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Derivatives and models



I have read only a few notes in the thread on derivatives. It seems
that some are upset at the prospect of having to deal with
discontinuous functions because it is felt that such functions are
somehow unphysical. No one has yet stated what physical principle is
being trampled when discontinuous functions are used to represent
physical phenomena, and I certainly can't provide such a statement.
Discontinuous functions are entirely fair game for modelling physical
phenomena so far as I am concerned.

It should be pointed out often that models are not identical with
physical reality. They are merely intellectual constructs of limited
utility and accuracy and circumscribed validity as descriptions of
natural phenomena. Confusing the model with its subject, reification,
is a cognitive error and a sin which cannot be deplored too often.

As balm for those who must invoke some physical reason for deploring
discontinuous acceleration let me suggest that all physical bodies
possess extent, and there must be some spatial separation between some
of the particles in a body and the point at which the instantaneous
force is applied. If that is the case then special relativity provides
an escape from discontinuous acceleration, since the force cannot be
applied simultaneously to all the constituent particles. To do so
would require that the force be transmitted at infinite speed, a
traditional no-no. As for forces which are applied simultaneously to
all the particles, well Einstein effectively demonstrated that
simultaneity is in the eyes of the beholder.

As for me, I have no difficulty whatever with using discontinuous
functions to describe physical phenomena or to pose problems for my
students. Often they are the easiest way to explain something in terms
of a model, and I use them. I think it is not a good idea to mystify
their use because that leads the student to worry about minutiae which
he might mistake for concepts of importance equal to that of the
concept which is the object of the exercise. If the student is
sufficiently well indoctrinated in the evils of reification and the
need to keep in mind the limitations of the modelling process.

Leigh

Oh, yes - derivatives! The advice I've been getting is "Don't trade in
derivatives".