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Re: [Phys-L] temperature +- thermometry +- interpolation +- abstraction

The original question combined /two/ issues:
-- thermometry, and
-- the definition of temperature.

It would be good to separate these. Alas my previous
answer didn't do a good job of that.

As for the definition of temperature:
If you ask me, there is only one definition. If
you want to be extra-explicit, "the" temperature
means the /thermodynamic temperature/.

As for thermometry:
That word can be applied to the business of making and
calibrating thermometers. In this business, there is
definitely a role for interpolation. That's because
we are faced with what's known as a bias/variance
tradeoff, or in this case a bias/variance/cost tradeoff.

1) It is common practice to calibrate thermometers
against a /secondary/ standard. This introduces
a small amount of systematic bias, due to the
built-in imperfections of the standard.
2) In contrast, calibrating things from first
principles would either:
2a) Introduce too much erratic variance, or
2b) Cost too much.

Consider in particular the ITS-90 temperature standard.

It does *not* say you can linearly interpolate willy-
nilly; instead they spent years and years identifying
14 specific things that you can interpolate -- and some
of the required interpolations are nonlinear.

Perhaps most importantly, nobody thinks ITS-90 defines
"the" temperature. It is unabashedly secondary. It is
an approximation to the thermodynamic temperature. When
there is found to be an appreciable difference between
ITS and the thermodynamic temperature, they change the
definition of ITS, not the definition of temperature.
This has happened before and will happen again.


Funny bias/variance story: Above 18,000 feet, airplanes
tend to cruise at some definite /pressure altitude/.
That's a bit of a misnomer; really it's just some
definite pressure, converted to units of altitude via
a peculiar formula.

As long as everybody plays by the same rules, this works
great for separating one aircraft from another. If the
other guy is 1000 feet above you, in terms of pressure
altitude, you're not going to run into him.

However, mountains do not play by the same rules. They
have a definite true altitude. Depending on temperature
and barometric factors, the pressure altitude could be
biased by more than a thousand feet relative to the true
altitude. So you'd better not rely on it for terrain
clearance, unless there is a huuuge safety margin.

This is an example of accepting a huge bias in exchange
for reducing the variance. It is mostly a good tradeoff
in this case, with a few well-understood limitations.

Similar words apply to thermometry. If your only goal
is to reproducibly compare one thermometer against
another, you can calibrate both of them to ITS-90,
without worrying how closely they conform to the
true thermodynamic temperature.


There is a pedagogy angle to this, too. Every so often
you see somebody fall in love with the idea of "operational"
definitions. That is, pressure is defined to be whatever
the dial on the pressure gauge says, temperature is defined
to be whatever the dial on the thermometer says, et cetera.

Usually this doesn't last very long. It seems conceptually
simple for about a millisecond, but then they discover
that they've got a mercury-in-glass temperature scale and
a methanol temperature scale and and and and ... which is
not simple at all, not conceptually, not operationally, not
in any other way.

The alternative is to embrace the abstractions. Consider
twelve carrots; that's something tangible. Twelve books
is tangible. Twelve chairs is tangible. But what about
twelve itself? That's an abstraction. It's really,
really abstract. And (!) the fact that it's abstract
makes it simpler and more useful. You get to apply it
to carrots and books and chairs and innumerable other

Students can handle abstractions just fine. Even a
two-year-old playing with a baby doll knows it's not
a real baby; it's just a symbol that represents a baby.

Every so often I run into Professor Schmoe who says we
need to find ways to make physics less abstract, because
students can't handle it. I don't buy that at all. It
may be that students find Shmoe's class to be useless,
but that's not because of the abstractions. Remember,
the fact that twelve is abstract makes it /more/ useful.

If Schmoe's students are so disabled that they have not
reached the stage where they can play with dolls, *then*
I'll believe they cannot handle abstraction and symbolism
... but I will wonder why they signed up to take physics.

So the agenda should be to make things more useful and
more relevant.
-- Twelve is an abstraction.
-- Energy is an abstraction.
-- Probability is an abstraction.
-- Entropy is an abstraction. It is defined in terms of
probability, but that doesn't make it any less abstract.
-- Temperature is an abstraction. It is defined in terms
of energy and entropy, but that doesn't make it any less
abstract. ITS-90 is somewhat less abstract, but it's not
"the" temperature.

Embrace the abstractions.