[FOM] Apartness topology and (constructive) nonstandard analysis

[Hendrik Boom]

On Tue, Sep 08, 2015 at 10:56:25AM +0300, katzmik at macs.biu.ac.il wrote:
> 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.

As far as I can tell, the infinitesimals apparently used by physicists 
have been adequately formalized by the theory of differential forms.

> 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.

Perhaps infinitesimals were discarded because the epsilon-delta 
definitions of limits and continuity were more than adequate.

> 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.

I really haven't seen the theoretical physicists abandoning mathematics.  
Nor do I see the current crops of quantum gravity theories languishing 
for a lack of infinitesimals.  They make heavy use of differential 
manifolds, where differentials manifest themselves as local linear 
approximations.

And when it comes to the very smallest scales of space, theories like 
loop quantum gravity seem to be coming up with a fine-grained structure 
rther than  a continuum.

The eigenstates of space are small grains; the space we are familiar 
with is made of these grains (made statistically ambiguous by the 
usual machinations of quantum mechanics) - about a googol of them i a 
teaspoon.

No infinitesimals in the sense of nonstandard analysis at all.

-- hendrik