ONT Re: Model Theory Unplugged
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MTU. Note 5
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I am taking the view that model theory, in practice, is really a body
of techniques for generating concrete examples of abstract concepts,
where the goal and the good of doing so is to clarify the concepts
in question. This quite clearly has to do with the "denotative"
or the "referent" dimension of sign relations, in other words,
with the 2-adic relation between objects and signs. If it
appears, on the contrary, that formal semantics has more
to do with the "connotative" or the "interpretive" face
of semiosis, in other words, with the 2-adic relation
between signs and interpretants, with the "Sinn" sans
the "Bedeutung", as it were, and in particular, with
that radically inscrutable business of translating a
less familiar (object) language into a more familiar
(mater) language, then I would suggest that this can
be explained by the fact that we thereby transit from
a language whose denotations and connotations are more
uncertain to us, we who find ourselves charged to play
the parts of interpreters, to a language whose referents
and interpretants are less uncertain to us, the aforesaid
interpreters, thereby amplifying the effective information
of the source text in question, and thereby achieving a gain
of informative value in the process of taking it to the target.
That'll have to stand as my best guess what's going on at present.
I shall continue with the view that model theory is a theory of models,
that is as much to say a theory of the "modeling relation" that exists
between models and theories, that is to say a little more, between the
things that a theory may be true of, if at all, and the theory itself.
Mathematicians, scientists, and other backward folks, commonly take up
this relation, at least "ab initio" or "da capo", in backward fashion,
from the models that they meet up with before they know what theories
they are models of, and only later coming to the theories themselves,
that may in time become stable enough platforms of axiomatic plancks
that they become able to "reverse-engineer" the models thereof, with
all of the Icharian dangers, if I may write it with-winged-words, of
the human tendency toward the hubris of thinking that craftily enough
wrought axioms literally dictate a lot that's woe-worth-while to Nature.
With all of these regards and cautions in mind, I think that it would
be useful to take up a simpler example of mathematical representation,
namely, the theory of group representations, or at least a few simple
examples of it. As it happens, the study of group representations is
one of the rare stations of the intellect where something akin to the
pragmatic maxim is applied on a work-a-day basis to clarify abstract
concepts by generating concrete examples of the concepts under study.
I think that philosophers who extol the benefits of this maxim could
learn a lot about how to put it into practice -- and as I could show
them the places where Peirce did before them -- by taking the theory
of group representations as a model for their practice of philosophy.
A longer pre-ramble than I had planned,
so I'll get to the nub of it next time.
Jon Awbrey
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