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Consider the situation where we are measuring the properties of, say, a
real spring. Not some fairy-tale ideal spring, but a real spring. It will
exhibit some nonlinear force-versus-extension relationship.
Now suppose that we do a really good job of measuring this
relationship. The data is reproducible to some huge number of significant
digits. For all practical purposes, the data is exact.
Next, suppose we want to model this data. Modelling is an important
scientific activity. We can model the data using a straight line. We can
also model it using an Nth-order polynomial. No matter what we do, there
will always be some "error". This is an error in the model, not in the
observed data. It will lead to errors in whatever predictions we make with
the model.
Proper error analysis will tell us *bounds* on the errors of the predictions.
Is this an example of "if it doesn't work, it's physics"? No! An inexact
prediction is often tremendously valuable. An approximate prediction is a
lot better than no prediction.
I mention this because far too many intro-level science books seem to
describe a fairy-tale axiomatic world where the theorists are always right
and the experimentalists are always wrong. Phooey!
It is very important to realize that error analysis is NOT limited to
hunting for errors in the data. In the above example, the data is
essentially exact. The spring is not "at fault" for not adhering to
Hooke's so-called law.
A huge part of real-world physics (and indeed a huge part of real life in
general) depends on making approximations, which includes finding and using
phenomenological relationships. The thing that sets the big leagues apart
from the bush leagues is the ability to make _controlled_ approximations.