Dave Berry wrote (on 10-04-01 13:17 +0100):
> You certainly can avoid currying in functional languages.  Currying is a
> hack that was created to keep the lambda calculus as simple as possible.

When you say "currying" you are talking about a syntactic matter which arises
due to positional application. When Daniel said that "currying" is
basic to the lambda-calculus, he was talking about a more fundamental,
semantic matter.

If you look at lambda-calculus from a sufficiently abstract perspective where
the syntax is immaterial, then the essential difference between a first-order
and higher-order language is that in the second case you can represent a
function of two arguments (or a function whose argument is a 2-tuple) as a
higher-order function of one argument (and be able to apply it), and vice
versa. This characterization says nothing about positional application, labels
or what-have-you, because they are only a means of notation.

> Daniel points out that you will always be able to return a function from
> a function.  But currying is a partly syntactic hack; it relies on
> function application being notated by juxtaposition.  Without this hack,
> you have to write:
> 	let f = g (x, y)
> 	f (z)
> instead of:
> 	g x y z

Whichever way you write it, the same thing is still going on. If your calculus
is really equivalent to lambda-calculus, I should be able to transform the first
example to:

  (g (x,y)) (z)

which still uses positional application. Otherwise, you are just forcing me to
name all my function terms.

And how do you define g in the first case if you don't have semantical currying?
You need lambda:

  let g = fun (x, y) -> fun z -> h(x,y,z)

In a traditional denotational model, every term t is modeled as a function f
from (the product of) its environment to t. To bind a free variable, say z,
you need to curry f w.r.t. to z. So you still need currying to define lambda.

-- 
Frank Atanassow, Information & Computing Sciences, Utrecht University
Padualaan 14, PO Box 80.089, 3508 TB Utrecht, Netherlands
Tel +31 (030) 253-3261 Fax +31 (030) 251-379
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