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## [uom-ontology-admin] Note on CLIF draft - approach to scale

 To: Pat Hayes uom-ontology-admin David Leal Tue, 05 Jan 2010 18:09:55 +0000 <1.5.4.32.20100105180955.0237638c@xxxxxxxxxxxxxxxx>
 ```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) ============================================================ David Leal CAESAR Systems Limited registered office: 29 Somertrees Avenue, Lee, London SE12 0BS registered in England no. 2422371 tel: +44 (0)20 8857 1095 mob: +44 (0)77 0702 6926 e-mail: david.leal@xxxxxxxxxxxxxxxxxxx web site: http://www.caesarsystems.co.uk ============================================================    (021) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/uom-ontology-admin/ Config/Unsubscribe: http://ontolog.cim3.net/mailman/listinfo/uom-ontology-admin/ Shared Files: http://ontolog.cim3.net/file/work/UoM/ Wiki: http://ontolog.cim3.net/cgi-bin/wiki.pl?UoM_Ontology_Standard    (022) ```
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