[FOM] Infinitesimal calculus

David Ross ross at math.hawaii.edu
Tue May 26 15:18:05 EDT 2009

> 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.

So can nonstandard analysis; you just write down a bunch of axioms governing 
the behavior of the hyperreal line and push on.  Just as a standard calculus 
book (even at the honors level) doesn't prove the Least Upper Bound property 
from the definition of the reals by Dedekind cuts, the infinitesimal 
calculus does not prove that every finite standard real number has a 
standard part; in both cases, the fact is treated as an axiom.  This is what 
the Keisler text does, and those of us who have taught out of it know that 
in practice this is not a burden for the students.

I don't really want to engage yet again in the argument as to whether it is 
better or not to teach calculus with infinitesimals, I just want to point 
out that some of the remaining arguments against doing so do not hold up to 
strong scrutiny.  When we made the switch in the US 100 years ago, it was 
for reasons of rigor; now, however, it is ultimately just a matter of taste.

David Ross 

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