FOM: Intuitions about infinitesimals

[Jon Barwise]

Is there an intuitive concept of infinitesimal?  Conserative exension
results can't answer THAT kind of question.  I think there is. In addition
to the historical evidence, starting with Archimedes "method," there is
also pedagogical evidence, formal and informal.  On the informal side, I
used Keisler infinitesimal calculus book two or three times, with happy
results.  The students, very typical Wisconsin freshmen, quickly came
accept and use infinitesimals in calculations and even in proofs.  On the
formal side, a grad student in math-education at Wisconsin years ago did an
empirical study, for her PhD, of students comparing the two approaches.  I
don't recall the details, but I recall that Keisler was pleased by them, so
students must have done at least as well using infinitesimals as with
epislon-deltas.  (Perhaps it needs saying, for those who have not looked at
Keisler's book, that he takes an informal axiomatic approach, he does not
construct the hyperreals.)

I am not interested in arguing that infinitesimal calculus is "better" than
epsilon-delta calculus (though I found it a lot easier and more fun to
teach).  I only want to suggest to Harvey that there is an intuitive
concept that people can learn to use correctly in doing mathematics
--without understanding Robinson's construction.  This point is relevant to
the philosophy of mathematics.

A philosophy of mathematics needs to be compatible with the way people do
correct mathematics. If a student comes across some function and uses
infinitesimals to correctly calculate its derivative or its integral, then
surely they are doing mathematics.