[Haskell-cafe] Re: [Haskell] Math behind Haskell

[Cale Gibbard]

On 23/09/2007, Neil Mitchell <ndmitchell at gmail.com> wrote:
> Hi
>
> The haskell-cafe@ mailing list is more appropriate for messages such
> as this. haskell@ is just for announcements (it should be called
> haskell-annouce@ !)
>
> >  * Lambda calculus - the basis of functional languages
> >
> >  * Category theory - where all these mysterious things like monads,
> > arrows, and functors come from.
>
> I'd add:
>
> * Discrete Maths - booleans, relations, functions etc.
>
> * Type theory
>
> * Logic programming (Prolog)
>
> * Semantics
>

Although it might technically be covered by "Discrete Maths", I'd like
to add algebraic and enumerative combinatorics to the list. That is,
the sort of mathematics where one deals with bijective decompositions
of combinatorial structures and generating series (generating
functions). The overall picture is that you put collections of
discrete structures in correspondence with algebraic objects of some
sort, typically the elements of a suitable ring, and make use of that
correspondence to move results back and forth.

I've found that a lot of the recent stuff like the theory of zippers
and differentiation of data structures directly reflects the sort of
combinatorial operations which go on in algebraic combinatorics. (See,
for instance, "Combinatorial Enumeration", by Jackson and Goulden, or
Joyal's combinatorial species.) No doubt there are more ideas which
could be fruitfully imported from there.