Subject: categories: Question

From: [email-masked] (Michael J. Healy 425-865-3123)
Date: Tue, 16 Jan 2001 16:17:19 -0800 (PST)

I'd like to ask category theorists how they would answer the attached message 
from a colleague here.  Both he and the person with whom he is corresponding 
are experts in the areas of knowledge representation within computer science 
(ontologies and the like).  I thought it best to hide their identities since 
I haven't asked permission to use them.  If you are interested, please respond 
to me privately if you would.

Thank you,
Mike Healy
------------------------------------------------------------------------------

Message I received:----

I would be delighted if there was no semantic conflict between
category theory and set theory.  I kind of flagged this as a
potential issue, but did not look  into it in detail, as it was
not my main concern at the time.  However, I remain unconvinced.
There has been some discussion of using set theory as the basis
for a semantics for SUOKIF. If this is true, then I think it may
be limiting to a CT based language. While it may be true that sets
are common example of a catagory, my sense is that CT is much more
powerful, and would be LIMITED if everything was forced into the
single catagory of sets.

Im a bit out of my element here, however, and need to defer to the
formal expertise of others on this issue.

Message to which the above was replying:---

I agree that category theory is very powerful and could be
an important basis for combining and sharing ontologies.
But I disagree with the following point:

>I think this idea has tremendous potential.  One problem is that the underlying

>formal semantics of category theory is NOT set theory (which is what KIF uses),

>furthermore, I think they may well be incompatible.

First-order logic (including any and all notations for it,
such as KIF, CGs, predicate calculus, existential graphs, etc.)
is completely neutral with respect to set theory or category
theory.  The version 3.0 of KIF did include a version of set
theory, but that was removed in the KIF'99 version because it
belongs to ontology rather than logic.

And for that matter, there is no reason why you can't use both
category theory and set theory together.  In fact, one of the
most common examples of a category is the category of sets.

Perhaps there may be incompatibilities between the methodology
associated with Ontolingua and category-based techiques, but
Ontolingua is not KIF.  Ontolingua simply uses KIF.
--

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---

From: "Joseph R. Kiniry" <[email-masked]>
Date: Tue, 16 Jan 2001 20:29:56 -0800

Hello Michael,

I believe that category theory is an excellent foundation for ontology 
representation and manipulation -- I use it myself.

However, choosing this foundation comes at a price.  Unless significant 
work is done to hide this unfamiliar foundation, many users (of the theory, 
the system, the language, &c) will be biased against the work from 
minute-one.  This has as much to do with the unfamiliarity of CT as it does 
with certain unfortunate negative biases regularly expressed by many 
mathematicians and computer scientists - biases that, IMHO, are founded in 
ignorance and not reason.

Many computer scientists, mathematician, and users of knowledge 
representation systems are quite familiar (at least in use, but probably 
not in foundations or related complications) and comfortable with set 
theory.  This familiarity is an incentive rather than an obstacle to using 
related work.

Personally, I chose not to pursue a set theoretical foundation because of 
theoretical and representational complexity issues (e.g. witness the use of 
a set theoretical foundation for the Z specification language) as well as 
the unfortunate binding to a particular formalism that isn't necessarily 
congruent with others that I work in and apply my work to (e.g. type theory 
and programming languages).  To rephrase, I find using CT to be more clear 
and tractable than set theory and I feel that my work can, as a result, say 
and do more than it could if it had a set theory (plus some extra 
formalisms) bases.

Note that I also chose not to build my work (solely) on type theory and 
order sorted algebras for the same reason, though my work has elements of 
both of those fields as well.

The comments about Ontolingua and KIF are on-target in my experience.  I 
see no obstructions to the representation of CKML (the variant related to 
KIF and RDF that I happen to know well) with my work.

Finally, I should point out that I am but an infant in CT - I'm much more 
comfortable with TT, OSA, PL, and others.  I've only been learning and 
using CT for a few months and, while there have been some objection to my 
choice, I feel that a dissertation founded in these three major fields (CT, 
TT, and OSA) has significantly broader application and, implicitly, more to 
say about its author. <grin>

Best,
Joe Kiniry
-- 
Joseph R. Kiniry                   http://www.cs.caltech.edu/~kiniry/
California Institute of Technology       ID 78860581      ICQ 4344804

--On Tuesday, January 16, 2001 04:17:19 PM -0800 "Michael J. Healy 
425-865-3123" <[email-masked]> wrote:

>
> I'd like to ask category theorists how they would answer the attached
> message  from a colleague here.  Both he and the person with whom he is
> corresponding  are experts in the areas of knowledge representation
> within computer science  (ontologies and the like).  I thought it best to
> hide their identities since  I haven't asked permission to use them.  If
> you are interested, please respond  to me privately if you would.
>
> Thank you,
> Mike Healy
> Michael J. Healy

---

From: Dusko Pavlovic <[email-masked]>
Date: Mon, 22 Jan 2001 21:55:00 -0800

it's interesting that almost no one took michael healy's bait (attached).
a couple of years ago, a similar question started a long battle between a
group of categorists from this list, and a group of set theorists on
another list. traces of that battle can still be found scattered on the
web.

i guess people got a bit tired. after all, for a working mathematician,
foundations are a bit like esperanto: ok, all math can be translated to
set theory, or to category theory, or to untyped lambda calculus --- so
what? do foundations help me calculate an integral? does anyone use von
neumann representation of numbers in arithmetic? no.

but guys, note that the question this time comes from cs.boeing.com! if
people at boeing think about categories, and sets, and foundations, then
this probably makes a difference for them, and helps them compute
something. might this not be worth your attention?

so let me say what i think. i think foundations mean different things for
different people.

for logicians, foundations are metamathematics: analyzing consistency,
independance etc of logical theories. this is what set theory was found to
be good for.

category theory, on the other hand, is not as handy for proving new
independence results, but it tells you to look for adjunctions everywhere,
or monads, or <the keyword from your last paper>. it is not
metamathematics, but perhaps *structuralist maths*: it displays abstract
structures... (i kno, this is getting to *just* the kind of philosophy you
were hoping to avoid by the synchronized silence. so let me make the
point.)

for software engineers, foundations are the link with the meaning of their
programs. having a slightly shorter history than math, they do not have
languages as natural as arithmetic, or calculus, but have to chose between
KIF and Ontolingua, and the various other versions of esperanto every day.
categories dam the flood of structure in software engineering, just like
they originally did in homology theory almost 60 years ago. some good math
may come out of it if taken from a good side.

-- dusko