Thinking of a number - a conversation with myself

bonkey

Think of a number.
What? A number? Any number?

Only one limitation. No imaginary component. No root-of-minus-one.
How big?

as big as you like, with as many positions after the decimal point as you fancy.
Okaaaaaaaay.

Now, without knowing this number I can predict the following:
Your number appears on the number line

Well Duh.

Not impressed yet, eh? Didn't think so. Lets continue though.
<shrug> I'm not up to much right now. Sure.

Now, what if I took all the information on my harddisk and read those 1 and 0 values as a number. I'd have a number which was a representation of my harddisk, wouldn't I?
Well, yeah, but its just, like, a number.

Indeed. And the numbers on the number-line which are near this one...they could be seen as representations of hard-disks containing almost the same information as mine, right?
Riiiiight.

Oh, hang on....doesnt that mean that every possible hard-disk that was, is and ever will be already exists in representational form on the number line?
<smiles> Now youre beginning to see where this is going.

In fact, anything we can model can be expressed as a number. And those numbers already exist.

Go on...

Hmmm.

If we could determine the most fundamental components of the universe, we could actually model the state of the universe at a moment in time, couldn't we.
Notionally, yes.

And given that the universe could be seen as a series of moments, each seperated by Planck Time, we could model all moments. So we could model time-space if we had somewhere to model it in.
Yes....

...And what would that model be?
A number.

And where would you find that number
On the number line!

Wow. Our universe might just be a representation of a number, or be representable by a number. I'm not smart enough to know if there's a difference though.

It might just be either or even both. I'm not smart enough to know the difference either.

But hang on a sec...

<smiles> Yes?

There's no end to those numbers. So sooner or later, everything which can be modelled appears in some form on the number line. So if the universe can be modelled, its just a point on a number line.

Indeed

So why is that point important?

Pardon?

I mean, its a line. Its an infinite set of points, only one of which would be a model of this universe, and all the other's, well, wouldn't. They'd model other stuff. Or sometimes they'd just appear to be models of garbage. Chaos.

That sounds reasonable, yes

And if something were to randomly pick points off this line, and do so forever, sooner or later it would pick our universe wouldn't it.

I'm no expert on what really happens when statistics and infinities cross swords, but I would say so..

So there's no need for intelligence-driven creation?

I mean...creation seems to suggest that the universe can be modelled, but if there's an infinite number of creations, each chosen randomly for no reason..... hey! We could be just another random number on a number line.

We could be.
Wow.


Well - as you said earlier - Duh

Hey. hang on a sec. You got me to think of a number earlier. Don't you want to know what it is?

Do you know what it is yourself, in order to tell me?
Pardon?


It might not be just a number. It could be a representation of a harddisk. Or of a universe.
Oh.

Son Goku

Woah!!

The Universe can in no way be represented as a number.
In fact your hard drive couldn't be either.

It's the specific 0,1 configuration of a hard drive at any given moment combined with Boolean Algebra that makes the hard drive what it is.

Turning this into a single number loses almost all the information.

bonkey

Son Goku wrote:

Woah!!

The Universe can in no way be represented as a number.
In fact your hard drive couldn't be either.

It's the specific 0,1 configuration of a hard drive at any given moment combined with Boolean Algebra that makes the hard drive what it is.

Turning this into a single number loses almost all the information.

But if you read the platter from start to finish, you get a single binary "string" which is no different to a binary number. Decimalising is simply changing the encoding. Or you create your numberline in binary.

OK, its only information, not the physical drive....but I'm willing to bet that someone could take my binary number, create a file out of it, manipulate that file to load it as a virtual drive, and have a full, virtual copy of my original physical hard-drive. So my hard-drive can be represented as a number, and that number can be manipulated as though it were my hard-drive as long as I can virtualise the interface.

As to whether or not the universe can be modelled...well...thats another thing I'm not smart enough to answer.

jc

Son Goku

But if you read the platter from start to finish, you get a single binary "string" which is no different to a binary number. Decimalising is simply changing the encoding. Or you create your numberline in binary.

Yes, but that string is the input, the food for the algebra.
You cannot compress the algebra into a number.

Additionally your hard drive isn't a single string.
Compressing the information into a single number wouldn't tell you where on the hard drive the 0s and 1s go.

If you "unzipped" the number back into binary, all you would have is a string of zeros and ones with no idea of where to place them.

You'd also get a different number depending on which way you read the drive.

bonkey

Son Goku wrote:

Yes, but that string is the input, the food for the algebra.
You cannot compress the algebra into a number.

Of course you can. All executables are is a large number (a string of 1s and 0s - which is transcribable as a number) which you put through an engine (the OS, and all that it is built upon).

Additionally your hard drive isn't a single string.

It can be represented as one. You can virtualise a hard-drive - just look at what programs like VMware do - at which point it is just a string of 1s and 0s....which can be transcribed as a number.

You'd also get a different number depending on which way you read the drive.

The same holds true of a physical disk drive. If I give you a HD from an early IBM mainframe and tell you "use this", the physical 1 and 0 combinations are useless to you, unless you know how the information on it was stored.

I've already acknowledged that you need to know how to manipulate the information stored within a number, but the fact remains that you need to know that about hardware as well. It doesn't tell you how to talk to it - you still need a seperate software/firmware driver to do the interfacing.

At the end of the day, your argument is basically boiling down to the notion that software modelling is impossible - you cannot replace a physical thing with a software model. I would suggest to you that you cannot tell whether a machine you connect to over the internet is a physical machine, or simply a virtual one run on something like Xen or VMWare. If you cannot tell whether or not it is physical or virtual, then you certainly can't tell that about the HD either. And if you cannot tell that a HD is actually a real HD while accessing it, or a virtual one, running in a program somewhere, pretending to be a real, physical hard-drive, then I cannot see how you can tell me its impossible to have such a thing.

jc

Son Goku

At the end of the day, your argument is basically boiling down to the notion that software modelling is impossible

My point is that a hard drive cannot be compressed into a single number.
That single natural/real number could be an infinity of different hard drives depending on what method/context you read it in.
I'm not arguing software modelling nor am I arguing that you cannot replace a physical thing with a software model.

However I shouldn't have indulged the hard drive analogy to begin with.

I'll tackle the universe one so that this doesn't become computer science.

The universe cannot be compressed into a single number, as its algebraic, geometric and group theoretic rules cannot be mapped straight into the number line.

Capt'n Midnight

Son Goku wrote:

Woah!!

The Universe can in no way be represented as a number.
In fact your hard drive couldn't be either.

It's the specific 0,1 configuration of a hard drive at any given moment combined with Boolean Algebra that makes the hard drive what it is.

Turning this into a single number loses almost all the information.

If you accept that the hard drive can have N different states, then at any time you can describe it a number between 0 and N. Actually if you ignore error checking and correction bits and treat empty space as 0's then you get a much smaller number for N.

The size of the observable universe is finite, excluding black holes isn't there a finite limite (Planck dimension) to dimensions and the speed of light would lead to a finite minimum time interval too. Also there aren't infinte levels of energy. So for every point of time we could describe the state of the universe in terms of the position ( or ) energy at every place as a number. Admittedly it's a big number..

Son Goku

If you accept that the hard drive can have N different states, then at any time you can describe it a number between 0 and N. Actually if you ignore error checking and correction bits and treat empty space as 0's then you get a much smaller number for N.

Yeah, that's true. If you are simply labelling Boolean states then it could be represented with a number.

Capt'n Midnight wrote:

The size of the observable universe is finite, excluding black holes isn't there a finite limite (Planck dimension) to dimensions and the speed of light would lead to a finite minimum time interval too. Also there aren't infinte levels of energy. So for every point of time we could describe the state of the universe in terms of the position ( or ) energy at every place as a number. Admittedly it's a big number..

Not every physical quantity is scalar/number valued.
Spatio-temporal curvature for instance is 4th rank tensor valued, compressing that into a number definitely loses information.
Also in chaotic systems, which depend very sensitively on their origins, a number expressing the current state of the system is useless and loses all the important information on the system.

bonkey

Son Goku wrote:

Not every physical quantity is scalar/number valued.

I don't believe its possible to make that statement definitively, in the absence of something like a ToE.

Which is why you'll note that I did base my argument on the supposition that
If we could determine the most fundamental components of the universe, we could actually model the state of the universe at a moment in time, couldn't we..

Your response seems to be that we haven't yet determined how to do this, ergo my argument is incorrect.

I can't see how that logic holds up. No-one has proven such a model cannot exist. I've never suggested that we will find the model, nor that we will be able to model the universe. I haven't even suggested the universe is modellable.

My entire original post sprang from a mix of thoughts, one of which would be what would be the implications were the universe modellable.

Your response seems to boild down to "its not modellable", which seems to be based on the reasoning of "We don't have a model", which I have to admit I don't see as being a particularly solid refutation.

Or perhaps what you're really saying is "its not modellable as a number, if it is modellable", then ask yourself if its modellable as a formula, or set of formulae. And if so, whether that is something you could save on a very large hard-drive. And if so, why that hard-drive isn't modellable as a series of 0s and 1s....which is a number.

jc


jc

Son Goku

bonkey wrote:

I don't believe its possible to make that statement definitively, in the absence of something like a ToE.

It's very possible to make that statement.
Progression in physics has been toward moving further from Natural number algebra, to vectors, to Tensors, to groups.
Of course I can't definitively say they won't be scalar valued, but it is unlikely.
For instance Quantum Gravity suggests a discontinuous manifold, without a continuous manifold you can't even define a scalar.

Your response seems to boild down to "its not modellable", which seems to be based on the reasoning of "We don't have a model", which I have to admit I don't see as being a particularly solid refutation.

No, my response is that it is cannot be a scalar model.

Or perhaps what you're really saying is "its not modellable as a number, if it is modellable", then ask yourself if its modellable as a formula, or set of formulae. And if so, whether that is something you could save on a very large hard-drive. And if so, why that hard-drive isn't modellable as a series of 0s and 1s....which is a number.

Certain Quantum Mechanical systems cannot be model by Boolean systems though.


I haven't even gone into Quantum Field theory yet.
And it certainly is not capable of scalar compression.

bonkey

Son Goku wrote:

Certain Quantum Mechanical systems cannot be model by Boolean systems though.

Look, any model which can be expressed in any language can be transcribed into a binary encoding. There isn't a single system of notation in existence which cannot be encoded in binary. If you can express the model in any mathematical notation, you can transcribe that mathematical notation to binary.

Look at how anything else is encoded in binary. the full ANSI 256-character set is recoded into a 2-character set, by use of an 8-bit encoding. The 32767-character Unicode set is recoded into a 2-character set by emans of a 16-bit encoding.

Exactly why can Quantum systems not be transposed into a binary encoding? Unless they cannot be modelled, they can be modelled using a base-2 encoding. You might need 32-bit words if its a large character set, or 64-bit, or n-bit where n is a very, very large number, but unless tehre's an infinite character set (and thus one which cannot be expressed in any language), it can be transposed into binary.

And it certainly is not capable of scalar compression.

Who said it had to be?

I'm saying it is re-encodable in binary format, and that once expressed in a binary format, it can be viewed without further transcription as a binary number, or with binary-to-decimal transcription as a decimal number.

jc

Son Goku

Look, any model which can be expressed in any language can be transcribed into a binary encoding. There isn't a single system of notation in existence which cannot be encoded in binary. If you can express the model in any mathematical notation, you can transcribe that mathematical notation to binary.

A lattice of N spin-1/2 particles with pairwise interactions cannot be modelled correctly with a classical computer.
There are many other systems, there is a classical (excuse the pun) problem surrounding boolean computers simulating local quantum system.

Exactly why can Quantum systems not be transposed into a binary encoding? Unless they cannot be modelled, they can be modelled using a base-2 encoding. You might need 32-bit words if its a large character set, or 64-bit, or n-bit where n is a very, very large number, but unless tehre's an infinite character set (and thus one which cannot be expressed in any language), it can be transposed into binary.

The problem is very different from n-bit encoding.
Quantum Computer Scientists like Seth Lloyd and David Deutsch work is largely concerned with this.

Who said it had to be?

I'm saying it is re-encodable in binary format, and that once expressed in a binary format, it can be viewed without further transcription as a binary number, or with binary-to-decimal transcription as a decimal number.

To be honest a Quantum Field theoretic systems can't be transcribed into binary correctly either.
This is the reason we have fields like lattice Quantum Chromodynamics.

bonkey

Son Goku wrote:

there is a classical (excuse the pun) problem surrounding boolean computers simulating local quantum system.

Where have I ever suggested that the simluation would be run on a boolean system.

All I have said is that if it is possible to create a model, it is possible to express the model in binary notation.

To be honest a Quantum Field theoretic systems can't be transcribed into binary correctly either.

Why not? They have an infinite character set?

Let me try it this way...

This stuff you're saying cannot be handled in binary form....

Could you write it up on a blackboard ? Indeed, can the people studying it write it in any form anywhere? If so, can I take a digital photograph of what they have written down? Do I not then have their model transcribed into a binary format.

I'm not saying I can run their model in a binary-based system.
I'm not saying I can evaluate their model on using any binary or otherwise-discrete-state system.
I'm saying I can store the model in binary form.

If I cannot store it in binary form, then I posit it cannot be stored in any form, and that there is no mechanism by which one scientist can convey knowledge of this model to another, as to do so would require langauge (which I can store in binary form) and/or diagrams (which I can store in binary form) and/or mathematical formulae expressed using appropriate notation (which I can store in binary form).

Son Goku

bonkey wrote:

Where have I ever suggested that the simluation would be run on a boolean system.

All I have said is that if it is possible to create a model, it is possible to express the model in binary notation.

But without time evolution, the system is meaningless, unless you can simulate the time evolution of the system, what you have is nothing.

Could you write it up on a blackboard ? Indeed, can the people studying it write it in any form anywhere?

I'm not saying I can run their model in a binary-based system.
I'm not saying I can evaluate their model on using any binary or otherwise-discrete-state system.
I'm saying I can store the model in binary form.

If I cannot store it in binary form, then I posit it cannot be stored in any form, and that there is no mechanism by which one scientist can convey knowledge of this model to another, as to do so would require langauge (which I can store in binary form) and/or diagrams (which I can store in binary form) and/or mathematical formulae expressed using appropriate notation (which I can store in binary form).

Yes, you can express the law in binary form, but you still cannot time evolve the system.
If you cannot time evolve the system you do not have a representation of the universe.
All you have is the laws transcribed into binary format.

I'm not applying this to just binary computers, I'm applying it to classical Turing machines in general.

If so, can I take a digital photograph of what they have written down? Do I not then have their model transcribed into a binary format.

You're taking this very far away from what you originally said:
"Our universe might just be a representation of a number, or be representable by a number."
This has now become
"Equations expressing behaviour in our universe can be stored in binary, upon which can be converted into a decimal number".

Look at the process:
1) You have a number: "x"
2) You unzip "x" back into binary
3) This binary stream stores the information on the mathematical expression of physical laws.

For instance if I have a piece of spacetime with a certain metric, that metric can be broken down into 0s and 1s so that it can be interpreted by a computer.
This stream of 0s and 1s can be read into a decimal system to give a number "x".

However "x" in no innate way expresses anything about our universe.
Now I could do these steps:
1)Convert the metric into binary.
2)Convert binary into a combinatorical expression.
3)Convert the combinatorical system into a complex number, "z".

So "x" and "z" both express the information about the metric not because they actually express any information about it, but because you have created an extremely convoluted encoding pattern.

Similarly I could encode this information into a number, a vector, a series of lines, a fluid flow. Where upon I could say:
"The Universe can be expressed as a fluid/vector/series of lines".

But it doesn't really mean anything, because each is just the result of an arbitrary encoding pattern.

bonkey

Son Goku wrote:

You're taking this very far away from what you originally said:
"Our universe might just be a representation of a number, or be representable by a number."

I haven't taken it away from that at all.

You've tried to infer something I haven't said, and then argued that this inferral isn't correct. I'm saying that its not what I was talking about.

I'm simply not addressing the question as to how the representation would be interpreted. I'm not addressing how it would be stored.

I am speculating that if it is modellable, then the information for that model can be represented as a number.

I am not discussing what type of model it needs to be, or what information one would need for such a model, nor how one would store a universe-worth's of information, nor how one would run the model.

Such issue are irrelevant to the thought experiment, unless to show that it is impossible to store certain models in binary form.

This has now become
"Equations expressing behaviour in our universe can be stored in binary, upon which can be converted into a decimal number".

We've gone down a tangent discussing certain behaviour of our universe because you've contended that there is no way to express such a model in binary form.

If you can express all behaviour of the universe - i.e. if the universe can be modelled, then what, exactly is the distinction between the two statements?

What is the difference between saying all behaviour of the universe and the universe.

What is the distinction, if not the same "you cannot virtualise" discussion we had about hard-disks earlier?

If all behaviour could be modelled, then that model could be stored and represented as a number.

With an appropriate virtualisation / interface, that model would be indistinguishable from the real thing, just as with an appropriate virtualisation / interface, you cannot tell the difference between a real hard-disk and a virtual copy of one.

What I'm talking about isn't that distinct from the question of whether or not the universe is deterministic, and what that would mean. From what I recall reading, its possible that the universe is deterministic, only at such a level of complexity that it is impossible that the determinism could ever be calculated by us. Am I wrong here? Has it been concluisively proven that the universe cannot be deterministic in this manner?

I guess if you look at it relatively, the universe may be deterministic, just not so from our perspective. Similarly, the universe may be modellable, but not so from our perspective.

The reason I brought this to philosophy is because its not a scientific question. Its not a question of whether or not the universe is modellable, or whether or not it is deterministic. Its a question of what would be the implications were it to be so.

People have long discussed the theory of transferring human consciousness into some artificial form of brain. The building of such an artificial construct may be inconceivable today, but I don't believe we've proven it to be impossible.

Allow me to speculate that it is possible.

Now, take that consciousness, and a virtualised input which allows it to perceive a limited virtual, spatial environment, which to all measurable intents and purposes will mirror how a real copy of that spatial environment would behave. Could the consciousness in question know whether it was a real consciousness in a real, physical closed environment, or a virtualised copy of a consciousness in a virtual, non-physical closed environment. Is there any way it could determine the difference?

What if that closed system could only process the passage of time at 1/10 "real-time". Could the consciousness inside the closed system be aware of this discrepancy, given that they have no way to detect it (as all detection mechanisms would equally run at 1/10 speed, as they too would be inside the virtual closed-space).

Now scale up.

jc

Son Goku

bonkey wrote:

Its not a question of whether or not the universe is modellable, or whether or not it is deterministic. Its a question of what would be the implications were it to be so.

If that’s what you're saying, then I did go on a tangent and I apologise, but you have to understand it seemed to me that you were arguing that the universe could be modelled as a number, rather than opening a discussion for what the implications of determinism and capability of modelling were.

If you can express all behaviour of the universe - i.e. if the universe can be modelled, then what, exactly is the distinction between the two statements?

None, but that isn't what my statement was about.
I was pointing out the difference between having the statement which contains the solutions and having an analytic solution.
So we can represent the equatons as a number, but not the actual state of the universe.

Now, take that consciousness, and a virtualised input which allows it to perceive a limited virtual, spatial environment, which to all measurable intents and purposes will mirror how a real copy of that spatial environment would behave. Could the consciousness in question know whether it was a real consciousness in a real, physical closed environment, or a virtualised copy of a consciousness in a virtual, non-physical closed environment. Is there any way it could determine the difference?

What if that closed system could only process the passage of time at 1/10 "real-time". Could the consciousness inside the closed system be aware of this discrepancy, given that they have no way to detect it (as all detection mechanisms would equally run at 1/10 speed, as they too would be inside the virtual closed-space).

This is just René Descartes' Demon in a different format.
So it's more a philosophical issue.

From a physical point of view there are certain things which make it doubtful.
For instance non-deterministic chaos cannot be computed.
We can have an equation which expresses it, but that does not mean it can be solved.

If all behaviour could be modelled, then that model could be stored and represented as a number.

Yes, but why?
You now have a number which expresses the same thing as the original equation, except it needs to be decompressed.
What's the point?

Is your main question "What implications are there if the universe can be expressed as a number?"

If so, I'd say none.
The number is the end result of an compression process and contains the same information(in a much harder to extract way) as the original equation and it doesn't contain analytic solutions.
As such it doesn't represent the universe, but is simply the equations encoded in a very arbitrary way.

You've tried to infer something I haven't said, and then argued that this inferral isn't correct. I'm saying that its not what I was talking about.

Dude, I'm not creating a straw man on purpose, I'm just finding it very difficult to understand your overarching point.

nobodythere

Interesting... I don't think we'll ever be able to model the universe in practical terms, but yeah, considering the universe as a three-dimensional array and assigning each part of this three-dimensional array corresponding with whether this space is filled with an electron/proton/neutron or is a vacuum, I suppose it is theoretically possible to represent the state of the universe as a number.

The probem being that on a hard-disk the distance between one bit on the platter is the same, whereas atoms can (presumably) move infinitely small distances. Hence it would require some sort of offset value from the points on your three-dimensional array.


Ah who cares, it's never gonna happen

fractal

Interesting discussion... (thought I picked up on it a little late)

What I think is:
With a hard drive you can of course boil its state down to a single (very large) number. As all a hard-disk contains is 1's or 0's, if you restore the same 1's and 0's back to the same hard-disk at some point in future you will restore the original state.

However this same model doesnt scale up to the idea of the universe being representable as a single number. And this is why i think not:
- For something to boil down to a number it would need to be measured to a certain degree of precision.
- As the universe by definition is everything that ishttp://dictionary.reference.com/search?q=universe.To measure it correctly would require infinite precision.. This would mean an infinite number. While im sure it can be argued that theoretically an infinite number is on the numberline. it can never be given a physical representation.This is problem 1.
- Even if it was possible to represent a specific infinite number we would still need a way to describle the universe in enough detail. Which again isnt possible due to uncertainty principle.. http://en.wikipedia.org/wiki/Uncertainty_principle This I think is problem 2...

Just my 2 cents...

wayfarer

But what about irrational numbers like pi and such?

Thsi universe you model could only ever be an approximation of the real one.

nobodythere

Pi isn't an absolute distance, only a relation between two lengths (diameter and radius).

wayfarer

grasshopa wrote:

Pi isn't an absolute distance, only a relation between two lengths (diameter and radius).

What I mean to say is that no actual distance between two points can be expressed as a rational number, it can only ever be an approximation, unless one point if defined as being a certain distance from the other, which isn't what we're talking about here.

And anyway, it is still impossible due to Heisenberg's Uncertainty Principle.

wayfarer

Just one last thing: If you could overcome the first two problems I suggested, how would you propose storing the number?

Capt'n Midnight

grasshopa wrote:

Pi isn't an absolute distance, only a relation between two lengths (diameter and radius).

YI'd suggest using non-eculidian geometry, very easy to get PI rounded up to 3 - think of a very flat cone.

http://en.wikipedia.org/wiki/Pi - look at the formulas to calculate PI they tend to suggest that it's more than a mere number.



but if you want a handy number 355/113 is good.

An how can you measure the distance between two points or around the outside when the pesky particles keep moving (if you stopped one then you'd know it's velocity so precisely that it could be anywhere in the universe.)