ONT Re: Model Theory Unplugged
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MTU. Note 3
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I will try to give a very rough sketch of what the
mathematical theory of representations is all about.
It would be best to view it under the heading of the
mathematical theory of categories, but I will tread
as lightly as I can on those grounds for the moment.
In mathematics, a "representation" is a particular kind of mapping
r : X -> Y, where X and Y are a couple of mathematical structures
and where the representation r is said to preserve some aspect
of the structure in X as it takes the elements of X over into
their "images" in Y.
Described that way, a representation does not seem to be any bit different
from the kind of mapping that we have previously and elsewhere called under
the names of an "arrow", a "morphism", a "homomorphism", a "linear mapping",
or a "structure-preserving map". Indeed, all of these diverse usages reflect
the founding idea of mathematical category theory, which could be described as
the study of mathematical analogies or mathematical metaphors.
But usually we find ourselves using the term "representation" to describe
a more specific type of structure-preserving map r : X -> Y, one in which
we view the domain X as being more "abstract" and the codomain Y as being
more "concrete" in some sense.
To give one example, the mathematical structures that one calls "groups"
are defined by a small set of axioms, and they can be studied either more
"abstractly", by way of the theorems that logically follow from the axioms,
or they can be studied more "concretely", by way of particular examples that
one "discovers" in nature or else "constructs" on an ad hoc basis. The link
between the abstract perspective and the concrete outlook is forged by way of
these kinds of mappings, that we now call "group representations" r : G -> A,
most commonly an arrow from an abstract group G to a "transformation group" A,
and this is usually something like a set of tranformations on a concrete set X.
For example, a "permutation group" on a set X is a set A of mappings f : X -> X,
each of which is one-to-one and onto, and a "permutation group representation" of
the abstract group G on the set X is a map r : G -> A that preserves the structure
of the group in a specified sense.
In the cases that interest us in the model-theoretic and sign-theoretic contexts,
the abstract realm is populated by "theories", which, taken at the tide of their
fullest generality, are just any old sets of sentences, and the question that we
keep asking ourselves, if we have any sense at all, is whether such theories are
true of anything, anything at all. One way to investigate such a question is to
hammer away at the concrete signs of the concrete sentences of the theory itself,
and to see if any sort of model of the theory can be constructed from these very
materials at hand. That, in essence and epitome, is what model theory is about.
Jon Awbrey
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