RE: [Axiom-developer] POLY INT =\\= UP(x, INT) (was: [Axiom-math] partialFraction behavi or in Axiom)

[Page, Bill]

Bertfried,

On Wednesday, October 08, 2003 4:49 AM you wrote:
> ... 
>       though I have very limited time resources, I nevertheless
> follow this interesting discussion. Perhaps I can add a few
> remarks from the math side of view (I am not a programmer at
> all, since I learned it with BASIC and assembler and my brain
> is hence damaged [Knuth] <grin>)

To be fair to Knuth and to yourself you should note that
Knuth was reacting against the then current trend of
introducing programming to novices by teaching them to
write simple programs in BASIC. Knuth on the other hand
is famous for his view that the first language a programmer
should learn is assembler. He even invented his own virtual
machine and an assembler to go with it (MIX) so that
students could learn to program in assembler on a well
behaved virtual machine instead of the often much uglier
real thing.

My first language was actually PL/1 (dah... ) and shortly
followed by too many assemblers, FORTRAN's, Basics, Algols,
SNOBOLs, COBOLs, Lisps ... (you get the idea). But I was
fortunate many years ago to also take one of the first
satellite two-way remote courses taught by Knuth himself.
I still remember MIX with mixed feelings!

So having learned *both* BASIS and assembler, we may still
have some hope that the "damage" is reversible ... <grin>

> 
> ... 
> c) philosophical
>       There is an ongoing movement in math for now 40 years
> to use categories as founding principle in mathematics
> (I like this). This is a revolution since Sets are no longer
> elementary objects but derived structures.

Yes! This is a very important issue. I am very much in
favour of this "ongoing movement". The application of
category theory to computer science is only at it's very
beginning stage.

> If AXIOM would have been based (is based?) on the idea
> that Set theory is the founding ground of math, it is quite
> clear that this will have an large impact on the type system.

Yes, AXIOM is clearly attempts to base its type system on the
older "set theory as foundation" approach. I said "attempts"
because I think one could argue that it does not wholly
succeed it this due to compromises introduced by the nature
of programming versus mathematical abstraction. This compromise
is at the root of the direct applicability of category
theory to computer science. So in other words, because of
these necessary compromises, what AXIOM actually implements
is closer (better described) by category theory then by
set theory itself.

> If now morphisms (functions) are seen as elementary and
> sets as derived, one will end up with an alternate type
> tower in AXIOM.
>       [Is there a possibility to reorganizes the type 
> structure in AXIOM or would that mean entirely reprogramming
> of the algebra?]

Originally I thought reprogramming (like Aldor) might be
best, but lately I am thinking that by taking a little more
care it may be possible for two rather different type
systems to coexist in AXIOM. Already the AXIOM type system
contains a number of type definitions which are either
obsolete or experimental and mostly not documented, e.g.
NEWPOLY, XPOLY etc.

I am very interested in participating in a joint project
(with people more knowledgeable about AXIOM then myself!)
to provide a much more "categorial" type system. There
has been some work along this line by the people involved
with Aldor as well. But I think that just as it was in
the past, AXIOM is a more comfortable "scratchpad" in
which to do these kind of experiments before casting
them in stone by compiling libraries in Aldor.

> ... 
> PPS: I had some discussions in Marseille with some
> french mathematicians.

I was in Avignon last week. Sorry to have missed you!

> They suggested to use a descendant of N.N. Bourbaki as
> AXIOM author, Thomas Schuecker suggested even to use
> Nicole Bourbaki, his `daughter` <grin>

Yes, I like that. I caste my vote for Nicole Bourbaki.

In fact I have begun thinking lately that the name AXIOM
itself it not all that appropriate - other than it's
historical relevance. For one thing, it is too "common".
Looking up "axiom" in a web search yields thousands of
irrelevant hits. I am inclined to suggest a name change
of the system itself to "bourbaki" (spelt lowercase). At
least then most of the search misses would be of some
interest...

> The Bourbaki group seems to be inexistent, but some are
> still alive and active and I will be contacted soon about
> the naming.
> 

Great. I think we should continue to pursue this.

Cheers,
Bill Page.