[FOM] Infinitesimals

[Jay Sulzberger]

On Thu, 12 Jul 2012, Sam Sanders <sasander at cage.ugent.be> wrote:

> Dear members of the FOM list,
>
> I would like your opinion on the following statements (Please declare them true/false, with a possible explanation why).
>
> 1) infinitesimals are used throughout physics and engineering
> (in some informal way, formalizable in Nonstandard Analysis).

There are many ways to formalize various arguments.  Before
Abraham Robinson's approach there were others.  For a crude one
embed the reals into the set of functions on some set with a
notion of "going to infinity".  For example the set F of
functions f: reals -> reals has such a notion, and we embed the
real a as the function f given by

   for all real numbers x, f(x) = a

Then "big and little o notation" suffices to "formalize" many
arguments.

Part of non-standard analysis may be considered as an example of
this crude approach, where we extend our set of reals to the
reals with many other sets, say all sets in a model of ZFC with
the reals a set of ur-elements, and the set on which our
functions are defined is some infinite set, with "at infinity"
meaning "at a non-principal ultra-filter".

>
> 2) when infinitesimals are used in physics and engineering, the choice of infinitesimal does not matter
> (i.e. a calculation involving an infinitesimal \e remains valid if \e is replaced with any other infinitesimal \e' .)

One of the most salient distinctions in "naive infinitesimals
theory" is the distinction between orders.  If we say "dx" is
first order, then "dx * dx" is second order.

The Kaehler differentials give one formalization of this distinction:

   http://en.wikipedia.org/wiki/K%C3%A4hler_differential
   [page was last modified on 6 January 2012 at 17:54]

Many "naive" arguments go through if one only uses this type
theory distinction of orders, without further refinement, and
without having to use another type theory for infinitesimals.

Early in the history of calculus after Newton and Leibniz
attempts were made to explicate the main theorems and techniques
by means of such a theory of orders of infinitesimals.

>
> 3) The aforementioned independence (of the choice of infinitesimal) in physics and engineering has been observed before by X (Please fill in X).
>
> With kindest regards,
>
> Sam

I do not know what metatheorems have been published here, but
clearly there are many.  In both Robinson's non-standard analysis
and in Anders Kock's synthetic differential geometry there are
such theorems, and also in various of the "crude" formalizations.

I just now happened on

   http://home.imf.au.dk/kock/real.PDF

which looks good.

oo--JS.