the terms in finite time, I hear a use of a metaphysical or physical context of events, speech and recitation which just isn't there in the definition of mathematical terms. And in that regard @Michael's response addresses physical or metaphysical possibility, rather than logical possibility.

The only reason I intervened is to pedantically point out the distinction, and that Michael let it slip for at least one post.

So when I hear Michael talking about the impossibility of a geometric series "completing" (so to speak) due to being unable to recite the terms in finite time... — fdrake

I don't think it impossible for a geometric series to complete. I think it impossible to have recited every natural number in descending order.

My issue is with supertasks, not with maths.

How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down. — noAxioms

There is no first natural number to start with. It is logically impossible to have started reciting the natural numbers in descending order.

I don't think it impossible for a geometric series to complete. — Michael

Good! Then it's logically possible for it to. An infinite number of things can complete without blowing up logic.