SUO: Re: Direct and Indirect Use of Axiom Systems
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JA = Jon Awbrey
MW = Matthew West
JA: After some thought, I think that it might be
good to elaborate on a few of these points,
especially to point out certain implications
of the indirect approach.
JA: There are two different ways of using axiom systems to describe an object system,
like common sense, domain knowledge, intelligent conduct, linguistic competence,
linguistic performance, and so on. It has taken three years for this recognition
to precipitate out of a suspension of nagging senses that we are talking in ways
that are radically skew to one another, but I think that I can safely describe it
now as a difference between the "direct" and the "indirect" use of axiom systems.
JA: For example, consider the case of common sense belief systems
and the performance systems that depend on them for guidance.
JA: 1. Common sense belief and behavior might be represented directly
as an axiom-based inference system, where the axioms give us
the starting points of belief and behavior and the inference
rules and proof chains provide us with literal simulations
of the processes that apply this knowledge in practice,
whether occurring "in the head" or in external action.
JA: 2. Common sense belief and behavior might be regarded as arising
from the states and transformations of an object system, and
this object system might be described by an axiomatic theory,
and yet the relationship of theory to model is more indirect,
in that the axioms and inferences occurring in the theory are
not the axioms and inferences occurring in the object system.
MW: This sounds closer to what we are attempting.
I muffed up the point a bit the first time, but think I said
it better on the second try below. But I get the impression
that many people are taking the "direct and literal" approach
for granted as being the only one that comes to mind. I don't
think this path has much chance of success with either creative
literature or full-blown scientific theories, mostly on account
of the unavoidability of indexical, metaphorical, and genuinely
symbolic modes of communication with multiple levels of meaning
or "poly-hermeneutic interpretation", if you will take the coin.
JA: As a matter of fact, the indirect approach allows us to deal with
a much wider range of object systems than just those that conduct
themselves like literal deductive systems. For instance, we might
be dealing with systems of belief and behavior that are configured
at several different levels and that are composed of many diverse
components, not all of which are consistent with each other, and
that interact in non-deterministic ways. Indeed, many of these
components may not even have all the structure of axiom systems
but may operate more like automatic associative complexes or
even very primitive reactive processes.
MW: But I think we are trying to produce one consistent system,
rather than many different views.
I think that our theories of what's going on in the object system have
to be empirical, rational, and as consistent as we can make them, but
it's kind of like abnormal psych -- one wants a theory that explains
the phenomenon, but the phenomenon itself may persist in appearing
inconsistent and irrational in all terms that make sense to us.
Closer to home, if we need a theory that describes and explains
the persistence existence of many different views of the world,
then we may hope to settle on one satisfactory theory about it,
but the objective phenomenon is the simple fact of diversity,
distribution of opinion, entropy, indeterminacy, uncertainty,
or whatever privative condition we have in view at the time.
JA: For instance, systems of heuristics often have this structure, with
one sage maxim contradicting the next deep truth, it all depending
on practical judgment, what the Greeks called "phronesis", I think,
to decide which rule of thumb applies in which immediate instance.
JA: In light of these possibilities, it might be better
to revise my description of the indirect approach
in something like the following way:
JA: 2. Common sense belief and behavior might be regarded as arising
from the states and transformations of an object system, and
this object system might be described by an axiomatic theory,
and yet the relationship of theory to model is more indirect,
in that the axioms and inferences occurring in the theory are
not literal representations of the initial conditions and the
transitional processes that take place in the object system.
MW: OK.
JA: I think that it would help to clarify many of our discussions
if we able to reflect on which implicit model we were using
in each case, and to make a point of making that explicit.
JA: Of course, it is also perfectly conceivable that we will be
using a mixture of both direct and indirect modalities in
our models and theories.
MW: I think what we are attempting is a single consistent indirect model --
from your description above.
I hope so, or I'd hope that we'd eventually get around to that,
as I think it's the only chance we have with complex phenomena,
but I'm not so sure that we'll get to that penultimate resort
until the slots in the back of the our medicine cabinets are
filled to overflowing with rusty occam's razor blades.
Jon Awbrey
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