Re: Misclassification of packages; "libs" and "doc" sections
Eray Ozkural <erayo@cs.bilkent.edu.tr> writes:
> This analogy didn't seem to me well adjusted since painting would seem
> to involve things other than fundamental questions.
It's a perfectly well adjusted relation. Geez, this is just not that
complex.
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.
> 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.
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 two things: a collection of objects and, for
each pair of objects, a collection of morphisms (sometimes called
"arrows") from one to another.
In most concrete categories over sets, an object is some mathematical
structure (e.g., a group, vector space, or differentiable manifold)
and a morphism is a map between two objects. The morphisms are then
required to satisfy some fairly natural conditions; for instance, the
identity map between any object and itself is always a morphism, and
the composition of two morphisms (if defined) is always a morphism.
One usually requires the morphisms to preserve the mathematical
structure of the objects. So if the objects are all groups, a good
choice for a morphism would be a group homomorphism. Similarly, for
vector spaces, one would choose linear maps, and for differentiable
manifolds, one would choose differentiable maps.
In the category of topological spaces, homomorphisms are usually
continuous maps between topological spaces. However, there are also
other category structures having topological spaces as objects, but
they are not nearly as important as the "standard" category of
topological spaces and continuous maps.
> 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.
Thomas
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