Re: Questions about MWI and mathematical formalism
This is a slight expansion on my previous post under the "simulation" thread. 1) The first step is to examine the act of definition. In this case the definition of a "Nothing". Any definition process simultaneously defines two entities. The definition is a boundary between an entity of interest and the leftover building blocks. In the special case of a "Nothing" the left over is an "Everything". Thus the two are dependent partners. Since the "Everything" contains all information the definition pair must itself specify all information and can be represented by a normal real. 2) A "Nothing" has an interesting logical problem: It can not answer any meaningful question about itself. Assuming there is a relevant meaningful question a "Nothing" would be incomplete. An inescapable meaningful question is its own stability. This is not only meaningful it is impossible to avoid answering. 3) To attempt to answer this question a "Nothing" randomly and spontaneously "decays" towards an "Everything" to resolve its incompleteness. But this is not sustainable since an "Everything" is not independent of a "Nothing". Therefore a "Nothing" rebounds from the decay. 4) Thus the definition or boundary between the "Nothing" and "Everything" pair is randomly dynamic equivalent to a random sequence of normal reals. 5) A universal computer is a good way to model a selector of a random sequence of normal reals. 6) Notice that the "Everything" also has a logical problem. Looking at the same meaningful question of its own stability it contains all possible answers. Just one would constitute a selection i.e. net internal information which is not an aspect of the "Everything". Thus the "Everything" is inconsistent. 7) Thus the entire system while being - apparently - the only game in town is also both incomplete and inconsistent. Hal
Questions about MWI and mathematical formalism
I'm a layperson fascinated with quantum mechanics and the MWI, and have
reached a point where to obtain a better understanding of the
qualitative descriptions (universes "splitting", "measure of a
universe", etc.) I must learn the mathematical formalism. It appears
that the popular descriptions of MWI use very loose terminology, and I
suspect much has been lost in translation.
Digging through online sources such as MathWorld, Wikipedia, and
CiteSeer, as well as reviving painful memories of matrix algebra from
university (CS), I think I've learned enough to be dangerous. Below is
a set of (possibly incorrect) statements and questions I have.
-=-=-=-
Let |phi> represent the quantum mechanical state of a system S as a
vector in Hilbert space. The state is determined by the angle of the
vector, not it's length. So any state multiplied by a constant is the
same physical state of the system. (Correct? Is this by decree or does
it fall out of something more fundamental?)
Let A represent a Hermitian operator corresponding to some observable of
the system S
Let {l} represent the set of eigenvalues for operator A such that
A|phi> = l|phi>
And finally:
{|An>} is the set of eigenvectors for operator A corresponding to {l}
This set of eigenvectors (if I understand correctly) form an orthonormal
basis for the possible states of S, such that if S is in a state phi
which is not an eigenvector of observable A, it may be represented as a
linear combination of such eigenvectors:
(1) |phi> = c1|A1> + c2|A2> + ... + cn|An>
In the case where |phi> is indeed an eigenvector of A, then one of the
constants cn is 1 while the remainder are 0.
So far so good (I hope.) Here are my questions:
A) What is the physical meaning of equation (1) above? Is this what is
meant when a system is described as being in a "superposition" of states
that are measured by A? Is "superposition" the accepted term in the MWI
or is there another?
B) In the Copenhagen Interpretation (CI), the collapse postulate states
that (somehow) as a result of a measurement, |phi> actually changes to
one of {|An>} with a probability related to {cn}, though I'm not sure of
the particulars. How do you describe the probability (within the CI) of
obtaining measurement l from state |phi> based on equation (1) ? This
is the Born rule, I think, but I haven't quite grasped the math.
C) In MWI, there is no collapse postulate. When a measurement occurs,
the quantum mechanical state of the measuring device (and ultimately the
observer) becomes a "superposition" as well, with each observer becoming
a linear combination of states corresponding the effect the measured
outcome has on the observer. Is this the technical meaning of "splitting
universes"?
D) Even in the case where the spectrum of A is discrete, the set of
constants {cn} in (1) can take on continuous values. When an observer
"splits" as a result of measuring A on S, how many "splits" occur? Is
there an infinity of them, each corresponding to a different set of
constants {cn}? Or, is there a split only into the number of
eigenvectors of A, since cn|An> represents the same physical state
regardless of the numerical value of cn?
E) What is the "measure" associated with each of the "observer states"
resulting from D? How is this "mathematically" related to the
probability values from B)?
F) What happens when you use a different observable B? How do the
answers to C), D), and E) change when observables A and B have different
sets of eigenvectors? Is this the "preferred basis" problem?
Struggling but determined to figure this out,
-Johnathan
The Map of the territory
Hi, I have put on my web page a link toward a "pdf" slide giving the main "map of the territory". http://iridia.ulb.ac.be/~marchal/MapOfTerritory.pdf It is a graph whose vertex give the main propositional logics used in the "derivation of physics from machine introspection", and the edges are just inclusion relations. For exemple CL = propositional classical logic, as seen as the set of all theorems of some (complete) presentation of it. Or as the set of all classical tautologies. IL = Intuitionistic logic; and QL = quantum logic. (Precise definitions will be given) Both IL and QL are sublogic of CL. (A v (not A)) is not a theorem of IL, for example, and A & (B v C) and (A & B) v (A & C) are not equivalent in QL. Above CL we find the main modal logic: the "well known" G and G*, and S4Grz, etc. That little diagram should help. Print and sleep with it ;-) More explanation asap. Bruno http://iridia.ulb.ac.be/~marchal/
Re: Private Minds in 3rd Person views?
Hi Stephen, At 20:00 30/04/04 -0400, Stephen Paul King wrote: Dear Bruno, I missed something that you wrote earlier! Do you truly think that the solution to the mind/body problem involves "explaining how a private mind can be attached to anything third-person describable"? I don't see how this makes any kind of sense! The mere fact that you cannot have a 1st person experience of "what it is like to be Stephen Paul King" unless you are, actually, Stephen Paul King tells me that it is impossible for a 3rd person description to exist. Ah Ah Ah ... OK. I agree, but it is not among the axioms, it is among the theorems. What is not yet clear to me is what you accept without proving and what you try to prove. What I see is that we have agreements and/or coincedances in the 1st person views of many SASs. Nobody knows. (Well we should say no-soul knows that). These give rise to the idea of 3rd person views, but such do not actually exist. We can postulate some 3rd person axioms. As you know the enterprise I advocate relies on accepting the notion of number, and accepting usual partial axiomatisation as third person correct. I do hope you accept that the proposition "17 is prime" is either true or false. It makes it 3-person well definite. At best we can associate an inferability of a private mind, ala Turing Test, or someother kind of justification of the belief in private minds, to some aspect of our individual experience. For example, I assume that you (and your private mind) are not merely a computational simulation generated by the same computation that generates my own experienciable actuality Too much ambiguity here. The expression "same computation" could have more than 1 interpretation. because if I did so I should be able to induce a transformation of my 1st person experience directly and smothly into yours. After all, the simulations would all be within the same repetuoir of possible simulations. This is classic problem of solipsism! Don't you agree? I am not sure. Arithmetic is the repertoir of all possible simulations, and this does not lead necessarily to solipsism. Best Regards, Bruno http://iridia.ulb.ac.be/~marchal/
Re: Are we simulated by some massive computer?
At 16:14 30/04/04 -0400, John M wrote: How about: "self"? is it a good enoug "1st person" "soul"? Here you put your finger on something quite important but rather hard to explain without saying more on the incompleteness phenomenon. We will certainly come back on this more than once. The idea is that there are many notion of selves for the sound machine. (I recall I am always talking about a machine which proves theorem of arithmetic, and by definition a machine is sound if she proves true theorems). Then the third person self is defined by a correct functional description of the machine at the right level (which exist for us by comp, and which can be made explicit for simple machine like Peano-Arithmetic, ...). It is a third person self, a little like when you say "I have a mouth" ... I remember now that I did use the term soul for this notion of *third* person (or self) in the provocative view of comp: "comp means you can save your soul on a disket". Stathis made me think that the word soul is perhaps better used for the first person self. In a nutshell, the third person self is the one which will be described by the Godelian beweisbar provability predicate Bew(x), (and then by the modal logical systems G and G* (for those who remember, I will re-explain later)). The first person self will be defined by applying the Theaetetus trick on the third person self, that is on bew. So the first person will be defined by a new predicate saying "Bew(x) and True(x)". But the predicate Truth(x), by Tarski theorem, cannot be defined in the language of the machine. Still, by using G (and G*) we can defined such a box (but detail will be given at time). Now the machine is sound, which means the machine proves only true proposition of arithmetic. So, obviously the first and third person are equivalent. But the incompleteness theorems will entail that neither the 3-machine self, nor 1-machine self can *prove* that equivalence. Such subtle nuances will be made cristal transparent by the explicit use of G and G*. I recall that G is a formal theory complete for the provable discours,by the machine, on the propositional provability logic of itself (the machine itself). G* is a formal theory complete for the true discours,by the machine, on the propositional provability logic of itself (the machine itself). That is: G* contains the true but unprovable sentences on and by the machine. What appears here, with the box [0] for the 3-person and [1] for the 1-person: G* proves [0] = [1], but G does not prove it. Well I guess this was difficult for those who doesn't know enough logic and my intend was to explain more before. So don't worry if you don't understand. Remember that the popular book by Smullyan "Forever Undecided" has been reedited, and is a not too bad introduction to the modal logic G. It could help. Old (in this list) definition of G and G* can be found here http://www.escribe.com/science/theory/m1417.html and in the neighborhood. Bruno John M - Original Message - From: "Bruno Marchal" <[EMAIL PROTECTED]> To: "Stathis Papaioannou" <[EMAIL PROTECTED]>; <[EMAIL PROTECTED]> Sent: Friday, April 30, 2004 9:37 AM Subject: Re: Are we simulated by some massive computer? > Ok Stathis, thanks for the precision. > Anyway you give me the temptation to identify the soul by the first person. > We will be able to prove (with the comp hyp) that not only the soul exists > but (I forget to say) also that from the *correct* soul point of view, the soul > is NOT a machine. > But perhaps the word "soul" is to charged with emotion, and perhaps > we should stick on the expression "first person". > 'course, it is just a matter of vocabulary. (But then humans are able > to fight themselves during centuries for matter of vocabulary ... :( > > Bruno > > > > > At 22:23 30/04/04 +1000, Stathis Papaioannou wrote: > > >On 29 April 2004 Bruno Marchal wrote: > > > >>At 23:16 28/04/04 +1000, Stathis Papaioannou wrote: > >> > >> > >>>There is a single idea underlying much of the confusion in discussions > >>>of personal identity: the belief in a soul. > >> > >>Indeed. > >> > >> > >>>I use this term for a quality or substance which resides in a person > >>>throughout his life and is somehow responsible for his identity, and > >>>which (here is the problem) is not captured by a complete description of > >>>the person's physical and psychological state. Often, it is a hidden > >>>assumption. > >> > >> > >>That's a nice definition of the soul, quite similar to the provable > >>properties > >>of the "first person", once we will define it precisely (in the Thaetetus > >>way). And comp will > >>entails, *as a theorem*, the existence of the soul, then! > > > >Actually, I didn't mean to use "soul" as a synonym for consciousness or > >subjective experience, which is why I said it was something not captured > >by a complete description of a person's physical *or psychological* state. > >Subjective experience differs from other empirical data in
