Ed B wrote: (01)
> The VIM speaks of a "scale" as a "sequence of marks", each "mark" being
> associated with a quantity value. (02)
The VIM does not in its defintion of scale speak of "marks". Point 1.27 says: (03)
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. (04)
Ingvar J (05)
> 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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