Matti Pitkanen (email@example.com)
Wed, 2 Jun 1999 08:33:31 +0300 (EET DST)
On Tue, 1 Jun 1999, Stephen P. King wrote:
> Dear Matti,
> Focussing into some detail..
> Matti Pitkanen wrote:
> > 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
> > > :) 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?
Purely p-adic universe obeying some field equations. Yes.
The situation TGD framework situation seems to be the
a) Real spacetime and also other geometric objects (imbedding
space, configuration space, configuration space spinor fields,...) are
mapped to their p-adic counterparts.
b) The mapping is unique from General Coordinate Invariance and is the
phase preserving canonical identification which I told about in earlier
posting (recall Pythagorean phases, parallel with quantum measurement
theory, etc...) but the direct image is not continuous p-adic surface.
c) One must replace the direct image with its minimal pinary cutoff
and continue this to p-adically smooth surface satisfying p-adic
counterparts of field equations. This completion is possible only because
of p-adic nondeterminism.
Consider c) more precisely:
a) The mapping of imbedding space to its p-adic counterpart is fundamental
and defined by phase preserving canonical identification. It induces the
map of spacetime to its p-adic counterpart somehow.
b) The p-adic counterparts of field equations determining spacetime
surface must be satisfied satisfied.
c) Canonical image must coincide with p-adic spacetime surface in
*maximal resolution* allowed by p-adic field equations.
c) means that *minimal pinary cutoff of the canonical image of the real
spacetime surface* (the preferred imbedding space coordinates h^k of
spacetime points are replaced with their pinary cutoffs) consisting of
discrete set of points coincides with the the pinary cutoff of p-adic
spacetime surface satisfying the p-adic field equations.
**This is made possible by p-adic nondeterminism!**
Phase preserving canonical identification map plus continuation of
minimal pinary cutoff of p-adic image to smooth p-adic object is general
recipe in the construction of p-adic counterparts of all real objects
(configuration spinor field basis, kernel of time development operator
satisfying Schrodinger type equation,..).
**Important: one must distinguish p-adic nondeterminism from the
classical nondeterminism of Kaehler action present also in real context:
also this feature forces 'coinductive' philosophy. One cannot predict
or retrodict everything from initial values for some snapshot.**
Thus it would seem that classical nondeterminism of Kaehler action
absolute crucial for cognition and possibility to have conscious
experience with contents localized in time, forces also
> > Perhaps p-adic physics could give realization for the 'co-induction'
> > philosophy?
> I think so! :)
> > > I would qualify the "either or" operation as to imply that it is only
> > > meaningful in a finite context with non-zero error terms.
> > 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.
This is similar to dissipative world of standard physics, which we
discussed in qmind recently. Basic physics
(definining quantum histories) is reversible but observed physics is
irreversible. The reason for dissipation is that quantum jumps between
histories change the history all the time. Dissipative effective history
replaces the sequence of quantum jumps between quantum histories
with *single dissipative history*. This definitely wrong
and mathematically ugly picture provides however practical effective
By the way, macroscopic irreversibelity can be regarded
as a direct signature for quantum jumps between quantum histories
and is visible to everyone since the only (as I believe)
to understand dissipation and reversibelity simultaneously at fundamental
level is based on this concept.
Growing old must be one of the basic irreversible processes. Even things
like chairs and desks grow old and must be performing quantum jumps,
perhaps also macroscopic ones now and then, and hence must have moments of
consciousness. Hydrodynamic flow dissipates rapidly and must involve
moments of consciousness (what it is to be a water flow
growing old and losing kinetic energy gradually but unavoidably?(;-))
> > 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! :)
> > > 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.
Yes. This is the standard physics with single objective reality.
In TGD \phi changes from quantum jump to quantum jump. Otherwise
pictures are identical.
> 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.
In TGD \phi would be completely determined by its values of ligthcone
boundary in strictly causal theory and one could say that everything
reduces to lightcone boundary: no psychological time. Classical
nondeterminism of Kaehler action changes the situation. In order to
specify nondeterminism one must introduce moments of multifurcations for
spacetime surfaces, and data like this and these parameters correspond
closely to time values around which contents of cs experience is
> 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."
I think that I understand this. System can have vanishing momentum even
when component systems have nonvanishing momenta.
If phi does not change it seems that universe has zero energy.
If one applies naively the idea that energy is additive one would
conclude that negative energies must be present. Gravitational binding
By the way, also in TGD zero energies might be possible if spacetime
surface is allowed to have time orientation opposite to that of
imbedding space. I do not really know whether to allow this or not.
The energy densities of various spacetime sheets could sum up to zero!
Everything would be created from emptiness!
> > > 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...
So you don't assume that different LS:s integrate to single spacetime
surface. In fact, manysheeted spacetime makes sense also in real
context: point is that different spacetime sheets allow effective p-adic
topology which is very useful in the construction of QFT limit:
in excellent approximation one can construct QFT in single region
of this kind forgetting what happens on boundaries.
> > 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.
Yes! But this might be its power! Only month ago I was ready to consider
the possibility of giving up GCI since it seemed that it simply does not
allow p-adicization of quantum TGD.
> 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.
This statement goes outside my mathematical intuitions (convex set,..).
Again this dangerous notion of 'observers': what about replacing
it with 'observations'?
> 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.
'Physics' is cognitive representation for what happens in external world?
> > > 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. :(
Inb TGD the inner product belongs to the Platonic Realm and is
observer/observation dependent. The inner
product is for the states of entire universe, for phi:s as you call them.
Jumps between quantum histories phi_1--> phi_2 pjio_2-->phi_3 ..!
The inner product for configuration space spinor fields reduces to inner
product ofm configuration space spinors integrated over entire
configuration space of 3-surfaces. Inner product of spinors is just Fock
space inner product for fermions (oscillator operators create the state).
In your case you have single phi and inner product must be inner product
for some subsystem (LS?). Hence situation is different from that
> > > 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".
> > 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.
I think that I disagree. The use of single phi means materialistic
(sorry!(;-)) world view with single objective reality. Materialism leads
to problems with inner product besides all these social problems(:-). In
TGD I allow all possible phis, quantum histories. TGD is nonmaterialistic
theory in strong sense.
> > 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. :)
>From TGD view point I see co-induction and bisimulation is higher
level concepts related to cognitive representations, which correspond in
TGD to cognitive spacetime sheets. Quantum jump is lower level concept.
Participation in same quantum jump with separate conscious experiences
mean experiences about different sub-Universes/tensor product factors of
overall state space. Objects of perception are different.
> > 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. :)
No limit is needed. Entanglement as binding solves the binding problem
of neurophysiology (how different components of conscious exoerience
fuse to form single experience and what this corresponds physically).
This is basic hypothesis of TGD inspired theory of consciousness.
When we are entangled, binding occurs and experience is 'we', moment
of successful communication(;-).
> > > 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!
> > 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!
> > > > 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.
> > > I think it is a local optimization! Thus TDG seems to be in the
> > > right track! :)
> > 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! :)
Yes. p-Adics are also necessary for evolution.
> > 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. :)
p-Adic ultrametricy leads naturally to hierarchical structures. Trees
in which each node has p branches. Second hierarchical structure are
p-adic spacetime sheets with various values of p glued on each other.
> 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. :)
There is strong similarity. But strong NMP is not like ordinary
variational principles. It does not imply deterministic time development
since each quantum jump/quantum measurement has several possible
outcomes. It only selects quantum jump. One cannot predict the future
using this variational principle since one ends up to a garden
of branching paths.
The interpretation of absolute minimization of Kaehler action as
maximation of classical nondeterminism<--> cognitive resources
is much nearer to Frieden's ideas. Note however that also now
nondetermninsm is involved!
There are strong reasons to believe that the most
interesting quantum jumps select between branches of classical
multifurcations: particle states being entangled with the branches
of multifurcation. Classical and quantum nondeterminism would be
very closely related!
Finally, principle what I call 'ontogeny repeats phylogeny'
states that nondeterminism time development at spacetime level
mimicks time development by quantum jumps at the level
of configuration space. This could perhaps mean that
p-adic nondetermism mimicks/simulates quantum nondetermism.
There would be kind of holy trinity of all three nondeterminisms.
> > For details see 'Strong form of NMP' at
> > http://www.physics.helsinki.fi/~matpitka/cbook.html
> I will read this more.
> > > > 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.
> > > 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."
OK, of course. There is long time from my school days and I am only a
poor physicist! Already p-adic topology is rough but not rough enough
to be Hausdorff.
> 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
> as contrasted with:
> 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":
> > > > 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.
> > > 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!
> > 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
Or no observers at all(;-)!
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