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.

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.

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.

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?

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?

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?

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?

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)?

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.

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?

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...

Motion cannot be continuous for the same reason that counting cannot be continuous. There cannot be a first coordinate to count to from a starting point and so there cannot be a first coordinate to move to from a starting point. Your position just seems to turn a blind eye to this. — Michael

Counting does not equal movement. Counting can only equal movement if you have a discrete topology (for example when you are explicitely ordered that you have to step from one point to another and nothing else exists in between). If the space is dense, however, you can still count it, but you can't order it. Rational numbers are dense (there's infinite number of them between any two rational numbers), they are countable, but they can't be ordered by size

Presumably there is no "in between". First it's at this discrete location and then it's at that discrete location. Maybe this involves discrete units of time as well. But this doesn't really matter. The logic still shows that continuous motion is impossible. — Michael

I'm afraid this is the end of the road, Michael. It does matter, you will have to be more precise here and make a solid counter argument. How would you define what distance is? Specifically, how would you define what distance is in a discrete topology and how would you define it in "normal" space of real numbers?

That makes no sense. — SophistiCat

I did clarify that I was talking about the Achilles racing turtle paradox, which is not the one from the OP. Are you still claiming that it makes no sense?

Edit: Oops, even when talking about the race paradox it really does not make sense. Lapsus, I see now. I will correct, thank you. :-*

At this point I am not sure which of the Zeno's paradoxes everyone is discussing. I've discussed both anyway. Michael, you are discussing the one from the original post? If so, I'm taking back my "spot on" comment, it's aletheist that's been spot on. The main reason why I thought this whole thread was about the first Zeno's paradox were Banno's posts that talk about the race and bring in the 1/2+1/4+... sequence, which is famously related to the turtle race paradox and not the "no motion/arrow" paradox.

Zeno's paradox: Anything moving from point A to pointB must first travel half of that distance. Before that it has to travel half of half of that distance and so on. It is possible to iterate this to infinity. Therefore, motion is impossible.
Yet, we can easily move from point A to B - we do it everyday.

Here math and/or logic claims motion is impossible. Reality is we can move. We have a contradiction. Therefore, either logic/math is wrong OR reality is wrong.

Possibilities:
1. Logic/math is wrong
2. Reality is wrong

Both cannot be wrong because that again leads to a contradiction.

How do we make sense of this paradox?

The key assumption is that space can be infinitely divided. Without this there is no paradox. And if space is infinitely divisible then motion would be truly impossible. However motion is possible. Therefore space is NOT infinitely divisible - the assumption that it is is false. So, logic/math is wrong - it rests on a false premise. There is no paradox; logic/math is wrong.

Paradox solved??? — TheMadFool

Ok, I actually forgot to reply to OP. Let's see... So it's a different Zeno's paradox, but logically it's almost identical to the turtle vs Achilles paradox. I would actually say that this one is even easier. Why? Well, in the Achilles-turtle problem, every step has a clear length, both in time and distance travelled from the last point. In the OP version, however, the first step is infinitesimally long and can be traversed in an infinitesimally short time interval. So the motion occurs. No paradox.
Another way to tackle it is also very simple and is pretty much what aletheist has been repeating in this thread over and over again. In fact maybe it is that this what caused all this misunderstanding - aletheist discussing the problem from original post and the rest of us the Achilles vs turtle version. If so, I was of course wrong to say that aletheist was not following the normal interpretation of the turtle-Achilles paradox. He was correct all the way.

Infinitesimal concept is something that Ancient Greeks had a lot of problems with and something that was discussed seemingly in half of their philosophical works. :) Well, to be honest, even in more "modern" times the infinitesimal concept was causing trouble and was even banned by Papal state in 17th century. :-O Dangerous stuff, be careful!

That it's the smallest measurable length is not that it's the smallest length. — Michael

I will quote several posts here but I will start with the only thing where I think Michael is wrong. Every other posts he writes is spot on.

The Planck length actually is the smallest possible length/size of anything in this universe. It is so much the smallest possible length, that the very moment universe came into existence, its size was exactly equal to Planck length.

I think that pretty much everyone with 5+ posts in this thread is correct about few things and incorrect about others. The main problem in my opinion is that everyone is talking about a different thing.

From your Wikipedia page Banno: "As n approaches infinity, sn tends to aproach 1." Does that means =1 to you? — Metaphysician Undercover

It actually does mean that, yes. It's exactly one. But this is surprisingly irrelevant in this case, because the problem doesn't seem to be mathematical at all.

The problem of the "paradox" is the way it's constructed.

You have it exactly backwards - the paradox only arises by insisting that space is made up of infinitely many points, and time is made up of infinitely many instants. When we recognize that both space and time are continuous, the paradox dissolves - there are no intermediate points that I have to "touch" while moving from defined point A to defined point B in a finite interval of time, just like I do not need to count any intermediate numbers in order to get from 1 to 2. — aletheist

You are of course correct that in real life we could just run and beat the turtle, but this trivial solution is not what drove all the philosophers/logicians to it for such a long time. Instead the most common interpretation of the paradox is that the runner MUST touch every point, as you worded it. So that's the version of the paradox that people try to discuss, including majority of posters in this thread. It is assumed that that's what Zeno was thinking.

Here is the problem in different words. Suppose Achilles is twice as fast as the turtle and turtle starts with 50m advantage. Achilles needs to "touch" the 50m point. At the moment of him touching the point, the turtle will be at 75m point. So the next point he needs to touch is the 75m. He always needs to touch the point where turtle is at the moment of him reaching the previous point. Since mathematically it's obvious that at 100m point Achilles will catch the turtle, we can just say "Screw the turtle, let's ask ourselves can Achilles actually reach 100m mark?" We can simplify the problem like this, because if he does reach 100m mark, then he had caught the turtle and if he doesn't, then turtle will always stay ahead of him. So that's where we get 1/2+1/4+...

Again, of course it's silly, but if we don't construct the problem this way, it becomes trivial and not worth a single keyboard press, let alone hundreds of books and articles written by some very smart people.

The sum of 1/3, 1/9, 1/27, 1/81...sequence is NOT 1. It is half(1/2).

You can try that with other fractions too. The sum doesn't equal 1.

Therefore the paradox remains unresolved as far as math is concerned. — TheMadFool

You misunderstand the problem and the reason why it's worded the way it is. Length of 1 is defined here at whatever point Achilles would mathematically catch the turtle. You want his first point to be at 1/3? No problem. So let's see... Turtle has 50m advantage and you want that to be 1/3 of the total length. Fine, so that means that turtle's speed is exactly 2/3 of Achille's speed. By the time Achilles reaches the 1/3 point, the turtle will be at 1/3 + 2/3*1/3 = 5/9, covering a distance of 2/9. Let's see the series:
1/3 + 2/9 + 4/27 + 8/81 + ... + 2^n / 3^(n+1) + ... = 1
There you go. Still 1. Let's move on.

Unfortunately if we want to look at the problem the way it was constructed such "mathematical" proof will not work. The rules made for Achilles are not fair. He can never catch the turtle, he will never reach the point 1. But since the construction itself assumes that he will never reach it, we don't have a paradox. It's all as expected.

Achilles is moving in steps. But he is moving in very special steps. If step n has length L, then step n+1 has length L/2. Again, not because of his choice, but because the rules are unfair. It's a fixed sport event. If I try to use a bit more mathematical language...

Edit: Initially made a terminology error, I am correcting this part.

The sequence 1/2, 1/4, 1/8, ... is a sequence with infinite number of terms. Each term corresponds to one step length. Let's make another sequence S, a sequence of partial sums:
S1 = 1/2
S2 = 3/4
S3 = 7/8
...

The reason why Achilles will never reach point 1 is because 1 is not a term of sequence S. 1 is the limit of the sequence, yes, but in order for Achilles to reach the 1, point 1 would actually have to one of the terms of the sequence.

For a more down to earth example, maths can quantify and measure music so that computers can produce music, but the vibe/soul/meaning/etc. of music (often resultant of indistinguishable variations that together harmonize into an expressive whole) can never be mathematically identified, even in principle. — javra

Can you elaborate on this? From what I know, music and its effect on us can be completely scientifically explained. You can play same tune in different scales and it will "sound" heavy, pop, oriental, happy, sad,...

For example, if you play a note at certain frequency together with another note at that frequency times X, the resulting sound will be "pleasant" only if X is sufficiently close (<1%) to a rational number with denominator lower than, let's say 15 (depends on how good our hearing is). This makes it easy to predict when two notes will sound harmonious, only by using mathematical language. Not only that, but we also have a pretty good idea, why it is like that.

Yes, I am almost certain that's it. Thank you.

Very good question.

The problems were not ethical dilemmas or paradoxes of physics. Instead they were presented in a way that we often see these puzzle books presenting problems. I might be wrong (I was 13-14 at that time, probably picked wrong book from the local library), but I seem to remember a problem involving envelopes and mail maybe?

I believe that the problems were ultimately analysed from perspective of philosophical logic. Recently I've been interested in this subject a bit more and this book came to my mind.

It's also worth mentioning, that the book was probably translated into many languages, including some languages spoken by relatively small number of people. That leads me to believe that the title is not very obscure.