So in reaching B from A the object has completed occupying infinitely many locations in succession. — Michael

Yes... however the time it needed to travel between those infinitesimally small distances is infinitesimally short as well. That's why when you "add" them together, they end up being less than infinity. Infinitesimally short means that whatever time you can imagine, that value is actually shorter.

Before some mathematician comes in and says I've butchered it - I know. But I'm trying my best to present it in a way that's understandable to non-mathematicians. Trying to find common ground here really.

We agree that it is not possible for us to plot actual coordinates at distances that correspond to all of the rational numbers. We thus agree that it is not possible for us to measure actual motion at that level of granularity.

You then draw the further conclusion - unwarranted, in my view - that the motion itself cannot occur at that level of granularity, such that space itself must be discrete. This has the consequence that all actual motion must involve somehow "jumping" from one discrete location to another, without ever occupying any of the space in between.

Since I find this patently absurd, I affirm instead that space must be continuous. Our inability to measure infinitesimal distances does not entail that they do not exist; objects can and do traverse infinitely many of them while moving from one arbitrarily defined coordinate to another. However, as @Svizec just explained, each such transit occurs in an infinitesimal interval of time. In the end, objects actually move finite distances in finite times, and that is what we can and do observe and measure.

Yes... however the time it needed to travel between those infinitesimally small distances is infinitesimally short as well. — Svizec

This doesn't address the logical problem with an infinite succession of events having being completed, or the logical problem with the notion of there being a first location to move to.

Since I find this patently absurd — aletheist

Less absurd than the notion of having completed an infinite succession of events or the notion of a first location to move to, which are logically absurd (whereas I think the "absurdity" you find with discrete motion is a different, lesser, kind).

In the end, objects actually move finite distances in finite times, and that is what we can and do observe and measure.

I know that they actually do. The problem is that the logic of continuous motion is incoherent, hence motion isn't continuous.

The problem is that the logic of continuous motion is incoherent, hence motion isn't continuous. — Michael

The problem is that the logic of discrete motion is incoherent, hence motion isn't discrete.

Like I said, we are at an impasse. Cheers.

The problem is that the logic of discrete motion is incoherent, hence motion isn't discrete. — aletheist

Then motion is logically impossible.

Which then means we have a genuine paradox in nature. Motion is logically impossible but physically actual. And so the first of MadFool's suggestions seems correct; our logic is faulty.

Motion is logically impossible but physically actual. And so the first of MadFool's suggestions seems correct; our logic is faulty. — Michael

That which is physically actual must be logically possible; and so it is only your logic that is faulty here, because you insist on applying the logic of finite/discrete mathematics to a problem that involves infinity/infinitesimals. Peirce said it well - "Of all conceptions Continuity is by far the most difficult for Philosophy to handle."

I have, with my example of a machine that counts each coordinate as it passes through them in order. — Michael

Such a machine would not be possible. But we are not talking about this machine specifically, we are talking about any thing that moves, so this is a red herring.

As I said, as long as you persist in conflating ordering with counting, your argument won't get off the ground. It's simply not logical, because there is no logical requirement for counting here. If ever you allow yourself to realize this (and I realize how hard it would be, given the effort you've put into defending your position), there is still an option left for you: you could try to stake out a metaphysical claim instead of a logical one. At least it wouldn't be obviously incoherent.

As I said, as long as you persist in conflating ordering with counting, your argument won't get off the ground. It's simply not logical, because there is no logical requirement for counting here. — SophistiCat

I don't know why you're comparing counting to ordering. The comparison is between counting and moving. And as explained here, there's no reason to suggest that they're fundamentally different.

Either space is infinitely divisible or it is not. Whether anyone can actually divide space into infinitely many parts is completely irrelevant - only whether it could potentially be divided into infinitely many parts. — aletheist

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. — aletheist

You're still making the same mistake. It is false to say that space is potentially infinitely divisible unless it actually is. It is false to say that an object could potentially move to coordinate A, unless the object can actually do this. You are allowing fiction into your perspective by claiming that actuality has no bearing on potential. This allows you to claim that all sorts of impossibilities are real potentials, because your notion of potential is not restricted by actuality. This indicates that you have a deep misunderstanding of the concept of "potential".

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

...

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. — Michael

How can something "jump" from one discrete location to another without ever occupying the space in between? This is pure nonsense to me. — aletheist

Michael is correct here, that is simply how motion is. Consider walking, your foot is on the ground at one point, then on the ground at the next. It is not on the ground at all points in between. When we see an object moving, we assume that it must occupy every point along its course, but this is an assumption only. The assumed "course" is an oversimplification of what is really happening, and this is well known in QM.

I don't know why you're comparing counting to ordering. — Michael

Because I was responding to your own line of argument, e.g. here.

The comparison is between counting and moving. And as explained here, there's no reason to suggest that they're fundamentally different. — Michael

You haven't argued that moving is somehow related to counting, you just imagined some impossible contraption and asserted without any argument that continuous motion necessarily involves something of the sort.

Because I was responding to your own line of argument, e.g. here. — SophistiCat

You seem to just be misunderstanding. What I'm trying to say there is that you can't answer the question "if we want to count every rational number between 1 and 2, what number do we count first?" with "pick any at random, and then pick the next one at random, and so on" (as Banno suggested). Each number must be greater than the previous, and we can't count a number if we haven't counted a smaller number.

And so by the same token, each coordinate an object passes through must be closer to the target than the previous, and it can't pass through a coordinate if it hasn't passed through one that's further away.

You haven't argued that moving is somehow related to counting, you just imagined some impossible contraption and asserted without any argument that continuous motion necessarily involves something of the sort.

I'm saying that the act of moving from one location to another can be considered an act of counting, like a clock counting the hours as the hand performs a rotation. Counting is just a physical act like any other. I don't know what you think it is.

So in this case, the ticks of the clock that are the count are the movements through each rational coordinate between two points (rather than just every 1/12th of a rotation as it is with a clock). It doesn't make sense for there to be a first tick, and it doesn't make sense for it to have already ticked an infinite number of times. So continuous motion doesn't make sense.

You seem to just be misunderstanding. What I'm trying to say there is that you can't answer the question "if we want to count every rational number between 1 and 2, what number do we count first?" with "pick any at random, and then pick the next one at random, and so on" (as Banno suggested). Each number must be greater than the previous, and we can't count a number if we haven't counted a smaller number.

And so by the same token, each coordinate an object passes through must be closer to the target than the previous, and it can't pass through a coordinate if it hasn't passed through one that's further away. — Michael

Yes, this nicely illustrates the very confusion that I've been talking about.

each coordinate an object passes through must be closer to the target than the previous, and it can't pass through a coordinate if it hasn't passed through one that's further away.

True enough, but this has nothing to do with counting.

I'm saying that the act of moving from one location to another can be considered an act of counting, like a clock counting the hours as the hand performs a rotation. — Michael

You are saying this, but you are not proving this.

Counting is just a physical act like any other. I don't know what you think it is. — Michael

True, but that doesn't imply that all physical acts involve counting.

Yes, this nicely illustrates the very confusion that I've been talking about.

True enough, but this has nothing to do with counting. — SophistiCat

I don't know what you're talking about here. I'm just explaining what I meant by sequentially. You seemed to take issue with that word. It was simply used to preempt any attempt to weasel out of the problem (as Banno did).

You are saying this, but you are not proving this. — SophistiCat

If I have to prove why each 12th or 60th rotation of a clock hand can be considered a count (or "tick", if you prefer) then I think the problem here is with your understanding.

True, but that doesn't imply that all physical acts involve counting.

It doesn't matter. The point is that, as with the example of a clock hand, the very act of moving from one point to another can be considered to be an act of counting. Therefore, if an object can move through every rationally-numbered coordinate then that object can be said to count every rationally-numbered coordinate. But as you say, it's impossible to count every rationally-numbered coordinate. Therefore it's impossible to pass through every rationally-numbered coordinate.

And to repeat what I said above, it doesn't make sense for there to be a first tick, and it doesn't make sense for it to have ticked an infinite number of times.

You're still making the same mistake. It is false to say that space is potentially infinitely divisible unless it actually is. — Metaphysician Undercover

You're still making the same mistake. It is false to say that space is potentially infinitely divisible only if it actually is.

This indicates that you have a deep misunderstanding of the concept of "potential". — Metaphysician Undercover

No, it indicates that I have a different understanding of the concept of potential. We have previously established in other threads that you and I have a fundamental disagreement about this, so there is really no point in discussing it further here.

You're still making the same mistake. It is false to say that space is potentially infinitely divisible only if it actually is. — aletheist

You seem to be confusing "divisibility" and "divided". Continuous space would actually be infinitely divisible.

Again from here:

While it is the fundamental nature of a continuum to be undivided, it is nevertheless generally (although not invariably) held that any continuum admits of repeated or successive division without limit. This means that the process of dividing it into ever smaller parts will never terminate in an indivisible or an atom—that is, a part which, lacking proper parts itself, cannot be further divided. In a word, continua are divisible without limit or infinitely divisible.

The point is that, as with the example of a clock hand, the very act of moving from one point to another can be considered to be an act of counting. — Michael

It can be, but it does not have to be. Your whole argument hinges on insisting that the very act of moving from one point to another must be considered to be an act of counting, and that this counting must include every single point corresponding to a rational number in the interval. Just because we can model it that way does not entail that it actually is that way.

You're still making the same mistake. It is false to say that space is potentially infinitely divisible only if it actually is. — aletheist

Well that's surely your problem not mine. You believe that something is possible (potentially doable) though it is actually impossible to do it. If you don't recognize this as a mistake, there's not much I can do to help you.

Space is actually infinitely divisible and potentially infinitely divided. — Michael

This is exactly the question brought up by the op. Is space actually infinitely divisible, or is this just a false assumption, a mistaken theory?

This is exactly the question brought up by the op. Is space actually infinitely divisible, or is this just a false assumption, a mistaken theory? — Metaphysician Undercover

Sorry, see my edit. I meant to say that continuous space would be infinitely divisible.

OK, that's why it comes down to the accuracy of our theories of "space".

You seem to be confusing "infinite divisibility" and "infinitely divided". — Michael

No, you are the one with that confusion, as I have stated before. Space only has to be discrete if it is infinitely divided, not merely infinitely divisible.

Space is actually infinitely divisible and potentially infinitely divided. — Michael

This illustrates your muddled thinking perfectly. "Divisible" means "potentially divided."

You believe that something is possible (potentially doable) though it is actually impossible to do it. — Metaphysician Undercover

No one is talking about doing anything. To say that something is infinitely divisible does not mean that a human being is actually capable of infinitely dividing it. It means that it is possible in principle to divide it infinitely.

I'm saying that the act of moving from one location to another can be considered an act of counting, like a clock counting the hours as the hand performs a rotation. Counting is just a physical act like any other. I don't know what you think it is. — Michael

No. Counting and moving is not the same and can not be considered the same. You can move "through" any number of uncountable points that you wouldn't be able to order at all. Obviously you can't count them. So it's not the same. It is so obvious that these two actions are not the same that it's outrageous to suggest otherwise without a proof. Basically your response to this paragraph will be "No, counting and moving is the same, you can't do the counting so you can't do moving". Without providing a shred of evidence. Basically your "argumentation" is identical to:
Travelling faster than light is the same as Travelling to China. Travelling faster than light is not possible ergo travelling to China is not possible.
When I tell you that travelling faster than light is NOT the same as travelling to China, you just say, yes it is.

Oh, and your clock analogy, it's wrong, because you are again making counting look too easy. In your example - yes it's easy. Let me spice that analogy a bit. Let's say it's exactly noon. What is the first unit of time this clock of yours will count/move into. What's the second? How about if you put liquid nitrogen on the clock and you count all the molecules that minute hand hits? How about you put all the rational numbers [1,2] between 00 and 12 and try to count those?

I will repeat one more time (probably we should just quit at this point, not sure that anyone is willing to listen to rational arguments): existence of Planck length does NOT imply discrete nature of space. Believe it or not, but it's actually possible to observe data in a way to look for evidence of space being discrete. The opposite was found - the data implies that the space is not discrete. It's called science. Experiments. Observation. Planck length is border of a resolution problem. And no, not resolution in sense of optical lenses or sensors, the resolution problem would manifest itself in quantum effects that would make anything else irrelevant, pointless.

You can move "through" any number of uncountable points — Svizec

This is the assumption that I'm showing to be false. Each movement from one point to the next is a tick. If the space between two points is infinitely divisible then it doesn't make sense for there to be a first tick and it doesn't make sense to have ticked up to the end as that would entail having completed an infinite succession of ticks. So it doesn't make sense to move through any number of uncountable points.

This is the assumption that I'm showing to be false. — Michael

Good luck, because it's a fact.

Each movement from one point to the next is a tick. — Michael

No, it's not. Once you will understand that it's not, you will see the light. I can't help you anymore.

Have fun, gentlemen, don't kill each other. We all know what the truth is, what the facts are. I am too old to try to "win" debates online.

No one is talking about doing anything. To say that something is infinitely divisible does not mean that a human being is actually capable of infinitely dividing it. It means that it is possible in principle to divide it infinitely. — aletheist

Right, and to divide something it is to do something. So to assume that it is infinitely divisible is to assume that something is capable of dividing it infinitely. If it is not possible for something to do this, then that principle is false.

No, it's not. Once you will understand that it's not, you will see the light. I can't help you anymore. — Svizec

Indeed.

No this is a misunderstanding; the moving object does not have to 'account for and check off' every point it moves through, not least because it does not actually move through any points.

No this is a misunderstanding; the moving object does not have to 'account for and check off' every point it moves through, not least because it does not actually move through any points. — John

Sure it does, as the motion is said to be continuous. It has to pass through the half way point and before that the quarter way point and before that the one eighth point, and so on ad infinitum.