ONT Re: Category Theory
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CAT. Discussion Note 7
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One way to get the motivation for category theory
is to look at some of the types of problems that
it was developed to solve. I will try to give
just my personal intuitions about the kinds of
settings where it becomes indispensable, and
why I feel like these are closely analogous
to the very sorts of problems that face us
in designing conceptual systems that are
capable of supporting communication and
co-operation among different views of
common realities, whether these are
embodied in people or in software.
I think that it all starts with the gap between
realities and representations, in other ways of
stating it, between terrains and maps, or maybe
the constrast between the role of an object and
the role of a sign. The big problem is that we
tend to think that the objective reality is one,
at least until there is good evidence otherwise,
whereas the big headache about the appearances,
datasets, maps, representations, sign systems,
or variant views of the terrain is that there
is just so darn many of them.
This is a problem that went critical quite some time
ago in mathematics, shortly after Descartes invented
analytic geometry, because instead of thinking about
geometric figures as unitary objects like most folks
intuit them to be in the synthesis of the mind's eye,
all of a sudden there is an embarrassing richness of
different coordinate systems, reference frames, and
points of view, assigning different coordinates to
objects, and all of which differing accounts have
to be reconciled among themselves if we want to
reconstruct the unity of the original figure.
So the first picture I get of the subject looks like this:
reality
?
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representations
The objective reality in question, marked by a question mark "?",
is not a total unknown to us, but it is known to us only in terms
of many different representations. These are typically expressed
in different coordinate systems, amounting to or being analogous
to data that's gathered from different points of view on things.
So the problem becomes a lot like that of stereoscopic vision,
to recover a more solid sense of the object from the mosaic
of different facets of data that is canvassed by a welter
of diverse reference frames. Reconstructing the object
depends on finding the proper correspondences between
the elements of the many splintered representations,
which involves us in contemplating the families of
transformations T that exist between each pair of
perspectives.
Anyway, that is how I see it getting started.
Jon Awbrey
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