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? 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 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

The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc.

The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc. — Michael

You continue to refuse to acknowledge the difference between the measurement and the thing measured. T1 and T2 are points designated by the measurer, therefore a feature of the measurement. The measurement is the difference between T1 and T2. However, the thing measured is the passage of time which occurs. Your confusion is due to your refusal to acknowledge a distinction between the measurement (the specified number of seconds) and the thing measured (the passage of time). You've been insisting that the thing measured is a number of seconds, rather than recognizing that seconds is the measurement, not the thing which is measured. And so I gave up trying to explain to you the difference.

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. — noAxioms

No one said free will has infinite capacity? Obviously we are limited by the circumstances we are in. But limitations are not absolutely either. So free will has it's own niche, to act according to a judgement of the circumstances.

In the circumstances you describe, an appeal to Newton would not help the poor soul, but a radio call to someone inside the spaceship, to please shoot me a line, might help. That demonstrates the benefit of free will, allowing one to act according to a judgement of the circumstances. And. it demonstrates how free will could actually get the person back to the space ship, in contrast to your suggestion of asking Newton to help, which of course, would be useless.

However, the thing measured is the passage of time which occurs. — Metaphysician Undercover

And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it.

The question makes no sense. You're asking for some second "level" of time to define the time between T1 and T2. There's no such thing. The only time is T1, T2, T3, etc. — Michael

The problem is adjacency. If object A is adjacent to object B on a finite grid, what is the distance from A to B? If it's 0 units, then A and B occupy the same space and A = B.

The problem is adjacency. If object A is adjacent to object B on a finite grid, what is the distance from A to B? If it's 0 units, then A and B occupy the same space and A = B. — Hanover

You seem to be imagining a model of discrete space overlaying some model of continuous space and then pointing out that in continuous space there is always more space between two discrete points.

That seems to be begging the question.

Best I can do is point you to something like quantum spacetime and quantum gravity.

There are physical theories that treat spacetime as discrete. They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false.

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity.

Let me start by saying my previous post here was poorly written. Now,

No. The mathematics is pristine. 1/2 + 1/4 + 1/8 + ... = 1 in the same sense that 1 + 1 = 2. Two names for the same thing. May be used interchangeably. Exactly equal. Denote exactly the same real number. — fishfry

There is no ∞-th item of a series. [...] But 1 is the limit, it's not a member of the sequence. — fishfry

The series of 1/2 + 1/4 + 1/8 + ... equals 1. The sequence $a_{n}=1-0.5^{n}$ converges to 1, yet 1 is not part of the sequence. As you agreed, there is no ∞-th item. Cool.

The issue that I see is:
1 – if we admit that time is infinitely divisible;
2 – and we admit that $a_{n}=1-0.5^{n}$ gives us the lenght covered by Achilles in the Zeno Walk at each step;
the walk only finishes if it accomplishes an infinite amount of steps. 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?; if so, is there not a corresponding state for the mechanism when the full time elapses?
In other words, by admitting that the result of an infinite series is necessarily true¹, how do you justify at the same time that the state is really undefined at 1 while also defending that Achilles can finish the run?

I want to emphasise that I am not arguing about the mathematics, but about the (meta)physical meaning of some mathematical concepts.

Does that make sense?

1 – Is that also the case for non-standard analysis and arithmetic?



–––––

How does it know where to go next, and at what speed? I think that's a more interesting puzzle. Where are velocity and momentum "recorded?" How does the arrow know what to do next? — fishfry

How much time elapses from travel to point a to point b and where is the object located during that time lapse?
Does the object leave existence between a and b and if it does, what maintains its identity during that interval? — Hanover

But there is that for an advantage of continuous time over discrete.

–––––

They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false. — Michael

Same here. General relativity is suspected to break down at high enough energies or small enough scales — where the quantum effects can't be ignored —, "like" Newtonian theory breaks down when v or Gm/R become large enough.

You seem to be imagining a model of discrete space overlaying some model of continuous space and then pointing out that in continuous space there is always more space between two discrete points. — Michael

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?

But one cannot use armchair philosophy to determine the smallest unit of space/time/movement. — Michael

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity. — Michael

I don't think you can have it both ways.

And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it. — Michael

I'm not quite sure what you are saying. Do you think that the passage of time occurs when we can't measure it? Analogously (if that's a word), if we can't measure the location or momentum of an object, it doesn't have them? Does that mean that it doesn't exist?
There are two philosophies that I can think of that would justify those views. One of them is Logical Positivism, which was developed precisely to justify both Quantum Mechanics and General Relativity. The other is Bishop Berkeley's idealism. Which do you hold?

There are physical theories that treat spacetime as discrete. They are not supported to the extent that General Relativity is, but given that quantum mechanics and General Relativity are known to be incompatible, it would seem that at least one of them is false, and my money is on General Relativity being false. — Michael

That means you think it is possible that space-time is continuous at the quantum level. Interesting. But I suppose it fits with your acceptance of continuous space-time in mathematics.
The empirical evidence for your position is the empirical fact that we can't measure very small units of time or space. I'm not sure that constitutes convincing empirical one way or the other. Or have I got something wrong?

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity. — Michael

Which ones do you have in mind? You mentioned the problems with a converging series. But that's a mathematical problem, not an empirical one. How does empirically non-continuous space and time solve those issues?

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

I don't think the question makes sense, but you'll have to ask a physicist who knows more about quantum gravity to explain it. I can only point out to you that there are physical theories that take spacetime to be discrete.

And the passage of time that we would measure as being 60 seconds occurs even when we don't measure it. — Michael

Yes, I agree with that. But, there is no "seconds" inherent in that passage of time, nor does it appear like there are any natural points for division within that passage of time, which appears to us to be absolutely continuous. This is why we assume principles which allow for infinite divisibility of time, because we see no reason for any real restrictions to its division. Therefore we tend to believe that we can simply insert a point (T1) at any random place, and another point (T2) at another random place, and determine the amount of time that has passed between those two arbitrarily assigned points.

Now, in your opening post in the thread, you concluded a "metaphysically necessary smallest period of time", and you used reference to the empirically based principle "60 seconds will pass" to support this conclusion. Therefore you've exposed inconsistency between two empirically based principles. The one principle being the assumption that the passing of time is continuous, as it appears, and the consequent principle that we can arbitrarily insert points, and divide it in absolutely any way that we please. The other principle being that "60 seconds will pass". There is inconsistency because the former leads to the example of the stairway to hell in the op, in which there is always more steps, and more time to pass, before 60 seconds can pass.

Given the logical paradoxes that continuous space and time entail, I think that discrete spacetime is not just a physical fact but a necessity. — Michael

Since you dismiss general relativity as probably false, then there is no need to maintain "spacetime". When we analyze space and time separately, then one might be discrete, and the other continuous. Logically, motion, which is a change of spatial location (place) requires the passing of time. We cannot conceive of a change in place without time passing because that implies the thing is in two different places at the same time. However, when time is separated from the constraints of spatial change we can conceive of time passing without spatial change. This allows that spatial change occurs as discrete 'quantum leaps', position at T1, to position at T2, without any spatial continuity between them. Between T1 and T2 there would be time passing, but no spatial change until that time has passed. That passage of time during which spatial change does not occur, is justified by activity at a deeper level, non-spatial, or immaterial activity, which determines the relationship between the spatial positions at T1 and the spatial positions at T2.

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.

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. — noAxioms

I told you how the person gets back to the ship using free will. That's one point for free will, zero for you.

A robot has the same capacity to make such a call, and robots by definition lack it. — noAxioms

A robot cannot decide whether or not to make the call, a person can. The person could decide not to, if perhaps the release of the tether was intentional. Two for free will, zero for you.

This is also utterly off topic to this discussion, but I took the easy bait anyway. — noAxioms

It's not off topic, because there is an issue of what is "physically possible", and whether physical possibility" is limited by the laws of physics. My argument is that there is a number of physical activities such as the effects of dark matter and dark energy, which violate the laws of physics. Furthermore, free will violates Newton's first law, and it causes physical movements. Therefore physical possibility is not limited by the laws of physics.