[time 1118] Re: [time 1113] interactions, windows and Monads (Re: [time 1105])

Stephen Paul King (stephenk1@home.com)
Tue, 14 Dec 1999 16:59:26 -0500

Dear Hitoshi and Friends,

        Oh, I have been creative. I hope this makes some sense! :-)

Hitoshi Kitada wrote:
> Dear Stephen,
> Thanks for your opinions and information. I am not clear yet how the western
> philosophy comes to the problem of dualism between mind and matter or its
> negation by Leibniz. Let me make some elementary questions below.
> Stephen Paul King <stephenk1@home.com>
> Subject: [time 1111] Re: [time 1109] Monads (Re: [time 1105])
> > > > It seems to me that Leibniz would lose nothing even if his monad is
> > > > allowed to have a tiny window through which to see the outside nearby.
> > I am afraid that the allowance of windows, however small, would bring
> > into the model of Local Systems a problem that would ruin it
> > consistency. We need to look carefully what it means to make
> > observations!
> Could you explain what inconsistencies arise in more detail?

        If we consider that the allowance of "windows" through which causal
connections could occur, e.g. the exchange of substances, it would seem
to be tantamount to allowing iterations (e.g. observations) to occur
between quantum mechanical systems that are by definition in a "pure"
state, we would be causing ourselves problems. It seems that one is
trapped in the tar pit of mechanistic explanations. :-( For a possible
alternative see: http://www.hpl.hp.com/techreports/97/HPL-97-122.html
        Is it necessary to have actual physical contacts between Local Systems?
What if it were possible to show that all of the properties of
iterations could be given by a method that is used to model how
computational systems interact in a distributed asynchronous concurrent
network? Perhaps I am being a bit idealistic, but if we consider how it
is that our minds "seem" to be able to have causal influences on each
other, even though our minds can not "touch" each other, this may not be
so confusing and difficult. Also, this might give us an explanation for
the strange phenomena that seems to be hinted at by resent studies of
"Consciousness-Related Anomalies In Random Physical Systems".


        The idea I have is to consider that the interaction of minds is more of
a sort of "morphic-resonance" type of interaction. I am considering LSs
as a good representation of minds in that they are indivisible wholes
and are not mechanical "windmills". :-) David Bohm has written about
        Minds that are similar in their external observational behavior would
be able to simulate each other's internal behavior if there exist some
consistent means of relating internal scattering dynamics with external
classical motions. We see this occurring tacitly in the interactions of
humans. The key is the possibility of an equivalence relation between
> > I would like to direct our attention to the following web site:
> >
> > http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/
> I read this page.
> The last paragraph:
> "He seems to think that causal interaction between two beings requires the
> transmission or transposition of the parts of those beings. But substances are
> simple unextended entities which contain no parts. Thus, there is no way to
> explain how one substance could influence another. Unfortunately, however, this
> line of reasoning would seem to also rule out one case of inter-substantial
> causation which Leibniz allows, viz., God's causal action on finite simple
> substances. "
> seems to be an explanation that Leibniz' monads do not have windows. And this
> seems a natural consequence of Leibniz' definition of monads. I do not see
> problems here insofar as we neglect the following two points raised in
> http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/:
> "Here Leibniz gives a reason tied to his complete concept theory of substance,
> according to which "the nature of an individual substance or of a complete being
> is to have a notion so complete that it is sufficient to contain and to allow us
> to deduce from it all the redicates of the subject to which this notion is
> attributed" (Discourse on Metaphysics, ec. 8). But there are, it seems, at least
> two problems with this explanation. First, Leibniz moves rather quickly from a
> conceptual explanation of substance in terms of the complete concept theory, to
> the conclusion that this consideration is sufficient to explain the activity of
> concrete substances. Second, even if conceptual considerations about substances
> were sufficient to explain their apparent causal activity, it does not seem to
> follow that substances do not interact--unless one is assuming that causal
> overdetermination is not a genuine possibility. Leibniz seems to be assuming
> just that, but without argument. "
        Yes, this is one reason why I was very happy that I found this article
on-line. :-) Leibniz may have been very happy with the implications of
Quantum Mechanics as it allows for "acausal" behavior in the classical
local sense...
        But, is this "acausality" truly randomness or is it perhaps merely the
local trace of a global causal situation such as illustrated by the
"secondary observers" that discussed by Peter Wegner?

> > The discussion is directed at the issue at hand! :-) The one comment
> > that I have of it is that the negation of dualism that Leibniz espouses
> > is a bit misguided. The use of an ab initio "pre-established harmony"
> > ("created minds and bodies are programmed at creation such that all
> > their natural states and actions are carried out in mutual
> > coordination.") to explain the facts of psycho-physical parallelism is
> > subject to the same criticisms as the notion of a priori Cauchy
> > hypersurfaces (cf. http://xxx.lanl.gov/abs/gr-qc/9310031)used in GR to
> > fix the initial conditions of the Universe.
> >
> > "Leibniz's account of mind-body causation was in terms of his famous
> > doctrine of the preestablished harmony. According to the latter, (1) no
> > state of a created substance has as a real cause some state of another
> > created substance (i.e. a denial of inter-substantial causality); (2)
> > every non-initial, non-miraculous, state of a created substance has as a
> > real cause some previous state of that very substance (i.e. an
> > affirmation of intra-substantial causality); and (3) each created
> > substance is programmed at creation such that all its natural states and
> > actions are carried out in conformity with all the natural states and
> > actions of every other created substance."
> > http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/#Noin
> >
> > If we are to mind the consequences of the Uncertainty Principle (UP),
> > we must dismiss this assumption on absolute initiality.
> I did not find time to see the pages you quoted below (I might have seen them
> before but am not sure). But if the initiality were an "absolute" initiality, it
> might be free from UP.

        Umm, I disagree. It would make the situation far worse since the notion
of absolute initiality would necessitate a unique initial point of time
for all existence, e.g. a metaphysical "ex nihilo" creation. Koichiro's
discussion of the problems of absolute synchronization is another
example of the problems that such implies. This is why I say that the
Universe, as the totality of existence, merely exists. It is without
beginning or end or extension or duration, it is merely itself. The
duality of observer and object only enters when we allow for a division
of the whole and this is implicit in the definition of an observer.
        The initiality is a matter of "fixing" a point in time as a lower bound
to the observers possible observations, as you delineate in the
explanation of the Hubble expansion in

"We remark that the `expansion' in this classical sense is different
from the stationary universe $\phi $ in our context of quantum
mechanical sense. The former `expansion' is the result of an observation
activity with fixing one observer's coordinate system, e.g., in the
above explanation we have assumed a synchronous coordinate system, which
explains why the universe looks expanding for all observers. The latter
quantum mechanical stationary universe $\phi $ is the inner structure
of its own and is independent of the observer's coordinate system.
Theorem 2 guarantees that these two views are consistent with each
other, and Axiom 6 predicts that this framework would explain and
resolve the problems related with the actual observations. In the
present problem of Hubble's law and `expansion' of the universe, these
phenomena are the consequences of the {\bf observation} with one
coordinate system fixed. In other words, they are 'appearance,' so to
speak, which the universe makes under the `interference' of the observer
to try to reveal its figure or shape. More philosophically, the past or
the future does not exist unless one fixes the time coordinate. The `Big
Bang' is an imagination under this {\bf assumption} of the {\it a
priori} existence of time coordinate. Unless it is observed with
assuming the existence of a time coordinate, the universe is no more
than a stationary state, which does not change and is correlated within
itself as a whole."

        The finitude of the "actuality" that any given LS could experience is
tied, I believe, to the finite scattering dynamics of the LS and,
specifically, the amount of information that it can encode. What I need
to find out is how is it possible to quantitate the amount of
information that an LS is capable of encoding with its scattering state!
:-) The fact that the precise amount of information is not decidable a
priori is not a problem! All we need is some form of upper bound, or
"maximality" and some iteration condition, which I believe, would be
given by the scattering equation of the LS and some way of quantifying
the information content thereof. Perhaps the BREMERMANN'S LIMIT
(http://pespmc1.vub.ac.be/ASC/Bremer_limit.html) or the
Bekenstein-Hawking formula
(http://www.math.ucr.edu/home/baez/week111.html) would help.

        On a parallel note, here is a quote from Peter's paper that applies:

"Mathematical Models of Interactive Computing 26/44
        Mappings m_t : S ­> G(A, S), where A are transactions that span
multiple time intervals, correspond to an extension of dynamical systems
from Markov to non­Markov processes that view time as an active rather
than passive variable in specifying system evolution. The dependence of
the mapping on uncontrollable inputs of other streams introduces both
nonlinearity of nondeterminism (section 4.2).
        Multi­agent behavior cannot be unfolded by iteration of a stationary
mapping m such that m^infinity = M. Temporal decomposition of behavior
is not in general possible either for Multi­agent computers or for
histories of a distributed interconnected world. Though time progresses
linearly, multi-agent (distributed) behavior cannot be linearly
described). Nondecomposition of behaviors into mappings for discrete
time steps corresponds loosely to nonlinearity and could in principle be
specified by set­theoretic axioms that specify solutions of nonlinear

> > > My interpretation is that a monad in the context of Leibniz is a
> > > local system without disturbance in my context. Simpleness
> > > which Leibniz requires monads does not contradict the plurality
> > > of the elements in a local system: A local system becomes a
> > > different local system if it is divided, so it is indivisible as
> > > local systems and is an elementary unit of existence. "Monads
> > > have no windows, by which anything could come in or go out." is
> > > true for local systems in the sense that: a local system becomes
> > > a different local system if "anything could come in or go out"
> > > with respect to the local system, and therefore, as far as a
> > > local system remains the same, it has "no windows."
> > Yes, it is very important to note that any observation whatsoever of a
> > Quantum Mechanical Local System (LS) implies that it is perturbed by
> > the act and thus we could consider such as implying a change of the LS.
> > We might consider that the act of observation of a LS is an act of
> > selection from an equivalence class (defined using ZFC- set theory. cf.
> > http://bugs.cs.wcupa.edu/~lizhang/Thesis/thesis/abstract.html) of
> > permitted LSs.
> >From the page you quoted:
> "In 1917, Mirimanoff first stated the fundamental difference between
> well-founded and non-well-founded sets. He called sets with no infinite
> descending membership sequence ordinary, and others extraordinary. In 1988,
> Peter Aczel introduced a uniform terminology. He replaced the Foundation Axiom
> (FA) with the Anti-Foundation Axiom (AFA). Aczel's AFA states that every graph,
> well founded or not, pictures a unique set. This results in Hyperset Theory or
> ZFC-. In ZFC-, a bisimulation determines whether two hypersets are equivalent
> and consequently makes the classification of hypersets possible. "
> What does the equivalence mean here, i.e. how is "bisimulation" defined and how
> does it determine the required equivalence relation? And how is that
> equivalence relation related with the following?:

        Ah, this is were we must get into the dirty mathematical details! :-)
Since I am not a mathematician, I will try to explain this the best I
can using metaphors and quotes over the course of our on-going
The best on-line definition that I have found is:

"An equivalence relation, defined in the context of process algebras,
which is a finer equivalence relation than trace equivalence and
distinguishes states based on branching properties."
from http://hissa.ncsl.nist.gov/~black/CRCDict/HTML/bisimulequiv.html

        I am trying to find a better definition! Section 3.4 of
http://www.cs.brown.edu/~pw/papers/math1.ps has the best definition.

"3.4 Bisimulation

        Equivalence is a subtle concept that may be progressively specialized
from equality (of all properties) to similarity (equivalence of some
properties) and simulation (dynamic equivalence of behavior). Symmetry
of equivalence gives rise to bisimilarity and bisimulation that capture
two-way step­by­step simulation of processes. Bisimulation captures
mutual two­way dynamic behavior simulation between two systems,
and is the natural extension of static equivalence to dynamic sequential
interaction. Bisimulation is a coinductive equivalence relation between
non­well­founded sets that models the behavioral equivalence of
streams. The mathematical question ``when do two equations have the same
solution?'' models the computational question ``when do two systems have
the same behavior?''. For coalgebras, this question becomes
``when do two coalgebras have the same final coalgebra?''
Equivalence for sets is specified by the $principle of extensionality$.
Two sets S, T are equal (S = T) if:

a) for every s \element S there is a t \element T such that s = t
b) for every t \element T there is an s \element t S such that s = t

        Equality of sets is recursively defined in terms of equality of subsets
down to an arbitrary recursive level. This recursion always terminates
for well­founded sets, giving us an inductive approach to proving set
equality. For non­well­founded sets, extensionality yields a circular,
coinductive form of extensionality called strong extensionality [BM]
that transcends inductive extensionality of finite structures. Strong
extensionality of non­well­founded sets determines equivalence of
infinite structures by interactive dynamic simulation processes.
        Two sets S, T are equivalent if there exists a $bisimilarity relation$
R \subset S x T involving all members of S and T; R is recursively
defined as follows:
for all s \element S and t \element T, R(s,t) iff s and t are atomic and
s=t, or
a) for every s' \element s there is a t' \element t such that R(s',t')
b) for every t' \element t there is an s' \element s such that R(s',t')
        The primary difference between bisimilarity and the earlier definition
of equivalence is the replacement of extensional (inductively defined)
equality "='' by a coinductively defined relation R. More than one
bisimilarity R \subset S x T may exist for a given pair of sets S,T.
However, the union of all such R is unique, and is the greatest
bisimilarity. When bisimilarity is interpreted as equivalence of system
behavior for all states s \element S and t \element T, the greatest
bisimilarity includes all pairs of states that preserve behavior for
every possible action a \element A. This greatest bisimilarity expresses
coinductive maximality and specifies coinductive equivalence for
non­well­founded sets and the systems that they model.
        When S and T are state sets of systems and R(s, t) means that s and t
have equivalent behavior, then bisimilarity expresses simulation of each
system by the other, and bisimilarity of sets becomes bisimulation of
systems. Bisimulation relations R model mutual on-line simulation of
sequences of actions in one system by sequences of actions in the other.
Bisimulation of systems is a specialized form of bisimilarity for
behavior equivalence between evolving systems about which we have
incomplete knowledge.
Bisimulation for coalgebras is defined by mutual simulation of their
system evolution functions.
        Bisimulation of coalgebras: Two coalgebras CS = (S, m:S­>\Lamda(S)) and
CT = (T, m':T­>\Lamda(T)) are related by a bisimulation relation R
\subset S x T if for each s \element S and each evolution step of CS
there is a t \element T and evolution step of CT that preserves R, and
some other online sources:


        Umm, it appears that what I need to find out is if it there exists an
isomorphism between the scattering dynamics of an LS and an automata:

        I apologize, but there is so much here that is very technical! I will
have to proceed very slowly!
This on-line essay by Onar Aam may be a good intuitive starting point!


        Let us meditate on how each LS "reflects" the Universe onto and into
each other! How is it that the LSs can make concrete representations of
each other? Consider the Turing Test that Peter Wegner discusses in his
papers. For example:

        "The key intuition is that the class of things that a finite agent can
observe is greater than the class of things that an agent can construct.
We can formalize this intuition by showing that the class of things that
an agent can construct is enumerable, while the set of situation that an
agent can observe is nonenumerable. More over, we can show that
constructible sets can be specified by induction, while observable sets
require a stronger inference rule called conduction.
        Bertrand Russell in the 1900s and Hilbert and Goedel in the 1920s made
a fundamental mathematical mistake in assuming that induction was the
strongest form of definition and reasoning. They were misled by the
paradoxes of set theory and mistakenly thought that circular reasoning
needed to describe observation processes was inconsistent. In fact,
circular reasoning is consistent and allows stronger forms of definition
and reasoning than is possible through induction. Though induction is
sufficient to describe construction processes stronger forms of
reasoning are needed to express observation processes. Turing machines
turn out to be the strongest form of computation possible by inductive
reasoning but are not strong enough to express interactive computations
of finite agents that observe an incompletely known environment, which
are modeled by circular reasoning.
        Inductive definitions of set theory and logic define minimal fixed
points which exclude everything that is not explicitly definable, while
coinductive definitions of non-well-founded set theory include
everything that is not explicitly excluded.
        Observers who consider all possible worlds not ruled out by
observations are using the coinductive maximal fixed-point principle.
The maximal class of things not explicitly excluded by a set of
observations is fundamentally larger than the minimal class of things
constructible from a set of primitives and allows us to build richer
kinds of models.
        Constructive models that employ induction can cope with only enumerable
situations, while observation-based models that employ circular
reasoning can cope with nonenumerable situations. Turing machines have
only enumerable input strings and can perform only an enumerable number
of computations, while interaction machines just like people can make
nonenumerable distinctions about their environment.
Interactive systems can handle nonenumerable environments while
noninteractive systems can handle only enumerable environments. The
existence of a mathematical foundation for interactive computing
provides a mathematical basis for interactive models of objects and
distributed systems."
from: http://www.cs.brown.edu/people/pw/papers/ecoop99_speech.pdf

> > > I agree with Koichiro:
> > > > It seems to me that Leibniz would lose nothing even if his monad is
> > > > allowed to have a tiny window through which to see the outside nearby.
> > > in the point that no local system is observable if it does not change by the
> > > perturbation associated with the observation. In so far as we consider
> > > observation of local systems, they have windows. However, being a true atom
> > > remains valid in the internal world of each local system, where no outside
> > > is considered and no disturbance is from the outside.
> > Yes, but here we are approaching the difficult issue! :-)
> I should appreciate it if you would explain the difficulties.
        Is it possible for LSs to observe each other without the perturbation?
They can not! So what is they merely are "guessing" or "simulating" each
the state other's inner world?

        This involves my idea that we should model interactions using the
notion of bisimulations between independent Local Systems. To do this we
will need some way of thinking of an LS as a computational system, e.g.
a system that is capable of "simulating" the behavior of other systems.
Perhaps the idea that one could define a surjective (?) isomorphism
(into mapping) between some subset of the configuration space of one LS
and another. (This is called an "infomorphism")
        The idea here is to show that some aspect of the evolution of one LS_i
is identical to some aspect of the evolution of another LS_j which is in
general disjoint to LS_i.
        Using the metaphor of persons making observations, we would say that
there exist a pair of observers that have some subset of their class of
observables that can be smoothly transformed into each other using a
Lorentzian transform, even though there exists another observer that
would not have such. More simply, there exist a pair of observational
agents (LSs) that can "kick a stone" that has properties that they can
agree upon, but there also exist another pair that can not agree with
either of the first pair.
        When we move from the observations of one LS to another in this
situation, we might notice that their is a change in the "meaning giving
context" that corresponds beautifully to Koichiro's notion of migrating
inconsistencies. :-)

> > It appears to me that the habituation of Western thought following a
> > materialistic and reductionistic paradigm is one root of this problem.
> I am interested in how this "materialistic and reductionistic paradigm" came
> into the western thoughts.

        Perhaps the emphasis on the replication and implementation of
industrial mechanisms... We see the concepts of "time is money" and
"consumerism" as exemplifying this idea...
> > So, what do we do in order to proceed? :-) Perhaps Leibniz offers a
> > clue:
> > (http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/)
> >
> > "...it is his view that the world consists solely of one type of
> > substance, though there are infinitely many substances of that type.
> > These substances are partless, unextended entities, some of which are
> > endowed with thought and consciousness, and others of which found the
> > phenomenality of the corporeal world. "
> > The question them becomes: What quality is it that distinguishes those
> > monads that are "endowed with thought and consciousness" and the "others
> > of which found the phenomenality of the corporeal world"?
> Given the problem this may be a solution, but I need to understand how the
> western philosophy comes to the problem of mind and matter, without
> understanding which I think we could not go further.

        This will take some effort that may be beneficial to all of us! Is the
key question: "How do remote objects, situations and events carry
information about one another without any substance moving between
This is the subject of Information Flow: The logic of distributed
systems by Jon Barwise and Jerry Seligman, Cambridge Univ Pr; ISBN:
0521583861 (July 1997)
        And guess what, it is the formalism of non-well funded sets that comes
to the rescue! :-)

        Consider the "Other Minds Problem" for example:
http://members.home.net/stephenk1/Outlaw/othermind.html; the use of
coinductive abduction could help us break free of the prison of the
argument from inductive analogy. See:
http://www.cs.brown.edu/~pw/papers/math1.ps section 1.3.
Kindest regards,


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