[Axiom-developer] natural isomorphisms (was: Re: 1/x x = 1 is a bug!)

[Ralf Hemmecke]

On 02/24/2006 01:14 PM, William Sit wrote:
> A big big question in computer algebra should be how to handle "canonically
> isomorphic" mathematical objects. Mathematicians assume these identifications
> without a thought in most cases, but there are occasions where distinction is
> important. For example, they would rarely identify two vastly differently
> defined finite sets just because they have the same cardinality.

I very much agree that computer algebra is harder than mathematics in 
the sense that one cannot simply identify certain objects. One must give 
the corresponding map explicitly.

As William said, in some context one wants to identify and in some other 
context it would be better to treat the objects as different things. 
That basically says, in order to do mathematics with a computer there 
should not be a program (like the interpreter) that automatically 
identifies objects for the user. Well, for beginners it would be nice to 
have such a tool, but when one wants to do serious mathematics and 
automatic coercion would forbid to express certain mathematical 
constructions.

William. recently, I have posted a little program

http://wiki.axiom-developer.org/address@hidden

That shows how to implement in Aldor that FRAC R has the same 
representation as R if R is a field.

Now, I don't believe that it is a good idea to do this in general. I 
rather think that it would be better to open up a new wiki page, for example

http://wiki.axiom-developer.org/AxiomDesignDecisions

and list such design issues as William gave in his mail. That would be a 
common denominator that all Axiom developers should agree on.

Ralf