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Re: Non-Euclidean Arithmetic
Posted:
Sep 11, 2012 5:11 PM
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On Tue, Sep 11, 2012 at 1:33 PM, Joe Niederberger
<niederberger@comcast.net> 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)).
Kirby
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