[FOM] Infinitesimal calculus

[Brian Hart]

Gödel sent a letter to Robinson commending him on his fine work on
non-standard analysis and he stated that he thought that it would
become much more important than it has become since.  Why is this?  Is
it because it just isn't as useful as Gödel thought it would be or
that it could become much more useful in the future but we just don't
know yet?  Interestingly enough from the perspective of the history of
math, Robinson was good enough of a mathematician that he was
seriously considered to replace Gödel at the IAS but tragically died
shortly thereafter.

On Tue, May 26, 2009 at 3:18 PM, David Ross <ross at math.hawaii.edu> wrote:
>> 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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