[Inquiry] Re: Prospects for Inquiry Driven Systems

[Jon Awbrey]

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PRO.  Note 25

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1.2.1.  Functions of Observation (concl.)

At this point the notion of observation put forward above would appear
to be identical to the notion of representation that is usual in AI and
cognitive science.  But mathematicians and physicists reserve the status
of representation to maps that are homomorphisms, in which some measure of
structure preservation is present.  And if these two notions are confounded,
what sort of observation would enable the detection of whether maps preserve
structure or not?  Therefore, it seems necessary to preserve a more general
notion of observation, one that permits arbitrary transformations, not just
the linear mappings or morphisms that properly constitute representations.

It has been appreciated in mathematics and physics for at least a century
that an isomorphism is almost totally useless for the purposes that motivate
representation, and further that a single representation is hardly ever enough.
Representations are analogous to coordinate projections or spectral coefficients
in a fourier analysis.  It is a necessary part of their function to severely reduce
the data, and this engenders the complementary necessity of having a complete set of
projections in order to be capable of resynthesizing the data to the extent possible.

The extent to which a representation found embodied in a system is an
isomorphic representation of its object system is the extent to which
that information has not really been processed yet.  Only a piecemeal
reductive, jointly analytic form of representation can supply grist for
the mill that applies rational knowledge to making incisive judgments
about action.  To object that the reality itself does not exist in the
analyzed form created by a system of representation is like objecting
to changing the form of bread in the process of digesting it.  It is
only necessary to remember that representations are supposed to be
different from the realities they address, and that the nature of
one need not existentially conflict with the nature of the other.

In exception to the general rule, there are beginning to appear a few instances
of work in AI and cognitive science that have reached the verge of applying the
homomorphic idea of representation, although the sense of the functorial arrows
may be reversed in some of these presentations.  Notable in this connection are
the concept of "structure-mapping" that is applied in (Gentner & Gentner, 1983)
and (Prieditis, 1988), and the notion of "quasi-morphism" that is introduced in
(Holland, Holyoak, Nisbett, & Thagard, 1986).  One of the software engineering
challenges that is implicit in this work is to provide the kind of standardized
category-theoretic computational support that is needed to routinely set up and
to test whole parametric families of such models.  An off-the-shelf facility
for categorial computing would of course have many other uses in everything
from theoretical mathematics to specialized engineering applications.

Jon Awbrey

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