Re: Categories ridiculously abstract

["DR Mawanda" <mm.mawanda@nul.ls>]

My understanding of the relation between  category theory and  set  theory
is that category theory  is a formal theory built on abstract concepts
(objects and morphisms). The way of defining category theory need a
metalanguage which is closed to the logic of set theory language (a
particular case of what is called boolean logic). There is a sort of
dichothomy between logic behind the two theories. This dichothomy come from
our limitation of talking about category theory. We use  already two-valued
logic (true and false)  which we cannot avoid if we need to talk about
identity of objects and morphisms. Now a kind of Godel's arguments about
natural numbers (If N is consistent, then there is no proof of its
consistency by method formalizable within the theory ) is  what is going on.
This doesn't stop the category theory 'game'. When you give birth to a child
you will never know in advance if the child will be an honest person or a
criminal. Category theory have generated many structures which can help us
to understand why many mathematicians have work differently to describe a
same mathematical concept in different ways. As an example we know, from
category theory,  that Cauchy and Dedekind were defining real numbers from
rational numbers but the two definitions are not saying the same thing.

----- Original Message -----
From: "Michael MAKKAI" <makkai@scylla.math.mcgill.ca>
To: <categories@mta.ca>
Sent: Saturday, December 02, 2000 12:19 AM
Subject: categories: Re: Categories ridiculously abstract


> In "Towards a categorical foundation of mathematics" (Logic Colloquium
> '95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic
> no.11, 1998; pp.153-190) and in subsequent work, I am proposing an
> approach to a foundation whose universe consists of the weak n-categories
> and whatever things are needed to connect them. This is done on the basis
> of a general point of view concerning the role of identity of mathematical
> objects. Readers of said paper who have followed developments on weak
> higher dimensional categories will note that much has been done since
> towards fleshing out the program.
>
> Michael Makkai
>
>
> On Thu, 30 Nov 2000, Tom Leinster wrote:
>
> >
> > Michael Barr wrote:
> > >
> > > And here is a question: are categories more abstract or less abstract
than
> > > sets?
> >
> > A higher-dimensional category theorist's answer:
> > "Neither - a set is merely a 0-category, and a category a 1-category."
> >
> > There's a more serious thought behind this.  Sometimes I've wondered, in
a
> > vague way, whether the much-discussed hierarchy
> >
> > 0-categories (sets) form a (1-)category,
> > (1-)categories form a 2-category,
> > ...
> >
> > has a role to play in foundations.  After all, set-theorists seek to
found
> > mathematics on the theory of 0-categories; category-theorists sometimes
talk
> > about founding mathematics on the theory of 1-categories and providing a
> > (Lawverian) axiomatization of the 1-category of 0-categories; you might
ask
> > "what next"?  Axiomatize the 2-category of (1-)categories?  Or the
> > (n+1)-category of n-categories?  Could it even be, I ask with tongue in
cheek
> > and head in clouds, that general n-categories provide a more natural
> > foundation than either 0-categories or 1-categories?
> >
> >
> > Tom
> >
>