Stephen P. King (stephenk1@home.com)
Sat, 03 Apr 1999 21:19:12 -0500
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
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