> By the way (he asked innocently), what do you think infinitesimals are?
That sounds like you're laying a trap for me. But I'll give
it my best shot anyway.
As I see it, Newton and Leibniz used infinitesimals without
having a very clear idea of what they were doing. That
was what Berkeley and others complained about, and why
Bolzano, Cauchy, Dedekind, Weierstrass worked so hard to
expunge infinitesimals from the theory. But the modern
view of people in Nonstandard Analysis (Robinson et al) and
Internal Set Theory (Nelson et al) is that Newton and
Leibniz had some very good intuition which can now be
made rigorous. In fact, some of Leibniz's writing sounded
a lot like modern model theory -- he said something about
how there should be some method or rule for extending
statements about ordinary numbers to statements about "all"
numbers, or something like that. I don't have the exact
quotes handy; I'll try to track them down if you wish.
To really carry out Leibniz's program would require
model theory, I guess, but some of the deeper aspects
can be swept under the rug if one is willing to state
some things without proof. That was the motivation of
Keisler's calculus book, "Elementary Calculus: An
Infinitesimal Approach." First edition 1976; second
edition 1986. I have a copy of the second edition.
It apparently didn't catch on -- I would guess because
it represented too radical a departure for most
teachers of calculus.
Anyway, to teach about infinitesimals does not really
require model theory. If you don't insist on carrying
out Leibniz's full program, and you just want to give
some examples of infinitesimals and illustrate their
properties, any non-Archimedean ordered field will do.
One such field, quite easy to describe, is the
field of all rational functions (with real coefficients,
with one variable, with the convention that two
rational functions are considered to be the same
if they differ at only finitely many points of R).
For the ordering, say that p is greater than q if
there exists a real number t such that, whenever
s > t, then p(s) > q(s). You can state this fact
without proof, if you're teaching to undergraduates:
It can be shown that if p and q are any rational
functions, then either they are the same (in the
sense above) or one is greater than the other
(in the sense above). Hence the rational functions
form a chain-ordered field.
In any ordered field, an infinitesimal is a number
that lies between 1/n and -1/n for all positive
integers n. An ordered field is Archimedean if
and only if 0 is the only infinitesimal. In
the field of rational functions, if x is the variable,
then 1/x is an infinitesimal.
This gives us an easy example of several other
equivalent definitions of Archimedean, as well as
an interesting example of a field that is not
Dedekind-complete.
When I taught about fields this semester, I gave
only a few examples. I included an example of
a finite field (5 elements) because I felt that
that emphasizes that the abstract axioms may be
satisfied by something that doesn't even remotely
resemble familiar things.
Well, you did ask. :)
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Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/
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