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

Obviously, the described process has no start, that is implied by the description. So your conclusion that it is logically impossible to have started such a process is irrelevant, as what is already known. You need to show that such a process, one without a start, is logically impossible.

That's what "first cause" arguments attempt to do. They describe the temporal aspect of "a process", "a thing", or similar term, in such a way that it necessarily has a beginning and an end in time, then they produce a logical argument from that description. It's an attempt to bring the realm of material (physical, or temporal) reality to bear on the realm of logical possibility, by stating premises which are supposed to represent the essence of material (physical) reality, and restricting logic with them. Another example of a similar restriction is the law of identity, and the other two fundamental principles.

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? — Metaphysician Undercover

Great. Then show the logic that concludes this, without resort to another premise.

Following from the described premises, the supertask cannot be completed.

That logic has not been shown. It's a non sequitur until it is spelled out.

It is logically implied that there is always further distance for Achilles to cover before overtaking the tortoise.

No such implication exists, and no such statement is made. Asserting this would be another premise, and one that makes no sense either. And yes, it would follow that the tortoise cannot be overtaken if this additional premise is added.

It clearly does not have a start. — Metaphysician Undercover

Your usage of 'clearly' implies you are referencing a second premise based on perhaps your intuition. What you may find 'clear' seems to be in direct contradiction with the first premise, I am presuming your 'clear' assumption is something on the order that there must be a first step, equivalent to asserting a bound to something explicitly defined as not being bounded. Of course you're going to run into contradictions if you add a second premise that directly denies the first premise. It isn't a paradox then, it's just wrong.
If that isn't your 'clear' premise, then state what you find so clear.

Michael also makes this mistake despite it being pointed out so many times.

There is no first natural number to start with. — Michael

Totally predictable response. We're like over 400 posts into this topic and you're you're stuck on the same fundamental mistake. You (as well as Meta above) seem to insist on an additional premise of the necessity of a bound to something explicitly defined to be unbounded. My method for performing the task made no mention of doing a first step, but it can be mathematically shown that any given step is done, and that the steps are done in order.

It is logically impossible to have started reciting the natural numbers in descending order.

An unbacked assertion, especially when I showed how to do it. Your presented 'logic' seems to be the argument above, declaring a second premise that happens to contradict the thing you want to find impossible. The logic to which you refer is only valid for finite sets, but you cannot learn this.

You can disprove it by naming a number that isn't covered by my procedure, or by naming a pair of numbers that are recited out of sequence, or some other such demonstration of a violation of the task as described. That's how you go about it when dealing with unbounded sets. Hilbert's hotel is a great educational exercise showing how mathematics deals with infinities.

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). — fdrake

I don't think it is the extension that is ill defined with that case, but rather a leveraging of the fact that the pieces are made of infinite points each, and you don't need 'more natural numbers' to count each one of them twice. I don't understand the Banach Tarski thing enough to know why 5 is a lower limit of the number of pieces.
Anyway, I chalk it up to another illustration of why the logical rules of bounded sets don't necessarily apply to unbounded ones. The posters above clearly cannot accept this, and so we go in circles.

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.

OK, here you seem to use 'metaphysically possible' to mean 'possible in a universe with different physical laws'. But I don't find that very distinct from logically 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 don't think he says that time is the issue. It is his insistence on the need to eventually recite the highest number, after which there are no more. That number doesn't exist, so the task cannot be done because it missed at least that one.

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

But we're talking about supertasks, not geometric series. That a geometric series is possible isn't that a supertask is possible.

Given that there is no largest natural number it is logically impossible to even start reciting all the natural numbers in descending order.

I don't know why you think the existence of a geometric series proves otherwise.

You (as well as Meta above) seem to insist on an additional premise of the necessity of a bound to something explicitly defined to be unbounded. — noAxioms

No, I'm saying that something with no start cannot start and something with no end cannot end.

Your argument is effectively "by definition it has no start therefore it can start without a start." You're trying to take the very thing that makes it impossible as proof that it's possible.

No, I'm saying that something with no start cannot start and something with no end cannot end. — Michael

This seems to be playing language equivocation games. You introduce the word 'start' here, undefined twice, once as a noun and once as a verb. Given certain definitions of both usages, I may or may not accept this additional premise you state.

The noun definition I will call Sn.
Sn1: The "start" of a sequence is a first step. This is exactly a bound, but you say 'No, ..." above, denying my calling it a bound, so you must mean a different defintion Sn2 of 'start' that this sequence doesn't have. I can't think of one that is distinct from a bound, so you need to help me here.

The verb is also used (cannot 'start'). Here I can think of at least two definitions:
Sv1: To start is to initiate the first step. This again is a reference to the bound.
Sv2: To to start is to commence the steps. There is a duration during which none of the steps has been performed, even in part. There is a duration when steps are being executed, and a duration when all steps have been completed. To start is to transition from the first to the second duration.

I suspect that you are actually equivocating multiple definitions of the verb to make your point. I mean, if you go with Sn1 and Sv1, then I actually agree with your added premise. A sequence lacking a specific step cannot execute the nonexistent step. That holds water. But then you equivocate to Sv2 and conclude that the existing steps cannot commence.

Your argument is effectively "by definition it has no start therefore it can start without a start" which is ridiculous

You are clearly using Sn1 as your noun definition here, which is a direct reference to the bound that we both acknowledge doesn't exist. This usage of the noun contradicts your opening word "No" in your post where you imply that your argument is something other than "an additional premise of the necessity of a bound to something explicitly defined to be unbounded". You contradict yourself.

Given the Sv2 definition for the verb, my argument is pretty much that, yes. It isn't ridiculous because I gave a precise description of how to do it. Your expressed ridicule isn't valid logic.

I cannot start reciting the natural numbers in descending order because there is no first natural number for me to start with.

That a geometric series has a finite sum is irrelevant to this very simple self-evident fact.

I don't think it is the extension that is ill defined with that case, but rather a leveraging of the fact that the pieces are made of infinite points each, and you don't need 'more natural numbers' to count each one of them twice.. — noAxioms

I'm pretty sure that one comes down to being able to split the pieces up into pieces that aren't measurable - IE can't be assigned a size - in a clever way, then applying some cool transformation to them that blow them back up into the sphere. But that's by the by.

OK, here you seem to use 'metaphysically possible' to mean 'possible in a universe with different physical laws'. But I don't find that very distinct from logically possible. — noAxioms

I think that's a species of metaphysical possibility - a different physics. What would distinguish that from logical possibility, in my book, is that there are simply more ways of being noncontradictory than being unable to exist in our universe. Like flibbertygibbets. And nonmeasurable sets. And, maybe, abstract categories.

(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. — noAxioms

. That's a relief. I suspect that there are still people around who have difficulty with the difference between "not fully applicable" and "false". I still wonder (when I haven't anything more important to wonder about) whether Aristotelian physics is not fully applicable or not physics or false. I don't think anything important hangs on the answer, but still, that doesn't usually bother philosophers much.

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

All I was pointing to was the conceptual explosion that happened when we finally split the atom. (Which, you will remember, was by definition unsplittable).

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. — fdrake

This is a fascinating issue, mostly swept under the carpet in philosophy. I don't say that you are wrong.
I think it was Aristotle who first articulated the idea that a fictional story must be at least plausible. (Does that mean "possible"? - possibly). The idea that it requires "suspension of disbelief" was, apparently first articulated by Coleridge in 1817. There's a distinct tension between these two requirements. Both high-light that the audience/reader needs to collaborate with the author. It seems to me that the collaboration is at least sometimes secured by "arm-waving" by the author at awkward moments to distract the audience's attention and the audience not pressing questions that would be irresistible in other circumstances. The concept of magic is a good example. Science fiction stories usually put up a better front than that, but nonetheless... The issue comes home to bite philosophers when we offer examples - thumbnail stories. (I won't give examples for fear of setting off a hare and distracting us all). The difficulty for us is to distinguish arm-waving from actual possibilities (!).

The conclusion that Achilles cannot overtake the tortoise does contradict empirical evidence, that's the reason it's called a paradox. — Metaphysician Undercover

Yes. Disagreements between logic and experience are not unfamiliar. Experience usually wins, because logic is more adaptable than it seems. (I realize that may seem like heresy in a philosophical concept, but doesn't experience support it?)

I think that's a species of metaphysical possibility - a different physics. What would distinguish that from logical possibility, in my book, is that there are simply more ways of being noncontradictory than being unable to exist in our universe. Like flibbertygibbets. And nonmeasurable sets. And, maybe, abstract categories. — fdrake

Surely a different physics will have to be consistent and complete - when it is finished. That looks very like "logically possible", doesn't it?
As for the rest, you seem to understand existence as a single category. Perhaps you believe the same of reality. That is not how I understand either term. Existence has many different modes? categories? which are defined contextually. Ditto reality. I understood a flibbertygibbet to be a silly person who talks too much, so they very much do exist in my universe and I envy you yours. Things like non-measurable sets and abstract categories exist all right, but not in the same way/mode/category as tables and chairs. So do fictional things like Achilles and his tortoise and the Gorgon's mirror.

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? — Metaphysician Undercover

I'm afraid that if you condescend to use ordinary arithmetic, one can predict exactly when Achilles will overtake the tortoise, given data about how fast each contestant moves and the size of the handicap.
Not being a fully trained mathematician, I'm not sure about it, but I suggested this earlier and no-one has contradicted me - yet. Perhaps it is just too boring.

Neglecting acceleration, let's say Achilles gives the tortoise a head start of 100 units of length and that Achilles runs at 11 units per second and the tortoise at 1 unit per second. So, at time t seconds after the tortoise is at 100 units from the start, the tortoise will be at 100 + t units from the start, and Achilles at 11t units. These will be the same - 110 units - at time t = 10 seconds. — Ludwig V

That's what "first cause" arguments attempt to do. They describe the temporal aspect of "a process", "a thing", or similar term, in such a way that it necessarily has a beginning and an end in time, then they produce a logical argument from that description. It's an attempt to bring the realm of material (physical, or temporal) reality to bear on the realm of logical possibility, by stating premises which are supposed to represent the essence of material (physical) reality, and restricting logic with them. Another example of a similar restriction is the law of identity, and the other two fundamental principles. — Metaphysician Undercover

That is a very interesting take on the argument, though I don't understand how this applies to the law or identity. But then, I don't understand the law of identity, either. What are the other two principles?

Great. Then show the logic that concludes this, without resort to another premise. — noAxioms

I don't see the need for any other premise. Achilles is moving, and described as doing this in a way in which he will always have to move further before he can overtake the tortoise. Since he will always have to move further before he will overtake the tortoise, we can conclude logically that he will never overtake the tortoise in that described activity. Why do you see the need for another premise?

I'm afraid that if you condescend to use ordinary arithmetic, one can predict exactly when Achilles will overtake the tortoise, given data about how fast each contestant moves and the size of the handicap. — Ludwig V

Sure, but those mathematical principles are not the premises described by Zeno.

Sure, but those mathematical principles are not the premises described by Zeno. — Metaphysician Undercover

Achilles is moving, and described as doing this in a way in which he will always have to move further before he can overtake the tortoise. Since he will always have to move further before he will overtake the tortoise, we can conclude logically that he will never overtake the tortoise in that described activity. — Metaphysician Undercover

Case closed, then.

Case closed, then. — Ludwig V

I think so, but we'll have to see what noAxioms is talking about with the reference to a requirement for further premises. I think noAxioms looks at Zeno in a different way.

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

You've been perfectly clear, and I've clearly responded to your points several times already.

If you want to argue that the first supertask can end ... — Michael

I have never made any such statement. I've repeatedly challenged you to name the first number not verbalized when we count forward 1, 2, 3, ... at successively halved intervals of time.

I ask you once again to tell me whether you appreciate the point that any interval containing the limit of a sequence must necessarily contain all but finitely elements of the sequence.

If you understand that, it addressed your counting backward argument. If not, let's discuss it.

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

Halving the time is your own thought experiment. It's not mine. Once I accept your own premise, I then work out the logical consequences. Halving the time is not my premise.

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. — noAxioms

Consciousness could perfectly well be a physical process, but not computable. So, what kind of process is physical but not computable? A task for some future genius to elucidate. FWIW Penrose believes that consciousness is not computable. He may be wrong, but he's Sir Roger and the rest of us are not.

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. — noAxioms

That's my point. The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers. The historical contingency is an argument against the theory, not for it.

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. — noAxioms

You're agreeing with my point.

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. — noAxioms

I agree with you that IF consciousness is a computation, then it could be implemented with pencil and paper. I regard that as an argument against the premise.

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.[/quoet]

I don't know about dominos. The pencil and paper argument is stronger.
— noAxioms

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. — noAxioms

I've seen Searle argue that consciousness is physical but not computational. Some kind of secret sauce found in living things and not in digital circuits. Don't know much about analog computation with respect to consciousness.

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). — noAxioms

The illusion can. But my consciousness can't. As Descartes noted, I may be deceived, but there is an I who is being deceived. So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is.

It's always been unclear to me which aspect of simulate/VR Bostrom is arguing.

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. — noAxioms

There is never a final tick in an infinite sequence, even if the sequence has a limit.

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. — Ludwig V

No worries. Like a certain Supreme court justice, I am not a biologist.

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. — Ludwig V

Ok. I figured that out, just couldn't remember anything about a horse. I agree that a hare or a tortoise or Achilles does just as well. Thanks.

I cannot start reciting the natural numbers in descending order because there is no first natural number for me to start with. — Michael

Given your reluctance to clarify the definition of the verb 'to start', I cannot respond appropriately to this statement. I gave a pair of options, or you can supply your own, so long as it isn't open to equivocation.

I'm pretty sure that one comes down to being able to split the pieces up into pieces that aren't measurable — fdrake

Your confidence in your own understanding is then stronger than my confidence in mind.

I still wonder (when I haven't anything more important to wonder about) whether Aristotelian physics is not fully applicable or not physics or false. — Ludwig V

Some of both, I'm sure. The impetus thing had to go (survived until Newton, not bad...), but one could argue that it is a poor description of inertia.

when we finally split the atom. (Which, you will remember, was by definition unsplittable).

The smallest thing still is. Unfortunately the word got applied to something that was a composite object, and they kept that instead of renaming the assembly and keeping 'atom' for anything fundamental.

I don't see the need for any other premise.Achilles is moving, and described as doing this in a way in which he will always have to move further before he can overtake the tortoise. — Metaphysician Undercover

Not always. Just a minute. I know, Zeno doesn't give the time, but we've been using a minute. The way the scenario is described has no effect on the situation compared to a different way of describing it.

Anyway, I deny that Zeno in any way suggests that the overtaking will never take place. He just says that another step always follows any given step.

Michael has added the verbalizing of the natural number count, but that doesn't change it taking only a minute.

The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers. — fishfry

They can't both be right?

You're agreeing with my point.

I think I am, yes.

I've seen Searle argue that consciousness is physical but not computational. Some kind of secret sauce found in living things and not in digital circuits. Don't know much about analog computation with respect to consciousness.

Anything analog can be approximated with digital. But anything digital can be perfectly implemented with analog. Searle is perhaps referencing property dualism? I don't know if I got that right. Can't seem to articulate the differences between the variants.

As Descartes noted, I may be deceived, but there is an I who is being deceived.

I guess I'm even more skeptical than Descartes. I win! I didn't pick my handle for no reason. I try not to leave anything unquestioned.

So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is.

VR says that all you know is potentially lies. You are not of this universe, but rather you are experiencing it. All very dualistic. The 'brain' in the body (if there is one at all, have you ever checked?) is not what's making any of the decisions.
If you think about it, the view can be empirically tested. Not so much with the simulation hypothesis.

It's always been unclear to me which aspect of simulate/VR Bostrom is arguing.

Definitely the former. But Elon musk is arguing for VR, and references Bostrom's paper to support it, so he has no idea what he's talking about.

The comment above (and my reply) belongs in the other topic. I see you posted more or less the same question there.

There is never a final tick in an infinite sequence, even if the sequence has a limit.

or not a first tick. Zeno's dichotomy very much has a final tick. I can make a scenario that has a first and last, and gets singular in the middle somewhere. Just illustrating the classical snippet: Never say never.

The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers.
— fishfry
They can't both be right? — noAxioms

Ok. We are ALL of whatever our latest technology is. Well ... maybe so. Something to be said for that.

You're agreeing with my point.
I think I am, yes. — noAxioms

Ok. But I'm arguing that the simulation theory, or the computational theory of mind, is suspect because of its very timeliness. We invent computers and the philosophers all go, "Ooh we're computers." That's a point against the idea IMO.

Anything analog can be approximated with digital. But anything digital can be perfectly implemented with analog. Searle is perhaps referencing property dualism? I don't know if I got that right. Can't seem to articulate the differences between the variants. — noAxioms

I think Searle was arguing against dualism in this instance. He was saying that mind is not a computation; but it's not something non-physical. Rather, there's something physical about living things that implements consciousness, in a way that rocks and digital circuits can't. I only saw him mention this on video, so perhaps he's added more detail in his writings.

I guess I'm even more skeptical than Descartes. I win! I didn't pick my handle for no reason. I try not to leave anything unquestioned. — noAxioms

Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right?

VR says that all you know is potentially lies. You are not of this universe, but rather you are experiencing it. All very dualistic. The 'brain' in the body (if there is one at all, have you ever checked?) is not what's making any of the decisions.
If you think about it, the view can be empirically tested. Not so much with the simulation hypothesis. — noAxioms

Yes but everyone agrees with that. There's a world "out there," and we experience it through our senses. Not sure what you mean by empirical testing here.

It's always been unclear to me which aspect of simulate/VR Bostrom is arguing.
Definitely the former. But Elon musk is arguing for VR, and references Bostrom's paper to support it, so he has no idea what he's talking about. — noAxioms

Right. And I saw a TED talk where George Smoot, the guy who discovered the cosmic background radiation anisotropy, was enthusiastically advocating simulation theory. Neil deGrasse Tyson too. A lot of people who should know better say trendy things for no reason at all. More arguments against simulation IMO. The pronouncements of celebrity scientists outside their expertise are always suspect.

The comment above (and my reply) belongs in the other topic. I see you posted more or less the same question there. — noAxioms

Yes. Let's talk about this over there.

There is never a final tick in an infinite sequence, even if the sequence has a limit.
or not a first tick. Zeno's dichotomy very much as a final tick. I can make a scenario that has a first and last, and gets singular in the middle somewhere. Just illustrating the classical snippet: Never say never. — noAxioms

Ok. Just talking about standard mathematical sequences. It's a common misunderstanding in this thread. The sequence 1/2, 3/4, 7/8, ... has a limit, namely 1, but no last element.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of @Michael's example of counting backwards.

I've repeatedly challenged you to name the first number not verbalized when we count forward 1, 2, 3, ... at successively halved intervals of time. — fishfry

I accept this:

P1. If we can recite forward 1, 2, 3, ... at successively halved intervals of time then we can recite all natural numbers in finite time

But I reject these:

P2. We can recite forward 1, 2, 3, ... at successively halved intervals of time
C1. We can recite all natural numbers in finite time

If you want to claim that C1 is true then you must prove that P2 is true. You haven't done so.

I think Thomson's lamp and similar examples prove that P2 is false. See here.

Given your reluctance to clarify the definition of the verb 'to start', I cannot respond appropriately to this statement. I gave a pair of options, or you can supply your own, so long as it isn't open to equivocation. — noAxioms

Just the ordinary meaning of "start", e.g. "begin".

You ask me, right now, to recite the natural numbers in descending order. How do I begin to perform this?

I think it's self-evident that I cannot begin because there is no first (largest) number for me to begin with.

I see that I misunderstood your idea. You are counting time backward. Ok I'll respond to that. But just wondering, when you realized I misunderstood you earlier, why didn't you point that out?

Ok. Suppose that I start at 1 and count backward through 1/2, 1/4, 1/8, ... — fishfry

It is Achilles' run but with time reversed: https://plato.stanford.edu/entries/spacetime-supertasks/#MissFinaInitStepZenoWalk

Ok. Just talking about standard mathematical sequences. It's a common misunderstanding in this thread. The sequence 1/2, 3/4, 7/8, ... has a limit, namely 1, but no last element.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of Michael's example of counting backwards. — fishfry

This is what I mean by reciting backwards:

If I recite the natural numbers <= 10 backwards then I recite 10, then 9, then 8, etc.
If I recite the natural numbers <= 100 backwards then I recite 100, then 99, then 98, etc.

If I recite all the natural numbers backwards then I recite ... ?

It's self-evidently impossible. There's no first (largest) natural number for me to start with.

Anyway, I deny that Zeno in any way suggests that the overtaking will never take place. He just says that another step always follows any given step. — noAxioms

According to this reasoning, Achilles will never catch the tortoise, says Zeno. — Internet Encyclopedia of Philosophy

https://iep.utm.edu/zenos-paradoxes/

The paradox is like this. Both Achilles and the tortoise are moving, but the tortoise has a head start. So at t1 Achilles is at location A and the tortoise is at location B. At t2, Achilles reaches location B, but the tortoise has moved to location C. At t3, Achilles reaches location C, but the tortoise has moved to location D. As this procedure will carry on without end, Zeno concludes that the faster runner cannot overtake the slower.

Zeno Paradox 1: Achilles and the Tortoise
Achilles is a lightening fast runner, while the tortoise is very slow. And yet, when the tortoise gets a head start, it seems Achilles can never overtake the tortoise in a race. For Achilles will first have to run to the tortoise's starting point; meanwhile, the tortoise will have moved ahead. So Achilles must run to the tortoise's new location; meanwhile the tortoise will have moved ahead again. And it seems that Achilles will always be stuck in this situation.

https://personal.lse.ac.uk/robert49/ebooks/philsciadventures/lecture24.html

Some of both, I'm sure. The impetus thing had to go (survived until Newton, not bad...), but one could argue that it is a poor description of inertia. — noAxioms

Well, one could argue that it isn't a description of inertia, but of certain phenomena which are better described by inertia. Either way, impetus proved unhelpful and alternative conceptualizations proved more helpful.

That's my point. The Romans thought mind was a flow, because they had great waterworks, and so forth. We live in the age of computation so we think we're computers. The historical contingency is an argument against the theory, not for it. — fishfry

You are right that the historical contingency should make us suspicious. (Descartes, by the way, has a description of statues "animated" by a hidden hydraulic system - I think in Versailles). But I don't think the process is simply over-enthusiastic. It seems reasonable to try to apply a new discovery as widely as possible. That way, one discovers its limitations.

The smallest thing still is. Unfortunately the word got applied to something that was a composite object, and they kept that instead of renaming the assembly and keeping 'atom' for anything fundamental. — noAxioms

Oh, I don't know. Given the conceptual revolution that happened when sub-atomic physics arrived, it's not a bad idea to signal the change by leaving atoms where they were.

So the VR theory doesn't solve anything at all, it leaves the mystery of what my own consciousness is. — fishfry

That's more or less one Ryle's favourite arguments against dualism.

The sequence 1/2, 1/4, 1/8, ... also has a limit, namely 0, and no last element. But if you put the elements of the sequence on the number line, they appear to "come from" 0 via a process that could never have gotten started. This is my interpretation of Michael's example of counting backwards. — fishfry

Clearly "<divide by> 2" is not applicable at 0. Would it be right to say that "+1" begins at 0 and has no bound and no limit, and that "<divide by> 2" begins at 1 and has no bound, but does have a limit? But they both they have a defined start and no defined end.

Not always. Just a minute. I know, Zeno doesn't give the time, but we've been using a minute. The way the scenario is described has no effect on the situation compared to a different way of describing it. — noAxioms

That's true. But different descriptions of the same situation can affect how we think about that situation. An additional difficulty, I suspect, is that our descriptions are fictional (sorry, thought-experiments), which means that the context is limited and evaluations of descriptions much more difficult. They need to be assessed in a different way - as useful or not.

I think @noAxioms looks at Zeno in a different way. — Metaphysician Undercover

Yes. You cannot necessarily decide that just one way of looking at things is true and all others false. They are better evaluated as useful or not. I think that applies here.

Without axioms it's difficult to get reasoning off the ground. You have to start somewhere, right? — fishfry

Yes. The difficulty is how to evaluate a starting-point. True or false isn't always relevant. Which means that it can be difficult to decide between lines of reasoning that have different starting-points.

The paradox is like this. Both Achilles and the tortoise are moving, but the tortoise has a head start. So at t1 Achilles is at location A and the tortoise is at location B. At t2, Achilles reaches location B, but the tortoise has moved to location C. At t3, Achilles reaches location C, but the tortoise has moved to location D. As this procedure will carry on without end, Zeno concludes that the faster runner cannot overtake the slower. — Metaphysician Undercover

So are you going to conclude, with Zeno, that motion is impossible? or that Zeno is analyzing the situation in a misleading way?

So are you going to conclude, with Zeno, that motion is impossible? or that Zeno is analyzing the situation in a misleading way? — Ludwig V

Yes, Zeno is analyzing in a misleading way, but only because the axioms of continuity and infinite divisibility are themselves misleading. So Zeno simply demonstrates how standard conventions are actually misleading us.

And here we are. a couple of millennia later, still being misled by the same conventions. This is because we have not yet determined the natural points of divisibility. And so, fundamental particles take every possible path when they move from A to B, because the direct spatial route does not allow them to get ahead of the tortoise.

but only because the axioms of continuity and infinite divisibility are themselves misleading. — Metaphysician Undercover

You mean because they allow the convergent infinite series?
Mathematically? Physically? (I'm inclined to think you mean physically, because of your reference to fundamental particles.)

So Zeno simply demonstrates how standard conventions are actually misleading us. — Metaphysician Undercover

Well, we've caught them out misleading us before, so I suppose they may be doing it again.

And so, fundamental particles take every possible path when they move from A to B, because the direct spatial route does not allow them to get ahead of the tortoise. — Metaphysician Undercover

Is the direct spatial route not available because it contains a convergent regress?
What path does Achilles take? (I assume he is not a fundamental particle.)

Yes. Let's talk about this over there. — fishfry

I replied to much of your post, but all over there.

Just the ordinary meaning of "start", e.g. "begin". — Michael

In that case I reject your premise. The lack of a first step does not prevent the beginning of the task, which is simply the transition from the time prior to any of the steps being taken, to the time during which steps are being taken.

You ask me, right now, to recite the natural numbers in descending order. How do I begin to perform this supertask? — Michael

I described exactly how to do that, and you found no fault with it, choosing instead to try a different wording of your additional premise. Why does my description fail? What step is missed? None, and it's done in finite time, so you apparently cannot find fault except by asserting additional premises, all of which take the form of asserting a need to perform a step that by definition doesn't exist.

The paradox is like this. — Metaphysician Undercover

I know the story. You seem to have reworded it for your purposes, since the quote you give does not come from that site, but the site also seems to be conveying the story in its own words, not as reported by Aristotle.

Zeno concludes that the faster runner cannot overtake the slower.

Yes, and without justification, or at least without explicitly stating the additional premise that makes the conclusion valid.

Other quote:
... And it seems that Achilles will always be stuck in this situation.

Same thing. Does not follow.

Funny that the animation on that site below that quote shows 'lightning fast' Achilles moving at only twice the speed of the tortoise. It also shows Achilles slowing down to pretty much a halt, which is why he never passes the tortoise in the animation.

Are we resorting back to the beginning of the discussion here? If you cannot counter my posts, you just start over with the original story? Yes, you can keep the topic going a long time this way, but you're not helping your case.

But different descriptions of the same situation can affect how we think about that situation. — Ludwig V

Yes, it affects how we think of them. It doesn't effect the situation, despite all the assertions to the contrary by several members.

An additional difficulty, I suspect, is that our descriptions are fictional (sorry, thought-experiments)

A thought experiment is a valid method of deriving conclusions from premises. They only get fictional if the premises are faulty, such as the lamp, a device which cannot physically operate as described.[/quote]

The lack of a first step does not prevent the beginning of the task — noAxioms

It literally does.

I described exactly how to do that — noAxioms

No you didn't. You ignored it and just said "when the time comes I say the next number". That doesn't explain how the recitation can begin without a first number to say.

I am right now trying to recite the natural numbers in descending order but am silent because I cannot begin. It's been 60 seconds. Not only have I failed to recite them all, but I have failed to recite even one. Help me out here.

Consider the infinite sequence {0, 1, 0, 1, 0, 1, ...}.

Now consider reciting its terms in reverse.

To recite its terms in reverse I am only allowed to say "0" or "1" but I cannot start by saying "0" and I cannot start by saying "1". Therefore I cannot start.

No appeal to a geometric series of time intervals can save you from this.

Yes, it affects how we think of them. It doesn't effect the situation, despite all the assertions to the contrary by several members. — noAxioms

Yes - unless it is a fictional situation - whether in the philosophical or the literary sense.

A thought experiment is a valid method of deriving conclusions from premises. They only get fictional if the premises are faulty, such as the lamp, a device which cannot physically operate as described. — noAxioms

That may explain why I have been confusing them. Thanks for that.
I have wondered whether one could replace the Thompson lamp with a question, such as whether the final number was odd or even. That would work if you start with an odd divisor and don't express everything in decimals. Perhaps it would work for all examples if you ask whether the number of steps taken is odd or even when the minute is up. I think.

I have wondered whether one could replace the Thompson lamp with a question, such as whether the final number was odd or even. That would work if you start with an odd divisor and don't express everything in decimals. Perhaps it would work for all examples if you ask whether the number of steps taken is odd or even when the minute is up. I think. — Ludwig V

P1. When the letter X is given a definition it retains this definition until it is redefined.

A1. At t₀ X = 0
A2. Therefore, at t₁ X = 0

B1. At t₀ X = 0 and then at t_1/2 X = 1
B2. Therefore, at t₁ X = 1

C1. At t₀ X = 0 and then at t_1/2 X = 1 and then at t_3/4 X = 0, and so on ad infinitum
C2. Therefore, at t₁ X = ?

In all cases the definition of X at t₁ must be a logical consequence of what occurs between t₀ and t₁.

Given that in C2 X cannot be defined as either "0" or "1" but must be defined as either "0" or "1" then C1 is necessarily false. The supertask described in C1 is impossible.

This addresses the very logic of a supertask without some dependency on a physical performance.

You mean because they allow the convergent infinite series?
Mathematically? Physically? (I'm inclined to think you mean physically, because of your reference to fundamental particles.) — Ludwig V

I meant, that they can mislead us when we apply the principles to the activities of the physical world. That's what Zeno's paradoxes show.

Is the direct spatial route not available because it contains a convergent regress?
What path does Achilles take? (I assume he is not a fundamental particle.) — Ludwig V

What is evident, is that we do not know how things move, and the exact "path" through space, that things take, whether they are big planets, stars and galaxies, small fundamental particles, or anything in between.

I know the story. You seem to have reworded it for your purposes, since the quote you give does not come from that site, but the site also seems to be conveying the story in its own words, not as reported by Aristotle. — noAxioms

Here's what Aristotle reported:

The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. — Aristotle Physics 239b 14-17

How is this different from what I said? I gave a full explanation, as did the site I quoted. Aristotle just said "it amounts to this...", providing a shortened version, probably because the specifics were well known at that time.

Yes, and without justification, or at least without explicitly stating the additional premise that makes the conclusion valid. — noAxioms

I'm still waiting for you to explain how the conclusion is not justified, and why you think there is a requirement of an additional premise.