Stephen P. King (stephenk1@home.com)
Tue, 01 Jun 1999 12:53:02 -0400
Dear Matti,
Focussing into some detail..
Matti Pitkanen wrote:
snip
> There is perhaps slight misunderstanding here. Each subsystem is
> characterized by a density matrix and defines a potential measurement.
> Strong NMP selects one these potential measurements. There is competion
> among potential measurements/among potential conscious experiences and
> only the most informative (I should not use this word!) measurement
> occurs. The winner can of course decompose to a set of indepenent
> sub-measurements (separate conscious experiences) and in general does.
Can we discuss NMP more. Could we start with your most concise
definition?
snip
[SPK]
> > :) Are you understanding how Peter uses non-well-founded sets to
> > do this?
>
> I studied the paper. I have impression that getting rid of inductive
> approach (forgive me for my loose use of terms) means getting rid of
> initial value problem and hence of the problem of the initial state. One
> cannot solve time development by starting from initial values at given
> moment.
We replace "absolute" initiality with the idea of finite windows that
are functions of the observer's ability to distinguish properties. See
Peter's discussion of expressiveness...
> As a matter fact, in p-adic context the possibility of pseudoconstants
> (piecewise constant functions have vanishing derivative) leads to just
> this situation when one tries to solve field equations. One must fit the
> solution to go through a set of points most naturally chosen
> by using all points with given pinary cutoff:
> x = SUM x_np^n --> x_N = SUM(n<N) x_np^n .
> One cannot predict future or retrodict past from recent in p-adic
> universe.
Is this why you say that p-adic "universe" is non-deterministic?
> Perhaps p-adic physics could give realization for the 'co-induction'
> philosophy?
I think so! :)
snip
[SPK]
> > I would qualify the "either or" operation as to imply that it is only
> > meaningful in a finite context with non-zero error terms.
[MP]
> I think that this would be choice of philosophy with accuracy of epsilon.
> I am not very enthusiastic about philosophy with accuracy of epsilon(;-).
I understand, but it "works"! We can appeal to Platonic Ideals that are
Absolute truths, but we both understand that these are unknowable from a
local stance and thus we are left with the reality of non-zero epsilons
in our wfft's statements. This is, explicitly, the message of fuzzy
logic and, implicitly, the message of probability theory. I prefer to
have the "uncertainty" out in the open, where I can keep track of it.
snip
[MP]
> By the way, I can explain very shortly how entanglment with vansishing
> entanglement entropy emerges. The entropy is in real context given by
>
> S=-SUM p_n log(p_n)
>
> One must generalize this to p-adic context. The problem is that p-adic
> log(x)
> exists only provided x has unit p-adic norm: x= x_0 +x_1p +.... ,
> x_0 in 1,...,p-1.
> One must generalize the definition of logarith so that it exists always
> and has the required additivity property to garantee the additivity of
> entropy.
>
> The idea comes from p-adic thermodynamics in which Boltzmann weight
> exp(-E/T) is replaced by p^(E/T) where E/T must be integer for the power
> to exist (this condition is satisfied in the applications to the
> calculation of elementary particle masses).
>
> Let x be of form x= p^n (x_0+x_1p+..)
>
> p-Adic counterpart Log_p(x) for p-based logarithm log_p x is defined as
>
> Log_p(x) = Log(p^n(x_0+...)) = log_p(p^n)= n
>
> and is integer valued and satisfies standard formulas for logarithm.
> Log_p(x) depends on the *p-adic norm of x only* and most importantly:
>
> It vanishes if x has unit p-adic norm (n=0 in previous formula).
>
> Entropy reads as
>
> S= -SUM_r p_r *Log_p(p_r) = SUM_r p_r* n(r)
>
> Hence, if probabilities p_r have unit p-adic norm, that is
>
> p_r = m/n such that n and n are not divisible by p,
>
> entropy vanishes identically and one has entanglement without
> entanglement entropy.
I would like to discuss this notion separately! I am reading an
Information Theory book that covers the Real version and your writting
here helps be gain a better intuition of your thinking! :)
[SPK]
> > This would be the case for single observers? The basis of the phase may
> > be different for another observer! This is like having more than one
> > convex vector space (subsets of Universe) in which to embed the "overall
> > phase".
>
> In TGD there is only single huge state space describing the states of
> universe. Strong form of NMP selects subsystem-complement pair as
> decomposiotion of state space to two tensor product factors.
> In p-adic context situation becomes more practical since NMP applies
> separately to subsystems with vanishing entanglement entropy.
That "single huge state space" looks just like Hitoshi's \phi in, for
instance: http://www.kitada.com/time_III.html:
"Our axiom 1 which asserts that the total universe, which will be
denoted \phi, is stationary means in its mathematical formulation that
it is an eigenstate of a total Hamiltonian H. This means that the
universe \phi is an eternal truth, which cannot be explained in terms of
duration or time. In fact, the eigenstate in itself contains no
reference to time, as may be seen from its definition: H\phi=\lambda\phi
for some real number \lambda. The reader might think that this
definition just states that the entire universe \phi is frozen at an
instant which lasts forever without a beginning or end. However, as we
will see, the total universe \phi has
infinite degrees of freedom inside itself, as internal motion of finite
and local systems, and never freezes. Therefore, as an existence itself,
the universe \phi does not change, however, at the same time, it is not
frozen internally. These two seemingly contradictory aspects of the
universe \phi are possible by virtue of the quantum mechanical nature of
the definition of eigenstates."
snip
[SPK]
> > It is this "ordering problem" that is what concerns me. The way that
> > events are assumed to be ordered in spacetime "trajectories" in
> > classical GR is a very wrong notion. Your way of talking about quantum
> > histories, seems much better, but, sadly, I need to understand the
> > details better...
> >
> What you say about GR is certainly true. In TGD situation is different.
> The very definition of configuration space geometry forces to associate
> to given 3-surface X^3 spacetime surface X^4(X^3): otherwise one could
> not realize four-dimensional coordinate invariance: Diff^4 must
> have something to act on! Thus classical theory becomes in a well
> defined sense an exact part of quantum theory. Configuration space spinor
> fields can be regarded as quantum superpositions spacetime surfaces
> and multiverse picture is realized in this sense. This picture is
> quite different from that provided by path integral approach.
The role of Diff^4 is localized in LS theory such that it is not
assumed to cover \phi at the totality level! We only have Diff^4 over
the set of simultaneous observables (time-like hypersuface?) of
individual observers. Thus it is not a single Diff^4 group for all
observers, but one Diff^4 for each observer, and thus a uniquie
space-time for each observer. Of course, when we generalize this notion,
as you have done by using p-adics, we get the "many sheeted" spacetimes
and can have overlapping and underlapping of the sheets...
> The power of General Coordinate Invariance is remarkable: it has
> practically fixed the general form of the theory totally. Configuration
> space geometry; quantum jump between quantum histories concept fixing the
> general structure of TGD inspired theory of consciousness; and finally the
> mapping of real spacetime surfaces to their p-adic counterparts
> and p-adicization of entire TGD, which I told in some earlier posting.
Yes, GCI is powerfull but it is far to restricive in the usual form. It
is necessary to say "the laws of physics look the same to all observers"
but this assumes that "all observers" form a convex set (complete graph)
and that there is only one such set. I claim that there is not, there an
an unenumerable number of such sets that are "almost convex" in that
they have a fuzzy boundary instead of a crisp binary boundary. This
notion is part of the "window" notion that represents the sampling of
the stream in my discussion of Peter's work.
In sort I say, "all observers that have similar enough perceptions of a
*set of physics* can communicate with non-zero *expressiveness*". Thus
this implies that observers that have different physics can not
communicate anything to each other other than noise! But, given
sufficient "interactive computational" time, ways to decode messages
from the noise become possible.
snip
[SPK]
> > I would not assume a unique metric (inner product norm) for the
> > integration, such assume that the configuration spaces of observations
> > all have the same size "parts", this is wrong! My comments about using
> > Weyl's geometry speaks to this. We solve the indefinite spectra problem
> > by showing that observers can only sample discrete partitions of the
> > continuous spectra and thus, just as Weyl said, the smearing is
> > unobservable just like "pure" states!
> >
> In TGD approach the 'physics as geometry' philosophy
> fixes the inner product to very high degree. p-Adicization
> seems however necessary. Real valued S-matrix elements simply do not
> exist mathematically. Integration in infinite-dimensional context
> is extremely tricky. Consider only volume of infinite-dimensional sphere:
> it is typically zero or infinite.
But is this "fixing" necessarily unique for all possible observers and
strictly not definable relative to finite subsets of communicating
observers? If it is, then there is a serious problem with my notion. :(
snip
[SPK]
> > Making "'our minds' as outsider" is modeling our minds, it does not
> > give a complete knowledge of the subjective stance, but we can use it as
> > information from which to infer sets of observables and the
> > superselection rules that order them. I call this "contextual
> > definiteness". I can not say with probability 1 what you see, but I can
> > calculate what you might see that I can also see. Does this make sense?
> > It is like figuring out if a distant observer that I can talk to on a
> > radio can observe something similar to what I do. I can not "see" what
> > he sees, but I can say with high certainty (low error) that we observer
> > "the same thing".
[MP]
> Your argument certainly makes sense. What I am however troubled is the
> introduction of observers as fundamental (the concept is of course very
> practical approximation). Introduction of observers at fundamental
> level leads to consistency conditions on the observations if they
> correspond to quantum jumps.
Neither the "observer" nor the "jumps" are "fundamental", as I see it;
they are complementary. Having one without the other renders them
meaningless! Existence is the grundlagen.
[MP]
> Introducing only observations one can avoid this problem.
> The point is that *You and I only rarely do we both participate same
> moment of consciousness*. If we participate the same moment of
> consciousness and have separate experiences (are unentangled) then what
> we see, are not views about the same landscape: no consistency problem.
I am thinking about how it is that we can "participate [in] the same
moment of consciousness"! :) I think of this as a correlation between
the observation (= "quantum jump"). I am identifying correlations with
co-inductions (and/or bisimulations?) between stream, which are "quantum
histories" to me, just in different clothing. :)
> When we are entangled we see the same
> thing but our conscious experiences fuse together so that there is only
> single experiencer 'we'! Consistency problem disapppears in all these
> three cases!
This situation describes what happens in the infinite limit only! This
is the level of the Grundlagen and there is no duality of subject and
object here, thus you are correct. :)
[SPK}
> > This is very different from the traditional notion that "we all
> > experience one and the same finite universe". We do not, we just happen
> > to be have subsets of our sets of observables that are very similar and
> > thus we have an "illusion" of a common finite universe!
[MP]
> I agree with what you say about 'we all experience....':
> we experience different worlds. My point, which I am
> perhaps boringly repeating, is that one
> should go even further and ask whether it makes sense to speak about
> observers in the fundamental theory: this Cartesian assumption is perhaps
> too strong assumption leading to insurmountable difficulties if one
> wants to have consistency with quantum jump concept.
We need something to use as a starting point in our model of QGR; thus,
yes, it is am "assumption", but we make it clear what we mean by
"observer": an observer is defined as a poset (partial ordering) of
quantum jumps over an ensemble of quantum histories . This wording is
insufficient for the final version, of course; I am just trying to hone
in on it. :) We need to be able to model concurrency!
snip
[MP]
> > > In a more general framework there is still one question making
> > > sense if state function collapse is identified as moment of
> > > consciousness.
> > > What principle determines which subsystem suffers wave packet
> > > collapse.
> > > Strong NMP answers this question in TGD approach.
[SPK]
> > I think it is a local optimization! Thus TDG seems to be in the
> > right track! :)
[MP]
> Strong NMP as such is formulated for entire universe. It reduces to local
> optimization in p-adic context: this is very important result. One can
> apply it to brain/neuron, etc. forgetting the rest of the universe.
> In real context this does not occur.
Thus we agree on the necessity of p-adics! :)
> The reason for localization in p-adic context is following.
> If universe decomposes to mutually unentangled sub-Universes (which
> can have even finite size) then also general subsystem participating in
> quantum jump has similar decomposition. The real counterpart of
> entanglement entropy must (I leave it as an exercise why!) be defined as
> sum for the real counterparts of p-adic entropies for unentangled parts
> of subsystem. Hence maximization of negentropy gain effectively reduces
> to that occurring separately in each unentangled sub-universe and one
> obtains the desired localization.
Hitoshi is proposing that Local Systems are "mutually unentangled
sub-Universes" composed of a finite number of parts which he calls
"quantum particles". They become LSs themselves when we shift to a frame
of observation that "focusses" on them. I believe that the hierarchical
nesting that this manifests is a clear example of p-adic orderings! Thus
my interest in your thinking. :)
We do not have a clear definition of entropy in Hitoshi's papers, in
my opinion, so I am very interested in your reasoning here. :) I see
this "maximization of negentropy gain" as an example of Frieden's "EPI"!
It is local to individual LSs (as "unentangled sub-universe[s]") and
thus your conclusion follows. :)
> For details see 'Strong form of NMP' at
> http://www.physics.helsinki.fi/~matpitka/cbook.html
I will read this more.
snip
[MP]
> > > If you assume that there are infinite number of
> > > observables giving rise to state function collapses, you could have
> > > consistency problems: if the state function collapses associated with
> > > different observers occur simultaneously, they might not be consistent
> > > with each other.
[SPK]
> > No, not infinite; _finite_. I think of the observed properties as
> > selected in a way that is very similar to how the traits of organisms
> > are selected for in Darwinian evolution! Local optima given finite
> > environments, the key is to recognize that these environments "overlap"
> > and "underlap" thus the non-Hausdorf property! The difficulty of
> > distributivity is a key property not a fatal flaw! The NP-completeness
> > of the decision problem of choosing a shortest route from a traveling
> > salesman well illustrates this! The reason also is intimately connected
> > with uncertainty, as explained by Hitoshi's paper. It takes an Eternity
> > to verify that the subsets of the Universe are in minimum "distance"
> > relations with respect to each other. The n-body problem is another
> > consequence of this!
>
> There is consistency problem also in finite case. I actually discussed
> this point already above. I must admit that I do not understand the
> non-Hausdorf property. This is probably something trivial. Please explain
> from basic definitions: I have forgotten what Hausdorff means!(here
> I should have a symbol of American sign language experssing deep shame!)
Here is an example that is very similar to what I am thinking:
http://ftp.esi.ac.at/Abstracts/abs356.html :)
The Hausdorff (sorry about my previous typo) property is: "A topological
space is 'Hausdorff' if any two disjoint points are contained in
disjoint neighborhoods." See [time 64] I am thinking that the pasting of
"pieces of plane glued to larger pieces of plane glued to larger" has a
deviation from the strict ultrametric as in:
http://www.unipissing.ca/topology/c/a/a/f/73.htm (this is very close to
Pratt's formalism!) and
http://www-i2.informatik.rwth-aachen.de/Forschung/MCS/Mailing_List_archive/con_hyperarchive_1995/0076.html
as contrasted with:
http://labhjk.life.nthu.edu.tw/~mjhsieh/document/bcd/Curric/MathAn/node16.html
or more generally http://at.yorku.ca/t/a/i/c/26.htm This one references
you! http://www.elec.uq.edu.au/~annis/doubt_misc/janet3.html
This whole paper is awesome, I just found it searching "ultrametric":
http://labhjk.life.nthu.edu.tw/~mjhsieh/document/bcd/Curric/MathAn/mathan.html
;)
snip
[MP]
> > > By the way, one could consider the formulation of the idea about
> > > representation independence also in TGD context.
> > > Quantum entanglement is characterized by the
> > > properties of density matrix (the eigenvalues of density matrix).
> > > The hypothesis would be that entanglement coefficients
> > > and the final state of quantum jump would completely determine the
> > > contents of conscious experience. This would mean that all kinds of
> > > quantum subsystems would give rise to same conscious experience
> > > since everything would reduce to the level of abstract Hilbert space.
[SPK]
> > Not if we have more than one convex Hilbert space!
I am talking about the finitesimal case here... ;)
> > This idea is
> > analogous to how there are many "consistent" set theories and
> > geometries! I am identifying a convex Hilbert space of state vectors
> > with a Complete Atomic Boolean Algebra, read Pratt's papers, he explains
> > this well! This notion is more general that Hitoshi's discussion of
> > Hilbert space and needs to be worked out further.
> > Suffice it to say that different "kinds of quantum systems" would have
> > different (but not completely disjoint!) conscious experiences!
[MP]
> There would be also loss of contact with geometry. Conscious experiences
> should present information about configuration space geometry and
> thus about imbedding space geometry and spacetime geometry. Otherwise
> it seems impossible that conscious experience leading to discovery of
> TGD could ever occur(;-). If contents of experience are
> determined by mere entanglement matrix, conscious experiences give no
> information about the fact that configuration space spinor fields
> provide the concrete realization of quantum states.
> This world would be perhaps the world experienced by pure linear logicians
> building their formal systems and not enough(;-).
I think that this is why I am thinking that we have to consider at
least two observers when modeling conscious experience as you say above.
We have non-linear situations since this is equivalent to n > 2 body
problem!
Onward to the Unknown,
Stephen
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