[FOM] foundational?

[Harvey Friedman]

Reply to Baldwin 3:01PM 7/3/03 and 8:54AM 7/4/03.
>
>I don't have a big quarrel with this characterization.  I think it
>does miss the psychological leanings of most model theorists
>towards foundational issues (as opposed to FOM -- check the
>archives long ago for a too exhaustive discussion of fom vrs FOM).
>To stoke the controversy very briefly, I'll outline in more detail
>a `foundational' view of model theory.

I am of course quite interested in seeing model theory take a more 
foundational viewpoint, and in fact the explicit form of the upward 
Lowenheim Skolem theorem that I have been talking about is an example 
of model theory taking a more foundational viewpoint. However, you 
have taken the position that this is no longer model theory. So you 
must have a quite different view of what "foundational model theory" 
should look like.

I would like to explore just what kind of foundational viewpoint you 
think model theory could take, or has taken.

The word "foundational" means something rather focused, and is not a 
synonym for "good" or "deep" or "interesting". But rather than start 
by arguing about just what is meant by "foundational", let us examine 
some examples and see what we can see. 

>
>One of the important discoveries of the middle 20th century is the
>futility of a trying to find a general foundations of mathematics.

I have tried to figure out just how to read this sentence so that it 
is even partly true. This has proved difficult.

In fact, one of the most important discoveries in all of mathematical 
thought of all time is just how productively successful the general 
foundations of mathematics through set theory really is.

To reconcile this, perhaps you are referring to the incompleteness 
phenomena. If we interpret mathematics very broadly, then no single 
formal system is sufficient to derive all mathematical statements 
that we WOULD accept as established, IF we considered them.

However, that is a rather idiosyncratic notion of "general foundations".

Furthermore, if that is what is meant by "general foundations", then 
moving to specific branches of mathematics does not help very much - 
except in certain very interesting but highly limited cases where one 
has at least quantifier elimination, or something like it.

In fact, under this standard, at the moment it appears "futile" to 
even focus on the field of real numbers with exponentiation added. It 
appears to be "futile" to even show that one has "general 
foundations" for the solvability of all equations in this restricted 
language. The transcendental number theory involved is just too 
awesomely difficult.

>One of the essential contributions of the model theoretic view
>point is to investigate the foundations of different mathematical
>subjects via different specific formalisms.

This is already anticipated, to some extent, by earlier work in 
standard f.o.m., from the standard f.o.m. viewpoint.  E.g., 3 
separate groups of formalisms for arithmetic, analysis, and set 
theory. This already goes back to the 1930's, I believe, with Godel.

>In a trite way one
>studies algebraic geometry as the first order theory of
>algebraically closed fields.

Of course, isn't even classical algebraic geometry full of non first 
order constructions? Or do these have interesting first order 
replacements in the hands of model theorists?

>But this viewpoint has led to ways
>to understand related topics in diophantine geometry.

What does "understand" here mean? The reason I am asking such a 
perhaps annoying question is that I want to see just what you mean by 
"foundations" here.

>Model
>theory studies both the comparisons between various systems (thus
>the stability hierarchy, o-minimality etc) and the analysis of
>particular systems - e.g. the intensive work on expansions of the
>real numbers or groups of finite Morley rank.

What does "systems" mean here? I don't think you just mean "first 
order systems in predicate calculus with equality". Even o-minimality 
cannot be reduced to a set of first order sentences.

>
>The key step in this analysis is to determine the subject. Thus,
>in studying algebraically closed fields we are studying sets with
>two binary operations and some constants; we are NOT studying the
>representation of the complex numbers on what ever cardinal
>happens to be the cardinality of the continuum.

Under the standard model theoretic viewpoint, alg closed fields are 
rather special in that the uncountable ones can be lumped together 
for any purpose, under this standard model theoretic viewpoint.

However, consider real closed fields. There is intricate set 
theoretic information at every uncountable cardinal, because of the 
order. Would you consider a necessarily set theoretic study of the 
structure of uncountable real closed fields part of model theory?

By the way, an underdeveloped subject you would reject as not model 
theory, is that of Borel algebraically closed fields, and Borel real 
closed fields. I.e., the uncountable ones that can be rather 
explicitly defined.

>
>Another important insight is that to study the structure of
>definable subsets of a single object (e.g. the complex field C) it
>is useful to study the family of all structures elementarily
>equivalent to C.

I suspect that you are calling such insights "foundational"? Any explanation?

>
>A third is that certain similarities appear when studying
>different systems - thus the study of all aleph_1 categorical
>structures led to detailed investigation of the geometry of
>strongly minimal sets.  These investigations then had specific
>applications in Diophantine geometry.  These developments are
>symbiotic not teleological.

This sounds like the standard good thing that happens throughout all 
of mathematics. One nice development leads to another, which leads to 
some application not expected in yet another. But is the term 
"foundational" appropriate for this? Under such terminology, perhaps 
most good mathematicians do foundations of mathematics.

>Different model theorists (at
>different times) focus on the specific theories (what Friedman
>calls applied model theory) and the development of the general
>theory.  These different foci enrich one another.

I like the term "applied model theory", since it suggests a 
difference with "pure model theory".

Model theory began as classical foundations of mathematics, done for 
a purely foundational purpose. The basic setup was Frege with his 
setup of predicate calculus with equality. The original spectacular 
(foundational) result was Godel's completeness theorem, implying the 
compactness theorem.

Model theory driven by classical foundations of mathematics appears 
to be largely extinct. I am looking towards a revival. 

>The 90's were a
>high point in the study of specific theories.  There seems to be
>more of a movement to the general situation in the last few years
>for two reasons.  The work of e.g. Buechler, Lessmann, Pillay, Ben
>Yaacov ... has tied together new insight into Shelah's pioneering
>work of the 70's on nonelementary classes with developments in
>simple theories, and studies of the expansions of models.  Zilber
>has discovered that infinitary logic and Shelah's theory of
>excellent classes provides a fruitful framework for investigation
>of the complex exponential field.

But now that the general mathematics community has a taste for at 
least some cases of actual applications of model theory to core 
mathematics - even if the mathematicians have figured out how to 
eliminate the model theory after the fact - it seems that model 
theory is destined to be evaluated by non model theorists almost 
entirely in terms of its applications to core mathematics. I.e., 
model theory may become a victim of its success?

Harvey Friedman