Bostrom is saying that one of these is almost certainly true:

1. Almost every intelligent civilisation is incapable of creating simulations — Michael

Bostrom does not say this. We create simulations today. He calls the state 'posthuman', and it apparently means a device capable of simulating all of human civilization to a level sufficient for the full consciousness of the humans, and also a full simulation of more complex things like the simulation hardware itself.

2. Almost every intelligent civilisation doesn't want to create simulations

He doesn't say that either. He says that nobody will run 'ancestor simulations', which is defined as simulations (however long or brief) of our own evolutionary history. But such a simulation is impossible since no intiial state they give it would evolve anything like our actual history. They can run a sim of an arbitrary alternate outcome from the initial state, but that won't be our ancestry history, it will just be a simulation of fiction. Depending on where they put the initial state, there might not ever be humans at all.

So two premises, each of which has odds of being almost exactly 1.

3. Almost every conscious person is living in a simulation

That is a valid suggestion if the odds of the above two are small.

Point is, you are misstating Bostrom's premises. Item 3 doesn't follow from the premises as you word them.

He doesn't say which of the three is most likely to be true.

He does. Most of the paper focuses on rationalizing low probabilities for the first two premises to the point of 3 being likely.

Therefore, if simulated persons do not greatly outnumber non-simulated persons then most civilisations are either incapable of or unwilling to make simulations. — Michael

Incapable or unwilling to simulate a lot of them. I see purpose in simulating one person, or a very small group in a closed environment. There's value to that. But not to simulating that group that has decided to have its own simulating machine and running the same simulation.

I don't see why you say that. I think you are assuming at least a soft determinism? — Ludwig V

Scientific discover is sort of inevitable. Einstein stated somewhere that relativity theory was totally ripe after M&M experiment showed the apparent frame invariance of light speed. Minkowski would have come up with SR, but not GR. Others would have had to finish it.

The progress of physics is yes, a sort of fatalistic thing, much like Asimov's foundation series: It will happen inevitably, presuming there is the means to make progress. Much of progress hinges on the political state of things, which cannot be fully predicted.

Remember, there were times during the Cold War when nuclear holocaust hung by a thread.

Oh yes. That's what I mean above by 'presuming there is the means to make progress'. Plenty of viable outcomes have us all nuked away, or a pandemic or something. Asteroid is not likely since that isn't a chaotic function over times as short as centuries.

You say that you wouldn't necessarily run detailed simulations of everything at the same time, but switch to closer simulations when necessary to maintain the illusion.

Bostrom suggests that, yes. It's a necessary thing for an open system. Most simulations we run today are not open. Not always the case. I used to run computer chip simulations which has to be an open system since (most) chips need external input to drive them. We needed to see how the chip would function before going to the great expense of actually manufacturing a batch.

That's all very well, though it imposes an extra burden on the machinery because it will have to be aware of what people are attending to at all times.

You got it. Also what their devices are attending to, even when the people are not around.

What if the human decides to dig at location X? That location was trivially simulated up until now, but suddenly the machine has to decide if there should be a dinosaur there or something, even when the digging is not being done for purposes of looking for them.

it wouldn't be easy to fool them all the time.

Nope. It would be dang difficult, which is a decent reason why nobody would attempt such simulations, simulations good enough to fool its occupants, even the very smart but skeptical ones.

Didn't you say something to the effect that quantum mechanics and general relativity couldn't be simulated?

QM can't easily be simulated, but it can be done. My example of the cc of water was an example beyond some limits, but it depends on the interpretation being simulated.
I don't see much difficulty with relativity theory being simulated. They do that all the time in astronomical simulations like what it looks like to fall into a black hole, or a sim of our collision with Andromeda.

The difficultly in simulating QM is not in any way evidence that it is wrong. It's just evidence that it isn't classical, and most simulations as we know them are classical simulations.

There are two physics involved. One is the physics of the simulated world, which would need to be quite like ours.

If we are simulated, then the physics of the simulated word IS our physics, by definition. They can't be wrong. They might be only an approximation of what the runners of the simulation actually wanted.

For instance, any simulation run by us is discreet. Humans and machines only have access to a countable set of numbers, leaving most of the real numbers inaccessible. For instance, a typical floating point number is but 64 bits in a computer, more if you want more precision. There are only so many values that a finite number of bits can represent. The rest are off limits.
Real physics seems to work with real numbers, not these discreet numbers. But we can't prove that.

You believe in limits, you said so. And if you believe even in the very basics of set theory, in the principle that I can always union two sets, then I can adjoin 1 to {1/2, 1/3, 1/4, 1/5, ...} to create the set {1/2, 1/3, 1/4, 1/5, ..., 1}.

It's such a commonplace example, yet you claim to not believe it? — fishfry

I said I had no problem with any of that.
Is it a belief thing, like it is some kind of religious proposition or something? "Hey, I'm going rogue here and will suspend belief that 7 is a factor of 35".

Or what is your objection, exactly?

Treating infinity as a number, something you didn't do in your unionized set above

It's an infinite sequence. I stuck the number 1 on the end.

Yea, when it normally is depicted at the beginning. From what I know, a set is a set regardless of the ordering. There must be a different term (ordered set?) that distinguishes two identical sets ordered differently, sort of like {1, 3, 5, 7 --- --- 8, 6, 4, 2}

The entire set is ordered by the usual order on the rational numbers. So why is it troubling you that I called 1 the "infinitieth" member of the ordered set?

It violates thebijunction. You can't say what number comes just before it, which you can for any other element except of course the first. You can do that with any other element.

It's a perfect description of what's going on. And it's a revealing and insightful way to conceptualize the final state of a supertask. Which is why I'm mentioning it so often in this thread.

OK, but what problem does it solve? It doesn't solve Zeno's thing because there's no problem with it. It doesn't solve the lamp thing since it still provides no answer to it.

In terms of known physics as of this writing, we can not sensibly discuss what might be going on below the Planck length.

Nobody's asking the particle to meaningfully discuss (mathematically or not) the step. It only has to get from one side to the other, and it does. Your argument is similar to Michael wanting a person to recite the number of each step, a form of meaningful discussion.

Maybe we live in a discrete grid of points -- which would actually resolve Zeno's paradoxes.

It would falsify the first premise. Continuous space falsifies the second premise. Zeno posits two mutually contradictory premises, which isn't a paradox, only a par of mutually contradictory premises,.

But you can't say "you can traverse the space of that step, even when well below the Planck length" because there is no evidence, no theory of physics that supports that claim.

But I can say "for all we know, ....", and then there's no claim. I'm not making the claim you state. I'm simply saying we don't know it's not true. I even put out my opinion that I don't think it's true, but the chessboard thing isn't the alternative. That's even worse. It is a direct violation of all the premises of relativity theory (none of which has been proved).

IMO the final state is simply not defined by the premises of the problem, — fishfry

Spot on, yes.

A supertask is "a countably infinite sequence of operations that occur sequentially within a finite interval of time." — Michael

Yea, I don't know how that could have been lost. I don't think anybody attempted to redefine it anywhere.

The first is that the whole of our world could not be simulated, because the hardware would have to be bigger than the whole (real) world. — Ludwig V

Yes, the world would have to be bounded, probably more than once. Bostrom for instance suggests the detailed simulation be bounded at human brains (all of them). A less detailed simulation of bodies, animals (all animals will apparently be NPCs), purposeful devices and such. Probably at least 5 levels of this, ending with 'everything else' which simulates the stars in the sky and such, more in detail only when purposefully being paid attention to.
This makes it hard to see physics be different here from there since the physics of a thing changes whenever you try to investigate it.

The second is that exact simulation of even a small part of the real world, down to sub-atomic and near-light-speed events could not be constructed, for the same reason.

It has to be done at that level if someone is paying attention to it. But you choose an easy interpretation like Copenhagen, and it's usually only one particle (like the electron being sent through the double slits) that has to be simulated.
Consider smoke detectors, which very much depend on quantum indeterminism to work. Those I suppose can be classically simulated when nobody is observing them closely.

So it would not be possible to simulate the progress of research in physics over the last 100 years or so?

That isn't an isulated system. One could put together an approximation of the state of Earth in 1924 and simulate it from there. That (the setting up of a plausible world) would require for instance a full understanding of physical consciousness and how memories work so that each person is created will a full memory of his past and has no idea that he just came into existence. The people there pushing the view of 'Last Tuesdayism' would be correct without knowing it.

Then you run the simulation for a simulated century and you get some world that differs completely from ours, but if they haven't killed themselves, the physics would likely have developed more or less at pace with our own history. All the names of the main contributors that hadn't been born by 1925 would be different, and the contributions of those that did exist would change.

I think you'll have to say that the hardware of this simulation we live in must be much, much more powerful than anything we can conceive of and that QM and GR are false. No?

Bostrom makes some outlandish suggestions that say otherwise, like for instance that Moore's law will continue indefinitely.
Don't know what you mean by QM and GR being wrong. They're not, but they're not necessarily the physics of whatever is simulating us.

The paradox of the situation is that believers in it have to put more faith in their fancies than in their experience — Ludwig V

You got it. I also see no motivation for our simulators to run this simulation. Bostrom suggests the 'ancestor history' thing, but it wouldn't be our history being simulated, just 'a' history, and a very different one. The only purpose of that might be to see how things might otherwise have turned out. How lucky are we to have survived to the point of being able to put together these simulations?

So I agree: the far simpler model is to presume our experiences are valid evidence of how things are, since the alternative is making up conclusions from zero evidence.

I don't beleive we are in a simulation, but this is my reaction to your points. — Tom Storm

As I've pointed out already, you're speaking to air. jasonm doesn't contribute to his own topics.

If we are a simulation and there is a world outside ours, how would we know what is possible? Since we know nothing of the world outside the simulation, we don't even know if it is done via computers. — Tom Storm

Exactly. Everybody online that pushes something like this presumes unreasonably that the world simulating us has similar physics.

Bostrom's Simulation Argument is that one of these is almost certainly true:

1. The fraction of human-level civilizations that reach a posthuman stage (that is, one capable of running high-fidelity ancestor simulations) is very close to zero, or
2. The fraction of posthuman civilizations that are interested in running simulations of their evolutionary history, or variations thereof, is very close to zero, or — Michael

I find both these to be highly unlikely, for the reason stated in this topic and mine. Bostrom of course has motivation to rationalize a higher probability for both of these, but rationalizing is not being rational.

OK. Is that because [points] have no dimension - are not a part of the line? — Ludwig V

They are part of the line. Yes, a point is dimensionless, size zero. Any sum of a finite bunch of zeros is zero. But the number of points on a line segment isn't finite.

Ok. Perhaps you and Michael could hash this out. He thinks supertasks are metaphysically impossible — fishfry

Perhaps he does, but he fallaciously keeps submitting cases that need a final step in order to demonstrate the contradiction. I don't.

I say they're conditionally physically possible, but the condition is unreasonable. There seems to be a finite number of steps involved for Achilles, and that makes the physical case not a supertask. I cannot prove this. It's an opinion.

Do you have a hard time with 0 being the limit of 1/2, 1/3, 1/4, 1/5, 1/6, ...? It's true that 0 is not a "step", but it's an element of the set {1/2, 1/3, 1/4, 1/5, 1/6, ..., 0}, which is a perfectly valid set. — Ludwig V

I have no problem with any that.

You can think of 0 as the infinitieth item, but not the infinitieth step.

OK, that's probably a problem. It is treating something that isn't a number as a number. It would suggest a prior element numbered ∞-1.

Even if space is continuous, we can't cut it up or even sensibly talk about it below the Planck length.

But you can traverse the space of that step, even when well below the Planck length.


In math? Via the standard limiting process. In physics? I don't know,
— fishfry
In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation.
— noAxioms

Yes ok ... math and physics are human inventions that bear some mysterious relation to reality. I agree with that, if that's what you meant.

The closed unit interval [0,1] has a first point and a last point, has length1, and is made up of 0-length points.

So it does. Zeno's supertask is not a closed interval, but I agree that closed intervals have first and last points.

I'm not sure I fully understand. Forgive me, but are these simulations not the ones where they put crash test dummies in a model of car and ram it into a brick wall? How is that not crashing actual cars?

Or do you mean studying thr aftermath of incidental crashes on the road? Not sure how often this actually happens as there would be a lot of legal red tape with ongoing investigations into real victims. — Benj96

No, none of those cases are examples of simulations. Yes, they're are crashing real cars. I'm talking about a computer model of a car crashed into a virtual brick wall, another car/truck, whatever... Yes, those simulations have occupants in them. Much of the point of the simulation to to find a design that best protects those occupants. The auto industry has huge computers dedicated to doing this sort of thing continuously.

I have myself run plenty of simulations, but not being in the auto industry, most of mine didn't have living things simulated in them.

Perhaps I am wrong about determinism tho. I always figured if variables were fully predetermined then the outcome would be invariably predetermined and fully predictable.

That's what determinism means, yes. I don't think 'predetermined' is a distinct concept from 'determined'.

There are valid interpretations of physics that are fully deterministic: Relativity theory, Bohmian mechanics come to mind. There are interpretations that are not deterministic, such as RQM or Copenhagen interpretations. Bottom line: jury is out on the subject.

You'd think a simulation of reality would choose to simulate one of the deterministic models, but if I was tasked with implementing one, I would choose the nondeterministic ones since it is far less work. But it means things occurring without cause, such as the decay of some unstable particle.

I figured that nothing is fully predetermined in real life experiment because there is almost certainly extraneous variables interacting to make the outcome for example 1+1 + X variable + Y variable + Nth variable = 2?

It is unpredictable because the initial conditions of the system fundamentally cannot be known, but given a deterministic model and perfect initial conditions, the (closed) system will do the same thing every single time.

That highlights a different issue with simulations: No system can be closed, so it is at the boundaries of the non-closed systems where one looks for the evidence of being simulated. The car crash thing is usually a closed system. There is no environment. There's the car and its target, and sufficient road to run the scenario. Nothing else.

Do you mean that no-one living in our world could create a simulation of our world? — Ludwig V

Of course not. There would for one be a need for more data than there is medium on which to store it. You you need to simulate a small system, with far less effort put into simulation of the interaction of that small system with the part outside the system.
For example, imagine an atomic simulation of a cc of water just sitting there in a tube, doing that under MWI, a hard deterministic model. I don't think any technology could simulate the water at that level for even one second. So you cut corners and don't simulate at that level unless something intentionally is paying attention to that level. Any you choose something like Copenhagen which is easier to simulate.

That's just a posh way of saying that the battle seems real to those in the simulation.

OK, 'seems' is a better word. But to us, we typically presume reality to be whatever 'seems' real to us without explicitly defining it that way.
Heck, that's why I favor a relation interpretation, which explicitly says exactly that. X is real to Y. Being real is a relation, not a property, so by that definition, the battle IS real to those partaking in it.

Reality, by definition, is not "in" the simulation, but outside it.

By another definition (one very appropriate for this topic, yes), I agree. Reality might not be the world simulating us. We might be 27 levels down, but there's a base reality up there (as is typically presumed), and that one is 'the reality' by the definition implied by a topic like this.

The OP has some problems. Most importantly, he has a reputation for posting an OP and then never returning to the topic. Hence I won't bother with direct replies.

He doesn't seem to know the difference between the simulation argument (Bostrom is a good example of this) and a virtual reality argument (the Matrix is the typical example). The difference is spelled out in my fairly recent topic here https://thephilosophyforum.com/discussion/15060/what-is-simulation-hypothesis-and-how-likely-is-it

Almost all of the arguments in favor of such things posit violations of the laws of physics. The VR hypothesis does by definition. Any simulated body being controlled by a real eternal mind/person operates under different physics than bodies that are not (called NPCs by the VR community).
Bostrom similarly suggests that physical law changes depending on the simulation's determination of intent, which means that the simulation is tasked with gleaning meaningful intent out of particle motion.

If this world is simulated, the "real" world must be very like this one - as in the "Matrix" — Ludwig V

Not true. We would have zero empirical access to the level that is running the simulation, so we can know nothing about it. It might not be a 3 dimensional space world with physics as we know it. That's kind of likely actually since our physics cannot be self-simulated. At the classical level, maybe, but not beyond that.

Therefore, everything cannot be a simulation. — jkop

Good argument, but nobody asserted that 'everything is a simulation'. The argument still is valid that if we're 'probably' simulated, and if the simulating world is similar to ours, then they're also 'probably simulated'. But that's a lot of 'if's.

In a VR, the frog, and even yourself is just a collection of colored shapes, as you put it. A VR is nothing but a simulated experiential feed to a real external entity (you). You would have no way of knowing if other people (or the frog) are similarly avatars under external control, or if they're just part of the simulation (an NPC).
Yes, the apparent frog would have to apparently attempt to evade capture for it to be convincing. A simulation that doesn't do that would be simulating physics contradictory to the physics being presented to us.

if the simulation (e.g an emergent property within a network of electrical circuits) — jkop

Just FYI, there are countless ways to run simulations. Networks of electrical circuits is but one, and those might not even be a thing in the world simulating us.
There was a suggesting that it be done via domino chains falling down. That's a tough one, but I couldn't falsify the suggestion that a Turning machine can be made thus from dominos, so it apparently works. That means (presuming physicalism) consciousness can arise from falling dominos.

If the universe is simulated or in part simulated, it doesn't make it any less real, it just means the product of the universe came about through non-conventional means — Barkon

Agree with this, but not sure what conventional is here. Adding a more fundamental layer to the model, especially a more complicated one, just makes the problem harder, very similar to positing that God created it all. The god is harder to explain than the simpler universe.

I agree that being simulated doesn't really make anything 'fake'.

"Universe" is a bit slippery here. If it means "everything that exists", — Ludwig V

Definitions vary. In this topic, it is helpful to say 'world'. We are one world, and the level simulating us is another. Maybe they're simulating a bunch of them and we are running several simulations of our own. Those are all different worlds, all part of one 'everything that exists', which is a defintion I never liked anyway.

The idea of "real" is also slippery here - or better, it's meaning is contextual. A simulation of a battle isn't a real battle, but it is a real simulation — Ludwig V

The battle is real to those in the simulation, but not real to those running the simulation.

If our world is a simulation, violations of the laws of physics would be bugs. — Lionino

Apparent violations would be bugs. Actual violations are seemingly necessary, to the point where I've never seen a hypothesis that didn't suggest fully consistent phsical laws. For instance, do we simulate the quantum interactions between a pair of protons in a star in some other galaxy? Or do we just simulate an occasional photon reaching Earth?

Minds/consciousness can't come from matter, therefore simulation theory is false. — RogueAI

So the alternative has been falsified? News to me.

If a simulation is wholly deterministic, there is no added value to run it in the first place. For all variables throughout the simulations play are already known by the creators. — Benj96

Lionino correctly points out the error here. Deterministic doesn't mean predictable. Simulations are run today precisely for the purpose of learning something unknown despite being fully determined. Car crashes are a great example of this, a far more cost effective method of testing automobile designs than crashing actual cars.

-----
Again, all this is covered in my other thread linked above.

The problem I was trying to point out that is that, if we admit a ∞-th step, this step should be associated with a state in one of those mechanisms Michael made up. — Lionino

Michael's mechanisms (some of which he made up) are not resolved by addiing a single step task to the supertask. The supertask reaches 1 when all the steps are completed. It isn't sort of 1, it's there since where else would it be? The arguments against that suggest some sort of 'point immediately adjacent to, and prior to 1', which is contradictory. There are no adjacent points in continuums.

I agree with fishfry that there is no step that gives us 1 since by definition, any given step gets us only halfway there
— noAxioms

But I don't agree that 1 is not reached by the completion of the supertask. Only that 1 is not reached by any step.

I take it you are talking about physical space, not mathematical space? — Ludwig V

Yes. 'Planck [pretty much anything] is a physical concept, not a mathematical one. In mathematics, there is no number smaller than can be meaningfully discussed.

But there are 3-dimensional figures in physics, aren't there? It's the solidity that's the problem, isn't it?

Sure. A rock, at a given time, is a 3 dimensional thing. A rock, it's entire worldline, is a 4 dimensional thing. Correct. It isn't a solid. You can measure a piece of it at a sort of 4D 'point', an event. The rock worldline consists of a collection of such point events, a huge number, but not infinite. They're not really points since position and momentum cannot be both known, so you can know one or the other or an approximate combination of both.

One can measure or calculate the length of a circumference, can't one? Or is uncountability a consequence of the irrationality of "pi"?

Yes, one can calculate the circumference. No, the irrationality of pi is irrelevant. It could be a line segment of length 1. You know the length, and it isn't irrational, but the segment still consists of an uncountable number of points. There's no part of the segment that isn't a point (or a set of them), and yet points have no size, so no finite number of them can actually fill a nonzero length of that segment.

Just checking - by "step" do you mean stage of the series. If I am travelling at any spead, I will complete more and more steps in a given period of time, and that number (of steps) will approach (but not reach) infinity.

Yes, a step is a finite (nonzero) duration, like the first step is going halfway to the goal. Each step goes half the remaining way to the goal. Those are steps. You complete all the steps by time 1, so the task is then complete. No contradiction so long as we don't reference 'the highest natural number' which doesn't exist.

So is the cutting up of the path into standard units. It's just a question of choosing the appropriate mathematical calculation for the task at hand.

One must define how the task is divided into steps in order to tell Zeno's story. There are multiple ways to do it, but to be a supertask, the steps need to get arbitrarily small somewhere, and the most simple way to do that is at the beginning or the end of the task. How one abstractly divides the space has no effect on the actual performance of the task. One can argue that all tasks of any kind are supertasks because one can easily divide any finite duration into infinite parts, but the much of the analysis of doing so relies on the mathematics of countable infinities.

So I can go from 0 to 1 and assign a 'step' to every zero-duration point between those limits. That can be done, and can be completed, but since the steps are not countable, it is hard to draw any conclusion from it all.

Then you say. — Lionino

That's me saying something, not fishfry.

I personally don't like the ∞-th step, but it works. The supertask is completed, then the ∞-th step is taken after that. The supertask had all nonzero duration steps, and this additional step has no duration. I don't find it wrong, but I find it needless.

Is there not a contrast between these two sets of statements?

I agree with fishfry that there is no step that gives us 1 since by definition, any given step gets us only halfway there. If fishfry wants to add an addition single step after the supertask completes, that's fine, but it isn't a step of the supertask.

I don't see how you could count all the natural numbers by saying them out loud or writing them down. Is this under dispute? — fishfry

No. Nobody seem to have suggested that was possible. It simply isn't a supertask.

Do you mean the fact that I can walk a city block in finite time even though I had to pass through 1/2, 3/4, etc? I agree with you, that's a mystery to me.

Yes, I mean that, and it's not a mystery to me. If spacetime is continuous, then it's an example of a physical supertask and there's no contradiction in it.

The lamp could turn into a pumpkin too.

No, the lamp changes things. It introduces a contradiction by attempting to measure a nonexistent thing. That in itself is fine, but the output of a non-measurement is undefined.

I looked up [Bernadete's Paradox of the God], didn't seem to find a definitive version.

Nicely stated by Michael in reply 30, top post of page 2 if you get 30 per page like I do.

Ah the ping pong balls. Don't know. I seem to remember it makes a difference as to whether they're numbered or not.

It's important to the demonstration of the jar being empty, so yes, it makes a difference.

If you number them 1, 2, 3, ... then the vase is empty at the end, since every ball eventually gets taken out. But if they're not numbered, the vase will have infinitely many balls because you're always adding another 9. Is that about right?

The outcome seems undefined if they're not numbered since no bijection can be assigned, They don't have to have a number written on them, they just need to be idenfifed, perhaps by placing them in order in the jar, which is a 1-ball wide linear pipe where you remove them from the bottom.

It nicely illustrates that ∞*9 is not larger than ∞, and so there's no reason to suggest that the jar shouldn't be empty after the completion of the supertask. Again, it seems that any argument against this relies on a fallacious assumption of a last step that sooo many people are making in this topic.

So I believe I've been trying to get across the opposite of what you thought I said. There is an ∞-th item, namely the limit of the sequence. — fishfry

That can't be a step, since every step in a supertask is followed by more steps, and that one isn't. I have a hard time with this ∞-th step.

The common explanation that calculus lets us sum an infinite series, I reject. Because that's only a mathematical exercise and has no evidentiary support in known physics. — fishfry

The cutting up of the path into infinite steps was already a mathematical exercise. The fact that the physical space can be thus meaningfully cut up is true if the space is continuous. That latter one is the only barrier, since it probably isn't meaningfully, despite all our naïve observations about the nice neat grid of the chessboard.

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? If not, how is the walk ever completed — Lionino

As has been stated so many times, by performing all the steps, which happens in finite time no problem. There is a final step only in a finite sequence, so using a finite definition of 'complete' is inapplicable to a non-finite task.

In math? Via the standard limiting process. In physics? I don't know, — fishfry

In physics, the same way as math, except one isn't required to ponder the physical case since it isn't abstract. One completes the task simply by moving, something an inertial particle can do. The inertial particle is incapable of worrying about the mathematics of the situation.

Physics doesn't support these notions since we can't reason below the Planck length.

Which is to say that space isn't measurably continuous, so the walk isn't measurably a supertask. I would agree with that.

How do dimensionless points form lines and planes and solids? — fishfry

Mathematics: by not having a last one (or adjacent ones even). Physics: There are no solids.

(But the converging series does not consist of points, but of lengths, which are components.) — Ludwig V

Yes. The latter is a countable set of lengths. The set of points on say a circle is an uncountable set

A robot cannot decide whether or not to make the call, a person can. — Metaphysician Undercover

That's quite the assertion. Above and beyond the usual conservative stance.

The point of my example with the ship was to counter your assertion of Newton forces not being necessary to move and free will being enough. I said you'd need help from Newton. Asking for a line to be thrown to you is you admitting the help from Newton was necessary. That's what the tether is: a way to do it by exerting an external force, since the free will couldn't do it itself.

Assuming at the most microscopic level, the object is on an 8x8 chessboard. The pawn moves from e2 to e3. There is no e2.1 or other smaller increments in this finite world. At T1 it's at e2 and T30 it's at e3. The assumption is that at some point in time, it was no where while transitioning (moving?) from e2 to e3. — Hanover

I discussed that in my post, but you quoted the bit at the bottom which abandons the chessboard model in favor of quantum mechanics, calling the former model a naïve

What empirical evidence is there that observations have been made of there being no object for some length of time and then it suddenly reappearing?

None, but there's also no evidence that it is there when not being measured. It's all about measurement and not about discreetness.

If it's at L-1 at T-1 and L-2 at T-2, how long did it take to get from L-1 to L-2? — Hanover

In that frame, it took time 1 to get from T-1 to T-2. That's pretty obvious, no? In natural units, that's light speed.

If the answer is 0, then it was at L-1 and L-2 at the same time because if T-2 minus T-1 = 0, then T-1 = T-2.

If the answer is zero, then T-2 is no-t when it is at L-2.
In computer jargon, what you are describing is 'jaggies', the tendency of 'straight' lines to appear jagged when displayed on say your computer screen, a discreet array. An object that moves fast (faster than one L per T) will either be at multiple locations at the same time, or it will skip all the locations between and only be at one location per time.
I've played a game with the latter physics. I could get my ship to go super fast and go straight for the enemy blocking my way. If I did it right, I would be in front of him at one time unit, and beyond him the next time unit, apparently passing right through without collision because there was no time 1.5 where I was where he was.

More problems with that model: If the particle is moving at 0.7 per time unit, it is never at a location in space except every 10 time units where you find it 7 units from where it was before. It can't be anyplace between since it is never at a space location at the same time as a time quanta. This is silly. You probably need to fill in the dots between, but then the motion is erratic rather than sporatic.

I'm only asking how far 1,1 is from 1,2 in a discrete space system. As far as I can tell, it's 0 units, right? — Hanover

No, they're 0,1 from each other, which isn't zero. One of the coordinates is different.

Anyway, you seem to see the sorts of contradictions that arise from such a naive model. If space and time is discreet, quantum mechanics describes it far better than the chessboard model.

the walk only finishes if it accomplishes an infinite amount of steps. Right? — Lionino

Right

If it is indeed accomplishing an infinite amount of steps, is there not a step where the sequence gives us 1? If not, how is the walk ever completed?

By completing all the steps. This is not a contradiction.

if so, is there not a corresponding state for the mechanism when the full time elapses?

Not any more than there is a last natural number. I'm presuming you're talking about the state of something like the lamp. The state of Achilles is easy: He's where the tortoise is.

I don't see a problem until the premise of a last step is introduced, which is by definition contradictory.

given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, — Michael

They're both incomplete, just like
Newtonian mechanics was incomplete, but not false. OK, parts of it were outright false, but it's still taught in (pretty much) any school. GR definitely breaks down at small scales.

No one said free will has infinite capacity? — Metaphysician Undercover

I didn't say infinite capacity. I denied that your free will has any capacity at all, since even the most trivial capacity would get you back to your ship 2 meters away, even if not quickly.

I think, and then I do. The "force" which moves me comes from within me, and therefore cannot be described by Newton's conceptions of force. — Metaphysician Undercover

The spaceship example shows this to be nonsense. It would be a revolution indeed if anybody could do that.

a radio call to someone inside the spaceship, to please shoot me a line, might help. That demonstrates the benefit of free will — Metaphysician Undercover

Free will isn't necessary to do any of that. A robot has the same capacity to make such a call, and robots by definition lack it. This is also utterly off topic to this discussion, but I took the easy bait anyway.

If I stand in a parking lot and call out "one, two, three, ..." and keep going .. — fishfry

OK, that other meaning of 'count'.

I think we're talking past each other. When asked for the difference between a mathematical and physical supertask, you seem to focus on two different definitions of countable: The assignment of a bijection, and calling or writing down each of the numbers.

I'm talking about a physical supertask as described by Zeno, which arguably has countably (first definition) steps performed in finite time. Nobody is posited to vocalize the number of each step as it is performed.

It's just like Cinderella's coach. It's a coach at midnight minus 1/2, midnight minus 1/4, etc. At at exactly midnight, it turns into a coach.

Bit off on the lore. It turns into a pumpkin, and at the 12th stroke, where presumably midnight is the first stroke, but I googled that and could not find an official ruling on the topic.

The Planck-scale defying lamp circuit is every bit as fictional as Cinderella's coach. Since the state at 1 is not defined, I'm free to define it as a plate of spaghetti. That's the solution to the lamp problem.

No argument. That seems to be a valid way out of most attempts to assign a count to the nonexistent last/first step, or to simply assert the necessity of the nonexistent thing.

I like Bernadete's Paradox of the Gods because it doesn't make those mistakes, and thus seems very much paradoxical since motion seems prevented by a nonexistent barrier.

For educational purposes concerning how infinity works, I like Littlewood-Ross Paradox because it is even more unintuitive, but actually not paradoxical at all since it doesn't break any of the above rules. It shows a linear series (effectively 9+9+9+...) being zero after the completion of every step.

If I could, say, produce an equation based on the one in my earlier post that could calculate the last time interval given a smallest stipulated chunk of time, would that be a valid final step in the summation? — ToothyMaw

If you stopped the summation there, then yes, there would be a final step, but it wouldn't have infinite steps defined then. It wouldn't be a supertask.

And would that sum not eventually terminate given a smallest sliver of time exists

If there's a smallest quanta of time, then there can be no physical supertasks.

I think, and then I do. The "force" which moves me comes from within me, and therefore cannot be described by Newton's conceptions of force. — Metaphysician Undercover

LOL. Tell that to the guy stranded 2 meters from his space ship without a tether. No amount of free will is going to get you back to it. You're going to need a little help from Newton.

The use of "physical" in this thread has gotten so ambiguous, that equivocation abounds everywhere. — Metaphysician Undercover

Yea, I noticed.

How much time elapses from travel to point a to point b and where is the object located during that time lapse? — Hanover

I'll attempt this. Michael talks about motion from A to B without there being a between. This can happen two ways.
1) Space is quantized. There simply isn't a location halfway between A and B. For a slow particle, this might mean that it spend quite a bit of time at A, and then suddenly is at B. That seems rather contradictory since one might ask what changes during all those times when the thing was at A. If it is at A twice, it is stationary and has no obvious state anywhere to go to B after some nonzero time.

2) Time is quantized, which is troublesome for fast particles. You have time 1 where the thing is at A, and time 2 where it is at B, quite a ways away. There are valid locations between A and B that the particle never visited since the time it should have been there is nonexistent.

All the above sort of presumes a naïve finite-automata sort of view of a quantized space and or time. It presumes a particle has a location (A, B) at a given state and time. Well that presumption was pretty much thrown out of the window with quantum theory.
A more realistic answer to your question comes from there. It says that you measure the particle at A, and later at B (maybe hours later). Where was the particle between those times? If not measured, it doesn't have a location. It does exist, but needs to be measured to have a location or (not and) a momentum.

what maintains its identity during that interval?

Physics has no concept of identity of anything. It is a human convention, a pure abstraction. Any given convention seems falsifiable by certain examples.

a set like N = {30, 15, 15/2}? Does that not include a first step? — ToothyMaw

Yes, that series has a first step, but not a last one. You can number the steps in the series if you start at the big steps. Similarly, you can number the dichotomy steps in reverse order, since the big steps are at the end.

And would that sum not eventually terminate given a smallest sliver of time exists

If there's a smallest sliver of time, there is no bijection with the set of natural numbers since there are only a finite number of steps.

or continue indefinitely given time is infinitely divisible?

'Continue indefinitely' is a phrase implying 'for all time', yet all the steps are taken after only a minute, so even if time is infinitely divisible, the series completes in short order.

Can we not count the intervals starting with 1 — ToothyMaw

No. In the dichotomy scenario, there is no first step to which that number can be assigned.

To count a set means to place it into bijection with: — fishfry

OK, that meaning of 'count'. In that case, I don't see how mathematical counting differs from physical counting. That bijection can be done in either case. In the case with the tortoise, for any physical moment in time, the step number of that moment can be known.


I am saying that Zeno describes a physical supertask, that Achilles must first go to where the tortoise was before beginning to travel to where the tortoise is at the end of that prior step.
Zeno goes on to beg the impossibility of the task he's just described, so yes, he ends up with a contradiction, but not a paradox.

Depends on the exchange rate. — fishfry

I also would hate to have to talk about the poor kilometerage that Bob's truck gets.

It [the even-oddness of ω]is neither, and who's asking such a thing? — fishfry

The lamp scenario asks it, which is why the comment was relevant.

Some supertasks are coherent and consistent, therefore logically logically possible. In this case, that is the proof that they are "possible" — Metaphysician Undercover

I think the person to whom I was replying was suggesting that somebody had asserted a proof that a physical supertask was possible. But I did not recall anybody posting such an assertion.

I have more or less dropped out due to the repetitive assertions not making progress, but thank you for this post.

the set {1/2, 3/4, 7/8, ..., 1} — fishfry

Interesting. Is it a countable set? I suppose it is, but only if you count the 1 first. The set without the 1 can be counted in order. The set with the 1 is still ordered, but cannot be counted in order unless you assign ω as its count, but that isn't a number, one to which one can apply operations that one might do to a number, such as factor it. That 'final step' does have a defined start and finish after all, both of which can be computed from knowing where it appears on the list.

This is not radical. The rational numbers are countable, but not if counted in order, so it's not a new thing.

If Zeno includes 'ω' as a zero-duration final step, then there is a final step, but it doesn't resolve the lamp thing because ω being odd or even is not a defined thing.

and we inquire about the final state at ω

Which works until you ask if ω is even or odd.

Using mathematics to try to prove that supertasks are possible is a fallacy. — Michael

Totally agree, but I'm not aware of anybody claiming a proof that supertasks are possible. Maybe I missed it.

Well physics is of course exempt from math and logic. The world does whatever it's doing. We humans came out of caves and invented math and logic. The world is always primary. Remember that Einstein's world was revolutionary -- overthrowing 230 years of Newtonian physics. — fishfry

The relativity thing was more of a refinement and had little practical value for some time. Newtonian physics put men on the moon well over a half century later.
QM on the other hand was quite a hit, especially to logic. Still, logic survived without changes and only a whole mess of intuitive premises had to be questioned. Can you think of any physical example that actually is exempt from mathematics or logic?

QM is also the road to travel if you want to find a way to demonstrate that supertasks are incoherent.
Zeno's primary premise is probably not valid under QM, but the points I'm trying to make presume it is.

in math I can invoke the axiom of infinity, declare the natural numbers to be the smallest inductive set guaranteed by the axiom, and count it by placing its elements into order-bijection with themselves. The former is a physical activity taking place in the world and subject to limitations of space, time, and energy. The latter is a purely abstract mental activity.

What is this 'the former'? The physical activity of making a declaration? There's definitely some abstraction going on there, as there is with any deliberate activity.
The latter seems to be the expression of a rule that maps the two halves of the bijection, which seems to be about as physical of an activity as was the declaration.

if thoughts are biochemical processes; are not our thoughts of infinity a kind of physical manifestation?

No argument here.

So bottom line it's clear to me that we can't count the integers physically

Depends on what you mean by count, and especially countable, since plenty of equivocation is going on in this topic.
If you mean mentally ponder each number in turn, that takes a finite time per number, and no person will get very far. That's one meaning of 'count'. Another is to assign this bijection, the creation of a method to assign a counting number to any given integer, and that is a task that can be done physically. It is this latter definition that is being referenced when a set is declared to be countably infinite. It means you can work out the count of any given term, not that there is a meaningful total count of them.

but we can easily count them mathematically

Sorry, but what? I still see no difference. What meaning of 'count them' are you using that it is easy only in mathematics?

And the reason I say that we can't physically do infinitely many things in finite time "as far as we know," is because the history of physics shows that every few centuries or so, we get very radically new notions of how the world works.

That doesn't follow at all since by this reasoning, 'as far as we know' we can do physically infinite things.
I never made the claim that a supertask is physically possible. I simply followed through with it as a premise, which, unless falsified, can be physically true 'as far as we know'.

Nobody can say whether physically instantiated infinities might be part of physics in two hundred years.

They've been a possibility already, since very long ago. It's just not been proven. Zeno's premise is a demonstration of one.

You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so?
— noAxioms

Not a flaw, of course, any more than general relativity revealed a flaw in Newtonian gravity. Rather, I expect radical refinements, paradigm shifts in Kuhn's terminology, in the way we understand the world. Infinitary physics is not part of contemporary physics. But there is no reason that it won't be at some time in the future. Therefore, I say that supertasks are incompatible with physics ... as far as I know.

We split the atom, you know. That was regarded as a metaphysical impossibility once too.

QM does very much suggest the discreetness of matter, but Zeno's premise doesn't rely on the continuity of matter. It works best with a single fundamental particle moving through continuous space and time, and overtaking another such particle.

The next shift just may well incorporate some notion of infinitary set theory; in which case actual supertasks may be on the table.

They were never off the table since current physics doesn't forbid them. Maybe future physics will for instance quantize either space or time (I can think of some obvious ways to drive that to contradiction). Future findings take things off the table, not put new ones on. The initial state of physics is "I know nothing so anything is possible'.

I analogize with the case of non-Euclidean geometry; at first considered too absurd to exist

Heh, despite the detractor standing on an obvious example of such a geometry.

then when shown to be logically consistent, considered only a mathematician's plaything, of no use to more practical-minded folk; and then shown to be the most suitable framework for Einstein's radical new geometry of spacetime.

Octonians shows signs of this sort of revolution.

eternal inflation. That's a theory of cosmology that posits a fixed beginning for the universe, but no ending.

Actually, the big bang theory already does that much.
Yes, I know about eternal inflation, and something like it seems necessary for reasons I gave in my prior post.

Physicists are vague on this point, but if time is eternally creating new universes, why shouldn't there be infinitely many of them.

It is a mistake to talk about 'time creating these other universe'. Time, as we know it, is a feature/dimension of our one 'universe' and there isn't that sort of time 'on the outside'. There is no simultaneity convention, so it isn't meaningful to talk about if new bubbles are still being started or that this one came before that one.

All that said, the model has no reason to be bounded, and infinite bubbles is likely. This is the type-II multiverse, as categorized by Tegmark. Types I and III are also infinite, as is IV if you accept his take on it. All different categories of multiverses.

And two, the many-world interpretation of quantum physics.

That's the type III.

In Everett's many-world's interpretation, an observation causes the thing to be in both states.[/quote]Ouch. Is that a quote? It did not match any google search.
Observation for one is a horrible word, implying that human experience of something is necessary for something fundamental to occur. This is only true in Wigner interpretation, and Wigner himself abandoned it due to it leading so solipsism.

In some other universe I didn't write this. I know it sounds like bullshit,

I don't buy into MWI, but bullshit is is not. It is easily the most clean and elegant of the interpretations with only one simple premise: "All isolated systems evolve according to the Schrodinger equation". That's it.

These are just two areas I know about in which the idea of infinity is being taken seriously by speculative physicists.

Everett's work is technically philosophy since, like any interpretation of anything, it is net empirically testable.
I would have loved to see Einstein's take on MWI since it so embraces the deterministic no-dice-rolling principle to which he held so dear.

Well I can walk a mile

Ah, local boy. I am more used to interacting with those who walk a km. There's more of em.

But let me riddle you this. Suppose that eternal inflation is true; so that the world had a beginning but no end, and bubble universes are forever coming into existence. — fishfry

That wording implies a sort of meaningful simultaneity that just doesn't exist.

And suppose that in the first bubble universe, somebody says "1".

The universes in eternal inflation theory are not countable.

Yes, each step in a supertask can and does have a serial number. That's what countably infinite means.

P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum. — Michael

You're not going to get past step 10 at best. I just takes longer than the step duration to recite a syllable. I don't think this is your point, but it's a poor wording due to this. Yes, step 13 has a defined duration at known start and stop times. The duration simply isn't long enough to recite anything.

P2. 30 + 15 + 7.5 + ... = 60

C1. The sequence of operations1 described in P1 ends at 60 seconds without ending on some final natural number.

But given that ad infinitum means "without end",

No. It means 'without final step'. You're apparently equivocating "without end" to mean that the process is incomplete after any amount of time.

What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?

There we go with the finite definition again.
"The sequence of operations ends" means that "all operations in the sequence are performed".

This is a great example of the endless repetition of assertions/bad-definitions I'm seeing in this topic. Surely you know this answer is coming from me.

Calculating the limit does not entail a process that reaches that limit. This is a misinterpretation of the concept of limit.This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value... — Relativist

Good source. It says that the limit is approached as the input approaches the specified value.
This means that the limit isn't reached at some finite point in the series, exemplified by the comment:
"This means that the value of the function f can be made arbitrarily close to L, by choosing x sufficiently close to c"
The approaching goes on while x is still at some finite step.

Since x reaches infinity at time 1, all steps are completed at that time, so the task is complete.

Are you saying that you believe that there would still be an April 29, even if there never was any human beings with their time measuring techniques, and dating practises? — Metaphysician Undercover

Read carefully. I didn't say that.

I said
1) that discussion of the question above and your personal beliefs in the matter is off topic
and
2), that you [cannot / choose not to] understand what others mean when they presume what Michael conveyed better than I could:

That we coin the term “X” to refer to some Y isn’t that Y depends on us referring to it using the term “X”. This is where you fail to make a use-mention distinction. — Michael

You (M-U) seem to either not be able to separate "X" and Y, or you refuse to communicate with those that do.

And do you believe that...

My personal beliefs in this matter are irrelevant. I simply know what somebody means when they treat Y as something independent of "X".

I agree that it's impossible to do infinitely many physical thinks in finite time according to present physics. — fishfry

What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something.

You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so?

I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time. By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask.

We seem to be talking in circles, with all logic from the 'impossible' side being based on either there being a last infinite number, or on non-sequiturs based on the lack of said last number.

The goal is not unreachable. That simply doesn't follow from arguments based on finite logic, and it is in defiance of modus ponens. It's just necessarily not reached by any specific act in the list.
— Relativist

There is a bijection yes. It does not imply that both or neither completes.
— noAxioms
Why not?

You defined the second task as a non-supertask, requiring infinite time. That's why not.
I can play that game with a finite list of three steps, with the middle step of one task requiring one to make a square circle. It does not follow that the other list of three steps cannot be completed in a short time just because there exists a bijection between the steps of the two tasks.

That's like saying today would be April 29 even if there was never any human beings to determine this. — Metaphysician Undercover

Exactly so.
Your disagreement with views that suggest this is a subject for a different topic. Your displayed lack of comprehension of what the person means when he says things like that is either in total ignorance of the alternatives or a deliberate choice. Being the cynic I am, I always suspect the latter. It's my job as a moderator elsewhere.

I do thank you for verifying my earlier assessment.

I'm not the one advocating for supertasks — fishfry

For the record, I am personally advocating that they have not been shown to be physically impossible. All the 'paradoxes' that result are from inappropriately wielding finite logic in my opinion.
Thomson's lamp is a wonderful example of this, but other examples seem to have more bite.

I would invite you to read up on eternal inflation, a speculative cosmological theory that involves actual infinity. — fishfry

Does it? It seems to be a more complex model that suggests stupid sizes for 'what is', but not 'actual infinite' more than the standard flat model that comes from the cosmological principle. Yes, I know the page you link mentions 'hypothetically infinite' once. I have a deep respect for the eternal inflation model since something like it is necessary to counter the fine-tuning argument for a purposeful creation.

I agree with Michal that the sort of infinity suggested by eternal inflation is not representative of a supertask. I do realize that some people just deny 'actual infinity' of any kind, but that is not justified, hence is not evidence.

No, you didn't. You merely asserted: "The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal." There is no direct violation. — Relativist

1. A given halfway step cannot reach the goal.
2 There is a specific step that reaches the goal (per PSA)
3 Therefore this final step is not a halfway step (1 & 2)
4 Any given step is halfway (per Zeno)

You don't find this contradictory?

Here's valid logic:
1. A halfway step cannot reach the goal.
2. All steps are halfway
3. Therefore the goal cannot be reached.

This shows that no specific halfway step reaches the goal, which is the same as saying that the goal cannot be reached in a finite number of steps.

It seems that every post seems to attempt finite logic on an unbounded situation. If you accept that motion is possible, there is a flaw in at least one of the premises.
— Relativist

You merely asserted the goal is reached (directly contradicting #3) but didn't explain how the sequence of halfway steps somehow reaches the goal.

Yea, I do, don't I? I'm not enough of the mathematician to regurgitate all the axioms and processes involved in the accepted validity of the value of a convergent series. Attack them if you will. The do require some axioms that are not obvious, so there's a good place to start. Nevertheless, I can do more than just handwave, by several unrelated methods.

Demonstration that immediate contradictions arise from denying either of the premises or presuming your conclusion 3 is also more than just handwaving. For instance, given the usual scenario, where is Achilles at time t=1? If he's not at the goal then, then where else is he?

There are those that deny an object falling past the event horizon of a black hole by suggesting that 'time stops' in a somewhat similar manner that some posting here have suggested. But that's just an abstract coordinate effect (and the leveraging of finite logic). Change the coordinate system to one that isn't singular at the point of contention and the object falls in, no problem. Similarly, Achilles is stuck in an abstract sense due to a deliberate choice of coordinate system that is singular at the goal. The impediment is entirely abstract and not physical at all.

Per modus ponens, empirical observation shows that motion is possible, as is the overtaking of a slower object. One need not accept that empirical evidence (keystone attempted this avenue), but I choose to start with acceptance of empirical evidence. There are a few places where it is inappropriate to do so, and this isn't one of them.

Also, no impediment to the reaching of the goal has been identified, so in a similar way, your stance (what is your stance? Supertasks are nonexistent, even given continuous assumptions?) is also achieved by handwaving when it is not just being flat out contradictory. You do seem to heavily rely on definitions that come only from finite logic. A definition that is being leverage outside its range of applicability is

The process does not continue forever, however there is no end to the process.

There is a temporal end to it, a final moment if not a final step.

But this process has a 1:1 correspondence to the supertask -- for every step taken in one scenario, there's a parallel step taken in the other. This suggests that either they both complete, or neither completes.

There is a bijection yes. It does not imply that both or neither completes.
This reminds me of some of the discussion behind Gabriel's horn, and attempting to suggest that its infinite area implies that it has infinite volume.

Yes, your example here very much illustrates how a deliberate abstraction can be made to be singular at any chosen point, in this case tying infinite time to a finite duration. Yes, this works even in uncountable infinities: There is a bijection between the space from 0 to 1 and the space from 1 on up, by the simple relation of y = 1/x. This in no way implies that 1 meter cannot exist.

The number line in question is an interval that is open on the right: i.e. it includes all points <1, but not including 1. There are infinitely many points in this interval, but the point "1" isn't one of them. So the process cannot reach 1, and 1 is the goal of the process.

The 'process' can go beyond the end of the line despite it ending before the goal. This is sort of a different issue since you're putting an uncountable set of points between 0 and 1. Why not just 1/2, 1/4, ...

The goal is therefore unreachable by the kinematic process.

Disagree. The kinematic process isn't restricted to only points on the number line.

Or the PSA is correct, and the goal can't be met. — Relativist

I showed that for a supertask, the PSA is not correct. So no, this cannot be for a supertask.

Why would it matter if the number of steps is infinite? — Relativist

Because a contradiction results from making that additional assertion. In the example given, it is a very direct contradiction.

What does it even mean for a kinematic process to be infinite? My answer: it means the process continues forever and does not end. What's your answer. — Relativist

If the process continues forever, by definition it isn't a supertask. It's a different process than the one being discussed.
I don't have an answer because I don't understand what 'kinematic' adds to the issue.

Points on a number line exist concurrently (in effect). — Relativist

I don't know what is meant by this. 'Concurrently' means 'at the same time' and there isn't time defined for a number line.
A number line seems to be a set of ordered points represented by a visual line. It can be defined otherwise, but functionally that seems sufficient. It being a visual aid, it seems physical, but a reference to the simultaneity of the positions along the line seems irrelevant to the concept.

Steps in a kinetic process do not: they occur sequentially, separated by durations of time. — Relativist

OK. I buy that. But this works mathematically as well, so 'kinetic' doesn't add anything. I can draw the worldlines of Achilles and the tortoise on some medium and all you get is two lines that cross at some point. The axes on the plot are x and t, so in this mathematical representation, the steps do not occur simultaneously, but are separate durations of time. What did 'kinetic' add to that?
I'm trying to understand your point about how the word somehow is relevant.

the Achilles/tortoise problem ... just clouds the issue with the stairway supertask. — Relativist

OK, this has been about the stairway. There is no objective kinematics about that since it involves a space-like worldline, so the steps are not unambiguously ordered in time. The ordering of the steps becomes ambiguous due to relativity of simultaneity, and it becomes meaningless to use the word 'sequential' in this context.

Hence my always referencing the tortoise example since it hasn't any physical ambiguities like that. There are still frame dependent fact, so for instance in another frame, it is the tortoise trying to overtake Achilles, both of whom are facing backwards.

Those are my thoughts on 'kinematics'.

Show the PSA is false. — Relativist

The PSA statement (that there is a step that reaches the goal) directly violates the premise that any given step gets only halfway to the goal.
Either PSA is wrong or the premise is. In neither case is PSA valid for a supertask.

Simply denying a final step is necessary doesn't make it so

Simply asserting that such a step is necessary doesn't make it so, especially when it being the case directly violates the initial premise. That violation does very much demonstrate not only the lack of necessity of a final step, but the impossibility of it, given the premise.

you have to explain why it's not necessary for a kinetic task to require a final step in order to be completed.

I don't know how the task being 'kinetic' changes the argument. You can phrase it as a n inertial object overtaking a slower one in frictionless space.

Issues that I see: The problem is 1 dimensional as phrased: The position of Achilles is given only in x. To overtake the tortoise, he'd have to collide with it, so he has to be off to the side,. If he's off to the side, there's at least two axes x and y. If they're in 2D+ space, then which of the two is in front is dependent on the chosen orientation of the axes. If you hold the orientation stable throughout the exercise, then the scenario still holds as described.

None of that seems to have any relevance to your reqirement of a rephrasing around 'kinetic'. Why does that word somehow invalidate the premise?

And so it is meaningless to claim that such a supertask can complete. — Michael

The lack of a defined number for the last task does not prevent completion (by the all-tasks definition), so I regard your statement as a non-sequitur.
It does prevent completion if completion is defined as the removal of the green ball in the bag of a dozen non-green balls (see green ball example above), so I agree with you there.

Maybe I've misunderstood what a supertask is. Are there not different kinds of cases? — Ludwig V

Several here have been defining completion effectively as measuring the value of the final task, and that instance I suppose differs from Zeno's that specifies no such requirement.

In the case of Achilles, we know that the task can be completed, but it is presented to us in a form in which it cannot be completed.[/quote]Only because he posits a second premise incompatible with the first. 1) Supertasks are possible (by demonstration). 2) Supertasks are impossible, a second premise that isn't in any way justified.

I mean that we know that Achilles will pass the tortoise

Well, keystone suggested that Zeno denies this, and M-U suggests that time somehow stops due to the offense we've given it. Anyway, I agree with you, but it requires that implied premise that empirical evidence is valid.

But then the same problem, presented in a different way, seems to suggest that it cannot.

This suggests fallacious reasoning in the second presentation. Most of the fallacies I've seen posted seem to be based on the premise of there being a limiting step. It's why I like Bernadete's Paradox of the Gods (see post ~30) which explicitly leverages the lack of there being a limiting step, and drives that to a seemingly paradoxical result. That's a harder one to wave off.

The staircase ... gives us a task (going down the infinite stairs) that cannot be completed

Just not physically. Mathematically it can, but then the story mentions 'the bottom' which implies something final that 'no more stairs' does not. So it lacks rigor.


The Littlewood-Ross Paradox illustrates an interesting way to treat infinities that highlights the dangers of treating infinities as numbers. From the SEP supertask page:
"We have a jar and a countably infinite pile of balls, numbered 1, 2, 3, 4, …. First we drop balls 1–10 into the jar, then remove ball 1. (This adds a total of nine balls to the jar.) Then we drop balls 11–20 in the jar, and remove ball 2. (This brings the total up to eighteen.) Suppose that we continue in this way ad infinitum, and that we do so with ever-increasing speed, so that we will have used up our entire infinite pile of balls in finite time. How many balls will be in the jar when this supertask is over?"

The answer is, as is argued by just about everybody: Zero. At any finite step n, there are n*9 balls in the jar. But after the supertask is complete, there is no final step Z with a state of Z*9 balls. In fact, every ball is numbered, and we know when it went in and when it went out. There is no exceptions to this, so the jar is empty at the end. Totally not intuitive, but not necessarily contradictory. Arguments against it have been attempted.

But I'm making the stronger claim that it is logically impossible. — Relativist

I'm trying to get a justification of that claim without the addition of the necessity of a final step, which would by definition be contradictory.

PSA

Has always meant 'prostate specific antigen' to me. I get my PSA checked at least once a year.

Taking a single step is an act. The acts are performed in a sequence (from step n to step n+1)..

OK, 'act' is a step (go half the remaining way to the goal). 'task' is a goal (pass the tortoise).
It makes sense now, thanks.

Doing successive steps does not get you past the tortoise unless the passing of the tortoise is done by one of the steps. That's the same as suggesting a final step, which suggests that infinity is a number. I cannot buy into that PSA statement. It is just a rewording of the 'do the last step' definition of completion, a definition which only works for tasks requiring a finite number of steps.

My point is that the stairs are countably infinite. — Relativist

Countably infinite means that any step can be assigned a number. It does not in any way mean that there is a meaningful count of steps.

[/quote][The physical process of descending stairs] fits this definition:
"a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time."[/quote]Physical (fixed size) stairs are of infinite length, and such a distance cannot be traversed in finite time. If the stairs get smaller as we go, then we get into the physical problem of matter being discreet, not continuous. Hence the steps have a minimum size. That's what I mean about physical stairs not qualifying as a supertask.
We seem to have lost @keystone, and the stairs thing was his. I prefer Zeno's scenario which doesn't seem to be plausibly physical so long as we take a classical continuous view of both time and space.

The article discusses the issue

It does, I'm quite aware. Just not in Zeno's argument.

Max Black (1950) argued that it is nevertheless impossible to complete the Zeno task, since there is no final step in the infinite sequence...

I pretty much quoted exactly Black's remarks just above. Yes, the task is not complete by this finite definition despite every step having been taken, and that final step must be taken for your counter to have a defined value after a minute.

The mathematical series completes, but this is an abstract, mathematical completion. The kinetic activity of descending the stairs does not complete.

Again, the stairs is utterly abstract. There's no kinematics to it. Not so with the tortoise. I can pass the tortoise, thus completing (by the 'all steps' definition) the supertask.

The SEP article leaves it there, but the implication seems clear: the abstract mathematics does not fully account for the kinetic activity.

How does the abstract mathematics not account for the physical ability of me passing the tortoise?

PSA:
The performance of a sequence of successive acts does not complete a particular task unless it is completed by the performance of one of the acts in the sequence.

I cannot parse this. What is an 'act' that is distinct from a 'task'? The word 'sequence' seems to refer to the entire collection.
A 'task' (what, one of the steps??) is not completed by a performance unless 'it' (what, the performance?, the task?) is completed I cannot follow it at all.
I cannot take a bite of an apple unless perhaps the bite taking is completed by the performance of the taking of a bite? Presumably the same bite??

Kindly translate. Perhaps it points out some error I'm making, but only if I can parse it. I can come back to your statement and respond more intelligently.

That's what I see going on with the posters who focus only on the mathematical series.

I'm trying to focus on the completion of all tasks and not on the measurement of a nonexistent value.

I agree we can't treat infinity as a number, and haven't suggested you should.

But I think you have. Your attempted counter (or the color change thing in the recent post) treats it as a number, and suggests taking its modulus relative to base 10 or 3. What is the lowest digit of the number of the final step? If there is no such number, then the output of your scenario is undefined, which is very differnt from the digit counter displaying a value of 'undefined', or an undefined lamp state somehow violating the law of excluded middle by being in some state between on and off.

But for the supertask to be meaningful, you have to identify where infinity fits in the kinetic task description. I'm saying it entails a never-ending sequence of tasks. Identifying the limit doesn't make this disappear.

Infinity means unbounded, which means there is a physical location and time interval.for any task n That's what makes it meaningful, and it only works if physicality is presumed not discreet.
Also, there is no violation of physics like faster-than-light movement as suggested by the OP.

I'll add that supertask scenarios actually are NOT coherent- because they entail a contradiction.

I can pass a tortoise without contradiction. That shows that at least one of three (two explicit, one implicit) premises are false. But it doesn't necessarily have to be the premise you just mentioned there, that supertasks are impossible.

You seem to be avoiding the contradiction by ignoring the incompleteness of the infinitely many kinematic steps. The presence of the contradiction implies supertasks are logically impossible (not merely physically impossible).

I'm ignoring it because those contradictions arise from a 4th premise (that there is a final step), one which I don't accept.

What puzzles me is why they are not dismissed out of hand. — Ludwig V

Why is the passing of a tortoise necessarily not a supertask, as described by Zeno, and given a presumption of continuous physics?

A white box turns red when the Earth completes a half-orbit, turns blue when it completes another quarter-orbit, turns back to white when it completes another eighth-orbit, and so on.

What colour is the box when the Earth completes its orbit around the Sun? — Michael

Undefined by the description. That is to say, the color of the box afterwards is not a defined thing, which is different than it displaying the color of 'undefined'.

If someone would explain to me, in a way which makes sense, a better perspective, then I'd happily switch. — Metaphysician Undercover

Michael did very nicely with his first line in his reply.

Your reading comprehension skills are also off. I never suggested converting you to some opinion other than the one which you hold. I simply suggests that you seem incapable of understanding alternatives, to the point where you don't understand people who presume one of these alternatives.

Your most recent reply demonstrates this, as does the frustration evident in Michael's reply just above.
Yes, it can be explained in a way that makes sense, but apparently only to others.

if a physical process ends, there has to be a final step.
— Relativist
This is equivalent to asserting that 'infinity' is the largest integer.
— noAxioms
Wrong. The statement applies universally to the physical process of descending stairs. — Relativist

The physical process of descending stairs is not a supertask. I couldn't think of a way to make it a supertask, even by making each step smaller. A supertask has no final (or first, respectively) step, so by counterexample, the assertion "there has to be a final step." is incorrect.

A contradiction is introduced when this statement ("a completed step counting entails a final step)

I had not mentioned a completion of a count. The supertask is to complete all steps, not to count them, and not to complete a specific step that is nonexistent.
The series (say the time needed to complete all tasks) converges. The count does not.

Cheap example: You have a bag with a modest quantity of red, blue and yellow marbles in it. The goal is to remove them all. The task is deemed to be complete when the green marble is removed. Such a task cannot be completed by that definition of complete

The SEP article says:
"... From this perspective, Achilles actually does complete all of the supertask steps in the limit as the number of steps goes to infinity"

I notice the SEP article correctly doesn't claim that the last step is taken.

As I noted above, a physical, step-counting process that completes must entail a final step.

Agree. But the only attempted step counting processes are examples like the lamp or Michael's digit counter, and those examples are not physical. The Achilles example can be physical, but it isn't counting anything.

Your preferred perspective ignores this - or pretends there can't be a final step because that introduces a contradiction.

There being a final step leads directly to contradiction, and you say I'm copping out by pretending there isn't a final step?

I agree with this, but this simply ignores the implication of the physical process of step-counting.

Kind of like I ignore the green ball in the bag, yes.

For the scenario to be coherent, BOTH view of completeness have to be true.

I cannot accept this assertion. I cannot accept a view of completeness that treats infinity as a specific number.

No they mustn’t. — Michael

:up:

Once again, M-U cannot comprehend a view outside his own idealistic assumptions.

if a physical process ends, there has to be a final step. — Relativist

This is equivalent to asserting that 'infinity' is the largest integer. Does nobody else see that making such an assertion is going to lead to contradiction? It doesn't mean that there cannot be an unbounded thing.

I'm asserting that an infinite process is necessarily never completed - by definition. — Relativist

This depends on one's definition of completing a process. The SEP article on supertasks has this to say about it:
"But as Thomson (1954) and Earman and Norton (1996) have pointed out, there is a sense in which this objection equivocates on two different meanings of the word “complete.” On the one hand “complete” can refer to the execution of a final action. This sense of completion does not occur in Zeno’s Dichotomy, since for every step in the task there is another step that happens later. On the other hand, “complete” can refer to carrying out every step in the task, which certainly does occur in Zeno’s Dichotomy."
The definition you appear to be using is the former, which is why Michael's one-digit counter doesn't have a defined output after the minute expires.

I've been using Zeno's definition of complete: That every step has been taken. Given that definition, the supertask can be completed.

Good. Then we're on the same page! — keystone

And a different page than me.

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists, motion is impossible. — keystone

Zeno's argument is that X is possible, and another that X is not possible.
I see no mention of presentism in his arguments. I cannot follow your arguments here if you don't show how he presumes any such thing, or why it matters. Motion is defined under either view, and the argument can be made in either view of time. By modus ponens, at least one of Zeno's premises must be false unless empirical evidence is entirely dismissed as invalid.

I think you are under the impression that motion is not meaningful under eternalism, and that this somehow absolves Zeno's conclusion, but all his arguments still apply, and are still self contradictory.

The cuts themselves are the points (think Dedekind cuts).

OK, so now we have point cuts separating shorter strings, each with nonzero extension.

One can observe a superposition directly? Please share a link.

Any interpretation that denies wave function collapse has everything in superposition at all times. One simply finds ones self in superposition with the observed state. So I observe both the dead and the live cat, presuming that "I" dong the observing is the same person as the person a moment ago with the closed box.

No, I'm not don't personally accept MWI, but the simplicity of it is elegance itself.

in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).

Moton is change of postion over time. The block universe very much has that for any moving object. The worldline of that object is a different spatial locations at different times. All of Zeno's arguments still apply, and are still contradictory.

Yet again, the only difference between the view is the positing of the preferred moment, which is irrelevant to the subject at hand. Both are effectively block view, but presentism assigns different (at least four kinds of) ontological states to different events based on its relation to the preferred moment, and eternalism assigns identical ontological states to all events.

The kind of motion you are referencing (the changing of the block (over what??)) is not suggested by either view, nor by Zeno.

If the universe is discrete, then Zeno's paradoxes cannot occur as he described them

The first premise would be demonstrably false. The second premise (that supertasks are impossible) would be moot, but arguably true then.

What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.

You seem to do this by reducing the universe to a point (your 'photo'), which is not something that is continuous. A point in time at least, which is the same as denial of time at all.

ZENO'S PARADOX
Instead of presentism vs. eternalism, let's talk about the photo vs. movie reel. For the photo and every frame of the movie reel the characters believe they're in the present. — keystone

There are no empirical differences, agree. Presentism is the movie reel being played (a sort of literal analogy of the moving spotlight version of presentism). The reel by itself is eternalism (even if it still represents a preferred frame, which eternalists typically deny). The photo is just a frame, and not even that, since it is just a mental state since nothing in the present can be detected. If the state is all there is, then all memories are false and do not constitute evidence of anything.
The film analogy is discreet by nature, but doesn't have to be if the 'frames' are stacked instead of arranged side by side.

I suppose that if Zeno actually accepts his (unreasonable) conclusions, then you get something like just that one state.

Reconciling general relativity with presentism is quite challenging.

There is a way to disprove GR, but it is similar to proving/disproving an afterlife: You cannot report the findings in a journal. Both premises of SR contradict presentism, so different premises must be used to take that stance. This has been done, but the theory was generalized about a century after GR came out. It necessarily denies things like black holes and the big bang.

Plus, adopting eternalism helps to render Zeno's Paradoxes largely non-paradoxical.

I beg to differ, but again, the addition of a premise of a preferred moment has nothing to do with the validity of Zeno's assertions. He makes no mention of the present in any of them. If you disagree, then you need to say how the additional premise interferes with Zeno's logic.

Consider reversing this perspective: adopt a parts-from-whole approach. Start with a single continuous line and then, as if it were a string, cut it to create discrete points (which correspond to the gaps). I encourage you to explore this mindset; I'm eager to discuss it more with you.

Not sure of the difference. If I cut a string, I don't get points, I get shorter strings.

While my explanation might differ from how Zeno would phrase it, I believe it aligns with his philosophical approach. He is quoted to have said “My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the One.”

You cannot directly observe a particle in a superposition state

You can under some interpretations.

I bring in QM, not to sound fancy, but there is an analogy here between observed states (which are like points)

I don't think QM states are like points. The analogy is going way off track it seems.

I believe you are discussing whether time is discrete or continuous.

It's one of the things I'm discussing. Zeno's arguments are of the form (quoted from the Supertask Wiki page):
"1 Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
2 Supertasks are impossible
3 Therefore, motion is impossible"

If motion is discreet, then premise 1 is demonstrably wrong. If it isn't, then premise 2 is demonstrably wrong, unless one just begs the conclusion and adopts the 'photo' interpretation.

In the context of Zeno's Paradoxes, it's necessary to consider space and time as continuous (as you later noted).

Necessary only if the first premise is to be accepted.

I'm not sure what you're referring to with time being continuous or discrete from a presentist perspective, especially since Zeno's arguments suggest that time does not progress in a presentist's view of the world.

Yet again, one's interpretation of time isn't relevant to the above analysis.

I explicitly wrote abstract string.

Fine, Then it's a mathematical line segment.

let’s say that adopting an eternalist perspective allows someone to reframe the impossibility of supertasks, turning it's non-existence from having unacceptable consequences to acceptable consequences.

You're going to have to spell out exactly how an eternalist stance makes a difference here. All I see is an assertion that it makes a difference, but I don't see how.

Additionally, none of the paradoxes explicitly rule out (experience of each task) as a possible solution.

It takes some minimum time to explicitly comprehend/experience a step in a series of steps. Hence the explicit experience of each step of a supertask cannot be completed in finite time.

If there is a continuous film reel capturing the ticking counter, the limits of observation dictate that there are just some frames that we cannot see.

Hence needing to see them being irrelevant.

Zeno contends that change is impossible, leading to stark implications depending on one's philosophical stance on time. Under presentism, this translates to an unchanging, static present—life as nothing more than a photograph. — keystone

That sounds like a Boltzmann Brain, a mere state from which all is fiction and nothing can be known. Under this sort of presentism, there is nothing but a mental state and no experience at all, so no Achilles, Tortoise, stairs, or whatever. Just a mental state with memories of unverifiable lies.
This is not the usual presentism where that state was caused by prior ones, and will cause subsequent states. I don't think what you describe can be validly categorized under the term 'presentism'.

In contrast, the eternalist perspective views this as a static block universe, a continuous timeline that encompasses past, present, and future

There is no 'past, present. future' defined under eternalism. All events share equal ontology. The view differs fundamentally from presentism only in that the latter posits a preferred location in time, relative to which those words have meaning.

So there is still motion and change over time under eternalism, and the 'paradox', as worded, works under either since no reference to the present is made. Hence my suggestion that the topic has nothing to do with whether or not one posits a preferred moment in time.

Which view do you think is more reasonable?

Irrelevant, but I prefer the one that doesn't posit the additional thing for which there is zero empirical evidence. This is my rational side making that statement.

Consider whether it is easier to draw a one-dimensional line by assembling zero-dimensional points consecutively or to cut a string (akin to dividing a line into segments).

That sounds like Zeno's arrow thing, a attempted demonstration that a nonzero thing cannot be the sum of zeroes, a sort of analysis of discreet vs continuous. Under the discreet interpretation, there are a finite number of points making up a finite length line segment. Under the continuous interpretation, no finite number of points can make up a line segment, but a line segment can still be defined as (informally) all points from here to there.
About the only practical difference is that for two non-identical points, they can be said to be adjacent only in the discreet view.

Zeno would argue that the first option is impossible: a timeline cannot be constructed from mere points in time.

But he cannot indicate a time that isn't represented by such a point, so I don't think he's shown this.

Instead, modern Zeno would suggest that the entire timeline already exists as a block universe

Irrelevant, per above. The block universe can still be interpreted as discreet or not, just like the presentist view. The difference between the two has nothing to do with any of the scenarios Zeno is describing.

However, there's a twist: abstract strings, like time, are infinitely divisible. No matter how many cuts we make (one after another), we never actually reduce the string to mere points.

You do if it is discreet. A physical string is very much discreet, but that is neither space nor time. Zeno seems to favor the continuous model since all his paradoxes seem to presume it. E.g: "That which is in locomotion must arrive at the half-way stage before it arrives at the goal", a statement that simply isn't true under a discreet view.

the eternalist perspective reframes the impossibility of supertasks from an unacceptable notion—that motion itself is impossible

Nonsense. It says no such thing. It is only a difference in the ontology of events.

that observing every instant in history is impossible.

This also seems irrelevant since none of his paradoxes seem to reference observation or comprehension. Surely it would take forever to comprehend the counting from 1 on up. Michael's digital counter runs into this: the positing of something attempting to measure the number of steps at a place where the thing being measured is singular.

If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. — keystone

Non sequitur. It presumes the length of the staircase is a number, which is contradictory.


Everybody here seems to be attempting to introduce a premise that there is a number that represents the number of steps, despite the immediate contradiction with the premise to which it leads.

Presumable it would be at (the number of steps in the first staircase divided by 2) — Ludwig V

Case in point.

But the last step down is not defined, which means it can't be reached. — Ludwig V

Doesn't follow, since clearly I can overtake the tortoise in a universe that is continuous.
I can also do the reverse (the dichotomy version), which is the equivalent of counting down from infinite steps.

So instead of the assertions, show formally how this contradictory conclusion follows from the premise. Nobody has done that.

It does follow that the journey cannot start. — Michael

This seems to be an assertion, not a logical consequence of the premise. In fact it leads to a contradiction of the premise, hence demonstrating that the journey being able to start very much does follow from the premise, unless you can also drive that to contradiction, in which case the premise has been shown to be false.

Therefore given that the journey can start then the premise that there is no first division is false.

I swear you changed this. You had something that logically followed from your assertion. The conclusion that movement is discreet contradicts Zeno's premise that "That which is in locomotion must arrive at the half-way stage before it arrives at the goal". So by contradiction, the journey not being able to start doesn't follow from the premise.

If movement is discreet, there is a finite number of steps. The first (smallest possible) step cannot be divided, and the inability to do so violates Zeno's premise.
As for Bernadete's Gods, there would be a first God and there would actually be a barrier preventing motion. No paradox at least. That is decent evidence that Zeno's premise is false. But suppose space and time is continuous. Then the journey can start without contradiction, unless you can find one. You say above that it implies a first division, but nobody has suggested that such a journey must begin with a first segment. It only needs to take finite time (1 unit in this case). The simple example is me going from here to there, which you apparently assert is impossible if space/time is continuous. A bold assertion.

Given that each division is some 1/n then such a movement is akin to counting all the real numbers from 0 to 1 in ascending ordering. Such a count cannot start because there is no first number to count after 0. — Michael

No, the reals are not countable. The example we've been using is. There is no final count of steps in Zeno's dichotomy, so there is no demonstrated requirement of a 'first step' or any kind of final count of steps. Insistence otherwise seems to be leading to contradictions.

As for the OP, its triad of premises are inconsistent. For only two of the three following premises can be true of a sequence

i) The length of the sequence is infinite.
ii) The sequence is countable
iii) The sequence is exhaustible — sime

Applying this to Zeno's cases, or to the OP: All three seem to be true. I disagree that only two can be.

OK, the OP has infinite length to deal with, but finite time to do it, which is just a different way of expressing the same mathematics, totally discarding physics.
Zeno doesn't violate physics. If space/time is continuous, then the number of steps is countably infinite, and it is exhausted in finite time, as illustrated by my ability to move and/or to overtake something slower. Zeno makes no mention of 'point at infinity'. The OP kind of does ('bottom of it'), but also doesn't since there's no 'bottom step' apparent from the post-1-minute state. The poetry obfuscates what's actually going on, so I mostly am in denial of Zeno's conclusions.

If I overtake the tortoise, I have also reached the 'bottom of the supertask', so I don't find the wording necessarily contradictory.


What am I missing? A formal proof that it leads to contradiction would be nice, but all I seem to get is assertions.

Bernadete's Paradox of the Gods: — Michael

Ah, thank you for that. I sort of remembered the story but not the name/author.
It seems far more paradoxical than Zeno's thing since motion is prevented despite the lack of any actual barrier.

It's the same principle as Zeno's dichotomy, albeit Zeno uses distance markers rather than barriers. Given that each division must be passed before any subsequent division, and given that there is no first division, the sequence of events cannot start.

But I've been arguing that the above reasoning is fallacious. Yes, each division must be passed, and each division is preceded by other divisions (infinitely many), and yes, from that it can be shown that there is no first division. All that is true even in a physical journey (at least if distance is continuous).
But it doesn't follow that the journey thus cannot start, since clearly it can. By such a method, one can count from negative infinity to zero. You just need to not take some minimum time to do a given count.

The solution, similar to my proposed solution above, is that movement is not infinitely divisible

Mathematically it is, and mathematics seems to have no problem with it. Yes, I believe certain axioms must be accepted, but I'm no expert there.
As for physics, the assertion that motion is infinitely divisible seems to be a counterfactual assertion, not necessarily false, but unjustifiable. Such is the nature of quantum mechanics. But a journey can begin in either case, whether or not motion is continuous. Zeno does not illustrate otherwise in my assessment.
Zeno did his thing well before QM made us all question our classical notions of motion, so we can for the sake of argument make classical assumptions for this topic. If there's a limit to divisibility, then the problem goes away since there are finite steps.

If movement is continuous then an object in motion passes through every marker in sequential order, but there is no first marker, so this is a contradiction. — Michael

I don't find that to be a contradiction.

The false premise for Zeno is that each distance, and each time period will always be divisible. — Metaphysician Undercover

OK, if you deny the continuous nature of both space and time, then the number of iterations is finite, and the argument falls apart. My arguments presume a more mathematical interpretation: the continuous nature of both. If space is discreet, Achilles passes the tortoise after finite iterations. There would be a last one, after which the tortoise is passed. The conclusion of the inability to overtake doesn't follow because the premise upon which it is based becomes false.

Your assumption of discrete space is interesting, given that space (and everything else) is abstract to you, and thus any abstract space can be halved.

In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. — keystone

Presentism is still presentism even if time is continuous. You seem to describe a discreet view there, which runs into problems.
I don't see how Zeno's paradoxes work any differently under presentism than under eternalism. Eternalism doesn't resolve the problems with any of them.
I was unaware of Zeno's 'eternalist' leaning. Yes, the term didn't exist back then (not until perhaps the 11th century)

n this comprehensive perspective, motion is impossible. — keystone

Block view also defines motion as change in position over time, and thus motion is very much meaningful under the view.

rip from 0 to 1-I don't get it. — keystone

All these are trips from beginning to end. Zeno's initial state (0) to the point where the tortoise is passed (1). In your OP, 0 is time zero, and 1 is time 1-minute.

Yes, that is the point. Your expressed conceptualization "60 seconds will pass in the universe" is not consistent with the conceptualization prescribed by the OP. But this conceptualization — Metaphysician Undercover

This seems to contradict yourlelf. You say time is discreet, in which case the number of digit changes is finite, and there is an answer. You also seem to deny that the sum of the converging series is not 1, or that time somehow is obligated to stop, which is the same thing.

Michael: The output of the counter is undefined. I can think of no better answer than that.

Surely even if halfing the time with every step, a minute will still eventually be exceeded somewhere along the infinite steps and before this so called "finite bottom" to an infinite staircase?!? Doesn't make sense mathematically either. — Benj96

The mathematics is clear. The sum of the infinite series 1/2 + 1/4 + 1/8 ... is 1, not more, not less. Nobody has claimed 'under a minute'.

The most interesting thing I found about this is the unidirectional counting. You can count from 1 toward infinity but you can't begin counting from infinity toward 1. — Benj96

Well, the counterexamples have shown otherwise. I can subdivide the trip from 0 to 1 the other way around, with the smallest steps coming first, thus showing that it can be physically traversed in either direction.

What's not defined in either case is a last or first step respectively, just like there is no highest integer.

From the description there is always further distance for Achilles to move before he overtakes the tortoise. — Metaphysician Undercover

This is not true. Perhaps you are reading a different account of the story than I did, which is the one on wiki, which says simply:
"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". The 2nd bolded part is the non-sequitur, and the first bolded part follows from the 2nd. None of it makes the assertion you claim. The non-sequitur makes the argument invalid. There are ways (such as with the light switch) that make it seem more paradoxical.

In the OP [...] the premises imply that a minute cannot pass for Icarus, who always has to take more steps before a minute can pass.

Same non-sequitur. It is not true that Icarus always has more steps to take, only that he does while still on a step, but the time to complete all the remaining steps always fits in the time remaining in his minute.

So, in the OP, the false premise is the description of acceleration.

Sort of. I agree It has no basis in physical reality like Zeno's examples do. The OP poetry is only mathematical in nature and isn't meaningfully translated into physics. No amount of physical acceleration can traverse an infinite physical distance in finite coordinate time.

there cannot be an end to pi.

Then it concludes, that after a minute has passed, the end has been reached.[/quote]No. It concludes that all of the steps have been traversed. It does not assert that there is a last one. In this suggestion, the OP at least does not commit the fallacy that Zeno does.

Zeno on the other hand, concludes that Achilles cannot overtake the tortoise, which is the valid conclusion. And the absurd conclusion reveals the falsity of the premises.

OK, which premise then is false in the Zeno case? The statement is really short. One premise that I see: "the pursuer must first reach the point whence the pursued started", which seems pretty true to me.

I don't think that this is representative of the OP at all.

No, it is more the reverse of Michael's digit counter, just like Zeno's dichotomy scenario is the Achilles/tortoise thing in reverse.

What digit does the counter show after 60 seconds? — Michael

You have changed the divisibility of time in the OP to a divisibility of space in your interpretation. — Metaphysician Undercover

Yes, my example is more on par with Zeno dividing space than the OP dividing time. It has the same problem as Michael's counter: Measuring something where the thing being measured is singular, which makes the whole thing invalid.

it's actually a variation of Thomson's lamp. — Michael

I'm interested in your take on the nonexistent 'barrier' thing described at the lower half of my prior post in this topic. It also is a variation on something somebody else authored, but I cannot remember what it was originally called.
Side note: Would be awful nice if the site put numbers on the posts.

Is there another source for this paradox? Or did you just invent this yourself? — flannel jesus

It was unclear if this was addressed to the OP, or to me since this question was asked immediately after I posted the thing about the barriers. Anyway, not mine, but I can't find a link.

However, I recommend that Icarus stops looking for the last step down and starts looking for the first step up. He should find that as easily as he found the first step down. — Ludwig V

It is indeed unexplained why the guy, after taking the first step, is somehow compelled to continue his journey after 30 seconds and not just turn around. Mathematically it has some meaning, but it never has physical meaning, as several have pointed out.

How is it possible for him to ascend the stairs if there isn't a first step? — keystone

This is nicely illustrated by Zeno's 'dichotomy paradox'. Per wiki:
"Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on."

Each 'step' of the path from here to there must be preceded by a prior step. The fallacious conclusion is that no journey can be taken anywhere since there can be no first step when they're set up as you have done. Likewise, Zeno fallaciously concludes that Achilles cannot overtake the Tortoise due to the journey being divided into infinite steps. Both are a non-sequitur fallacy since it simply does not follow that the goal cannot be reached just because there exists a way to slice it into an unbounded quantity of segments.

The specifications do not allow for a minute to pass, — Metaphysician Undercover

By what is stipulated, yes, Achilles cannot surpass the tortoise. — Metaphysician Undercover

What do you mean stipulated? That Achilles cannot overtake is a non-sequitur. It simply doesn't follow from there being a way to divide the journey into infinite segments. This isn't a stipulation, it is merely a fallacious conclusion. Time not being allowed to pass was never a specification in the OP. Of course the lack of the stairs back up was actually a specification, and I find that contradictory.


The dichotomy thing was better illustrated by something that actually seems to be a paradox.
You are at location x < 0. The goal is to traverse the space between x=0 and x=1.
Thing is, a magic barrier appears at x=1/2 if you are at x <= 1/2, but x > 1/4.
A second barrier appears at x=1/4 if you are at x <= 1/4, but x > 1/8.
And so on. Each barrier appears only if you're past the prior one.
Furthermore, for fun, the last barrier is red. The prior one blue, then green, then red again. Three colors in rotation, all the way up the line.

Per the dichotomy thing (and Keystone's stairs), there can be no first barrier. So you walk up to x=0 and are stopped, despite there not being anything there to stop you. I mean, if there's a barrier, you'd see it and know its color, which is like suggesting a remainder if you divide infinity by three.

So paradoxically, you are prevented from advancing despite a total lack of a first barrier. You can see the goal. But you can't move.

That's a far better wording, and less fallacious than the way Zeno is reported to have worded it.

Each step takes a discernible amount of time which is a different time from the prior step. — Metaphysician Undercover

Exactly. Step n takes 60/2**n seconds. That's very much a nonzero duration for any n.

You say he reaches the bottom

After a minute, yes. Do you contend otherwise, that the sum of 60/2**n is not 60?

yet there is not "last step".

Just like there is no last natural number, yes. There is no last step to 'be' at.

How do you think it is possible that he got finished with all the steps, in the described order, yet there was no last step?

It's pretty clear from the mathematics. Where do you expect him to be then at 61 seconds if not 'past them all'?

according to the prescribed formula for figuring the increments, there can be no finish time — Metaphysician Undercover

OK, so mathematics is not your forte. The sum of this infinite series is not 60 according to you.

The infinite staircase appears to only allow one to traverse it in one direction. — keystone

Your poetry asserts this, but the reverse can be done There is simply no first step in the process, just like there wasn't a last step on the way down. The sum of the same series in reverse order is also 60 seconds.

I get shades of Zeno's paradox going on here, except Zeno get's there.

Despite the staircase being endless, he reached the bottom of it in just a minute. — keystone

He reaches the bottom of something with no bottom. It taking a minute is fine, but there being a bottom is contradictory. Hence I think resolution. Just as there is no first step to take back up, there is no last step to reach, even if it is all reached in a minute.

By sentient I mean conscious. Philosophical zombies behave as if they are conscious but are not actually. — Truth Seeker

Then you seem to define 'conscious' as having one of those 'self' thingys as defined by the quoted book.
I don't consider myself to be conscious then, in the same way that a roomba isn't. Sure, it reacts to its enviroment, but it's only an automaton and lacks the external control that would give it the free will that actual consciousness would.

an internal individual who resides inside our bodies, making decisions, authoring actions and possessing free will — Quoting the description of the book

That seems to be a straight assertion of dualism, but a non-dualist can also have a sense of self, so I must disagree with the book's definition.

I am sentient but I can't prove to you — Truth Seeker

You can react to external stimuli, which is perception, and sentience is perception or feelings. Perhaps you cannot prove qualia (what you might designate as feelings), but it's hard to deny that you have perception. Perhaps a different definition of sentience is being referenced. It wasn't given.


I suppose I consider the self to be an illusion (I did not vote since I'm using a different definition of self) since there is no way to demonstrate the persistent identity of anything, be it sentient or not. It is a very pragmatic illusion, without which any complex biological being would not be fit, so the illusion goes incredibly far back in our evolutionary history. I make choices (like draw breath) for the benefit of some future material state, which I consider to be my self. The choice yields no benefit to that which actually makes the choice. It's a sort of pay-it-forward system, and it works.

Firstly, I am NOT referring to linguistic definitions: I am referring to conceptual definitions. — Bob Ross

OK, I accept that,and retract the bit about the dictionary.

Firstly, not all definitions are about causality.

In general,no,but I gave an alternate definition that is very much about causality. It solves the circularity problem. It is analyzable,and it works for how most people use the word, even if the typical person would reach for the circular definition you reference.

Secondly, I don't see how this would provide non-circular definitions for concepts like 'being'.

I didn't define 'exists' in terms of 'being'. I used something far less circular. 'Being' is just a synonym, and can be defined similarly if you choose.

I'm not saying it's the correct definition. It's just one that I find far more useful, and avoids a lot of the problems that arise with the more circular definition.

First of all, all definitions are essentially circular, as evidence by somebody not being able to immediately glean a language simply by by being handed a dictionary. But with some ideas, the circularity of the definition becomes very short, such as in your example.

Do you have others? The one you selected is so very loaded with opinions and varying but valid interpretations, as illustrated below.

This pecularity indicates, by my lights, that ‘being’ is a primitive concept and, as such, is absolutely simple, unanalyzable, and (yet) still perfectly valid. — Bob Ross

That peculiarity renders the chosen definition rather empty in my opinion. I shy from such definitions and prefer something more pragmatic such as a relational definition. A exists to B if A in any way has a causal effect on B. Hence the nonexistence of unicorns because no unicorn seems to have a causal effect on humans, despite the legends to the contrary.

I've seen several articles about interesting ways to do logic and computation. Scientific American had one that designed a Turing machine consisting of only tracks and one train. Being a Turing machine, it could compute anything. Same with the domino computer. I'm pretty sure one can be built from dominoes, but unlike the tracks, no path can be taken twice, so time needs to be a spatial dimension.

As for the question in the stackexchange post:

Why did the last domino fall?
Answer 1: Because the penultimate domino made it fall.
Answer 2: Because the number 7 is prime

Both correct answers of course, which simply illustrates that there is never just one correct answer to 'what caused X to happen?"
But 2 is debatable. It requires the assignment of meaning to the input, and that meaning can in theory only come from whatever set up the dominoes for this particular purpose. In the absence of the meaning that the purpose gives, the last domino falls only because of it being a causal outcome of some prior state.

Why was my house destroyed?
1) because a hurricane swept through
2) because the strain capacity of pine is below a certain threshold
3) because a butterfly in the Amazon wiggled its wings 6 months ago

How can a physical system, in which each piece is truly only following local physical rules, be said to produce a certain output "because the number 7 is prime"? — flannel jesus

Well, because the physical system produces that output for certain input values, which in this case happens to correspond to only inputs that, when represented by some standard, happen to encode prime numbers.

Then there is a bigger question: Given the domino computer that lets the last domino fall only if the input is prime, suppose no input is ever made to it. The thing sits in its unfallen state awaiting the initial push. It is already determined that the last domino will fall if the encoding of 7 is entered. It being deterministic, the actual falling of the dominoes need not happen for the result to be determined, which means instantiation of the computation is not necessary for 7 to be prime.

Dominos can make logic gates, which means the domino system is turing complete. — flannel jesus

I agree that it can be Turing complete, but it's hard to implement a normal gate since the domino one can only be used once, and gates need to be used multiple times. So it gets complicated, but I think it can be done anyway. The train track thing was easier since the same track and switch could be traversed multiple times.

We already have algorithms to calculate if a particular position is check mate, so it's possible, in principle, to set up a series of dominos such that the last domino will only fall over if, say, Queen to D4 is checkmate.

Yes, that's just like the prime detector. There's no need even to describe the move. It matters not from whence the queen came, only that the board position includes it being at D4 as part of the state to be evaluated. Apparently you envision the prior state as the fixed setup, and the move in question as the input to be tested.

You've seen it fall, so you say "that domino fell because queen to D4 is checkmate".

Yes, just like 'because 7 is prime'. You don't need to see it fall. You need only to realize that it would fall if Q-D4 is entered to know that the move would be checkmate. And if you take epistemology away, Q-D4 is still checkmate even though no move is ever entered and nobody knows about the dominoes. It is still checkmate because the last domino would fall if that input were entered.

And yet... how computers work already is not too far removed from that, don't you think?

Turing machines are deliciously inefficient. Computers are simply far more optimized than these deliberately inefficient devices that accomplish the same thing.

You never asked if the domino setup can be conscious. Answer: Depends who you ask.

I didn't vote on the poll since 'none of the above' wasn't a choice.

Is determinism true? How can we know for sure? — Truth Seeker

Contrary to the popular belief, determinism has nothing to do with this. It has to do with the physics of our universe being causally closed. If it is (deterministic or random), then there can be no objective morality, or as 180 puts it:

There cannot be a vantage point for us outside of this causal nexus to differentiate right or wrong about assigning "actual moral culpability — 180 Proof

:100:

I don't take it for granted that determinism means you shouldn't hold someone culpable. — flannel jesus

That's the common mistake. Determinism (or any closed physics) means that one cannot be held objectively culpable, which is very different from being held culpable.

So while there are valid interpretations of physics that are deterministic and ones that are not, the difference is moot. The question is if physics is causally closed. Those that posit objective morality require the cause of one's choices to come from elsewhere than physics, hence the need for physics to not be causally closed.


I find all this a side track to what moral culpability is. Morals are a product of a society with expectations on its members. If something is not a member of such a society, then moral culpability is meaningless. If you disagree (which I'm sure many do), then provide a counterexample.

If morals are objective, then the rules of all societies anywhere must be based on these objective rules. So say you're playing dungeons and dragons. The dungeon is not closed. The choices made are made by people outside. There is a moral code that you don't betray your team members. That can only be meaningful if the player has some control over his character in the game. Else the character's actions are determined by the closed physics of the game and those outside the game cannot hold him moral culpable.

couldn't one adopt the kind of approach that the weather forecasters (and, I believe, physicists trying to work out fluid dynamics, which is probably the same problem) have adopted? — Ludwig V

The weather is closer. Fluid dynamics of a system in stable state (say water moving through a pipe, dam spillway) needs a description of that state, a calculus task. If it is dynamic (simulation of water waves), then it's more complicated, closer to the weather.

No simulation of the weather will produce an accurate forcast a month away, no matter how accurate the initial state is measured. Trends can be predicted, but actual weather at location X at time T is not going to happen. Similarly, no simulation of people is going to predict them doing what history says actually happened, no matter how accurate the initial state.

One does not improve weather forecasting by simulating the formation of individual rain drops, but nothing else at that level of detail, yet Bostrom is suggesting that such an inefficient choice would be made on a regular basis, for seemingly no purpose except that his argument depends on it. He clearly isn't a programmer.

Comment - this possibility high-lights for me a question about Bostrom's first two hypotheses.

The entire paper is one hypothesis. There are not more that I am aware of.
Your following description doesn't help in me trying to figure out what you're considering to be 'the first two of presumably more than two hypotheses'.

That would require us to define what is meant by "post-human" and "extinction".

I posted his definition of 'posthuman', which is, in short, a level of technology capable of running the numbers he underestimates, and far worse, capable of simulating a posthuman set of machines doing similar simulations.
As for extinct, there would only be two possible definitions: 1) No being in the universe is biologically descended from what is the Human species today. This of course is totally undefined, since if we're simulated, the actual humans of 2024 may not appear human at all to us. Much depends on what era the simulation uses for its initial state.
2) The other definition is that no entity in the universe has the human race of today as a significant part of the causal history of its existence. In short, if there are human-created machines that have replaced us, then humans are technically still not extinct. This is very consistent with his choice of the term 'posthuman'. One can imagine the machine race actually getting curious about their origins, and knowing about humans and presumably having some DNA still around, they might run simulations in attempt to see how machines might emerge from that. Of course, the simulations would produce a different outcome every time, sometimes with humans going extinct quickly, or losing all technology and reverting essentially to a smart animal, much like how things were before people started digging metals out of the ground.

Then we would have to deal with the difference between two different possibilities. We may go extinct and be replaced (or ousted) by some other form of life or we may evolve into something else (and replace or oust our evolutionary predecessors).

There you go. You seem to see both routes. The third path is extinction, or simple permanent loss of technology.

Given that inheritance is not exact copy and the feed-back loop of survival to reproduction works on us just as surely as on everything else, can we exactly define the difference between these two possibilities?

What two possibilitie? Humans that evolve into something we'd not consider human by today's standard? Many species do that all the time. Other possibility is 'ousted' as you put it. Our biological line is severed, as happens to nearly all biological lines given time.

They say that birds evolved from dinosaurs, and that mammals took over as dominant species from dinosaurs.

Good example. There are no dinosaurs (which, unlike humans, is a collection of species). The vast majority of those species were simply ousted. They have no descendants. But some do, and the alligators and birds are their descendants. They are not dinosaurs because none of them is sexually compatible with any species that was around when the asteroid hit. They are postdinosaur.

Which possibility was realized for dinosaurs?

It depends on the species, or the individual. Mom has 2 kids. One of those has children of his own, and the other is ousted, a terminal point in the family tree.

Another problem. Given that a feed-back loop is at work on these phenomena, can prediction ever be reliable?

Prediction of what? A simulation of history makes no predictions. A simulation of the future is needed for that, hence the weather predictors.
To guess at the question, no simulation of any past Earth state will produce 'what actually happens', especially if that simulation is of evolutionary history. There is for instance no way to predict what children anybody will have, or when, so none of the famous people we know will appear in any simulation. Again, Bostrom seems entirely ignorant of such things, and of chaos theory in general.

The third hypothesis suffers, for me

You really need to tell me what these hypotheses are, because I know of only the one. Two if you count the VR suggestion, but that doesn't come from Bostrom. i know of several that support a VR view, but none that has attempted a formal hypothesis around it.

Anyway, Bostrom posits nothing that is equivalent to a brain in a vat. That is more appropriate to a VR discussion.

The second premise - any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof) - seems obviously true to me. — wonderer1

It the second possibility. He says one of the three must be true. It's not a list of three premises.
I agree that granted this super-improbable posthuman state, that indeed, nobody is going to run a simulation of the history that actually took place. It just cannot be done, even with the impossible technology required.

The simulator would need to consist of more particles than the system which is being simulated.

If it is simulating at the particle level, yes. I can run an easy simulation of the planetary motions without simulating each bit. Each planet/moon/asteroid can effectively be treated as a point object, at least until they collide.

That's a rather fundamental problem. In practice, only things that are simpler than the simulator (or things treated simplistically) can be simulated.

Yes, and Bostrom claims several levels of depth, meaning the simulation is simulating the machines doing simulations.

It seems to me that the person who would seek to disprove the second premise would need to prove that consciousness can arise in a simulation of something much more simplistic than the world we find ourselves in,

Yes. If the goal was to simulate consciousness, they'd probably do one person, or a small isolated community (a closed system). And it wouldn't be a simulation of anybody real, but rather just a learning tool to show that a simulated person behaves like we do. If it worked, it would be a big blow to the dualists, but I'm sure they'd find a way to explain the results away.
The dualists can similarly deal a pretty fatal blow to the physicalists, but they choose not to pursue such avenues of investigation, which to me sounds like they don't buy their own schtick.

So I wanted to address the Simulation Hypothesis from Bostrom directly.
I quote only the abstract and a few parts of the intro.

This paper argues that at least one of the following propositions is true:
(1) the human species is very likely to go extinct before reaching a “posthuman” stage;
(2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history (or variations thereof);
(3) we are almost certainly living in a computer simulation. It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor‐simulations is false, unless we are currently living in a simulation. A number of other consequences of this result are also discussed. — BostromSimHypothesis

Posthuman is defined here:

The simulation argument works equally well for those who think that it will take hundreds of thousands of years to reach a “posthuman” stage of civilization, where humankind has acquired most of the technological capabilities that one can currently show to be consistent with physical laws and with material and energy constraints. — BostromSimHypothesis

The trichotomy is reasonable, but worded in a misleading way. Point 1 makes it sound like this preposterous posthuman state is somehow inevitable if the human race doesn't meet an untimely demise along the way. This is nonsense since the posthuman state described is totally unreasonable, and human technology seems heavily dependent on non-renewable resources upon which this gilded age depends.
The computer envisioned is a molecular machine that isn't electronic, but works with levers and gears and such, very small. But it needs a huge structure to supply energy and dissipate heat. The latter isn't problem, but a mechanical computer made of individually placed atoms would be phenomenally unreliable, and would be very size constrained. How does one fetch data from distant locations, using levers and shafts and such? The data set required by the description would require far more molecules than the described device would have.

The third point seems to suggest that all this fictional processing power would be regularly pressed into service doing what he calls 'evolutionary history', a simulation of our ancestors. This is not just unlikely, but actually impossible.
Say the people from 100 centuries in the future wants to simulate the world of today. To do that, they'd need to know the exact state of the world today, down to almost the molecular level, and I know for a fact that nobody has taken such a scan. Furthermore, any simulation of that state would last a few minutes/hours at best and then diverge significantly from what actually happened. So a simulation of one's own history cannot be done. At best, to simulate 'evolutionary history', one might set the world to the state of 20 million years ago with many of the species known to exist at that time, and see what evolves. It won't be themselves, but if those running the simulation are not human, then we're the unexpected things that evolves instead of them. That's plausible, but it isn't a simulation of their own history.

More problems when they claim to simulate the high performance machines that run the simulations later in time. He is after all claiming that there are simulations being run by the simulations. He seem to have no idea how inefficient this would be, that it takes millions of instructions to simulate a single interaction, coupled with all the side effects. I've written code to simulate the running of code (for profiling purposes). It didn't simulate transistors or anything, it just needed to assume that the processor works correctly and simulate at the intstruction level. It still took thousands of instructions to simulate one instruction.

That's just me tearing apart the abstract. The article goes on to suggest impossilbe future computer speeds, and tasks that are more than even that fictional processor could handle. There's a section specifically about substrate independence, with which I agree. It essentially says that doing it with paper and pencil, electronics, mechanical, etc all work the same. The outcome of the simualtion in no way depends on what substrate is used.


He does an estimation of 10³³-10³⁶ instructions needed to do one of his simulations of human history. Apparently only the people are simulated, and the rest (animals, plants, geology, and much worse, all the computers) are only imitated, not simulated. He justifies this small number with 100 billion humans, 50 years per human, 30M seconds per year, 10¹⁴-10¹⁷ brain ops per second, which comes to 15 times the figure stated above.

OK, it takes a lot of instructions to simulate all that goes on during a single brain op, and all that goes on between them. To simulate world history, it seems far more than just brains need to be simulated. At 100 billion people, only about a century or so of history can be simulated, nowhere near enough to get to the point of them running such simulations of their own.
Why 50 years? Is life expectancy expected to go down? What's the point of simulating minds at all when imitating people (as is done for everything else) is so much more efficient? The only reason is that Bostrom's idea doesn't hold water if you don't presume this needless complication.

Given future technology, simulation of a small closed system (maybe a person, or an island village) can be done. Actual world history? No actual history of any person, let alone all of them, can be done. Why does Bostrom choose to ignore this?

So I have to imagine myself as being a sim and not knowing it? — Ludwig V

Yes. That's Bostrom's whole point. He says we're probably all simulated, but it's based on the anthropic reasoning above, which makes many many unreasonable assumptions.
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