We could just go for the Planck length as you suggested earlier. This is the smallest possible unit of space. There is no half-a-Plank length of space. So space wouldn't be continuous but composed of discrete Plank-length "tiles". And the same too with time. At 1 Planck time the object is at the 1 Planck coordinate and at 2 Planck time the object is at the 2 Plank coordinate, and it doesn't make sense to talk about the half way point (even in space or in time). — Michael

I still think we are missing the point though, because the discussion is mostly about a space of numbers, there is nothing quantum or physical in Q or R. Planck constant is purely physical stuff, but the logical discussion (the way I see it) here is about movement in abstract space.

Also, if I am really nitpicky here, existence of Planck length doesn't necessarily make space discrete. It's just a ... line between "normal" and quantum. Experiments so far have always confirmed continuous space-time, although who knows...

No, if space is discrete, then you need to capture every actual location; i.e., you need there to be an infinite number of actual locations (e.g. the rationally-numbered coordinates). — aletheist

I'm addressing your claim of continuous space. You seem to think that in saying that there are spaces between my rationally-numbered coordinates you somehow disprove my argument. You don't. You admit that it makes sense to consider an object passing through every rationally-numbered coordinate between two points. You're just adding to this that an even larger infinite series can be considered. But the point still stands that given the infinite number of coordinates, an object must have passed through an infinite number of prior coordinates to get to any arbitrary point, which is like saying that a person must have counted an infinite number of prior numbers to get to an arbitrary point. It doesn't make sense. You're saying that an infinite series of events has been completed. But an infinite series of events cannot be completed, by definition. And this, incidentally, is reasoning against an infinite past.

Also, if I am really nitpicky here, existence of Planck length doesn't necessarily make space discrete. It's just a ... line between "normal" and quantum. Experiments so far have always confirmed continuous space-time, although who knows... — Svizec

This is why I said earlier "that it's the smallest measurable length is not that it's the smallest length". If space is continuous then there's a length of space that's smaller than the Planck length. It's just that it's impossible in principle to measure (and, in the same vein, it might be that space is discrete but at a length smaller than the Planck length).

If the distance between adjacent locations is zero, then by definition they are the same location, not adjacent locations at all. — aletheist

This still doesn't seem right. That there is zero distance between adjacent locations only seems to entail that there is no boundary of any breadth between them. I don't see how it follows that they would be the same location.

This still doesn't seem right. That there is zero distance between adjacent locations only seems to entail that there is no boundary of any breadth between them. I don't see how it follows that they would be the same location. — Arkady

You're right. Conside two 1cm lengths with (hypothetically) 0 space in between. Is there 1cm length or 2cm length? 2cm. No space in between does not entail that there's just one location.

The sequence is the rational coordinates between two points. — Michael

I think I need clarification, because I don't know anymore. Suppose that you are right and that motion can only be discrete (this IS what you are arguing, correct?). So assume that there was a motion achieved, from point 2 to point 3. Since motion was discrete, there should be no problem making the list of all rational numbers you passed, in order you passed them. What was the first step you made? If you are not able to make that list, what exactly is your explanation that you are not able to make the list and how does that argument prove your point?

Consider a machine that counts each coordinate as it passes through it. If it can pass through all rational coordinates then it can count all rational coordinates. It can't count all rational coordinates, therefore it can't pass through all rational coordinates. — Michael

This is simply false. In order to build a machine that counts coordinates, you have to set it up using a particular (arbitrary) coordinate system, and that coordinate system will necessarily have finite intervals between coordinates. The fact that the machine cannot count all rational coordinates has no bearing whatsoever on whether it can pass through all rational coordinates; it merely reflects the machine's inability to measure distance at such a small interval. If the machine breaks, and thus cannot count any coordinates at all, does this mean that it cannot move?

This ridiculous assumption that all motion requires counting, or that motion is directly analogous to counting, is the whole basis of your entire argument. Anyone (like me) who rejects that particular assumption has no reason to take your argument seriously.

Suppose that you are right and that motion can only be discrete (this IS what you are arguing, correct?). — Svizec

Well, I'm saying that continuous motion is impossible and so if motion is possible then it must be discrete. It could also be (although seemingly absurd) that Zeno's conclusion is correct and motion is impossible.

Either that or we have a genuine paradox in nature where continuous motion is logically impossible but nonetheless the case, which would then suggest a fundamental flaw with logic (as the OP suggested).

Since motion was discrete, there should be no problem making the list of all rational numbers you passed, in order you passed them. What was the first step you made? If you are not able to make that list, what exactly is your explanation that you are not able to make the list and how does that argument prove your point?

If motion is discrete then the object didn't pass through every rational number. It made jumps from one coordinate to another without passing any coordinate in between. Could be Planck-length teleportation, as an example.

But the point still stands that given the infinite number of coordinates, an object must have passed through an infinite number of prior coordinates to get to any arbitrary point, which is like saying that a person must have counted an infinite number of prior numbers to get to an arbitrary point. — Michael

This right here is the mistake that you keep making. There are two statements here.

• An object must have passed through an infinite number of prior coordinates to get to any arbitrary point.
• A person must have counted an infinite number of prior numbers to get to an arbitrary point.

You treat these as equivalent, or at least analogous. They are not.

You're saying that an infinite series of events has been completed. But an infinite series of events cannot be completed, by definition. — Michael

Here we see your mistake from a different angle. You insist on treating the passing of each coordinate as a separate (i.e., discrete) event, just like counting. It is not - motion itself is continuous; only measuring distance is discrete, like counting.

The fact that the machine cannot count all rational coordinates has no bearing whatsoever on whether it can pass through all rational coordinates — aletheist

Why? What's the difference between a physical tick that is a count and a physical tick that is a movement? Counting can't simply be reduced to, say, speaking the coordinate. We can count by tapping the table, or by clicking our fingers. In this scenario, the machine performs a count by moving to a different point in space. So there is no fundamental difference between moving and counting.

I don't see how it follows that they would be the same location. — Arkady

Well, I don't see how two locations separated by zero distance can be different locations.

Conside two 1cm lengths with (hypothetically) 0 space in between. Is there 1cm length or 2cm length? 2cm. No space in between does not entail that there's just one location. — Michael

You are talking about length (i.e., measurement), rather than location. Consider two dimensionless points with zero space in between them. How can they correspond to different locations?

Well, I don't see how two locations separated by zero distance can be different locations. — aletheist

I suspect that there's a failure of imagination on one (or both) of our parts. Consider:

[X][Y]

[X] and [Y] are discrete regions of space. There is no boundary of any breadth between them, and no distance separating them. Does it then follow that [X] and [Y] are the same region of space? It does not appear so to me.

Well, I don't see how two locations separated by zero distance can be different locations. — aletheist

This is possible if we are talking about discrete sets. It's a bit cheeky definition of distance there, though.

If motion is discrete then the object didn't pass through every rational number. It made jumps from one coordinate to another without passing any coordinate in between. — Michael

Which is absurd.

In this scenario, the machine performs a count by moving to a different point in space. So there is no fundamental difference between moving and counting. — Michael

No, all we can say is that there is no fundamental difference between measuring movement/distance and counting, which I have acknowledged all along. Measurement is not a prerequisite for motion.

Does it then follow that [X] and [Y] are the same region of space? It does not appear so to me. — Arkady

Well, it does appear so to me. So we have contradictory intuitions, which just goes to show that intuitions are not infallible guides to truth.

Edit: Wait, I see it now. You said that [X] and [Y] are adjacent regions of space, not adjacent locations in space. So I understand that two regions can have zero separation between them, yet not be the same region. But what we were discussing was whether two (dimensionless) locations can have zero separation between them, yet not be the same location.

If motion is discrete then the object didn't pass through every rational number. It made jumps from one coordinate to another without passing any coordinate in between. — Michael

Ok... but did it pass through ANY rational number? Are you saying that it was at point 2 at the beginning and after that it ended at point 3 without passing a single rational number in between (like 5/2 for example)?

Ok... but did it pass through ANY rational number? Are you saying that it was at point 2 at the beginning and after that it ended at point 3 without passing a single rational number in between (like 5/2 for example)? — Svizec

For example, the first coordinate would be the one at 1 Planck length. The second coordinate would be the one at 2 Planck length. And so on. But at no point does it pass through the coordinate at 0.5 Planck length or at 1.5 Planck length.

For example, the first coordinate would be the one at 1 Planck length. The second coordinate would be the one at 2 Planck length. And so on. But at no point does it pass through the coordinate at 0.5 Planck length or at 1.5 Planck length. — Michael

You don't think that we can leave Planck constant out of this trip between points 2 and 3 on the real number axis though? If not, are you basically saying that the trip didn't visit any rational number, unless it happens to be at n x Planck length? But Planck length is a physical distance, while [2,3] is just an interval. How much is one meter on real number axis?

Consider:

[X][Y]

[X] and [Y] are discrete regions of space. There is no boundary of any breadth between them, and no distance separating them. Does it then follow that [X] and [Y] are the same region of space? It does not appear so to me. — Arkady

Could you provide an actual example of that? Are you talking about 2 sets of discrete points?

You don't think that we can leave Planck constant out of this trip between points 2 and 3 on the real number axis though? If not, are you basically saying that the trip didn't visit any rational number, unless it happens to be at n x Planck length? But Planck length is a physical distance, while [2,3] is just an interval. How much is one meter on real number axis? — Svizec

I don't really understand what you're trying to get at here. The point is that if movement is discrete then one doesn't have to consider an object moving first to the half way point ad infinitum. So there are a finite number of coordinates that it must pass through, which doesn't run into the logical problem that having to have completed an infinite series does.

For example, the first coordinate would be the one at 1 Planck length. The second coordinate would be the one at 2 Planck length. And so on. But at no point does it pass through the coordinate at 0.5 Planck length or at 1.5 Planck length. — Michael

This is a good example to show why we are at an impasse. Your claim is that all actual objects that actually move go from one Planck length coordinate to the next without ever occupying any intermediate locations. My claim is that all actual objects that move occupy infinitely many intermediate locations between any two arbitrary coordinates, even if the interval between them is one Planck length. You thus take the Planck length to be a limit on actual events themselves, while I take it to be a limit only on our measurement of actual events.

I don't really understand what you're trying to get at here. The point is that if movement is discrete then one doesn't have to consider an object moving first to the half way point ad infinitum. So there are a finite number of coordinates that it must pass through. — Michael

Ok but what are these coordinates if the movement is just between point (2,0) and (3,0) in real number space. You were mentioning something about Planck lengths, but you don't have physical units of measurement in space of real numbers. If the number of coordinates it passed is finite, what are they?

Your claim is that all actual objects that actually move go from one Planck length coordinate to the next without ever occupying any intermediate locations — aletheist

Well, it's only that something like this must happen if motion is to be possible.

My claim is that all actual objects that move occupy infinitely many intermediate locations between any two arbitrary coordinates, even if the interval between them is one Planck length.

Yes, and this runs into logical problems. Given that it has occupied an infinite number of prior locations in succession, it has completed an infinite series of events. But an infinite series of events cannot, by definition, be completed. And given that there's no first location for it to move to (as there's always a half-way point that it must travel through first) this infinite series cannot even be started (this being Zeno's paradox).

Ok but what are these coordinates if the movement is just between point (2,0) and (3,0) in real number space. You were mentioning something about Planck lengths, but you don't have physical units of measurement in space of real numbers. If the number of coordinates it passed is finite, what are they? — Svizec

I really don't understand your question. We just have some distance that an object is to travel and we plot a coordinate at Planck-length intervals. There are a finite number of coordinates for the object to pass through, "jumping" from one point to the next without passing through the space in between. This differs from continuous movement in which the object must also pass through the half-Plank length intervals (that we haven't plotted), and also the quarter-Planck length intervals, and so on, leading to an infinite number of coordinates for the object to pass through.

I really don't understand your question. We just have some distance that an object is to travel and we plot a coordinate at each Planck-length interval. There are a finite number of coordinates for the object to through, "jumping" from one point to the next without passing through the space in between. — Michael

Ok I'll try one more time. I am not talking about moving through physical space, like travelling from Earth to Mars. I am only talking about moving in R:
https://en.wikipedia.org/wiki/Real_coordinate_space

This space has defined distance and an object is moving from one point to another.

1. Do you agree that this has nothing to do with Planck constant, meters, centimetres? It's completely abstract.

2.In this space, do you still think that movement can be only discrete?

Ok I'll try one more time. I am not talking about moving through physical space, like travelling from Earth to Mars. I am only talking about moving in R:
https://en.wikipedia.org/wiki/Real_coordinate_space

This space has defined distance and an object is moving from one point to another.

1. Do you agree that this has nothing to do with Planck constant, meters, centimetres? It's completely abstract.

2.In this space, do you still think that movement can be only discrete? — Svizec

Sorry, I don't know what that is.

Given that it has occupied an infinite number of prior locations in succession, it has completed an infinite series of events. — Michael

No. The motion from one potential location to the next is not a discrete event. Only the motion from one actual location (i.e., arbitrarily defined coordinate) to the next is a discrete event. We can only define a finite number of distance coordinates, so we can only measure motion in discrete units. However, the motion itself is continuous between those discrete coordinates that we use to measure it.

Your position is the one with logical problems from my point of view. How can something "jump" from one discrete location to another without ever occupying the space in between? This is pure nonsense to me.

We can only define a finite number of distance coordinates, so we can only measure motion in discrete units. However, the motion itself is continuous between those discrete coordinates that we use to measure it. — aletheist

This doesn't matter. It still occupied the infinite number of spaces that we could have plotted as coordinates.

You even admit this yourself:

My claim is that all actual objects that move occupy infinitely many intermediate locations between any two arbitrary coordinates — aletheist

So in reaching B from A the object has completed occupying infinitely many locations in succession. This has logical problems. And, again, so does the notion of even starting the movement, as there isn't a first location (whether plotted or not) to move to.