> I can sum up my final view by saying that I came to
> believe that a complete linearly ordered (therefore Archimedean) field,
> augmented with some limit theory in the spirit of Cauchy and Weierstrass
> was simpler (that word "simple" again!) than using a non-Archimedean
> field, and also easier to relate to applications, especially in connection
> with the sort of approximations one works with when using computers.
Oh, I quite agree. When I mentioned the non-Archimedean fields,
I didn't mean to suggest that I prefer them. Personally, I would
be reluctant to try using a book like Kiesler's. I don't know
nonstandard analysis very well, but for the little bit of it that
I do know, epsilons and deltas are just as good as infinitesimals
for doing calculus, and they don't require quite as much explanation.
I actually see them as being nearly the same thing, anyway. In the
epsilon-delta argument, you have a variable delta which varies through
smaller and smaller, "ordinary" (not infinitesimal) positive numbers;
delta eventually passes and gets smaller than any particular fixed
positive ordinary number that you might compare it with. In the
infinitesimal argument, you have a constant (not variable) infinitesimal
number delta, which is positive but is *already* smaller than any
particular positive ordinary number that you might compare it with.
In my mind, the real difference is how "time" enters our intuition.
But perhaps there are more substantial differences that I am not
aware of.
> Yes, but I didn't find this level of abstraction workable in elementary
> calculus courses of the sort I taught in a number of different
> universities. I used to use examples of this sort for various purposes
> in real variable courses.
We're discussing two different courses here. In my history of
math class, I spent a few days talking about infinitesimals.
In my calculus class, I spent only a few minutes talking about them.
> Perhaps your history of mathematics course is for upper undergraduates
> who have a fairly good grounding in algebra and analysis? Also, do you
> do anything with infinitely large numbers, as reciprocals of infinitesals?
I mentioned them. I actually didn't do a lot with the whole subject
-- infinitesimals or their reciprocals -- except state a few basic
definitions and properties and examples. Even doing that much took me
two or three lectures. Next year I'd like to get the students a bit more
actively involved with the subject, if I can think of how -- e.g., I'd
like to teach some sort of computational technique using infinitesimals,
and then assign homework problems that use that technique.
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Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/
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