Re: [Axiom-developer] The Axiom Library and Category Theory

[Bill Page]

Quoting Gabriel Dos Reis wrote:

> ... 
> On the other hand, Category Theory (or the Theory of Empty Set),
> does not require set -- you can do things with small and large
> categories. 

Just out of curiosity I did a quick web search for the phrase
"Theory of Empty Set" and only turned up your paper for LCSD'05:

  "What is Generic Programming?"
  http://www.cs.rpi.edu/research/pdf/06-12.pdf

where you and Jarvi wrote:

  Category theory - also occasionally referred to as "abstract
  nonesense" or "the theory of empty set" -- has found an
  unreasonably  effective application in Computer Science. 

The phrase "abstract nonsense" is well known, e.g. 

http://en.wikipedia.org/wiki/Abstract_nonsense

but I had not heard the phrase "theory of empty set" before. 
Can you tell where else you have seen this phrase used in
reference to category theory?

> It only cares about *forms and structures*.  From an implementation
> point of view, it means that you don't need to require all computational
> objects to derive from a single universal base.  That gives flexibility for
> composition -- something much harder and clumsy with OO paradigms
> (all incarnations that have been tried so far). 

I think you are right. Although the Axiom language might have the
ability to avoid this "crisis", the current Axiom library seems to at
least partly embody a more conventional OO approach. 

> You can failures of OO thinking in the current library in forms of the
> curious Abelian Monoids that are not Monoids.  No undergrade in
> math will get away with that.  But, Axiom apparently does :-(

Yes, this is a serious problem with the Axiom library that has been
recognized for some time. I think it is time we tried a little harder
to do something about it. 

In the thread:

http://lists.nongnu.org/archive/html/axiom-math/2007-05/msg00015.html

we were talking about the possibility of defining the monoid unit as
a higher order functional. Have you had any more thoughts about
this?

Regards,
Bill Page.