1a) A person's weight is 1b) A person's weight is
proportional to proportional to
that person's height. that person's height,
but the constant of
proportionality changes
from person to person.
2a) A person's weight is 2b) A person's weight is
proportional to the cube of proportional to the cube of
that person's height. that person's height,
but the constant of
proportionality changes
from person to person.
Now, theory 1a is an honest theory. It plays by the rules. Alas,
experiment will show that it is a rather weak theory, i.e. it makes rather
inaccurate predictions.
Theory 2a is similar, but it is a stronger theory, i.e. it makes rather
more accurate predictions. Of course it is by no means perfect either.
If theory 1a were the only thing available, we would probably use it,
within limits, with caution. We would not reject it out of hand. A weak
theory is better than no theory. However, as soon as theory 2a comes
along, it will quickly supplant 1a.
===============
What about theory 1b? In Fermi's immortal words, it is "not even wrong".
Indeed, theory 1b is *not* wrong. It's *true* that each person has a
weight-to-height ratio. But this is not physics. It's not science. It
has no predictive power.
In certain hokey circumstances, one might claim that theory 1b has
*mnemonic* power, in this sense: If, after calculating the
weight-to-height ratio, you forget the weight but remember the ratio, you
can rederive the weight from the height and the weight-to-height ratio.
But still this is not a prediction; it is a postdiction, because you had
previously used the weight to calculate the weight-to-height ratio.
Theory 2b is not an improvement over theory 1b.
================
I hope everybody can see the crucial distinction between the
weak-but-honest theory 1a and the unscientific theory 1b.
This distinction crops up in other contexts.
In data analysis and curve-fitting, for example, you may have heard the
witticism "Given enough parameters, you can fit an elephant. With a few
more, you can wiggle its tail." I first heard that in a chemistry lab,
where the data was not supposed to resemble an elephant, and it was not
supposed to wiggle. The teaching assistant was warning against using too
many adjustable parameters.
Popper and Kuhn and others have addressed this from the philosophical and
historical angles.
In statistics, you can find this discussed under the heading "bias versus
variance tradeoff". In the field of machine learning, you can find very
powerful no-nonsense analyses of this under headings such as
"probably-almost-correct models" and "Vapnik-Chernovenkis dimensionality".
=================
Pedagogical ramifications:
ISTM it is important for everybody (every student, every citizen, and
certainly every scientist) to be able to recognize the distinction between
theory 1a and theory 1b. That chem-lab course I took wasn't about
chemistry -- it was about science. In 20 years I haven't used a single one
of the laboratory techniques I learned there... but I have greatly
benefitted from the lessons I got about strong theories, weak theories, and
not-even-wrong theories.