[FOM] Apartness topology and (constructive) nonstandard analysis

[S. S. Kutateladze]

Dear Mikhail,

I've heard this argument about ultrafilters many from those mathematicians who think that they are physicists as well, but the real physicists   are ignorant of ultrafilters and use infinitesimals when this is relevant, in hydrodynamics for instance. 
Best regards,
S. Kutateladze


08.09.2015, you wrote to me:

katzmik at macs.biu.ac.il> Dear Frank,

katzmik at macs.biu.ac.il> Thanks for your message.

katzmik at macs.biu.ac.il> 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.

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

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

katzmik at macs.biu.ac.il> MK


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