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.
No pretence, I assure you. Notice that I am not asking for a tutorial
on category theory (not do I propose to give anyone else such a
tutorial, though I will hold people to their promises on foundational
issues), but a reasoned account (if only brief) of what its perceived
relevance is to the overall topic/project of the SUO. Just being
nifty mathematics that all good men should know isn't good enough
(and one example of one person's work isn't good enough, either, Jon,
just to save you the cutting and pasting.)
>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.
Well, again, I'd like to see a case made. The case may be there, but
it isn't obvious; it needs to be actually stated. A little brisk
canter through the foothills of Birkhoff and McLain might be good
mental exercise, but I need a lot more convincing to base my entire
ontological metatheory on foundational distinctions which are only
meaningful in circles that are even more restricted than the FOM
mailing list.
>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".
It hardly involves getting all concerned with distinctions like
proper class versus set, however. These are distinctions in the
foundations of topos theory, not the usual fodder of the working
mathematician.
>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.
I would agree that particular 'due appreciation' is rather important,
and that it isn't found widespread in nature. But that has nothing
specially to do with topos theory. If anything, topos theory is
rather cavalier about the the map/territory distinction, like most of
mathematics, since it is almost entirely concerned with the maps (in
the case of topos theory, literally so); and one certainly does not
*need* topos theory to understand these distinctions or to state them
properly. In fact, I find the topos METAtheory to be quite opaque,
c.f. my comments on the interpretational issues arising from
first-order axiomatizations of topos theory. I certainly can see no
good argument for adopting such strongly-worded slogans as the
'categorial principle', or for assuming that our first business
should be to give a foundational ontology for categories.
Pat Hayes
PS. BTW, perhaps it would aid communication if I said why I tend to
be rather cynical about basing ontology activity in topos theory, or
indeed almost any other piece of 'neat' mathematics. This cynicism
comes from many years trying to apply elegant mathematics to
real-world ontologizing, and finding again and again that the
structures one wants most to capture are precisely the ones that do
not fit well into the mathematical theories. Reasoning with notions
of approximation would be easy if approximations were equivalence
relations, or metric open spheres or closed algebras of one kind or
another; but they aren't, which is where the ontology gets
interesting. Real-world 'fractal' spaces arent quite genuinely
fractal: they typically are only in some ad-hoc ranges of scale.
Ontological spatial reasoning isnt supported well by conventional
topologies (which are too 'rubbery') or by conventional metric-space
geometry (which is too 'stiff'); and so on. Topos theory is of such
wonderful utility in mathematics precisely because it is a general
framework for all kinds of otherwise disparate mathematical
structures (and so it greatly simplifies and rationalizes the process
of proving theorems about them), but these structural families all
have the characteristic closure properties that realistic spaces, of
the kind that physical ontologies must often try to tackle, usually
conspicuously lack. Mathematics studies the universals; ontology is
often trying to do a better job on a smaller territory.
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