Is a "smallest distance" self-contradictory?

[ryan]

A surprising result came to me when trying to picture a smallest possible distance. Here is an attempt to falsify the logic of a smallest distance using a proof by contradiction.

Because this is a proof by contradiction, we will start off assuming a smallest distance exists and see what happens.

Assume there is a smallest distance between 2 points, x1 and x2. Oxford dictionary has distance as, "The length of the space between two points.". If you can agree with this definition for this purpose, continue reading.

If all of the possible options are listed below, then a smallest distance must cause a contradiction:

1) an object will travel through space from x1 to x2

2) the object will jump from x1 to x2 taking no time

3) or it will jump taking some amount of time.

For (1), the object is obviously breaking the assumption of a minimal distance. For (2) and (3), the object travels no space; therefore, it was never actually a distance.

 

[Juma]

A surprising result came to me when trying to picture a smallest possible distance. Here is an attempt to falsify the logic of a smallest distance using a proof by contradiction.

Because this is a proof by contradiction, we will start off assuming a smallest distance exists and see what happens.

Assume there is a smallest distance between 2 points, x1 and x2. Oxford dictionary has distance as, "The length of the space between two points.". If you can agree with this definition for this purpose, continue reading.

If all of the possible options are listed below, then a smallest distance must cause a contradiction:

1) an object will travel through space from x1 to x2

2) the object will just jump from x1 to x2 taking no time

3) or it will jump taking some amount of time.

For (1), the object is obviously breaking the assumption of a minimal distance. For (2) and (3), the object travels no space; therefore, it was never actually a distance.

That doesnt foliow:
If space consists of "tiny cubes" of miinimal position, each position is still a distance apart. Think of squares on a chess board.

 

[ryan]

That doesnt foliow:
If space consists of "tiny cubes" of miinimal position, each position is still a distance apart. Think of squares on a chess board.

That's the space that the position occupies. The minimal distance between the 2 cubes or squares next to each other wouldn't have a space less than a square. There would just be one or more squares between them or nothing at all.

 

[beero1000]

You need a more specific definition for distance. Dictionaries are fine for everyday use, but technical topics need more formal descriptions.

"Object travels from one point to another without occupying the intervening space" is not the same thing as "Object travels no space". If I teleported from New York to Sydney, would you say that that means there's no distance between them?

 

[ryan]

You need a more specific definition for distance. Dictionaries are fine for everyday use, but technical topics need more formal descriptions.

"Object travels from one point to another without occupying the intervening space" is not the same thing as "Object travels no space". If I teleported from New York to Sydney, would you say that that means there's no distance between them?

Ah, but the single dimension that the space of the position takes up (like a cube or a sphere, or whatever shape) must be equal to or larger (in the same single dimension) as the space/distance between the two positions, in order to fulfill the assumption.

I assumed the shortest distance between the two points. So if the distance in R1 is larger than the R1 distance of the smallest position, then the two points are not separated by the smallest distance. If the distance between points is equal to the largest R1 dimension of the position, then that is an empty space that the position would have to move into to fulfill the assumption.

 

[beero1000]

You need a more specific definition for distance. Dictionaries are fine for everyday use, but technical topics need more formal descriptions.

"Object travels from one point to another without occupying the intervening space" is not the same thing as "Object travels no space". If I teleported from New York to Sydney, would you say that that means there's no distance between them?

Ah, but the single dimension that the space of the position takes up (like a cube or a sphere, or whatever shape) must be equal to or larger (in the same single dimension) as the space/distance between the two positions.

I assumed the shortest distance between the two points. So if the distance is larger than the smallest position, then the two points are not separated by the smallest distance. If the distance between points is equal to the largest dimension of the position, then that is an empty space, and the position then have to move closer to the other to fulfill the assumption.

Must it? Why? Weren't you talking about point positions which take up no space at all?

Like I said, definitions are important. You seem to be conflating the point locations with all the space in their 'cells'. Firstly, there are distance models where you won't be able to define geometric cells like that and secondly, you are contradicting your assumption that the point locations are the only locations there are. It's probably better to think of a graph model, with nodes and edges connecting the nodes that have given distance weights. Movement goes from node to node, where the distance traveled is the weight of the edge traversed.

 

[ryan]

Ah, but the single dimension that the space of the position takes up (like a cube or a sphere, or whatever shape) must be equal to or larger (in the same single dimension) as the space/distance between the two positions.

I assumed the shortest distance between the two points. So if the distance is larger than the smallest position, then the two points are not separated by the smallest distance. If the distance between points is equal to the largest dimension of the position, then that is an empty space, and the position then have to move closer to the other to fulfill the assumption.

Must it? Why? Weren't you talking about point positions which take up no space at all?

I cleaned up my last post, but it wouldn't make a difference to your response.

It seems as though it doesn't matter if the positions are points or real volumes. I'll explain below.

Like I said, definitions are important. You seem to be conflating the point locations with all the space in their 'cells'. Firstly, there are distance models where you won't be able to define geometric cells like that and secondly, you are contradicting your assumption that the point locations are the only locations there are.

If positions can exist between points, then the two positions in the assumption were not the smallest distance apart.

It's probably better to think of a graph model, with nodes and edges connecting the nodes that have given distance weights. Movement goes from node to node, where the distance traveled is the weight of the edge traversed.

I honestly don't see a problem with my OP.

 

[untermensche]

A surprising result came to me when trying to picture a smallest possible distance. Here is an attempt to falsify the logic of a smallest distance using a proof by contradiction.

Because this is a proof by contradiction, we will start off assuming a smallest distance exists and see what happens.

Assume there is a smallest distance between 2 points, x1 and x2. Oxford dictionary has distance as, "The length of the space between two points.". If you can agree with this definition for this purpose, continue reading.

If all of the possible options are listed below, then a smallest distance must cause a contradiction:

1) an object will travel through space from x1 to x2

2) the object will jump from x1 to x2 taking no time

3) or it will jump taking some amount of time.

For (1), the object is obviously breaking the assumption of a minimal distance. For (2) and (3), the object travels no space; therefore, it was never actually a distance.

Are you assuming space is continuous or quantized?

 

[ryan]

A surprising result came to me when trying to picture a smallest possible distance. Here is an attempt to falsify the logic of a smallest distance using a proof by contradiction.

Because this is a proof by contradiction, we will start off assuming a smallest distance exists and see what happens.

Assume there is a smallest distance between 2 points, x1 and x2. Oxford dictionary has distance as, "The length of the space between two points.". If you can agree with this definition for this purpose, continue reading.

If all of the possible options are listed below, then a smallest distance must cause a contradiction:

1) an object will travel through space from x1 to x2

2) the object will jump from x1 to x2 taking no time

3) or it will jump taking some amount of time.

For (1), the object is obviously breaking the assumption of a minimal distance. For (2) and (3), the object travels no space; therefore, it was never actually a distance.

Are you assuming space is continuous or quantized?

I am assuming your smallest possible distance and attempting to prove that it isn't logical.

 

[beero1000]

Must it? Why? Weren't you talking about point positions which take up no space at all?

I cleaned up my last post, but it wouldn't make a difference to your response.

It seems as though it doesn't matter if the positions are points or real volumes. I'll explain below.

Like I said, definitions are important. You seem to be conflating the point locations with all the space in their 'cells'. Firstly, there are distance models where you won't be able to define geometric cells like that and secondly, you are contradicting your assumption that the point locations are the only locations there are.

If positions can exist between points, then the two positions in the assumption were not the smallest distance apart.

It's probably better to think of a graph model, with nodes and edges connecting the nodes that have given distance weights. Movement goes from node to node, where the distance traveled is the weight of the edge traversed.

I honestly don't see a problem with my OP.

That's because you aren't thinking through the implications of your assumptions and haven't clarified your definitions. It seems to me that your model is essentially just the standard continuous space partitioned into cells. Using the standard distance function obviously leaves no shortest distance, but you haven't considered other models of space and distance that do have a shortest distance.

 

[untermensche]

Are you assuming space is continuous or quantized?

I am assuming your smallest possible distance and attempting to prove that it isn't logical.

But what is the nature of the medium the object is moving in?

And my position is there no smallest movement, that is an imaginary concept not a real world possibility, only a smallest movement possible.

And anything that moves makes the smallest movement possible first when it moves.

 

[ryan]

I cleaned up my last post, but it wouldn't make a difference to your response.

It seems as though it doesn't matter if the positions are points or real volumes. I'll explain below.

Like I said, definitions are important. You seem to be conflating the point locations with all the space in their 'cells'. Firstly, there are distance models where you won't be able to define geometric cells like that and secondly, you are contradicting your assumption that the point locations are the only locations there are.

If positions can exist between points, then the two positions in the assumption were not the smallest distance apart.

It's probably better to think of a graph model, with nodes and edges connecting the nodes that have given distance weights. Movement goes from node to node, where the distance traveled is the weight of the edge traversed.

I honestly don't see a problem with my OP.

That's because you aren't thinking through the implications of your assumptions and haven't clarified your definitions. It seems to me that your model is essentially just the standard continuous space partitioned into cells. Using the standard distance function obviously leaves no shortest distance, but you haven't considered other models of space and distance that do have a shortest distance.

I could have been more thorough.

(2) and (3) would result from the object being larger than the minimal distance that it would otherwise have to travel to go from x1 to x2. And if the object is equal to the distance between spaces x1 an x2, then the object jumps to a space between x1 and x2 which contradicts the assumption of minimal space between x1 and x2. And the final option is that the object has a shorter dimension of space (in the direction it travels) that it would occupy between the minimal distance x1 to x2, thus contradicting the assumption again.

 

[ryan]

I am assuming your smallest possible distance and attempting to prove that it isn't logical.

But what is the nature of the medium the object is moving in?

And my position is there no smallest movement, that is an imaginary concept not a real world possibility, only a smallest movement possible.

And anything that moves makes the smallest movement possible first when it moves.

What's the difference between the smallest movement and the smallest movement possible?

 

[DBT]

Probably relative, the smallest movement may relate to what one object can achieve but not another object, which may move in even smaller increments. While ''the smallest possible movement'' may relate what is possible to an object that is capable of making the smallest possible movement, Planck length or whatever...

 

[ryan]

Probably relative, the smallest movement may relate to what one object can achieve but not another object, which may move in even smaller increments. While ''the smallest possible movement'' may relate what is possible to an object that is capable of making the smallest possible movement, Planck length or whatever...

But UM, said, "And anything that moves makes the smallest movement possible first when it moves.". There was no qualifying the object.

 

[DBT]

Yeah, well, I won't comment on that.

 

[ryan]

Yeah, well, I won't comment on that.

Then what is the point of answering a question way out of context like that?

I don't mean to be harsh, but I don't understand how this could happen.

 

[DBT]

Yeah, well, I won't comment on that.

Then what is the point of answering a question way out of context like that?

Just on a whim.

I don't mean to be harsh, but I don't understand how this could happen.

As far as harsh things go, it hardly even rates. Maybe launching a lawsuit would ramp it up the harshness scale somewhat?

 

[ryan]

Then what is the point of answering a question way out of context like that?

Just on a whim.

I don't mean to be harsh, but I don't understand how this could happen.

As far as harsh things go, it hardly even rates. Maybe launching a lawsuit would ramp it up the harshness scale somewhat?

I'm just in a crappy mood. My life is a nightmare lately, yet it has to be this way.

 

[DBT]

Just on a whim.

I don't mean to be harsh, but I don't understand how this could happen.

As far as harsh things go, it hardly even rates. Maybe launching a lawsuit would ramp it up the harshness scale somewhat?

I'm just in a crappy mood. My life is a nightmare lately, yet it has to be this way.

No problem. Hope things improve, Cheers.