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

[Bertfried Fauser]

Dear Bill,

        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 assebler and my brain is hence damaged [Knuth] <grin>)

        For me arise several very general issues with AXIOMs type system:

a) A technical point is to get an overview on the sheer hughe amount of
possiblilities and sometimes very alike types. To use AXIOMs facilities
effiently one has to be *aware* of the types possibly available not to
restrict later applications (or to be forced to coerce)
        This is a problem of documentation and might be solved as soon as
the hyperdoc is available.

b) math versus prog
        As you point out with the types `Real` versus `Float`, there is a
shift in the point of view. The type of a programmer (I guess) is tied to
the *data structure* hence Interger, Float etc are logical, as
Poly(Int) is too. A mathematician will most likely not want to care about
internal representation of data, but wants to deal with mathematical
objects. Hence things like `Abelian Semigroup` (PI), `Ring` (Integer,
Rationals, Reals, lambda-rings, finite rings) etc might be wanted. But as
this example shows, one needs a structuring system to `order` these
notions which is tremendously delicate.

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. If AXIOM would have been based (is based?) on the idea that
Set theory is the founding groung of math, it is quite clear that this
will have an large impact on the type system. If now morphisms (functions)
are seens 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 teh algebra?]

d) The 4Ms
        Eg Maple is going to be sold to students, even high school
students and ingeneers most likely. Hence the await not mathematics but
just an symbolic pocket calculator. Tha is seen by looking at the
improvement of floating point abilities of Maple/Mathematica in the lates
releases (eg maple 9) and was told to me by a maple reseller literally
this way. Hence systems like the 4Ms have to face teh fact, that the
averagde user will not even have seen the words ring, field or module but
wants to manipulate polynomials, matrices, numbers and sole equations
(without asking if thats a particular good idea, e.g. maple comes up with
complex solutions for problems formulated over the reals...)
        IFF AXIOM wants to be sucessfull on a larger scale (say in the
academic area and partly in teaching) it is a *must* to come with a plain
description of the mathematics behind the types. Example: Which of your
students can say spontanously what the difference betrween Ring and
Euclidean Ring is? But without this distinction two such types make no
sense at all (to the user however <grin>)

e) Rafal Ablamowicz and myself called the CAS based approch to mathematics
`experimental mathematics` which might be as missleading as `computational
matehmatics` since the first looks probably not seriouse, while the second
might miss computer ALGEBRA. Indeed the algorithmic aspects of mathematics
are the quint essence of its usability in concrete problems and a better
documentation of this link would be extraordinary desirable.

I hope to be able to contribute a little bit to such a point of view, but
I do also know hom *much* time is swallowed by documentation, if thats
done seriously. Furthermore I am still a novice with AXIOM and on the way
to explore whats already there. Even that is quite a torture ... looking
up types and their relation is still not easy, it might be extraordinary
helpfull to have a graph of dependencies in the types. Eg Poly(Int)
depends on Integer, Euclidean Ring depends on Ring etc. This might even
helpfull for mathematics to see in which ways such a type system can be
coherently build up, probably in *various ways!* (that goes in teh theory
of matroids under the name crypto morphism, since matroids can be
axiomatixes in a tremendously large number of ways.

cheers
BF.

PS: I am still hunting for teh Springer book, ....

PPS: I had some discussions in Marseille with some french mathematicians.
They suggested to use a decendant of N.N. Bourbaki as AXIOM author, Thomas
Schuecker suggested even to use Nicole Bourbaki, his `daughter` <grin>
The Bourbaki group seems to be inexistent, but some are still alive and
active and I will be conatcted soon about the nameing.

% |   | PD Dr Bertfried Fauser    Fachbereich Physik    Fach M 678  |
%  \ /  Universit"at Konstanz     78457 Konstanz        Germany     |
% (mul) Phone : +49 7531 883786   FAX : +49 7531 88-4864 or 4266 (comul)
%   |   E-mail: address@hidden                   / \
%   |   URL   : http://clifford.physik.uni-konstanz.de/~fauser    |   |

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