ONT Re: Data Models, Ontologies, Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Note 19
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
The Oath of Eratosthenes, also known as "The Sifter":
| I will never forget that the most important thing about the datum
| is how it refers to the objective world from which it is gathered,
| in the way of all signs to be abstract partial samples of objects,
| by virtue of which sampling they denote the vaster reality beyond.
From a forged letter, of course.
O! What a tangled web we sieve!
After being up til 6 am trying to recover from a power outage,
you would think I'd know better than to try and think de-caff.
Anyway, I got a little bit ahead of myself on that last Figure.
What I really need to illustrate in the immediate setting is a
couple of "fibers", "level sets", or "pre-images" -- just some
of the different strokes that different folks use for the idea --
of different functional values within the same data dimension.
To this end, I introduce the following notions and notations:
Given the "coding function" or the "measuring mapping" m : X -> U,
and the "data to data projection" u_j : U -> U_j let us determine
the "object to data projection" m_j : X -> U_j in the usual way:
| m_j
| X o---------->o U_j
| \ ^
| \ /
| m \ / u_j
| \ /
| v /
| o
| U
|
| m_j (x) = u_j (m(x))
I worry a bit about this formulation,
for reasons of its empirical reality,
but that is the general idea we need,
so I will use it until I wear it out.
Let U_j_a, the a^th 'fiber' of u_j,
be the inverse image (u_j)^(-1)(a).
Let X_j_a, the a^th 'fiber' of m_j,
be the inverse image (m_j)^(-1)(a).
Here is the amended Figure that I should've drawn at this stage:
o-----------------------------------------------------------o
| X |
| |
| o-------------o o-------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | X_j_a | | X_j_b | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| | |
o-----------------------------|-----------------------------o
|
| m
|
o-----------------------------|-----------------------------o
| U v |
| |
| o-------------o o-------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | U_j_a | | U_j_b | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-----------------------------------------------------------o
I hope I finally got it right this time, but I'll
need to take a break to think about it for a while.
Jon Awbrey
Incidental Musement:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Eratosthenes.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o