Paradox or not, I think this one passed the mathematical conundrum standard, and it generated far more discussion than I expected.

:grin: :up:

It shows that no matter how fast you go you can never finish. — Michael

Suppose Zeus is reciting all of the natural numbers infinitely quickly.

What does it mean to say that he never finishes?

Does he ever recite the largest integer? No. There isn't one. That's a point for not finishing.

But is there any integer he never gets to? Nope. That's a point for finishing.

Suppose we do it differently: Give Zeus one second to recite "1", half a second to recite "2", a quarter to recite "3", and so on. Is there any natural number he hasn't recited after 2 seconds have passed?

But is there any integer he never gets to? Nope. That's a point for finishing. — Srap Tasmaner

He will get to any integer but at no point has he ever gotten to every integer.

Suppose we do it differently: Give Zeus one second to recite "1", half a second to recite "2", a quarter to recite "3", and so on. Is there any natural number he hasn't recited after 2 seconds have passed? — Srap Tasmaner

This is Thomson's lamp paradox, except the question in this case will be "was the last number odd or even?" I suppose the paradox shows that in principle there must be some fundamental division of time that cannot be halved.

It really doesn't make sense for this program to ever terminate, and so it must be that one of the premises that leads to the conclusion that it does is necessarily false. That’s a proof by contradiction (which is exactly what Zeno’s paradox is).

Anyway, that's why I suggest that Planck units solve the paradox--space is not infinitely divisible.
Kant explains in the Critique of Pure Reason why it's hard for us to accept finite divisibility--it's outside of anything humans ever experience, so we can't wrap out heads around it. — NKBJ

Planck UNITS don't solve the "problem".

So to reiterate, what can and what cannot happen in Reality cannot be deduced from mathematics. In particular, the assumption that our understanding of the abstract infinity tells us anything about what is finite or infinite or possible in Reality is a mistake.

But of course, it is interesting to look at what our best theories tell us about motion, and how it happens.

It might be useful to consider a similar scenario. Zeus counts backwards to 1, getting slower as he counts. It took him 1 second to count from 2 to 1, half a second to count from 3 to 2, and so on. We can use calculus to show that he must have started 2 seconds ago. But given that there wouldn't have been a first number to count, he can't actually have even started (much like counting the rationals in order), and the fact that we can sum a geometric series doesn't avoid this. There isn’t in principle a relevant difference between my count and your count such that mine is contradictory and yours isn't (they both are).

I think these calculus solutions are just a bewitchment. They work as pure maths, but they can't apply to an actual supertask.

This is Thomson's lamp paradox, except the question in this case will be "was the last number odd or even?" — Michael

I suggested there are two criteria for "having finished a task":

(1) Having performed the last step;
(2) Having performed all of the steps, in some specified order.

For finite tasks, these are the same: "last" can just be defined as "all but one have already been performed."

But what about infinite tasks?

I see your argument as something like this:
1. If you have recited all the members of a set, there is some member of the set that is the last one you recited.
2. Zeus has recited the natural numbers in order. At step n, he recited the natural number "n".
3. By (1) and (2), there is a natural number z that was the last one Zeus recited.
4. By (2) and (3), z is the largest natural number.
5. Since there is no largest natural number, (2) is false.

I'm questioning step (5). We have the option of discarding premise (1) instead of (2).

Look at how criterion (1) works with finite tasks. Each time you perform a step, the number of steps remaining to be performed is one smaller. You're done when that number is 0. But this is just not true for infinite tasks. The number of natural numbers remaining to be recited is the same after reciting any finite number.

In fact, it looks to me like (1) is derivative of (2). We need a closer look at what it means to specify a task.

Suppose I give you a jar of marbles and tell you to count them. I come back half an hour later to find you haven't even started. Your explanation is that I didn't tell you what order to count them in. Fine. I know order doesn't matter, but evidently you don't, so I instruct you to pick one, take it out of the jar, add 1 to your running total, then pick any remaining marble as the next one. Go on until there are no marbles left.

Is it reasonable now to say you cannot count the marbles because I didn't tell you which one is the last one? No, of course not, because my recursive specification is enough. Here's how to start; here's how to continue; here's how to know when you're done.

It might be useful to consider a similar scenario. Zeus counts backwards to 1, getting slower as he counts. It took him 1 second to count from 2 to 1, half a second to count from 3 to 2, and so on. — Michael

I think we could play around with "first" as I have been with "last", but for many cases recursive specifications are exactly what we want, so I can just as well say that what you describe here is not a task at all.

I think these calculus solutions are just a bewitchment — Michael

I think if there's an intuition pump in the room, it's not calculus but Thompson's lamp.

I think it could be that some tasks we specify by specifying the last step -- maybe that's all we care about and are indifferent about what steps are or aren't taken. Really that seems more like a direction just to bring about a certain state of affairs.

But some tasks we naturally specify using recursion, and the infinite tasks we're talking about are clearly that kind. (Counting all the marbles is not the same as making the jar empty; the jar being empty is just how you know you're done.)

So is there an argument for (1), or an argument that it is not just a special case of (2)?

I think we could play around with "first" as I have been with "last", but for many cases recursive specifications are exactly what we want, so I can just as well say that what you describe here is not a task at all. — Srap Tasmaner

Counting up from 1 is a task but counting down to 1 isn't? Why is that?

So is there an argument for (1) — Srap Tasmaner

I don't know. It seems a truism that if one has completed a series of consecutive tasks then some task was the final task. I know it's not much of an argument, but I honestly can't make sense of it being any other way.

Surely "I finished all the tasks" is the same as "I completed the final task"? But you're saying that one can have finished all the tasks without having completed a final task?

But this is just not true for infinite tasks. The number of natural numbers remaining to be recited is the same after reciting any finite number. — Srap Tasmaner

Which is why it can't ever be finished. So the simple answer to Thomson's lamp and Zeus counting is that it is incoherent for a supertask to occur in a way described by a convergent series.

Counting up from 1 is a task but counting down to 1 isn't? Why is that? — Michael

Just in the sense that I don't know how to specify that task recursively. Is there a way? If not, is there some other way?

I don't know. It seems a truism that if one has completed a series of consecutive tasks then some task was the final task. I know it's not much of an argument, but I honestly can't make sense of it being any other way. — Michael

And my argument is that you're generalizing something that happens to be true of finite tasks. In effect your claim is that "infinite task" is a contradiction. My claim is that this is false, because (1) is a side-effect, and we needn't consider it part of the very definition of "task".

In effect your claim is that "infinite task" is a contradiction. — Srap Tasmaner

No, my claim is that completing an infinite task is a contradiction. If a task has been completed then, by definition, it wasn't infinite.

Just in the sense that I don't know how to specify that task recursively. Is there a way? If not, is there some other way? — Srap Tasmaner

Can you specify the task of movement recursively? “Move to the first half-way point”? It’s a lot like counting the rational numbers between 0 and 1 in order.

Can you specify the task of movement recursively? “Move to the first half-way point”? It’s a lot like counting the rational numbers between 0 and 1 in order — Michael

Those are not the same.

Moving by half the remaining distance can be specified recursively; doing the rationals between 0 and 1 in order cannot be if by "in order" you mean "smallest to largest". There's no smallest rational > 0 to be the first.

Moving by half the remaining distance can be specified recursively; doing the rationals between 0 and 1 in order cannot be if by "in order" you mean "smallest to largest". There's no smallest rational > 0 to be the first. — Srap Tasmaner

There's also no smallest half-way point (by which I mean $\frac{1}{2^n}m$). You can't move to 0.5m before moving to 0.25m, just as you can't count to 0.5 before counting to 0.25.

Oh I see -- my spec is "move each time by half the distance remaining to be covered" and that works recursively. Your way makes it impossible to start. I'm content for the moment with defining task as something

1. I know how to start,
2. I know what the next step is, and
3. I know when I'm done.

Just off-the-shelf recursion.

Edit: nevermind. I see your argument. Hang on.

You've switched back to talking about movement, where there is a strong intuition that each step in the task of moving from A to B can be subdivided into just as many steps as the original task. (I.e., a lot.)

I was talking about Zeus reciting the natural numbers (with geometrically increasing speed).

If you now want to say that each step can be divided into, let's say, "starting" and "finishing", then we'd be back in your regress of being unable to start. Do we like that argument? Now the problem is that every task is an infinite task, and we needn't worry about whether any of them can be finished because none of them can even be started.

I'm still leaving aside movement. My point for the moment is that if anyone can do anything, then Zeus can recite all the natural numbers in 2 seconds.

My point for the moment is that if anyone can do anything, then Zeus can recite all the natural numbers in 2 seconds. — Srap Tasmaner

So you're saying that because we can sum a geometric series then we can coherently talk about this program terminating (although again; what is the value of $$state$ after the 2 seconds)?

This is where I think we have to treat this as a proof by contradiction to show that it doesn't make sense to suggest that speed can increase geometrically without end. Convergent series are necessarily inapplicable to supertasks.

Convergent series are necessarily inapplicable to supertasks. — Michael

Oh I think that's probably true, even though I'm feeling a bit uncertain about how supertasks should be analyzed.

(One reason I've been going through this is to get clearer about what your thinking is. Sometimes your objection is that a given task can't be finished, sometimes that it can't be started, sometimes, as here, that if it could be completed then something else you don't like could also be possible.)

Zeus, it should have been clear, is here as a stand-in for the power of mathematics itself, which isn't bound by many of the usual considerations. He is by stipulation magic. Thus if you encountered this on a test in a math class

17. If Zeus takes 1 second to say "1", one half a second to say "2", one quarter to say "3", and so on, how many seconds does it take Zeus to say all the natural numbers?

you'd answer "2 seconds", and you'd be right. I'm not arguing for informal pedagogy as serious philosophy, but I am interested in how the Zeus story does make perfect and uncontroversial sense in the right context.

*

Getting back to Zeno ... What turns out to be wrong with this family of arguments? It's not just about movement, for instance, but about there being any sort of change at all, about anyone, as I said before, ever doing anything. So what's going wrong here?

@Michael There's a fundamental difference between the number line example and movement. Disregarding the numbers between, it takes time to count any amount of numbers, but doing the same with distances the movement between two points would be instantaneous. Thus, if you don't stay at any location, the movement between two locations does not take infinite time.

As to the lamp, its speed of switching would be infinite which could be interpreted to mean multiple things. Thomson had the premise that at any given moment the lamp is either on or off that he never questioned. Is the lamp on or off the moment it's switched? That's the answer to whether it's on or off at two minutes.

Sometimes your objection is that a given task can't be finished, sometimes that it can't be started, sometimes, as here, that if it could be completed then something else you don't like could also be possible. — Srap Tasmaner

Yes, if you're counting up the natural numbers then it can't be finished; if you're counting down the natural numbers (or up or down the rational numbers) then it can't be started. The math of a convergent series can tentatively be used in either case to show how it could take 2 seconds despite the fact that there's still the fundamental problem of not being able to start or finish, and so the takeaway is that a convergent series can't actually apply to these supertasks.

What turns out to be wrong with this family of arguments? It's not just about movement, for instance, but about there being any sort of change at all, about anyone, as I said before, ever doing anything. So what's going wrong here? — Srap Tasmaner

Infinite divisibility is the problem, which was Zeno's target all along (although in his case he wanted to argue that all is one, whereas I'm suggesting that there must be some fundamental unit of space/time (or at least movement) that cannot be halved).

There's a fundamental difference between the number line example and movement. Disregarding the numbers between, it takes time to count any amount of numbers, but doing the same with distances the movement between two points would be instantaneous. Thus, if you don't stay at any location, the movement between two locations does not take infinite time. — BlueBanana

It's not about time. It's about there not being a first position to move to. Even if you assume you could count at infinite speed (whatever that would mean) you can't count the rational numbers between 0 and 1. There's no first number to count to after 0. And even if you could move at infinite speed, there's no first $\frac{1}{2^n}m$ point to move to after the start.

So it's not that it would take too long to finish but that starting at all is impossible.

Why does there need to be the first point?

For the same reason there has to be a first number to count to.

The difference between time taken is relevant again. If one number takes a time to count, in that time you'll have to have counted one number, and that number should be the first one. With locations, in any given time, you'll have moved that distance and all the distances between that point and the starting point, and there's no need for the existence of the first point. The first point would only be needed if it took time to travel to it.

The first point would only be needed if it took time to travel to it. — BlueBanana

It needs to be there regardless. Even if we considered the idea of instantaneous counting then it would still be impossible to count the rational numbers in order because there's no first number. This has nothing to do with speed or time but the very logic of starting a task that has no start.

Then the first number would be 0, was it included. There's no need for the first number with instantaneous counting between two numbers.

The first number after 0.

the movement between two points would be instantaneous — BlueBanana

Also, this is false, as movement isn't instantaneous. It takes time to travel from one point to the next.

Not when

Disregarding the numbers between — BlueBanana

Or, in the case of locations, the locations between the two points, i.e. teleportation.

So, to clarify, if I count the natural numbers from 1 to 2 and say "one, two," that takes time, but if I move from point A to point B, without moving through the locations between, it takes no time, assuming I don't stay at the locations.

My understanding of Zeno's paradox is that across infinite timelines were are in a solid state of vibrations.

I say solid state in a metaphorical sense.

Pretty much this.