Re: [Axiom-developer] Time for another crazy idea...

[Gabriel Dos Reis]

C Y <address@hidden> writes:

| I know my understanding of categories and domains is massively
| imperfect, so perhaps this idea is not worth much, but since it might
| be one of those cases where an explanation of why its a bad idea might
| be helpful, here we go...
| 
| My understanding of Axiom's concept of categories and domains is that
| it is similar to and at least somewhat compatible with category and/or
| set theory, but it was not explicitly designed to implement and rely
| these theories as a foundation.

An operational view, and I would say actually close to the practice,
is that Axiom/Aldor categories are abstract classes or interfaces, and
Axiom/Aldor domains are implementation classes.  A key distinction
from most OO systems is you can (to some extent) manipulate them as
first class citizens.

|  If that's true, I was wondering what
| sort of effort would be involved in tuning Axiom's use of categories to
| be in line with category theory mathematics, along the lines of things
| like Categories for the Working Mathematician.

I think it would require to turn Axiom/Adlor into a macro system.
I believe Axiom's terminology is very confusing, even if the analogy
with "Category Theory" can be catchy.  Axiom does not rely on
"structures", it relies on OO tagging.  

|  If Axiom is already
| close to this anyway, perhaps the next step could be taken and Axiom
| could relate its category structure directly to research in category
| theory, and turn its own core implementation of mathematical
| fundamentals into a literate document summarizing the research in those
| areas and how it is applied.

I don't know how many students will be attracted that way and how many
researchers will join the movement :-)
I believe the emptyset theory, or in mundain terms Category Theory, is
a good language to talk about structures one *already* understands.
It requires lots of "background" knowledge and examples.  I'm deeply
skeptical about it being well-suited for introducing Computer Algebra.

| Is that a) an insane amount of work and I just don't realize it yet b)
| a non-practical idea in terms of having an actual working CAS or c) all
| of the above?  I always thought the best way to lead off the Algebra
| volumes of Axiom was with a discussion of the fundamental theories of
| mathematics encoded in Axiom and the research they are based off of,
| but perhaps that's not a good approach for Axiom?

You can have a look at Generic Haskell and PolyP for how and what it
takes to get a first approximation of categorical datatypes in a
programming language.

-- Gaby