FOM: ReplyToDavis

[Harvey Friedman]

This is a reply to Davis, 11/7/97.

In a way, I agree and welcome this posting. However, the choice of words
does reveal an undercurrent here that I take some issue with. In
particular, the posting contains at least the suggestion that mainline
f.o.m. may not be as fruitful and active as I know it to be. And also, I
have some constructive reservations about the f.o.m. importance of
nonstandard analysis.

>Something I miss in many of the postings here is the realization that
>foundational work tends to be in response to specific foundational problems.

The problem of "what (kind of) interesting mathematical information can
only be obtained by expanding the usual axioms for mathematics" is very
mainline f.o.m., clearly discussed in various ways by Kurt Godel. However,
I'm not sure that it is "in response to a specific foundational problem" in
the sense that you mean later in your e-mail. What do you think?

Another very mainline foundational problem: "what kind of coherent
justification can one give for the generation of the large cardinal axioms"
or for that matter ".... for higher set theory?"

Yet another very mainline foundational problem: "what is so special or
perhaps canoncial about the currently studied formal systems of f.o.m.?"

There are many more, and at a later time I want to make a systematic
posting of them. These are all topics of great promise and current activity
at various levels. They are not dead subjects in any way shape or form.

>There are various things in mathematical practice that lead mathematicians
>beyond what they are able to justify in terms of their contemporary
>understanding. When this happens, a foundational problem arises. In the
>happiest instances, foundational work will not only provide a satisfactory
>explanation, but also will provide new directions for mathematical research.

I agree that there are fruitful interpretations and reinterpretations of
foundational developments along these lines. However, what specifically do
you propose as a foundational problem that fits this mold now?

>Analogy is another force leading mathematicians beyond what they are able to
>justify. (This has great contemporary relevance for set theory: the infinite
>by analogy with the finite, the huge infinite by analogy with the countably
>infinite.)

Now this is something that is currently being adressed in f.o.m. E.g.,
there are transfer principles from discrete to transfinite that generate
some large cardinals. This seems to fit into your mold.

>The obvious success of infinitesimal methods, used freely by engineers and
>physicists long after mathematicians had insisted they were illegitimate,
>was an example of a foundational problem not resolved until Abraham Robinson
>explained it.

Frankly, I have not been a big fan of the f.o.m. aspect of nonstandard
analysis - at least in its present form. Here are two extreme positive
views of nonstandard analysis:

a. There is a fundamental notion of infinitesmial and/or infinitesmial
reasoning that is not captured by epilson-delta, and has independent
significance. The interpretation of this notion and/or reasoning in modern
mathematical terms is an important development of f.o.m.

b. There is no fundamental notion of infinitesmial and/or infinitesmial
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. It is of some heuristic value, especially in teaching. It is
however undoubtedly somewhat useful in giving new proofs or discovering
proofs in certain contexts. After all, first order predicate calculus,
compactness, and ultraproducts are themselves sophisticated mathematical
constructions, and it is not surprising that they are on occassion somewhat
useful.

Of course, you can tell by comparing the lengths of these statements that I
tend to side with b. There is the middle of the road view that
infinitesmials make no sense, but infinitesmial reasoning makes sense and
is worth formalizing. As early as 1967, I had already looked at formal
systems of nonstandard arithemtic and proved that they were conservative
extensions of standard arithmetic - a topic to which I never returned.
Others such as Ed Nelson have worked on this.

Notice how some interesting f.o.m. comes out of justifying b:

***The reals are unique. E.g., they are the only complete dense linear
ordering without endpoints which has a countable dense subset. However, the
nonstandard reals cannot be unique.***

Now this is really too vague. So let's make some precise statements. First,
let's back off and consider the more elementary case of Arithmetic. Let us
consider nonstandard models of arithemtic. NOTE: I don't have the space
here to move on to nonstandard analysis. But many of the points already can
be made for nonstandard arithmetic.

***All nonstandard models of arithmetic are pathological. There is nothing
unique about any of them.***

We know many forms of this. E.g., the only countable model of even PA that
is recursively presentable in very weak senses is the standard model. Also,
there are continuumly many countable elementary extensions of the standard
model of arithmetic. This is because you can fool with the "standard parts"
of elementary extensions of the standard model. The standard part in this
special context is defined as those sets of actual natural numbers that are
defined by a formula over the model (just looking at standard solutions to
the formula). By (essentially) a theorem of Scott, this can be any
countable model of WKL (Scott sets) which contains the true sentences of
arithmetic.

[Actually, there may be a technical question here of exactly what the
standard parts of models of the true sentences of arithemtic are, or more
generally, given any consistent extension X of PA, what are the countable
standard parts of models of X? This is given by Scott in the case that X is
recursive. I just don't remember what happens in general - even in the
countable case; can anyone on FOM tell us if this was completely analyzed?]

In fact, it appears that all "standard parts" of countable models of
arithmetic are perhaps themselves pathological. However, there is the
following possibility: that one could still claim that there is a preferred
countable nonstandard elementary extension of the standard model, where the
standard part would presumably be exactly the arithmetic sets of natural
numbers.

However, I have never seen such a construction. I can imagine that one
would take, say, certain functions of natural numbers - perhaps
arithmetical functions of natural numbers - and then perhaps factor in some
way or another, and come up with some sort of canoncial construction of a
nonstandard countable elementary extension of the standard model. But I
don't see how to do this.

$$$ I remember years ago, such a construction was proposed by Kochen (and
also perhaps Kripke) building on the Paris/Harrington indiscernibles
construction. I don't know if this was fully pursued. $$$

I assume that it is well known that there are continuumly many countable
elementary extensions of the standard model whose standard part is the
arithemtic sets of natural numbers. But can one get a "preferred" one?

Moving on to uncountable models of arithmetic, it is in the spirit of A.
Robinson's treatment to consider elementary extensions of the standard
model whose standard sets are all of the sets of integers. Let us call such
models of true arithmetic FULL.

Now I believe it is known that there are 2^c nonisomorphic full models of
arithemtic of power c. Also, it is clear that there is no Borel measurable
full model of arithmetic.

We can attempt to more directly address the question of a canoncial or
preferred full model of arithemtic. Is there a full model of arithmetic
that is set theoretically definable? The answer is that this is independent
of ZFC. However, if there is to be a canoncial full model of arithemtic
then we would want:

***an explicit definition that provably defines, within ZFC, a full model
of arithemtic.***

This can be shown to not exist - and is well known. Also the following is
known:

"an explicit definition that provably defines, up to isomorphism, within
ZFC, a full model of arithmetic.***

cannot exist. This holds even if, in the appropriate sense, parameters for
real numbers are allowed. What about just defining the order type?

>I would like to urge that thinking about fom in terms of foundational
>problems needing to be addressed, rather than in terms of which concepts or
>theorems are truly "foundational", is  likely to be a more fruitful approach.

I personally get a lot out of considering what is truly "foundational" -
although not in the argumentative sense of some of the FOM interchanges. I
use such considerations in choosing focused topics such as those mentioned
at the beginning of this e-mail. Some of the mainline f.o.m. is obviously
of timeless importance - especially, "what (kind of) interesting
mathematical information can only be obtained by expanding the usual axioms
for mathematics?"

I have to admit that many people don't think that the particular arguments
over the FOM about whether or not some specific topics are foundational or
not is fruitful. Is that what you are criticizing?

And it would be nice if you could list some foundational problems "needing
to be addressed." Do you think that

"the free use of algebraic manipulation of
operators in solving ordinary differential equations (see George Boole's
classical text for masterful use of these techniques without a shred of
justification), the Heaviside operator calculus, the Dirac delta function"

are still timely topics for f.o.m.?