SUO: Re: IFF Comments Requested
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JA = Jon Awbrey
JF = Jim Farrugia
PH = Pat Hayes
JF: Please submit your comments by October 18, 2001, replying
to this subject line ("IFF Comments Requested"), so that
we can easily gather all comments. (At some point later,
we may suggest other subject lines to group together
related comments.
PH: OK, I have a few.
PH: First, I fail to see the utility of the emphasis placed on category theory.
This is not motivated anywhere, but it badly needs to be motivated if you
expect anyone to take it seriously enough to even read the sources to
find out what you are talking about.
JA: I think that the following is a fair statement:
A modest amount of category theory, along with
a modest amount of set theory, is indispensable
to understanding what mathematics is about and how
mathematics is done today. This is important, not
just for representing the ontology of mathematical
objects, structures, and systems, but further, and
more importantly for applications, because these
objects, structures, and systems are used in
modeling most other objects, processes, and
situations of any complexity that anyone
might happen to care about.
PH: Well, maybe. I would still like to see a bit more detail, however.
Be careful what you (pretend to?) wish for.
PH: This response really amounts to saying that Category Theory is a Good Thing.
I am saying that a modest amount of category theory is indispensable
to several of our objectives here, which, though qualified, is still
a slightly stronger statement.
PH: In fact, however, I think that its influence has almost entirely been
in pure mathematics (where indeed it is part of the general competence
expected of a professional mathematician these days) but hardly at all
outside pure mathematics (and even within large parts of mathematics,
it really amounts to little more than a style of terminological usage.)
Yes, there are people who do category theory for its own sake, just as there are
people who do descriptive set theory for its own sake, but the word "modest" was
meant to set aside those further reaches of both subjects. I'm talking about the
part of category theory that is a standard working tool in almost every branch of
math that I can remember taking. It ain't all that much, but it is "indispensable".
PH: Most ontological modelling is not mathematical modelling,
and category theory plays virtually no significant role
in mathematical modelling in any case. Fractal theory
would be far more germane, for example.
By "mathematical model" I meant not just the sort of elaborate model that
usually comes to mind, but the sort of thing that we are doing as soon as
we use "abstract objects" like numbers and sets as "intermediary fictions",
if you prefer to think of them that way, to describe a world that is made
of more concrete things like numbers of apples, sets of oranges, and even
on upward to the hydrodynamic flow of oil through a pipeline from Seville.
In my bemused observations of the discussions of elementary set theory
that have belabored this group for more than a year now, I sense that
a due appreciation of difference between the territory being modeled
and the mathematical objects forming the model, even in such a basic
form of modeling activity as counting and "setting" them, might just
be, well, indispensable to our progress here.
PH: Ontology is supposed to be concerned with what there is, and the best theories
we have about what there actually is are general relativity and quantum theory.
One could make out exactly this kind of case for an approach to ontology based
on those disciplines, and it would be just as poor a case.
Actually, the problems that had to be solved in formulating these two theories,
which depended to a large extent on the mathematics that was produced somewhat
earlier by Galois and Riemann and a host of others, is very analogous to the
types of problems that we need to solve here, in a more logical, linguistic,
or qualitative vein, in order to carry home the Grail of intercomm-&-ops.
In physics, the problem had to do with how the "piles of numbers" (PON's)
that one observer gathers by means of his or her measuring apparatus and
relative to his or her "frame of reference" (FOR) can be matched up with
the PON's that another observer gathers in such a way that they both can
recognize just how much of an invariant, common, real, objective cosmos
they might or might not be cudgeling their PON's about. From a proper
aesthetic distance, we have pretty much the same kind of problem here,
only with "heaps of words" (HOW's) instead of numbers.
Jon Awbrey
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