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

From: C Y
Subject: Re: [Axiom-developer] Time for another crazy idea...
Date: Tue, 21 Feb 2006 06:02:36 -0800 (PST)

--- Gabriel Dos Reis <address@hidden> wrote:

> |  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.  

Hmm.  macro in the sense of lisp macros or in the sense of a
"metamathematics" system?

> |  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 :-)

Heh.  Maybe we could get some of the CMU guys interested - I think
they've got some folks who are into that kind of stuff.

> 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.

Do you mean introducing Computer Algebra to students or starting out
the Algebra volumes?  Most subjects are taught "top down" - we start
out with integers in mathematics, and action-reaction in physics, but
both are actually small pieces resting on more complex foundations.  I
think when implementing a powerful CAS system, the logical approach is
bottom up - build your foundations, then build on top of them.  Insofar
as Category Theory or Set Theory respresent foundations on which all of
mathematics can be built, they seem like a logical foundational layer
for the mathematics in Axiom, which strives to be as close to the math
as possible.  B-natural will be the layer that hides the complexity and
provides the normal "CAS" environment, but under the hood would be as
much fundamental mathematical research and power as possible, ready for
use by experts and hard core mathematical researchers.

> | 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.

OK, I'll take a look.  Thanks!  Do you mean an implementation of
categorical datatypes on top of a language or the datatypes as part of
the language definition?


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