I don't know if there's a single answer to this, as @jozefg mentioned. And so purely out of speculation, here's one possible explanation:
I suspect the reason is because the type -- (a -> b -> b) in Haskell's foldr, for example -- is more meaningful than any name one could come up with. Since the a and b type parameters could be anything, the function could do pretty much anything as well. So you're left with names like function, combiner, morphism, and argument, which aren't especially meaningful either. Might as well use a short, non-distracting name.
Another example is the id :: a -> a function: what should you call its argument? Again, I think id's type is more descriptive than its argument name.
However, I agree with you -- it seems like there should be a common name, perhaps in mathematics. I hope somebody can correct me on this.
Some examples of its name in real code:
In Haskell's libraries, it's mostly called f (and sometimes operator in the comments):
class Foldable t where
-- | Map each element of the structure to a monoid,
-- and combine the results.
foldMap :: Monoid m => (a -> m) -> t a -> m
foldMap f = foldr (mappend . f) mempty
-- | Right-associative fold of a structure.
--
-- @'foldr' f z = 'Prelude.foldr' f z . 'toList'@
foldr :: (a -> b -> b) -> b -> t a -> b
foldr f z t = appEndo (foldMap (Endo . f) t) z
-- | Right-associative fold of a structure,
-- but with strict application of the operator.
foldr' :: (a -> b -> b) -> b -> t a -> b
foldr' f z0 xs = foldl f' id xs z0
where f' k x z = k $! f x z
-- | Left-associative fold of a structure.
--
-- @'foldl' f z = 'Prelude.foldl' f z . 'toList'@
foldl :: (a -> b -> a) -> a -> t b -> a
foldl f z t = appEndo (getDual (foldMap (Dual . Endo . flip f) t)) z
-- | Left-associative fold of a structure.
-- but with strict application of the operator.
--
-- @'foldl' f z = 'List.foldl'' f z . 'toList'@
foldl' :: (a -> b -> a) -> a -> t b -> a
foldl' f z0 xs = foldr f' id xs z0
where f' x k z = k $! f z x
-- | A variant of 'foldr' that has no base case,
-- and thus may only be applied to non-empty structures.
--
-- @'foldr1' f = 'Prelude.foldr1' f . 'toList'@
foldr1 :: (a -> a -> a) -> t a -> a
foldr1 f xs = fromMaybe (error "foldr1: empty structure")
(foldr mf Nothing xs)
where
mf x Nothing = Just x
mf x (Just y) = Just (f x y)
-- | A variant of 'foldl' that has no base case,
-- and thus may only be applied to non-empty structures.
--
-- @'foldl1' f = 'Prelude.foldl1' f . 'toList'@
foldl1 :: (a -> a -> a) -> t a -> a
foldl1 f xs = fromMaybe (error "foldl1: empty structure")
(foldl mf Nothing xs)
where
mf Nothing y = Just y
mf (Just x) y = Just (f x y)
-- instances for Prelude types
instance Foldable Maybe where
foldr _ z Nothing = z
foldr f z (Just x) = f x z
foldl _ z Nothing = z
foldl f z (Just x) = f z x
instance Ix i => Foldable (Array i) where
foldr f z = Prelude.foldr f z . elems
foldl f z = Prelude.foldl f z . elems
foldr1 f = Prelude.foldr1 f . elems
foldl1 f = Prelude.foldl1 f . elems
-- | Monadic fold over the elements of a structure,
-- associating to the right, i.e. from right to left.
foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b
foldrM f z0 xs = foldl f' return xs z0
where f' k x z = f x z >>= k
-- | Monadic fold over the elements of a structure,
-- associating to the left, i.e. from left to right.
foldlM :: (Foldable t, Monad m) => (a -> b -> m a) -> a -> t b -> m a
foldlM f z0 xs = foldr f' return xs z0
where f' x k z = f z x >>= k
-- | Map each element of a structure to an action, evaluate
-- these actions from left to right, and ignore the results.
traverse_ :: (Foldable t, Applicative f) => (a -> f b) -> t a -> f ()
traverse_ f = foldr ((*>) . f) (pure ())
-- | Map each element of a structure to a monadic action, evaluate
-- these actions from left to right, and ignore the results.
mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m ()
mapM_ f = foldr ((>>) . f) (return ())
It's also called f in Clojure:
(def
^{:arglists '([f coll] [f val coll])
:doc "f should be a function of 2 arguments. If val is not supplied,
returns the result of applying f to the first 2 items in coll, then
applying f to that result and the 3rd item, etc. If coll contains no
items, f must accept no arguments as well, and reduce returns the
result of calling f with no arguments. If coll has only 1 item, it
is returned and f is not called. If val is supplied, returns the
result of applying f to val and the first item in coll, then
applying f to that result and the 2nd item, etc. If coll contains no
items, returns val and f is not called."
:added "1.0"}
reduce
(fn r
([f coll]
(let [s (seq coll)]
(if s
(r f (first s) (next s))
(f))))
([f val coll]
(let [s (seq coll)]
(if s
(if (chunked-seq? s)
(recur f
(.reduce (chunk-first s) f val)
(chunk-next s))
(recur f (f val (first s)) (next s)))
val)))))
foldis a higher order function. The argument to fold is just a that, an argument – Daniel Gratzer Dec 20 '13 at 3:21