# Space is discrete?

• B
I be grateful for any feedback on this argument:

- First assume space is continuous
- Then there is an actually infinite amount of information in a spacial volume of 10000 cubic units
- There is also an actually infinite amount of information in a spacial volume of 1 cubic unit
- So space must be discrete.

mjc123
Homework Helper
Infinity * 10000 = infinity is not a logical contradiction

Infinity * 10000 = infinity is not a logical contradiction

- A little confused.
- Infinity * 10000 = infinity
- Implies a grain of sand contains the same information as the whole universe?

Thanks. I'm a little confused though:

If space truly is continuous then each particle has a infinite amount of information in a sense:
- The particle has position (x, y, z) in space.
- If space is continuous then the positional co-ordinates have infinite precision thus infinite information?

(I may not be using the term 'information' in the conventional physics sense... sorry)

Staff Emeritus
2021 Award
This boils down to "I don't like infinities". Infinities certainly have different properties than we are used to - for example, there are the same number of odd natural numbers as natural numbers in total.

jbriggs444
Homework Helper
If space truly is continuous then each particle has a infinite amount of information in a sense:
- The particle has position (x, y, z) in space.
- If space is continuous then the positional co-ordinates have infinite precision thus infinite information?
That could apply if "position" was a property of a particle and if it were precisely knowable. It isn't.

rrogers
That could apply if "position" was a property of a particle and if it were precisely knowable. It isn't.

I thought particles had a position when you measure them?

If space is truly continuous each particle must have infinite precision for it position (we could not measure it accurately, but the information would still be there in the system).

- So there is infinite information for one particle
- And also infinite information for all particles in the universe
- The two infinities are in one-to-one correspondence so have same cardinality (N0)
- So maths says the same amount of information in a particle as for the whole universe. Reductio ad absurdum
- So either mathematics treatment of infinity is wrong or space is discrete (or both I suspect)

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jbriggs444
Homework Helper
I thought particles had a position when you measure them?

If space is truly continuous each particle must have infinite precision for it position (we could not measure it accurately, but the information would still be there in the system).
First, you cannot measure position with infinite precision. Second, the assumption that the position you measure is the position that existed before you measured is not justified.

First, you cannot measure position with infinite precision.

- But the particle still has infinite precision (its just we can't measure it accurately) so we could say the particle has infinite (but unmeasurable) information?

Second, the assumption that the position you measure is the position that existed before you measured is not justified.

- I don't think this effects my argument (it does not matter if we can't measure accurately)

russ_watters
Mentor
I'm more interested in the math issue here, so I have two questions:

1. Is x<2x true for all values of x even as x tends to infinity?

2. Does pi, being an irrational number, contain an infinite amount of data?

sophiecentaur
jbriggs444
Homework Helper
I'm more interested in the math issue here, so I have two questions:

1. Is x<2x true for all values of x even as x tends to infinity?
That one is easy. Yes. For all real-valued x > 0, x < 2x.

Note that this does not imply that ##\lim_{x \rightarrow +\infty} {x} < \lim_{x \rightarrow +\infty} {2x}## is true or even well defined.
2. Does pi, being an irrational number, contain an infinite amount of data?
That one is more difficult. One would have to specify the information content of pi. One way would be to measure its Kolmogorov complexity: The length of the smallest program that can compute pi. That is finite. Definitely finite.

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PeterDonis and russ_watters
Drakkith
Staff Emeritus
If space is truly continuous each particle must have infinite precision for it position (we could not measure it accurately, but the information would still be there in the system).

Infinite precision doesn't really make sense in this context. Or at least it leads to confusion with the terminology. A better term might be perfect precision. Precision also isn't the same thing as information. Precision is related to our ability to measure things, it is not something inherent to an object, so it doesn't automatically follow that precision implies any amount of information. If a particle truly has a single position at a certain time, then it would seem that this actually only requires a very small amount of information. Just a single number in fact, if you define information in this way.

- So there is infinite information for one particle

I see no reason that this is true.

phinds
Staff Emeritus
2021 Award
We could take all the books in all the libraries, and concatenate their binary representation, e.g. 0.1101011010111001001... That is a real number, so we take a stick and mark it at exactly that point. Presto...all of mankind's knowledge in a stick!

DannyTr and jbriggs444
jbriggs444
Homework Helper
we could say the particle has infinite (but unmeasurable) information?
If you cannot measure it, then its existence is a matter of interpretation rather than of physical fact.

Precision also isn't the same thing as information. Precision is related to our ability to measure things, it is not something inherent to an object

As well as our measurement ability, I think precision (of position) might relate to the nature of the universe:

- Imagine a very simple discrete universe which contained one particle that could be in one of two possible positions. Then position is 1 bit of information.
- A slightly more complex discrete universe with 8 'positional slots' for particles. In this universe, position is 4 bits of information.
- A continuous universe with infinite positional slots for particles, then position has infinite bits of information.

When the amount of information for a particle is infinite then the maths starts to break down (particle has same info content as universe). So we either need different maths - different sizes of countable infinity (to reflect the fact the universe has more info than the particle) or a discrete universe with finite maths.

A similar argument could be made for the particle's velocity I think.

Matterwave
Gold Member
I be grateful for any feedback on this argument:

- First assume space is continuous
- Then there is an actually infinite amount of information in a spacial volume of 10000 cubic units
- There is also an actually infinite amount of information in a spacial volume of 1 cubic unit
- So space must be discrete.

How are you defining "information" to come up with these results? All of the "information" definitions that I am aware of involves matter and how much "information" we have on that matter (which is closely related with the entropy of that matter). How are you assigning information to space itself? Admittedly, I do not know of every definition of information there is out there.

How are you defining "information" to come up with these results? All of the "information" definitions that I am aware of involves matter and how much "information" we have on that matter (which is closely related with the entropy of that matter). How are you assigning information to space itself? Admittedly, I do not know of every definition of information there is out there.

If you check #16 above, I'm using my own definition of information: a region of space contains information on the positions of particles within it.

Drakkith
Staff Emeritus
- Imagine a very simple discrete universe which contained one particle that could be in one of two possible positions. Then position is 1 bit of information.
- A slightly more complex discrete universe with 8 'positional slots' for particles. In this universe, position is 4 bits of information.

I don't agree with this definition of information. How we choose to represent the position of the particle should not be part of the information about the particle.

Matterwave
Gold Member
If you check #16 above, I'm using my own definition of information: a region of space contains information on the positions of particles within it.

But then why would you expect your definition of information to give rise to actual physics/mathematics? I can define "a number Z which is equal to both 1 and 2 simultaneously" - this number breaks transitivity of the equals sign in mathematics - something that is pretty fundamental. I can certainly make such a definition if I so choose, but then how can I then demand that mathematics - as practiced by the larger mathematics community - be modified in some way to accommodate my definition? Especially if I don't show any merit to my definition?

Imager, phinds and Dale
I don't agree with this definition of information. How we choose to represent the position of the particle should not be part of the information about the particle.

OK, lets instead make the particle's position information belong to the region of space that contains the particle. Then it follows that both a small region of space and the whole universe contain the same countable infinity of information (my original argument).

Drakkith
Staff Emeritus
OK, lets instead make the particle's position information belong to the region of space that contains the particle. Then it follows that both a small region of space and the whole universe contain the same countable infinity of information (my original argument).

So? Why is that a problem? A 1x1 square and a 2x2 square both contain an infinite number of points. Yet the latter is clearly larger than the former.

But then why would you expect your definition of information to give rise to actual physics/mathematics? I can define "a number Z which is equal to both 1 and 2 simultaneously" - this number breaks transitivity of the equals sign in mathematics - something that is pretty fundamental. I can certainly make such a definition if I so choose, but then how can I then demand that mathematics - as practiced by the larger mathematics community - be modified in some way to accommodate my definition? Especially if I don't show any merit to my definition?

It makes intuitive sense to me that a finite region of space should contain a finite amount of information (implying discrete space).

With continuous space, a finite region always contains an infinite amount of information (by my definition)... which seems contradictory.

Then the math from set theory seems to lead to a contradiction... if the universe is continuous its seems different math is required...

So? Why is that a problem? A 1x1 square and a 2x2 square both contain an infinite number of points. Yet the latter is clearly larger than the former.

A point is defined to have length=0. So the number of points on a line segment length 1 is: (segment length)/(point length) = 1/0 = undefined.

Similarly, a 1x1 and 2x2 square both contain an undefined number of points.

IMO the definition of a point in maths is not too great (defined to have length 0 IE points do not exist. That definition leads to contradictions).

If we use a non-zero point size (IE a discrete universe) then a 1x1 and 2x2 square do contain different numbers of points, which makes sense.

IMO the definition of a point in maths is not too great (defined to have length 0 (...))

That is not a definition, it's a property.

Drakkith
Staff Emeritus
Then the math from set theory seems to lead to a contradiction... if the universe is continuous its seems different math is required...

Standard mathematics describes a continuous universe just fine. It may not make 'intuitive' sense, but it works just fine with no mathematical contradictions.

A point is defined to have length=0. So the number of points on a line segment length 1 is: (segment length)/(point length) = 1/0 = undefined.

Similarly, a 1x1 and 2x2 square both contain an undefined number of points.

IMO the definition of a point in maths is not too great (defined to have length 0 IE points do not exist. That definition leads to contradictions).

If we use a non-zero point size (IE a discrete universe) then a 1x1 and 2x2 square do contain different numbers of points, which makes sense.

I think you need to look into the finer points of geometry. The concept of a point with no size is extremely well accepted by mathematicians. If there were contradictions it wouldn't be accepted.

Matterwave
Gold Member
It makes intuitive sense to me that a finite region of space should contain a finite amount of information (implying discrete space).

With continuous space, a finite region always contains an infinite amount of information (by my definition)... which seems contradictory.

Then the math from set theory seems to lead to a contradiction... if the universe is continuous its seems different math is required...

Something making intuitive sense does not imply that something is right - just as not making intuitive sense doesn't imply something is wrong either. It's hard to argue rigorous math and physics "from intuition". The only response I can really give you is your definition of information doesn't appear to match any definition that I am aware of. Its usefulness is suspect to me.

berkeman
berkeman
Mentor