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Re: Integer residue-classes [was: Questions about srfi-77 Generic Arithmetic]

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Dr Sebastian Egner wrote:
> Will wrote:
> > Denying the ad hockitude doesn't make it go away. 
> > [...]
> > I therefore raise this as an issue:  Should we choose the
> > representatives based on the sign of the second argument,
> > as in Egner et al and in SRFI 77, or should we redefine
> > div and mod to choose the representatives consistently in
> > all cases, and define a second pair of procedures that use
> > the second most important choice of representatives?
> I would really appreciate if you read my posting before you
> repeat parts of it in a rather annoying and personal fashion,
> and put them forward as your own inventions.

I'm sorry, Sebastian, but the idea of defining a second
pair of Scheme procedures was so obvious that it entered
even my mind as soon as I understood Dr Lucier's objection,
which was several days before you mentioned the idea in your
recent post.

I admit to being annoyed by your up-front denial ("No, it is
not "ad hoc.") coupled with your later admission, buried as
it was within all the number-theoretic background.  ("Of course
there is no mathematical need for using the sign of the modulus
in this way---and in this sense this choice is 'ad hoc.')

When you finally admitted the ad hockitude, you immediately
said "But it turns out to be useful."  The only argument
you ever gave against the separate procedures and in favor
of the ad hockitude was this:

> This makes the system
> of representatives a static property of the program,
> but I doubt that this is helpful in any way. The down
> side is that there is no representation readily available
> for the choice of system of representives.

That first sentence was not very convincing.  I don't know
what you meant by the second sentence, so it just went over
my head.

> And, yes, I do understand "Dr. Lucier's objection" and it
> is not worth a lot, as I explained.

Brad had two objections.  You don't seem to understand his
objection to the ad hockitude of using an argument's sign
to select between two sets of representatives.

Brad's other objection is that the general property you want,
that "x ~ y <=> m divides (x - y)" where m is the modulus,
uses the notion of "divides" that is usually defined to mean
there exists an integer n such that n*m=(x-y).  That is why
you made an exception to that property for m=0, "for reasons
that will become clear later."  You did explain later, and I
accept your explanation as an argument for having (div q 0)
return 0.

Unfortunately, you dismissed Brad's objection as a "reflex"
and wrote that he "confuses residue-class operation 'div'
with the *field* operation '/'."  As I explained in the
paragraph immediately above this one, Brad's objection was
indeed based on the usual number-theoretic definition that
you yourself cited.  For you to attribute his objection to
mathematical incompetence in "a rather annoying and personal
fashion" was rather annoying.

> > The background material you provided will help me to explain
> > how little is at stake here to those who are wondering
> > what this is about, and I thank you for that.
> I see. You are probably right, discussing this with you
> is indeed a waste of my time.
> Sincerely,
> Dr. Egner

Thank you for wasting your time with us.