Dear Pat, (01)
Thank you for this useful and clear note. I am not clear what the procedure
will be for raising issues or suggesting clarifications, but I have a few
comments on your approach to scale. (02)
Classification of a scale
-------------------------
Taking a scale to be a mapping between numbers and "quantity values of a
particular kind", there is only a little we can say about the nature of the
mapping before assuming something about the structure of the "quantity
values of a particular kind". (03)
Cardinality
-----------
Without structure we can just talk about cardinality. It is reasonable for
us to insist that in a scale, a number maps to only one quantity value.
However a quantity value can map to more than one number. An example of this
the mapping between real numbers and angle. If the range of the real
numbers is [-180, +180], then there is an angle which maps to both -180 and
+180. We can get the problem to go away in this simple case by making the
interval of numbers open at one end, but this may be inconvenient. (04)
In more complicated cases, even this does not work. Consider angle in 3D -
i.e. a direction defined by the Euler angles theta and phi as in
http://conventions.cnb.uam.es/Submit/Euler.jpg . (We have ruled non-scalar
quantities out of scope, but it is useful to consider a more general case.)
The same direction is defined for theta=0, whatever the value of phi. (05)
The source of the problem is the same for angle in 2D or 3D. It is because
the space of quantity values is topologically equivalent to a circle or
sphere, whilst the space of the numbers is topologically equivalent to a
line or square. (06)
Order
-----
You cannot classify a scale as an OrderScale unless the quantity values have
an order. The property that we are looking for in a scale is that the order
of the numbers is consistent with the order of the quantity values that they
represent. If we formalise scale as a function, then there is already a term
for this - "increasing function" see
http://mathworld.wolfram.com/IncreasingFunction.html . (07)
2D angle is almost certainly within scope but is only ordered locally. Hence
the scale for angle is not an increasing function everywhere. However we can
choose a function which looks locally like an increasing function almost
everywhere. (08)
3D angle is not ordered. We still have a requirement for a "good function",
so this must be expressed in terms of continuity. (09)
Continuity
----------
We cannot define continuity for a scale unless we have a concept of
closeness between quantity values. This relies neither upon an order for the
quantity values nor upon a "zero quantity value". Intuitivly a scale is a
continuous function if closeness between numbers (or vectors of numbers)
implies closeness between corresponding quantity values. (010)
There is a definition of continuous function which works for any dimension
of the domain and range - see
http://mathworld.wolfram.com/ContinuousFunction.html . The definition relies
upon our ability to define an "open set" of quantity values. Hence we assume
a topological structure for the quantity values. (011)
Linearity
---------
Whether or not a quantity value can be multiplied by a real number is
nothing to do with scale. The statement that Ta = 2.Tb , where Ta and Tb are
values of thermodynamic temperature, is well defined irrespective of scale. (012)
The concept we require is "linear function", in the narrow sense - see
http://mathworld.wolfram.com/LinearFunction.html. Hence if x = 2.y, then
(the quantity value represented by x) = 2.(the quantity value represented by
y). Unfortunately the term "linear" is sometimes used to include affine
functions - see http://mathworld.wolfram.com/AffineFunction.html . Kelvin
and Rankine are linear scales. Celsius and Fahrenheit are affine but not linear. (013)
The definition of linearity relies upon our ability to define addition
between quality values and the multiplication of a quantity value by a real
number. Hence we assume an algebraic structure for the quantity values. (014)
Conversion between scales
-------------------------
Because scales are functions, a scale conversions is not expressed by an
equation, such as (= (meter 1)(foot 3.281)). The counter example is (=
(celsius -40)(fahrenheit -40)). This is true but not enough. (015)
The relationship between two scales is itself a function. Consider celsius:R
-> T and fahrenheit:R -> T . (016)
We can define a function X:R -> R, such that X(x) (5/9)*(x-32) for all real x. (017)
Hence we can say that Fahrenheit(x) = Celsius(X(x)). Or Fahrenheit =
Celsius*X, where * is the composition operator - see
http://mathworld.wolfram.com/Composition.html . (018)
Best regards from a conservative applied mathematician,
David (019)
At 02:47 04/01/2010 -0600, you wrote:
>I have put up some preliminary notes at
http://ontolog.cim3.net/cgi-bin/wiki.pl?UoM_Ontology_Standard_CLIF_Draft
>
>Pat (020)
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CAESAR Systems Limited
registered office: 29 Somertrees Avenue, Lee, London SE12 0BS
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tel: +44 (0)20 8857 1095
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