FOM: Pure and Applied Model Theory

[Harvey Friedman]

Marker 6/16/00 11:06AM writes:

>I have no objection to the the phrase "applied model theory"
>provided it is used in a meaningful way.  Certainly it would make
>sense to call my Urbana lectures on o-minimal expansions of the real
>field or Hrushovski's proof of Mordell-Lang applied model theory.
>My objection is when you use it a blanket to cover almost all of the work
>going on in model theory.
>
>	* Unlike 20 years ago the distinction between "pure" and "applied"
>is often not very clear. Algebraic considerations arise naturally in pure
>problems
>	* Even a statements like "o-minimality is part of applied model
>theory" is misleading as it ignores the exciting work of Peterzil and
>Starchenko carrying out the analog of the Zilber program for o-minimal
>structures.
>	* The label "applied model theorist" is even more problematic for
>many people as their work spans a variety of subjects. Pillay's work
>in Differential Galois Theory is applied, but his work on forking
>in simple theories or his Geometric Stability Theory book is
>not. Hrushovski's work on the Manin-Mumford conjecture is applied, but his
>recent work with Hart and Laskowski on uncountable spectrum functions is
>not.

Virtually all subjects have both pure and applied aspects. In virtually all
subjects, applied work is used in pure work, and pure work is used in
applied work. In varitually all subjects, many people do applied work as
well as pure work.

So nothing here is out of the ordinary. It is also true about, say, PDE, or
the theory of algorithms.

What I have in mind is the statements of Theorems and the stated
motivations of programs - definitely not underlying proof techniques. It is
normally reasonably clear what this means, but, as is to be expected,
entirely unclear in many borderline cases. So what?

In the case of Model Theory, one normally means "Applied to mathematics."
However, there is also model theory that connects up with its roots in
philosophy (e.g., my Complete Theory of Everything). One should probably
call this Philosophical Model Theory. And then there is finite model theory
aimed at issues in computer science.

Also, there is the good old fashioned classical model theory, dear to the
hearts of such people as Grossberg and Shelah. It's not for math and it's
not for philosophy and it's not for computer science.

What I object to is the idea that Philosophical Model Theory and Classical
Model Theory as indicated above are either

i) no longer model theory at all;
ii) or are just uninteresting model theory.

Of course, we know that we are moving towards a situation where it is

iii) unsupportable by the National Science Foundation.

Baldwin writes 6/21/00 9:15AM:

>In his thesis, Morley posed the question.  Can an aleph-1 categorical
>theory be finitely axiomatizable?
>
>This question has some philosophical content.  Bill Tait put it, do we
>know all the  ways a single sentence can demand infinity:
>  discrete linear order, dense linear; a pairing function.  (I think these
>basically remain the only examples.)

Is there a precise theorem or conjecture that says that "these basically
remain the only examples"?

There is no question in my mind that Morley's question and related matters
is pure model theory. That applied model theory might be useful, or that
normal mathematics might be useful, is not unexpected, but does not affect
the attribution. See my response to Marker above.

>(Note that even a requirement of completeness complicates
>the issue:  Asserting that a unary function is 1-1 and only one element
>does not have a predecessor forces an infinite model without dealing
>with how many finite cycles are permitted.  Morley asks more strongly
>that the sentence be aleph_1 categorical.)
>
>...This work is
>certainly a blend of model theory and group theory.  But the problems grew
>up within the area so the description `applied model theory' is strained.

Not strained at all. Just an example of how work in other fields can be
used to answer pure questions.

>...One could imagine the answers to these questions
>being characterized as either pure or applied model theory (depending in
>some cases on which way the questions were answered).

This situation is no different than for most subjects.