A concurring opinion here, and an example.

iff :: Bol -> a -> a -> a
iff True x _ = x
iff False _ x = x

f, g :: Bool -> Bool
f x = x
g x = iff x True False

Are these two functions equal?  I would say yes, they are.  Yet once you can pattern match on functions, you can easily tell these functions apart, and create a function

h :: (Bool -> Bool) -> Bool
such that h f => True but h g => False.

  -- ryan

On Thu, Apr 5, 2012 at 8:52 AM, Dan Doel <dan.doel@gmail.com> wrote:
On Thu, Apr 5, 2012 at 10:14 AM, Grigory Sarnitskiy <sargrigory@ya.ru> wrote:
> First, what are 'functions' we are interested at? It can't be the usual set-theoretic definition, since it is not constructive. The constructive definition should imply functions that can be constructed, computed. Thus these are computable functions that should be of our concern. But computable functions in essence are just a synonym for programs.

This is a flawed premise. The point of working with functions is
abstraction, and that abstraction is given by extensional equality of
functions:

   f = g  iff  forall x. f x = g x

So functions are not synonymous with programs or algorithms, they
correspond to an equivalence class of algorithms that produce the same
results from the same starting points. If you can access the source of
functions within the language, this abstraction has been broken.

And this abstraction is useful, because it allows you to switch freely
between programs that do the same thing without having to worry that
someone somewhere is relying on the particular algorithm or source.
This is the heart of optimization and refactoring and the like.

There are places for metaprogramming, or perhaps even a type of
algorithms that can be distinguished by means other than the functions
they compute. But to say that functions are that type is false, and
that functions should mean that is, I think, wrong headed.

-- Dan

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