On Tue, Jan 27, 2009 at 3:48 AM, Matthew West <dr.matthew.west@xxxxxxxxx> wrote: (01)
> We seem to be more or less on the same page. (02)
OK. I think I was following the trend of other threads here of being
over exact. So I just wanted to clarify for others in case they get a
misdirected idea from correct but unexplained statements. (03)
>> > > I was unaware that Category Theory was distinct from logic.
>>
>> It's all math. Category theory is usually considered coming out of the
>> subdomain of algebra (though its primary examples are in topology;
>> logic (mathematical logic that is) is usually considered a quite
>> separate subdomain of mathematics. That said, you can make meaningful
>> connections between any two subdomains (e.g. Categorical logic).
>>
>> > Category Theory is an alternative foundation. (04)
Ah...my apologies, an unfinished statement on my part. Category theory
is a foundation in terms of explaining many mathematical concepts by
similarities with other mathematical objects, with great heights of
abstraction. Logic is a foundation in terms of explaining how
mathematical knowledge is managed (how things are proven, how things
are inferred). When I say 'is', I mean 'from my limited perspective'. (05)
>> Yes, but that's at a different idea of foundation. Logic (and set
>> theory is considered a foundation of mathematics with respect to
>> truth. Category theory provides a foundation
>>
>> > You can for example describe logic in terms of category theory.
>>
>> And you can describe category theory in terms of logic.
>
> [MW] Indeed, that is what I meant by being an alternative foundation. (06)
There's a lot of controversy here but mostly on the part of category
theorists. There has been much discussion over the years within the
category theory community claiming foundational status (but little
acceptance of that from the logicians). More recent discussion can be
found in the FOM (Foundations of Mathematics) mailing list (search on
'category theory'). But that is mostly from the point of view of
logicians, not category theorists. (07)
>> I think category theory is a red herring here (as well are probably
>> most of my comments). Category theory is a fascinating unifying theory
>> of mathematics, and is also quite useful in its applications to
>> programming language semantics, I would think that any benefit that
>> you could get out of it would be way beyond the learning curve. And
>> that benefit could be gotten with, well, simplifying to semantic
>> networks, though I'm sure there are some nice tricks that category
>> theory provides that logic (logical theories?) can use.
>
> [MW] Actually I think Category Theory may provide some useful ways to do
> inferencing in a more structural way, but that is really just a hunch (or
> conjecture if you want to be a bit more formal). (08)
To be pedantic, category theory does not provide mechanisms of
inference (taken as technical term). One might be able to pick up some
ideas that will be useful (higher order relations are like functors).
But then just by the suggestion, I can already draw a chain of loose
analogies: category theory ~~ abstract data types (in computer
programming theory ~~ data descriptions and ER diagrams ~~ ontologies.
But that is fairly loose (PhD topic anyone?) (09)
(partial disclaimer: my training is in computer science specializing
in algorithms, complexity, combinatorics, and logic) (010)

Mitch Harris (011)
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