FOM: 'Constructive' infinitesimals and Calculus

[David Ross]

The first interesting result in Calculus which can't be treated
easily with an axiomatic description of the hyperreals (i.e., as in
Keisler's Calculus text) is the statement that a continuous function on
a compact interval attains a max.   I suggest this result as a kind of
pons asinorum for those who are advocating series, surreal numbers, or
other 'concrete' extensions of R.  I don't mean this as an attack
on any of these ideas. (I feel obliged to say this, as there does seem
to be an unexpected amount of interspecies, I mean interspecialty, 
antagonism on this list.)

BTW, nonuniqueness of the nonstandard reals doesn't seem to me to be a
problem unless you want to believe that they are the 'real' reals.  Some
nonstandard approaches *do* adopt this ontology (e.g., Nelson's
internal set theory), and then there is indeed a problem, since it is
possible to state and prove deep-looking theorems in such a framework
with no real standard meaning.  (Nelson and some of the other people
using this framework avoid this by dint of being very smart and very
careful.) Of course, this ontology is probably the one most interesting
for FOM issues.

- David (ross at tarski.math.hawaii.edu)