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

From: Ralf Hemmecke
Subject: [Axiom-developer] natural isomorphisms (was: Re: 1/x x = 1 is a bug!)
Date: Fri, 24 Feb 2006 14:36:49 +0100
User-agent: Thunderbird 1.5 (X11/20051201)

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

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

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


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