----- Original Message -----
From: "Greg Woodhouse" <gregory.woodho...@sbcglobal.net>
To: "Hardhats" <hardhats-members at lists.sourceforge.net>; "Openhealth"
<openhealth at yahoogroups.com>
Sent: 2006-04-10 11:54 AM
Subject: [Hardhats-members] Archetypes for mathematicians?
> Forgive the cross-post, but the topic of archetypes has come up
> Hardhats, and once again, I find the language used to describe the
> concept vague and (at times) mysterious. If you'll forgive me for the
> use of some mathematical jargon here, I'd like to revisit the concept
> of abstracting measurements away from a particular choice of units. A
> familiar mathematical structure used to discuss the concept of distance
> is the metric space, defined as follows:
>
> A metric space (X, d) consists of a set X and map d: X x X --> R
> satisfying
>
> 1) d(x, y) >= 0
> 2) d(x, y) = 0 if and only if x = y
> 3) d(x, y) = d(y, x)
> 4) d(x, z) <= d(x, y) + d(y, z) (triangle inequality)
>
> Now, there are many metrics, not all of them arrising from Euclidean
> geometry, but certainly if X = R^2, the normal Euclidean metric
> provides a metric that we want to think of as independent of any
> particular scaling transformation. In particular, if alpha > 0, then
>
> d2(x, y) = alpha * d(x, y)
>
> is a perfectly good metric giving rise to a geometry that is
> essentially the same. In fact, a useful interpretation of the
> difference between d and d2 is that each represents a different set of
> units for the *same* metric space. But note that we "cheated" by
> introducing *two* metric spaces and then declared them to be the same,
> only having different units of measurement.
>
> Is there any other way to capture this sameness? Well, in truth, if you
> change the function d, then you have defined an entrirely new metric
> space, but the are not unrelated!
>
> Given spaces (X, d) and (Y, d) we say a function f: X --> Y is an
> isometry if
>
> 1) it is one to one and onto
> 2) for all x1, x2 in X, f(d(x1, x2) = d(f(x1), f(x2))
>
> If there exists an isometry (any isometry) between two spaces, we say
> they are *isometric*. It turns out that isometry is an equivalence
> relationship; i.e., it satisfies
>
> a) a ~ a
> b) if a ~ b then b ~ a
> c) if a ~ b and b ~ c, then a ~ c
>
> (where = is just a name for the relation in question).
>
> Mathematically, it seems natural to abstract distance away from a
> particular choice of units by passing to the equivalance class of
> spaces equivalent under isometry (written X/~). But there is a problem:
> The map pi : X --> X/~ associating earch metric space with its
> equivalence class destroys any privileged status a metric space in the
> equivalence class might have. If you like, for x in X/~ the fiber over
> x (pi^-1(x)) is just a set with no distinguished elements.
>
> So, what can be done? How can we hold onto the idea that there are
> parti cular units of measurement in which we might be interested
> (inches, centimeters, cubits) without sacrificing the idea that no
> particular system of units is privileged above the others?
>
> It seems a little formal, but one possibility is to define 1 as "the"
> set with one element. In fact, there are many, many such sets, but all
> of what follows can be shown to be independent of that choice. A
> "pointed" set is just a nondegenerate map 1 --> X, that is a function
> that picks out a distinguished element of the category of sets.By
> fixing a map 1 --> X, you are essenrially selecting units for your
> metric space. Now, if 1 in R is the number 1, then you can actually
> draw a communtative diagram
>
> X x X ----> R
> | |
> v v
> 1 x 1 ----> R
>
> Now, given an isometry X --> Y, is it possible to choose units in a
> natural way, so that change of units will give you a new isometry? The
> answer is yes and, in fact if f : X --> X is a change of unit, what is
> required is a *functor* T tranforming f to T(f) (making the above into
> a commutative cube!) in such a way that an isometry between metric
> spaces can be naturally viewed as an isometry between pointed spaces
> (spaces with units). Such a thing is called a natural transformation
> and it gives content to the vague assertion that the choice of 1 really
> doesn't matter.
>
> Intuitively, what this all means is that it's not enough to just choose
> units of meansurement, you need to be able to describe how that choice
> fits into the rest of your system. In practice, you need to fix a
> representation for storage, but that representation must not be
> privileged, and that can be avoided by describing in mathematical terms
> how a concrete model is transormed when the choice of representation
> changes.
>
> I do not know if I've missed the point of archetypes, but it does seem
> to me that functorial language can at least clarify what it means to
> get a value "through an archetype". To an outsider, this makes little
> sense because it seems to be mixing categories, and because an
> archetype should not really depend up on particular choice of
> represntation (units).
>
> ===
> Gregory Woodhouse <gregory.woodhouse at sbcglobal.net>
>
> "It is foolish to answer a question that
> you do not understand."
> --G. Polya ("How to Solve It")
>
>
> -------------------------------------------------------
> This SF.Net email is sponsored by xPML, a groundbreaking scripting
language
> that extends applications into web and mobile media. Attend the live
webcast
> and join the prime developer group breaking into this new coding
territory!
> http://sel.as-us.falkag.net/sel?cmd=lnk&kid=110944&bid=241720&dat=121642
> _______________________________________________
> Hardhats-members mailing list
> Hardhats-members at lists.sourceforge.net
> https://lists.sourceforge.net/lists/listinfo/hardhats-members
>
