[time 515] Re: [time 513] Rational mechanics and payoff matrix

Stephen P. King (stephenk1@home.com)
Wed, 04 Aug 1999 22:12:30 -0400

Dear Matti,

Matti Pitkanen wrote:
> Dear Stephen,
> Thank You for the posting. I think I understand now something
> about the basic ideas of Pratt's theory. Some critical comments
> below.
> I read also the posting about pay-off matrix.
> Did I understood correctly: payoff matrix tells how much
> either participant wins in the game in which the 'initial
> states' of the players are M and N and both
> players have selected their strategies so that
> the end result of the game is predictable.
> Perhaps the interpretation of M and N is not correct.
> What kind of game is basically in question and what is the
> interpretation of the indices of the payoff matrix?
> ***********

        One aspect that we must be carefull about is that the information as to
the possible moves of the "other" player are usually subject to
uncertainty. We see in Peter's work how interactions change the systems
and thus change the "best" stratergies as the games are played. Remember
that this is not deterministic in the sense that all information is
given a priori, information is what the game produces! This is the basic
idea of Frieden's work and why I advocate it.
        I think that it would help to use Peter's notions to consiter the
"initiality" notion! We are dealing with "windows" not absolute
pre-givens, we see this in the way that person to person interactions
work. We are dealing with the situation where what is a game in one
framing is a player in another! This follows along the same lines as the
subject-object situation.
        (a note added in editing) It seems that the "size" of the "window" is a
measure-like term, it relates to the sample from which the relative
frequency terms are calculated. It looks like the \epsilon term that I
had mentioned earlier on that you think is related to the prime... Umm,
I am thinking hard about this!

> [SPK]
> 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.]
> [MP]
> I am objecting from the very beginning! The summation over histories
> is mere formal representation. One can quite well do without it!
> The calculation of all these histories would involve huge amount
> of computational work since even in case of single point like
> particle the space of paths is infinite-dimensional and one must
> allow also nondifferentiable paths. Using simple Schrodinger
> equation one avoids all this huge computational activity!
> The fact is also that whereas Wiener integrals are well defined
> objects (measures), path integrals do not simply allow rigorous
> mathematical formulation as measures.

        Matti! Are not the formal representations of Heisenberg's matrix
mechanics and Schrodinger's Waves and Feynman's path summations
equivalent? Your point that "calculation of all these histories would
involve huge amount of computational work" holds for all of these
methods and is exactly why I am making a big fuss about computational
issues! Your point here is an example of my reasoning!
> Path integral is the only manner to get connection with classical
> theory but my belief is that it is just here were standard
> QFT fails. Classical world (spacetime) is much more than stationary phase
> approximation of mathematically non-existing path integral!

        The "classical world" is a mental construction. It is NOT "out there",
it is in our head! So we are faced with the question: How does the self
"calculate" the path that minimizes the action? We know that the q-jump
is just such a computation; now we need to look more carefully how the
"measures" (Wiener integrals, etc.) are defined by the "way" that the
observation is made.
> ********
> [SPK]
> 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 can be fixed. 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.
> [MP]
> I understood the idea. The idea of finding model for how
> particle measures the action for all possible paths
> is natural in computationalistic framework. But is this
> really needed? Only the sum over paths appears and one can avoid
> this representation totally. If the predictions of the theory would
> involve also quantities defined by single path, the situation
> would be different.
> *********

        Yes, it is necessary. The problem is not completely intractable since
we only need to model finite subsets of the Universe, it is only when
one tries to create a model for the Totality that the intratibility
appears! The problem with 'quantities defined by single paths" is that
one needs a context (albeit finite) for such quantities to be
meaningful, for instance, for the transporation of a vector along a
single path, it is al least necessary that there be a basis against
which the vector could be compared at each point on the path...
        After talking to my friend David Woolsey, it has become evident that we
need to look at how finite limitations are placed on the "neighbors" of
points in the spinoral configuration space! This is equivalent to saying
that a given local system can only observe (interact) with a finite
number of other local systems. It is my opinion that the requirements of
bisimulational equivalence are what are involved...

> 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.]
> [MP] I am not sure whether I understood: I blame
> my formal philosophical training(;-).
> ******

        That is ok, we learn from each other as we go! :-) Could you point to
a particular statement here that is difficult?
> [SPK]
> 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.
> ****
> [MP]
> I tried to understand this in following manner.
> a) Physical causality corresponds to the causation
> physical event--> mental event, impression: Sensory experience.

        No, Physical causality is what traditional "forces" are representing.
Mental events are the information concerning such.
> b) Mental causality of inference: Thought of a ---> inference of
> a:s occurrence. I would identify this typically as motor activity.
> I decide to raise my hand and it raises.
> But it seems that mental causation and physical causation are understood
> to be duals of each other.

        Here you are confusing mental and physical causalities. The neuro-motor
activity follows physical causality, the particular action that results
follows from the selection of one particular physical movement out of
the many possible by the action of mental inference, the mental
(information) state dual to the neuronal pattern "consults" all possible
physical events and select the one that closest one to the movement
required. (We need to frame this in the mathematics of bisumulation, my
wording here is merely metaphorical!)
> c) One could argue that experiments of Libet about active
> aspects of consciousness (EEG activity starts before
> I decide to raise my hand) shows that mental event is later
> than physical event and mental to physical goes backwards in
> time. On the other hand: we experience these two kinds of causations
> as different. How this difference is explained if impression and
> inference are identified? What in these events make matter
> or mind active agent?
> **************

        Umm, this is in accord with the "cognito, ergo erram" (I think,
therefore I was) aspect of Pratt's notion! The "identification of
impression and inference" is equivalent to the identification of wave
and its Fourier transform, they are mathematically dual! Both matter and
mind are "active"! The notion of invariants and "static" quatities are
merely the result of synchronizations or, more generally, like the
situation seem in dissipative structures. WE derive "Being" from
"Becoming"! :-)
> [SPK]
> 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)..." ...
> [MP]
> I think I have understood the idea. Interaction within each plane is
> derived from primary interaction between matter and mind
> by some kind of conscistency requirements.
> But how matter-mind interaction is defined?
> *********

        This is the key question! We look at the behaviour of matter and notice
that its dynamics can be considered as transforming the information
content encoded in the configuration that are changed, likewise, we can
look at a database of information and consider how changes in it can be
used to identify the dynamics of the matter that encodes the
        I will try an example, I may be very wrong here! Consider the
configuration space X formed by identifying a material dynamical
system's events to points and consider a space I where each point
represents a different description of the system. We can see that when
we consider the path through X defined by the particular transformation
of the system, there would also be a "path" defined through I, but how
is it given? The "connection" that links points in I is one of logical
entailment, and the connection that links points in X is one of
"physical" causality.

> [SPK]
> "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)
> [MP]
> I understand time-space but what would information space mean
> mathematically?
> Are these spaces isomorphic?
> *******

        No, they are not isomorphic. Pratt says that the graphs of functions
and antifunctions are opposites, anti-parallel(?). See page 4 of
> [SPK]
> 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
> (http://www.cs.brown.edu/~pw/papers/math1.ps)
> "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
> Goldin)
> [MP]
> I think that simulation is what selves are doing all the time.
> They must do so. Higher selves can have only abstracted
> average experiences about the life of subsubselves
> and cannot catch at all what it is to be subsub...subself.

        Yes, we agree on this. I am saying that the mathematics of bisimulation
that Peter discusses are the best way to think of this. Peter's
"expressiveness" captures your thinking about "Higher selves can have
only abstracted..." :-)

> This forces selves to build simulations of the subsub...selves
> on their on spacetime sheets. We are indeed doing busily
> Monte Carlo calculations on elementary particle physics.

        That does this tell us, that we resort to random samples to model
particle properties?!

> I am pondering concrete mechanisms of how this could
> be possible. This leads immediately to the question
> how selves could replicate. DNA replication would be
> only special case of this and perhaps induced
> by the replication of cognitive spacetime sheets. A more refined
> possibility to is to generate messages generating desired cascades
> of selves of subselves of... on receiver. These
> self-cascades would be memes/ideas/thoughts. Nerve pulses,
> written and spoken language, movies, art..., email discussion
> groups...
> **********

        The idea that Natural Selection is a Universal Principle is becoming
more tenable. Lee Smolin and Loius (?) Kauffmann and advancing it in
their books... Again, we see this reflected also in Peter's model of
computation; interactive computers "evolve"!
> [SPK]
> 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.
> [MP]
> I understand now this better. You have matter and mind and
> must have two different causations. I am somewhat confused
> of terminology. Why properties are events/states? I have been
> accustomed to assign properties with states.
> ************

        I understand. Pratt uses "events' to represent physical aspects and
"states" to represent mental, or more generally, informational aspects.
> [SPK]
> 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.)
> [MP]
> I already protested about the idea of doing the huge calculational
> work involved in calculating the action of every path in
> Feynmann integral.
> Classical nondeterminism of Kahler action however leads to
> discrete version of Wiener integral in
> the calculation configuration
> space integral. In this case the representation
> is *not* formal and real calculation of the contributions
> of various spacetime surfaces with degenerate
> Kahler action is needed. This seems to require the
> calculation of all possible spacetime surfaces consistent
> with the initial data.
> ********

        To be honest, I looked at the first chapter of your (on-line) book and
I wonder how phenomena can be calculated easily with it! I am looking at
the philosophical aspect of Quantum gravity, I firmly believe that
Nature is performing computations that make our miserable machines look
like stone age tools! :-) I believe that physics is really about
figuring how Nature works so that we can aim the cause-effect in a
direction more friendly to our human needs.
> [SPK]
> 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
> inference.
> [MP]
> Anti-set corresponds to information space and set to space?
> There is somekind of one-one correspondence.
> ********

        Of sorts, but we are taking category theory, so we have equivalence
classes of each. Infomorphisms (or residuations, same thing) are
identifications between subsets of the the two equivalence classes. So
we match a particular mind with a particular body by considering the
duality: what mind best describes the body that best implements the
mind. So the notion of MNP is very close to my thinking! :-)
> [SPK]
> 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
> monistically.
> [MP] I did not quite understand what state transitions meant in this
> context.

        Each physical event (including a space-time framing!) has a particular
state associated; when we consider the evolution of physical systems by
transitions of events, for example: E_1 -> E_2 -> E_3, there is a
corresponding evolution of the associated states, for example: S_3 ->
S_2 -> s_1.
> [SPK]
> 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."
> [MP] Certainly monism is not enough. But is even dualism enough? There
> is also the possibility of tripartism. Matter as geometric form,
> ideas and subjective existence.

        This is a very good question. I think that you are right, but we do
need to synchronize our thinking/wording further... I see "subjective
existence" as the dynamic union of the duality, so the self reflects a
finite image of the Universe. And this tends to infinity in the limit of



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