if I move from point A to point B, without moving through the locations between, it takes no time — BlueBanana

So you're agreeing with me that movement is possible if it is discrete rather than continuous. The issue is where one argues for continuous movement, and so that in moving from point A to point B one must move through all the locations between.

As for taking no time, I disagree. If one can move from A to B instantly and from B to C instantly and from C to D instantly and so on then one can move from one point to any arbitrarily distant point instantly, which empirically isn't the case. So even with discrete movement it takes time to move from one point to the next.

with a conditional: movement could be possible if the space was discrete.

You don't have any empirical proof that movement is either discrete or continuous; however, mathematically, calculus can show net change and rate of speed across differences on a continuous line. Your supertask, on the other hand, is nothing but a subjective classification of ill fitting paradoxes.

Infinite divisibility is the problem, which was Zeno's target all along (although in his case he wanted to argue that all is one, whereas I'm suggesting that there must be some fundamental unit of space/time (or at least movement) that cannot be halved). — Michael

What law of motion are you using? If you are using Newton's laws, then you might want to reconsider.

Anyway, when we consider even the classical equation for the time evolution of the state variables (p,q) or any function of them F(p,q,t), one is not immediately struck by the impossibility of motion, or the need to render space as discrete. Everything takes place in the continuum.

$\begin{align}\frac{\mathrm{d}F}{\mathrm{d}t} &= \{F,H\} + \frac{\partial F}{\partial t}\end{align}$

If we leave Hamilton's equation behind and take a look at the quantum equivalent, the continuum is still there, but there are also some other features that might be worth noticing.

$\begin{align}\frac{\mathrm{d}\hat F}{\mathrm{d}t} &= -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t}\end{align}$

This is an equation for the time evolution of operators, so, instead of the law of motion being about a particular value of interest, it is about a matrix of values. This indicates, to me at least, that what is going on in reality is quite different from what our classical intuition tells us.

Perhaps the most famous implications of the Heisenberg equation are the uncertainty relations, which have to have some bearing on Zeno's paradox.

As I have mentioned before, you can make deductions about Reality from physical laws, you are mistaken if you think you can make similar deductions from abstract mathematical ideas.

You don't have any empirical proof that movement is either discrete or continuous; — Jeremiah

Our best theories tell us space(-time) is continuous.

It really does not matter the discrete being discussed here is really effectively discrete.

It really does not matter the discrete being discussed here is really effectively discrete. — Jeremiah

I see. The proposal makes even less sense than I had previously thought. Thanks.

Infinite divisibility is the problem, which was Zeno's target all along (although in his case he wanted to argue that all is one, whereas I'm suggesting that there must be some fundamental unit of space/time (or at least movement) that cannot be halved). — Michael

Are you saying that Zeno's argument is sound, and that it shows that if space-time is continuous, then motion is impossible?

What about other variants, like the "starting and finishing" one I proposed?

Are you saying that Zeno's argument is sound — Srap Tasmaner

Zeno's argument is mathematically consistent, there is a path from 0.1 to 0.9...n, and if I do walk this path, I will never reach the end of it. But it requires that I walk a very specific path. Most paths are consistently sequential in nature, meaning if you didn't subdivide n+1 by 1/2 by the second step, you have no reason to believe you are going to do so at the last step.

I'm totally not following this. Could you take another run at it?

Are you saying that Zeno's argument is sound, and that it shows that if space-time is continuous, then motion is impossible? — Srap Tasmaner

Can someone remind me why anyone should assume that the abstract properties of an abstract idea (in this case infinity) should have any bearing on physical Reality?

Only the laws of physics can tell us what is physically infinite, and they tell us that when motion occurs, nothing physically infinite happens.

Think of it as a barrier to understanding, if you like. Until I know what's wrong with Zeno's argument, I don't really understand the physics.

Think of it as a barrier to understanding, if you like. Until I know what's wrong with Zeno's argument, I don't really understand the physics. — Srap Tasmaner

There is nothing "wrong" with Zeno's argument beyond the PRESUMPTION that the properties of abstract entities are identical the the properties of real entities that bear the same name.

If Zeno is complaining about the mathematical notion of infinity, then we can refer him to Cantor. If he is complaining that motion does not make sense, then we can refer him to the laws of physics. That only leaves the complaint that what is mathematically infinite and what is physically infinite are not the same thing. Why should they be the same?

Only the laws of physics, whether you understand them or not, can tell us what is finite or infinite in Reality.

It's a mild disappointment though, that no one seems interested in examining what the laws of physics tell us about motion. I suppose they already know.