Re: Question

[Dusko Pavlovic <dusko@kestrel.edu>]

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



"Michael J. Healy 425-865-3123" 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
> ------------------------------------------------------------------------------
>
> 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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