[Haskell-cafe] New to Haskell

[Henning Thielemann]

On Tue, 18 Dec 2007, Benja Fallenstein wrote:

> Hi Henning,
>
> On Dec 18, 2007 5:17 PM, Henning Thielemann
> <lemming at henning-thielemann.de> wrote:
> > The mathematical definition of "function" I know of, says that functions
> > are special relations, and relations are sets of pairs. Their is nothing
> > about intension.
>
> That's the standard definition in set theory, but it's not the only
> mathematical definition of function. It also doesn't suffice for
> defining all Haskell functions-- consider
>
> data T = T (T -> Int)
>
> fn :: T -> Int
> fn _ = 7
>
> We have (fn (T fn) == 7), so in the graph of 'fn' we must have a pair
> (T fn, 7). But if 'fn' is the same mathematical object as its graph,
> that would mean that the graph of 'fn' would have to contain a pair
> whose first element indirectly contains... the graph of fn!
>
> This sort of circularity is not allowed in standard ZFC set theory, so
> if we're going to be precise, we will have to choose a different
> representation for functions than their graphs.

I see.

 I'm also wondering what 'total function' and 'partial function' might
mean in Haskell, since values can be partially defined. Is
   Just undefined
 defined or undefined and is
   const (Just undefined)
 a total or a partial function?