A classical reference regarding concepts in metrology is Krantz et al
http://www.amazon.com/Foundations-Measurement-Volume-Representations-Mathematics/dp/0486453146 (01)
. (02)
Since it is not downloadable, I'll attempt to give a summary of the
points I think are relevant to the development of ontologies of measurement. (03)
The foundational concepts employed in this reference are
* a "relational structure", consisting of a set of mathematical entities
(such as positive real numbers) and one or more relations among them.
* A "numerical assignment" is a homomorphism that maps physical entities
and measurement procedures into a relational structure in a way that the
properties of the procedures are preserved. (04)
From these foundations, it can be shown that measurement theories
corresponding to particular relational structures may be grouped into
families according to the "permissible transformations" that may be
applied to a numerical assignment that result in another numerical
assignment. The familiar Stevens scale types correspond to certain families: (05)
* Ratio scale: entities are 1-dimensional (e.g. real or positive real
numbers), and permissible transformation are only multiplication by a
positive scalar constant. Extensive measurements such as length fall
into this family. (06)
* Log-Interval scale: entities are again 1-dimensional, but typically
only the positive real numbers, and permissible transformations include
raising to a power in addition to multiplication by a positive scalar
constant. Intensive measurements such as absolute temperature and
density fall into this family. It's called log-interval because taking
the log gives a theory in the "interval-scale" family, and Stevens
recognized its existence in his original work, although it is usually
not included in elementary treatments today. (07)
and so on. For ordinal-scale measurements, permissible transformations
include any strictly monotonic function. (08)
However, there are plenty of measurement theories that do not fit into
these families, notably, multi-dimensional theories. (09)
Tara (010)
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