What does it mean for something to be "logically possible"?

[untermensche]

So there must be some movement that is greater than zero that is the smallest possible movement something can make.

Noop. No logical reason to assume there has to be a smallest possible movement. You can divide that movement into 2 smaller movements, ad infinitum. Unless you claim to possess knowledge of reality that is unavailable to current generation humans? Are you a subatomic scale being?

Only in the imagination can you divide the smallest possible movement.

In the real world there has to be a movement such that nothing can possibly move a shorter distance.

There has to be a smallest possible movement that something real can actually make.

Perhaps it's a different distance depending on the makeup of the object moving.

There is no "smallest amount of time elapsed since a movement began".

If time remains at zero there is no movement. The smallest movement must take some time.

But you'd need to know the velocity of the movement and the length of the movement to know the time.

 

[Kharakov]

Like I said in another thread a minute ago, you can change the momentum of an asteroid with a single photon. If, for example the asteroid is at rest compared to another body in space, the asteroid will no longer be at rest compared to the other body.

So now the asteroid is probably moving at 10^-57 meters per second away from the other body. If it's a very low energy photon (emitted out of a gravity well), the asteroid might now be moving at 10^-12414 meters per second away from the other body. Say it's an even lower energy photon?? You should get the direction this is going (well, unless you know the momentum with extreme precision).

You can always have something with a bit less momentum. If energy was disappearing, we'd see evidence that it was when looking at large scale interactions (galaxy formation, etc.). It doesn't appear to be. This means that things in spacetime have absolutely smooth/continuous momentum because there isn't a whole bunch of energy disappearing.

If momentum/velocity can be any arbitrarily small amount (this is what prevents energy from disappearing), then the minimum length movement in a period of time can also be an arbitrarily small amount. So we're left with the idea that if there isn't a whole bunch of missing energy in the universe, then spacetime must be continuous.

 

[fromderinside]

I'm wondering about untermenche's measuring stick, the one with which he says one can't measure beyond it's minimum, whether the numbers on it are readable between his least distance and zero. Has he figured out how the rabbit passed the tortice?

 

[untermensche]

Tortoise?

Try to imagine there is no shortest possible movement.

So when something moves you always can say it could have made a smaller movement.

Does this lead to some insight?

Or to absurdity?

 

[fromderinside]

There was this problem. Someone conjectured that if a hare halved the distance between itself and the turtle the hare would never pass the turtle. Bad logic, just as is your current logic. But, whoa, right over your head. Who'da thunk.

 

[untermensche]

There was this problem. Someone conjectured that if a hare halved the distance between itself and the turtle the hare would never pass the turtle. Bad logic, just as is your current logic. But, whoa, right over your head. Who'da thunk.

Yes it leads to absurdity.

You are on the side of absurdity.

Defending absurdity against what must be.

A movement must have length.

Therefore a smallest possible movement must exist.

 

[ryan]

Then these "movements" would be just jumps that jump 0 distances (if we assume the smallest movement is the smallest unit of space); it becomes a paradox.

That would be because you have the wrong, and indeed paradoxical, conception of distance.

In a discrete or quantified space, the distance between any two neighbouring space locations is not zero but one. So, a zero distance movement would be staying put on the same space location and then we can agree it's no movement at all. The smallest movement then becomes any movement where the distance covered is one, i.e. from any one location to any neighbouring location.

Okay, call it a movement of one, but it moves through 0 space to get to that position.

In other words, distance can only be a function of the space locations attained by the moving thing different from the starting location. In particular, in a discrete space, there's nothing in between neighbouring space locations and that 'nothingness' cannot be allowed to count in how we compute the distance covered.

There seems to be a difference between nothing and 0, and I am not just talking about the empty set versus {0}. 0 space on the real number line can take up a point/number/element; it can cause a discontinuity. Nothing on the real number line doesn't even have a point.

 

[ryan]

Then these "movements" would be just jumps that jump 0 distances (if we assume the smallest movement is the smallest unit of space); it becomes a paradox. Or we can accept infinitesimals even though they aren't easily imagined but are logically consistent.

They don't "jump" anywhere.

They slide right into that smallest possible movement smoothly. The only problem is they can't stop sooner.

So now you are have just regressed the scale. Before the argument was that something had to make a first movement between any distance b to c. Nobody specified the scale of what b and c they were using; this was about any movement of b to c. That means it could have been for what you are calling your first movement. You would be in a conflict with your previous argument.

 

[Speakpigeon]

That would be because you have the wrong, and indeed paradoxical, conception of distance.

In a discrete or quantified space, the distance between any two neighbouring space locations is not zero but one. So, a zero distance movement would be staying put on the same space location and then we can agree it's no movement at all. The smallest movement then becomes any movement where the distance covered is one, i.e. from any one location to any neighbouring location.

Okay, call it a movement of one, but it moves through 0 space to get to that position.

If that was the correct way to talk about it then what would be the relevance of a notion of space where, to move over a distance of 1, you'd have to go through 0 space, suggesting that to mover over a distance of 2 you just repeat and move twice through zero space?

Rather, you have to think of space as the set of space positions. There's not position in between two space neighbours so there's just no space in between.

So what matters when moving isn't whatever is between space positions, be it zero space, but the number of positions the moving ting has to go through.

In other words, distance can only be a function of the space locations attained by the moving thing different from the starting location. In particular, in a discrete space, there's nothing in between neighbouring space locations and that 'nothingness' cannot be allowed to count in how we compute the distance covered.

There seems to be a difference between nothing and 0, and I am not just talking about the empty set versus {0}. 0 space on the real number line can take up a point/number/element; it can cause a discontinuity. Nothing on the real number line doesn't even have a point.

It's just absurd to insist on reasoning at the same time in terms of discrete space and continuous space. These two notions seem contradictory. It's one or the other, not both.

If there's nothing between two neighbouring locations then there's no space. If there's no space in between, then space in between can't be allowed to count when measuring the distance between neighbouring positions. So zero space is an irrelevance here. So something else has to be relevant. One simple way to count, for example, is the number of new space positions attained by the moving thing, which effectively excludes the starting point. There are other conceptions possible but this one is good enough to have a coherent view of how it could work.
EB

 

[Speakpigeon]

There was this problem. Someone conjectured that if a hare halved the distance between itself and the turtle the hare would never pass the turtle. Bad logic, just as is your current logic. But, whoa, right over your head. Who'da thunk.

Yeah, right over, but still, a very, very close shave.

I'd say a nice hare cut.
EB

 

[ryan]

Okay, call it a movement of one, but it moves through 0 space to get to that position.

If that was the correct way to talk about it then what would be the relevance of a notion of space where, to move over a distance of 1, you'd have to go through 0 space, suggesting that to mover over a distance of 2 you just repeat and move twice through zero space?

Rather, you have to think of space as the set of space positions. There's not position in between two space neighbours so there's just no space in between.

So what matters when moving isn't whatever is between space positions, be it zero space, but the number of positions the moving ting has to go through.

In other words, distance can only be a function of the space locations attained by the moving thing different from the starting location. In particular, in a discrete space, there's nothing in between neighbouring space locations and that 'nothingness' cannot be allowed to count in how we compute the distance covered.

There seems to be a difference between nothing and 0, and I am not just talking about the empty set versus {0}. 0 space on the real number line can take up a point/number/element; it can cause a discontinuity. Nothing on the real number line doesn't even have a point.

It's just absurd to insist on reasoning at the same time in terms of discrete space and continuous space. These two notions seem contradictory. It's one or the other, not both.

If there's nothing between two neighbouring locations then there's no space. If there's no space in between, then space in between can't be allowed to count when measuring the distance between neighbouring positions.

That's not true with infinity. My reasoning for this comes from the calculation of an area under a curve. The integral of f(x) uses 0 widths and still can get a width > 0:

Area = lim(n-->infinity) sum (1 to n)f(xi)(b - a)/n. The widths (b - a)/n become 0 for their values. But because we simplify f(xi)(b - a)/n before finding the limit, usually the bottom n cancels out so the area will usually have a value > 0 depending on the function.

Your discrete steps are as though we zoomed into the integral. Your segregated surface is what we would expect to see if infinitely magnified in any area under f(x). Each position matters and is unique, label them with the naturals, but there is 0 space/distance between them. Any finite number of them will have 0 area unless you use an infinite number of discrete steps and make their positions point positions so that we can fit an infinite number of them in a finite segment.

So zero space is an irrelevance here. So something else has to be relevant. One simple way to count, for example, is the number of new space positions attained by the moving thing, which effectively excludes the starting point. There are other conceptions possible but this one is good enough to have a coherent view of how it could work.
EB

 

[Speakpigeon]

That's not true with infinity. My reasoning for this comes from the calculation of an area under a curve. The integral of f(x) uses 0 widths

That's just not true.

And ∑ 0 = 0.

and still can get a width > 0:

Area = lim(n-->infinity) sum (1 to n)f(xi)(b - a)/n. The widths (b - a)/n become 0 for their values.

It never gets to infinity and it never gets to 0. We look at the tendency to suss the limit without having to get there.

Your discrete steps are as though we zoomed into the integral. Your segregated surface is what we would expect to see if infinitely magnified in any area under f(x). Each position matters and is unique, label them with the naturals, but there is 0 space/distance between them. Any finite number of them will have 0 area unless you use an infinite number of discrete steps and make their positions point positions so that we can fit an infinite number of them in a finite segment.

You seem to be partial to analogous reasoning.

You seem unable to think in terms of properties specific to discrete spaces.

You seem unable to drop continuum as the default position and look at discreteness in its own right.

You seem to have to imagine a discrete space to even be able to think about it, and them you can't help but imagine it as included in a sort of default continuous space backdrop. You can't think discreteness if you can't drop the default continuum backdrop.

You should try conceptual thinking. Lots of people do it.
EB

 

[untermensche]

They don't "jump" anywhere.

They slide right into that smallest possible movement smoothly. The only problem is they can't stop sooner.

So now you are have just regressed the scale. Before the argument was that something had to make a first movement between any distance b to c. Nobody specified the scale of what b and c they were using; this was about any movement of b to c. That means it could have been for what you are calling your first movement. You would be in a conflict with your previous argument.

In theory there could have been a smaller movement.

In reality no such movement is possible.

 

[ryan]

and still can get a width > 0:

Area = lim(n-->infinity) sum (1 to n)f(xi)(b - a)/n. The widths (b - a)/n become 0 for their values.

It never gets to infinity and it never gets to 0. We look at the tendency to suss the limit without having to get there.

We use 0 widths to get the exact area. Anything more than 0 will not be the exact area most of the time. The widths diminish to 0, and that gets us the exact area, where any finite width usually won't.

The proof for this is shown by adding 1 to n: f(a+i(b-a)/(n+1))((b-a)/(n+1)) = f((a*i*b-a^2i)/(n+1))(b/(n+1)*a/(n+1)). So we are adding one more rectangle. You will see that we added 0 area to the integral of f(x); or just:

Area = lim(n-->infinity) sum (1 to n)f(xi)(b - a)/n = lim(n-->infinity) sum (1 to n)f(xi)(b - a)/(n+1). The bold +1 is the only difference.

 

[ryan]

They slide right into that smallest possible movement smoothly. The only problem is they can't stop sooner.

So now you are have just regressed the scale. Before the argument was that something had to make a first movement between any distance b to c. Nobody specified the scale of what b and c they were using; this was about any movement of b to c. That means it could have been for what you are calling your first movement. You would be in a conflict with your previous argument.

In theory there could have been a smaller movement.

In reality no such movement is possible.

You said, "They slide right into that smallest possible movement smoothly. The only problem is they can't stop sooner.", but they still slid at least halfway in reality. That means that there is a smaller movement.

 

[untermensche]

They slide right into that smallest possible movement smoothly. The only problem is they can't stop sooner.

So now you are have just regressed the scale. Before the argument was that something had to make a first movement between any distance b to c. Nobody specified the scale of what b and c they were using; this was about any movement of b to c. That means it could have been for what you are calling your first movement. You would be in a conflict with your previous argument.

In theory there could have been a smaller movement.

In reality no such movement is possible.

You said, "They slide right into that smallest possible movement smoothly. The only problem is they can't stop sooner.", but they still slid at least halfway in reality. That means that there is a smaller movement.

I wasn't all that serious.

But as I said there can always be a theoretical smaller movement just not a smaller movement something real can make.

How something gets from zero to the smallest possible movement would have to be answered in other ways.

If there is such a thing as an object then the object must have functional capacities on a scale greater than the smallest possible scale. In other words there must be a limit to the smallest possible way it can interact with other things.

But that smallest possible scale may be way beyond our current understandings which are limited to our abilities to break matter apart and record what happens.

 

[fromderinside]

There was this problem. Someone conjectured that if a hare halved the distance between itself and the turtle the hare would never pass the turtle. Bad logic, just as is your current logic. But, whoa, right over your head. Who'da thunk.

....

A movement must have length.

Therefore a smallest possible movement must exist.

If there is a continuum and one has a cursor riding on that continuum what is to limit the cursor from being anywhere on the continuum? We're not talking stepper motors - rack and pinion here. Even if we were we could hook up a hydraulic system to it and drive pistons continuously making use of elasticity and momentum.

 

[untermensche]

....

A movement must have length.

Therefore a smallest possible movement must exist.

If there is a continuum and one has a cursor riding on that continuum what is to limit the cursor from being anywhere on the continuum? We're not talking stepper motors - rack and pinion here. Even if we were we could hook up a hydraulic system to it and drive pistons continuously making use of elasticity and momentum.

Non responsive.

 

[ryan]

I wasn't all that serious.

But as I said there can always be a theoretical smaller movement just not a smaller movement something real can make.

How something gets from zero to the smallest possible movement would have to be answered in other ways.

If there is such a thing as an object then the object must have functional capacities on a scale greater than the smallest possible scale. In other words there must be a limit to the smallest possible way it can interact with other things.

But that smallest possible scale may be way beyond our current understandings which are limited to our abilities to break matter apart and record what happens.

But math gives a logical explanation on how a continuum would work (whether or not it is in accordance with the universe is another issue). There shouldn't be an issue with what math says because it is based on logic, inductive logic.

 

[untermensche]

I wasn't all that serious.

But as I said there can always be a theoretical smaller movement just not a smaller movement something real can make.

How something gets from zero to the smallest possible movement would have to be answered in other ways.

If there is such a thing as an object then the object must have functional capacities on a scale greater than the smallest possible scale. In other words there must be a limit to the smallest possible way it can interact with other things.

But that smallest possible scale may be way beyond our current understandings which are limited to our abilities to break matter apart and record what happens.

But math gives a logical explanation on how a continuum would work (whether or not it is in accordance with the universe is another issue). There shouldn't be an issue with what math says because it is based on logic, inductive logic.

It may be "logical" in mathematical formulations.

But I'm trying to make real world sense of the idea.

I find no real world example where you try to apply infinity to something and you are not reduced to absurdities.