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 [EMAIL PROTECTED]
It is foolish to answer a question that
you do not understand.
--G. Polya (How to Solve It)
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