David & Pat, (01)
May I suggest that this thread will conducted under the
[uom-ontology-std] mailing list (which is for technical discussions
like this, and has a much wider subscription base) rather than having
it on the [uom-ontology-admin] mailing list (as you are doing now.
The [uom-ontology-admin] is more for administrivia and for getting
things coordinated.) (02)
Will the next person who respond to this so put this discussion thread
(by posting to <uom-ontology-std@xxxxxxxxxxxxxxxx> instead). To
provide context (to the wider audience), kindly add the link to
<http://ontolog.cim3.net/cgi-bin/wiki.pl?action=browse&id=UoM_Ontology_Standard_CLIF_Draft&revision=2>
somewhere, which this conversation is referencing. (03)
Thanks in advance. =ppy
-- (04)
On Tue, Jan 5, 2010 at 3:47 PM, Pat Hayes <phayes@xxxxxxx> wrote:
>
> On Jan 5, 2010, at 12:09 PM, David Leal wrote:
>
>> 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.
>
> Email is fine. I'm not a great believer in procedures :-)
>
>>
>> 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".
>
> 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.
>
>> 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.
>
> 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.
>
>> 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.
>
> 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]
>
>> 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.
>
> 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
>
> (forall (x y)(= (Euler 0 x)(Euler 0 y))
>
> thereby defining the actual scale as a mathematical object with the
> correct topology.
>
>>
>> Order
>> -----
>> You cannot classify a scale as an OrderScale unless the quantity
>> values have
>> an order.
>
> 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.
>
>> 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.
>
> 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.
>
>> 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.
>
> ? 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.
>
>> 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.
>
> 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.)
>
>> 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.
>
> 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?
>
>> 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.
>
> 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.
>
>>
>> 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.
>
> 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.
>
>>
>> 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.
>
> 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?
>
>> 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.
>
> The latter depends upon the former, in fact. 3 x foo is (foo + foo +
> foo)
>
>> Hence we assume an algebraic structure for the quantity values.
>
> 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.
>
>>
>> Conversion between scales
>> -------------------------
>> Because scales are functions, a scale conversions is not expressed
>> by an
>> equation, such as (= (meter 1)(foot 3.281)).
>
> This is sufficient when the scales are both unit scales. I will spell
> this out in full.
>
>> 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 .
>
> These are not unit scales. The general conversion is an equation with
> a zero shift, yes.
>
>>
>> We can define a function X:R -> R, such that X(x) (5/9)*(x-32) 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/cgi-bin/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
>> e-mail: david.leal@xxxxxxxxxxxxxxxxxxx
>> web site: http://www.caesarsystems.co.uk
>> ============================================================
>>
>>
>>
>
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>
>
>
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