Hitting two targets: OO + group theory (pedagogy)

[David C. Ullrich]

On Sat, 03 Mar 2001 09:16:14 -0800, Kirby Urner
<urner at alumni.princeton.edu> wrote:

>ullrich at math.okstate.edu (David C. Ullrich) wrote:
>
>>Some people would say that the reference to Genesis
>>is out of place here... probably you know that some
>>people would say that. Might be regarded as just
>>generally inappropriate - some other people might
>>think that the bit about Abel is misleading, since
>>"Abelian" _does_ refer to a man named Abel, but
>>not the one in Genesis.
>
>True, but it's a useful mnemonic connected to a 
>well-known story.  Making allusions to Biblical 
>characters need be no more or less churchy than 
>alluding to Greek mythology or to cartoon characters
>like Bugs Bunny (or Coyote in Native American lore)
>-- just taking advantage of some shared cultural 
>heritage.  

Right. I didn't say it was inappropriate. Although in fact
something about the bit about "If you've studied Genesis"
or however you put it seems a little off-putting; if
you were referring to Bugs Bunny or Zeus you'd
just refer to them.

Never mind - could be I've been in Okllahoma
too long.

>Anyway, I learned this mnemonic from a very strait-
>laced group theory book (which I've since returned 
>to the library), plus saw it mentioned on another 
>list frequented by math heads -- sort of assumed 
>this clever CAIN/Abel thing was cited routinely 
>by those more familiar with the literature.  Agreed 
>though: we should make clear who the Abel in Abelian 
>was.
>
>>But why not start with a class Group instead
>>of special cases? (Would be a class with a
>>__mul__ method that does nothing but raise
>>an exception "must implement __mul__ in
>>a subclass", an Id attribute, etc. I can't think
>>of anything offhand that Group would actually
>>_do_, but if we're bent on this OO-math thing
>>it seems like groups _should_ be descendants
>>of a class Group...)
>>
>
>In my example, it's the elements of a group which
>get instantiated as separate objects, with each 
>element inheriting the same meaning for *, whatever
>that might be. 

Right. That's actually what I meant, a class Group
instances of which would represent elements
of a generic group - of course there's no such
thing so you'd never create instances of Group.
Then subclasses of Group represent actual
groups; instances of the subclasses being
elements of actual groups.

Not important, just seems to me like what it
"should" be.