From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: nonstandard analysis
Date: 15 Dec 92 20:34:46 GMT
	<1g3lfiINNcmf@usenet.INS.CWRU.Edu> <Bz0Lyp.Mu7.1@cs.cmu.edu>
Organization: M.I.T. Artificial Intelligence Lab.

In article <Bz0Lyp.Mu7.1@cs.cmu.edu> asp+@cs.cmu.edu (James Aspnes) writes:

   Regarding Robert's _Nonstandard Analysis_ vs. Nelson's _Radically
   Elementary Probability Theory_:

   I own copies of both of these books.  Nelson's is, as was stated by
   another poster, an amazing little book.  It's a little too dense to be
   read easily by someone who's not used to heavy math.  Robert is a
   better introduction in general, especially if you haven't had a lot of
   analysis background or are more interested in the set-theoretic
   aspects of the system.

   Note that both books are using a particular axiomatization of
   nonstandard analysis called IST (which Nelson invented).  There are
   other forms of nonstandard analysis out there that use different
   axiomatizations and which are not necessarily equivalent to IST.  It's
   also worth remembering that beautiful though this stuff is, it's a bit
   like learning Esperanto.  You'll still need to be able to translate
   anything you do with it back into more standard mathematics to explain
   it to most people.


I personally have a warm spot for nonstandard analysis and I'm glad
other people share this feeling. I'm not entirely pleased that the
emphasis in recent postings seems to be on Nelson's foundations IST.

What is wrong with the beautiful work of Abraham Robinson
in which he invented the subject, including his book Nonstandard
Analysis?

Another aspect of nonstandard analysis which is sometimes neglected in
these discussions is that the external objects which arise in nonstandard
analysis are sometimes explored for their own sake, and doing so
goes beyond the framework of what can be translated back into "standard"
mathematics. For example, in nonstandard analysis, one can consider the
field of p-adic numbers where p is an infinite prime and one can consider
the field of formal power series over the integers mod p where p is
an infinite prime. Ax and Kochen showed that these fields are isomorphic
(I don't know if this is true for all models of nonstandard analysis.
They proved it for ultraproducts based on the same ultrafilter) as valued
fields, and as a corollary they concluded that the Artin conjecture
is almost true. In this case, the external object is the isomorphism.

Another example is the practice of taking ultraproducts of Banach
spaces and obtaining a new Banach space from the bounded elements of
the ultraproduct modulo the infinitesimal elements. In terms of nonstandard
analysis, one can view this as starting with a collection {B_i | i in I}
of Banach spaces, considering the collection in the nonstandard enlargment,
hence an enlarged index set I^* and for each i in I^* something B^*_i which
the nonstandard world considers a Banach space, taking an element i of
I^* not in I and considering the bounded elements of B^*_i modulo
infinitesimals. This is an external construction which produces
mathematical objects of interest in their own right.

I'm not sure, but I think that in the applications of nostandard
analysis to stochastic processes (as in the book of Albeverrio et al
published by Academic Press), they are working with external constructions
on the integers modulo N, where N is an infinite integer.

As long as we are on the subject, let me ask a question: am I mistaken
in thinking that there is a well-developed theory, possibly using
nonstandard analysis, of brownian motion and stochastic processes
either on or with values in noncommutative Lie groups. And if so,
where is the most painless account of this theory written.

I looked at Nelson's book in the Annals of Math studies and
I found that he didn't take enough care to maintain a
distinction between standard mathematical objects and nonstandard ones.
This made the book a lot harder to read. I understand that one of the
points he would like to make is that the difficulties and
unappealing complexities of the subject of stochastic processes
lie in the translation between standard and nonstandard mathematics
and that greater simplicity is therefore attained by staying in the
nonstandard world and treating it as the proper object of study,
forgoing the translation, but it still makes the book harder to read.

I have never had an opportunity to read his description of IST. If
someone has a copy of his original Annals of Math. Logic (?) paper on the
subject  and wouldn't mind sending it to me, I would like to look at it.
Efforts to obtain it from Nelson have been unsuccessful.

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math,comp.os.msdos.misc,comp.sys.ibm.pc.misc,sci.engr,sci.physics,sci.math.stat,sci.math.num-analysis,sci.edu