[FOM] Apartness topology and (constructive) nonstandard analysis

[katzmik at macs.biu.ac.il]

Dear Hendrik,

Thanks for your message.  (Parts of what follows include responses to Frank
Waaldijk's comments).

On Wed, September 9, 2015 11:28, Hendrik Boom wrote:
>> > 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.

My book http://www.ams.org/books/surv/137/ exploits the differential graded
algebra defined by differential forms to explore further invariants of Massey
type. Differential forms are certainly a powerful tool and have been so at
least since de Rham's theorem. There is no doubt about their utility. Surely
many mathematical physicists in our post-Witten generation exploit them
routinely, but many physicists don't, and arguably don't need to.

I am reliably informed that Kutateladze (pere) achieved many breakthroughs in
hydrodynamics without ever using a differential form. The key adjective in
your comment is "apparently." It is a metatheorem in Robinson's framework that
in certain well-defined sense, any result proved using infinitesimals can also
be proved in a Weierstrassian framework through a suitable paraphrase, but at
what price? Recall that Goldbring's proof of the local version of Hilbert's
5th problem still does not have a noninfinitesimal proof.

I would like to propose the following thought experiment, extending your
argument. Every proof using the real numbers can be paraphrased using the
rational numbers only. I have it on Harvey's authority that this is a feasible
project.  One can push this further and reformulate every such argument in
terms of natural numbers.

This is what Kronecker tried to do for arithmetic (though he wisely stopped
before claiming any such paraphrase for geometry and physics). There is a
parallel discussion currently at FoM about the achievements of Euclid and
greek geometry.  From our modern point of view, they may have been attempting
just such a paraphrase (though of course they did not think of it as a
paraphrase of anything).

In fact, infinitesimals can be adequately formalized in terms of the integers,
as well, as Borovik, Jin, and myself proved in this article:
http://dx.doi.org/10.1215/00294527-1722755

To return to Frank's original comment, infinitesimals occur naturally in
physics and have been used there for over three centuries. I agree with you
(Hendrik) though that the mathematical possibility of paraphrazing arguments
in terms of narrower number systems is interesting in its own right, and is
close to the spirit of Reverse Mathematics.

MK