> Date: Wed, 21 Dec 2005 09:42:44 -0800 (PST)
> From: Greg Woodhouse <[hidden email]>
> Subject: Monads and pedagogy (Functions with side-effects?)
> To: David Barton <[hidden email]>
> --- David Barton <[hidden email]> wrote:
> > Wolfgang Jeltsch writes:
> > ----- Original Message -----
> > > Am Mittwoch, 21. Dezember 2005 13:15 schrieb Creighton Hogg:
> > >> [...]
> > >
> > >> Monads, I believe, can be just thought of as containers for
> > >
> > > I would say that you are talking especially about the I/O monad
> > here. A
> > > monad
> > > as such is a rather general concept like a group is in algebra.
> > While this is correct, I'm afraid that for most of us it is a
> > flavorless
> > answer. I wish I had the mathematical mind that made the word
> > "group" in
> > this context instantly intuitively recognizable, but I don't.
> Groups are actually relatively intuitive structures, representing (no
> pun intended) the symmetries of some type of object. So, in teaching
> group theory it is easy to immediately provide a number of easy to
> grasp examples: the symmetric group (any kind of rearrangement), the
> cyclic group (cyclic permutation), dihedral group (geometric
> of a "square"), SL(n, R) ("area preserving" linear transformations),
> SO(n, R) (transformation preserving angles), Isom(M) (all geomteric
> motions of an object so that in the end it occupies the same "space"
> without any stretching or other deformation), etc., etc.
> Rings are a little more tricky, but R[x] (polynomials over R) is the
> universal example, with homomorphic images like Z[i] just being
> "polynomials in i" where you "reduce" according to i^2 = -1, and so
> Matrices over a ring give you the prototypical example of a module. I
> could go on, but the point is that in abstract algebra it is usually
> easy to provide simple intuitive examples of the structures
> illustrating most of the basic properties of the category. (Okay,
> topoi and coherent sheaves may be an exception! But even then,
> functions go a long way.)
> With monads, it is not hard to give a formal definition, just as is
> case with groups, but much harder (for me, anyway) to find a simple
> intuitive set of examples that capture the essence of monads in such
> way that the definition is obvious. Think about it: If groups are
> symmetries, then "doing nothing" is clearly a symmetry (1), "doing
> nothing" before or after a symmetry doesn't change anything.
> a symmetry gives you an inverse. Associativity is less obvious,
> you think of groups realized as permutations of a set, so that the
> group product just becomes function composition.
> Gregory Woodhouse <[hidden email]>
> "Interaction is the mind-body problem of computing."
> --Philip L. Wadler