However, the viewpoint “everything can be seen a function (in addition to whatever else it is)” seems quite reasonable. As others have pointed out, n-ary (curried) functions are a very helpful concept, where formally an n-ary function can be defined as a term of type “A_1 -> … -> A_n -> B”, for some types A_1 … A…n. So “map” can be seen both as a unary function from “A -> B” to “[A] -> [B]”, and also as a binary function from “A -> B” and “[A]” to “[B]”.

But then by the same token, the integer 5 is a nullary function to “Int”; it’s also still an “Int”, but that doesn’t stop it being a nullary function as well.

So “everything is an n-ary function, for some n” is surely acceptable? In the context of Haskell, I don’t know how useful it is; but in some mathematical contexts, it’s very fruitful indeed (e.g. the insight that constants could be seen as 0-ary functions was essential for universal algebra, and its cousin the theory of operads).

Let X and Y be sets and let the cartesian product of X and Y, denoted X x Y be the set {(x,y) | x belongs to X, y belongs to Y}. A function from X to Y, written as f : X -> Y, is a subset of X x Y with the property that for all (x1,y1), (x2,y2) in f, x1 = x2 implies that y1 = y2.

This is the mathematical definition of of a function and is the one used by Haskell and probably the other functional programming languages as well. The defining property of the function as stated above is usually referred to as ‘referential transparency’. The C programming language started the confusion, and other such languages have perpetuated it, by using the term function to mean something quite different. In C a function is not necessarily an association at all – it may not take an input and it may not produce an output – it may be pure side effect and even if it does define an association there is no guarantee of referential transparency.

Adopting this definition, how do you make sense of the phrase ‘a function with 0 arguments’?

It seems more natural to express Vect as a symmetric bimonoidal category (http://ncatlab.org/nlab/show/bimonoidal+category) using the tensor product and direct sum. (***) is basically the tensor product, but I think the direct sum of transformations would be more like (+++).

in what way is this a “reimagining”? That matrices represent linear transformations is absolutely fundamental.

Thanks for asking. What I mean by “reimagining” is (a) packaging of linear maps via the Category & Arrow vocabulary (more explicit in the library), (b) structuring the representation and semantics to match the algebraic structure of dot and (&&&), and (c) derivation of operations from semantics.