Revamping the numeric classes

[William Lee Irwin III]

On Thu, Feb 08, 2001 at 08:30:31PM +0000, Marcin 'Qrczak' Kowalczyk wrote:
> Signum doesn't require Ord.
>     signum z = z / abs z
> for complex numbers.

I'd be careful here.

\begin{code}
	signum 0 = 0
	signum z = z / abs z
\end{code}

This is, perhaps, neither precise nor general enough.

The signum/abs pair seem to represent direction and magnitude.
According to the line of reasoning in some of the earlier posts in this
flamewar, the following constraints:

	(1) z = signum z <*> abs z where <*> is appropriately defined
	(2) abs $ signum z = 1

should be enforced, if possible, by the type system. This suggests
that for any type having a vector space structure over Fractional
(or whatever the hierarchy you're brewing up uses for rings with
a division partial function on them) that the result type of signum
lives in a more restricted universe, perhaps even one with a different
structure (operations defined on it, set of elements) than the argument
type, and it seems more than possible to parametrize it on the argument
type. The abs is in fact a norm, and the signum projects V^n -> V^n / V.
Attempts to define these things on Gaussian integers, p-adic numbers,
polynomial rings, and rational points on elliptic curves will quickly
reveal limitations of the stock class hierarchy.

Now, whether it's actually desirable to scare newcomers to the language
into math phobia, wetting their pants, and running screaming with
subtleties like this suggests perhaps that one or more "alternative
Preludes" may be desirable to have. There is a standard Prelude, why not
a nonstandard one or two? We have the source. The needs of the geek do
not outweigh the needs of the many. Hence, we can cook up a few Preludes
or so on our own, and certainly if we can tinker enough to spam the list
with counterexamples and suggestions of what we'd like the Prelude to
have, we can compile up a Prelude for ourselves with our "suggested
changes" included and perhaps one day knock together something which can
actually be used and has been tested, no?

The Standard Prelude serves its purpose well and accommodates the
largest cross-section of users. Perhaps a Geek Prelude could
accommodate the few of us who do need these sorts of schenanigans.


Cheers,
Bill
-- 
<j0][nD33R:#math> Excel/Spreadsheet Q: What is the formula for finding
	out the time passed between two dates and or two times in the same day?
<MatroiDN:#math> excel/spreadsheet? Hmm, this is math? Is there a GTM on
	excel or maybe an article in annals about spreadsheets or maybe
	there's a link from wolfram to doing your own computer work, eh?
<danprime:#math> jeeem, haven't you seen "Introduction to Algebraic Excel"?
<danprime:#math> or "Spreadsheet Space Embeddings in 2-Manifolds"
<brouwer:#math> i got my phd in spreadsheet theory
<brouwer:#math> i did my thesis on the spreadsheet conjecture