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Re: [Axiomdeveloper] Time for another crazy idea...
From: 
Gabriel Dos Reis 
Subject: 
Re: [Axiomdeveloper] Time for another crazy idea... 
Date: 
21 Feb 2006 13:11:44 +0100 
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 wellsuited for introducing Computer Algebra.
 Is that a) an insane amount of work and I just don't realize it yet b)
 a nonpractical 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