Joe Collins wrote: (01)
> There is an inherent mathematical structure to the SI units and dimensions
> beyond scale which are defined in the SI.
>
> 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. (02)
In my paper "Two Changes in the International System of Units?", which I
mentioned in my former mail, I argue that it is important not to conflate
*metrological* multiplication and division of dimensions with
*arithmetical* multiplication and division. If one does, one ends up in
proposing (as VIM and the SI system do) the dimension 'dimension one'. (03)
Ingvar (04)
> 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).
>
> If you prefer Quantity Dimensions raised to rational or real exponents
> instead
> of simple integer exponents, it's a little different.
>
> The following mappings are important:
>
> 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].
>
> The Numerical Value and the Measurement Unit have the relation, {Q} =
> Q/[Q]
>
> A conversion scale factor for a non-SI unit, U, which is defined in terms
> of SI
> units is just U/[U].
>
> Addition of quantities is often allowable: when it is not allowable it is
> generally because two quantities are not of the same Kind.
>
> 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.
>
> Regards,
> Joe C.
>
> 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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>
> --
> _______________________________
> Joseph B. Collins, Ph.D.
> Code 5583, Adv. Info. Tech.
> Naval Research Laboratory
> Washington, DC 20375
> (202) 404-7041
> (202) 767-1122 (fax)
> B34, R221C
> _______________________________
>
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