For anything to be a Monoid (or any type class with laws), you implicitly have a definition of equivalence you want your laws to use.  And for many classes, those laws are using an equivalence not definable using Eq.  A good example Is monad! You can not define Eq instances for arbitrary a-> m b.  We can still define and talk about lawful monads. 

Point of order though, no Num  a instance has a Monoid  a instance.  Instead Sum a and Product a are the Monoid Instances.

On Sep 26, 2014 9:46 PM, "Noon Silk" <noonslists@gmail.com> wrote:
> for equational laws to be sensible requires a sensible notion of equality, 
> the Eq for Floating point numbers is

could it not be then that for floating points to be a monoid you must specify a satisfying notion of equality? (well, i guess nothing is stopping anyone from doing that themselves; and your point is that simply not having floats as a monoid is somehow "bad"?)



On Sat, Sep 27, 2014 at 5:41 AM, Carter Schonwald <carter.schonwald@gmail.com> wrote:
for equational laws to be sensible requires a sensible notion of equality, the Eq for Floating point numbers is
meant for handling corner cases (eg: am i about to divide by zero), not "semantic/denotational equivalence"

Exact equality is fundamentally incorrect for finite precision mathematical computation. 
You typically want to have something like 

nearlyEq  tolerance a b = if distance a b <= tolerance then True else False

Floating point is geometry, not exact things 
is a package that provides an approx equality notion.

Basically, floating points work the way they do because its a compromise that works decently for those who really need it.
If you dont need to use floating point, dont! :) 



On Fri, Sep 26, 2014 at 9:28 AM, Jason Choy <jjwchoy@gmail.com> wrote:
subject to certain caveats.  It's not unfair to say that
floating point multiplication is (nearly) associative
"within a few ulp".

I'm not disputing this.

However, you can't deny that this monoid law is broken for the floating point operations:

mappend x (mappend y z) = mappend (mappend x y) z

Perhaps I'm being pedantic, but this law should hold for all x, y, z, and it clearly doesn't.


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Noon Silk, ن

https://sites.google.com/site/noonsilk/

"Every morning when I wake up, I experience an exquisite joy — the joy
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