Dear John, (01)
> MW> Another obvious question is the relationship between a scale
> and
> > a unit of measure. They are not the same thing, but just how are
> > they related?
>
> In an earlier note (but I forget which one), someone pointed to
> a nicely organized ranking of terms, each of which includes the
> previous as a special case:
>
> 1. Set: A set of values with no ordering.
>
> 2. Linear order: A set of values with a relation such as "less
> than"
> but no significant metric between members of the set.
>
> 3. Interval: A linear order with a metric, such as the real
> numbers,
> for which the distance between two points is significant, but
> there is no preferred origin (or zero).
>
> 4. Scale: A linear order with a metric for which the distance
> from a preferred origin (or zero) is significant.
>
> The question of what is significant is determined by the kinds
> of axioms, but different axioms may have different implications.
>
> The temperature scale is a good example, because it has several
> different kinds of axioms. The following two determine a zero
> point:
>
> Gas law: The pressure times the volume of a gas is proportional
> to the temperature T (as measured from absolute zero).
>
> Boltzmann's law: The amount of energy radiated from an object
> per unit of time is proportional to T^4 (where T is measured
> from the same zero point determined by the gas law).
>
> But there are also interval axioms that involve temperature:
> Heat transfer by conduction is proportional to (T1 - T2), where
> T1 and T2 are the temperatures at the end points of the transfer.
>
> For conduction, that implies temperature behaves like an interval
> measure. But heat transfer by radiation is proportional to T^4,
> which depends critically on the absolute temperature.
>
> The first Google hit for heat transfer is a good summary:
>
> http://sol.sci.uop.edu/~jfalward/heattransfer/heattransfer.html
>
> Bottom line: It's important to be aware of these issues, but
> the detailed axioms for each kind of measure would belong in
> appropriate microtheories. In fact, a system for weather reports
> would use different microtheories than one for designing ovens
> or insulating houses. (02)
MW: I don't see what this tells us about the relationship between UoM
and scales. (03)
Regards (04)
Matthew West
Information Junction
Tel: +44 560 302 3685
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>
> John
>
>
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