I still think you're just reading motion into that graph that isn't necessarily there.

However, I don't think there's much more to say on the subject. Thanks for taking the time. I have very much enjoyed it. :smile: :up:

No worries dude.

Sorry, I can't leave this alone.

But nothing's moving along the time axis; it's all just laid out. — Kenosha Kid

Motion or change along the time axis is required by the definition of motion (v=dx/dt). If there is no change in temporal position, then there is no motion. This is not according to a Presentist view of motion, but according to the definition of motion.

To repeat my argument against the 3D-4D analogy, a 2D cross-section of a mountain cannot change/move in the 3rd dimension, and neither can a 3D cross-section change/move in the 4th dimension. You might say that the radius of a 2D cross-section of a 3D mountain changes wrt altitude, and, likewise, that the spatial position of a 3D cross-section (body) of a 4D object changes wrt time. But none of this implies any motion of a 2D or 3D cross-section. By analogy, if there is motion in the 4D object, then there is motion in the mountain. I trust you would agree that there is no motion in the mountain.

Motion or change along the time axis is required by the definition of motion (v=dx/dt). — Luke

The 4D object spans time. It occupies multiple temporal positions, a whole continuous span of them. At one one its temporal positions, its spatial position is different than the spatial position it has at a different spatial position. Change which temporal position you're asking about, and the corresponding spatial position will be different.

Just like, on my south-facing coast here, the main road has a higher elevation in the north than it does in the south. The road spans many north-south locations. At some of those locations, its elevation is different than at others. So its elevation changes with latitude.

Just like a 4D object's spatial position changes with time.

But the road isn't moving north over time, and the 4D object isn't moving later through... something.

Change which temporal position you're asking about, and the corresponding spatial position will be different. — Pfhorrest

Asking about it causes it to change?

the main road has a higher elevation in the north than it does in the south...So its elevation changes with latitude — Pfhorrest

Elevation might change wrt latitude, but nothing about the (3D) road changes at a time, including the position of any of its 2D cross-sections (per elevation).

But the road isn't moving north over time, and the 4D object isn't moving later through... something. — Pfhorrest

That's not analogous. In the case of your 3D road, it's the elevation of 2D cross-sections of the road changing wrt latitude/length. In the case of a 4D object, it would be (some attribute of) 3D cross-sections of the 4D object changing wrt time. It is not analogous for the 4D object to change/move. We are only considering the motion of a 3D object over time.

According to Eternalism, a 3D object does not move from here to there over time; it is always here and there over time. This is no different to the 2D cross-sections of your road, which do not change position either.

Motion or change along the time axis is required by the definition of motion (v=dx/dt). — Luke

It is merely x that need differ. The value at one time being different to that at another. This does not depend on something moving wrt something else along either x or t.

To repeat my argument against the 3D-4D analogy, a 2D cross-section of a mountain cannot change/move in the 3rd dimension, and neither can a 3D cross-section change/move in the 4th dimension. — Luke

This is worth clarifying: it is not an analogy. Everything that is true about mountains in 3D is true about mountains in 4D.

4D is a generalisation of 3D that implements time as a dimension like space. That which is true of space in 3D remains true of spacetime in 4D. Just as a mountain has a slope in altitude wrt radius in 3D, it does so in 4D. It may also have a slope in altitude wrt time (erosion or formation).

It is merely x that need differ. The value at one time being different to that at another. This does not depend on something moving wrt something else along either x or t. — Kenosha Kid

Sounds like there is motion even though nothing moves.

This is worth clarifying: it is not an analogy. Everything that is true about mountains in 3D is true about mountains in 4D.

4D is a generalisation of 3D that implements time as a dimension like space. That which is true of space in 3D remains true of spacetime in 4D. Just as a mountain has a slope in altitude wrt radius in 3D, it does so in 4D. It may also have a slope in altitude wrt time (erosion or formation). — Kenosha Kid

Do the 2D cross-sections of a 3D mountain move? Is there motion in the 3D mountain?

Sounds like there is motion even though nothing moves. — Luke

I would say that, if there is motion, by definition it moves.

Do the 2D cross-sections of a 3D mountain move? Is there motion in the 3D mountain? — Luke

On geological timescales, sure. On hiker timescales, not so much.

Asking about it causes it to change? — Luke

No, but what answer you get changes with what questions you ask. “What is the position at time t?” has a different answer than “What is the position at time t+1?”

Elevation might change wrt latitude, but nothing about the (3D) road changes at a time, including the position of any of its 2D cross-sections (per elevation). — Luke

Exactly, now you’re talking sense.

The elevation changes with latitude but not with time.

A similar gradient in 4D changes with time (which we’re picturing as a dimension just like space here) but not with... hypertime or something.

In 4D, a change over time is simply a slope of a 4D object, just like in 3D a change of elevation over latitude is just a literal slope in the usual sense.

In the case of your 3D road, it's the elevation of 2D cross-sections of the road changing wrt latitude/length. In the case of a 4D object, it would be (some attribute of) 3D cross-sections of the 4D object changing wrt time. — Luke

Some attribute like their position in space? So you have a change in position over time. That’s exactly what motion is.

Do the 2D cross-sections of a 3D mountain move? Is there motion in the 3D mountain?
— Luke

On geological timescales, sure. On hiker timescales, not so much. — Kenosha Kid

I may have been unclear. Here you are talking about motion of the mountain over time (over the fourth dimension), whereas I was asking about motion of the mountain in three dimensions only (over the third dimension).

You stated that: "4D is a generalisation of 3D that implements time as a dimension like space." If time is space-like, or to be treated as another spatial dimension, then it seems reasonable that space is also time-like (i,e. they are the same type of dimension). You also stated that: "Everything that is true about mountains in 3D is true about mountains in 4D", so it seems reasonable that the converse is also true. My question was intended to be whether there is motion in a mountain in three dimensions only.

I may have been unclear. Here you are talking about motion of the mountain over time (over the fourth dimension), whereas I was asking about motion of the mountain in three dimensions only (over the third dimension). — Luke

Oh, sorry, I see. We call the gradient motion if its with respect to time only. And one wouldn't call a gradient in time a hill either. In 4D, purely spatial gradients and velocities are generalised together as one thing. In relativity this is particularly important because, for other generalisations like this, it is the generalisations that are invariant, not the individual coordinates.

For instance, position generalises to (x, y, z, t). What we call the spatial parts are still there. What we call the temporal part is still there. The values of these depend on your frame of reference. As such, spatial position and temporal position separately are always frame-dependent. But the 4D position as a whole is not. (Really, length not position, but same principle.)

So what you're suggesting is about generalising the concepts of 3D shape and motion to a higher-order concept that encompasses both.

So you have a change in position over time. That’s exactly what motion is. — Pfhorrest

Except that no 3D part ever changes its temporal or spatial position.

So what you're suggesting is about generalising the concepts of 3D shape and motion to a higher-order concept that encompasses both. — Kenosha Kid

I don't believe so. I'm saying that if there is motion within a 4D object, then there should also be motion within a 3D object, given that time is just another dimension like space (time is "space-like"). That is, I'm asking about generalising from 4D to 3D.

I don't believe so. I'm saying that if there is motion within a 4D object, then there should also be motion within a 3D object, given that time is just another dimension like space (time is "space-like"). That is, I'm asking about generalising from 4D to 3D. — Luke

Well, there is something like motion: spatial gradients. Given that motion is by definition with respect to time, just as spatial gradients are by definition with respect to space, you can't use them interchangeably any more than you can measure the radius of a mountain and say that's how tall it is. However you can move to a higher-order and just consider them as the same general thing: gradients with respect to 4D position.

Except that no 3D part ever changes its temporal or spatial position. — Luke

With respect to time, they change their spatial position.

Pick a particle. Step outside of time and look down at a 4D model of the universe. That particle will look like some crazy string zig-zagging its way the 4D universe. At some point in time, that string is at one point in space. At other points in time, it is at other points in space. But there is no meta-time across which they can "previously" have been at one place at time t, but "now" they're in a different place at the same time t.

Given that motion is by definition with respect to time, just as spatial gradients are by definition with respect to space, you can't use them interchangeably any more than you can measure the radius of a mountain and say that's how tall it is. — Kenosha Kid

But you say that the radius changes wrt altitude (in a 3D object) just as spatial position changes wrt time (in a 4D object). They both "change" in the same respect, and time is just another spatial dimension, so why is there no motion in the mountain?

Pick a particle. Step outside of time and look down at a 4D model of the universe. That particle will look like some crazy string zig-zagging its wave through the universe. — Pfhorrest

Or it will look like a motionless string merely existing at all points.

At some point in time, that string is at one point in space. At other points in time, it is at other points in space. But there is no meta-time across which they can "previously" have been at one place at time t, but "now" they're in a different place at the same time t. — Pfhorrest

In other words, it doesn't move.

It moves relative to time, which you would be perceiving as another spatial dimension. It doesn't more relative to something else that you're perceiving as time.

I'm getting bored of repeating the same point.

It moves relative to time, which you would be perceiving as another spatial dimension. — Pfhorrest

The same way that radius moves along a mountain relative to altitude?

Basically, motion requires that an object changes its temporal position, but this does not and cannot occur in an Eternalist 4D universe. The object needn't change its temporal position with respect to anything else; it simply changes temporal position (or "moves over time"). You and @Kenosha Kid are both assuming that a 3D part of the 4D object changes temporal position and/or that the same 3D part changes spatial position, but it doesn't really.

The other thing I think you are both illicitly assuming is that it is the same 3D object/part over time.

You and Kenosha Kid are both assuming that a 3D part of the 4D object changes temporal position and/or that the same 3D part changes spatial position, but it doesn't really. — Luke

Any object is extended over time, from its creation to its destruction. I'm pretty sure I repeatedly denied that an object "changes" temporal position in classical kinematics. It has a 4D shape. If that shape is not just comprised of the same 3D slice for all times, it is moving. In the same way, a mountain is not comprised of the same 2D slice at all altitudes and thus has a spatial gradient. If you understand the latter, there's no obvious barrier to understanding the former other than, which seems evident here, insisting on features that break that symmetry. I think you are a presentist in denial. You insist on presentist notions being true in 4D for motion to occur.

The other thing I think you are both illicitly assuming is that it is the same 3D object/part over time. — Luke

Yes, but that is true of any kinematics.

If that shape is not just comprised of the same 3D slice for all times, it is moving. — Kenosha Kid

The 4D shape is not moving. In what sense are the different 3D slices moving?

In the same way, a mountain is not comprised of the same 2D slice at all altitudes and thus has a spatial gradient. — Kenosha Kid

Are the 2D slices moving?

I think you are a presentist in denial. You insist on presentist notions being true in 4D for motion to occur. — Kenosha Kid

That's irrelevant if we agree to a definition of motion.

The other thing I think you are both illicitly assuming is that it is the same 3D object/part over time.
— Luke

Yes, but that is true of any kinematics. — Kenosha Kid

It may be true and even required for motion, but that doesn't mean it's true or possible in Eternalism.

The 4D shape is not moving. In what sense are the different 3D slices moving? — Luke

It is, as in "it is sometimes moving".

Are the 2D slices moving? — Luke

As already stated, "moving" refers to gradients wrt time not space, in the same way that "altitude" refers to height of the 2D slice, not the radius. 4D shape is a generalisation of all of these.

It may be true and even required for motion, but that doesn't mean it's true or possible in Eternalism. — Luke

I think I covered this point comprehensively a long time ago. I'm happy to clarify anything but, as Pfhorrest said, merely repeating myself is tiresome. TL;DR version: the concept of motion is recoverable even without continuity of identity, however I consider such flipbook notions of reality contra to eternalism.

The 4D shape is not moving. In what sense are the different 3D slices moving?
— Luke

It is, as in "it is sometimes moving". — Kenosha Kid

This contradicts your earlier statement:

It is not a condition in eternalism that a 4D object need move within a 4D space to have motion, since that would be a new kind of motion (hypermotion, I guess) in an even higher-dimensional space that would be hard to conceive of. — Kenosha Kid

TL;DR version: the concept of motion is recoverable even without continuity of identity — Kenosha Kid

How so? (I trust that your explanation will be about continuity of identity, rather than just saying "the gradient!" ("first base!")).

Having a gradient appears to mean no more than that a 3D part has a different spatial position than the spatial position of its temporal (3D part) predecessor. That they are the same 3D part (they're not) or that there is some sort of change/motion between parts (there isn't) is what you and @Pfhorrest appear to have simply assumed without argument.

This contradicts your earlier statement: — Luke

It is exactly in concord. "has motion", "is sometimes moving" appear to be equivalent expressions. Again I feel a sense that your focus is not on making any sense of a 4D body having geometry but not motion -- the paradox you'd need to resolve to support your claim -- and more on trying to catch me out on thinkos and inconsistencies. It's not an argument in good faith.

Having a gradient appears to mean no more than that a 3D part has a different spatial position than the spatial position of its temporal (3D part) predecessor. That they are the same 3D part (they're not) or that there is some sort of change/motion between parts (there isn't) is what you and Pfhorrest appear to have simply assumed without argument. — Luke

As I said, I have already been through this in quite some detail. I am not inclined to do it again to save you scrolling up, especially knowing that anything I say will just be ignored.

It is exactly in concord. "has motion", "is sometimes moving" appear to be equivalent expressions. — Kenosha Kid

You didn't say "has motion". You were apparently mocking the idea that a 4D object moves, saying it would require a "higher-dimensional space that would be hard to conceive of." You're now saying that a 4D object "sometimes moves", but this seems like an attempt to avoid the issue I've raised regarding the existence of different 3D parts vs. your assumption of a single 3D object changing position over time. I'm not trying to "catch you out", I'm arguing for my position.

You didn't say "has motion". You were apparently mocking the idea that a 4D object moves, saying it would require a "higher-dimensional space that would be hard to conceive of." — Luke

I was rejecting the idea that the 4D object moves wrt the 4D universe, an idea that would require some other dimension of time to make sense of. I frequently said that if the 4D object has slopes or wiggles, "it is moving", i.e. has motion, i.e. is sometimes moving. Your claim that I am saying this "now", as if I hadn't repeatedly said it already, is a non-starter.

this seems like an attempt to avoid the issue I've raised regarding the existence of different 3D parts vs. your assumption of a single 3D object changing position over time. — Luke

You're acting as if a) this was your angle from the start and b) I haven't already spent quite a bit of time addressing this very point (after which you dropped it for a while), so you'll forgive me for feeling your interpreting my reluctance to repeat myself ad infinitum is another mode of your bad faith. Refer to my previous explanation of how motion is recovered without assuming the 3D object at t' is the same as that at t, if you're interested, but don't expect people to feel obliged to endlessly repeat themselves on demand. You're not paying for my time.

I was rejecting the idea that the 4D object moves wrt the 4D universe, an idea that would require some other dimension of time to make sense of. — Kenosha Kid

You have said that the 4D object does "sometimes move". Since you reject "the idea that the 4D object moves wrt the 4D universe", then with respect to what universe (3D? 5D?) does the 4D object "sometimes move"?

I frequently said that if the 4D object has slopes or wiggles, "it is moving", i.e. has motion, i.e. is sometimes moving. — Kenosha Kid

Is it the 4D object moving or a 3D part/s?

Refer to my previous explanation of how motion is recovered without assuming the 3D object at t' is the same as that at t, if you're interested — Kenosha Kid

You previously provided the same argument in about three different ways as far as I can tell, including this one:

motion may still be recovered in this eternalism, even if we assume the object at t' to be different to the object at t, so long as there exists another continuity connecting the objects at t and t'. This is at least sensible: we do not see an object disappear then be replaced by a different but indistingushable object.

Then we can define a new kinematics over that continuity, identical in mathematical form to the previous kinematics except maybe from some replacement of dummy variables (e.g. t -> i), and giving exactly the same net result. This thing would look identical to what motion looks like in normal eternalism, where the object at t' is just another part of the same object at t. — Kenosha Kid

My response is that: If the objects at t and t' are different, then you are no longer talking about the motion of a single object from t to t'. What's moving in that case? Given that the 4D object can be infinitely divided into 3D parts, then no single 3D part ever moves. Maintaining that they are different objects at t and t' doesn't help to prove that an object has motion from t to t' (or between t and t'), because it's not the motion of the same object (or 3D part) from one point to another (or between one point and another).

Sorry, I missed this.

You have said that the 4D object does "sometimes move". Since you reject "the idea that the 4D object moves wrt the 4D universe", then with respect to what universe (3D? 5D?) does the 4D object "sometimes move"? — Luke

Its own temporal axis. The ground sometimes grades up. Up with respect to what? Some fixed plane that doesn't. Likewise, motion in 4D is manifest as a deviation from, say, a purely cylindrical shape (for the case of a 3D ball).

If the objects at t and t' are different, then you are no longer talking about the motion of a single object from t to t'. — Luke

This is, again, a presentist notion with no business in eternalism .