Dear Matthew and Ed, (01)
The distinction of three kinds of linear ordering is a useful
classification, but by itself it does not specify all significant
relationships. (02)
MW> I don't see what this [a discussion of temperature] tells us
> about the relationship between UoM and scales. (03)
It addresses some of the issues that Ed mentioned: (04)
EB> An absolute zero _alone_ tells you nothing about "intervals".
> As Pat pointed out, you have to have some notion of "increment"
> (Matthew's +1). It doesn't mean anything to "add an extra 10
> degrees" if "10 degrees" only means an absolute temperature. What
> you are assuming is that the absolute meaning of "10 degrees" is
> 10 unit amounts of temperature change from absolute zero, where
> the "unit amount of temperature change" from 10 degrees to 20
> degrees is the same as the "unit amount of temperature change"
> from 0 to 10 degrees. There are scales that have absolute zero
> but do not have that property. (05)
The temperature scale is an important example because it behaves
like an interval measure with respect to some axioms, but not others. (06)
For the axiom of heat conductivity, a one-degree interval (either
C or K) is sufficient to determine the rate of heat flow. (07)
But for the axiom of heat radiation, a one-degree interval at
the temperature of a hot oven determines a much higher heat flow
than a one-degree interval for a cool oven. (08)
This observation implies that an interval measure that is
truly independent of the origin (or zero point) is the
only kind for which you can depend on an interval measure
(such as a yard stick) to be consistently reliable. (09)
For any kind of measure that has axioms that determine an
absolute zero, there may be some axioms that behave like
an interval measure. But any axioms that are critical to
determining the zero point are likely to be sensitive to
the placement of zero. (010)
John (011)
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