Hans Aberg haberg at math.su.se
Thu Apr 3 11:09:36 EDT 2008

```On 3 Apr 2008, at 16:07, Chris Smith wrote:
>> This problem is not caused by defining f+g, but by defining
>> numerals as
>> constants.
>
> Yup.  So the current (Num thing) is basically:
>
> 1. The type thing is a ring
> 2. ... with signs and absolute values
> 3. ... along with a natural homomorphism from Z into thing
> 4. ... and with Eq and Show.
>
> If one wanted to be perfectly formally correct, then each of 2-4
> could be
> split out of Num.  For example, 2 doesn't make sense for
> polynomials or n
> by n square matrices.

Or ordinals.

> 4 doesn't make sense for functions.  3 doesn't
> make sense for square matrices of dimension greater than 1.  And, this
> quirk about 2(x+y) can be seen as an argument for not wanting it in
> the
> case of functions, either.  I'm not sure I find the argument terribly
> compelling, but it is there anyway.
>
> On the other hand, I have enough time already trying to explain Num,
> Fractional, Floating, RealFrac, ... to new haskell programmes.  I'm
> not
> sure it's an advantage if someone must learn the meaning of an
> commutative semigroup in order to understand the type signatures
> inferred
> from code that does basic math in Haskell.  At least in the U.S., very
> few computer science students take an algebra course before getting

It is probably not worth to change Num, as code already depends on it.

But by inserting some extra classes, and letting it derive, makes it
possible to use operators such as (+) without tying it to Eq, Show, etc.

>> In mathematical terms, the set of functions is a (math) module
>> ("generalized vectorspace"), not a ring.
>
> Well, I agree that functions are modules; but it's hard to agree that
> they are not rings.  After all, it's not too difficult to verify
> the ring
> axioms.

It is the set of function that may be a module or ring, and what it
is depends on how you define it. - It may be different in different
contexts.

And there is a different question finding unambiguous notation. How
would a explicit constant like 2 be defined in HAskell as a constant
so that 2(sin) becomes possible? INs't OK to write (const 2)?

Hans

```