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

[Weiss, Juergen]

I think one has to discern the AXIOM algebra libraries and
the AXIOM interpreter. The library is strongly typed. It's 
conceptually not so different to the ALDOR libraries. And
the libraries are not about abstract mathematics but about
efficient computational mathematics. With a stress both
on computational and efficient. So it's not enough to have
a ring, but you need effective algorithms for the 
operations. And if you have a ring with special structure,
which allows faster calculations of some of the operations,
you will define and use it in AXIOM as well. This makes
the library ugly in some places, but there is not much of
a choice.

On the other side, you have the user, who wants to work
interactively. You can not force him to give the type of 
every number or symbol he types in. Besides, even
mathematicians are used to change the interpretation
of an expression from one line of a computation to
the next without even thinking about it. So you need
the AXIOM interpreter as a mediatior between the user
which expects something like "do what I mean" and the
strongly typed library. This part is certainly a weak
part of the system, because it's hard to predict
from a user's perspective.

There are domains in the library which are probably
written with the interpreter in mind (POLY for example,
which handles arbitrary indeterminates, as opposed to
DMP domains with fixed indeterminates). By the
way, AXIOM discerns programming variables from indeterminates,
which was the cause of a problem raised in a previous
mail. 

The expression domain in a way simulates the way 
the classical computer algebra systems handle mathematics
(don't flame me for this statement). If you have to handle
symbols with their algebraic depenencies only partly known,
it's difficult to do any better. 

Regards

Juergen Weiss

Juergen Weiss     | Universitaet Mainz, Zentrum fuer Datenverarbeitung,
address@hidden| 55099 Mainz, Tel: +49(6131)39-26361, FAX:
+49(6131)39-26407


> -----Original Message-----
> From: Page, Bill [mailto:address@hidden 
> Sent: Wednesday, October 08, 2003 5:41 PM
> To: 'address@hidden'
> Cc: address@hidden; 'Tim Daly'
> Subject: RE: [Axiom-developer] POLY INT =\\= UP(x, INT) (was: 
> [Axiom-math]partialFraction behavi or in Axiom)
> 
> 
> 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.
> 
> 
> 
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