[FOM] Infinitesimal calculus

Monroe Eskew meskew at math.uci.edu
Mon May 25 23:21:05 EDT 2009

If the idea is just to teach calculus techniques to high school
students, without regard to whether they understand the
justifications, then I don't see what the controversy is about.  We
can just ignore justification altogether, teach neither epsilon-delta
nor infinitesimal, and just give them intuitive notions of limits and
continuity, via "what happens as this approaches that."  As I recall,
this is basically what goes on in high school classrooms.  There are a
few mentions of the formal epsilon-delta definitions, but the only
applications of this that the students are expected to make, is to
repeat it by rote memory for quiz points.

If on the other hand we are talking about teaching mathematical theory
to undergraduates, then I think standard analysis is more accessible
than nonstandard analysis since it can be developed from elementary
principles.  In any case, a well-rounded education should include
both.  But probably standard analysis should be studied first.

Just my opinion,

On Mon, May 25, 2009 at 4:07 PM, David Ross <ross at math.hawaii.edu> wrote:
> Whether nonstandard analysis "works" is a foundational problem which need
> not appear in Calculus textbooks (as Keisler's text demonstrated); the fact
> that it is somewhat harder than showing that real numbers "work" is not very
> important.  The failure of modern infinitesimal calculus to catch on in the
> curriculum is fundamentally a matter of mathematical sociology and inertia,
> not practicality or correctness.

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