FOM: Smooth infinitesimals
David Ross
ross at math.hawaii.edu
Tue Feb 22 03:12:20 EST 2000
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
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