There is an inherent mathematical structure to the SI units and dimensions
beyond scale which are defined in the SI. (01)
The Base Quantities, length, mass, time, electric current, temperature, amount
of substance, and luminous intensity, whose dimensions, L, M, T, I, Θ, N, J,
form a generating set, using the operations of multiplication and
multiplicative
inverse, for the Quantity Dimensions. The Quantity Dimensions constitute a
representation of an abelian group. The Base Dimensions map 1:1 to the Base
Units (metre, kilogram, second, ampere, kelvin, mole, and candela). The
Quantity
Dimensions map 1:1 to the Coherent Derived Units. The Coherent Derived Units,
of
course, also form a representation of the same abelian group. This group is
isomorphic to the group of integers under addition, raised to the 7th power
(Z^7). (02)
If you prefer Quantity Dimensions raised to rational or real exponents instead
of simple integer exponents, it's a little different. (03)
The following mappings are important: (04)
Dim(Q) maps a quantity to its Quantity Dimension.
Numerical Value of a quantity, Q, is represented using braces, i.e., {Q}.
Measurement Unit (Coherent Derived Unit) of a quantity, Q, is represented using
rectangular brackets, i.e., [Q]. (05)
The Numerical Value and the Measurement Unit have the relation, {Q} = Q/[Q] (06)
A conversion scale factor for a non-SI unit, U, which is defined in terms of SI
units is just U/[U]. (07)
Addition of quantities is often allowable: when it is not allowable it is
generally because two quantities are not of the same Kind. (08)
I would think this all matters in the ontology. I have captured much of this
using OpenMath, but I haven't a clear idea how to do it in Owl. (09)
Regards,
Joe C. (010)
ingvar_johansson wrote:
> Ed B wrote:
>
>> The VIM speaks of a "scale" as a "sequence of marks", each "mark" being
>> associated with a quantity value.
>
> The VIM does not in its defintion of scale speak of "marks". Point 1.27 says:
>
> quantity-value scale (measurement scale) = ordered set of quantity values
> of quantities of a given kind of quantity used in ranking, according to
> magnitude, quantities of that kind.
>
> Ingvar J
>
>
>
>> That is, the "mark" that is "5" or
>> "5cm" is associated with "5 centimetres" and therefore with the
>> magnitude of length that is expressed by "5 centimetres". In a simlar
>> way, the second of four little lines on my thermometer between the line
>> labeled "20" and the line labeled "25" is a mark that is associated with
>> the quantity value "22 degress C" and refers to the magnitude of
>> temperature that is expressed by "22 degress C".
>>
>> The fundamental idea of VIM scales is that they are _sequences_ of marks
>> -- the marks are ordered -- and the mark ordering corresponds to an
>> ordering of quantity magnitudes.
>>
>> David's idea that a scale is a _set_ of symbols (marks) is too weak.
>>> [DL]
>>>> Hence re-expressing the consensus in these terms we have:
>>>>
>>>> scale: a mapping f from Q (set of magnitudes of quantity) to S (set of
>>>> symbols - commonly numbers), such that:
>>>>
>>>> f(q1) = f(q2) if and only if q1 = q2
>>>>
>>> [PH]
>>> Iff is surely too strong. And if true, this makes the two sets S and Q
>>> exactly equivalent, so why are we talking about two of them?
>>>
>>>
>> Whether this is too strong or not depends on what S is and what f is.
>>
>> In particular, a scale may be continuous or discrete. It only has the
>> marks it has. A "continuous" scale has one mark for each real number in
>> some interval, which makes the quantity values real numbers and creates
>> a nominal 1-to-1 and onto mapping from symbols/marks to quantity
>> magnitudes. A discrete scale has marks that are associated with
>> quantity values whose number parts are integers, or rationals with a
>> fixed denominator.
>>
>> For example, my temperature scale distinguishes 21 degrees from 22
>> degrees, but it has no mark that corresponds to 21.5 degrees -- that
>> magnitude will be interpreted as either 21 degrees or 22 degrees,
>> according to some convention. (If I talk about interpolating between
>> the scale marks, I really mean I have a conceptually continuous scale
>> that happens to have a physical implementation that only looks discrete.)
>>
>> So there are two mappings involved here. The principal mapping is:
>> m maps the set of marks S into the set of quantity magnitudes Q
>>
>> m is a function (OWL:functionalProperty), and it is 1-to-1
>> (OWL:inverseFunctionalProperty), but it is not onto unless m is
>> continuous (and S is 1-to-1 with Real quantity values).
>>
>> David refers to a function f such that
>> f maps the set of quantity magnitudes Q to the set of marks S
>>
>> What David says is only true if m is continuous.
>>
>> If f is understood to be the inverse of m and S is discrete, then f is
>> only defined on a countable subset of Q. When q1, q2 are in the image
>> of m, then David's axiom above holds:
>> f(q1) = f(q2) iff q1 = q2.
>> But for arbitrary q1 in Q, f(q1) is almost always undefined. (A
>> mathematician would say: f is undefined "almost everywhere".)
>>
>> Now in the discrete case, f may be an "extended inverse" of m, which
>> maps every q in Q to some s. For example, f may be defined to map q to
>> the mark s in S such that:
>> m(s) <= q and m(successor(s)) > q
>> (On my thermometer, this f maps every temperature in the half-open
>> interval [21 C, 22 C) to the 21 mark.)
>> Such an f is a function from Q to S that is well-defined everywhere,
>> that is:
>> for all q1, q2 in Q, if q1 = q2, then f(q1) = f(q2)
>> but David's axiom does not hold: many different q's can be mapped to the
>> same mark.
>>
>> David wrote:
>>> [DL]
>>>> ordinal scale: a scale where both Q and S are ordered, such that:
>>>>
>>>> f(q1) > f(q2) if and only if q1 > q2
>>>>
>> According to the VIM, S (the sequence of marks) is always ordered, and
>> quantity magnitudes of the same 'kind of quantity' are always ordered
>> (that is what is meant by "comparable").
>>
>> The VIM requires:
>> for all s1, s2 in S, s1 < s2 iff m(s1) < m(s2)
>>
>> And it is safe to assume that for an extended inverse f
>> for all q1, q2 in Q, q1 <= q2 iff f(q1) <= f(q2)
>> But unless m is continuous, and thus f is 1-to-1, we cannot conclude the
>> inequality:
>> for all q1, q2 in Q, if q1 < q2 then f(q1) < f(q2)
>>
>> [DL]:
>>>> ratio scale: a scale where ratios can be defined for both Q and S,
>>>> such that:
>>>>
>>>> r.f(q1) = f(q2) if and only if r.q1 = q2
>>>>
>>> [PH]
>>> What is r here? Any real number? What does it mean to multiply a
>>> number and a symbol? And do we really want to say that multiplication
>>> can be done directly to a magnitude?
>>>
>>>
>> Pat is right. The above is badly misstated. What a ratio scale
>> requires is:
>> (1) S is a quantity scale, a sequence of marks denoting quantity values
>> So each mark is associated with a (number, unit) pair.
>> (2) The magnitude q expressed by each such quantity value (n, unit) has
>> the property: m/unit = n
>> Which means that the notion "ratio of magnitudes" must be well-defined
>> for the kind of quantity measured by the scale. That is, there is a
>> function "ratio" that maps any pair of quantities (q1, q2) in Q into
>> some number field, usually the real numbers. And instead of
>> ratio(q1,q2), we use the notation q1/q2. This ability to construct a
>> ratio of magnitudes is a stronger requirement than the ability to
>> compare them. (Note that m and unit are both magnitudes of the same
>> quantity kind, which makes m/unit meaningful.)
>>
>> All the consequences for the scale fall out of these two requirements.
>>
>> It supports the idea of the scale as a kind of vector space over the
>> real numbers R in which, for r in R and q in Q, r*q is interpreted to
>> mean that quantity q1 in Q such that q1/q = r. But that model assumes
>> that the scale is continuous, i.e., that the number part of the quantity
>> values is Real, not Rational or Integer, and that m is a continuous
>> mapping from the number part of each s in S into Q. There are weaker
>> forms of this that work for rational scales.
>>
>> [PH]
>>> In this formulation, the (sets of) magnitudes and symbols are so
>>> similar that it hardly seems worth distinguishing them. They would be
>>> isomorphic, considered as mathematical categories. One could for
>>> example swap them without changing anything. Surely this cannot be
>>> right.
>>>
>> It isn't.
>>
>> Also, we have to be careful about the meaning of "ratio" for scales that
>> are logarthmic instead of affine. This is why we need an ontology that
>> covers the problem space, even if most of our interest is in continuous
>> scales based on a linear real axis. Such an ontology would allow us to
>> say that "hardness" satisfies axioms 1 thru 6, but not 7, and therefore
>> most of the rest of the ontology doesn't apply.
>>
>> -Ed
>>
>> --
>> Edward J. Barkmeyer Email: edbark@xxxxxxxx
>> National Institute of Standards & Technology
>> Manufacturing Systems Integration Division
>> 100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
>> Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694
>>
>> "The opinions expressed above do not reflect consensus of NIST,
>> and have not been reviewed by any Government authority."
>>
>>
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>
>
>
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> (011)
--
_______________________________
Joseph B. Collins, Ph.D.
Code 5583, Adv. Info. Tech.
Naval Research Laboratory
Washington, DC 20375
(202) 404-7041
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