**Stephen P. King** (*stephenk1@home.com*)

*Sat, 03 Apr 1999 21:19:12 -0500*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 128] Re: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality"**Previous message:**Stephen P. King: "[time 126] Re: [time 109] Spacetime& consciousness"**In reply to:**Ben Goertzel: "[time 120] Re: [time 109] RE: [time 104] Re: [time 102] Re: [time 99] Spacetime& consciousness"**Next in thread:**Hitoshi Kitada: "[time 128] Re: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality"

Dear Hitoshi,

I apologize for the length of this... :) BTW, I think that Prugovecki's

formalism already has Chu_2 spaces built in, he just does not understand

the implications! More on this later... ;)

Hitoshi Kitada wrote:

*>
*

*> Dear Stephen,
*

*>
*

*> -----Original Message-----
*

*> From: Stephen P. King <stephenk1@home.com>
*

*> To: Time List <time@kitada.com>
*

*> Date: Sunday, April 04, 1999 12:33 AM
*

*> Subject: [time 115] On Pratt's Duality [time 81] Entropy, wholeness, dialogue,
*

*> algebras
*

*>
*

*> >Hi All,
*

*> >
*

*> > A brief note. The duality that Pratt speaks to can is exeplified by the
*

*> >duality between the set of vectors in a tangent space and its cotangent
*

*> >space of linear functionals. Each of these can be seen as a 4-space. It
*

*> >could be that octonions are involved, but I do not know the details. But
*

*> >I do notice that this looks like Matti's thoughts:
*

*> >
*

*> >> "One one results on my side was the realization that 8-dimensional
*

*> >> imbedding space H of TGD allows octonionic structures as tangent space
*

*> >> structure: one could say that H is locally a number field. Future
*

*> >> lightcone of Minkowski space allows very natural quaternion structure.
*

*> >> This was a surprise for me since I had used to think quaternions are
*

*> >> as inherently Euclidian space: one can however define also Minkowksi
*

*> >> metric: inner product is defined as the real part of xy instead of
*

*> >> real part of x^*y."
*

*> >
*

*> >and
*

*> >
*

*> >> "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."
*

*> >
*

*> > I am interested in how we pair-wise partition the Universe into, to use
*

*> >MAtti's terms, material and cognitive spacetimes. I think the former are
*

*> >modeled by Pratt's Body and the latter by Mind. There is an optimization
*

*> >involved that looks like Matti's ideas, but I am missing the details of
*

*> >its mechanism. Is it like a step-wise adaptation or a converging
*

*> >tournament of games or a darwinian group selection, or all of the above?
*

*>
*

*> I think here you miss the subjectivity. Who does play the tournament games?
*

*> Who does decide the darwinian group selection? Or Who does divide the universe
*

*> into material and cognitive spacetimes. Or whose body and mind are you
*

*> discussing? Who does make the step-wise adaptation? I think there is no model
*

*> valid without specifying for whom/which the model is.
*

You raise the essential question! :) "who" can be either the object as

an observer *xor* the environment as an observer, hence the duality. The

division of the universe into perceived and perceiver, object and

subject, is not assumed to "exist" a priori, such a construct is

non-informative, zero Shannon entropy. It is constructed. When we ask

such a question we fix a frame of reference, we are also fixing a scale

and an ordering and orientation with which to "frame" an observation.

There is no Absolute "Who" asking questions. We all are and we all are

the Universe, kind of like government: "for the people, by the people".

This is one point that Wheeler got right! :) When we think of time's

flow, we are seeing a drama unfolding, written one moment prior and

checked for logical consistency by the dual of time, logic. Here are

quotes from Vaughan Pratt adressing this:

Return-Path: <pratt@coraki.Stanford.EDU>

To: David Wolf <greywolf@ptolemy.arc.nasa.gov>

cc: Vaughan Pratt <pratt@CS.Stanford.EDU>,

Stephen Paul King <spking1@mindspring.com>, rotwang@ricochet.net

Subject: Re: more things, if measured?

Date: Mon, 27 Apr 1998 13:12:36 -0700

From: Vaughan Pratt <pratt@CS.Stanford.EDU>

*>QL was motivated by the well-established fact that objects in quantum
*

*>mechanics behaved in non-local fashion,
*

Is this historically accurate? While it's true that EPR pointed out

the apparent conflict between causality and locality a year before

QL was introduced, few if any back then even took EPR seriously,

let alone focused on locality as a quantifiable concept, which only

happened thirty years later. The inventors of QL motivated it only in

terms of Heisenberg uncertainty and noncommutativity of observations.

There is no mention of locality in the original QL paper.

*>yes, but it is not more than standard probability theory can manage or
*

*>describe. and that is just a mathematical tool. do you ascribe
*

*>simpler derivations of results to your approach, or perhaps a conceptual
*

*>breakthrough?
*

A bit of both. Here's one insight. A certain simple phenomenon that

arises with matrices (over any set, not necessarily a field) accounts

for an "uncertainty" that arises between time and information. In the

case where this uncertainty is the Heisenberg uncertainty of time and

energy these matrices are over the complex numbers. Here I'll treat

the much simpler case of matrices over {0,1}, or {A,D} as I'll call it

for Across and Down.

The phenomenon is that the rows and columns of a matrix cannot

simultanously enjoy the same closure properties. Consider a matrix

over the symbols A and D, and view the row (resp. column) indices as the

members of the Across (resp. Down) team. An A at row a and column d

means

that a beats d (team A wins that match), and vice versa for D. Assume

for

simplicity that all rows are different, and likewise for all columns.

Claim 1. (Zeroary case) There cannot be an invincible

player on both teams.

Proof. This is an obvious variant of the

irrestible-force/immovable-object paradox.

Claim 2. (Binary case) Refer to player a as a *perfect replacement*

for a1 and a2 when for every opponent d, a loses to d if and only if

both

a1 and a2 lose to d, and likewise for the other team (for every oppenent

a, d loses to a iff d1 and d2 lose to a). I claim that if every pair

of players on team A has a perfect replacement, and likewise on team D,

then for every pair of players on either team one of that pair is a

perfect replacement for the pair.

Proof. Suppose a is a perfect replacement for a1 and a2 and is neither

a1

nor a2. Then there must exist players d1, d2 such that a1 and a2 each

win

one match with d1 and d2, and d1 and d2 each win one match with a1 and

a2

(2x2 checkerboard of 2 A's and 2 D's). Let d be a perfect replacement

for d1, d2. Then d beats both a1 and a2 (and hence is neither d1 nor

d2)

but loses to a. But a being a perfect replacement for a1 and a2 must

lose to d, contradiction.

Corollary. When every pair of players on either team has a perfect

replacement, each team can be linearly ordered by inclusion on the set

of players on the other team beatable by a given player.

When the players on team A are events and those on D states, with an A

at (a,d) meaning that event a is in state d's past (state d "knows of"

event a), the meaning of Claim 1 is that you can't simultaneously have

an

initial event (one which is known to every state) *and* an initial state

(one which knows of no events). In cosmological language the big bang

was either an event or a state, but it cannot be both.

The interpretation of Claim 2 is more involved. When events are

linearly

ordered it means there is no concurrency---for every pair of events one

is

in the past of the other. When states are linearly ordered it means

there

are no alternatives, i.e. no branching, the future is fully determined.

Hence in any dynamic universe with either some concurrency or some

branching, either there exist events a,b such that every prior event

is known to a state that is ignorant of one of a or b, or there exist

states d,e such that every prior state is ignorant of some event known

to both d and e.

All this is admittedly pretty abstract, yet it evidently says

something about the sort of time-information interactions occurring

with computation. I'd like to think it wasn't so abstract as to be

completely uninteristing to physicists, but clearly that's not for me

to say, you physicists have to be the judge of that.

*>do you see / feel that there is a way to "fix this" deficiency as a
*

*>physical theory? i.e. do you see a way that these structures could
*

*>be seen as arising naturally in your framework?
*

Nothing more than a prospector's hunch. Even with Chu spaces over 3

that are as small as 6 points and 6 states there is a rich and diverse

structure that will take a very long time to explore (3^36 is a big

number, and 5^36 is bigger still if there is stuff lurking in Chu(Set,5)

that is not in Chu(Set,K) for smaller K). On the one hand I can't

promise

anyone that any structures in there will connect up with structures and

constants encountered in the particle zoo, on the other I would not be

at all surprised to find some given the universality of Chu spaces.

Privately I confess to being somewhat of a neo-Eddingtonian, in that

I fondly imagine that much of the combinatorial structure of nuclear

physics will be traceable to structure residing in small Chu spaces over

some small alphabet. Ditto for nonnuclear forces, but for larger

spaces.

At the moment however I have *no* evidence at all for such a wild

belief.

Many prospectors pan sterile soil for the few that strike it rich,

and there's no particular reason I should be among the latter.

Vaughan

Return-Path: <pratt@coraki.Stanford.EDU>

To: Stephen Paul King <spking1@mindspring.com>

cc: Vaughan Pratt <pratt@CS.Stanford.EDU>

Subject: Re: more things, if measured?

Date: Tue, 28 Apr 1998 10:23:46 -0700

From: Vaughan Pratt <pratt@CS.Stanford.EDU>

*>I have been looking into graph theory for way to better understand Chu
*

*>spaces. Is it possible to consider the "games" represented by your matrix
*

*>example by an n-ary tree(s)? Is there any connection to "knock out
*

*>tournaments?"
*

Not really. They are much better understood as payoff matrices in the

sense of Morgenstern and von Neumann. In the example a strategy

for a player (in the technical sense, namely what I was previously

calling a team) consists simply of selecting a team member (what I was

calling a player). When the matrix is

D

d e f

a A D D

A b D A D

c A A A

D's best strategy seems like it ought to be to pick f, but obviously

this is wishful thinking since A is sure to pick c. But if we eliminate

those two players to yield

D

d e

a A D

A b D A

then the situation is entirely symmetric. In that case either player

can be sure of winning with probability one half by selecting the team

member at random.

Note that probability here is with respect to just one player's state of

knowledge, an observer who knows both players choices knows the outcome

nonprobabilistically.

This relationship between Chu spaces and games is why Lafont and

Streicher

called them "games" in their LICS'91 paper. They only briefly mentioned

this connection however and did not go any further into why Chu spaces

could be understood as games, rightly assuming that people could read

about payoff matrices elsewhere in the game theory literature.

The point of my example is that it makes a connection between games and

partial distributive lattices (PDL's) such as posets, semilattices, and

distributive lattices. Via this connection one observes an interference

phenomenon between the points and states (filters) of PDL's.

Chu(Set,3) and higher extends the theory beyond these partial

distributive

lattices. While the same uncertainty principle holds I have not yet

explored instances of it in any detail, and I have so far made only the

completely trivial yet nevertheless meaningful observation about finite

Chu spaces, that Planck's constant for them is the reciprocal of their

size (number of entries), as described in the Budapest lecture notes.

Vaughan

Return-Path: <pratt@coraki.Stanford.EDU>

To: Stephen Paul King <spking1@mindspring.com>

cc: Vaughan Pratt <pratt@CS.Stanford.EDU>

Subject: Re: more things, if measured?

Date: Tue, 28 Apr 1998 17:05:37 -0700

From: Vaughan Pratt <pratt@CS.Stanford.EDU>

*>But, when faced with the situation of complementarity of information,
*

*>do we not get into a situation were there would be different matrices
*

*>representing the knowledge of the players after each "move?"
*

In game theory points and states are called strategies rather than

moves. When each player has chosen a strategy, a sequence of moves is

thereby defined. The matrix gives only the payoff at the end of the

game, which could be after either finitely or infinitely many moves.

The actual moves are not individually represented in the matrix, at

least not as indices of rows and columns.

There is however a way of interpreting rows and columns not as

strategies

but as moves. This interpretation is the event-state time-information

process interpretation states and information reinterpreted as events

and time of an opponent. Here we are thinking of the Player P as

particle-like and forward-looking and the Opponent O as wave-like and

backward-looking, as though time were a football field and the big bang

and the apocalypse were the goal posts. At some point I will write this

up.

The sort of alternating sequence of moves you have in mind is what

recent

work in game semantics by Blass, Abramsky, Jagadeeson, Curien, Hyland,

Ong, and others has been about. Like Chu spaces this notion of game

semantics ties in well with linear logic, though the models despite

having games as a common motivation are very different in character.

Vaughan

Return-Path: <pratt@coraki.Stanford.EDU>

To: David Wolf <greywolf@ptolemy.arc.nasa.gov>

cc: Vaughan Pratt <pratt@CS.Stanford.EDU>,

Stephen Paul King <spking1@mindspring.com>, rotwang@ricochet.net

Subject: Re: more things, if measured?

Date: Sat, 02 May 1998 11:53:47 -0700

From: Vaughan Pratt <pratt@CS.Stanford.EDU>

*>what connection did you make between the chu space stuff and particle
*

*>properties?
*

Chu spaces being universal for mathematics, if some or all of the

particle properties are of mathematical origin then those properties

must necessarily be found in Chu spaces.

The flip side of this is that if it is shown somehow that particle

properties are extra-mathematical, this will surely be a temporary

matter

fixed by suitably extending mathematics to model particle properties.

But while physics has demonstrated no lack of creativity in extending

mathematics in the past, I have no idea what form any future extension

might take. Chu spaces as presently defined will definitely not be

universal for any such extension, but the underlying Chu principle could

conceivably be extended concomitantly, though there is no guarantee

of this.

*>what would you be identifying in the chu spaces as "fundamental constants
*

*>of nature".
*

I don't know how these would arise. For one thing I don't know how

mass,

charge, etc. would manifest themselves as features of Chu spaces. I can

account for energy, and mass is energy to be sure, but I don't know how

how bound energy would distinguish itself from radiant energy.

I'm hoping this might answer itself by the discovery of some class or

other of finite Chu spaces having general structural characteristics

very reminiscent of those of the particle zoo. Then one could see

how far that connection could be pushed.

I'm painfully aware that this is all extremely vague, sufficiently so

that I rarely write or talk about it---what's the point of talking about

things one doesn't know anything about?

*>how did particles enter, much more details of their properties?
*

The thesis that nature is of mathematical origin is not as I see it tied

specifically to particles per se, but rather is equally applicable to

properties of atoms, inorganic molecules, organic molecules, materials,

cells, organisms, ecologies, planets, etc. I presently believe that

this

thesis should stand or fall for all entities of nature simultaneously.

That said, the associated combinatorial search space for properties of

objects grows exponentially with size of object. Hence any search of

that space for evidence for the thesis is likely to hit paydirt first

for particle-sized objects. Leptons, quarks, and the fundamental forces

between them are especially promising.

Vaughan

Return-Path: <pratt@coraki.Stanford.EDU>

To: David Wolf <greywolf@ptolemy.arc.nasa.gov>

cc: Vaughan Pratt <pratt@CS.Stanford.EDU>,

Stephen Paul King <spking1@mindspring.com>, rotwang@ricochet.net

Subject: Re: more things, if measured?

Date: Sat, 02 May 1998 21:20:56 -0700

From: Vaughan Pratt <pratt@CS.Stanford.EDU>

*>do you mean that chu spaces are isomorphic to mathematics?
*

Basically, yes.

A potential obstacle to such a claim is that people don't agree

on what constitutes mathematics. For example those who believe

that all mathematics can be expressed in first order logic make the

tacit assumption that all mathematical domains are representable as

sets equipped with a family of relations, these being the models that

interpret any first order theory. But any such domain can be

topologized,

a notion that is beyond the scope of first order logic, unless done

indirectly via ZF set theory, a domain having only one relation, the

binary relation of membership. Then there are the category theorists,

who view category theory rather than first order logic as the proper

foundation for mathematics.

What I find so remarkable about Chu spaces is that they overcome that

obstacle by subsuming all relational structures, with or without

topology, as well as all categorical structures. They don't do so

uniformly---relational structures are represented by Chu spaces using a

different approach to how categorical structures are represented---but

that would be asking too much. Furthermore the representation is in

every

meaningful case concrete, independently of the origin of the represented

objects' concreteness. Concreteness means that the Chu space

representing

the mathematical object in question has the same underlying set. This

is

a first for objects making any such claim for universality.

In short, no matter which of the mainstream mathematical frameworks you

choose to work in, the objects of that framework are all *realized* by

Chu spaces, where "realize" abbreviates "concretely, fully, and

faithfully

represents." No other mathematical structures are this capable.

*>in this case you would be saying, effectively, mathematics is capable of
*

*>describing whatever mathematically describable properties particles have.
*

*>is there a technical use of the term "universal" here? unless you mean
*

*>something very special by "universal" then this is a vacuous statement.
*

Yes, "mathematical description" is too vague a concept to be meaningful.

A more specific, hence meaningful, concept is realization in the above

sense, the idea of capturing both the static and the dynamic attributes

of mathematical objects: static via concreteness, dynamic via full and

faithful representation. The former gets at the "stuff" of an object,

the latter at its transformational characteristics, the glue that binds.

*>i feel that there some thing about your formalism that has convinced you
*

*>that it somehow must represent the very fundamental foundational levels of
*

*>the material and perceptual world, that it must be the very bottom "stuff"
*

*>somehow.
*

Exactly so. I believe that realization is close enough: once you have

a class of objects that realize all known mathematical objects you have

realized known mathematics itself. Note that this says nothing about

thus

far unknown mathematical objects which we might discover in the future.

*>one begins to argue over the definition of mathematics at this point.
*

Quite right. My definition is in terms of realizability of objects,

combined with my thesis that realizability is close enough: I claim that

you understand a mathematical object well enough for any purposes of

science once you have one realization of it, and that other realizations

add nothing further to the role of mathematics in describing and

understanding nature.

*>we should agree to not banter about whether mathematics is all math "known"
*

*>or all math "possibly knowable", as if such things were definable.
*

"Mathematics" is certainly a very open-ended term. My usage of the

term is the narrow one in which the mathematical universe is viewed as

consisting of its transformable objects. While I certainly would not

claim that this brand of mathematics is what science has to use, I do

claim that it suffices for science. All mathematics used in physics

can be presented in terms of transformable mathematical objects such

as locally compact abelian groups, noncommutative rings, Lie algebras,

vector spaces, Hilbert spaces, Banach spaces, etc. etc.

*>what do you identify as energy?
*

Energy is the distance between states, for a suitable notion of

distance.

Dually time is the distance between events (points), for the same notion

of distance. This makes both energy and time relative notions.

When K=2 there is a binary notion of distance that amounts to temporal

precedence for the events and information subsumption (in the sense of

the

information order in a Scott-domain information system) for the states.

These partially order their respective domains, and continuous functions

(aka Chu transforms) are automatically monotone with respect to those

partial orders.

For any K there is a metric notion of distance: two states are separated

by the (cardinality of the) set of events separating them, and dually

two

events are separated by how many states separate them. This metric is

symmetric and satisfies the triangle inequality, and continuous

functions

are automatically Lipshitz (contracting maps) with respect to this

metric.

This metric is mainly of interest for finite Chu spaces, beyond which

its value is dubious.

There are also other useful notions of distance, some symmetric and some

not, but all satisfying some sort of triangle inequality. Of particular

interest are those obtained by furnishing the set K with the structure

of

a quantale (done implicitly above for K=2), which yields a metric whose

distances are elements of K (what one could call an internal metric).

(This metric is equally useful for finite and infinite spaces, thanks

to quantales being complete semilattices.)

Thus far none of these have struck me as *the* candidate for an energy

metric, but I believe that some such metric will turn out to behave very

much like energy, much more so than its competitors.

*>there are a host of examples on why this approach is inappropriate: for
*

*>instance, the binary logic seems to crop up in a lot of ways, perhaps
*

*>one might even say it is ubiqitous, since any function is approximable
*

*>on machines using binary logic throughout.
*

Indeed. And moreover I don't share Wheeler's bit-as-it view that

physics

will ultimately be boiled down to bits, not because I can disprove it

but

because I can clearly see limits to the world of bits which physics

seems

almost certain to transcend. I identify the bit world with Chu(Set,2)

(i.e. K=2, Chu spaces whose entries are 0 or 1), which before I knew

about Chu spaces I was calling partial distributive lattices. These are

good for logic but are less obviously useful for concrete geometry,

where Chu(Set,3) is needed at a minimum and Chu(Set,5) seems needed for

more serious geometry. (But if one drops concreteness as a requirement

then Chu(Set,2) becomes universal for mathematics, i.e. if concreteness

is not important then Wheeler *is* right. Proving this is not too hard,

2-3 pages.)

What I do sense is that one doesn't have to go very far beyond the bit

to

get into serious mathematics. In particular one doesn't have to replace

bits by complex numbers, somewhere between three and five values seems

to

me likely to suffice for nature. Thus homology becomes possible at K=5

(so Mike Barr tells me), while group theory complete with the "monster

groups" emerges at K=8.

If I were to assign confidence levels to all these wild claims of mine,

these numbers like 3-5 as the "right" value of K would have rather lower

confidence than what I said about energy, which I'm relatively sure of.

It is perfectly conceivable that K has to keep pace with the size of

object in order to model physics, I'm currently betting it doesn't,

but more cautiously than some of my other bets. If it turns out that

the central role currently played by Pontryagin duality and traditional

Fourier analysis in quantum mechanics is necessary then I will have been

completely wrong because Pontryagin duality calls for K being the set

of complex numbers, or at least those on the unit circle.

*>but this does not make the bit a fundamental quantity, nor does
*

*>discovering yet another way in which bits seem related to the physical
*

*>world make them a good place to start building a theory. having just
*

*>an interesting subgroup or math structure is insufficient for building
*

*>good theory.
*

I fully agree.

*>the work to be done is in the cross-identification of mathematical and
*

*>physical objects. reminiscent structures have a nasty habit of occurring
*

*>all over creation, and the pursuit of such has usually been without fruit
*

*>without sound theoretical basis. getting onto a sound theoretical basis
*

*>necessarily involves making the appropriate object identifications.
*

Indeed. It's a very quixotic search, like Eddington's but based on a

promise of mathematical universality. I'm proposing to search a

haystack

for a needle. The haystack may well be too big and the needle may well

not be there. And even if it is there, and even if we stumble across

it, there remains the risk that we may not recognize it. A situation

not unlike that of SETI.

Vaughan

and from an old post of mine:

About your questions: I find Pratt's paper "Rational Mechanics and

Natural Mathematics" to be a good starting point since it has the most

intuitive expressions about the usefulness of Chu spaces (and the

associated mathematics). On page 3 we find the following statement: "We

propose to reduce complex mind-body interaction to the elementary

interactions of their constituents. Events of the body interact with

states of the mind. This interaction has two dual forms. A physical

event a in the body A 'impresses' its occurrence on a mental state X,

written a =| x. Dually, in state x the mind 'infers' the prior

occurrence of event a, written x |= a. States may be understood as

corresponding more or less to the possible worlds of a Kripke structure,

and events to propositions that may or may not hold in different worlds

of that structure. [I would like to think of this as a relationship

between possible worlds and possible minds, each "represented" within

each other. SPK note]

"With regard to orientation, impression is 'causal' and its direction

is that of time [whose arrow is that of thermodynamic entropy/ Hubble

expansion. SPK note] Inference is 'logical' and logic swims upstream

against time. [a search along a n-ary tree to find a consistent path -a

historical president- has an arrow defined by a monotonical (?)

reduction in the Shannon entropy of the tree. SPK note] "Prolog's

backward-chaining strategy dualizes this by viewing logic as primary and

time as swimming upstream against logic, but this amounts to the same

thing. The basic ideas is that time and logic flow in opposite

directions."

I propose that the action of transitioning from one event to another

that we perceive as the "flow" of time is dual to the transitioning from

one state to another of the logical computation of "what event a' can

happen next." The future is "out there" but since it must be thought of

in terms of all possible events that could happen next and similarly the

past as all possible events that could of happened before a', which is

the event experience "now." The computation via interactive competition

of possible states follows a similar yet dual chain.

"The Duality of Time and Information," dti.ps, describes this in

detail.

*> >:) Could the act of communication be represented by "wormholes"? How do
*

*> >we quantitate the information flow allows by wormholes?
*

*> >
*

*> >Later,
*

*> >
*

*> >Stephen
*

*> >
*

*>
*

*> Best,
*

*> Hitoshi
*

**Next message:**Hitoshi Kitada: "[time 128] Re: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality"**Previous message:**Stephen P. King: "[time 126] Re: [time 109] Spacetime& consciousness"**In reply to:**Ben Goertzel: "[time 120] Re: [time 109] RE: [time 104] Re: [time 102] Re: [time 99] Spacetime& consciousness"**Next in thread:**Hitoshi Kitada: "[time 128] Re: [time 127] Re: [time 121] RE: [time 115] On Pratt's Duality"

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