Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 2 Apr 1999 22:07:05 +0300 (EET DST)
On Fri, 2 Apr 1999, Stephen P. King wrote:
> Dear Matti and Friends,
>
> Matti Pitkanen wrote:
> >
> > On Thu, 1 Apr 1999, Ben Goertzel wrote:
> >
> > >
> > > >The amount of entanglement is measured by entanglement entropy
> > > >
> > > >S= Tr(rho*log(rho))= SUM p(m)log(pm)
> > > >
> > > >and is of same form as Shannon entropy characterizing how far subsystem
> > > >is from pure state.
> > >
> > > OK, I understand. Thanks.
> > >
> > > >TGD based QM measurement theory postulates that density matrix rho is the
> > > >universal observable measured in quantum measurement and that subsystem
> > > >goes in quantum jump to eigenstate |m> of rho with probability p(m)
> > > >and thus ends up to pure state without any entanglement.
> > >
> > > This is really not far from the standard von Neumann interpretation
> >
> > Actually the recent picture about quantu jump provides generalization for
> > von Neumann's intuitions about brain as ultimate reducer.
> > In TGD framework cognitive spacetime sheets, which are nearly vacuum and
> > have finite time duration. [Energy and other conserved quantities flow
> > from material spacetime sheets to cognitive sheets when they are formed
> > and back to material spacetime sheets when cognitive spacetime sheets
> > disappear.] The entanglement of cognitive spacetime sheets, 'Mind' with
> > spacetime sheets carrying matter, 'Matter' is reduced in allowed quantum
> > jumps.
>
> Question: What is the mathematical relationship between "cognitive"
> spacetime sheets and "matter carrying" spacetime sheets?
I would rather describe the difference between these spacetime sheets.
Material spacetime sheets have infinite duration with respect to Minkowski
time: this follows from the conservation of energy momentum (Poincare
invariance is exact in classical theory). One can construct cognitive
spacetime sheetes by taking vacuum extremal of finite duration: there
are huge number of them as I explained in previous posting. The gluing of
this vacuum extremal to material spacetime sheets generates interaction
between these spacetime sheets (via wormholes connecting them).
Vacuum spacetime sheet is deformed to slightly nonvacuum spacetime
sheets providing cognitive representation for the material spacetime
sheet. For instance, oscillation frequences of
Kahler fields material spacetime sheet appear as oscillation
frequences of very weak Kahler fields of the cognitive spacetime
sheet.
Besides this the role of cognitive spacetime sheets is to provide the
necessary nondeterminism to make quantum jumps with nondeterminism located
to finite time interval and hence also the localization of conscious
experience to this interval.
The presence of cognitive spacetime sheets forces to generalize the
concept of 3-surface in order to achieve classical determinism in
generalized sense. One can select uniquely onbe of the degenerate
absolute minima X^4_i(X^3) going through X^3 by fixing some minimum
number of spacelike 3-surfaces X^3_k from X^4_i(X^3) with timelike
separations: thus generalized 3-surraces are of form X^3UX^3_1U....X^3_n.
This is nothing but fixing particular multifurcating orbit by
selecting some minimum number of points from the multifurcating orbit.
These sequences of 3-surfaces with timelike separations are like
movies giving simulation of classical history: classical model of
thought. Conscious thoughts are quantum jumps
selecting one thought from a superposition of the classical thoughts.
Entanglement of particle degrees of freedom with the branches of
multifurcations makes this quantum jump possible.
The fact that the time resolution of our visual experience
is .1 seconds, suggests that the time separations between sequences
of 3-surfaces appearing in brain is .1 seconds.
I try to put it quite generally: classical mind/cognition is made
possible by classical nondeterminism and this results from the
transformation of unphysical U1 gauge degeneracy to physical spin glass
type degeneracy, when ordinary Maxwell field is replaced with a Maxwell
field expressible in terms of CP2 coordinates. It is also absolutely
essential that U1 gauge invariance is in question: for more general
nonabelian YM action degeneracy is dramatically reduced: only 4-surfaces
with 1-dimensional CP_2 projection are vacuum extremals now.
>Vaughan Pratt
> posits a Stone-Birkhoff-Pontryagin duality to exits between the
> equivalent in this formalism. In Peter Wegner's work, "interagive
> agents" are connected by "infomorphisms." Which are identical to what
> Pratt uses ("right and left residuation" identified with inner and outer
> products) to identify individual features 'between' the two dual
> aspects. He defines a category Set with matter and category Set^op (or
> antiSet) with mind. The properties that we usually associate with
> spacetimes would be well represented by Chu spaces, including clocks,
> scales and orderings.
What about variational principles and dynamics? Could one build the
analog of classical nondeterminism in this context? What about the analog
of Maxwell field and of induced Maxwell field expressed in terms of
primary dynamical variables? And the analog of submanifold.
Chu manifolds with Abelian connection? Submanifolds of Chu manifolds
with induced connection? Could these concept make sense?
> But this is all formalism; I am interested in what these "words" mean,
> what they "point" to. We assume that we are speaking about objects that
> are not solely represented in our minds and/or text files.
> Could we look carefully at what it is we are talking about
> independently of any particular formalism? It seems obvious to me that
> Nature's "objective" qualities and quantities are invarient with respect
> to changes in language, it is only our "expressiveness" that is
> dependent. Please read Wegner and Pratt. :)
I will look at this. In my approach the origin of Mind as cognitive
representation (not yet consciousness) is classical nondeterminism. As I
already said, in more general context the realization of this idea would
require classical dynamics with nondeterminism.
>
> > >
> > > >> I don't understand this. How do we get from this mathematical measure
> > > >> to "cognitive resources"??
> > > >
> > > >This is a long story told in my homepage
> > > >(http://www.physics.helsinki.fi/~matpitka/cbook.html).
> > > >In TGD quantum states are replaced with quantum *histories* and moments
> > > >of consciousness correspond to quantum jumps between them. Contents of
> > > >conscious experience are assumed to localize into region where
> > > >nondeterminism of quantum jump is localized: consciousness is where the
> > > >free will is.
> > >
> > > I think this ties in with my theory that "consciousness is randomness"
>
> Could you elaborate on this, Ben? I don't understand. I see
> consciousness as the act of adaptively mapping between possible maps
> which represent objects and their interactions, to barrow a mental
> picture from Edelman. Spacetimes are examples of these maps! The idea of
> maps of maps of maps of... might seem like an infinite regress, but this
> is avoided since any given map can only "sample" or span a finite number
> of elements of another. There are mathematical properties involved that
> we need to elaborate on. :)
About objects: my view is that the idea about continually existing objects
and subjects is perhaps too strong extrapolation: only moments of
consciousness exist basically. 'I' as continuous stream of consciousness
is illusion. 'I' exists only at the moment of consciousness just as the
separation to subsystem and complement.
>
> > I would not use the world randomness. After all, in quantum jump selection
> > between eigenstates of subsystem density matrix occurs: final state is not
> > at all random. Secondly, the probabilities for final states are not
> > in general same. And thirdly, strong form of Negentropy Maximization
> > principle selects unique subsystem which can perform quantum jump: the
> > system is the one giving maximum negentropy gain and thus having largest
> > entanglement.
>
> Could it be that there is a "pruning the tree" effect involved? I think
> that the selective process is a competition, like a tournament. I have a
> little formal model involving the "gossiping" of information between
> nodes in an arbitrary graph that appear to model both a selection and a
> entropy. It is my own "pet project". :)
In neurophysiology (which I know very little about)
sensory rivalry provides concrete example of competition. Strong NMP
indeed implies effective competion (conscious competition as at the level
of human society?).
>
> > [Entanglement corresponds to attention 'psychologically' so that the
> > most alert subsystem has moment of consciousness). The most alert
> > subsystem can of course decompose to mutually unentangled subsystems
> > having their own separate conscious experiences].
>
> Would this be similar to saying that entanglement is equivalent to the
> ability of a pattern recognition system to map many different inputs to
> outputs, if so that I think we are dealing with a "subse
> ystem's" ability to generalize and to be creative in its predictions of
> "future" behaviour of its neighboring subsystems. This speaks to the
> "adaptive" aspect of consciousness. But, again, we need to look beyond
> the words to the meanings. :)
>
Quantum entanglement is not all what is needed: I
described above the concept of the cognitive spacetime sheet:
cognitive spacetime sheets make possible the formation of
cognitive representations and prediction of what will probably happen in
the external world.
Entanglement is the physical counterpart of (unconscious) attention
and as such does not make cognition possible: it forms one-one
correlation between some states of
subsystem and those of external world (dead cat-bottle open, cat
alive-bottle closed...). I have considered also quantum pattern
regognition and classical pattern recognition in spirit Haken theory is
part of quantum pattern recognition: the quantum part of model takes care
of making the pattern recognition conscious: here sensory rivalry
emerges: the most catchy 'featureä (the attractor generating largest
entanglement with 'observing' subsystem) is consciously recognized.
Nondeterminism makes possible also entanglement between systems (cognitive
spacetime sheets) with timelike separation. Intentionality and plans
could correspond to time like entanglement.
Attention is not the only psychological aspect of entanglement:
entanglement provides quantum description of binding: the conscious
experiences of two unentangled subsystems performing simultaneous quantum
jump are separate. If these subsystems are entangled their consciousnesses
fuse together to form larger consciousness: there is single 'I' instead of
two separate 'I':s. This would provide general quantum model for binding,
telepathy, group consciousness and transpersonal consciousness.
> > > >c) The concept of local system has as its TGD analog spacetime sheet of
> > > >finite size. The idea of local system is however realized quite
> > > >differently in TGD. Hitoshi introduces clock at every point (I
> > > >apologives if I have not understood correctly!). In TGD approach
> > > >spacetime sheets representing elementary
> > > >particles, nuclei, atoms,...ourselves,.... , galaxies,... are
> > > >local systems realized as spacetime sheets which have contact to larger
> > > >spacetime sheets via extremely tiny wormholes.
>
> It looks like TGD defines its Local Systems as GR systems and the
> relations as quantum, while Hitoshi defines his LSs as quantum and the
> relations as GR!
Might be! General Coordinate Invariance is basic building bricks of TGD.
Einstein's equations however result only as structual equations at long
length scale limit of the theory. I am not able to say anything definite
about the concept of relation: I am not sure about the precise meaning
you give to this concept.
This should be expected! I say that the two "aspects"
> are dual, just as an independent graph is dual to a complete graph! When
> we fix one aspect as our basis with which to make observations and
> predictions, we are restrained to using that aspects objects and
> relations. For example: wormholes to TGD what observations or
> connections are to Local Time theory.
Wormholes seem to provide fundamental classical information transfer
mechanism between different spacetime sheets/local systems at different
levels of hierarchy: they are needed to form cognitive representations.
But this is all classical: consciousness involves quantum jump and this is
something totally irreducible, not modellable by any dynamics.
> The "diameter" of the wormhole, I
> think, speaks to the "size" of the information/causal influence that
> LSs/spacetime sheetes can have on each other. We would then expect a
> density and/or flux quantity for the information/causal influence.
> I feel that the infomorphism consept would help understand this by a
> quantum leap. (Pun intended! ;) )
>
I should mention an interesting connection with blackholes. Wormhole
tube connecting two spacetime sheets has induced metric with Euclidian
signature: the signature must change to Minkowskian at some 3-surface
which is like blackhole horizon. I have proposed the generalization of
Hawking-Bekenstein area law to p-adic context for similar elementary
particle horizons. This generalization might make sense for wormhole
horizons, too!
> > > Perhaps the wormholes constitute a clock in some way?
> > > Just daydreaming ;)
> >
> > I think that any periodic phenomen provides a clock: the basic
> > problem is to find someone to perceive the reading of the clock(:-).
> > In quantum jumps between quantum histories picture the nondeterminism of
> > Kahler action comes at rescue and makes possible conscious experiences
> > with time localized contents.
>
> We need more than a periodicity to constitute a clock, we need a
> periodicity and a scale and a mapping relation between them, e.g. we
> have the moving hands of the analogue clock as the periodicity, the
> face as the scale, and an observer as a mapping relation between.
> Assumed in this thought is that we can somehow compare the clock's state
> at t_0 with its state at t_1. We must be VERY careful here, we have only
> "memory" to aid us in this comparison and Live =/= Memorex! The idea of
> parralel transport used in differential geometry is purely ideal, as
> well explaind by Weyl. I think it would benefit us to look at this issue
> carefully.
For me the big problem is how to explain physically the generation of
conscious experience about the ticking of the clock. Since I am doing
theory of consciousness, I cannot assume fixed external
observer sitting outside the system and looking at the clock. Observer is
created only by quantum jump. What kind of quantum jumps can give
information about finite time interval and thus also about
several 'ticks' of the clock: this is the problem. And I believe that
here the classical nondeterminism is they key to the problem.
> > The oscillations of Josephson junctions formed by wormhole super
> > conductors indeed generate clocks if one believes that EEG is a clock.
> > Amusingly, simplest EEG clock corresponds to sequence of solitons of Sine
> > Gordon, which is mathematically nothing but gravitational pendulum
> > rotating. Also EEG oscillations equivalent with oscillating
> > gravitational pendulum are possible. In latter case EEG is equivalent
> > with the clock in the wall!
> >
> > > > Quantum jumps between
> > > >quantum histories give rise to moments of consciousness creating
> > > >the experiences of separation.
> > >
> > > I think that this general concept is compatible with Hitoshi's theory, but
> > > he realizes the evolution of histories by a different formalism, and he
> > > defines the "jump leading to a moment of consciousness" as the classification of a
> > > collection of quantum particles as a local system (center of mass, category)
> > >
> > Subsystem corresponds to local system clearly.
> >
> > > >There is no unique objective reality/whole
> > > >as in materialistic world view since quantum jump replaces the cosmology
> > > >with a new one: as conscious beings also we are (mini)Gods(;-).
> > >
> > > Agreed
>
> Not only are we finite Gods, but we also exist in the center of our own
> finite universe, we don't experience it directly because we are
> "perfectly adapted" to it. I think of it as the frame of reference that
> describes the complete lack of inertia with respect to the LS at the
> origin. This idea gives us the equivalent to Einstein's equivalence
> principle! I hope to discuss this more latter, and to hope you guys with
> critique me well. :) Remember that there are at least an uncountable
> number of such God/universe pairs and they might "share" objects...
>
> > > >A different aspect to whole/part distinction is related to the
> > > >manysheeted spacetime concept. Different spacetime sheets correspond
> > > >to different branches of physics: at nuclear spacetime sheets nuclear
> > > >physics applies and at atomic spacetime sheets atomic physics is
> > > >satisfactory description. The reason why these physics are practically
> > > >separate is that interactions between different spacetime sheets are
> > > >weak.
> > >
> > > In a discrete view your sheets become lattices, and we have multiple weakly
> > > coupled > lattices
> > >
> >
> > p-Adicity leads in a natural manner to lattice like structure. You can
> > form from real axis 1-dimensional lattice by cutting, say decimal
> > expansion, from n:th decimal. In p-adic context cutting of pinary
> > expansion of pinary number so that O(p^n) part of p-adic number is put to
> > zero is analogous procedure but defines equivalence relation in p-adic
> > context. Hence one can define entire hierarchy of discrete coset spaces
> > R_p/E_n by this equivalence relation (denoted by E_n).
> > This hierarchy of lattices defines extremely rapidly converging
> > approximation procedure for physically interesting primes p (p=2^127-1 for
> > electron!). Various physical fields become in this approximation fields in
> > lattices.
>
> Can we think of these "lattices" in terms of logical inference and/or
> implication structures, like Hasse diagrams or Kripke structures. Please
> read Pratt!
Might be. I try to read Pratt! I see p-adic discretization as elegant
and natural approximation method: not anything deeper. In anycase,
ultrametericity is the key concept: it gives rise to hierarchical tree
and spin glass property has for this reason been suggested to be
essential for working of brain (Parisi).
But I must mention here peculiar coincidence which I found
with help of Tony Smith (I think he had to do a lot of work before I
finally realized this(;-)): primes associated with integer octonions and
quaternions have euclidian length squared R^2=p, p prime. p-Adic length
scales hypothesis states the same thing: p-adic length scales are
proportional to square roots of primes. So: it might be that lattices are
something more deeper (note that octonions and quaterions fit
very naturally with dimensions of imbedding space and spacetime).
> There is evidence from many sources that there is a duality
> between logical implication and material causality, enbodied in the
> phrase: A causes B iff B implies A. When will we look at the logical
> aspect of physical structures?
I have analogous view: LOGOS = COSMOS: objective realities =ideas.
More concretely: many fermion Fock state basis forms structure
isomorphic to Boolean algebra: 2^N states can be regarded as all possible
statements about N basic statements. My interpretation is that fermionic
degrees of freedom corresponds to reflective level of consciousness:
statements about statements/experiences about experiences/thoughts about
thoughts...
>We can assume that spacetimes or LSs are
> a priori and not constructable? The idea of evolution of the universe,
> like the evolution of any system involves selection/mutation, like
> Wheeler's Surprise 20 Questions game, the Universe has no meaning in
> itself so any order or meaningfull information embodied in one of its
> subsets required a finite duration/extention for such to be constructed.
>
> > What is especially nice is that p-adic counterpart of, say, Poincare group
> > respects these lattice structures. I told about how p-adic Poincare
> > group leaves finite p-adic spacetime cube invariant in some earlier
> > posting few weeks ago. One can quite well say that p-adics are Taylor made
> > for lattice approximation.
> >
> > Personally I however believe that geometry is continuous at basic level.
> > The basic reason for this is that infinite-dimensional geometry is highly
> > unique: in TGD case the sole requirements that Riemann connection exists
> > mathematically + some other general requirements fix the entire
> > geometry and also imbedding space itself essentially uniquely. In TGD
> > framework this means unique physics also since physics is just
> > infinite-dimensional spinor geometry. The inability of physicists to
> > find divegence free QFT:s reflects also this high uniquess of
> > infinite-dimensional mathematics.
>
> The Universe, as a totality, is necessarily Infinite, in all possible
> aspects and as such it is pure noice and pure order simulataneously. It
> is both Being and Nothingness.
Agree. Even in Cantorian sense: entire hierarchy of completely physical
infinities is present.
> > > Different sets of links on the same set of nodes, perhaps?
> > >
> > > if you could present a discretized version of the many sheeted theory it
> > > would make it more clear to
> > > everyone and might make correspondences with Hitoshi's and other theories
> > > more clear --
> > > just a pie-in-the-sky suggestion ;)
> > >
> > > >I do not believe in mathematical tricks (although I have tried them
> > > >occasionally(;-)). My basic philosophy has been to construct quantum TGD
> > > >using only the basic classical spinor-geometry generalized to
> > > >infinite-dimensional context.
> > >
> > >
> > > But it's all bits and bytes ultimately, Matti. Infinite dimensional math is
> > > a shortcut for some purposes,
> > > but also obscures things sometimes IMO. I say this as a mathematician who
> > > spent many years studying
> > > functional analysis etc.
> >
> > p-Adic approach fits very naturally with bits and bytes philosophy.
> > For instance, even infinite-dimensional configuration space integral
> > reduces to a discrete sum. My own view is that
> > objective realities=quantum histories are continuous object but that our
> > consciousness is able to work with bits and bytes
> > only. TGD however leads naturally infinite primes and p-adic number fields
> > associated with infinite primes (which are actually very much like reals):
> > also infinite hierarchy of consciousnesses is predicted. Perhaps these
> > Godlike consciousness above us are not limited to play with bits and bytes
> > only(;-).
>
> This is soo interesting to me. :) There has been a lot of speculation
> that primes hold a secret. ;)
>
I really believe, I would almost say 'I know', that they do so(;-).
That is might be the case is suggested by the fact that construction of
infinite primes is formally like repeated quantization of supersymmetric
QFT with states labelled by primes: many particle states of previous level
become one particle states of new level and quantization is performed
again and again and new level of infinity is reached.
The connection between number theory and basic quantum physics is
something wonderful.
MP
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