Fun with numbers

[horvack]

I got my 2nd vaccination shot today, so while waiting around for the required 15 minutes, I did a little research.

• Age of the universe in seconds: 4.355 * 10¹⁷

• Number of atoms in the universe: 10⁸²

• Number of cubic plank lengths in the observable universe: 4.65 * 10¹⁸⁵ (this is probably the weakest estimate, it comes from https://davidpratten.com/2008/02/11/add ... -universe/.)

So, I multiply those 3 numbers to get 2.025075 * 10²⁸⁵.

Convert that to binary and I get 1.10101110011... * 2⁹⁴⁶, so 947 bits.

This implies to me that we could map, in excruciating detail, the precise shape and location of everything we can observe (that's not subatomic)...
for all time up to now...
on my smartphone...
using only working memory (RAM)...
and have space left over for another 3 million universes of equal size! (in 4GB).

Someone please correct me.

[Utisz]

I got my 2nd vaccination shot today, so while waiting around for the required 15 minutes, I did a little research.

• Age of the universe in seconds: 4.355 * 10¹⁷

• Number of atoms in the universe: 10⁸²

• Number of cubic plank lengths in the observable universe: 4.65 * 10¹⁸⁵ (this is probably the weakest estimate, it comes from https://davidpratten.com/2008/02/11/add ... -universe/.)

So, I multiply those 3 numbers to get 2.025075 * 10²⁸⁵.

Convert that to binary and I get 1.10101110011... * 2⁹⁴⁶, so 947 bits.

This implies to me that we could map, in excruciating detail, the precise shape and location of everything we can observe (that's not subatomic)...
for all time up to now...
on my smartphone...
using only working memory (RAM)...
and have space left over for another 3 million universes of equal size! (in 4GB).

Someone please correct me.

All you have done is encoded the number you computed (atoms × planck-cubes × seconds) in binary. So you've stored the number 2.025075 * 10²⁸⁵ on your phone in plain binary. You've not actually encoded any state, or anything like that, so you have no shape, no location, etc., just how many elements there are.

I think where you were heading was towards encoding a (bit) state for each element you identified, but that would require 2.025075 * 10²⁸⁵ bits (uncompressed), one for each element.

Not sure why you multiply seconds, atoms and planck-lengths. I guess you're trying to one-hot encode which atoms are occupying which planck-lengths, but didn't really understand where you were going with that part? It's not clear that what you find at a planck-length can be discretised so easily into bits. We cannot observe anything below that length, but there's still some funky quantum shit going on at that scale.


I use a distinct but related counter-argument to folks who like to say "the universe is so big that we can't be alone". The argument is to consider a hypothetical switchboard associated with each planet. Each switch needs to be turned on for a planet to become habitable. So one switch would be for the planet to be in a habitable zone of its star. Another would be for it to have carbon. Another would be ... To simplify, assume that each switch has a 50/50 chance of being turned on or off (if some condition for life is more improbable than 50/50, it might require four or five switches to be on; conversely we might group several probable things into one switch).

Now even if the standard switchboard for life on a planet contains just 300 switches, and even if every atom in the observable universe were a planet, the odds of finding an atom with all switches on would be 1 in several billion.

[horvack]

I got my 2nd vaccination shot today, so while waiting around for the required 15 minutes, I did a little research.

• Age of the universe in seconds: 4.355 * 10¹⁷

• Number of atoms in the universe: 10⁸²

• Number of cubic plank lengths in the observable universe: 4.65 * 10¹⁸⁵ (this is probably the weakest estimate, it comes from https://davidpratten.com/2008/02/11/add ... -universe/.)

So, I multiply those 3 numbers to get 2.025075 * 10²⁸⁵.

Convert that to binary and I get 1.10101110011... * 2⁹⁴⁶, so 947 bits.

This implies to me that we could map, in excruciating detail, the precise shape and location of everything we can observe (that's not subatomic)...
for all time up to now...
on my smartphone...
using only working memory (RAM)...
and have space left over for another 3 million universes of equal size! (in 4GB).

Someone please correct me.

All you have done is encoded the number you computed (atoms × planck-cubes × seconds) in binary. So you've stored the number 2.025075 * 10²⁸⁵ on your phone in plain binary. You've not actually encoded any state, or anything like that, so you have no shape, no location, etc., just how many elements there are.

I think where you were heading was towards encoding a (bit) state for each element you identified, but that would require 2.025075 * 10²⁸⁵ bits (uncompressed), one for each element.

Not sure why you multiply seconds, atoms and planck-lengths. I guess you're trying to one-hot encode which atoms are occupying which planck-lengths, but didn't really understand where you were going with that part? It's not clear that what you find at a planck-length can be discretised so easily into bits. We cannot observe anything below that length, but there's still some funky quantum shit going on at that scale.


I use a distinct but related counter-argument to folks who like to say "the universe is so big that we can't be alone". The argument is to consider a hypothetical switchboard associated with each planet. Each switch needs to be turned on for a planet to become habitable. So one switch would be for the planet to be in a habitable zone of its star. Another would be for it to have carbon. Another would be ... To simplify, assume that each switch has a 50/50 chance of being turned on or off (if some condition for life is more improbable than 50/50, it might require four or five switches to be on; conversely we might group several probable things into one switch).

Now even if the standard switchboard for life on a planet contains just 300 switches, and even if every atom in the observable universe were a planet, the odds of finding an atom with all switches on would be 1 in several billion.

Wait... What the difference between knowing how many elements there are (of 4 dimensional space-time), and posting a 0/1 flag for each of those elements, which can be evaluated to show which atoms are present in which cubic plank lengths?

[Senseye]

Further to Utiz point about not encoding any information, you would need 947 bits to simply identify (label, address - call it what you will) each atom at a point in time. That's about 118 bytes per address. So if you want to store all 2**946 of them in your phone, you are going to need considerably more than 4GB.

4,096,000/118 = 34,711 (rounded) so that is how many unique atoms you can reference in your phone's address book, but again, that just gives them a label, you still don't know where (or when) to find them.

[Utisz]

Wait... What the difference between knowing how many elements there are (of 4 dimensional space-time), and posting a 0/1 flag for each of those elements, which can be evaluated to show which atoms are present in which cubic plank lengths?

Now sure I completely understood the question, but assume we have eight coins.

2³ = 8, so to store the number 8 in binary we need 3 bits (more or less).

But if we want to encode which coin lands heads and which lands tails in a coin flip, we cannot do that in 3 bits, rather we need 8 bits.

Storing the number of things is not the same as storing state for those things.

[Utisz]

Further to Utiz point about not encoding any information, you would need 947 bits to simply identify (label, address - call it what you will) each atom at a point in time. That's about 118 bytes per address. So if you want to store all 2**946 of them in your phone, you are going to need considerably more than 4GB.

4,096,000/118 = 34,711 (rounded) so that is how many unique atoms you can reference in your phone's address book, but again, that just gives them a label, you still don't know where (or when) to find them.

You could map labels to addresses using a fixed-length encoding, so they become one in the same. Then you don't need to label explicitly as the offset is the label.

But yep, you're going to need a bigger boat if you want to actually encode something besides the number itself.

[Ferrus]

What if the universe is a Turing machine (or a quantum variant thereof) and we used a standard numbering system to definite it? Surely in that case the encoding number of said machine could be stored in much lower memory than storing its entire state? Although to actually run it you would need that capacity.

[Utisz]

What if the universe is a Turing machine (or a quantum variant thereof) and we used a standard numbering system to definite it? Surely in that case the encoding number of said machine could be stored in much lower memory than storing its entire state? Although to actually run it you would need that capacity.

You mean a Gödel number for the universe? It would be the same thing.

The bits you use to encode the state just become the Gödel number in binary (well, I guess it depends on the exact encoding used, but you cannot beat entropy in this way).

edit: maybe you are referring to the Kolmogorov complexity of the universe, meaning that you do not store the state, but rather you store a way to compute the state. In this case it might be possible to compress the state of the universe, but that would require some sort of fundamental theory of everything regarding the computability/regularity of the universe. (And computing Kolmogorov complexity is undecidable in the general case.)

[Ferrus]

edit: maybe you are referring to the Kolmogorov complexity of the universe, meaning that you do not store the state, but rather you store a way to compute the state. In this case it might be possible to compress the state of the universe, but that would require some sort of fundamental theory of everything regarding the computability/regularity of the universe. (And computing Kolmogorov complexity is undecidable in the general case.)

Yeah that is what I meant - I had heard of Kolmogorov complexity but hadn't investigated it before, it looks a really interesting topic, I will definitely have to read up on it - though I was thinking ignoring computing it more as a hypothetically existing number that could potentially be stored. It is of course speculative but AFAIK not entirely ruled out empirically.

Indeed couldn't it be argued that physics since Newton is a (perhaps futile) attempt to create the logical building blocks (or routines) of what would be the code of any theoretical 'universe program'? Modern quantum mechanics with quantum fields seems to also allow the possibility that state could emerge spontaneously from a running simulation as there is no longer the Newtonian idea of an initial condition made up of every particle's position and velocity, although the individual probabilistic decisions that make up the path that our universe took would not be compressible.

[horvack]

Ah, I see my mistake now. Yes, I was trying to calculate a Gödel number for the universe, but instead I stopped at the state of the smallest of my arbitrarily chosen units (does atom X occupy the cubic plank length Y at time Z seconds).
So, yeah. I was only off by a couple hundred orders of magnitude!

How would you approach getting the state of the universe in the smallest possible compression?