# The Universe of Discourse

Sat, 08 Sep 2018

Here's something else that often goes wrong when I am writing a Haskell program. It's related to the problem in the previous article but not the same.

Let's say I'm building a module for managing polynomials. Say Polynomial a is the type of (univariate) polynomials over some number-like set of coefficients a.

Now clearly this is going to be a functor, so I define the Functor instance, which is totally straightforward:

      instance Functor Polynomial where
fmap f (Poly a) = Poly \$ map f a


Then I ask myself if it is also going to be an Applicative. Certainly the pure function makes sense; it just lifts a number to be a constant polynomial:

       pure a = Poly [a]


But what about <*>? This would have the type:

    (Polynomial (a -> b)) -> Polynomial a -> Polynomial b


The first argument there is a polynomial whose coefficients are functions. This is not something we normally deal with. That ought to be the end of the matter.

But instead I pursue it just a little farther. Suppose we did have such an object. What would it mean to apply a functional polynomial and an ordinary polynomial? Do we apply the functions on the left to the coefficients on the right and then collect like terms? Say for example

\begin{align} \left((\sqrt\bullet) \cdot x + \left(\frac1\bullet\right) \cdot 1 \right) ⊛ (9x+4) & = \sqrt9 x^2 + \sqrt4 x + \frac19 x + \frac14 \\ & = 3x^2 + \frac{19}{9} x + \frac 14 \end{align}

Well, this is kinda interesting. And it would mean that the pure definition wouldn't be what I said; instead it would lift a number to a constant function:

    pure a = Poly [λ_ -> a]


Then the ⊛ can be understood to be just like polynomial multiplication, except that coefficients are combined with function composition instead of with multiplication. The operation is associative, as one would hope and expect, and even though the ⊛ operation is not commutative, it has a two-sided identity element, which is Poly [id]. Then I start to wonder if it's useful for anything, and how ⊛ interacts with ordinary multiplication, and so forth.

This is different from the failure mode of the previous article because in that example I was going down a Haskell rabbit hole of more and more unnecessary programming. This time the programming is all trivial. Instead, I've discovered a new kind of mathematical operation and I abandon the programming entirely and go off chasing a mathematical wild goose.

[ Addendum 20181109: Another one of these. ]