Stephen P. King (stephenk1@home.com)
Fri, 02 Apr 1999 10:34:21 -0500
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? 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.
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 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. :)
> 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". :)
> [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. :)
> > >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! 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. 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! ;) )
> > 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.
> 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! 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? 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.
> > 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. ;)
> MP
Onward to the Unknown!
Stephen
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