On Mon, Oct 5, 2009 at 10:36 AM, Miguel Mitrofanov
Of course, it's OK to call anything "numbers" provided that you stated explicitly what exactly you would mean by that. But then you have to drop all kind of stuff mathematicians developed for the usual notion of numbers. In the same way, you shouldn't use the "Num" class for your "numbers".
On the other hand, people can (ab)use the "Num" class as they wish, and it's their business until they ask a question about it somewhere outside - which makes the business not only theirs. [...]
The Num class has `negate' as part of its definition. Natural numbers are numbers, but I don't believe there is any sensible definition of `negate' for them. Haskell 98's numeric hierarchy combines many operations which should be separate. As further evidence, every bit of Haskell I have seen that does symbolic manipulation of numeric expressions either leaves out instances that would make the syntax more convenient, or else defines partial instances because certain class functions have no sensible definition for symbolic expressions. Sincerely, Brad