Re: categorical foundations

[Colin McLarty <colin.mclarty@case.edu>]

2009/11/12  <Andre.Rodin@ens.fr>:

writes

>  ETCS is the formal basis of CCAF.

This is simply false.  On some versions ETCS is a part of CCAF but
even then it is in no sense prior to other parts.

> ETCS relies on a pre-formal
> notion of set or collection just like ZF or any other axiomatic theory built
> with Hilbert-Tarski axiomatic method.

Do you mean that every formalized axiom system uses arithmetical
notions such as "finite string of symbols."  This is why that formal
axioms cannot be the real basis of our knowledge of math, but it has
no more bearing on categorical axioms than any others.

Or do you think that pre-formal notions of "set" or "collection" are
all based on iterated membership and Zermelo's form of the axiom of
extensionality, so that CCAF is less basic than ZFC?  That is a common
belief among logicians who have not read Zermelo's critique of Cantor
(where Zermelo points out that Cantor did not hold these beliefs) and
who know a great deal more of ZFC than of other mathematics.

In fact, long before mathematicians could analyze the continuum into a
discrete set of points plus a topology, they were well aware of
collections like the collection of rigid motions of the plane -- and
that "collection" is a category.  It is not just a ZFC set of motions
but comes with composition of motions and with an object that the
motions act on.

> Elements of a new properly categorical
> method of theory-building are present in the "basic theory" (BC) that follows
> ETC. (I mean, in particular, the "redefinition" of functor in BC as 2-->A, etc.
> The standard definition of functor given earlier in ETC never reappears in BC.)

The "standard" definition of functor appears as the definition of a
small category in the category of sets.

> However in CCAF these new features are not yet developed into an autonomous
> axiomatic method - or into a new way of formalisation of pre-formal concepts,
> if you like.

Well, yes, they are developed into one.  That was Bill's achievement
with CCAF.

best, Colin


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