Re: Misclassification of packages; "libs" and "doc" sections
"Thomas Bushnell, BSG" wrote:
>
> Mathematicians needed something that was a vague synonym for "set",
> and picked "category". They had already used both "set" and "class"
> for different things. Just like "group", "ring", "field" and on and
> on, there is no real significance, other than vague allusion, to the
> word they chose.
Well, no. Category Theory has radical differences from the rest of
Algebra. As you may know Category Theory is used in various ways,
one of which is meta-mathematics, connecting theories of seemingly
distinct subjects into a uniform framework. Thus, at least
for a philosopher of Mathematics, it has serious implications about
the fundamental questions. [Considering it as a replacement for
Set Theory for fundamentals] As I said before, the theory seems to
me to be the single plausible formalization of the notion of "Category",
which would would be appreciated in the tradition of analytic philosophy.
You don't have to resort to metaphysics while talking about Category
Theory :)
> > The most obvious relation is from the Philosophy of Mind. Hold the
> > typical analytical view that thinking is essentially computational,
> > and posit that there exist categories in your mind, and that categorical
> > objects exhaust the definition of "reference". Not very different
> > from what formal semantics community has been doing with logic, still
> > the philosophy of language is intimately interested in "Predicate"
> > as it pertains to philosophical theories of meaning. I simply think
> > we have a difference in tradition.
>
> This is all perfectly rational. It's all about the philosophy of
> categories. Formal semantics, logic, reference, and categories are
> certainly all connected.
>
I'm asking you to assume that the Category Theoretical "Categories"
and of course objects and arrows of those categories do exist in
your mind. Approaches of this vein would connect Category Theory with those
fields of thought. [That is, you can use just Category Theory to
reason about these matters. Nothing more, only a pure analytical
reasoning which defines "Category" in terms of Category Theory only]
I hope I've been able to communicate my thought this time.
> But, of course, this has nothing to do with the mathematical topic of
> Category Theory. A category is a particular kind of mathematical
> structure. Since apparently you didn't know that, I quote from Eric
> Weisstein's World of Mathematics, an excellent resource for this kind
> of thing:
>
> A category consists of ....
I'm aware of the mathematical theory itself [1], I don't like to talk
about a formal theory without having studied it first. Of course
don't forget that "Category has been defined in order to define
functor, and functor has been defined in order to define natural
transformation" as S. Mac Lane has said. I suggest you to read a more
thorough exposition of Category Theory for it is the most abstract
theory ever devised by mathematicians and is of general interest
to philosophers. Since you also happen to be involved with programming,
perhaps you'd like to have a look at Barr's excellent "Category
Theory for Computer Science"; which you can find at any science library. [2]
> > On the other hand, it would not be correct to talk about the philosophy
> > of "Categories" without mentioning Aristotelian or Kantian views of
> > the matter. I'd read that Husserl had a general theory of Categories
> > but I haven't had the chance to study it.
>
> Right. But Category Theory, a part of mathematics, has nothing at all
> to do with anything like the philosophy of categories, which is what I
> started out saying, and what I still say.
Okay, we're doing the most offtopic discussion on this list ever,
but I anticipate that I reserve the right to continue ;)
The developments in the last 20 years would make Category Theory
related to semantics, cognitive science and as result of this Philosophy
of Mind [and Philosophy of Mathematics which it has been related
to all the time].
Categorial Grammar and its variations which include formal semantics
theories are in fact based on the bi-closed monoidal Categories
which is a subject of Category Theory. Anyone interested in the
philosophical significance of Categorial Grammar has to regard
philosophical status of Category Theory, too.
Categorical Logic is also a recent development which has found
application in a theory of denotation. Categories and functors
as in Category Theory have been used in a philosophical work
on universals. [So perhaps those works are vacuous if what
you claim are true]
Category Theory has been used to formalize and extend possible
world semantics. For instance, we use that semantics for programming
languages. Because of its nice properties, it'd be possible
to adopt this more recent formalism instead of logic languages.
You may still assert that Category Theory has nothing to with
the philosophy of Category, but that would imply that logic
has nothing to do with semantics, and so on. Which is a position
that I could not take as I've studied philosophy following the
analytic tradition. It's a rather personal choice I guess.
Thanks for this nice discussion,
[1] I've been involved with Computer Science and Linguistics for far too
long to avoid it.
[2] I'm writing the title from memory, but it should be right. Just the
first 2-3 chapters are enough.
--
Eray (exa) Ozkural
Comp. Sci. Dept., Bilkent University, Ankara
e-mail: erayo@cs.bilkent.edu.tr
www: http://www.cs.bilkent.edu.tr/~erayo
Reply to: