ONT Re: Qualitative Understanding Of Complex Systems
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
QUOCS. Discussion Note 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Matthew,
I made this attempt to summarize the remainder
of our previous series of dialogues on systems:
JA: When I think about these things in systems theory terms,
it always starts with some state space X that contains
all of the logically possible states of the system you
want to think about. Your first stab at dealing with
process will then be in terms of a "path" p : R -> X,
where R is a real line that you think of as the time
domain. Physical laws and practical constraints are
then expressed as subsets of X and subsets of the
set {p : R -> X} of all possible paths through X.
MW: The states you describe are essentially complex properties of a system.
JA: This is where I begin to have trouble making translations
into a language that I understand. Let me think about a
state x in a state space X -- some people will be fussy
here and want to say "configuration space", but because
I start with non-determinstic systems from the outset
I don't need to make make that distinction.
MW: So far so good.
JA: One thing that we are talking about when we say "the system"
is a thing that can be in various states, and a thing that
can pass through a line or a sequence of states.
MW: Agreed.
JA: In that case, we can say that the state is a complex property of this
system-thing. We say that the system is in the state x in the set X,
and there are projection maps p_j : X -> R, just taking the reals R
as a typical example, and that we think of as "measuring" x in X
with regard to the property that corresponds to p_j, and giving
us a real number in R as a result of the measurement.
MW: I think I'm still with you.
JA: I think that I now have a way of understanding the statement
that a state is just a complex property of a system. Some
people will call that system-thing the "reified system".
MW: It is that which takes on the state.
JA: Yes, the picky point that I think some people are trying to make
is that the data of experience, or the results of measurement,
are what we really have on hand, whereas the system-thing is
in the bush, as it were, at some remove from our immediate
impressions, and thus it is the object of possibly many
competing hypotheses that we form to explain why the
data are as they are.
JA: They will say that it is nothing but a figment of our
hypothetical imaginations, a "hypostatic abstraction"
that personifies the measurements that we observers
make, that enter into our experience, and that are
the only real things that we ever know about this
system, but I don't see the harm of positing such
objects, so long as we stay conscious of what we
are doing.
MW: I'm not one of them, as you have probably guessed.
As noted below you can have any individual object
you want as long as you can demonstrate it has
a spatio-temporal extent.
JA: Yes, I have no qualms about pretending a hypothesis here and there,
but the question is whether the specifics of a given hypothesis
explain the data better than the many competing alternatives.
This is a stricter test than demonstrating mere consistency
or the projection of a possible extension in space and time.
JA: Morever, data can vary widely with changes among different bases
or frames of reference, whereas real objects are associated with
functions that remain invariant through transformations from one
frame to another, so it is a non-trivial step to relate the data
to the object.
MW: One of the things I like about 4-dimensionalism is that
it is very unfussy about what objects there are. If it
is some extent in space-time, you can have it. Why you
are interested is up to you. So we are allowed systems.
JA: Seems eminently reasonable to me.
MW: The time chunk of the system that possesses the state is the relevant
possible individual, that is itself a temporal part of the whole life
of the system.
JA: Almost. Let me try. Instead of a reified-system-at-an-instant,
the thing that is in the state x in X, you want to talk about
a "time chunk" as the thing that has the properties p_j that
afford a basis for defining the state x in X, along with all
of the other properties that the system has at that moment?
MW: Sounds good. In the limit it can be reified-system-at-an-instant.
MW: You can of course decompose the system and the state.
Your "of course", of course,
we will be having, perforce,
a recourse to in due course.
JA: Just as a subset of X corresponds to an indicator function q : X -> B,
where B is the boolean domain -- here, q is the sort of thing that is
convenient to think of as a "proposition about X" -- constraints on
paths p : R -> X can often be expressed through the intermediary
of other functions that one builds up from X, R, and B.
JA: Now, my question is: Can I think of a possible_individual as being
associated with such a path p : R -> X through such a state space X?
MW: I'm not quite sure if you are talking about a continuous changing
function here or not. We can do that to. We talk about a spatio-
temporal extent having a property distribution, which can perhaps be
described by a function.
JA: In practice, both empirical and theoretical practice,
we commonly approximate continuous trajectories with
discrete sequences of states, and computational work
pretty much forces us to start with discrete and even
finite cases, so it's best to have concepts that cover
both indifferently, at least, until push comes to shove.
MW: Indeed. I think we have this covered.
JA: Remark. In the beginning, X is really the state space of the whole system.
It can be a tricky question to say whether X decomposes into some form of
composite, but let's start with the easy case, where X = X_1 x ... x X_k,
a cartesian product of component state spaces X_j for j = 1 to k, and X_j
is the state space of the j^th component subsystem. Then let us suppose
that we are talking about one of these ostensibly independent components,
and read what I asked above with X_j in for X.
JA: Rephrasing the question: Can I think of a possible_individual as being
associated with such a path p : R -> X_j through such a state space X_j?
MW: Not sure if I understand the difference here. It looks like
X is a state space, and X_j is a state space, and a possible
individual could be seen as taking a path through it, i.e.,
temporal parts of the possible individual would take up
states in the state space.
JA: Will get back to this later in more detail.
The short schrifft is that the structure
of the most fitting state space for the
phenomenon or problem has to be teased
out of the data in discovery fashion
and not just imposed ad hoc. But
not every possible state space
factors into a direct product
of component spaces quite
so facilely. There are
things that are called
"semi-direct products"
that come up a lot,
just for example.
JA: In physics this often comes up under the heading of "many-body systems",
say, 3-body systems, and whether their dynamics factors into separable
pieces where you can just think about independent 2-body interactions,
working in pairwise fashion toward the whole, or not. Sound familiar?
JA: As often as not --
you cannot.
MW: You seem to have some idea that there are some "real" objects.
What do you consider these to be?
JA: I am working on the hypothesis that there is a reality.
I have not always been so realistic, but reality is
a persistent, if not always patient teacher, and
a sequence of recurring impressions of the type
that is commonly referred to as brute reactions,
the "dent" or "dint" of recalcitrant experience,
or just plain "hard knocks", have willme, nillme,
conduced me to adopt the hypothesis that there is
some kind of objective reality that produces these
impressions on my sphere of pathic, felt experience,
and which Reality or Nature ought to be my objective
to know better, if I have a clue what is good for me,
and so I am following out the consequences, practical
and theoretical, of that hypothesis, at least, until
some more viable alternative commands my attention.
JA: After that, pretty much everything is up for grabs:
where to draw the boundaries reciprocally defining
the terms of "self" and "other", or more remotely,
"organism" and "environment", whether the spaces
of the boundary, the exterior, and the interior
split into many "objects" in the conventional
acceptance of the word, all that stuff I've
wondered about at different times, doing
my best to suspect the usual suspects,
and also to grill a few novel perps,
but things muddle up very quickly
after such clear beginnings,
as everyone is well aware.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o