On Wed, Jul 9, 2008 at 9:05 PM, Bill Page <[EMAIL PROTECTED]> wrote: > On Wed, Jul 9, 2008 at 8:43 PM, Gabriel Dos Reis > <[EMAIL PROTECTED]> wrote: > >> ... >> In the field of fraction of polynomial, the fact that 1/(2d) is well >> defined seals no doubt. It does not matter whether d can take >> on the value zero. That is just a simple fact. That is one of >> reasons why simplification of expressions that assume >> fraction of polynomials prove inadequate some times. >> > > If "d can take on the value zero" doesn't that imply we are thinking > of '1/(2d)' as a function?
No, absolutely, not! In one step, there is the construction of the formal objects 2*d and 1/(2*d). Any sound computations must mention the domain of computation (which therefore embodies the applicable rules). Those are the ring of polynomial over integers (or preferably rational numbers), and the field of fractions of such polynomials. Then, some people go on *evaluating* those objects. But then, again, one has to define that evaluation formally. In most algebra texts, the evaluation from R[X] to R is defined a ring morphism (assuming R is a ring) without ever mentioning polynomial as functions! Then, there is also the Analysis point of view, where people want to interpret polynomials as functions, but then you have a different interpretation function and domain of computations. And applicable rules are not the same. For example, equality in the fraction of polynomials does not carry to equality of functions. You need to be careful about the domain of interpretation of computation. > > Isn't the following result another example such an error? > > (1) -> q:=((x^2-1)/(x-1)) > > (1) x + 1 > Type: Fraction Polynomial Integer It is NOT an error. The result is absolutely correct within that domain of computation. > (2) -> q(1) > > (2) 2 > Type: PositiveInteger > > Surely the fact that we can write (2) means this simplification is > unjustified. Absolutely not -- again, go back to the domain of computation and the definition of evaluation or interpretation. > >> Have a look at this wonderful paper by James Davenport. >> >> "Equality in Computer Algebra and Beyond" >> > > Thanks, I will review it again. You have referred to this paper before: > > http://sourceforge.net/mailarchive/message.php?msg_name=87od8i6aej.fsf_-_%40gauss.cs.tamu.edu > > In the abstract from which you quote Davenport writes: > > "For example, we refer to Q(x) as "rational functions", even though > (x^2-1)/(x-1) and x+1 are not equal as functions from R to R, since > the former is not defined at x=1, even though they are equal as > elements of Q(x)." > > But as I recall he does not deal with this issue in greater > specificity in the rest of the paper. He does make the excellent point that being precise about the domain of computation is fundamental (if not a trivial task). And that is exactly the fundamental issue underlying the current discussion. > > For the accessibility challenged here is an alternate (free) link: > > http://www.calculemus.net/meetings/siena01/Papers/Davenport.ps > > to this paper. > > Regards, > Bill Page. > ------------------------------------------------------------------------- Sponsored by: SourceForge.net Community Choice Awards: VOTE NOW! Studies have shown that voting for your favorite open source project, along with a healthy diet, reduces your potential for chronic lameness and boredom. Vote Now at http://www.sourceforge.net/community/cca08 _______________________________________________ open-axiom-devel mailing list open-axiom-devel@lists.sourceforge.net https://lists.sourceforge.net/lists/listinfo/open-axiom-devel
