Oops...meant to send this to the list, too. --- Greg Woodhouse <[hidden email]> wrote: > 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 > state. > > > > > > 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 > symmetries > 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 > on. > 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, > okay, > topoi and coherent sheaves may be an exception! But even then, > analytic > functions go a long way.) > > With monads, it is not hard to give a formal definition, just as is > the > 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 > a > way that the definition is obvious. Think about it: If groups are > just > symmetries, then "doing nothing" is clearly a symmetry (1), "doing > nothing" before or after a symmetry doesn't change anything. > "Undoing" > a symmetry gives you an inverse. Associativity is less obvious, > unless > 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 > _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Powered by Nabble | Edit this page |