On Jan 5, 2010, at 12:09 PM, David Leal wrote: (01)
> Dear Pat,
>
> 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)
Email is fine. I'm not a great believer in procedures :) (03)
>
> 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". (04)
True, which is why I am avoiding saying very much about these
mappings, other than that they exist and what their domains and ranges
are. (05)
> 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. (06)
Hmm, not sure about that. There can be 'coarse' scales, for example,
which do not make distinctions between different quantity values.
Qualitative reasoning often uses {negative, zero, positive} as a crude
measure scale, and yet is capable of quite surprisingly sophisticated
analyses. (07)
> 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. (08)
It is arbitrary, but it can always be done, I think. (Do you agree?) I
would prefer to say that in cases like this, the scale is simply
underspecified as stated, and the actual scale is either [180, 180)
or (180, 180] (09)
> 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.
>
> 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. (010)
We should describe the structure of our scales so as to eliminate this
kind of ambiguity. Perhaps the results will not be conventional
numbers, but then scale values are often not treated as purely
numerical in any case. For example, we can simply declare that in the
Euler angle scale (writing (Euler theta phi) for such a 3d angle) that (011)
(forall (x y)(= (Euler 0 x)(Euler 0 y)) (012)
thereby defining the actual scale as a mathematical object with the
correct topology. (013)
>
> Order
> 
> You cannot classify a scale as an OrderScale unless the quantity
> values have
> an order. (014)
Sure I can :). I can simply assign things to points on an order scale
according to some fixed but arbitrary rule. Rockwell hardness is
pretty arbitrary, in fact. (015)
> 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. (016)
Surely that is about the appropriateness of the scale to measure the
quantity, rather than about the structure of the scale itself. A
scale, considered in isolation, is simply a mathematical abstraction. (017)
> If we formalise scale as a function, then there is already a term
> for this  "increasing function" see
> http://mathworld.wolfram.com/IncreasingFunction.html .
>
> 2D angle is almost certainly within scope but is only ordered locally. (018)
? I do not understand this. Seems to me that angles have no boundary
in either direction. Engineers talk of rotations through angles
greater than 360. Angles used to measure relative orientation of
lines in space are of course in the range (0 360) or perhaps (180,
180), but angle has other uses. (019)
> 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.
>
> 3D angle is not ordered. We still have a requirement for a "good
> function",
> so this must be expressed in terms of continuity.
>
> Continuity
> 
> We cannot define continuity for a scale unless we have a concept of
> closeness between quantity values. (020)
Again, the *scale* can be continuous even when the quantity is not
(though I agree this would be a highly improper use in most
circumstances.) (021)
> 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. (022)
But how is one to express this independently of any scale? I am
reluctant to assert that quantity values themselves satisfy any
particular algebraic or topological axioms. What is the meaning of
subtracting one Rockwell hardness from another? (023)
> 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. (024)
Right, but again, it seems like hubris to assert topological
properties of real things like quantity values. (?) Bear in mind that
at this level of generality, a set of quantity values can be almost
anything, including things like the pharmaceutical 'amounts of
dilution' and other unscientific exotica. (025)
>
> 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. (026)
True, but that is because we have a sophisticated understanding of
exactly what underlies the concept of temperature. But the Fahrenheit
and Centigrade scales were invented and used long before this
understanding was developed, so they cannot depend upon it for their
basic ontological description, surely. (027)
>
> 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. (028)
How is it known that the 'degree' between, say, 0F and 1F, is the same
"amount" of temperature as that between, say, 1000F and 1001F? Does
this assertion even have a clear physical meaning? (029)
> Celsius and Fahrenheit are affine but not linear.
>
> 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. (030)
The latter depends upon the former, in fact. 3 x foo is (foo + foo +
foo) (031)
> Hence we assume an algebraic structure for the quantity values. (032)
Yes, we could do that, for some quantity values. I had thought to
characterize the value types by referring to the mathematical
structures of the scales used to measure them, rather than the values
themselves. (033)
>
> Conversion between scales
> 
> Because scales are functions, a scale conversions is not expressed
> by an
> equation, such as (= (meter 1)(foot 3.281)). (034)
This is sufficient when the scales are both unit scales. I will spell
this out in full. (035)
> The counter example is (=
> (celsius 40)(fahrenheit 40)). This is true but not enough.
>
> The relationship between two scales is itself a function. Consider
> celsius:R
> > T and fahrenheit:R > T . (036)
These are not unit scales. The general conversion is an equation with
a zero shift, yes. (037)
>
> We can define a function X:R > R, such that X(x) (5/9)*(x32) for
> all real x.
>
> 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 .
>
> Best regards from a conservative applied mathematician,
> David
>
> At 02:47 04/01/2010 0600, you wrote:
>> I have put up some preliminary notes at
> http://ontolog.cim3.net/cgibin/wiki.pl?UoM_Ontology_Standard_CLIF_Draft
>>
>> Pat
>
> ============================================================
> 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
> email: david.leal@xxxxxxxxxxxxxxxxxxx
> web site: http://www.caesarsystems.co.uk
> ============================================================
>
>
> (038)

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