SUO: Representation Invariant Ontology
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RIO. Note 1
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Recent discussions have led me to see that some basic ideas about
the relationship between scientific ontologies and the properties
of mappings between syntactic systems, often confused with object
realities, are far less well understood than I would have wished
to believe. In the spirit of starting with first things first,
let me introduce what may well be one of the simplest cases
where these sorts of issues arise.
"Labels Are Fables" is a piece of folk wisdom,
among mathematicians and many other wise folk.
The distinction between mathematical objects and their representations
is a case in point, and one of the first places where we run up against
this theme is in the distinction between points in some objective space,
mathematical or physical, it doesn't matter much at this point, and their
coordinate images with regard to a particular choice of coordinate systems.
For example, consider the point p of the space M that is illustrated in Figure 1.
o-----------------------------------------------------------o
| M |
| |
| o-------------o o-------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | p | | |
| | U | o | V | |
| | | | | |
| o o o o |
| \ \ W / / |
| \ \ / / |
| \ o / |
| \ | / \ | / |
| o------|------o o------|------o |
| | | |
| | | |
o--------------------|-----------------|--------------------o
| |
f | | g
| |
o--------------------|-----o o-----|--------------------o
| X v | | v Y |
| | | |
| o----------o | | o----------o |
| / \ | | / \ |
| / o | | o \ |
| / / \ | | / \ \ |
| / / \ | | / \ \ |
| o o o | | o o o |
| | | f(p)| | | | g(p)| | |
| | f(U) | o | | | | o | g(V) | |
| | | | | | | | | |
| o o o | | o o o |
| \ \ / | | \ / / |
| \ \ / | | \ / / |
| \ o | | o / |
| \ / | | \ / |
| o----------o | | o----------o |
| | | |
| | | |
o--------------------------o o--------------------------o
Figure 1. Overlapping Coordinate Maps f : U -> X, g : V -> Y
By way of concrete illustration, we may imagine that two observers
have mapped overlapping pieces of territory, one of them assigning
labels to points according to the mapping f : U -> X and the other
assigning labels to points according to the mapping g : V -> Y.
The objective domains of the surveys are U, V c M.
The coordinate codomains are respectively X and Y.
The focal point p is contained in the intersection
W = U |^| V, and so p is assigned what may well be
the different coordinates f(p) in X and g(p) in Y.
The point is this: There is nothing very "essential" or "ontic"
about any observer's coordinate labels -- they can in general be
almost wholly accidental, arbitrary, ephemeral, or phenomenal in
in their basic character. If you want to get at the "ontology"
of geometric points, mathematical or physical, you have to look
a lot deeper than the superficialities of coordinate systems.
Jon Awbrey
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