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The exponential RC discharge opportunity



Long ago (in the 60's) I decided that the most useful insight I could
contribute to a student's education was a critical appreciation of our
intellectual efforts as the willful creation of useful (as opposed to
"true") models of reality in human terms. This (and its many
implications) has not always been easy to sell - to students, other
faculty or editors. In class, I try to take every opportunity to sing
that song - it is the answer to many otherwise vexing student questions
with which I see other profs struggle.

I find that the exponential RC decay model shows in an especially
transparent way that our models speak as much about us and our modes of
thinking, as they speak about the reality being modeled. Even more, it
is an instance where our model is based on a fundamentally erroneous (but
useful!) premise, leads to a ridiculous conclusion, and yet is
unsurpassed in its usefulness to this application and leaves nothing to
be desired!.
I think it is difficult to find another such irresistible lesson in
"cognition", "epistemology", "the scientific method", etc.

What others have already said is of course true: the "flaw" is in the
granularity of Q; the "true" curve consists of steps of discontinuities;
and when the last e is gone, it's over! Besides the sense of wonder at
physical reality there is any even more mysterious awe to be enjoyed at
the behavior of mathematical models of reality. Most students will see
no problem; they need to be told - it is part of their training in
critical thinking, reading and listening. (And even after being told,
few will identify the flaw - ask and see.)

I would be pleased to collect your examples of other (non-exponential)
phenomena where our calculus model parallels the behavior:

1) divide the process/quantity into an infinite number of infinitesimals
(loose language for a limiting process).
2) state the governing physics in the form of a differential (rather than
a difference) equation.
3) put the pieces back together by integrating the DE (adding an infinite
number of infinitesimals), producing: as a result:
4) a calculational model (result) that is useful and also makes a
patently absurd assertion traceable to the intrinsic granularity of the
process/quantity.

-Bob

Bob Sciamanda sciamanda@edinboro.edu
Dept of Physics trebor@velocity.net
Edinboro Univ of PA http://www.edinboro.edu/~sciamanda/home.html
Edinboro, PA (814)838-7185

"I wanted certainty in the kind of way in which people want religious
faith. I thought that certainty is more likely to be found in
mathematics than elsewhere. But I discovered that many mathematical
demonstrations, which my teachers expected me to
accept, were full of fallacies, and that, if certainty were indeed
discoverable in mathematics, it would be in a new field of
mathematics, with more solid foundations than those that had hitherto
been thought secure. But as the work proceeded, I was
continually reminded of the fable about the elephant and the tortoise.
having constructed an elephant upon which the
mathematical world could rest, I found the elephant tottering, and
proceeded to construct a tortoise to keep the elephant from
falling. But the tortoise was no more secure than the elephant, and after
some twenty years of very arduous toil, I came to the
conclusion that there was nothing more that I could do in the way of
making mathematical knowledge indubitable."

Russell, Bertrand (1872-1970), Portraits from Memory.