FOM: Replies/Nonstandard Analysis

[Harvey Friedman]

Baldwin has provided us with the following quote from Godel:

>"Arithmetic starts with the integers and proceeds by successively
>enlarging
>the number system by rational and negative numbers, etc.
>But the next quite natural step after the reals, namely the introduction
>of infinitesimals, has simply been omitted.  I think in coming centuries it
>will be considered a great oddity that the first exact theory of
>infinitesimals was developed 300 years after the invention of the
>differential calculus \cite{godel-remnon}."

>This is from page 311 of Volume II of Godel's collected works.  I have
>it on hand thanks to Matt Moore who indicates that Godel made such remarks
>after a talk by Robinson at IAS in 1973.

Let me firstly say right off that I strongly disagree with this statement
of Godel, mainly because I reject the idea that there is a prior notion of
infinitesmial. However, I am willing to agree that there is a notion of
"infinitesmial reasoning." As this is usually construed, there are many
many results that indicate how this is a conservative extension of ordinary
reasoning. As far as f.o.m. is concerned, this is by far the most
interesting and relevant aspect (at this time).There are a series of
conservative extension results, starting, perhaps, with a result of mine
from my student days, which I formulate at the end of this posting. The
result was new to A. Robinson, with whom I discussed it, and Kreisel told
me he was not pleased with the result.

Kanamori writes:

>... it is intriguing to consider
>how Godel actually regarded the infusion of the infinitesimals into
>analysis through logic, as a formal device or as something more
>actual.

At this point, let me say something about the context in which Godel wrote
this. This was in "Remark on non-standard analysis", 1974, reprinted on
page 311 of Godel's collected works as Baldwin says. It's just a couple of
paragraphs, and I consider it only a minor annoyance that I have to
strongly disagree with Godel about this. Only Popes are infallible.

I feel better about disagreeing with Godel in these two paragraphs, since
Godel goes on to say:

"I am inclined to believe that this oddity has something to do with another
oddity relating to the same span of time, namely the fact that such
problems as Fermat's, which can be written down in ten symbols of
elementary arithmetic, are still unsolved 300 years after they have been
posed. Perhaps the omission mentioned is largely responsible for the fact
that, compared to the enormous development of abstract mathematics, the
solution of concrete numerical problems was left far behind."

This is of course laughable in the face of current developments and the
advent of conservative extension technology.

Some related statements were also made by Godel with regard to the large
cardinal axioms, which are demonstrably not conservative. My view on this
can be summarized as follows:
********
1. I do not believe it likely that there are particular outstanding
concrete open problems of contemporary mathematics that require the use of
large cardinal axioms.

2. However, I do believe that there are a large number of interesting
subareas of concrete mathematics that can (demonstrably) only be developed
with the use of large cardinals, and that these areas are of sufficient
genuine mathematical interest to be welcomed additions to the contemporary
mathematical scene. In particular, they will be developed in such a way
that they form attractive areas of research for mathematicians
independently of their spectacular metamathematical status.

3. The nature of the ongoing initial results in this direction strongly
suggest this to be the case. The matter is in its infancy, and will
radically change the general view of the nature of mathematics.
********

Sommer writes:

>In e-mail of Tue, 11 Nov 1997 05:35:35 +0100, Harvey Friedman
>indicates that he tends to side with the following view of
>non-standard analysis:
>
>   "There is no fundamental notion of infinitesimal and/or infinitesimal
>   reasoning. It is simply a confused way of thinking which people can get by
>   with at an elementary level. It was replaced by something more coherent and
>   powerful...."
>
>But, intuitions about infinitesimals and infinite natural numbers had
>extremely important roles in the development of analysis, and those
>intuitions are put to frequent use by applied mathematicians,
>physicists and engineers. A careful look might reveal that the
>"coherence and power" of epsilon-delta arguments is actually inferior
>to the power the mathematician has to reason about infinitesimals
>all-the-while being consistent with observed reality. This can't be
>overlooked.

Sommer has not fully quoted me, where I conceded a notion of "infinitesmial
reasoning," (see above), which Sommer should like. However, I submit that
"a careful look might reveal" but a careful look will actually reveal the
opposite. And "consistent with observed reality" seems difficult to defend.
Rick - do you literally mean this, or is this just an overeaction?

Let me also ask this pointed question of the people on FOM who are so
impressed by nonstandard analysis: which major mathematical analysts do you
view as using nonstandard analysis, or being sympathetic to nonstandard
analysis, and of the major analysts who are not, why aren't they
sympathetic? Also which major theoretical physicists use nonstandard
analysis, or are sympathetic to nonstandard analysis, etcetera? Please -
I'm not asking a question that I definitely know the answer to.

Barwise writes:

> On my
>view there is, or at least was for many, many years, an intuitive concept
>of an infinitesimal, a positive number that is less than 1/N for N = 1, 2,
>3,...  Robinson's work showed that this intuitive concept is consistent
>with the standard assumptions about the real numbers by giving us a model
>of it.

No, there is not such a well defined concept. There is at most
"infinitesmial reasoning" which is a known conservative extension.

>But there is a richer
>concept, corresponding to countable saturation.  The richer concept is more
>useful than the core concept alone, as various applications have shown.
>See, e.g. Keisler's AMS Memoir.  Perhaps there is no final unique concept
>here, but a family of increasingly rich notions.  That seems to me a more
>fruitful response to the phenomenon than to dismiss infinitesimals because
>they are not unique.  We don't reject set theory simply because there is
>not a unique conception.  As Godel has taught us, we keep looking for ever
>richer conceptions, conceptions that might shed light on open questions.

There is no prior concept of "saturated infinitesmials." This is a
(interesting and useful) purely technical concept. The comparison with set
theory is compeltely invalid. To make this clear, at some appropriate
level, the normal view is that we do have a unique conception of set - at
least V(w+w) - i.e., set of limited rank. This is a restricted concept of
set - not a murky expansion of the concept. Compare the clarity of V(w) and
V(w+w) - with its myriad of uniqueness theorems - with "infinitesmials" or
"saturated systems of infinitesmials." This is an absurd comparison. And
what is open ended about V(w) and V(w+w)? And is there a restricted notion
of infinitesmial that fares better in terms of uniqueness than "general
infinitesmial" whatever that is? The whole thing makes no definite sense.

But I will repeat - there is an interesting notion of infinitesmial
reasoning - at least up to a point. The actual formalization of actual
thinking like this is interesting, the main point (at this time) being the
conservative extension results.

Having said this, let me say that it appears to be to be absurdly unlikely
that the pursuit of these ideas - even the ones I acknowledge are
foundational - has any chance of leading to the kind of radical rethinking
of the nature of mathematics that is going to come out of the NECESSARY use
of large cardinals.

**************

A conservative extension result (approx. 1967).

Add a predicate St(x) for "x is standard" to the language of Peano
Arithmetic (PA). Consider the following axioms:

1. Usual equational axioms and axioms for successor.
2. Induction for all formulas with respect to the Standard integers only.
3. Induction for all formulas that don't mention St with respect to all
integers.
4. The successor of every standard integer is standard.
5. There exists a nonstandard integer.

THEOREM. Every sentence without St that is provable is already provable in PA.

PS: Now that my interest in conservative extension results of this kind has
been rekindled, maybe Rick and some others on FOM can state some other
conservative extension results.