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.
JA: Be careful what you (pretend to?) wish for.
PH: 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.
I think that I already said this in my initial statement,
and I limited myself to facts that are fairly obvious to
anyone who looks around at what tools are actually being
used, where and for what. I am trying to get out of the
habit of repeating the obvious to an unneccesary degree.
PH: 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.)
This is hardly a matter of a few people's eccentricities.
PH: This response really amounts to saying that Category Theory is a Good Thing.
JA: 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: 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 McLane 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.
What you say here does not give an accurate picture of the situation in the field.
Category theory is a basic tool in widespread use, for basic algebra and analysis,
discrete math and computer science, automata and formal languages, modeling and
simulation, yea, verily, unto all the sinful vales of impure math and engineering.
In many applications it is preferred to set theory for its utilities in getting
the subject off the ground far more quickly than would otherwise be possible.
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.)
JA: 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's
"indispensable".
PH: 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.
I tend to agree with this. It has always been my sense that this distinction
is more of a non-small deal to the set theorist, though, and that the working
class of category-theory-users bother with it only so that set theorists will
continue taliking to them. Just my sense of the sociodynamics -- I am not up
to talking topoi/toposes just yet. For my part, I use the distinction mostly
to mark off the more overweening stuff that I do not care all that much about.
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.
JA: 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: 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.
I did not address the topic of topos theory in my initial remark.
Some people include it in their modest set, some do not. Given
the number of times in my life that I have had to readjust my
sense of modesty, I think that I will bow out on this for now.
But a modicum of category theory has been found by very many people
to be of utility in organizing their studies of modeling relations --
this is only natural when you consider the fact that a morphism is
just a mathematical metaphor. Of course, people are free to avoid
the use of any tool that they are dead-set against -- the meaning
of the word "indispensable" is not designed to cover outliers like
that -- but if the tool is really "necessary" in a practical sense
the demands of the problem domain itself will usually require them
to re-create it , however obscurely transmongrelfied.
My own crisis of foundations is decades behind me now -- from the pragmatic
perspective that I take today I tend to think of the whole thing as having
been yet another undergraduate hangover of post-cartesian dis-illusionment --
and now there is just too much of a practical nature that can be done and
demands to be done while yet I have the time to do it. I am not all that
concerned with telling people where they should live, foundation-wise, or
even how they should talk -- human nature is just plain against it -- but
if I can string a few lines of communication between diverse perspectives,
I think that it'd be worth the trouble to do so, and non-obtrusive enough
for almost anybody -- though not for the Shakers, bless their souls.
PH: 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.
I think that different people have different ideas about the scope of "neat" mathematics.
Most of the areas that I consider the "neatest" were not on your list. Closure properties
are a convenience appropriate to maps -- how else would you fold them? -- and were never
meant to be taken as properties of the landscape, nor indeed of mathematics itself.
Jon Awbrey
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