[FOM] Dependence relations in model theory, 2

[Harvey Friedman]

Reply to Baldwin, Dependence relations in model theory, July 19, 2003, 
at 10:55 AM.

The reply is in two parts. This is the second part. In this part, I 
respond to Baldwin more directly.

In general, Baldwin hints at what I call a semblance of some 
"foundational" work.

Baldwin wrote:

> Dependence relations
>
> This note is a short survey of the role of dependence relations in
> model theory. It continues the series I have been asked to write
> discussing various fundamental concepts of model theory.  I will
> discuss 3 developments.  ...

> The properties of linear independence and algebraic independence
> were isolated by Van der Waerden (I think in the 30's).  The
> investigation of closure systems continues as a minor area--
> sometimes called matroid theory.  Much of the effort goes to the
> representation of abstract closure systems as vector spaces or
> fields or .....  There was (is?) also active development of these
> notions in universal algebra.

The notions of linear independence and algebraic independence are well 
known as very important - in fact essential - mathematical notions 
which share crucial properties. There is also a general feeling that 
these notions share properties that are perhaps fundamental even 
outside mathematics, perhaps in the sciences, both quantitative and 
descriptive, and in philosophy, including social and political 
philosophy. A truly foundational program would be to establish (as 
strong) a completeness theorem (as is feasible) for the laws that 
Baldwin writes down (credited to Van der Waerden), which would not only 
shed light on the mathematical notions, but also on such notions of 
dependence that transcend mathematics. I suspect that there is valuable 
research along these lines already, and would be rather pleasantly 
surprised if such research was fully mined out for significant 
findings. I am happy to become educated.

Incidentally, I note that Baldwin writes above:

> The
> investigation of closure systems continues as a minor area--
> sometimes called matroid theory.

By this standard, would Baldwin regard model theory as a minor area? 
What kind of "minor" is used here?

>
> Marsh noticed that defining a in acl(X) if a is in a
> FINITE set which is first order definable with parameters from X,
> then on a strongly minimal structure, one has a closure system in
> the above sense.  Baldwin and Lachlan generalized this by letting
> D be any strongly minimal set in a model.
>
> The key point is that if there is a closure system on D then D is
> determined up to isomorphism by the cardinality of a basis (a
> maximal set of independent points.

Can such developments be reworked and/or augmented to give them  a  
clear foundational purpose?
>
> Shelah discovered the notion of forking provides a relation
> on a model M of a stable theory such that for any subset B of M
> tp(a/BX) forks  over B  is a relation_B which satisfies all
> the axioms except transitivity 1).
>
> The big surprise here is that `exchange' is global; unlike the
> prototypic notions the difficulty is to establish transitivity.

Can such developments be reworked and/or augmented to give them  a  
clear foundational purpose?

I have the same question for the rest of Baldwin's posting.

> CONCLUSION:  Perhaps it was to avoid wrangling about the meaning
> of `foundations' that led Tarski to his less than  mellifluous
> phrase `methodology of the deductive sciences'.

I find it doubtful that Tarski was concerned "about the meaning of 
'foundations' ". If he did, he didn't need to be. For there is one 
particular development of Tarski, aside from formalization of truth 
predicates, that is of clear general intellectual interest.

He established that in real algebra, or in Euclidean geometry, one can 
circumvent Godel's incompleteness theorem. In particular, the axioms of 
real closed fields are not only complete, but a great many important 
mathematical statements can be readily formalized as statements in real 
closed fields, subject to his completeness theorem.

In fact, this is the most obviously foundational aspect of model 
theory, as far as I can tell. This doesn't mean that it is the only 
foundational aspect, but just the one that is most obvious. Namely, 
that there are substantial fragments of mathematics for which one can 
give a (natural, clear, and) complete axiomatization. This fulfills 
Hilbert's dream in certain rich mathematical contexts. Hilbert's dream 
is of obvious general intellectual interest, and the extent to which it 
can be fulfilled and cannot be fulfilled is also of obvious general 
intellectual interest, at least to the extent that appropriately 
illuminating answers are given.

> In this spirit,
> let me point out that it is already an astonishing convergence of
> diverse methods when Morley's combinatorial definition of rank
> yields the same object as Krull dimension (defined in terms of
> chains of prime ideals) for the rank of a variety in algebaically
> closed field.  The unifying role of `forking' as providing a
> universal definition for notions of dependence  in the study of
> diophantine geometry, algebraic geometry, vector space,
> differential fields,compact complex manifolds etc.  is an
> important contribution of model theory to the `methodology of
> mathematics'.
>
>

Again, I see a semblance of "foundations" hinted at here. Can this be 
capitalized on in order to create general intellectual interest?

What I have in mind is this. Some clear general theory of "dependence 
relations". A clear set of examples of "dependence relations" in 
diverse areas of mathematics. Even better would be to include some 
outside mathematics. A clean theorem to the effect that these notions 
of dependence can be uniquely defined from certain simple parameters 
clearly associated with the mathematical context. Perhaps a theorem to 
the effect that under suitably general conditions, there is always a 
unique good notion of dependence.

Whether this rises to the level of general intellectual interest 
depends on just how bogged down this gets into technical definitions. 
However, in situations like this, there is a real prospect of further 
reworking in order to replace technical definitions by less technical 
definitions.

But I have confidence that a distinctly foundational attitude to all of 
this should greatly increase the prospects of casting this in a way 
that maximizes its general intellectual interest.

Having said this, let me caution that this is still unlikely to reflect 
a major aspect of some of our legacy. Some of the great results in 
f.o.m. are not only of great general intellectual interest, but they 
also change our view of the very nature of mathematics. This is 
particularly true of the independence results. It is also true, to some 
extent, of the analysis of algorithms, as it was not at all clear that 
there was even a clear and robust notion of (discrete deterministic) 
algorithm at all.

Of course, the enormous pre Godelian advance that mathematics could 
even be formalized is also is of that character.

I don't see anything in present day model theory that has a chance of 
changing our view of the very nature of mathematics, in such profound 
ways. For that kind of advance, the best prospect still seems to be 
through independence results.

Let me close by noting that Simpson, Dependence relations in model 
theory, July 17, 2003, 3:20PM, has written a very thoughtful and 
thought provoking response to an earlier posting of Baldwin, and 
Baldwin has not directly addressed any of Simpson's points. I would 
like to see if Baldwin, Simpson, and Friedman are, or can get on, the 
same page.

Harvey Friedman