[FOM] Nonstandard Methods

Wed Jul 30 23:17:46 EDT 2003

Reply to Lemberg Nonstandard Methods.

First of all, this topic of smooth analysis has already appeared on the FOM.
See the tremendously effective FOM Archives at

http://www.cs.nyu.edu/pipermail/fom/2000-February/003843.html
http://www.cs.nyu.edu/pipermail/fom/2000-March/003919.html
http://www.cs.nyu.edu/pipermail/fom/2003-January/006197.html
http://www.cs.nyu.edu/pipermail/fom/2000-February/003838.html
http://www.cs.nyu.edu/pipermail/fom/1999-April/003103.html
http://www.cs.nyu.edu/pipermail/fom/1998-February/001306.html

I compiled this using the Google search Martin Davis installed on the FOM
Information Page. 

It is now clear that there are rival theories of infinitesimal analysis, and
there is not going to be any much of an agreement about what approach is
most illuminating or most real or most appropriate.

I have a radical idea about what to do about such an impasse, and that is to
get much more radical than everything else, in a setting where there cannot
be any funny business going on.

I.e., do FINITE ANALYSIS ONLY. In fact, do FINITE MATHEMATICS ONLY.

A great deal of the issues that offbeat analysis tries to deal with in
various ways, concerning infinitesimal quantities must be handled, squarely
and fairly, with no hocus pocus, if one is doing FINITE MATHEMATICS ONLY.

I do not currently have the time to initiate this in a serious way, as I am
up to my eyeballs in finishing Equational Boolean Relation Theory for
publication, using nonstandard models on every line of every page.

But - **I will be back on the FOM in a serious way soon.**

See my posting 

164:Foundations with (almost) no axioms, 4/22/0  5:31PM

where I laid out some basics. But in the present discussion, what I need to
do is to start developing some fundamental real analysis and other topics,
in strictly finite terms.

Coming back to standard, analysis, nonstandard analysis, and smooth
analysis, I do see some differences at this stage.

1. Standard analysis has clear axiomatic foundations in terms of clear
formal systems that have survived the test of time.

2. Nonstandard analysis also has clear foundations in terms of standard
analysis, as supplied by A. Robinson.

3. But nonstandard analysis also has, independently of 2, rather clear
axiomatic foundations also in terms of clear formal systems. These are not
nearly as well known as the axiomatic foundations of standard analysis.

4. In axiomatic nonstandard arithmetic, one takes the notion of standard
integer as primitive. As I worked it out in 1967, one has full induction for
all formulas over the standard integers, and induction over all integers for
formulas that do not mention "being standard" and have only standard integer
parameters. One also has that the ring of all integers is an elementary
extension of the ring of all standard integers. I proved that this was a
conservative extension of Peano Arithmetic.

5. I believe that this result of mine has been looked at from the point of
view of fragments of PA, but probably not exhaustively, and it should be
pursued exhaustively. I think that Avigad has written about this.

6. Also, I don't know how much detailed work of this kind, with conservative
extension results, have been done for more elaborate nonstandard notions,
such as nonstandard reals, etc.

7. There is also constructive analysis in the sense of Bishop, which is
analogous to what the intuitionists call "lawlike analysis". There is the
out of print book by Errett Bishop, Foundations of Constructive Analysis.
McGraw Hill, 1967.

8. A particularly simple, rich, relatively robust axiomatic foundations of
constructive analysis (lawlike) is given by my system B (for Bishop) in

H. Friedman, Set Theoretic Foundations for Constructive Analysis, Annals of
Mathematics, Vol. 105, (1977), pp. 1-28.

This foundations is in the style of constructive set theoretic foundations,
and is in some analogy to the classical set theoretic foundations of
classical analysis. There it is proved that B is a conservative extension of
HA = Heyting's Arithmetic = Peano Arithmetic with intuitionistic logic.

9. I have never seen any clear kind of axiomatic foundations for "smooth
analysis" in the sense that Lemberg is talking about, and Bell, etc. This is
needed in order for it to take hold foundationally. It is nowhere near
enough to simply give some involved category theoretic "models".

Here is a quote from Connes that I found on the web.

"... The answer given by nonstandard analysis, a so-called nonstandard
real, is equally DECEIVING.  From every nonstandard real number one can
construct canonically a subset of the interval [0, 1], which is NOT Lebesque
measurable.  No such set can be exhibited (Stern, 1985).  This IMPLIES that
NOT A SINGLE nonstandard real number can actually be exhibited."  .....   A.
Connes "Noncommutative Geometry and Space-Time", Page 55 in "The Geometric
Universe", Huggett et al (Tribute to Roger Penrose), CUP 1999.

BY THE WAY!!!: I got the quote from a "moderated newsgroup" with an archive,
which can be accessed at

http://www.lns.cornell.edu/spr/

called Sci.Physics.Research. Their home page is at

http://math.ucr.edu/home/baez/spr.html

Are any FOM subscribers familiar with it?

They also mention these groups:

*    sci.physics.fusion
*    sci.physics.accelerators
*    sci.physics.cond-matter
*    sci.physics.electromag
*    sci.physics.plasma
*    sci.physics.relativity
*    sci.med.physics
*    alt.sci.physics.new-theories
*    sci.math 
*    sci.math.research
*    sci.astro 
*    sci.astro.research
*    sci.chem 
*    sci.energy 
*    sci.environment
*    talk.environm

On 7/30/03 7:55 PM, "Alexander M Lemberg" <sandylemberg at juno.com> wrote:

> On Wed, 30 Jul 2003 09:43:15 -0400 Harvey Friedman
> <friedman at math.ohio-state.edu> writes:
> 
> "Is "galaxies" a technical term now in nonstandard analysis?"
> 
> I don't know about now or technical but it is term which has been widely
> used in this context. I am surprised that you ask this, maybe I'm missing
> the point.

Reference or explanation?
> 
> Well, my interest is from a broadly philosophical viewpoint and so the
> heart of the matter for me may be quite different than for
> mathematicians. Briefly, I am trying to understand spatial and temporal
> continua and motion from the physical, mental, and logical (or
> mathematical) perspectives. For me, this is what the originators of
> calculus and infinitesimals were trying to do. Their work was grounded in
> trying to make their intuitions precise. So, for me, the so called
> "smooth analysis" is ontologically relevant in this context. In a
> nutshell, the heart of the matter for me is ontological relevance. Does
> that make sense?

Yes, but I would think that you would not be satisfied, as a philosopher,
with an involved category theoretic theory unless it had an axiomatic
foundation in the sense I discussed above. Recall that I said above that
standard analysis and also nonstandard analysis have clear axiomatic
foundations. 

Harvey Friedman