On the discretization of space

[Nugatory]

Why do you require from the points to be physical for my argument to be correct?

If they aren't something physically real, if they're just mathematical abstractions, all you have a is a formal mathematical structure and a correspondence with another abstraction, namely the real numbers. Sure, in that structure there's always a point between any other two points... But why does it necessarily follow that that property of your mathematical model describes the motion of physical objects?

Ibix has already given you a pretty good hint when he pointed out that your mathematical model doesn't include the passage of time, which seems like <understatement>something of a defect</understatement> in a model of motion. Include time in the model, do some calculus, and you'll end up with phyti's answer.

 

[JK423]

Yes, it's not so different from discretizing space in this sense: assume you have the real numbers, but that you can only jump from integer to integer. The integers are discrete but are embedded in the reals, so since the reals contain discrete things, you can move.

Anyway, since hopefully now you agree we can move in continuous space, why don't you take a look a look at the calculus, which answers whether you can move smoothly in continuous space.

The fact that you impose discrete motion has consequences: you don't have to move from the closest point to the other. This is what you've said before. If you accept this, then you cannot also accept what i have in bold, that smooth motion is possible. If you accept smooth motion, then you have to go from closest point to the next, and if you accept the latter then my argument at post 11 says that motion is impossible.

Hence, you're believing in two incompatible things that cannot be true simultaneously.

 

[JK423]

If they aren't something physically real, if they're just mathematical abstractions, all you have a is a formal mathematical structure and a correspondence with another abstraction, namely the real numbers. Sure, in that structure there's always a point between any other two points... But why does it necessarily follow that that property of your mathematical model describes the motion of physical objects?

Ibix has already given you a pretty good hint when he pointed out that your mathematical model doesn't include the passage of time, which seems like <understatement>something of a defect</understatement> in a model of motion. Include time in the model, do some calculus, and you'll end up with phyti's answer.

My answer to the bolded: If you agree that space and real numbers are in one-to-one correspondence (definition of continuity), then the property that there is always a point between two other points holds for space as well. If you accept that motion is smooth without jumps, then in order to go from one point to the other you need to pass from all points in-between (if you don't pass through all intermediate points, then you don't move smoothly, there are jumps). That implies that motion is impossible from my argument in post 11.

Hence, if you accept ALL the following,

1) 1-to-1 correspondence between space and real numbers (i.e. space is continuous)
2) motion is smooth (i.e. in order to go from point x0 to point x1 you need to pass from all points in-between, otherwise there are jumps )

then motion is impossible.

About the concept of time: Of course there is time, time is implied. When i say, "go from point x0 to point x1" you need time to do that. The fact that when you are at point x0, there is no next point different than x0 in a continuous space (hence motion is impossible) has nothing to do with time. Time is still flowing, it's just that it's impossible to move in such a space.

 

[Ibix]

I assume your argument is the same with atyy's post 23. If yes then I've given my reply in my previous post 27.

Your argument is that, no matter how closely you look, there are always an infinite number of points between any two points in continuous space. I'm just pointing out that, however close you look, there are always an infinite number of slices of time to visit them in.

If you are uncomfortable with that kind of ininite regress I suggest you learn some calculus. The whole concept of integration (a sum of an infinite number of infinitely small areas), for example, would be impossible if your argument were valid.

 

[JK423]

yea I read the whole thing. Are there an infinite number colors in a rainbow? Perhaps it becomes a matter of defining a color (coordinates), as opposed to attempting to segregate the overlapping of frequencies. Much like how space & time "overlap" continuously. So depending on our "angle of view" a meter of length for you maybe a "meter" of time for me.

You are trying to conclude something about reality, based on a false premise (spacetime concept seems fairly agreed on).

The specific problem is not considering time to be a component of motion, let alone space.

I re-write part of my previous post about time:

About the concept of time: Of course there is time, time is implied. When i say, "go from point x0 to point x1" you need time to do that. The fact that when you are at point x0, there is no next point different than x0 in a continuous space (hence motion is impossible) has nothing to do with time. Time is still flowing, it's just that it's impossible to move in such a space.

 

[atyy]

The fact that you impose discrete motion has consequences: you don't have to move from the closest point to the other. This is what you've said before. If you accept this, then you cannot also accept what i have in bold, that smooth motion is possible. If you accept smooth motion, then you have to go from closest point to the next, and if you accept the latter then my argument at post 11 says that motion is impossible.

Hence, you're believing in two incompatible things that cannot be true simultaneously.

Let's say space and time are continuous. Discrete motion means the displacement time graph looks like a series of steps: you are at one spot at some time, then suddenly you are at another spot Δx away at a time that is Δt later. Your average speed is Δx/Δt. Now if you take Δx and Δt to be smaller and smaller, does your speed go to zero? If your speed goes to zero, them obviously you cannot move continuously. However, you can see that although the numerator Δx is getting smaller, the denominator Δt is also getting smaller, so the speed is not necessarily going to zero. What the calculus does is to make the argument rigourous that you can take Δx and Δt to zero, but Δx/Δt doesn't go to zero. So you have finite speed in infinitesimally small steps and you can move continuously.

 

[D H]

Thread closed. We do not discuss nonsense philosophy at this site, and Zeno's paradoxes rank right up there on the nonsense scale.