In the thread titled 'Units of an angle' Patrick Cassidy wrote: (01)
> For an ontology that is intended to represent meanings, I am very
> leery of oversimplifications that work fine in restricted contexts but
> may prove confusing in missed contexts. (02)
This reminds me of this paragraph in the Wikipedia definition of the
units for torque [1]: (03)
The joule, which is the SI unit for energy or work, is dimensionally
equivalent to a N m, but this unit is not used for torque. Energy
and torque are entirely different concepts, so the practice of using
different unit names for them helps avoid mistakes and
misunderstandings. (04)
Which to me implies they will be different "things" in our ontology,
have different labels, and be disjoint in the same way that linear
measures and area measures are disjoint, like apples and oranges. (05)
Dimensional analysis by itself doesn't seem to be enough. (06)
> I think it is a misleading oversimplification, when taking ratios of
> things that are not themselves pure numbers, to ignore the meanings
> of the measures that are being divided. (07)
Indeed. (08)
So in our ontology there needs to be a way to say that "these units
belong to the same family" (inches, feet, meters, ...) or (radians,
angular degrees, ...) and "these do not" (energy, torque). I don't
think calling them dimensions is going to work because the term is so
heavily overloaded. (09)
So here is an attempt to formalize the concept, I'll introduce a new term: (010)
A 'decorated value' is thing that is a value from the set of
non-negative real numbers bound together with a unit. (011)
The intent is to distinguish between "dimensionless values" and those
with units in a way that doesn't use the word 'dimension'. I would also
remove negative numbers because that implies something closer to a
'vector', and this ontology should limit itself to 'magnitude'. (012)
Now I can describe a family: (013)
Two units belong to the same family if there is exists some
bijective function that allows values decorated by one unit to be
mapped into values decorated by the other. (014)
The idea is to cover both cases where the function is a simple
multiplier (x 1000), and more complicated (degrees F -> degrees C). And
by specifying bijective means that it is both one-to-one and onto. (015)
> I would suggest that we promiscuously include all quantifiable "units"
> that carry meaning in any application, and not take as "dimensionless"
> any measures that are in fact distinguishable in their intended
> meaning. (016)
I would somehow make a stronger statement that once a value has been
decorated, mathematical operations on values should be restricted to
those that have the same unit or are in the same family. (017)
> In this view, a 'radian' is a unit of measure, as is a
> 'degree-of-angle', and if the dimension is represented separately from
> the unit of measure, the dimension in either case would be 'angular
> measure'. (018)
I would say they belong to the same family, and don't forget 'grad'. [2] (019)
Joel
[1] <http://en.wikipedia.org/wiki/Torque#Units>
[2] <http://en.wikipedia.org/wiki/Grad_(angle)> (020)
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