Macroscopic superpositions and pure states in the Everettian Interpretation

[Nick Prince]

 

(1)

If you are an adherent of the Everett interpretation, then it seems that you have to accept that macroscopic superpositions exist. For example Max Tegmark in his paper “Many worlds in context” http://arxiv.org/abs/0905.2182 argues that …. .“Everett’s MWI is simply standard QM with the collapse postulate removed, so that the Schrödinger equation holds without exception”.  Thus he also argues that from this we can deduce that not only microscopic-superpositions are inevitable, but also that macroscopic-superpositions (say of a cat being dead and of being alive) are perfectly legitimate quantum states. It seems that the whole  Everettian  ontology rests on this!

 

Now David Wallace on p5 of:

http://philsci-archive.pitt.edu/8888/1/Wallace_chapter_in_Oxford_Handbook.pdf

 

 argues the case that cats can be in  alive and dead superpositions but If I understand him correctly this must only be for an extremely small amount of time before decoherence rapidly renders the system to be in an improper mixture (equivalent to  tracing over the environment and hence globally the entire state vector remains in a superposition, with off diagonal terms in the density matrix being asymptotic to zero). Hence why he then argues on p5 that macroscopic superpositions do not describe indefiniteness  but rather multiplicity.

 

Have I understood this correctly? Is there actually initially indefiniteness – at least for a tiny instant? We don’t see macroscopic superpositions because they decohere so fast - leaving us in only one of the decohered branches.

 

 

(2)

In one representation of this scenario below:

 

 

|psi> = 1/sqrt2(|up>  +  |down>)|detector> |cat alive> |Rest of universe> --------(1)  

 

I have usually thought that a state like |cat alive>  can be taken to be pure. 

 

However, there are those who argue against the fact that states like |cat alive> or  |environment> can even be considered pure at all – for example see bottom of page 2 in:

http://www.rickbradford.co.uk/ChoiceCutsCh1.pdf

 

I’d like to know  whether my explanation below of how we can think of a state like |cat alive> is “a good story” or not. I presume the author above feels that he must specify the quantum state of the cat for it to be in a pure state, but as I argue, is this really necessary? Surely a cat is always in a pure quantum state? We can use the aliveness operator with eigenvalues 1, 0 to test which subspace of the Hilbert space the cat’s state vector is in - at least for that instant!

 

I agree that it would be impossible to generate an ensemble of identical pure states of a given cat which would mean it  is in some sense a member of a mixed state but I can also see things differently as below.

 

 

 

I can see that another way to think about this is roughly that the state  vector of the cat can be explained as the tensor product of all the (10^26?) particle states that make up the cat? E.g.  |cat> = |atom 1> ⊗ |atom 2> ⊗ |atom4> ⊗......., where |atom#>  would somehow incorporate appropriate quantum numbers to describe the atom’s state.  Obviously we could never actually intentionally prepare such a state specifically.  But do we need to?  Surely any of the states the cat can be in will do for the experiment above - provided it is alive at the start of the experiment! We only need to know that it has a quantum state for the infinitesimal time that it takes the atomic spin system (which we know can be in a superposition) to interact with the apparatus plus cat, In any case, by similar reasoning the cat will - dead or alive - continue to be able, in principal, to be represented by a vector in Hilbert space as required by the definition of purity - unless the vector strays far enough outside of the region of the Hilbert space to be considered no longer a cat! It seems to me this must be all part of the Everettian ontology that everything can be described by a vector in Hilbert space and as such all states in equation (1) would be considered pure, thereby ensuring the universal wave function was also pure.

 

David Wallace in the same reference discusses this scenario saying that states like |cat alive> live in a very big subspace of Hilbert space which can  have states in it that  don’t look anything like a cat (there are an awful lot of things that can be made out of the constituents of a cat!). As I see it, a cat is essentially an emergent object. Never the less he agrees that it is accurate to say of some states in this space that they would be a superposition of a live and dead cat.

 

I don't know how to reconcile the mixed/pure status.

Is my interpretation of pure states of a cat (macroscopic object) correct? At all instants can they be considered to have a pure state (which varies from instant to instant) and if so, what is wrong with treating it as a member of a mixed state? It surely cannot be both.

 

 

[Brett Hall]

On 10 Jul 2015, at 04:08, Nick Prince <nickmag...@gmail.com> wrote:

 

(1)

If you are an adherent of the Everett interpretation, then it seems that you have to accept that macroscopic superpositions exist. For example Max Tegmark in his paper “Many worlds in context” http://arxiv.org/abs/0905.2182 argues that …. .“Everett’s MWI is simply standard QM with the collapse postulate removed, so that the Schrödinger equation holds without exception”.  Thus he also argues that from this we can deduce that not only microscopic-superpositions are inevitable, but also that macroscopic-superpositions (say of a cat being dead and of being alive) are perfectly legitimate quantum states. It seems that the whole  Everettian  ontology rests on this!

 

Now David Wallace on p5 of:

http://philsci-archive.pitt.edu/8888/1/Wallace_chapter_in_Oxford_Handbook.pdf

 

 argues the case that cats can be in  alive and dead superpositions but If I understand him correctly this must only be for an extremely small amount of time before decoherence rapidly renders the system to be in an improper mixture (equivalent to  tracing over the environment and hence globally the entire state vector remains in a superposition, with off diagonal terms in the density matrix being asymptotic to zero). Hence why he then argues on p5 that macroscopic superpositions do not describe indefiniteness  but rather multiplicity.

 

Have I understood this correctly? Is there actually initially indefiniteness – at least for a tiny instant? We don’t see macroscopic superpositions because they decohere so fast - leaving us in only one of the decohered branches.

You don't see them because you occupy one universe. Not a universe with multiple cats. One cat. In the *multiverse* (i.e in other "parallel branches") there are cats in other states. 

*Note that David Deutsch explains the universes are neither parallel and nor do they branch. But loose language works well enough here.

 

 

(2)

 everything can be described by a vector in Hilbert space and as such all states in equation (1) would be considered pure, thereby ensuring the universal wave function was also pure.

 

David Wallace in the same reference discusses this scenario saying that states like |cat alive> live in a very big subspace of Hilbert space which can  have states in it that  don’t look anything like a cat (there are an awful lot of things that can be made out of the constituents of a cat!). As I see it, a cat is essentially an emergent object. Never the less he agrees that it is accurate to say of some states in this space that they would be a superposition of a live and dead cat.

Hilbert space is higher dimensional space. It's not space in this universe. It's the space in which the wave equation is waving. (Quantum waves don't exist in single universes. Stuff is made of particles. Not waves. That is also a great, simple, succinct conclusion of the Everett interpretation. No wave-particle duality nonsense.) 

 

I don't know how to reconcile the mixed/pure status.

Is my interpretation of pure states of a cat (macroscopic object) correct?

Seems ok. Except you seem to be thinking that superpositions can happen in single universes. That's impossible. A superposition is a combination of states. So you need more than one. But in a universe you have *instances* only. If you have two universes then you can have a superposition of at most 2 instances. But as you yourself are an instance in one universe you will only ever observe the instance in your universe. But you *know* when you observe something like a table other instances exist in other universes. That's what your "superposition" is kind of like. Whether anything happens due to this superposition (like interference) is rare...and requires other explanations about what particles are doing according to the laws of quantum theory.

At all instants can they be considered to have a pure state (which varies from instant to instant) and if so, what is wrong with treating it as a member of a mixed state? It surely cannot be both.

It's not so surprising to have what you're calling "macroscopic" superpositions. The cat really is alive and dead...just in different universes. Quantum mechanics provides a measure for the number of universes in which the cat is alive vs dead.

"Instant to instant" also means "universe to universe". Different times, in the multiverse picture, are just special cases of different universes. 

God's eye view (i.e: looking at the whole multiverse from outside it): you get a cat that is both alive and dead. But you and I don't have that perspective. We are only ever in a universe where the cat is alive XOR dead. It's one...or the other. Not both. Not in the universe we occupy.

But if we find an alive cat, we know there really does exist another universe where the cat is dead. Indeed more than this. Our universe is just one of many many many where the cat is also alive. And there are many many many where the cat is dead. And many where it's a dog. Or a hat. Or a bunch of neon gas etc etc.

Brett.

[Nick Prince]

On Friday, July 10, 2015 at 12:50:08 AM UTC+1, Brett Hall wrote:

On 10 Jul 2015, at 04:08, Nick Prince <nickmag...@gmail.com> wrote:

 

(1)

If you are an adherent of the Everett interpretation, then it seems that you have to accept that macroscopic superpositions exist. For example Max Tegmark in his paper “Many worlds in context” http://arxiv.org/abs/0905.2182 argues that …. .“Everett’s MWI is simply standard QM with the collapse postulate removed, so that the Schrödinger equation holds without exception”.  Thus he also argues that from this we can deduce that not only microscopic-superpositions are inevitable, but also that macroscopic-superpositions (say of a cat being dead and of being alive) are perfectly legitimate quantum states. It seems that the whole  Everettian  ontology rests on this!

 

Now David Wallace on p5 of:

http://philsci-archive.pitt.edu/8888/1/Wallace_chapter_in_Oxford_Handbook.pdf

 

 argues the case that cats can be in  alive and dead superpositions but If I understand him correctly this must only be for an extremely small amount of time before decoherence rapidly renders the system to be in an improper mixture (equivalent to  tracing over the environment and hence globally the entire state vector remains in a superposition, with off diagonal terms in the density matrix being asymptotic to zero). Hence why he then argues on p5 that macroscopic superpositions do not describe indefiniteness  but rather multiplicity.

 

Have I understood this correctly? Is there actually initially indefiniteness – at least for a tiny instant? We don’t see macroscopic superpositions because they decohere so fast - leaving us in only one of the decohered branches.

You don't see them because you occupy one universe. Not a universe with multiple cats. One cat. In the *multiverse* (i.e in other "parallel branches") there are cats in other states.

 

*Note that David Deutsch explains the universes are neither parallel and nor do they branch. But loose language works well enough here.

I guess you are referring here to the fungibility/differentiation picture here. I had an exchange on the BOI forum on this some time ago. see :

https://groups.google.com/forum/#!searchin/beginning-of-infinity/prince/beginning-of-infinity/o0hDyE_yrSg/GUFrDhGBdScJ 

I'll re read the whole branching versus differentiating again in BOI and review the exchange on https://groups.google.com/forum/#!searchin/beginning-of-infinity/deutsch$20branching/beginning-of-infinity/kmgG2M4uzz8/sKVNBi7Dq_0J

and see if this helps.  I have to say that as I remember I had difficulty in reconciling this picture. If macroscopic objects  are identical instances in some proportion of the universes in the multiverse then what makes them differentiate? Deutsch says on p265 of BOI that it is because they are fungible! Differentiation then spreads out in all directions at or near light speed!

OK so as I said, I guess I'll have to review all this again to see if it sheds any light on the matter.

Bearing in mind what I say above, In normal measurement "speak" I accept that in my experiment there is only one cat. Suppose I generate particles in the x direction and send them into a stern gerlach device aligned in the z direction to produce particles aligned in that direction. I could use spin down in the z direction to trigger the poison and spin up to keep my cat safe. I start the experiment and equation (1) will evolve unitarily into a superposition.  

|psi> = 1/sqrt2(|up>|cat alive>|detector(u)>  |Rest of universe(u)>   +  |down>|Cat dead)|detector(d)>  |Rest of universe(d)>) --------(2)   

I interpret this as, initially with equation (1) there was an infinite number of instantiations of me in the multiverse about to carry out my experiment. During the experiment roughly half of the infinite instantiations consist of me finding 1 dead cat in the universe I find me to be  in and roughly half of the universes will consist of me finding 1 live cat. I can't see all this birds eye view - only my frog view eg 1 universe and one cat! 

 If It were possible to fire identical cats in a pure state one at a time into a suitable double slit arrangement  then I won't get interference fringes where they strike because decoherence (of the cats) would occur so rapidly.

Do you agree?

 

 

(2)

 everything can be described by a vector in Hilbert space and as such all states in equation (1) would be considered pure, thereby ensuring the universal wave function was also pure.

 

David Wallace in the same reference discusses this scenario saying that states like |cat alive> live in a very big subspace of Hilbert space which can  have states in it that  don’t look anything like a cat (there are an awful lot of things that can be made out of the constituents of a cat!). As I see it, a cat is essentially an emergent object. Never the less he agrees that it is accurate to say of some states in this space that they would be a superposition of a live and dead cat.

Hilbert space is higher dimensional space. It's not space in this universe. It's the space in which the wave equation is waving. (Quantum waves don't exist in single universes. Stuff is made of particles. Not waves. That is also a great, simple, succinct conclusion of the Everett interpretation. No wave-particle duality nonsense.) 

I understand this clearly.

 

I don't know how to reconcile the mixed/pure status.

  OK so I'm still puzzled by the above mixed/pure status representation - are they both correct?

Is my interpretation of pure states of a cat (macroscopic object) correct?

 

Seems ok. Except you seem to be thinking that superpositions can happen in single universes. That's impossible. A superposition is a combination of states. So you need more than one. But in a universe you have *instances* only. If you have two universes then you can have a superposition of at most 2 instances. But as you yourself are an instance in one universe you will only ever observe the instance in your universe. But you *know* when you observe something like a table other instances exist in other universes. That's what your "superposition" is kind of like. Whether anything happens due to this superposition (like interference) is rare...and requires other explanations about what particles are doing according to the laws of quantum theory.

At all instants can they be considered to have a pure state (which varies from instant to instant) and if so, what is wrong with treating it as a member of a mixed state? It surely cannot be both.

It's not so surprising to have what you're calling "macroscopic" superpositions. The cat really is alive and dead...just in different universes. Quantum mechanics provides a measure for the number of universes in which the cat is alive vs dead.

"Instant to instant" also means "universe to universe". Different times, in the multiverse picture, are just special cases of different universes. 

God's eye view (i.e: looking at the whole multiverse from outside it): you get a cat that is both alive and dead. But you and I don't have that perspective. We are only ever in a universe where the cat is alive XOR dead. It's one...or the other. Not both. Not in the universe we occupy.

But if we find an alive cat, we know there really does exist another universe where the cat is dead. Indeed more than this. Our universe is just one of many many many where the cat is also alive. And there are many many many where the cat is dead. And many where it's a dog. Or a hat. Or a bunch of neon gas etc etc.

Brett.

Thank you 

Kindest regards 

Nick

[Brett Hall]

On 10 Jul 2015, at 23:02, Nick Prince <nickmag...@gmail.com> wrote:

On Friday, July 10, 2015 at 12:50:08 AM UTC+1, Brett Hall wrote:

On 10 Jul 2015, at 04:08, Nick Prince <nickmag...@gmail.com> wrote:

 

(1)

If you are an adherent of the Everett interpretation, then it seems that you have to accept that macroscopic superpositions exist. For example Max Tegmark in his paper “Many worlds in context” http://arxiv.org/abs/0905.2182 argues that …. .“Everett’s MWI is simply standard QM with the collapse postulate removed, so that the Schrödinger equation holds without exception”.  Thus he also argues that from this we can deduce that not only microscopic-superpositions are inevitable, but also that macroscopic-superpositions (say of a cat being dead and of being alive) are perfectly legitimate quantum states. It seems that the whole  Everettian  ontology rests on this!

 

Now David Wallace on p5 of:

http://philsci-archive.pitt.edu/8888/1/Wallace_chapter_in_Oxford_Handbook.pdf

 

 argues the case that cats can be in  alive and dead superpositions but If I understand him correctly this must only be for an extremely small amount of time before decoherence rapidly renders the system to be in an improper mixture (equivalent to  tracing over the environment and hence globally the entire state vector remains in a superposition, with off diagonal terms in the density matrix being asymptotic to zero). Hence why he then argues on p5 that macroscopic superpositions do not describe indefiniteness  but rather multiplicity.

 

Have I understood this correctly? Is there actually initially indefiniteness – at least for a tiny instant? We don’t see macroscopic superpositions because they decohere so fast - leaving us in only one of the decohered branches.

You don't see them because you occupy one universe. Not a universe with multiple cats. One cat. In the *multiverse* (i.e in other "parallel branches") there are cats in other states.

 

*Note that David Deutsch explains the universes are neither parallel and nor do they branch. But loose language works well enough here.

I guess you are referring here to the fungibility/differentiation picture here. I had an exchange on the BOI forum on this some time ago. see :

https://groups.google.com/forum/#!searchin/beginning-of-infinity/prince/beginning-of-infinity/o0hDyE_yrSg/GUFrDhGBdScJ 

I'll re read the whole branching versus differentiating again in BOI and review the exchange on https://groups.google.com/forum/#!searchin/beginning-of-infinity/deutsch$20branching/beginning-of-infinity/kmgG2M4uzz8/sKVNBi7Dq_0J

and see if this helps.  I have to say that as I remember I had difficulty in reconciling this picture. If macroscopic objects  are identical instances in some proportion of the universes in the multiverse then what makes them differentiate? Deutsch says on p265 of BOI that it is because they are fungible! Differentiation then spreads out in all directions at or near light speed!

OK so as I said, I guess I'll have to review all this again to see if it sheds any light on the matter.

Yes. So a single particle does something different -> differentiation of whole universes. 

Bearing in mind what I say above, In normal measurement "speak" I accept that in my experiment there is only one cat. Suppose I generate particles in the x direction and send them into a stern gerlach device aligned in the z direction to produce particles aligned in that direction. I could use spin down in the z direction to trigger the poison and spin up to keep my cat safe. I start the experiment and equation (1) will evolve unitarily into a superposition.  

|psi> = 1/sqrt2(|up>|cat alive>|detector(u)>  |Rest of universe(u)>   +  |down>|Cat dead)|detector(d)>  |Rest of universe(d)>) --------(2)   

I interpret this as, initially with equation (1) there was an infinite number of instantiations of me in the multiverse about to carry out my experiment. 

Not infinite. Just (very) large. But even if infinite...okay...

During the experiment roughly half of the infinite instantiations consist of me finding 1 dead cat in the universe I find me to be  in and roughly half of the universes will consist of me finding 1 live cat. I can't see all this birds eye view - only my frog view eg 1 universe and one cat! 

Okay...right.

 If It were possible to fire identical cats in a pure state one at a time into a suitable double slit arrangement  then I won't get interference fringes where they strike because decoherence (of the cats) would occur so rapidly.

Do you agree?

No. This is where Schrodinger's cat cannot be used. Particles can interfere. Cats do not. Cats are made of many particles...in this universe but their counterparts in other universes do not interfere.

It is not possible for cats to interfere when passing through a double slit arrangement. Because it doesn't happen at all, there's nothing to decohere. That is to say: the particles that make up a cat, and copies of it in parallel universes, are not already coherent in such a way as to lose that as a result of passing through the apparatus.

<I snipped the rest>

Brett

[Gary Oberbrunner]

On Sun, Jul 12, 2015 at 11:45 PM, Brett Hall <brha...@hotmail.com> wrote:

Particles can interfere. Cats do not.

Not so, cats just have a _much_ higher frequency due to their larger mass.  And yes, the temperature of a live cat increases the entropy drastically, so getting a coherent stream of cats is well beyond current technology (realistically it's probably beyond any practical technology, so in _practice_ Brett's right).  If you run the Schroedinger equation there is nothing there that says cats behave differently from photons; there is no hard quantum/classical boundary (unlike Copenhagen which assumes there is one).  In fact there have been interference experiments done with heavier particles (e.g. alpha particles and beyond); it's just considerably more difficult.

--

Gary

[Gary Oberbrunner]

On Thu, Jul 9, 2015 at 7:50 PM, Brett Hall <brha...@hotmail.com> wrote:

I don't know how to reconcile the mixed/pure status.

Is my interpretation of pure states of a cat (macroscopic object) correct?

Seems ok. Except you seem to be thinking that superpositions can happen in single universes. That's impossible.

True.  But I thought Nick's concern was about the "preferred basis problem", i.e. why do we only see "pure" states, live or dead and not a mixture (resp. spin up or spin down, or whatever).  Nick?

--

Gary

[Bruno Marchal]

But there is a steady progress.

http://arstechnica.com/science/2012/03/quantum-interference-with-big-molecules-approaches-the-macroscopic/

Actually, t is very easy to put a cat in a macroscopic superposition. That happens all the time with real cats. It is just difficult to observe them due to the rapidity of the decoherence/universe-differentiation. The problem is not the  macro-size of the cat, but the difficulty to isolate the cat from the environment and the observer to which the cat superposition contagiates very quickly.

Bruno

--

Gary

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[Nick Prince]

On Monday, July 13, 2015 at 3:07:58 PM UTC+1, Gary O wrote:

On Thu, Jul 9, 2015 at 7:50 PM, Brett Hall <brha...@hotmail.com> wrote:

I don't know how to reconcile the mixed/pure status

 I think I may understand now where I was going wrong.  A cat being made up of very many particles can easily change from one pure state to another. so it is pure in the sense that for any given instant I can write its density matrix as |cat alive(t)><cat alive(t)|. However, it could be expressed in the form Sum_i P_i|cat alive_i>< cat alive_i| where i runs over all possible pure states the cat might be in. Does this make sense?

Is my interpretation of pure states of a cat (macroscopic object) correct?

Seems ok. Except you seem to be thinking that superpositions can happen in single universes. That's impossible.

True.  But I thought Nick's concern was about the "preferred basis problem", i.e. why do we only see "pure" states, live or dead and not a mixture (resp. spin up or spin down, or whatever).  Nick?

No actually I wasn't thinking about the preferred basis problem in this instance. But now that you mention it  I've never quite grasped this problem either. By choosing to measure spin in the z direction say, I've already made the basis what I'll get results in? I know you can change the basis of the  particle's state from

 |psi> = 1/sqrt2(|+z>  +  |-z>)|  to 1/2[(1-i)|+y>+(1+i)|-y>] but it won't change the direction I've preferred to make a measurement in. I suppose if we are just considering  general interactions then  from what I've read one might expect a basis to emerge which, as they say is robust. I can't see why it should not be position as in the Bohm interpretation. Even my stern gerlach device actually records its values of up or down as a position (spot on a paper backing or one of two lights etc). I've never properly grasped what the problem is here - what am I missing?

Gary

[Nick Prince]

On Monday, July 13, 2015 at 4:45:51 AM UTC+1, Brett Hall wrote:

On 10 Jul 2015, at 23:02, Nick Prince <nickmag...@gmail.com> wrote:

On Friday, July 10, 2015 at 12:50:08 AM UTC+1, Brett Hall wrote:

On 10 Jul 2015, at 04:08, Nick Prince <nickmag...@gmail.com> wrote:

I guess you are referring here to the fungibility/differentiation picture here. I had an exchange on the BOI forum on this some time ago. see :

https://groups.google.com/forum/#!searchin/beginning-of-infinity/prince/beginning-of-infinity/o0hDyE_yrSg/GUFrDhGBdScJ 

I'll re read the whole branching versus differentiating again in BOI and review the exchange on https://groups.google.com/forum/#!searchin/beginning-of-infinity/deutsch$20branching/beginning-of-infinity/kmgG2M4uzz8/sKVNBi7Dq_0J

and see if this helps.  I have to say that as I remember I had difficulty in reconciling this picture. If macroscopic objects  are identical instances in some proportion of the universes in the multiverse then what makes them differentiate? Deutsch says on p265 of BOI that it is because they are fungible! Differentiation then spreads out in all directions at or near light speed!

OK so as I said, I guess I'll have to review all this again to see if it sheds any light on the matter.

Yes. So a single particle does something different -> differentiation of whole universes. 

I'm just re reading Deutsch's BOI to figure out how it is that in two identical universes the particle can become or do something different. He goes to some lengths to build up to this ( He uses the analogy of a transporter failures leading to voltage surges to say how things can become different). Essentially he argues that it is fungibility that allows the possibility of a single particle to do something different across universes, which then leads to differentiation. He argues that it is not random which universes have changed by say, the particle changing direction for some reason and those which might have that same particle continuing to move in the same direction, but due to a deterministic law of physics - the same law in all universes.  So what is this law he is talking about? If it is quantum mechanics then we are using it to explain itself. At a guess I might say it was the Born rule but we only get born probabilities from experimental statistics or symmetry principles. No one has really derived in a completely convincing way why the this rule works 

So how does fungibility allow the single particle to become different - I cannot see it that it is Schrodinger's equation. Yes the equation's deterministic but it does not give a measure over fungible universes to account for some differing evolutions.  

 

Bearing in mind what I say above, In normal measurement "speak" I accept that in my experiment there is only one cat. Suppose I generate particles in the x direction and send them into a stern gerlach device aligned in the z direction to produce particles aligned in that direction. I could use spin down in the z direction to trigger the poison and spin up to keep my cat safe. I start the experiment and equation (1) will evolve unitarily into a superposition.  

|psi> = 1/sqrt2(|up>|cat alive>|detector(u)>  |Rest of universe(u)>   +  |down>|Cat dead)|detector(d)>  |Rest of universe(d)>) --------(2)   

I interpret this as, initially with equation (1) there was an infinite number of instantiations of me in the multiverse about to carry out my experiment. 

Not infinite. Just (very) large. But even if infinite...okay...

 

Yes I see what you mean although Deutsch does seem to be inferring on p276 that an uncountably infinite number of universes are involved. He seems to be saying the same on p279 of the FOR(1998)When I talk to finitist mathematicians they are very scathing of how physicists just can't base physical theory on this. For what it's worth though I DO think there may well be actual infinities. 

What I really want to do is to be able to tell a convincing "story" of what happens from beginning i.e.  my equation (1) to the end of the measurement process - my equation(2) in the language of fungibility. which explains how that particle can do something different.

Thank you.

Kindest regards

Nick

[Gary Oberbrunner]

On Tue, Jul 14, 2015 at 7:28 PM, Nick Prince <nickmag...@gmail.com> wrote:

What I really want to do is to be able to tell a convincing "story" of what happens from beginning i.e.  my equation (1) to the end of the measurement process - my equation(2) in the language of fungibility. which explains how that particle can do something different.

I'm not a mathematical physicist but it seems to me that there _is_ necessarily randomness.  E.g. the time at which a particle decays and emits a photon.  It's different in different universes; multiversally it's "deterministic" (summed over all universes, the photon is emitted, in such a way that the SWE is satisfied) but individually we can't predict when the decay will happen.  I guess another way to look at it is you can't tell which universe you'll end up in, so that's another source of at least perceived randomness.

--

Gary

[Nick Prince]

Thanks for that Gary. Deutsch says on p269 that a phenomena appears unpredictable to observers for three reasons. First is fundamental randomness which he says is excluded because there are no such variables in real physics. Second is that factors affecting the phenomena, though deterministic are unknown or to complex to account for. Third is that two or more instances of the observer become different - he then says that it makes their outcomes strictly unpredictable despite being described by deterministic laws? 

Would not the first of these rule out what you call necessary randomness? The third does not explain 'What' makes the two instances become different. on p278 He says .."the origin of QM randomness ... is due to the measure that the theory provides for the multiverse, which is in turn due to what kinds of physical processes the theory allows and forbids."

Maybe this is what you are getting at but the meaning he gives here eludes me.  Is he talking about the Born rule here?

Regards

Nick

[Brett Hall]

Yeah. David Deutsch makes exactly that last point: subjective randomness is explained entirely by objectively deterministic laws of physics. Although your first point about necessary randomness, I don't think that's right. Unless you put "subjective" in there somewhere. Which is what you were getting at. I think.

Have people seen DD's talk on probability in light of the multiverse? Conclusion: probability theory is largely misdirected, misapplied, it's a misconception and leads to nonsense - in physics (and therefore the physical world generally, by extension). Nothing ever "probably" happens. Things happen, or they do not happen. (This is one of the motivations for his new constructor theory).

Here's the talk: http://youtu.be/wfzSE4Hoxbc

Brett.

[Bruno Marchal]

On 16 Jul 2015, at 06:16, Brett Hall wrote:

On 16 Jul 2015, at 02:36, Gary Oberbrunner <ga...@oberbrunner.com> wrote:

On Tue, Jul 14, 2015 at 7:28 PM, Nick Prince <nickmag...@gmail.com> wrote:

What I really want to do is to be able to tell a convincing "story" of what happens from beginning i.e.  my equation (1) to the end of the measurement process - my equation(2) in the language of fungibility. which explains how that particle can do something different.

I'm not a mathematical physicist but it seems to me that there _is_ necessarily randomness.  E.g. the time at which a particle decays and emits a photon.  It's different in different universes; multiversally it's "deterministic" (summed over all universes, the photon is emitted, in such a way that the SWE is satisfied) but individually we can't predict when the decay will happen.  I guess another way to look at it is you can't tell which universe you'll end up in, so that's another source of at least perceived randomness.

Yeah. David Deutsch makes exactly that last point: subjective randomness is explained entirely by objectively deterministic laws of physics. Although your first point about necessary randomness, I don't think that's right. Unless you put "subjective" in there somewhere. Which is what you were getting at. I think.

Yes, that third form of indeterminacy is just a particular case of the more general computationalist first person indeterminacy. But then it occurs also in arithmetic, and, without giving matter a special magical (non Turing emulable) role, it extends the quantum indeterminacy (on the relative quantum computations) on *all* computation. But the schroedinger equation needs to be a derivable consequence on arithmetic and arithmetical self-reference. And that is nice because such self-reference makes it possible to explain the incorrigibility of consciousness and its non justifiability in the third person way. 

Have people seen DD's talk on probability in light of the multiverse? Conclusion: probability theory is largely misdirected, misapplied, it's a misconception and leads to nonsense - in physics (and therefore the physical world generally, by extension). Nothing ever "probably" happens. Things happen, or they do not happen. (This is one of the motivations for his new constructor theory).

Things happens, but only in the 1p view. In the 3p view, we need only Turing universal relations, and those exist already in a tiny segment of arithmetic. Physicalness is one of the machine's way to view arithmetic from inside. Deutsch's constructor theory is still too much naive on the mind-body relation, and uses implicitly non Turing emulable properties of matter for which we have no evidences.

Bruno

Here's the talk: http://youtu.be/wfzSE4Hoxbc

Brett.

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[John Clark]

On Wed, Jul 15, 2015 at 6:28 PM, Nick Prince <nickmag...@gmail.com> wrote:

 

 

Deutsch says on p269 that a phenomena appears unpredictable to observers for three reasons. First is fundamental randomness which he says is excluded because there are no such variables in real physics. Second is that factors affecting the phenomena, though deterministic are unknown or to complex to account for. Third is that two or more instances of the observer become different - he then says that it makes their outcomes strictly unpredictable despite being described by deterministic laws? 

​But would it really be legitimate ​to call such laws deterministic? It seems to me that to be deterministic a law must not only say that under condition X Y will always happen but also say that under condition X  Z will NEVER happen; but if Many Worlds is true everything will happen. 

 John K Clark  

 

[LizR]

The laws are globally deterministic, but random from the point of view of observers. However I don't think everything happens, does it? Only everything that's consistent with quantum mechanics.

[John Clark]

On Sat, Jul 18, 2015 at 6:35 PM, LizR <liz...@gmail.com> wrote:

​> ​

The laws are globally deterministic, but random from the point of view of observers.

​It would certainly be random from the point of view of any observer, the very laws of physics would ensure it, but I'm not sure it would be deterministic even from a global viewpoint. First there is the fact that nothing can have that global viewpoint, but even if something could when a particle is exposed to influence X and the particle then moves up and down and left and right and back and forth and every other conceivable direction could it really be said that X determined the way the particle moved? ​

 

​And if X turns into Z but Y can also turn into Z then even a mythical global observer (which can't exist) couldn't tell if Z came from X or Y.​

 

​> ​

However I don't think everything happens, does it? Only everything that's consistent with quantum mechanics.

​Well yes, but if everything that can happen does happen that's still a astronomical number of things and maybe a infinite number of things; deterministic laws are supposed to impose restrictions on what can happen and that's not much of a restriction.

  John K Clark  ​

 

[LizR]

On 19 July 2015 at 13:10, John Clark <johnk...@gmail.com> wrote:

On Sat, Jul 18, 2015 at 6:35 PM, LizR <liz...@gmail.com> wrote:

​> ​

The laws are globally deterministic, but random from the point of view of observers.

​It would certainly be random from the point of view of any observer, the very laws of physics would ensure it, but I'm not sure it would be deterministic even from a global viewpoint. First there is the fact that nothing can have that global viewpoint, but even if something could when a particle is exposed to influence X and the particle then moves up and down and left and right and back and forth and every other conceivable direction could it really be said that X determined the way the particle moved? ​

 

​And if X turns into Z but Y can also turn into Z then even a mythical global observer (which can't exist) couldn't tell if Z came from X or Y.​

As I understand it, the equations give a distribution of outcomes and explain why these appear to be random to any given observer. The lack of a global observer isn't necessarily important - it was one of the original objections to atomic theory, for example, but that survived. One could equally object that various quantum phenomena are unobservable because an observation would disturb the system - but in practice the theory still works, producing the correct results when something can be observed.

 

​> ​

However I don't think everything happens, does it? Only everything that's consistent with quantum mechanics.

​Well yes, but if everything that can happen does happen that's still a astronomical number of things and maybe a infinite number of things; deterministic laws are supposed to impose restrictions on what can happen and that's not much of a restriction.

No, deterministic laws are supposed to say what happens without invoking any intrinsic randomness.

[John Clark]

On Sun, Jul 19, 2015 at 2:57 AM, LizR <liz...@gmail.com> wrote:

 

​> ​

The lack of a global observer isn't necessarily important

​If you start saying that from a global viewpoint this and that happens then it is important if such a viewpoint does not exist. ​

 

​Maybe.​

​>> ​

​if everything that can happen does happen that's still a astronomical number of things and maybe a infinite number of things; deterministic laws are supposed to impose restrictions on what can happen and that's not much of a restriction.

 

​> ​

No, deterministic laws are supposed to say what happens without invoking any intrinsic randomness.

​Laws are also supposed to say what doesn't happen, Feynman said  "Science is imagination in a straitjacket" and by that he meant that a law that says anything can happen is almost the same as no law at all. For example,  all the laws of physics (with the important exception of the second law of thermodynamics) can be boiled down to saying that something is conserved, something can not be created or destroyed, which as Noether proved is equivalent to saying something is symmetrical; but if nothing (or at least very little) is conserved then nothing (or at least very little) is symmetrical and the fundamental laws of physics are starting to look a bit anemic. Maybe.  

  John K Clark 

 

[Bruno Marchal]

Assuming quantum mechanics, of course.

But the problem, or the solution of the problem, is that if we assume mechanism all possible machine subjective experiences "happens" in arithmetic. It is an open problem if some of them can glue enough coherently to define a particular physical realities. But the math of auto-reference already shows that there is a core fundamental physical reality, invariant for all machines (actually, a part is shared with many non-machine too, as long as they are self-referentially correct.

I agree that we should never say "everything" out of a theory of the thing we admit. We must also be clear if we talk on the ontology or the phenomenology.

Bruno

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[Bruno Marchal]

Well, you will see every position, if you have found the precise momentum, or vice versal, but in practive you have gaussians, large or thin or in between, and what the theory will provide are the relative proportion of consistent extension accessible to you, assuming you are part of the global determinisic process. That is the advantage of Everett (and mechanism in general hopefully), it makes randomness first person plural phenomenological. 

Bruno

 John K Clark  

 

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[Nick Prince]

I've just been re reading your answer here. So suppose you have an electron prepared in the |+z> direction, then  I can expect that if I perform an experiment where I send it to a SG apparatus aligned in the |+x> direction then, the fact that we can write the |+z> in the x- basis as |+z> =  1/sqrt 2(|+x> + |-x>) simply tell us that there is a huge number(infinite!) of similar experiments across the multiverse with fungible instances of the |+z> particle  being sent to similar apparatus arrangements - in which case the expression above also simply tells us the number of possible outcomes of the experiment (i.e. in this case 2 possibles viz. spin up in the x dirn. or spin down in the x direction)  and the proportions of worlds to expect with these outcomes in the experiments going on concurrently in all these worlds. Moreover it's not really correct to say that a |+z> particle state can be written as a superposition  of the |+x> and |-x> in a single world. Have I got this right here ??

Nick

[LizR]

Isn't there a problem where the probability of one event is (say) 0.1234567 and another event is 0.8765433 - this event has to generate (I think) 10 million universes, while an electron being up or down with 50% probability only needs to generate two. Or are these both actually subdividing continua?

--

[Nick Prince]

I'm not sure about that. If you rotate the SG device that receives the |+z> particle by theta degrees to point in the "n" direction laying in  then the x,z plane the expression becomes

|+n>  = cos(theta/2)|+z> + sin(theta/2)|-z>

so the respective probabilities for getting |+n> to be in the up(down) direction are just  cos(theta/2)^2 and sin(theta)^2.  These are normalised and there are just 2 outcomes? so the *infinite* amount of fungible  universes in which this experiment is carried out will then end up differentiating into two equally infinite branches, one of which gets spin up(down)  in the ratios roughly cos(theta/2)^2 for spin up, and sin(theta)^2  for spin down

Have I missed your point?

[Nick Prince]

sorry, I meant cos(theta/2)^2 and sin(theta/2)^2  throughout in the text.

[LizR]

I don't think you missed the point. I think you said - effectively - yes, the branches are continua. So you have a continuum which can be divided at any arbitrary (irrational number) point into two sub-continua (which makes sense to me - except that it makes the multiverse "classical" perhaps overall, rather than quantised?)

[Nick Prince]

On Sunday, August 2, 2015 at 9:00:35 PM UTC+1, Liz R wrote:

I don't think you missed the point. I think you said - effectively - yes, the branches are continua. So you have a continuum which can be divided at any arbitrary (irrational number) point into two sub-continua (which makes sense to me - except that it makes the multiverse "classical" perhaps overall, rather than quantised?)

This problem of irrationals and how they are dealt with has bothered me too, but it's above my pay grade to answer. I've searched around to try to get some answers but only get dead ends.  One explanation put forward was that different worlds *are* separated by a discrete amount - you can't make the difference arbitrarily small. See  

https://groups.google.com/forum/#!searchin/beginning-of-infinity/fungi

ble/beginning-of-infinity/lS632yjUKxM/f4nCbLhXecwJ

in which Alan Forester says "In any finite region of space time there is only a finite number of
histories. Probabilities being irrational isn't really directly
testable since you can always explain the results of any finite series
of experiments by a range of probabilities including rational ones."

 I don't really understand this! I've also read (somewhere?)

that all measureable quantities are really discrete - even position has to be measured within limits which admittedly can go to arbitrary levels, but still end up as a discrete value  so the continuously infinite original set of worlds can be  divided up into a countable set (i.e. two in this case) of different infinite sets of worlds. See http://math.stackexchange.com/questions/262988/infinite-set-as-union-of-disjoint-countable-sets which might be related but I can't understand any of it. As I have said,  these are not my ideas about the matter though.  I have to admit these explanations leave me decidedly uneasy and I would like know how irrational probabilities proportional to say 1/pi are dealt with. Can anyone help with this - I would welcome some  clarification?

Thanks Liz

Kind Regards

Nick 

[LizR]

Thanks, Nick.

I can only think about this sort of thing - assuming I manage it at all - in fairly metaphorical or perhaps pictorial ways (i.e. I lose track fairly quickly if I pursue the mathematics very far). So what follows will not be very rigorous.

It seems to me that the first part of the answer you've quoted comes down to "everything is quantised" (including space-time) and hence whatever underlies reality, and the assumed branching, can only have a discrete, if very large, number of permutations. This leaves me uneasy, as I think it does you, when we have a situation that gives rise to a probability described by some rational number with a large value above and below the line - can the multiverse really split into 1234567 identical branches in state A and 8765433 in state B simply to bring the probabilities out correctly? Perhaps that situation never arises - maybe at the quantum level we always have 50-50 splits, and it's only our macroscopic view that makes these multiply up into something that looks like 1234567-8765433. (But I don't feel completely happy with that explanation.)

The second part - which needless to say I also don't understand - appears to be saying that the multiverse contains a continuum (uncountably infinite number) of worlds, and that this continuum can be partitioned using discrete values. This would appear to also deal with irrational numbers, in principle, so I'm not sure where the need for "disjoint countable sets" comes in.

Or maybe I just misunderstood!

[Gary Oberbrunner]

Both are continua (or at least very large numbers) of "universes".  Otherwise probabilities that are rational numbers would be very, needlessly, special.  Think of it as slicing the "probability space" between 0 and 1.  0.5 just happens to slice it in half; 0.50001 is nearly the same with just a slightly higher measure on one side.  Multiple outcomes (>2) are the same.

--

Gary

[Gary Oberbrunner]

On Sun, Aug 2, 2015 at 7:40 PM, Nick Prince <nickmag...@gmail.com> wrote:

you can always explain the results of any finite series
of experiments by a range of probabilities including rational ones."

 I don't really understand this!

I think the point is that since you're only doing N (finite N) experiments, let's say they're binary, you'll always get a rational probability.  Hence there's no way to _test_ whether a given experiment actually has an irrational probability of a certain result.  The more tests you do, the closer you can estimate the actual probability, but you'll never _exactly_ get an irrational probability from a finite set of tests.  Right?

--

Gary

[Gary Oberbrunner]

On Sun, Aug 2, 2015 at 8:29 PM, LizR <liz...@gmail.com> wrote:

It seems to me that the first part of the answer you've quoted comes down to "everything is quantised" (including space-time) and hence whatever underlies reality, and the assumed branching, can only have a discrete, if very large, number of permutations. This leaves me uneasy, as I think it does you, when we have a situation that gives rise to a probability described by some rational number with a large value above and below the line - can the multiverse really split into 1234567 identical branches in state A and 8765433 in state B simply to bring the probabilities out correctly? Perhaps that situation never arises - maybe at the quantum level we always have 50-50 splits, and it's only our macroscopic view that makes these multiply up into something that looks like 1234567-8765433. (But I don't feel completely happy with that explanation.)

The second part - which needless to say I also don't understand - appears to be saying that the multiverse contains a continuum (uncountably infinite number) of worlds, and that this continuum can be partitioned using discrete values. This would appear to also deal with irrational numbers, in principle, so I'm not sure where the need for "disjoint countable sets" comes in.

As to your first question, absolutely!  This is what makes Harry Potter universes physically possible.  Or the thermodynamics 101 problem where you estimate the probability that all the coffee molecules in the coffee cup will happen to randomly be oriented up at the same time, making the coffee jump out of the cup spontaneously.  These are _very_ small numbers, and they represent a bifurcation of the multiverse into one tiny sliver (where the coffee hits the ceiling), and all the other slivers where it doesn't.  (Note those aren't exactly fungible either; there's lots of heat and thus thermal noise in all those coffee molecules.)  There is plenty of "thickness" to the multiverse, if you want to think about it that way.

As to whether it's just very, very thick... or an actual continuum: nobody knows for sure! (AFAIK)  It might come down to whether gravity is quantized.  I'm also unsure whether the direction of momentum is quantized, e.g. if a photon is emitted, can it go off in _literally_ any direction, or just a very large number of them?  But this latter is just my ignorance; the gravity question is I believe really unknown.

On the other hand if your question is how can an infinite set be partitioned into a discrete set of subsets with definite relations between them (like they're the same size), this is what's called measure.  https://en.wikipedia.org/wiki/Measure_(mathematics) will enlighten you.

--

Gary

[Russell Standish]

Quantisation is one answer, but an unsatisfactory one
IMHO. Particularly, when recent measurement put the possible scale of
quantisation well below the planck length.

This is, of course, only a problem for the Deutsch view that there is
a physically objective Multiverse, which decoheres at the speed of
light into a collection of countably infinite possibilities.

The alternative is "it's all in the head" idea of Wheeler's participative
universe, would have that there are as many universes as there there
are distinct observers. This automatically entails a countably
infinite number, but no fundamental scale of quantisation, as in
principle, universes can be ever more finely subdivided by making ever
more precise measurements.

Of course the latter point of view is what I strenuously argue in
favor of in my Theory of Nothing book. It solves this problem, the
"probability problem" (Born's rule), the "basis problem", but does a
few of its own, the biggest being how the anthropic principle is
supposed to work.

Cheers

--

----------------------------------------------------------------------------
Prof Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics hpc...@hpcoders.com.au
University of New South Wales http://www.hpcoders.com.au
----------------------------------------------------------------------------

[LizR]

On 3 August 2015 at 13:29, Russell Standish <li...@hpcoders.com.au> wrote:

Quantisation is one answer, but an unsatisfactory one
IMHO. Particularly, when recent measurement put the possible scale of
quantisation well below the planck length.

Ah yes, I remember that. I guess there were other assumptions involved in that experiment that might fall to later ones, but that's the current view I believe.

This is, of course, only a problem for the Deutsch view that there is
a physically objective Multiverse, which decoheres at the speed of
light into a collection of countably infinite possibilities.

The alternative is "it's all in the head" idea of Wheeler's participative
universe, would have that there are as many universes as there there
are distinct observers. This automatically entails a countably
infinite number, but no fundamental scale of quantisation, as in
principle, universes can be ever more finely subdivided by making ever
more precise measurements.

This makes me wonder .... (a) why there are any observers, and (b) why there are a countably infinite number of them. (Would this have anything to do with "comp" and observers being embedded in arithmetic? I have read TON about three times but I still can't recall the finer details.)

Of course the latter point of view is what I strenuously argue in
favor of in my Theory of Nothing book. It solves this problem, the
"probability problem" (Born's rule), the "basis problem", but does a
few of its own, the biggest being how the anthropic principle is
supposed to work.

I would imagine, "all the time" if the universe is only all in our heads. (Or would be, if we had heads.)

Cheers

--

----------------------------------------------------------------------------
Prof Russell Standish                  Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics      hpc...@hpcoders.com.au
University of New South Wales          http://www.hpcoders.com.au
----------------------------------------------------------------------------

[LizR]

On 3 August 2015 at 13:15, Gary Oberbrunner <ga...@oberbrunner.com> wrote:

On Sun, Aug 2, 2015 at 8:29 PM, LizR <liz...@gmail.com> wrote:

It seems to me that the first part of the answer you've quoted comes down to "everything is quantised" (including space-time) and hence whatever underlies reality, and the assumed branching, can only have a discrete, if very large, number of permutations. This leaves me uneasy, as I think it does you, when we have a situation that gives rise to a probability described by some rational number with a large value above and below the line - can the multiverse really split into 1234567 identical branches in state A and 8765433 in state B simply to bring the probabilities out correctly? Perhaps that situation never arises - maybe at the quantum level we always have 50-50 splits, and it's only our macroscopic view that makes these multiply up into something that looks like 1234567-8765433. (But I don't feel completely happy with that explanation.)

The second part - which needless to say I also don't understand - appears to be saying that the multiverse contains a continuum (uncountably infinite number) of worlds, and that this continuum can be partitioned using discrete values. This would appear to also deal with irrational numbers, in principle, so I'm not sure where the need for "disjoint countable sets" comes in.

As to your first question, absolutely!  This is what makes Harry Potter universes physically possible.  Or the thermodynamics 101 problem where you estimate the probability that all the coffee molecules in the coffee cup will happen to randomly be oriented up at the same time, making the coffee jump out of the cup spontaneously.  These are _very_ small numbers, and they represent a bifurcation of the multiverse into one tiny sliver (where the coffee hits the ceiling), and all the other slivers where it doesn't.  (Note those aren't exactly fungible either; there's lots of heat and thus thermal noise in all those coffee molecules.)  There is plenty of "thickness" to the multiverse, if you want to think about it that way.

As to whether it's just very, very thick... or an actual continuum: nobody knows for sure! (AFAIK)  It might come down to whether gravity is quantized.  I'm also unsure whether the direction of momentum is quantized, e.g. if a photon is emitted, can it go off in _literally_ any direction, or just a very large number of them?  But this latter is just my ignorance; the gravity question is I believe really unknown.

If space and time are quantised, I would think that a photon can only go in a finite number of directions, or to be more exact, can only trace a finite number of paths through, for example, some underlying network of nodes and edges, with a probability at each node of taking one edge leading from it (in roughly the direction of motion, I imagine) as opposed to another one. This could lead in a multiverse to a sum over histories which is equivalent to following every available path through a (very fine grained) web, each path being weighted by - I assume - the current position and momentum of the photon.

On the other hand if your question is how can an infinite set be partitioned into a discrete set of subsets with definite relations between them (like they're the same size), this is what's called measure.  https://en.wikipedia.org/wiki/Measure_(mathematics) will enlighten you.

That article ------->

My head -> O

But thank you anyway :-)

 

--

Gary

[Gary Oberbrunner]

On Sun, Aug 2, 2015 at 10:01 PM, LizR <liz...@gmail.com> wrote:

On the other hand if your question is how can an infinite set be partitioned into a discrete set of subsets with definite relations between them (like they're the same size), this is what's called measure.  https://en.wikipedia.org/wiki/Measure_(mathematics) will enlighten you.

That article ------->

My head -> O

But thank you anyway :-)

I agree, that's absurdly overcomplicated.  Sorry about that.  For our purposes, the measure of a subset of an infinite set, like the reals, is just the length of the line defining that subset.  So the subset of [0,1] from 0.2 to 0.5 is of length (and therefore measure) 0.3.  Measure is just length along the number line.  This means rational/irrational doesn't come into it.  So probabilities split nicely just as you'd expect; as the slices get thinner the next slice just partitions according to probability from that slice no matter how thin it is.  To take a concrete example, in the coffee example which is already absurdly unlikely, we can say that _in the universes in which that has already happened_,  half (or more) of the coffee will end up on one side of the table in about half the universes, and less than half the coffee in the other half of the already very tiny sliver of the multiverse.  This works fine because _if_ we find ourself in that tiny slice, it's all we can see -- so the slices that "branch" from it all add up to the size of that slice, but it looks like they sum to 1 to anyone in that sliver.  Does that help a little?

And as for Russell's "The alternative is "it's all in the head" idea of Wheeler's participative

universe, would have that there are as many universes as there there

are distinct observers."... all I can say is:

  eww.

(OK, I can say a little more: MWI elegantly removes all anthropormorphism, "observers" whatever they are, observations, collapse and all the other ugliness... and you want to put it back??  Seriously?  I thought I understood your book, I guess I have to go reread it.  It was a long time ago.)

And Liz, I'm not sure a node-graph is the only way to quantize space... I personally lean toward an infinite multiverse so I'd want some harder proof that quantization really works as you propose before I abandon that view.

--

Gary

[LizR]

On 3 August 2015 at 14:22, Gary Oberbrunner <ga...@oberbrunner.com> wrote:

I agree, that's absurdly overcomplicated.  Sorry about that.  For our purposes, the measure of a subset of an infinite set, like the reals, is just the length of the line defining that subset.  So the subset of [0,1] from 0.2 to 0.5 is of length (and therefore measure) 0.3.  Measure is just length along the number line.  This means rational/irrational doesn't come into it.  So probabilities split nicely just as you'd expect; as the slices get thinner the next slice just partitions according to probability from that slice no matter how thin it is.  To take a concrete example, in the coffee example which is already absurdly unlikely, we can say that _in the universes in which that has already happened_,  half (or more) of the coffee will end up on one side of the table in about half the universes, and less than half the coffee in the other half of the already very tiny sliver of the multiverse.  This works fine because _if_ we find ourself in that tiny slice, it's all we can see -- so the slices that "branch" from it all add up to the size of that slice, but it looks like they sum to 1 to anyone in that sliver.  Does that help a little?

Yes, thanks!

And as for Russell's "The alternative is "it's all in the head" idea of Wheeler's participative
universe, would have that there are as many universes as there there
are distinct observers."... all I can say is:

  eww.

:-)

(OK, I can say a little more: MWI elegantly removes all anthropormorphism, "observers" whatever they are, observations, collapse and all the other ugliness... and you want to put it back??  Seriously?  I thought I understood your book, I guess I have to go reread it.  It was a long time ago.)

I seem to recall that it's about what information an observer is likely to see. But maybe I need to re-read it too.

And Liz, I'm not sure a node-graph is the only way to quantize space... I personally lean toward an infinite multiverse so I'd want some harder proof that quantization really works as you propose before I abandon that view.

That was just my attempt at an example (based on vague memories of LQG and spin-foams). It could be all sorts of things - but I assume (in this view) it would have to be a finite, if unthinkably large, thing-of-whatever-type. Or at least a countably infinite thing in which the nodes or whatever go on forever, like an infinite game of life.

However I'm also sympathetic to the idea of an infinite multiverse - in many ways it seems to make more intuitive sense than the alternative of "vastly, hugely, mind-bogglingly big". However, I also have to put up with a gut feeling that nature won't allow uncountably infinite things to exist, because in a way they seem like magic (e.g. you can stretch space-time forever, and you just get more of it...)

(*sigh*)

[Russell Standish]

On Sun, Aug 02, 2015 at 10:22:50PM -0400, Gary Oberbrunner wrote:
>
> And as for Russell's "The alternative is "it's all in the head" idea of
> Wheeler's participative
> universe, would have that there are as many universes as there there
> are distinct observers."... all I can say is:
>
> eww.
>
> (OK, I can say a little more: MWI elegantly removes all anthropormorphism,
> "observers" whatever they are, observations, collapse and all the other
> ugliness... and you want to put it back?? Seriously? I thought I
> understood your book, I guess I have to go reread it. It was a long time
> ago.)
>

Anthropomorphism means assuming human motivations in non-human beings
or objects. The participatory idea makes no assumptions about humans
being the only possible type of observer. Indeed, it seems highly
likely that other type of observer exist elsewhere in the great
unwashed multiverse, and perhaps even here on Earth. To each of those
would correspond a universe.

What it does mean is that there is no observer independent notion of a
universe - each universe is defined by the full set of measurements
performed by the observers in it. It is what selects the volume from
the library of Babel - it is what hews Michaelangelo's statue in the
recent paper that Brent posted on Platonism.

As for collapse - there is no collapse. The collapse idea came from
assuming there is a single objective reality which observation must
collapse the wave function to. If anything, the participatory idea has
moved even further away from that notion than does Deutsch's objective
multiverse, with its observer independent decoherence mechanism.

[Bruno Marchal]

On 02 Aug 2015, at 22:00, LizR wrote:

I don't think you missed the point. I think you said - effectively - yes, the branches are continua. So you have a continuum which can be divided at any arbitrary (irrational number) point into two sub-continua (which makes sense to me - except that it makes the multiverse "classical" perhaps overall, rather than quantised?)

Is not the goal of the "MWI" to make the "universe" classical or as much classical as possible?  I am not sure I see a problem here, on the contrary, I see a solution (except that such a solution entails that we must explain the wave itself from the machine phenomenology).

Bruno

http://iridia.ulb.ac.be/~marchal/

[Bruno Marchal]

On 03 Aug 2015, at 03:56, LizR wrote:

On 3 August 2015 at 13:29, Russell Standish <li...@hpcoders.com.au> wrote:

Quantisation is one answer, but an unsatisfactory one
IMHO. Particularly, when recent measurement put the possible scale of
quantisation well below the planck length.

Ah yes, I remember that. I guess there were other assumptions involved in that experiment that might fall to later ones, but that's the current view I believe.

This is, of course, only a problem for the Deutsch view that there is
a physically objective Multiverse, which decoheres at the speed of
light into a collection of countably infinite possibilities.

The alternative is "it's all in the head" idea of Wheeler's participative

Do Wheeler use the expression "all in the head"?

universe, would have that there are as many universes as there there
are distinct observers. This automatically entails a countably
infinite number, but no fundamental scale of quantisation, as in
principle, universes can be ever more finely subdivided by making ever
more precise measurements.

This makes me wonder .... (a) why there are any observers, and (b) why there are a countably infinite number of them.

That is an open problem, but a countable universe is not really plausible, neither with QM, nor with comp (which should imply QM if QM is correct). The universal dovetailing, which is in arithmetic dovetail also on the (Turing) oracles. Do you see this? The random oracle is unavoidable, and it might help to get the measure right, but this is not yet extracted from the logic of self-reference. What is clear is that measure theory is simpler on finite or on non-countable sets. Infinite countable sets needs to weaken the sigma-additive axiom of measure theory. 

Bruno

http://iridia.ulb.ac.be/~marchal/

[Gary Oberbrunner]

On Mon, Aug 3, 2015 at 5:52 AM, Bruno Marchal <mar...@ulb.ac.be> wrote:

Is not the goal of the "MWI" to make the "universe" classical or as much classical as possible?  I am not sure I see a problem here, on the contrary, I see a solution

+1!

--

Gary

[LizR]

On 3 August 2015 at 21:52, Bruno Marchal <mar...@ulb.ac.be> wrote:

On 02 Aug 2015, at 22:00, LizR wrote:

I don't think you missed the point. I think you said - effectively - yes, the branches are continua. So you have a continuum which can be divided at any arbitrary (irrational number) point into two sub-continua (which makes sense to me - except that it makes the multiverse "classical" perhaps overall, rather than quantised?)

Is not the goal of the "MWI" to make the "universe" classical or as much classical as possible?  I am not sure I see a problem here, on the contrary, I see a solution (except that such a solution entails that we must explain the wave itself from the machine phenomenology).

The only problem, if it is a problem, is that "classical" (in the sense I'm using it here) involves uncountable infinities to be present everywhere in space-time and the multiverse, while "quantised" involves, at most, a single countable infinity. Please forgive me if I feel sufficiently uneasy with this concept of "untamed infinities" that I would welcome any evidence which proves it to be true, and hence overcomes my intuitive fear.

(Comp would appear to only assume the existence of one countable infinity, I think? Or have I got that wrong?)

[LizR]

On 3 August 2015 at 22:45, Bruno Marchal <mar...@ulb.ac.be> wrote:

(I said: This makes me wonder .... (a) why there are any observers, and (b) why there are a countably infinite number of them.)

That is an open problem, but a countable universe is not really plausible, neither with QM, nor with comp (which should imply QM if QM is correct). The universal dovetailing, which is in arithmetic dovetail also on the (Turing) oracles. Do you see this? The random oracle is unavoidable, and it might help to get the measure right, but this is not yet extracted from the logic of self-reference. What is clear is that measure theory is simpler on finite or on non-countable sets. Infinite countable sets needs to weaken the sigma-additive axiom of measure theory. 

Hi Bruno!

That was over my head like the empyrean ... so ... to start at the very beginning ... why is a countable universe (do you mean countably infinite, or just countable) not really plausible with QM?

[Nick Prince]

Gary

Yes that makes sense. So if you had a state vector like

|+n> = (1/sqrt )(1/pi)|+z> + Sqrt({pi+1}/pi})|-z>

You could perform countless tests (N) which in the limit would approach the irrational  probability but never achieve it. I guess it's similar to the fact that a line  in 2D say, cannot in practice be drawn if the length turns out to be irrational which as Liz points out comes down to when quantization of space starts. In the example of the SG I've given above it would be quantization of the angular measure.

kind regards

Nick  

[John Clark]

On Sun, Aug 2, 2015 at 8:29 PM, LizR <liz...@gmail.com> wrote:

​> ​

This leaves me uneasy, as I think it does you

​ [Nick]​

, when we have a situation that gives rise to a probability described by some rational number with a large value above and below the line - can the multiverse really split into 1234567 identical branches in state A and 8765433 in state B simply to bring the probabilities out correctly? 

​Maybe it shouldn't but I must admit ​

It makes me a bit uneasy too

​,​

mainly because if all

​ 

8765433 universes are identical then I don't quite understand in what sense they can be said to have split.

​ 

Deutsch

​ tries to solve this

with

​ an anthology ​

the with the fungibility

​ 

of money

​,​

one dollar is identical to another dollar but having 8765433

​ 

of them is different from having just one

​;​

and

​ 

8765433

​ 

identical universes are different than just one

​ universe​

. Maybe Deutsch is right, but sometimes I wonder if he's not just kicking the problem upstairs.

​  John K Clark​

 John K Clark

[Bruno Marchal]

On 03 Aug 2015, at 23:56, LizR wrote:

On 3 August 2015 at 21:52, Bruno Marchal <mar...@ulb.ac.be> wrote:

On 02 Aug 2015, at 22:00, LizR wrote:

I don't think you missed the point. I think you said - effectively - yes, the branches are continua. So you have a continuum which can be divided at any arbitrary (irrational number) point into two sub-continua (which makes sense to me - except that it makes the multiverse "classical" perhaps overall, rather than quantised?)

Is not the goal of the "MWI" to make the "universe" classical or as much classical as possible?  I am not sure I see a problem here, on the contrary, I see a solution (except that such a solution entails that we must explain the wave itself from the machine phenomenology).

The only problem, if it is a problem, is that "classical" (in the sense I'm using it here) involves uncountable infinities to be present everywhere in space-time and the multiverse, while "quantised" involves, at most, a single countable infinity.

I am not sure why you say this. takes the energy level of an electron interacting with a proton. The levels of energy are quantized: you have indeed a infinite discrete spectrum E0, E1, E2, etc.

But QM implies that all complex linear combination represents a possible physical state.

... and I don't think that even if space and time are quantized, why that would change. 

That is why I tend to agree with Deutsch's idea that there is a continuum of "parallel universe", or better "relative states". 1/sqrt(2)(up + down) describes 2^aleph_0 up, and down, with the same measure, particles.

QM might change, but the theory is currently strongly related to the usual theory of real (and complex) numbers.

Please forgive me if I feel sufficiently uneasy with this concept of "untamed infinities" that I would welcome any evidence which proves it to be true, and hence overcomes my intuitive fear.

(Comp would appear to only assume the existence of one countable infinity, I think? Or have I got that wrong?)

The problem is that we must distinguish what is explicitly assumed in the theory, and the assumption made at the meta-level and needed to interpret the theory. In usual math and physics, we are just careless on this, and it does no lead to trouble as such distinction are not relevant for most application. Bt once we want to be able to get some light on the mind-body problem, the, like in mathematical logic, such distinction becomes important (even crucial at some point).

So, comp does not even assume anything infinite. That's why Judson Webb called it a "finitisme" assumption. We assume only 0, s(0), s(s(0)) ... which are all finite objects. We assume the laws of addition and multiplication, which produces again only finite objects.

The set {0, s(0), ...} is just not part of the theory. It is only part of the informal meta-theory, used when we do meta-mathematics, and reason no more about numbers, but about that the theory of numbers itself. In that case, we assume the usual math with all its infinities (like infinite sets, Hilbert space, etc.).

Now, such real numbers will reappear *in* the theory, but only as epistemological constructs by numbers or machine to think about the natural numbers and machines. There we can prove that machine believing in some infinities (and perhaps neutral or not interested in the nature (ontological, epistemological, ...) of their existence) are able to prove more propositions on 0, s(0), ... than machine not believing in them. So such belief in infinity (at least epistemological infinity) is a "selective" advantage for machine trying to learn things even if only about numbers, a bit like the complex Riemann Zeta function can provide information on the discrete prime numbers. Since Gödel, we know that even to learn as much as possible in just arithmetic, we have to climb the hierarchy of the higher infinities, despite they do not necessarily needed to exist at the same ontological level than the machines and the finite things.

To be sure, in practice, we have up to now, always been able to eliminate the use of infinities in the "interesting" theorem of number theory, so that some conjecture that the whole of interesting math (excluding mathematical logic, category theory, set theory, classical analysis, ...) can be done in PA. But machine theology, or just any deep cosmological inquiry will require them at the meta-level, and some pure combinatorial problem have often been solved with the use of high cardinal long before others succeed in making the proof "elementary" (formalizable) in Peano Arithmetic.

So comp requires only the assumption of the finite things 0, 1, 2, ...

But both the machines and us needs much more, even to just understand how 0, 1, 2, ... behave, in their (additive/multiplicative) relations with each others.

Bruno

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[Bruno Marchal]

 I meant countably infinite.

, or just countable) not really plausible with QM?

See my other post. 

It can also be proved that once you have a countable model for a first order theory, you have models of all cardinalities. 

My intuition relies on my intuition of computationalism, where, by the FPI we are confronted, epistemologically, but also physically,  to the Turing (random) oracle.

I like comp because we need to postulate only finite things *ontologically.

But then universal numbers need to develop stronger belief to harness those finite things, and the machines cannot avoid them in the epistemologies.

Bruno

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[LizR]

On 5 August 2015 at 01:14, Bruno Marchal <mar...@ulb.ac.be> wrote:

On 03 Aug 2015, at 23:56, LizR wrote:

On 3 August 2015 at 21:52, Bruno Marchal <mar...@ulb.ac.be> wrote:

On 02 Aug 2015, at 22:00, LizR wrote:

I don't think you missed the point. I think you said - effectively - yes, the branches are continua. So you have a continuum which can be divided at any arbitrary (irrational number) point into two sub-continua (which makes sense to me - except that it makes the multiverse "classical" perhaps overall, rather than quantised?)

Is not the goal of the "MWI" to make the "universe" classical or as much classical as possible?  I am not sure I see a problem here, on the contrary, I see a solution (except that such a solution entails that we must explain the wave itself from the machine phenomenology).

The only problem, if it is a problem, is that "classical" (in the sense I'm using it here) involves uncountable infinities to be present everywhere in space-time and the multiverse, while "quantised" involves, at most, a single countable infinity.

I am not sure why you say this. takes the energy level of an electron interacting with a proton. The levels of energy are quantized: you have indeed a infinite discrete spectrum E0, E1, E2, etc.

I was assuming that space and time are quantised, and so the energy levels aren't really infinitely divisible. (This was following on from previous comments to the effect that if space-time is, say, a directed graph or spin foam or CDT or LQG whatsit, there is only a finite number of units of space and time per a given 4-dimensional volume, and everything embedded in space-time are similarly constrained.)

But QM implies that all complex linear combination represents a possible physical state.

... and I don't think that even if space and time are quantized, why that would change. 

OK, my intuition may have been wrong.

That is why I tend to agree with Deutsch's idea that there is a continuum of "parallel universe", or better "relative states". 1/sqrt(2)(up + down) describes 2^aleph_0 up, and down, with the same measure, particles.

If there is a continuum involved then we have an uncountable infinity. But it seems to me that this would involve space-time also not being quantised, otherwise how can a particle, say, have an infinite number of spin directions? Or maybe it can - I can see that it might be possible, with the quantised space-time itself being able to occupy all possible positions in an underlying continuum, or something like that. Or am I wrong?

QM might change, but the theory is currently strongly related to the usual theory of real (and complex) numbers.

Ah, well, that would give continua, I suppose, given that complex numbers are points in a plane which is a continuum.

Please forgive me if I feel sufficiently uneasy with this concept of "untamed infinities" that I would welcome any evidence which proves it to be true, and hence overcomes my intuitive fear.

(Comp would appear to only assume the existence of one countable infinity, I think? Or have I got that wrong?)

The problem is that we must distinguish what is explicitly assumed in the theory, and the assumption made at the meta-level and needed to interpret the theory. In usual math and physics, we are just careless on this, and it does no lead to trouble as such distinction are not relevant for most application. Bt once we want to be able to get some light on the mind-body problem, the, like in mathematical logic, such distinction becomes important (even crucial at some point).

So, comp does not even assume anything infinite. That's why Judson Webb called it a "finitisme" assumption. We assume only 0, s(0), s(s(0)) ... which are all finite objects. We assume the laws of addition and multiplication, which produces again only finite objects.

OK, but that does seem to assume implicitly that the sequence is unbounded. But I suppose any actual computation never requires any actual infinities.

The set {0, s(0), ...} is just not part of the theory. It is only part of the informal meta-theory, used when we do meta-mathematics, and reason no more about numbers, but about that the theory of numbers itself. In that case, we assume the usual math with all its infinities (like infinite sets, Hilbert space, etc.).

Now, such real numbers will reappear *in* the theory, but only as epistemological constructs by numbers or machine to think about the natural numbers and machines. There we can prove that machine believing in some infinities (and perhaps neutral or not interested in the nature (ontological, epistemological, ...) of their existence) are able to prove more propositions on 0, s(0), ... than machine not believing in them. So such belief in infinity (at least epistemological infinity) is a "selective" advantage for machine trying to learn things even if only about numbers, a bit like the complex Riemann Zeta function can provide information on the discrete prime numbers. Since Gödel, we know that even to learn as much as possible in just arithmetic, we have to climb the hierarchy of the higher infinities, despite they do not necessarily needed to exist at the same ontological level than the machines and the finite things.

This sounds very interesting, though I'm not completely sure I understand. The ontological level at which finite numbers exist is different from the level for countably infinite numbers - I suppose that means comp (or a person theorising comp) assumes finite numbers exist, but is neutral about infinite ones?

[Bruno Marchal]

On 05 Aug 2015, at 10:56, LizR wrote:

On 5 August 2015 at 01:14, Bruno Marchal <mar...@ulb.ac.be> wrote:

On 03 Aug 2015, at 23:56, LizR wrote:

On 3 August 2015 at 21:52, Bruno Marchal <mar...@ulb.ac.be> wrote:

On 02 Aug 2015, at 22:00, LizR wrote:

I don't think you missed the point. I think you said - effectively - yes, the branches are continua. So you have a continuum which can be divided at any arbitrary (irrational number) point into two sub-continua (which makes sense to me - except that it makes the multiverse "classical" perhaps overall, rather than quantised?)

Is not the goal of the "MWI" to make the "universe" classical or as much classical as possible?  I am not sure I see a problem here, on the contrary, I see a solution (except that such a solution entails that we must explain the wave itself from the machine phenomenology).

The only problem, if it is a problem, is that "classical" (in the sense I'm using it here) involves uncountable infinities to be present everywhere in space-time and the multiverse, while "quantised" involves, at most, a single countable infinity.

I am not sure why you say this. takes the energy level of an electron interacting with a proton. The levels of energy are quantized: you have indeed a infinite discrete spectrum E0, E1, E2, etc.

I was assuming that space and time are quantised, and so the energy levels aren't really infinitely divisible. (This was following on from previous comments to the effect that if space-time is, say, a directed graph or spin foam or CDT or LQG whatsit, there is only a finite number of units of space and time per a given 4-dimensional volume, and everything embedded in space-time are similarly constrained.)

But QM implies that all complex linear combination represents a possible physical state.

... and I don't think that even if space and time are quantized, why that would change. 

OK, my intuition may have been wrong.

That is why I tend to agree with Deutsch's idea that there is a continuum of "parallel universe", or better "relative states". 1/sqrt(2)(up + down) describes 2^aleph_0 up, and down, with the same measure, particles.

If there is a continuum involved then we have an uncountable infinity. But it seems to me that this would involve space-time also not being quantised,

I am not sure this would not refute at once String Theory, but even without this, I don't think that the existence of some continuum requires this or that observable to have a continuum spectrum, except for the limit of statistical ftrequency operators, like used by Hartle, Finkelstein, Preskill (but here density might be enough).

Quantum logic bears on those question. It is the type of thing we might get the universal machine opinion in a not too long future.

otherwise how can a particle, say, have an infinite number of spin directions? Or maybe it can - I can see that it might be possible, with the quantised space-time itself being able to occupy all possible positions in an underlying continuum, or something like that. Or am I wrong?

Just look at the mathematical description of one qubit. Of course it is classical QM, not QM+GR (which does not exist). 

QM + Gravity + Spin = my head is spinning :)

Difficult question. 

In logic we have a beautiful result: If a first order theory has an infinite model, it has a model for each cardinality (aleph_0, aleph_1, aleph_2, aleph_3, aleph_4, ...). Loewhenheim-Skolem theorem.

So a set theory like ZF has a model for each cardinality, including the a model with aleph_0 elements, that is enumerable (countable). This is amazing as it means that we can interpret Cantor theorem, which asserts the existence of very large uncountable cardinals in terms of relation between particular ebumerbale object in an enumerable structure. Viewed in the model, there are uncountable structure, but we can put ourselves out of the model, and see the bijection with N, but those bijection does not exist *in* the model, so that the ppor creature living in the models are just unaware of those bijection. 

This relativize the notion of infinities.

I have not that problem, as I use the separable part of mathematics: elementary arithmetic. 

I put the induction axioms already in the epistemology. 

QM might change, but the theory is currently strongly related to the usual theory of real (and complex) numbers.

Ah, well, that would give continua, I suppose, given that complex numbers are points in a plane which is a continuum.

Indeed. The cardinal of C is 2^aleph_0, like the real interval (0, 1). And so there are 2^(2^aleph_0)) complex functions and predicates, but there are important less vast subclass of them.

Please forgive me if I feel sufficiently uneasy with this concept of "untamed infinities" that I would welcome any evidence which proves it to be true, and hence overcomes my intuitive fear.

(Comp would appear to only assume the existence of one countable infinity, I think? Or have I got that wrong?)

The problem is that we must distinguish what is explicitly assumed in the theory, and the assumption made at the meta-level and needed to interpret the theory. In usual math and physics, we are just careless on this, and it does no lead to trouble as such distinction are not relevant for most application. Bt once we want to be able to get some light on the mind-body problem, the, like in mathematical logic, such distinction becomes important (even crucial at some point).

So, comp does not even assume anything infinite. That's why Judson Webb called it a "finitisme" assumption. We assume only 0, s(0), s(s(0)) ... which are all finite objects. We assume the laws of addition and multiplication, which produces again only finite objects.

OK, but that does seem to assume implicitly that the sequence is unbounded.

Yes, and PA can prove it. PA can prove that there is an infinity of prime number in the sense of proving that for all natural number whic is a prime there is a bigger natural numbers which is also a prime.

PA believes in only the finite things 0, s(0), .. but can express and sometimes prove complex relation involving infinities of numbers. RA can already talk about all partial computable functions and prove the existence of all the finite piece of computations, and "their" universal machines, and PA can get much deeper in the generalization. 

But I suppose any actual computation never requires any actual infinities.

I agree. It is the very nature of computations. They are born in the finite realm, and live there, sometimes without ever stopping, which does not require an actual infinity, only a growing finite thing.

But things are different for the soul, sorry, I meant the first person experience, whose fate relies on the infinity of computations supporting its current state.

It is a bit like in QM, people can hope and search for selection principles, but assuming Mechanism, I think this is like the God-of-the-gap strategy to hide a problem. Let us just see where the math drives us. 

The set {0, s(0), ...} is just not part of the theory. It is only part of the informal meta-theory, used when we do meta-mathematics, and reason no more about numbers, but about that the theory of numbers itself. In that case, we assume the usual math with all its infinities (like infinite sets, Hilbert space, etc.).

Now, such real numbers will reappear *in* the theory, but only as epistemological constructs by numbers or machine to think about the natural numbers and machines. There we can prove that machine believing in some infinities (and perhaps neutral or not interested in the nature (ontological, epistemological, ...) of their existence) are able to prove more propositions on 0, s(0), ... than machine not believing in them. So such belief in infinity (at least epistemological infinity) is a "selective" advantage for machine trying to learn things even if only about numbers, a bit like the complex Riemann Zeta function can provide information on the discrete prime numbers. Since Gödel, we know that even to learn as much as possible in just arithmetic, we have to climb the hierarchy of the higher infinities, despite they do not necessarily needed to exist at the same ontological level than the machines and the finite things.

This sounds very interesting, though I'm not completely sure I understand. The ontological level at which finite numbers exist is different from the level for countably infinite numbers - I suppose that means comp (or a person theorising comp) assumes finite numbers exist, but is neutral about infinite ones?

As a mathematician, he can have his own opinion, but most mathematicians have not too much problem up to kappa, and virtually all agrees on what is a constructive ordinals, or on Church's thesis (and all this means nothing, but is just factual).

When you have (by a sort of bad luck perhaps) interest in theology and question about the relation between consciousness and appearances, it helps to be as clear as possible on what is explicitly assumed. It is not always that much important, but it helps to prevent some misunderstanding.

Human invention, or machine's inventions are not arbitrary things. I would say that a real number can be seen as a necessary mindtool for  universal machine, even when just trying to grasp relations between natural number, or trying to understand itself. And those mindtools obeys the laws of the mind (computer science), which in this case can be said not to badly approximated by Cauchy theory of reals, although there are many other one (no equivalent of Church thesis for things like real numbers, or analog machines).

I am neutral on the "existence" of real numbers, but with computationalism, we don't need to assume them, and we can see them as numbers invention, even just to understand themselves. At some point we get something similar to the Loewenheim-Skolem "paradoxical" situation:  arithmetic, seen from inside is unboundably complex, not even nameable. So, as those bizarre big things are (partially) derivable from the interior of something rather "simple" conceptually, adding explicit axioms on them can only have a practical purpose, not a fundamental one. I think.

Like I said, with mechanism Kronecker's crazy sentence become: God created the natural numbers, and told them add and multiply, and all the rest are numbers inventions to understand what happens when you add and multiplies.

We can use the S and K combinators, or even discrete version of string theory, instead of natural numbers. The more neutral are our assumptions, the more genuine will be the information that we derived from it. Normally, if we are machine, the laws of physics are independent of the basic (turing) universal system assumed. 

Bruno

To be sure, in practice, we have up to now, always been able to eliminate the use of infinities in the "interesting" theorem of number theory, so that some conjecture that the whole of interesting math (excluding mathematical logic, category theory, set theory, classical analysis, ...) can be done in PA. But machine theology, or just any deep cosmological inquiry will require them at the meta-level, and some pure combinatorial problem have often been solved with the use of high cardinal long before others succeed in making the proof "elementary" (formalizable) in Peano Arithmetic.

So comp requires only the assumption of the finite things 0, 1, 2, ...

But both the machines and us needs much more, even to just understand how 0, 1, 2, ... behave, in their (additive/multiplicative) relations with each others.

Bruno

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http://iridia.ulb.ac.be/~marchal/