FOM: Smooth infinitesimals

[David Ross]

Todd Wilson wrote:

 "    As an aside, I'd like to mention in this regard that, more
     recently than Robinson's work, a number of researchers in category
     theory have made an important advance in the explication of these
     intuitions through what is called "smooth infinitesimal analysis",
     a theory that is inconsistent with classical logic but consistent
     with and very fruitful under intuitionistic logic.  An excellent
     introduction to this work, written by someone who is both a set
     theorist and a category theorist, is the book

         J.L. Bell, A Primer of Infinitesimal Analysis, Cambridge
         University Press, 1998."

 Strictly speaking, the intuitions explicated by the Moerdijk/Reyes models
 popularized by Bell's book are not those employed by, e.g., Kepler and
 Euler.  The reals the former construct contain
 nilpotent infinitesimals, which of course means that R is not a field, but
 does make the infinitesimals more like those that geometers (as opposed to
 analysts)  are wont to use nowadays.
 (A nice side effect of the intuitionistic logic is that the proofs are all in
 some sense constructive.)

 Closely related work which I think has more potential ramifications for FOM
 is the work of Palmgren et al on constructive models of Robinsonian
 nonstandard analysis.  This makes it possible to interpret nonstandard
arguments
 which seem to be  intrinsically nonconstructive (e.g. using a fair amount of
saturation) into a
 completely constructive model.  The net flavor is a bit like that of Skolem's
 paradox.

 --
 David A. Ross
 Professor of Mathematics
 University of Hawaii
 ross at math.hawaii.edu
 www.math.hawaii.edu/~ross