Is the distance between any object infinite?

[ujellytek]

Hi, I just finished grade 11 in high school and I have a question that is there in infinite amount of space in between objects?
Here is my logic:

***First off what is the minimum distance matter can move within the midst of space? I'm thinking there is no minimum.

On a 2-dimentional graph there is an object at (0,0) and (0,10) (any point would work) objects A (0,0) & B (10,0). Let's say every unit in between them is 1m so ultimately the distance between them is 10m.Let us say object A shoots a bullet at object B, it has to travel through about 10m of space to get there, right? Well the bullet could theoretically .1m , .08m, .002m, .0000000432m through space; so we could keep and keep on putting those zeroes in meaning that the bullet would have in infinite distance to cover to get to object B. Get it? First it needs to move 10m which means it has to cover 5m two times, or 2 meters, 5 times, or 0.00000000000078 x amount of times or 0,0000000000000000000000000000000098969584 m y amount of times and so on?

So is there an infinite amount of space/distance-to-cover between object A & B?

Now I've asked many people in school about this and no one knew the answer and would mumble something about asymptotes and one said something about Zola's or Zopha's dilemma (I know I'm getting the Z guys name wrong, sorry), I hope some thing there gets the gears turning.

~thanks for your time!

 

[jedishrfu]

You're looking for Zeno's paradoxes:

https://en.wikipedia.org/wiki/Zeno's_paradoxes

 

[phinds]

If two objects are X distance apart, X cannot be infinite because there is always X + 1.

The confusion comes from trying to apply numerical rules to something that is not numerical. It doesn't matter how many zeros you add, you STILL have a number and it is not infinity

 

[ujellytek]

@phinds so say a pixel is traveling across the screen of a computer, it moves from 0,0 to 0,1 to 0,2 and so on. Are you saying that an object moves like that through space? So it does not move smoothly but sort of teleports from 0,0 to 0,1 and so on? I'm sorry I'm not sure how to phrase what I have just asked, I hope it makes sense.

 

[phinds]

I understand exactly what you are asking but I have no idea how you inferred that from what I said. I made no comment on the granularity of space-time. I don't think there's any proof one way or the other, but my belief is that it is continuous.

 

[ujellytek]

What is "granularity" of space?

 

[ujellytek]

Guys, I'm thinking that space cannot be gridded, cause there would be minimum movement distances for objects. For example say that an object received 0.0000001N it would not be able to move to the next point on the "grid" of space but if it got 0.0000002N of energy if would move to the next point. I am thinking that this is not what happens on the space fabric, I'm thinking that the bullet mentioned above, moves smoothly from one point on the grid to the next, not suddenly appearing at the second point. Hope that makes sense.

 

[phinds]

What is "granularity" of space?

Granular means "grainy" as in "made up of grains" instead of smooth. My original post made no comment on whether space-time is smooth or not but as I said in my next post, I personally think (without any proof) that it is.

 

[ujellytek]

Hmm, I searched up granularity of space and I've figured out that this topic belongs in the quantum section

 

[phinds]

Hmm, I searched up granularity of space and I've figured out that this topic belongs in the quantum section

Does it matter, now that your question has been answered?

 

[ujellytek]

It hasn't been answered but that's because nobody knows the answer but thanks for your input, that's the furthers this conversation will probably go.

 

[russ_watters]

It hasn't been answered but that's because nobody knows the answer but thanks for your input, that's the furthers this conversation will probably go.

Your question was answered, it just doesn't seem like you read or understood the answers. The issue you are having is common, ancient and comes from some basic misconceptions about math. I encourage you to reread the above Posts and in particular read the link about Zeno's Paradox.

 

[mgkii]

Hi Ujellytek

I think you're putting two different concepts together and maybe getting a bit mixed. Here goes...

First the infinities. Infinity is a math concept pure and - I was going to say simple, but let's be real - pure and wonderfully complex. However, we don't see infinities in real word space/time - infinities pop up all over the place when trying to model the real world and we've got some fabulous tricks to deal with them (google why 1+2+3+4.. = minus 1/12). But that's just maths and models; there are no specific evidence of any infinite (or infinitesimal) distances, temperatures, densities or the real world - just areas where our models break down. For example, we can't accurately say that black holes have an infinite density, only that the density is tending towards infinity and then who knows... that's when the model breaks down!

Next Grainy space. You are absolutely spot on that this belongs in the quantum section! Quantum mechanics is possibly the most thoroughly tested and accurate theory we have; as we've been able to measure distances, time, magnetic moments and zillion other things to greater and greater accuracy, QM predictions have been shown to be accurate to within a few parts per many trillions of whatever is being measured. It's all good stuff! And if you extrapolate from the maths to real life (*warning*) then it certainly looks like life isn't continuous at the tiny (plank) scale, but quite digital!

Why the warning? For similar reasons that you can't take EVERY maths concept and map it to real world (infinity as a case in point), you can't be sure that our interpretation of QM is correct - in fact QM interpretations are akin to religious wars to some! Is life grainy? I've no idea... but spend some time in the QM threads and you'll quickly have a far better understanding than I do :-)

Regards
Matt

 

[tadchem]

This discussion just provoked a thought. General Relativity allows that space and time can be interchanged by motion. This is a continuous function of velocity (itself a continuous variable) as described by the Lorenz transforms. It is this continuity that precludes the 'granularity' of space or time. Both must be continuous variables.

 

[Drakkith]

So is there an infinite amount of space/distance-to-cover between object A & B?

No, the distance between A and B can be divided into an infinite amount of 'pieces', but there is NOT an infinite amount of distance between them because each piece also gets smaller as you divide the distance into more pieces.

 

[mgkii]

Hey - I got no beef with Einstein! If he say's it's a continuous function then my mortgage is on it being a continuous function!

That said (:-) continuous functions are maths concept and no matter how well the maths describes the world, maths and the world are not the same. And even the Duke himself didn't get his relativity and quantum mechanics to stack up at the tiny level.

There's a fabulous paper written by Eugene Wigner called The Unreasonable Effectiveness of Mathematics in the Natural Sciences and it can make your brain ache and pancake in equal measures; but the underlying message (to me anyway - Wigner was a bit before my time to ask) is as maths as a descriptor of reality, not as an equivalence.

 

[Drakkith]

There's no need to get into the details of the granularity of space. The concept is so far over the head of the OP (and most members for that matter) that it serves no purpose. Let's try to keep the discussion at a level appropriate for the OP.

 

[ujellytek]

@Drakkith I see what you mean. But wouldn't it mean that if an object could move less than 0.002 m or less than 0.0000000000000001m or less
than 0.(lots of zeroes)0001m that an object could theoretically move an infinite amount of times before moving 1m.
#13
@mgkii About the 1+2+3... = -1/12: My math teacher told me it is inpossible to add numbers infinitly because infinity is only a concept, not a variable

 

[mgkii]

@ujellytek you're getting infinity confused here; you can't move an infinite number of times - infinity isn't a number. Have a browse around wiki etc on infinity; the penny will drop.

 

[ujellytek]

mqkii>"you're getting infinity confused here; you can't move an infinite number of times - infinity isn't a number. Have a browse around wiki etc on infinity; the penny will drop. "

Imean that you could do if forever, always moving lesser and lesser amount and never reaching your destination

 

[mgkii]

@ujellytek Your maths teach was correct - you can't actually add all the numbers, but you can define a mathematical function that is the sum of the set of all integer numbers. For the same reason, you can't actually move (in real space) an infinite number of times, but you can make models that deal with infininte /infinitesimal distances.

 

[russ_watters]

@Drakkith I see what you mean. But wouldn't it mean that if an object could move less than 0.002 m or less than 0.0000000000000001m or less
than 0.(lots of zeroes)0001m that an object could theoretically move an infinite amount of times before moving 1m.
#13
@mgkii About the 1+2+3... = -1/12: My math teacher told me it is inpossible to add numbers infinitly because infinity is only a concept, not a variable

It should be easy to see that if you move in 1 step of 1 meter, 2 steps of 1/2 meter, 3 steps of 1/3 meter, the total is always the same.

You shouldn't need to take the logic any further to get the above, but introduction of infinities and infinitesimal is easily solved with calculus.

Imean that you could do if forever, always moving lesser and lesser amount and never reaching your destination

Nowhere in this analysis has time been included, so you have no basis for saying that. However, it should be obvious enough that if your speed is constant, the total time follows the same math i gave above and the total time is therefore always the same regardless of the number of steps you choose to split the trip into.

 

[ujellytek]

Wow thanks for everybody's input, this is all so mind blowing for me I'm getting tears in my eyes (anyone else get that when so many things all of a sudden make sense?)

 

[Drakkith]

@Drakkith I see what you mean. But wouldn't it mean that if an object could move less than 0.002 m or less than 0.0000000000000001m or less
than 0.(lots of zeroes)0001m that an object could theoretically move an infinite amount of times before moving 1m.

That implies that motion is discrete and an object moves X distance Y number of times between 0 and 1 meter, where X is an ever decreasing number as Y increases. As far as we know, this isn't how nature works.

Imean that you could do if forever, always moving lesser and lesser amount and never reaching your destination

Only if you also decrease the time step between each jump. The entire point is that time and distance are continuous, not discrete. An object traveling at 1 m/s moves 1 meter every 1 second. If you decrease the time, the distance also decreases. Nothing about this implies that distance is discrete.

 

[mgkii]

I'm getting tears in my eyes (anyone else get that when so many things all of a sudden make sense?)

Lol - 40 years old and it's still happening to me.

 

[ujellytek]

One more question I think, is there a minimum amount of space/distance an object is allowed to move?

 

[mgkii]

I think I'd have to point you to the realms of the quantum thread... hang on to your sanity whilst you're in there and also hang on to what's real, what's math and what's interpretation!

 

[Drakkith]

One more question I think, is there a minimum amount of space/distance an object is allowed to move?

There is not.

 

[Drakkith]

I think I'd have to point you to the realms of the quantum thread... hang on to your sanity whilst you're in there and also hang on to what's real, what's math and what's interpretation!

As far as I know quantum physics says nothing about a minimum distance an object is allowed to move.

 

[ujellytek]

Its weird to think that if an object traveled downwards a certain amount, it could have traveled an infinite amount of times, eh? or because of time it cannot move an infinite amount of times, geez my mind is going to explode

 

[mgkii]

There is not.

Is it really that definitive? The Planck length is the smallest distance, and I'm not sure whether the concept of moving a distance smaller than the Planck length is meaningful?

And when you get down to those sizes, we have to consider what it is that @ujellytek is moving? Heisenberg's uncertainty principle must play havoc in understanding whether you have actually moved that distance, no matter how slowly you attempt to do it!

 

[Drakkith]

Its weird to think that if an object traveled downwards a certain amount, it could have traveled an infinite amount of times, eh? or because of time it cannot move an infinite amount of times, geez my mind is going to explode

Don't think of it as 'traveled X number of times'. Think of it as a smooth, continuous motion. Just because an object travels X distance in Y time does not mean that that distance is 'different' or 'separate' from the rest of its motion.

Is it really that definitive? The Planck length is the smallest distance, and I'm not sure whether the concept of moving a distance smaller than the Planck length is meaningful?

The Planck length is not the smallest possible distance. It is the smallest distance in which you can theoretically measure. If a particle moves 1 Planck lengthin Y time, how far does it move in Y/2 time? Half a Planck length. Whether you can measure that distance is another question.

 

[ujellytek]

this thread boils down to that in space, an object either has an infinite distance to cover because before it can move/reach-its-destination-of-say 1m it must move .99m but before it could move that it needs to move .98 m and so on but that ill not work because it will never reach its destination, thus space must be gridded but that means that there is a minimum distance an object can move. Am I on the right track?

 

[mgkii]

Sorry @ujellytek I don't think you are. I think you've gone back full circle on infinites. What you're actually doing is almost calculus - looking at smaller and smaller movements in smaller and smaller units of time; both of which are tending to zero. Don't mix the maths with the real life - it does not follow that space must be "gridded"

 

[Doc Al]

this thread boils down to that in space, an object either has an infinite distance to cover because before it can move/reach-its-destination-of-say 1m it must move .99m but before it could move that it needs to move .98 m and so on but that ill not work because it will never reach its destination, thus space must be gridded but that means that there is a minimum distance an object can move. Am I on the right track?

No. As advised in the first response to your post, please look into Zeno's paradox and its resolution.

Having an infinite number of "steps" does not imply an infinite distance.

 

[mgkii]

@ujellytek to help you get your head around the infinities and why they don't map to real life, then as well as Zeno's paradox (and as Doc Al says - specifically its resolution), have a look at
- gabriel's horn (an object with infinite surface area but a finite volume)
- Kock's Snowflake (a shape with a perimeter of infinite length, but a finite volume)

 

[Thor90]

The Planck length is not the smallest possible distance. It is the smallest distance in which you can theoretically measure. If a particle moves 1 Planck lengthin Y time, how far does it move in Y/2 time? Half a Planck length. Whether you can measure that distance is another question.

Well there are theories in which the possible distances are quantized (I do not mean the energy levels of a particle in a given potential, I am saying theories in which you have a grid in the space time), maybe you are referring to validated teories? I think there is no special reason why you should not have this kind of configurations below the plank scale, as I see neither for the other point of view, since at length over the plank scale you would see them as a continuum anyway.

this thread boils down to that in space, an object either has an infinite distance to cover because before it can move/reach-its-destination-of-say 1m it must move .99m but before it could move that it needs to move .98 m and so on but that ill not work because it will never reach its destination, thus space must be gridded but that means that there is a minimum distance an object can move. Am I on the right track?

As stated by other before, no. Since you divide your path in a number of increasing intervals, these intervals must become tinyer as you increase their number. So, while the number of intervals goes to infinity, the size of these intervals actually goes to 0, so if you call l your total distance to cover, n the number of intervals and D their lenght, you have

l = n x D --> inf x 0

which is an indefinite form, but if you study a little bit of infinitesimal calculus you'll understand that it converges to a finite distance.
It's like dividing a square figure you have drown on a paper in more little squares: you can make them as small as you want, but the figure you have in your paper won't grow in size just because you are dividing it.

 

[Drakkith]

this thread boils down to that in space, an object either has an infinite distance to cover because before it can move/reach-its-destination-of-say 1m it must move .99m but before it could move that it needs to move .98 m and so on but that ill not work because it will never reach its destination, thus space must be gridded but that means that there is a minimum distance an object can move. Am I on the right track?

How could the distance it has to travel be infinite when you just said it needs to move a distance of 1 meter to get from point A to point B? Whatever number of 'grids' you try to add up, the sum of the combined distances will ALWAYS be 1 meter.

an object either has an infinite distance to cover because before it can move/reach-its-destination-of-say 1m it must move .99m but before it could move that it needs to move .98 m and so on but that ill not work because it will never reach its destination

At 1 m/s, an object takes 0.98 seconds to go from X = 0 to X = 0.98. It takes an additional 0.01 seconds fro the object to go from X = 0.98 to X = 0.99. The total time it takes to go from X = 0 to X = 0.99 is 0.99 seconds. There's really not much else to it.

 

[ujellytek]

Right, I'm not understanding how an object suddenly changes its position from 0.000001m to 0.000002m (w/e the distance is) there is always distance in between which must be covered. The object just jumps through space into the 0.000002m point. Is not it somehow supposed to pass through 0.0000011m, then0.0000012m, then 0.0000013m all the way to 0.000002m ? Then that means that on the objects way to its destination it must pass through 0.00000101m but before that 0.000001001m.

 

[russ_watters]

Right, I'm not understanding how an object suddenly changes its position from 0.000001m to 0.000002m (w/e the distance is) there is always distance in between which must be covered. The object just jumps through space into the 0.000002m point. Is not it somehow supposed to pass through 0.0000011m, then0.0000012m, then 0.0000013m all the way to 0.000002m ? Then that means that on the objects way to its destination it must pass through 0.00000101m but before that 0.000001001m.

Yes, you can cut up a number line into an infinite amount of segments. Or to say it another way, between any two points are an infinite number more points.

But do not make the mistake of thinking that the abvove implies that an objects must move in steps from one point to another.

 

[ujellytek]

So how does it move?

 

[Drakkith]

Right, I'm not understanding how an object suddenly changes its position from 0.000001m to 0.000002m (w/e the distance is) there is always distance in between which must be covered. The object just jumps through space into the 0.000002m point. Is not it somehow supposed to pass through 0.0000011m, then0.0000012m, then 0.0000013m all the way to 0.000002m ? Then that means that on the objects way to its destination it must pass through 0.00000101m but before that 0.000001001m.

An object moving from A to B passes through every point between them. There is no sudden jump.

 

[ujellytek]

isn't there an infinite amount of points in between them?

 

[Drakkith]

isn't there an infinite amount of points in between them?

Yes there are.

 

[ujellytek]

So an object passes an infinite amount of points in a finite amount of time? that sounds impossible to me.

 

[Drakkith]

So an object passes an infinite amount of points in a finite amount of time? that sounds impossible to me.

There are also an infinite amount of time steps between 0 and 1 second. That's really what we're dealing with. The object travels X distance in Y time, both of which are finite numbers. It also passes through an infinite amount of points in space and time as it does so.

 

[ujellytek]

Wow that`s mind blowing, thanks!

 

[russ_watters]

So how does it move?

Continuously. Not stopping at every point.

 

[russ_watters]

Wow that`s mind blowing, thanks!

This is a feature of the number systems we use. It is true for any range of numbers, that they are infinitely divisible.

Edit: wrong link.
https://en.m.wikipedia.org/wiki/Real_number

 

[mgkii]

Don't fall into the trap of thinking the math equals reality; it's just a model. Numbers are, in this context, discrete- I.e you jump from one to the next. It's the best model we have, but it's not perfect. Reality as far as we know is continuous.