On Tue, 12 Oct 2004 16:45:50 -0500, "Steve Nosko"
wrote:

Alan,

John Popelish got a good start with "e is a natural constant that has
some very sweet properties in many applications of mathematics, and
simplifying..."

Then, it looked as thought John Jardine was going to steal my thunder with
"the voltage knows nothing about how it's "supposed" to behave. "

This could resolve to a mater of faith Alan.

Indeed, the voltage/current "knows" nothing.

After observing what happens in such circuits, "we" (those who must
understand all things) very carefully examined what was going on and
"discovered" that there were mathematical expressions or equations which
would model what happens in nature. "We" came up with theories about what
was going on and what was causing it to happen. "We" then found ways to
make the math fit reality.

Reminds me of Galileo writing in "The Assayer," saying:

"Philosophy is written in this grand book-I mean the universe-which stands
continually open to our gaze, but it cannot be understood unless one first
learns to comprehend the language and interpret the characters in which it is
written. It is written in the language of mathematics, and its characters are
triangles, circles, and other geometric figures, without which it is humanly
impossible to understand a single word of it; without these, one is wandering
about in a dark labyrinth."

(By the way, to anyone who has NOT actually read The Assayer from beginning to
end, I highly recommend it!)

Mathematics is a wonderful world all of its own, independent of nature, yet
where it often turns out that insights in that world happen to happily suggest
relationships found in this world and where proper deductions there imply proper
deductions here. The language is sufficiently rigorous that someone two
millennia before me can describe a circle using it and I can read it today,
knowing absolutely nothing about their lives, their fads or interests, their
politics or style of dress, and come away with exactly the same image in mind
with exactly the same deductive power. In short, mathematics is a quantitative
language that speaks across culture, time, and place. And there is nothing we
have to compare with that.

The processes of science work to achieve a relatively objective process that
works well. It requires the use of objective language sufficient for rigorous
quantitative deductions (by anyone adequately trained in the language) to
specific circumstances, insists that such language both explain past results
well and (more importantly) also make accurate and repeatable predictions,
requires quantitative prediction for discernment, and requires time and patience
for the resulting critical opinion of others skilled in the field to arrive at a
consensus. But mathematics *is* a key part of this objective language used in
science because of its demonstrated congruencies with nature.

In the case of time constants, we have a natural
phenomena which is very nicely described by the equations stated elsewhere
in this thread (the 1/e thingy). It is just like the F=MA equation. "We"
discovered that the force applied to a mass is equal to the mass times the
acceleration. The Mass knows nothing about force, acceleration or
mathematics. We found that this math describes nature.

One thing to keep in mind is that ideas like "density," a useful relationship
between volume and mass, are truly discovered through hard work and through
trying to find some kind of useful discernment regarding sinking and floating.
One doesn't just naturally _know_ about density, as our direct senses tell us
nothing of the kind. It's discovered and then taught and learned. And such
relationships are about parsimonious tools for prediction.

And yes, we have been fortunate that some math describes some nature.

It is exactly like a model airplane (or whatever). We make the model to
look like the real thing. The real thing knows not of the model that we
built, but if we did a good job, I or you can now look at the model and
"know" just how the real thing looks.

It can also be that the model ignores some of the unimportant details of the
"real thing" and still be quite useful. Or that it ignores some important
details, but that so long as we keep those boundaries and limitations in mind
the model is still quite useful for many other things.

I like your example.

The math behind all of our sciences is just like this. *WE* found math
which models reality and because we did such a good job, we can now "do the
math" and "know" how the real thing should behave.

We can also disappear into the mathematical universe and discover brand new
relationships there and have some expectation that where such new territory is
true there, it will probably be found true in the real world as well. One can
make important discoveries using mathematics and use them to suggest what can be
searched out and found here. Surprising, at times.

To be a little more specific, in the case of the time constant. we have
theories about current flow, charge, capacitance, inductance magnetism and
resistance which are borne out by countless experiments and then by
subsequent usage. These theories have all had mathematics fitted to them,
and by golly everything fits. We can now plug-in values to equations till
the cows come home and holy-cripes! The real thing does just what the math
predicted. Based upon the properties we have observed for each type of
component, this math works out such that this 1/e thingy fits just right.

In other words, the answer is: "It just does!"

Yup. In the capacitor case, for example, I idealized it as a simple
differential equation. Real capacitors are more complex, but the ideal is often
close enough in practice to be useful.

Enjoyed seeing your thunder!

Jon