On Tue, Sep 11, 2012 at 1:33 PM, Joe Niederberger <email@example.com> wrote: >>There is nothing that these operations *are* as distinct from our many models thereof. Even computation is modeling. > > I'm not sure what you mean here. When we "prove" things about computation, sure, models of computation are used. But computation happens in the physical world, real computers natural or man made (and finite in all respects) do the work. Since computation seems always to involve representations of something, we might say modeling is always involved? > > Joe N
I'm saying there's no "essence of multiplication" that we're trying to get at in flipping through this coffee table book of exotic and not so exotic examples.
It's an interesting book, with lots going for it, but communicating "the essence of multiplication" as a reward is neither promised on the dust cover nor used as a teaser in the introduction.
That being said, we have lots of family resemblances and if this were theater I'd say: if we're doing a play about multiplication, we need someone to play One. One is the entity that when you meet One, you remain the same. Then everyone needs an opposite. When you meet your opposite, you beget One.
Yes I know it sounds mystical, philosophical, and if I bring in CAIN (for properties of a group) it sounds downright Biblical. Oh well, when in Rome...
So to say there's "no essence" doesn't isn't meant to demean the search for commonalities. We're just not pretending there's some Holy Grail called "the essence of multiplication" and it's our business to lead a treasure hunt.
A barrier is people think that to talk this way is putting on airs as some conceited snob-genius wannabe, because it sounds just a tad more removed and aloof than common classroom math banter.
However, with programming languages in the picture, we have the ability to make the above entirely concrete. For example, if I want one of those permutation objects that multiplies, I can just go:
>>> a = anyperm( ) >>> b = anyperm( ) >>> c = a * b
Then feed any lowercase plaintext to a perm, and get encrypted text out the other end, a substitution code, something to play with. Something fun (for a change). But noooooo.... we're all too busy "aligning with common core standards" (snicker, chuckle, guffaw (so sad)).