B Space is discrete?

1. Oct 1, 2018

DannyTr

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.

2. Oct 1, 2018

mjc123

Infinity * 10000 = infinity is not a logical contradiction

3. Oct 1, 2018

4. Oct 1, 2018

DannyTr

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

5. Oct 1, 2018

DannyTr

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)

6. Oct 1, 2018

Staff Emeritus
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.

7. Oct 1, 2018

jbriggs444

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

8. Oct 1, 2018

DannyTr

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)

Last edited: Oct 1, 2018
9. Oct 1, 2018

jbriggs444

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.

10. Oct 1, 2018

DannyTr

- 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?

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

11. Oct 1, 2018

Staff: 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?

12. Oct 1, 2018

jbriggs444

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.
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.

Last edited: Oct 1, 2018
13. Oct 1, 2018

Drakkith

Staff Emeritus
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.

I see no reason that this is true.

14. Oct 1, 2018

Staff Emeritus
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!

15. Oct 1, 2018

jbriggs444

If you cannot measure it, then its existence is a matter of interpretation rather than of physical fact.

16. Oct 1, 2018

DannyTr

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.

17. Oct 1, 2018

Matterwave

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.

18. Oct 1, 2018

DannyTr

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.

19. Oct 1, 2018

Drakkith

Staff Emeritus
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.

20. Oct 1, 2018

Matterwave

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?