A sane mathematical structure - or rather: sane description
of math structures in Haskell is something which worries
me for years. On this list, on funct. newsgroup and elsewhere
this is a recurring, cyclic theme.
-- And we have still this horrible Num hierarchy, which does
not correspond to anything serious. So, in my opinion such
initiatives as Docon of Sergey Mechveliani merit all our
attention, and objections written in the style
"... did I say that it is too complicated?"
are not very constructive, even if Docon is really
complicated. Personally I don't use it, I have my own,
private library with AdditiveGroups, Rings, Modules, etc.,
and I have used it to work with differental algebras,
forms, generators of parametric surfaces, lazy sequences,
power series, and other silly stuff I like. But other
people like other things, and will not submit my stuff
to any "jury" or "committee" for their benevolent consi-
deration, because it is intrinsically ill and incomplete.
And it will remain so, because Haskell doesn't seem ready
to permit a global approach to math. structure definition.
Jan Skibinski writes:
> It appears to me that we have reached some impasse
> in a design of basic mathematical structure for
> Haskell 2. Sergey's proposal [...] does not seem to
> reverberate on this group.
>
> Shouldn't we thus start with something more moderate,
> that does not offer a concrete solution as yet,
> but at least presents some framework for a serious
> discussion?
and later:
> If I could suggest [...]
>
> + Start with a big picture and forget details
> for a moment.
> Use standard naming convention from Mathematics
> Subject Classification, so we all could refer
> to it, check it, and compare notes. Graphic
> representation would be nice.
>
> + Justify the needs for all those elements
> from the big picture. What am I buying
> from this as a whole and why I need this
> particular structure? What can I do with it?
> [That's why I cited Tegmar's diagram yesterday:
> he evidently knew where he was heading]
Well, all this is ambiguous. A "big picture" and
"something moderate" contradict themselves IMHO.
Such diagrams as presented in the TOE paper are to be
found elsewhere. See the cover of the AXIOM manual for
example. The "object-like" classification of math.
structures *is not enough*.
Not only some properties of operations, such as the
commutativity cannot be expressed by such diagrams, but
several links: subsumptions, implicit inheritance etc.
will be always missing. For example:
Any additive group *must* be a Module over integers.
A Ring inherits twice a semi-group.
A modular Ring with N generators for N prime becomes
"miraculously" a Field.
The ordering generates an algebraic structure.
etc. In Axiom, Magma and MuPAD (and also GAP) there is
plenty of dynamics, the "types" (categories, domains,
axioms, hyla, callThemAsYouLike,...) combine the class
approach, only *partially* resolved statically, with
some constraint semantics.
===
I believe that a modest approach is really what we need,
but for me the modesty means - try to *apply* to concrete
problems whatever you have, and if you miss something -
CRY LOUD! (Perhaps in such a way I will see one day the
possibility to use my own *IMPLICIT* fromInt or fromDouble
conversion of constants, and not those inserted by the
compiler "who" naively thinks that I use the standard
preludes... /Hugs/)
===
Mad Max Tegmark "TOE" seems to ignore the theory of categories,
his approach to math structures in the Universe is a little
"Bourbakiste"...
I see why Jan Skibinski liked this paper, some physics
background becomes visible. This is also my case. But I
disagree with the statement that Tegmarks knows where he
is heading.
Getting back to categories, they began to appear in math.
physics as well, although I still remember when one of
my professors many years ago told us publicly that there
are some branches of mathematics which belong to a purely
speculative layer of science/philosophy, and will *never*
find any applications, for example the theory of categories,
or non-classical logic.
http://www.math.sunysb.edu/~kirillov/tensor/tensor.html
http://math.nwu.edu/~getzler/conf97.html
===
Jerzy Karczmarczuk
Caen, France
