[FOM] Computational Nonstandard Analysis

[Harvey Friedman]

On Mon, Aug 31, 2015 at 11:05 AM,  <katzmik at macs.biu.ac.il> wrote:
> Harvey's recent post on NSA (see below) seems to contain a number of
> inaccuracies.  Some NSA/IST experts that I contacted did not seem interested
> in
> providing a correction.  Therefore I will volunteer a reply.
>
> As I see it the inaccuracies are at least the following:
>
> (1) Contrary to Friedman's claim, Robinson did not use the ultrapower
> construction to build the non-standard reals.  Luxemburg popularized this
> construction but at any rate the experts in the field do not follow this route
> which is less powerful than the compactness theorem route.

Thanks! So A. Robinson used the compactness route and Luxembourg the
ultra power route? Might be convenient to have you mention a readable
history of nonstandard analysis for people who don't work in it, like
me and most subscribers.

> (2) Contrary to Friedman's claim, IST is not an obscure framework that "very
> few" people other than Sanders know about, but rather the main point of
> reference in contemporary NSA, such as the Kanovei-Reeken book or
> Gordon-Kusraev-Kutateladze.

Because Nelson was so involved in the "claim" that the exponential
function is not total on the natural numbers, I automatically assumed
that he must have an idiosyncratic setup for nonstandard analysis.
Especially, that he must be working with a construction that can be
done in a tiny fragment of arithmetic.
>
> (3) Contrary to Friedman's claim, a proper extension of R of the type
> constructed by Robinson CANNOT be exhibited in ZF.

This is definitely not a claim of mine, but a simple TYPO. Thanks for
pointing it out.
>
> (4) Contrary to Friedman's claim, NSA has NOT been used "mostly for doing
> rather abstract analysis".

What I meant was that for mathematical applications of note to
mathematicians, the target mathematics is rather abstract analysis.

Now this may still be false or misleading?
>
> (5) Harvey has requested a clarification with regard to Nelson's system in a
> SLOW mode.  Reproducd below are some relevant comments from a recent article.
>
> To elaborate on Nelson's IST approach to infinitesimals in a non-technical
> way, note that the general mathematical public often takes the
> Zermelo-Fraenkel theory with the Axiom of Choice (ZFC) to be THE foundation of
> mathematics.  How much ontological reality one assigns to this varies from
> person to person.  Some mathematicians distance themselves from any kind of
> ontological endorsement, which is a formalist position in line with
> Robinson's.  On the other hand, many do assume that the ultimate test of truth
> and/or verifiability is in the context of ZFC, so that in this sense ZFC still
> is THE FOUNDATION, though it is unclear whether Robinson himself would have
> subscribed to such a view.

Of course, I view ZFC is the at this time most convenient, robust,
coherent, useable, general foundational scheme for general
mathematics. However, I am deeply interested in any alternatives that
are based on at least a prima facie coherent idea. Taking
infinitesimals is such an idea. That doesn't mean that we are in a
position to offer it up as a good alternative to ZFC. Instead, it
seems to be an alternative of among others which needs to be explored,
and is being explored. There probably are many avenues of serious
foundational interest to go with it that have not yet been explored.
>
> Much of the mathematical public attaches more significance to ZFC than merely
> a formalist acceptance of it as the ultimate test of provability, and tends to
> accept the existence of sets (wherever they are to be found exactly but that
> is a separate question), including infinite sets, and particularly the
> existence of the set of REAL numbers which are often assumed to be built into
> the nature of REALITY itself, as it were.  When such views are challenged what
> one often hears in response is an indignant monologue concerning the
> categoricity of the real numbers, etc.  As G. F. Lawler put it in his
> "Comments on Edward Nelson's `Internal Set Theory: a new approach to
> nonstandard analysis'":

The lack of categoricity and the intrinsic undefinability issues are
both major drawback for NSA as any kind of prima facie replacement of
ZFC. However, for me, that does not mean that there aren't some very
good reasons for taking a hard look at NSA or NSM. After all, long
before we really had a decent grasp of epsilon/delta we (e.g., Newton
and Leibniz) were casting our mathematics (in and around calculus) in
these terms. Obviously it has a lot of conceptual attractions. There
are other things that we have wrung out of the mathematical setup that
also need to be revisited. I won't go into them here.
>
> "Clearly, the real numbers exist and have these properties. Indeed, many
> courses in elementary analysis choose not to construct the reals but rather to
> take the existence of an ordered field as given. This is reasonable: we are
> implicitly assuming such an object exists, otherwise, why are we studying it?"
>  (Lawler 2011).

Better take a complete ordered field.
>
> In such a situation infinitesimals may not thrive: the real numbers are
> thought truly to EXIST, infinitesimals emphatically NOT.  This is where Nelson
> comes in with his syntactic novelty and declares: Guess what, we can find
> infinitesimals WITHIN THE REAL NUMBERS THEMSELVES if we are only willing to
> enrich the language we speak!  The perceived ontological differences between
> the real numbers and infinitesimals are therefore seen to be merely a function
> of the technical choices, including syntactic limitations, imposed by Cantor,
> Dedekind, Weierstrass, and their followers in establishing the foundations of
> analysis purged of infinitesimals.

Now you are confirming my idea that Nelson's stuff is idiosyncratic
and the working mathematicians that I know who are familiar and
occasionally use NSA will have nothing to do with it.

You have stopped your account at the interesting place. Rather than
simply refer the FOM readers to an article, it would be interesting
for you to elaborate. For example, do you have an example of an
infinitesimal "within the real numbers themselves"? And if you must
enrich the language, you could explain briefly just what enrichment of
the language you have in mind?

Harvey Friedman