Re: [Axiom-developer] RE: learning in public

[Bertfried Fauser]

Dear Tim,

        I regard this page as a motivation to start with an actual
thinking on the design of a Clifford package. If less ambiguose, I can
help with the most peculiar algorithms, but I still do not see how to do
it categorial and in a full generality.
        This week I am technically still in vacations and next week partly
on a conference, so I will have a fast internet connection only from
pre-next week onwards. I will try to take this time to think about a start
implementation and what it should look like.

Categorical:

>From a categorical point of view, there is a functor from the spaces
having a quadratic (bilinear not necessarily symmetric) form into the
category of associative algebras. To characterize this functor would be
the most difficult thing. But this is a unsolved math problem.

Analytic:

Many people work in Clifford analysis and many physicisty use Clifford
algebras actually as Clifford modules. To cope with this generalization
(in a mathematical rigorous way) is a major obstacte which hangs around
and should be kept in ming if a wide aplicability is intended.

Engineering:

"Geometric Algebra" is the term promoted by Hestenes and followers for a
style of using Clifford algebras as a "unifying language for mathematics
and physics". I was some time also infected by this nicely looking vision
and Hestenes charismatic writing. However, threse poeple have different
goals in mind. They stick to 19th century geometry and to 19th century
mathematics. From a categorial point, there cannot be a geometric algebra,
but that is a question of defining a representation functor from some
algebra into some geometry. This is a very wide filed and needs also a
rectification of many concepts in the foundation of mathematics and in
particular to logic. Clifford logic is not boolean etc pp.

will write later more
BF.

PS: I am not unhappy that the wiki pages were refered ;-) but check out
there the paper on Geometric Algebra, which I was intended to change....
however, after typing in for 2 hours, mozilla crashed whene I pressed the
submit button, and I was not fit enough mentally to do a second attempt.

% PD Dr Bertfried Fauser
%       Institution: Max Planck Institut for Mathematics Leipzig 
<http://www.mis.mpg.de>
%       Privat Docent: University of Konstanz, Physics Dept 
<http://www.uni-konstanz.de>
% contact |->    URL : http://clifford.physik.uni-konstanz.de/~fauser/
%              Phone : Leipzig +49 341 9959 735  Konstanz +49 7531 693491