ONT Re: Differential Logic & Dynamic Systems
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Howard Pattee wrote (HP):
Jon Awbrey wrote (JA):
Amendment? I am not happy with the labels on this picture and
may need to experiment with a number of different possibilities.
The problem comes from trying to preserve as many of the customary
terms and natural ways of discussing the situation as one can, with
all of their ambiguities, while working toward a technical vocabulary.
In some respects, as soon as one starts to talk and to think consistently
about anything, no matter how abstract or concrete it may be, it becomes
an "object of discussion and thought" (OODAT). In spite of that, it is
also natural to think of the initial objects that one came in the door
with as the primary objects, while discussables and conceivables that
reside higher and lower in their ontological hierarchy are regarded
as their properties and their instances, respectively. I do not
have a perfect solution yet, but do not be shocked if I shift to
speaking of three levels: "instances", "objects", "properties",
or abandon absolute positions altogether and start to speak
of object-to-property relations solely in relative terms.
| Natural Property o---------------------------->o Model Property
| ^ ^
| | |
| | |
| | |
| Natural Object o---------------------------->o Model Object
| ^ ^
| | |
| | |
| | |
| Natural Instance o---------------------------->o Model Instance
HP: I am uncertain about your unlabeled edges.
How is the "natural property" or "natural object"
related to the "model object" and "model property"?
I would like to try to keep the "filling in" or the "fleshing out" of the scheme
as flexible and as generic as I possibly can, but just by way of my own concrete
intuitions, I start out by thinking of the horizontal dimension as passing from
a "source category" of objects, properties, and their relations in some portion
of the natural world to a "target category", comprised of what I usually regard
as their "analogical models", "functorial or homomorphic images", or "icons" in
sign-theoretic terms. (Later on, I will remind myself that the source category
could just as easily be something that lives in Plato's World Wide Web of Ideas,
but let's not go there, not just yet.) All of this abstract non-sense stuff is
still just a bit more abstract than the kinds of actual organa (implementations)
down to which we would all like to get, but it's a start in the right direction.
Short answer. For now, the instances and the properties in the model world
are still what many folks call "abstract objects" of discussion and thought,
and the arrows are abstract functors or morphisms, or something of that ilk.
A major segment of the work that I am having to do in my dissertation is to build
a bridge between the relatively abstract sorts of signs (icons, indices, symbols)
that we commonly have in mind when we carry on a discussion about types of signs,
even in our most tokenly concrete moments, and the points in the state space of
a real live agent, organism, or other species of dynamic system. Ultimately,
I need to refine the abstract relational domains, O, S, I, into state spaces
or the projective components of state spaces of realistic dynamic systems.
Short answer. Let's think of the horizontal arrows as analogous to perception.
HP: I would mean by this arrow an epistemic process of
detection, pattern recognition, perception, observation,
measurement (listed roughly in an evolutionary progression).
Most generally, I would call these edges the organism's
construction of an image, using these words in a broad,
partly metaphorical sense.
Okay, so far, so good. But remind me sometime to read you the Peircean revival
of an ancient problem about the relationship of "sensibles" and "intelligibles".
The gist of his solution is that a process very akin to abduction is present in
the most elementary of sensations and perceptions.
HP: I assume by "model object" you mean some pattern in our image of the world,
or at least something in the brain of an organism that is interpreted as
referring to some localized structure in the world.
Yes, eventually it would be good to refine things this far.
I would also keep a wary eye out for non-localizable states
of the brain, the whole body, or any natural or artificial
system, in addition to local patterns "in" the brain et al.
From a sign-theoretic perspective, all of these models are
just icons, that is, signs that reference their objects by
virtue of sharing a property with them.
Now, one of the other "cross-cultural" issues that we run into here
is the different ways that logicians, especially "model-theorists",
use the word "model". For these folks, at least, once upon a time,
a "model" is anything, real or imaginary, that "satisfies" a theory.
I have made a special effort to reconcile this usage with the more
ordinary "model as analogue" way of thinking, and think that I now
understand their proper relationship fairly well.
HP: A "model property" might be some sub-pattern or aspect of
that "model object". For example, we recognize a pattern
we call a "table" as an object. We also recognize in the
table (model object) a model property we call "hard". Or,
we recognize a pattern we call a "particle" and we recognize
in a particle (model object) a model property we call "spin."
HP: Would that interpretation fit your diagram's horizontal edges?
Well, things get tricky at this point. There are good reasons
to think that we will run into correspondences that preserve
information at some level, but that will not be so parallel.
For instance, it may be more like a Galois correspondence,
an order-inverting relationship between the two lattices.
Indeed, even more complex relationships are conceivable.
So it is necessary to proceed carefully as we go.
Which We Will,
Jon Awbrey
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
JA: So let me redraw an approximate sketch of this scene in the
manner to which I've become accustomed, and that makes me more comfortable
from a hierarchical perspective, with properties placed above their objects:
JA: | Natural Property Model Property
| o---------------------------->o
| ^ ^
| | |
| | |
| | |
| o---------------------------->o
| Natural Object Model Object
JA: I take this sort of commutative diagram as nothing more or less than
a pictorial representation of what we mean when we intone the jingle:
"The picture of a quality is a quality of the picture".
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤