Excellent comment, Hugh,
My viewpoint is that mathematical models, which assert a numerical
equality between functions of the readings of defined measurement devices,
are indeed testable and can be branded "right or wrong" as an empirical,
mathematical model.
The human mind is not satisfied to stop here. It goes on to account for
these device readings and their connections by erecting a conceptual
model of the object under consideration, the measuring devices and their
interactions. The device readings then acquire conceptual meanings within
the context of that conceptual model. These concepts come from our
library of sensory-provoked ideas.
Eg; the point particle model of the "reality" behind the empirical ideal
gas equation PV=RT. Another example is the meaning of the particle
velocity in the magnetic force F=qVxB. Common wisdom was to (delightedly)
take it as an absolute velocity (or at least, relative to an ether
medium}; Einstein's weird conception was that it was the particle
velocity in whatever inertial frame the physics was being done. One man's
(or generation's) weirdness is often another's beauty.
Perhaps the best example is the plethora of conceptual models being
invented to conceptualize the reality behind the (notoriously useful)
equations of quantum mechanics.
My favorite (lengthy) RPF quote:
Feynman on Conceptual vs Mathematical Models:
"Sometimes I wonder why it's possible to visualize or imagine reality at
all. . . . It's easy to imagine, say, the earth as a ball with people and
things stuck on it, because we've all seen balls and can imagine one going
around the sun - it's just a proportional thing, and in the same way I can
imagine atoms in a cup of coffee, at least for elementary purposes, as
little jiggling balls. But when I am worrying about the specific
frequencies of light that are emitted in lasers or some other complicated
circumstance, then I have to use a set of pictures which are not really
very good at all - they're not good images. But what are "good images"?
Probably something you're familiar with. But suppose that little things
behave very differently than anything that was big, anything that you're
familiar with?
Animals evolved brains designed for ordinary circumstances, but if the gut
particles in the deep inner workings of things go by some other rules, and
were completely different from anything on a large scale, there would be
some kind of difficulty, and that difficulty we are in - the behavior of
things on a small scale is so fantastic, so wonderfully and marvelously
different from anything on a large scale! You can say, 'Electrons behave
like waves' - no, they don't, exactly; 'they act like particles' - no,
they don't, exactly; 'they act like a fog around the nucleus' - no, they
don't, exactly. Well, if you would like to get a clear, sharp picture of
an atom, so that you can tell correctly how it's going to behave - have a
good image of reality, in other words - I don't know how to do it, because
that image has to be mathematical. Strange! I don't understand how it is
that we can write mathematical expressions and calculate what the thing is
going to do without actually being able to picture it. It would be
something like having a computer where you put some numbers in, and the
computer can do the arithmetic to figure out what time a car will arrive
at different destinations but it cannot picture the car. . . .
For certain approximations, it's okay. With the atom pictures, for
example, the idea of a fog around the nucleus, which repels you when you
squeeze it, is good for understanding the stiffness of materials; the idea
of a wave is good for other phenomena. The picture of atoms, for
instance, as little balls is good enough to give a nice picture of
temperature. But if you ask more, and you get down to questions like 'How
is it that if you cool helium down, even to absolute zero where there's
not supposed to be any motion, you find a fluid with no viscosity, no
resistance - it flows perfectly, and it isn't frozen solid?' Well, if you
want to get a picture of atoms that has all that in it, I just can't do
it. But I can explain why the helium behaves as it does by taking the
equations and showing the consequences of them is that helium will behave
as it is observed to behave, so we know we have the theory right - we just
don't have the pictures that will go with the theory.
I wonder whether you could get to know things better than we do today, and
as the generations develop, will they invent tricky ways of looking at
things - be so well trained that they won't have our troubles with the
atom-picturing? There is still a school of thought that cannot believe
that the atomic behavior is so different than large-scale behavior. I
think that's a deep prejudice, a prejudice from being so used to
large-scale behavior. They are waiting for the day that we discover,
underneath the quantum mechanics, some mundane, ordinary balls hitting
each other. I think nature's imagination is so much greater than man's
that she's never going to be defeated!"
- R. P. Feynman, quoted in "No Ordinary Genius ...", Christopher Sykes