[FOM] Apartness topology and (constructive) nonstandard analysis

[katzmik at macs.biu.ac.il]

Dear Frank,

Thanks for your message.

On Tue, September 8, 2015 00:21, Frank Waaldijk wrote:
> I'm sure that if you read my post again, and the ones preceding it, you
> will find that I do not conflate `structure' and `construction' nor assume
> that there is a non-trivial ultrafilter on N somewhere in the galaxy.
> Personally I even doubt I can find pi in the outside world, in that sense
> my thoughts tend to ultrafinitism.

My message was admittedly too brief.  I was merely trying to emphasize the
difference between structure/procedure/syntax on the one hand, and
construction/semantics/ontology, on the other.  The axiomatisations commonly
accepted as the foundations routinely eliminate infinitesimals from
consideration and in order to put them back in by construction within the
traditional foundational frameworks, one needs to resort to ultrafilters. 
However, arguably this is more of a problem with the traditional foundations
than with infinitesimals themselves.

> I read Nelson's clear paper on IST and the syntactic approach was never in
> doubt.
>
> I merely wished to clarify that although IST is not a constructive theory,
> I still like it a lot. I also like non-trivial ultrafilters (they are such
> fun), and would surely put one above a model train on my Santa Claus wish
> list. If some physicist manages to match a phenomenon in physics to a
> mathematical theory involving ultrafilter products, then I will enjoy that.

The physicists have been doing such matching for centuries already, starting
with Kepler and Galileo, passing via Cauchy in his theory of elasticity as
well as geometric probability, and until the present day when they routinely
use infinitesimals in their work.  It is not their fault that mathematicians
chose artificially to exclude infinitesimals from their foundational
frameworks thereby artificially creating a need to put them back in.  As I
think we both agree, Nelson corrects the problem by providing a foundation for
mathematics more hospitable to physics.  At the end of the 19th century, Frege
and Paul du Bois-Reymond had all the tools necessary to provide a foundation
of a preliminary Nelsonian type (though perhaps not all the details). 
Instead, the Weierstassians, breaking with Cauchy, carried the day with their
nominalist program of eliminating numbers they couldn't find a way of
formalizing properly. The result was a divorce between physics and mathematics
thas is still in place today.

MK