[Haskell-cafe] Function Precedence

[Hans Aberg]

On 3 Apr 2008, at 07:59, Henning Thielemann wrote:
>> But one should also be able to write (f+g)(x). - This does not  
>> work in Haskell, because Num requires an instance of Eq and Show.
>
> You could define these instances with undefined function  
> implementations anyway. But also in a more cleaner type hierarchy  
> like that of NumericPrelude you should not define this instance,  
> because it would open new surprising sources of errors:
>   http://www.haskell.org/haskellwiki/Num_instance_for_functions

This problem is not caused by defining f+g, but by defining numerals  
as constants. In some contexts is natural to let the identity be  
written as 1, and then 2 = 2*1 is not a constant. With this  
definition, a (unitary) ring may be identified with an additive  
category with only one object.

In mathematical terms, the set of functions is a (math) module  
("generalized vectorspace"), not a ring.

Anyway, Num is a type for unifying some common computer numerical  
types, and not for doing algebra. If its (+) is derived from an  
additive monoid (or magma) type, then defining f+g will not interfere  
with Num.

   Hans