[time 512] A new approach to Causality via Chu space residuation, Part 1

Stephen P. King (stephenk1@home.com)
Mon, 02 Aug 1999 21:46:05 -0400

Dear Friends,

        I have been talking a lot about the work of Vaughan Pratt on the Time
List and it has become evident that I need to write a post on his work
and how it is relevant to our work here. This post is only a rough
sketch that I hope is polished by the interactive discussions to follow.
        I will start with a quote from Richard Feynman speaking about the
Principle of Least Action:

        "We have lost the idea of causality that the particle feels the pull
and moves in accordance with it. Instead of that, in some grand fashion
it smells all the curves, all possibilities, and decides which one to
take (by choosing that for which our quantity is least)."
[Richard P. Feynman, The Character of Physical Law, The M.I.T. Press,
Cambridge, 1965.]

        In the papers of Vaughan Pratt we find an interesting way of thinking
about how a particle could "smell all the curves". He proposes that the
short-lived theory of mind-body dualism of Rene Descartes. Pratt finds
that the formalism of Chu spaces addresses the shortcomings of the
philosophy of mind-body dualism. The causal interaction of mind and body
is defined in terms of Chu spaces and residuation. First we need to set
up some preliminary concepts.

The concept of Linear Logic was used as the framework that lead to this
line of thinking. It is used construct ways to test the behavior of
computer programs in conjunction with hardware implementations of such.


The concepts of Mind and Body speak to the distinct aspects of the
Universe, matter and information. It is proposed that matter (qua
material configurations) and information (qua "meaning") are not
identical, they are complementary or "dual" aspects of finite subsets of
the Universe (which is identified with the totality of existence and is
non-dual in-itself (or "self-dual"?) and tenseless).

Pratt told me in a e-mail today when I asked him about how to define
Mind and Body:

"Perhaps "mental" and "physical" would have been better, illustrated by
such examples as "predicate" ("is red") vs. "subject" ("the ball"), or
"thought" vs. "thing", as instances of mental vs. physical."

A quote from ratmech.ps:
        "We interpret interaction as causality. Causality is directional, but
the direction depends on whether we have in mind physical or mental
causality. We interpret $x$ |= $a$ ambiguously as the time elapsed
between the occurrence of the physical a and its impression on the
mental state $x$, and as the truth value of $a$ as a proposition. [The
reader may be understandably concerned at this identification of
physical and ostensible mental propositions. However a Boolean
proposition about events A is of type 2^2^A and $each exponentiation$
dualizes, whence two of them return us to the physical plane. The truly
mental propositions are constituent descriptive clauses of a physical
DNF formula, each describing a possible world.] The former is physical
causality or $impression$, flowing forward in time from events to
states. The latter is mental causality or $inference$, flowing backwards
in time from the thought of $a$ to the inference of $a$'s occurrence. In
this way time flows forward (from the usual point of view) while logic
flows backwards. This is the $primary$ interaction, and it occurs only
$between$ the mental and the physical planes.
        We thus see that the seat of casual interaction in Cartesian duality is
not the pineal gland but the identification of impression and inference.
We write $x$ |= $a$ as expressing equally the impression of event $a$ on
subsequent state $x$ and the deduction by state $x$ of the prior
occurrence of event $a$. The Cartesian dictum $cogito, ergo sum$ is the
case of this where $x$ is the thinker's state and $a$ the event of his
or her existence." ... "Examined closely, our analysis shows that
Descartes' dictum properly tensed becomes $cognito, ergo eram$ (I
was)..." ... "We pass now to interaction $within$ each plane, whether
physical or mental, which we derive as $secondary$ interaction from the
primary form with the aid of $residuation$, a pair of operations on
binary relations that constitutes dynamic implications forwards and
backwards in time: For K = 2, =| as a matrix of 0's and 1's is an
ordinary binary relation: the event $a$ either is or is not related to
state $x$. This relation is understood ambiguously as a two-valued
distance in either time-space (a =| x, physical) or information space
(x|= a, mental)." (pg. 8)

        Here Pratt discusses how the $direction$ of causality depends on
whether one is considering the physical aspects of a subset of the
Universe (identified by Hitoshi as a Local System) or the informational
aspect of a LS. A LS is considered as a generic model of an observer or
measuring system, capable of registering the dynamical behavior of other
LSs via the mechanism of $ coinductive bisimulation$ between LSs.
Bisimulation is discussed in Peter Wegner & Dana Goldin's paper

"Symmetry of equivalence gives rise to the terms bisimularity and
bisimulation that captures the if-and-only-iff nature of relations
between equivalent objects. Coinductive bisimulation captures mutual
two-way behavior simulation of each system by the other." (pg. 16,
Mathematical Models of Interactive Computing, By Peter Wegner & Dana

        The physical configurations of a LS's observations are ordered via
physical $implication$ and the information of a LS is ordered via the
mechanism of logical $entailment$. Individual physical properties are
called $events$ and individual mental or informative properties are
called $states$ in Pratt's presentation.

Pratt says:
        "When we unravel the primitive causal links contributing to secondary
causal interaction we find that two events, or two states, communicate
with each other by interrogating $all$ entities of the opposite type.
Thus event $a$ deduces that it precedes event $b$ not by broaching the
matter with $b$ directly, but instead by consulting the record of every
state to see if there is any state volunteering a counterexample. When
none is found, the precedence is established. Conversely, when a Chu
space is in state $x$ and desires to pass to state $y$, it inquires as
to whether this would undo any event that has already occurred. If not
then the transition is allowed." (pg. 9, ibid.)

        Here we are getting closer to our opening remark by Feynman! We must
note that in the preceding quote from Pratt, idealistic conditions are
considered. In the situation of Local Systems, as modeled By Hitoshi, it
is assumed that only a finite number of states (events) can be
"consulted" in a finite amount of time and that the "record" is subject
to falsification unintentionally by noise or intentionally by secondary
"eavesdropping" observers. Going back to Feynman's notion, we must ask
how it that the curve that minimizes the action is selected? We are
considering the transition of events $x$ to $y$ and dually, of states
$a$ to $b$, to be defined by such minimized curves.
        Now, in traditional physical thinking we have a clear notion of that
the curves connecting physical events, they are called $geodesics$ and
are considered to be the paths that rays of light take in space-time.
The question of whether space-time is defined by the rays or has a
priori ontological status will not be addressed here. I merely say that
it seems to be enough to consider that a space-time is defined by the
Diff^4 group of a set of light-rays.
        We must note that traditional physics also assumes that a single
space-time exists and thus only a single unique set of geodesics, as a
light-cone structure, need to be considered. But as we have seen in
situations that consider the quantum mechanical properties, it is
impossible to define a single unique light-cone structure for all
subsets of the Universe since the definition of such implies that a
single $anti-set$ of curves exists connecting the information states
entailed by the single set of physical events and that this anti-set
exists a priori as a "pre-computed" Boolean lattice of logical
        The a priori existence of such a Boolean lattice must be considered
carefully! If such exists in the same sense that the Universe exists, we
are faced with the question as to how this "object", which would have
the status of a Platonic Form, would be $knowable$ by the infinity of
finite subsets of the Universe or LSs. It is obvious that there is a
deep difference between the existence of something and the ability to
have useful knowledge of it. The ability to gain useful information
about something takes into a discussion of thermodynamics and I wish to
reserve that topic to a latter date.
        It might be asked why am I proposing a dualistic model of interactions
instead of a monistic one; to answer this I shall again quote Pratt:
        "If one truly believed that the [subsets of the] universe proceeded via
state transitions, this might seem a rather roundabout and inefficient
way of implementing those transitions. However it seems to us ... that
the more likely possibility is that the [subsets of the] universe only
$seems$ to proceed via state transitions, due perhaps to our ancestors
having ill-advisedly chosen monism as the natural world view, perhaps
millennia before the rise of Cartesianism, perhaps only some years after
its decline. What we conjecture actually happens is that events signal
states forward in time, or equivalently that states infer events
backwards in time, and the world we imagine we live in is simply what
that process looks like to its inhabitants when interpreted
        Why this theory as opposed to any other? Well, certainly no other
theory has satisfactorily explained this causal interaction of real
mental and physical planes as conceived by Descartes."
        Next, Quantum Mechanics in the Chu space representation.



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