An analogy can only be useful for illustrating an argument, and you have yet to offer an argument. You assert that moving from place to place is possible if and only if one can utter all rational numbers between 1 and 2 in sequence and in finite time, but you haven't offered an argument for this assertion — SophistiCat

An object cannot move sequentially through every rationally-numbered coordinate between two points for the same reason that we cannot count sequentially every rationally-numbered coordinate between two points. There's no first coordinate and so no first thing to count/move through.

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.

It's not that logic/math are wrong, it's that people are getting wrong--or they're getting confused about, mislead about, etc.--what logic/math are. Logic and math are a way of thinking about relations, thinking about them via maximum abstraction, and the vast majority of math/logic is a sort of game based on extrapolations of relations we experience, which means that logic/math is not at all the same as the relations we experience.

Aside from that, the idea that space is anything like an independent substance that is infinitely divisible is wrong.

"This thing is just like that thing" is not an argument. The best that I can make of your attempt is basically the same as before: you are saying that moving from place to place is possible if and only if it is possible to put all rational numbers between 1 and 2 into an ordered sequence (which, of course, is an impossibility). But you are not offering any argument for this assertion.

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!

f I try to use a bit more mathematical language... The sequence 1/2 + 1/4 + 1/8 + ... is a sequence with infinite number of terms. Each term corresponds to one step. The reason why Achilles will never reach point 1 is because 1 is not a term of this geometric sequence. 1 is the sum, yes, but in order for Achilles to reach the 1, point 1 would actually have to one of the terms of the sequence. — Svizec

That makes no sense.

The best that I can make of your attempt is basically the same as before: you are saying that moving from place to place is possible if and only if it is possible to put all rational numbers between 1 and 2 into an ordered sequence (which, of course, is an impossibility). But you are not offering any argument for this assertion. — SophistiCat

What I'm saying is that continuous motion between one place and another is possible if and only if it is possible to sequentially pass through each coordinate between them (and for the number of coordinates to be infinite, of course). It seems to be that this is what it means for motion to be continuous (rather than discrete). But sequentially passing through each coordinate doesn't make sense if there's not even a first coordinate to move to (or a second coordinate after that, or a third after that).

It's the exact same reason why we can't sequentially count each coordinate. There's no first coordinate for us to start our count. Where do we go from the starting point of 0? Not 0.1, not 0.01, not 0.001, ad infinitum.

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.

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. :-*

What I'm saying is that continuous motion between one place and another is possible if and only if it is possible to sequentially pass through each coordinate between them (and for the number or coordinates to be infinite). It seems to be that this is what it means for motion to be continuous (rather than discrete). — Michael

The superfluous assumption here is sequentially. It would be reasonable to say that for motion to be continuous the position of the body must pass every rational (or real for that matter) coordinate in order. But you demand something on top of that: that all of these coordinates form an ordered sequence. That demand is not motivated by any reasoning (indeed, you will necessarily run into a contradiction if you try).

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? — Svizec

Yes. That "1 must be part of the sequence" came out of nowhere.

It would be reasonable to say that for motion to be continuous the position of the body must pass every rational (or real for that matter) coordinate in order. But you demand something on top of that: that all of these coordinates form an ordered sequence. That demand is not motivated by any reasoning (indeed, you will necessarily run into a contradiction if you try). — SophistiCat

Of course it has to pass through them in order. It doesn't pass the half way point then the quarter way point then the three-quarter way point and then reach the end, making discrete jumps back and forth.

That's what I said. But you are asking more than that. You must recognize the difference between there being an order and there being a sequence.

That's what I said. But you are asking more than that. You must recognize the difference between there being an order and there being a sequence. — SophistiCat

Not really. I'm using the term "sequentially" in the sense that before it can reach the half way point in must reach the quarter point, and so on. And when it comes to counting, we can't do as Banno suggested and start our counting as "3/2, 4/3, 5/3, 5/4, 7/4..."

And so what is the first potential rationally-numbered coordinate that an object must pass through in its movement from A to B? — Michael

This is a nonsensical question. The only discrete coordinates that an object must actually pass through are those that we arbitrarily establish. Spatial coordinates do not exist apart from our construction of them for specific purposes. What you have identified is the reason why no one ever uses the rational numbers as a spatial coordinate system.

If the counting is to be possible it must be that the number line (or just the counting) is discrete; we have some minimum fraction to work with (say, 1/18). — Michael

Agreed, counting is discrete. That is precisely why it is a false analogy to motion, which is continuous.

And so if movement is possible it must be that space (or just the movement) is discrete; the object has some minimum fraction to work with. — Michael

Again, this is backwards; movement is only possible because space and time are continuous. If they were discrete, then it would be impossible to traverse the finite distance between adjacent locations. Where would the object be during the finite interval of time between the instant when it left one point and the instant when it arrived at the other?

The only discrete coordinates that an object must actually pass through are those that we arbitrarily establish. Spatial coordinates do not exist apart from our construction of them for specific purposes. What you have identified is the reason why no one ever uses the rational numbers as a spatial coordinate system. — aletheist

Even if we only define three coordinates between A and B it must still pass through the space between those coordinates. There are an infinite number of potential coordinates than an object must pass through, and this is true even if we don't actually make use of such coordinates. Your criticism makes no sense.

Agreed, counting is discrete. That is precisely why it is a false analogy to motion, which is continuous.

Continuous motion is impossible for the same reason that continuous counting is impossible. The reason counting is possible is because it is discrete. And so the reason motion is possible is because it is discrete.

Like I said, passing all rational coordinates in order is not a problem. After all, there is a (total) order relation for rational coordinates, so that for every pair of coordinates a and b, either a = b or a < b or a > b. But counting is not part of that.

Like I said, passing all rational coordinates in order is not a problem. After all, there is a (total) order relation for rational coordinates, so that for every pair of coordinates a and b, either a = b or a < b or a > b. But counting is not part of that. — SophistiCat

What's the difference between moving from one coordinate to the next and counting from one coordinate to the next?

Saying that passing all rational coordinates in order is not a problem is akin to saying that counting all rational coordinates in order is not a problem. It is a problem. Given that there's no first coordinate to count to from a starting point, counting cannot even start. Given that there's no first coordinate to move to from a starting point, movement cannot even start.

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.

Even if we only define three coordinates between A and B it must still pass through the space between those coordinates. — Michael

Even if we define as many coordinates between A and B as there are rational numbers between 1 and 2, the object must still pass through the space between those coordinates. After all, there are infinitely many irrational numbers between any two rational numbers. My whole point throughout this thread is that there is always space between any two coordinates that you define. That is precisely what it means for space to be continuous; it does not consist of discrete locations. No coordinate system, no matter how finely grained, can capture every potential location.

And so the reason motion is possible is because it's discrete. — Michael

Then please answer my question that you conveniently ignored. Where would the object be during the finite interval of time between the instant when it left one point and the instant when it arrived at the other?

Even if we define as many coordinates between A and B as there are rational numbers between 1 and 2,
the object must still pass through the space between those coordinates. After all, there are infinitely many irrational numbers between any two rational numbers. My whole point throughout this thread is that there is always space between any two coordinates that you define. That is precisely what it means for space to be continuous; it does not consist of discrete locations. No coordinate system, no matter how finely grained, can capture every potential location. — aletheist

This doesn't contradict my claim; it confirms it. I don't need to capture every potential location. I only need for there to be an infinite number of potential locations (e.g. the rationally-numbered coordinates).

Then please answer my question that you conveniently ignored. Where would the object be during the finite interval of time between the instant when it left one point and the instant when it arrived at the other?

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.

Again, this is backwards; movement is only possible because space and time are continuous. If they were discrete, then it would be impossible to traverse the finite distance between adjacent locations. Where would the object be during the finite interval of time between the instant when it left one point and the instant when it arrived at the other? — aletheist

Perhaps I'm missing something, but why must there be a finite distance between adjacent locations (assuming that you mean a non-zero finite distance; if there's zero distance, then your objection is moot)?

What's the difference between moving from one coordinate to the next and counting from one coordinate to the next? — Michael

Simple - counting is discrete by definition, because it requires explicitly recognizing every intermediate step, but motion is not. You keep insisting that motion has to be discrete like counting, but have made no argument for this assertion.

Simple - counting is discrete by definition, because it requires explicitly recognizing every intermediate step, but motion is not. You keep insisting that motion has to be discrete like counting, but have made no argument for this assertion — aletheist

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.

Continuous motion is impossible for the same reason that continuous counting is impossible. The reason counting is possible is because it is discrete. And so the reason motion is possible is because it is discrete. — Michael

You are begging the question. You are essentially saying that motion is just like this impossible thing, therefore motion is impossible. You must show the necessary connection between motion and counting all rational numbers in an interval in order.

What's the difference between moving from one coordinate to the next and counting from one coordinate to the next? — Michael

When you are saying "the next" you are already implying a sequence.

Saying that passing all rational coordinates in order is not a problem is akin to saying that counting all rational coordinates in order is not a problem. — Michael

Nope. Order is not the same as sequence. Ordering is not the same as counting. Until you understand this you will keep running in circles.

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?

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. — Svizec

If I wanted to show that discrete motion is the case, sure. But I don't need to do this to show that continuous motion is impossible. And the issue here is whether or not Zeno's paradox shows continuous motion to be impossible, which I'm trying to show it does.

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?

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 (whether in space or in time).

But, again, this isn't really relevant.

I don't need to capture every potential location. I only need for there to be an infinite number of potential locations (e.g. the rationally-numbered coordinates). — Michael

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).

Presumably there is no time in between. First it's at this discrete location and then it's at that discrete location. — Michael

So the movement from one to the other is somehow instantaneous?

The logic still shows that continuous motion is impossible. — Michael

It should be no surprise to anyone that assuming motion to be discrete (like counting) renders continuous motion impossible.

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. — Michael

No, there is absolutely no need for there to be a first coordinate in order to move from a starting point. There is only a need for there to be a first coordinate in order to measure movement; and the distance to that first coordinate is completely arbitrary, so we can use any finite interval that we choose.

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

Perhaps I'm missing something, but why must there be a finite distance between adjacent locations (assuming that you mean a non-zero finite distance; if there's zero distance, then your objection is moot)? — Arkady

If the distance between adjacent locations is zero, then by definition they are the same location, not adjacent locations at all. If the distance between adjacent locations is infinitesimal, then by definition space is continuous, as I have been arguing all along.

You must show the necessary connection between motion and counting all rational numbers in an interval in order. — SophistiCat

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

When you are saying "the next" you are already implying a sequence.

The sequence is the rational coordinates between two points.