The largest numbers

[geordief]

12 minutes ago, Genady said:

If there are E events, then there are E² pairs of events, which represent, e.g., potential causal relations between the events. E² > E.

(Actually, I anticipated your line of thinking if I am not deluding myself)

So the set of these potential  causal relations is still finite in a finite universe  and if we call the new number derived from E , E² then there will be another number ,say N(new) where N(new)>E²

9 minutes ago, sethoflagos said:

The earth's surface is almost exactly 2x10^84 square Planck units. I vaguely remember reading that something unpleasant happens when you try storing that much information on a limited surface.

What happens?A black hole?

But just creating a number is not the same as storing information(in my mind the mapping was  just theoretical.)

Edited January 22, 2023 by geordief

[Genady]

7 minutes ago, geordief said:

So the set of these potential  causal relations is still finite in a finite universe  and if we call the new number derived from E , E² then there will be another number ,say N(new) where N(new)>E²

But then there are triplets of events, pairs of causal relations, quadruplets, combinations of events and relations, etc. which all have their various roles, and the count goes up and up, unbounded.

11 minutes ago, geordief said:

creating a number is not the same as storing information(in my mind the mapping was  just theoretical.)

Yes, I understand now much better what your OP is about. Not storing information or representing it, but about relation between the numbers and the real world.

[studiot]

27 minutes ago, geordief said:

But just creating a number is not the same as storing information(in my mind the mapping was  just theoretical.)

You guys are talking about the Heine- Borel finite covering or patching theorem.

https://en.wikipedia.org/wiki/Heine–Borel_theorem

[geordief]

15 minutes ago, Genady said:

But then there are triplets of events, pairs of causal relations, quadruplets, combinations of events and relations, etc. which all have their various roles, and the count goes up and up, unbounded.

Is it unbounded?If we start with ,say 3 events then the set of all relationships is still a finite number.

And if we increase 3  to a number representing the set of all events in the spacetime of a finite universe  then the corresponding set of relationships is still also another finite  number.

 

So ,if we have a theoretical number larger than that ,it will not have its "territory " will it?

Edited January 22, 2023 by geordief

[Genady]

Finite on every step, but unbounded. E^3, E^4, E^5, etc.

[joigus]

2 minutes ago, geordief said:

Is it unbounded?If we start with ,say 3 events then the set of all relationships is still a finite number.

And if we increase 3  to a number representing the set of all events in the spacetime of a finite universe  then the corresponding set of relationships is still also another finite  number.

 

So ,if we have a theoretical number larger than that ,it will not have its "territory " will it?

I'm rather confused by your use of the term "a number's territory." You obviously mean something you're finding difficult to define --I infer that from your repeated use of quote marks. Can you be more precise?

It's obvious there are numbers so big that you would be hard pressed to find anything physical that makes them relevant. Or so I think. Is something like that what you mean?

[sethoflagos]

36 minutes ago, geordief said:

But just creating a number is not the same as storing information(in my mind the mapping was  just theoretical.)

So your OP reduces to 'What is the best compression algorithm for big numbers?'.

[geordief]

19 minutes ago, joigus said:

I'm rather confused by your use of the term "a number's territory." You obviously mean something you're finding difficult to define --I infer that from your repeated use of quote marks. Can you be more precise?

It's obvious there are numbers so big that you would be hard pressed to find anything physical that makes them relevant. Or so I think. Is something like that what you mean?

Yes ,that is what I was asking.(perhaps I was trying to shoehorn my understanding of the map/territory ,it model/modeled idea into the discussion since I find it very important in it's own right)

22 minutes ago, sethoflagos said:

So your OP reduces to 'What is the best compression algorithm for big numbers?'.

Perhaps ,but I don't see it.I don't have that expertise.

[sethoflagos]

57 minutes ago, geordief said:

Perhaps ,but I don't see it.I don't have that expertise.

Neither do I, but compression ratios are still finite and therefore the Bekenstein bounds still limit the amount of information that can be stored in a given space.   

[geordief]

42 minutes ago, sethoflagos said:

Neither do I, but compression ratios are still finite and therefore the Bekenstein bounds still limit the amount of information that can be stored in a given space.   

I looked up Berkenstein  bounds (very cursorily as befits my weak mathematical and physical brain) and it seems to say that for a given finite system  it is possible to describe it (presumably with a number) in only a finite way. 

If that is a correct interpretation  then one can extrapolate that finding to a  system that is the entirety of a finite universe**  and so it is possible to create a number that is larger than that number.

I think that is what I was trying to say.(not in the OP  itself because my assumption there seems to have been incorrect  and I changed my tune thereafter)

 

**ie describe it with a finite number

[Genady]

14 minutes ago, geordief said:

the entirety of a finite universe

If "finite universe" means a universe with a finite number of states, then you are right: there is a largest number that describes all of it and any number larger than that one is unnecessary in that universe.

[geordief]

13 minutes ago, Genady said:

If "finite universe" means a universe with a finite number of states, then you are right: there is a largest number that describes all of it and any number larger than that one is unnecessary in that universe.

Yes.

(Not that I was claiming that the universe is actually either finite or infinite .I don't think there is any way to know)

Edited January 23, 2023 by geordief

[studiot]

13 hours ago, geordief said:

13 hours ago, Genady said:

I think I understand now. And tend to disagree. I think that any number can be mapped onto something in the real world.

Even if the real world is finite?

 

13 hours ago, Genady said:

Yes.

 

I think this very important conversation got lost in a welter of other extraneous things.

Genady is mathematically correct.

I pointed you at the maths of this.

It rather depends whether you want to regard Heine-Borel as a set-theoretic or  geometric or topological theorem .

Cantor started it with his definition of an infinite set as being a set that one can put part of the set (a subset) into one-to-one with the whole set.

The HB theorem says that you can 'cover' an infinite set with a finite number of subsets.

Loosely speaking that means you can find a partner for every member of the infinite set in a finite number of subsets.

 

The simplest example is one dimensional and we can incorporate infinite on a line within a single subset called an interval.

Quote

(Heine-Borel in One Dimension) Let a and b be real numbers satisfying a < b. Then the closed bounded interval [a,b] is a compact subset of R. Let C be a collection of open sets in R with the property that each point of the interval [a,b] belongs to at least one of these open sets.

https://www.maths.tcd.ie/~dwilkins/Courses/MAU23203/MAU23203_Mich2019_Slides/MAU23203_Mich2019_HeineBorel_Slides.pdf

 

 

The bottom line is that there is no largest number in our universe since it has at least one dimension, which is enough to provide the relevant subset to cover infinity.

 

Edited January 23, 2023 by studiot

[geordief]

3 hours ago, studiot said:

bottom line is that there is no largest number in our universe since it has at least one dimension, which is enough to provideThe bottom line is that there is no largest number in our universe since it has at least one dimension, which is enough to provide the relevant subset to cover infinity.

(I have to apologise first for my inability to go through your and others' links in anything like a rigorous way as I am not equipped for that degree of intellectual  inquiry-I may once have been and I hope I still retain some curiosity and openmindedness)

 

That said and in my own mind the question in my OP has been answered in the negative but you seem to be looking at it in a different  way and  to try and ddress  what you are saying (your bottom line)  you seem perhaps to be saying that the number of  relations between the set of finite spacetime events in a hypothetically finite universe can be infinite.

On reflection perhaps so .My first scenario was actually just the number of actual events.They would be (in my mind anyway) physical whereas the relations  would be mathematical or geometric.

Have I understood some of what you were saying?

 

 

[studiot]

1 hour ago, geordief said:

(I have to apologise first for my inability to go through your and others' links in anything like a rigorous way as I am not equipped for that degree of intellectual  inquiry-I may once have been and I hope I still retain some curiosity and openmindedness)

 

That said and in my own mind the question in my OP has been answered in the negative but you seem to be looking at it in a different  way and  to try and ddress  what you are saying (your bottom line)  you seem perhaps to be saying that the number of  relations between the set of finite spacetime events in a hypothetically finite universe can be infinite.

On reflection perhaps so .My first scenario was actually just the number of actual events.They would be (in my mind anyway) physical whereas the relations  would be mathematical or geometric.

Have I understood some of what you were saying?

 

 

As always I am trying to simplify the complicated.

 

All numbers are abstract.

Yet I maintain that they exist in our universe, whether man (or any other being) has enumerated them or not.

Pi has infinite digits, but its value is not infinite. It's value lies between 3 and 4.

 

[geordief]

28 minutes ago, studiot said:

As always I am trying to simplify the complicated.

 

All numbers are abstract.

Yet I maintain that they exist in our universe, whether man (or any other being) has enumerated them or not.

Pi has infinite digits, but its value is not infinite. It's value lies between 3 and 4.

 

Some people (I think I have heard) claim that the universe may be made of numbers.

Think that may be an ancient(pythagorean or similar ?) belief but I  think that the hologram view of the universe  says something similar.

If that were the case then it would be trivial to say that the universe "contained numbers"(well maybe numbers do contain other numbers, both larger and smaller,)

I prefer to think that numbers are abstractions and that the universe contains abstractions.

 

By the way ,(and back to the OP) if events and particles are very closely related(particles  only manifesting when interacting) and particles are not local but spread out like waves,then maybe events likewise are spread out and so are not finite even in a finite system?

[studiot]

12 minutes ago, geordief said:

By the way ,(and back to the OP) if events and particles are very closely related(particles  only manifesting when interacting) and particles are not local but spread out like waves,then maybe events likewise are spread out and so are not finite even in a finite system?

Scientifically 'events'  are defined bu relativity, which is pointwise  - the opposite od spread out.

Waves and to some extent particles are defined by quantum theory and extend, perhaps forever.

Riemann in his world famour lecture of 1863 introduced the 'concept of the n-fold extended quantity'.

Sounds posh but it really means the relationship between geometry and real world quantities and enabled us to calculate using graphs of as many dimensions as necessary.

 

 

[MigL]

Getting back to the OP and how much information is required to represent a certain number of events in space-time, Seth has already alluded to an upper limit, here 

21 hours ago, sethoflagos said:

21 hours ago, Genady said:

I think I understand now. And tend to disagree. I think that any number can be mapped onto something in the real world.

The earth's surface is almost exactly 2x10^84 square Planck units. I vaguely remember reading that something unpleasant happens when you try storing that much information on a limited surface.

The Bekenstein  bound "implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy are finite. In computer science this implies that non-finite models such as Turing machines are not realizable as finite devices"

See here    Bekenstein bound - Wikipedia

Sometimes Physicist have to bring Mathematicians back 'down to earth' from their 'flights of fancy'.
( no offence meant, Studiot 😄 )

[Genady]

32 minutes ago, MigL said:

the information necessary to perfectly describe that system, must be finite if the region of space and the energy are finite

I'd like to understand this better. For example: How much information is there in the number pi?

[MigL]

Question asked in the OP

 

On 1/22/2023 at 8:05 AM, geordief said:

If we calculate the number of events/interactions that take place within a defined spacetime volume   is it possible to represent  that in a conventional way ,like 10 to power of some finite number?

 

The number is finite, so definitely representable.

[studiot]

4 hours ago, studiot said:

All numbers are abstract.

Which means they are not subject to energy constraints.

 

How much energy is required to provide a perfect shadow in some region ?

Yet every point (and there is an infinity of such points) is 'in shadow'.

[MigL]

The OP did not deal with 'abstracts', but events in space-time, which are information.
The Bekenstein bound does not limit itself to energy/entropy, but applies to all information.

Bekenstein derived the 'bound' from entropy considerations of Black Holes, and it was  reinterpreted in the framework of QFT by Casini in 2008.

Would a 'perfect' shadow be  information, or absence of information ?
I grant that abstract  concepts, such as infinities, can be represented, and Mathematics needs to deal with such 'abstacts', but Physics has constraints; one being the Bekenstein bound.

[Markus Hanke]

14 hours ago, MigL said:

The Bekenstein  bound "implies that the information of a physical system, or the information necessary to perfectly describe that system, must be finite if the region of space and the energy are finite.

It’s interesting to ponder what this actually means in the case of black holes, because there is no obvious reason why a region of completely empty spacetime (which is what a Schwarzschild BH is, in classical GR) should have finite non-zero entropy associated with it at all. Even if we disregard the issues around singularities etc for the moment, if we consider an ordinary star of 1 solar mass, then its total entropy is on the order \(~10^{57}\). If we let that star collapse gravitationally, and ignore any energy losses during the collapse process, then the resulting 1 solar mass Schwarzschild BH will have entropy on the order \(~10^{77}\), which is 20 orders of magnitude larger. While it is interesting in itself that the total entropy of the system increases by orders of magnitude, what’s really surprising is that the resulting BH has finite, non-zero entropy at all - remember again that classical Schwarzschild spacetime is everywhere empty. 

Simply and somewhat sloppily put, entropy is a statistical property that reflects the number of ways the microstates of a system can be rearranged without affecting its overall macrostate. But a Schwarzschild BH is just empty spacetime - every point within that manifold is exactly the same as every other point, meaning no small local neighbourhood can be physically distinguished from any other small local neighbourhood. And because this is a classical model, the spacetime manifold is implicitly assumed to be smooth and differentiable everywhere (disregarding the singularity for now), or else the entire formalism of GR makes no sense. Thus we can just pick any arbitrary pair of events within the volume in question, and swap them - doing this will not change anything about the BH at all. Because the manifold is smooth and differentiable, there are infinitely many such pairs in any given volume of empty spacetime, so the total entropy of this system should diverge. But it doesn’t - it’s finite and well defined, and always >0. IOW, there is a finite, well defined number of distinct operations one can perform in such a volume that leaves the overall system unaffected. It would seem to me that this is possible only if spacetime within such a volume is not in fact smooth and continuous everywhere - there needs to be some kind of non-trivial structure present at least in some subregion of the volume enclosed by the event horizon. The mere fact of geodesic incompleteness in and around r=0 doesn’t seem to account for this (a point singularity has no degrees of freedom, and isn’t part of the manifold in any case) - it would take a very non-trivial kind of micro-structure to yield an entropy of the magnitudes mentioned above.

So what is this micro-structure? I for one would dearly like to know…

Edited January 24, 2023 by Markus Hanke

[studiot]

5 hours ago, MigL said:

The OP did not deal with 'abstracts', but events in space-time, which are information.
The Bekenstein bound does not limit itself to energy/entropy, but applies to all information.

Bekenstein derived the 'bound' from entropy considerations of Black Holes, and it was  reinterpreted in the framework of QFT by Casini in 2008.

Would a 'perfect' shadow be  information, or absence of information ?
I grant that abstract  concepts, such as infinities, can be represented, and Mathematics needs to deal with such 'abstacts', but Physics has constraints; one being the Bekenstein bound.

On 1/22/2023 at 2:57 PM, swansont said:

!

Moderator Note

Moved to mathematics because it sure as heck doesn’t belong in the Lounge

 

I note this was moved to Mathematics, not Physics.

 

I have noted before  that just because two physical phenomena follow the same mathematical models does not imple any connection between them.

For example the voltage transfer characteristic of an FET follows a square law, as does the trajectory of a ballistic object on Earth.

Yet I can see no connection between these phenomena.

 

Information entropy and thermodynamic entropy have the same disconnection.

17 minutes ago, Markus Hanke said:

Simply and somewhat sloppily put, entropy is a statistical property that reflects the number of ways the microstates of a system can be rearranged without affecting its overall macrostate.

I can't agree with that.

Phase change at constant temperature is purely entropy drive but the macrostate is definitely significantly affected.

[Markus Hanke]

44 minutes ago, studiot said:

I can't agree with that.

Phase change at constant temperature is purely entropy drive but the macrostate is definitely significantly affected

Well, perhaps there’s a gap in my own understanding then. 
So how would you physically interpret the notion of “entropy” associated with a region of spacetime (ie a region on a semi-Riemannian manifold)?