[Haskell-cafe] Re: Laws and partial values

[Jake McArthur]

roconnor at theorem.ca wrote:
> Anyhow, the big point is that monoid laws apply to the concepts, not to 
> the representatives.

I agree with this.

> Ah, but it is impossible to create such smart constructors because it is 
> undecidable if a particular CReal represents 0 or not.

Either way, we still have undecidability, and I don't think changing the 
representation can solve it in all cases. Of course, this leads us right 
back to how to treat _|_ in the context of the monoid laws:

>>    _|_ `mappend` nonzeroRealNum
> 
> Since _|_ is a represenative that has no corresponding concept, the 
> above program is simply an error.

I won't argue with the conclusion, but rather with the assumption that 
led you there. _|_ does have a corresponding concept. _|_ corresponds to 
the concept that the computation does not terminate. That is, _|_ is not 
a part of the concept itself, but is another concept altogether.

In one sense, I think you are right. _|_ is not a valid representation 
of the abstract concept. I don't think it is entirely out of place 
though. In the presence of nontermination, you still have to have an 
abstraction of nontermination. Haskell's "flaw" is that _|_ can have the 
same type as the representation of a nonzero real number, even though 
_|_ itself is not a representation of a nonzero real number.

- Jake