Dear David and Matthew, (01)
DL> I have proposed that a scale be defined as a function from a kind
> of quantity to a "set of elements with axioms", such as the real
> numbers. (02)
That is important, but there are further issues about the kinds
of axioms, which may include more than just a metric (a distance
measure between elements). (03)
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? (04)
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: (05)
1. Set: A set of values with no ordering. (06)
2. Linear order: A set of values with a relation such as "less than"
but no significant metric between members of the set. (07)
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). (08)
4. Scale: A linear order with a metric for which the distance
from a preferred origin (or zero) is significant. (09)
The question of what is significant is determined by the kinds
of axioms, but different axioms may have different implications. (010)
The temperature scale is a good example, because it has several
different kinds of axioms. The following two determine a zero
point: (011)
Gas law: The pressure times the volume of a gas is proportional
to the temperature T (as measured from absolute zero). (012)
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). (013)
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. (014)
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. (015)
The first Google hit for heat transfer is a good summary: (016)
http://sol.sci.uop.edu/~jfalward/heattransfer/heattransfer.html (017)
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. (018)
John (019)
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