No subject
Anand Pillay
pillay at math.uiuc.edu
Fri Oct 17 11:21:24 EDT 1997
I concur with Lou's views on "fom", logic and model theory.
As a rather minor point I note that Steve and Harvey often use the
expression "applied model theory" and Harvey in an earlier message even
discusses it as something in a conflicting or competitive relation with
"pure model theory". As Lou points out the exciting development over the
last 10 years in model theory has been the convergence of traditions from
stability theory and from the model theory of fields, and it is from within
this new conceptual unity that many interesting results are arising. So I
would prefer just "model theory".
Harvey and Steve appear to have taken over the notion of foundations of
math., so that it refers to only one kind of activity. By foundations of
math., I understand work on uncovering the "basic" math. concepts,
principles of mathematical reasoning, underlying relationships between
basic concepts and between basic objects, even the ontological status of
mathematical objects. Modern mathematical logic (i.e. over the last 100
years) has thrown up many fundamental notions and ideas in this regard
(set-theoretic axioms, formal proof, structure and truth in a structure,
definability, effectivity etc.) These "metamathematical" objects and
concepts are now themselves objects of mathematical study. A theorem in
mathematical logic may have a dual status; as a mathematical result, and as
something with foundational content. The same is true of results and
concepts produced by mathematicians outside the math. logic tradition. For
example, the Lang conjectures (mentioned several times in our discussions)
have, I believe, foundational content, relating notions from
geometry/topology (genus, hyperbolicity) to number theory (number,
behaviour and structure of rational soutions to systems of equations).
Rather paradoxically (and provocatively), from what I understand of results
in reverse mathematics, I tend to view the results as quite interesting
from the mathematical point of view, but just a curiosity from the
foundational point of view. But I'd be happy to revise my judgement after
looking at Steve's book.
Steve is right that model theory is concerned a lot now with trying to
understand concrete mathematical structures using methods from first order
logic. But such activity has also a foundational side, in so far as one
attempts to derive theorems from general logical properties of
definability, interpretability etc. In fact in at least a couple of places
(Introduction to my "Geometric Stability Theory" and review of Zilber's
"Uncountably categorical theories" in Bulletin AMS) I have explicitly
referred to the foundational content of some of this work: The Zilber
conjectures (now known to be false in full generality) were based around
the idea that rather basic mathematical objects, such as algebraically
closed fields, could be recovered from very general
logical-model-theoretic-combinatorial notions, such as that of a strongly
minimal set.(It should be said that the analogous problem for real closed
fields has been successfully solved.) The Russel_Whitehead program was to
recover all of mathematics from basic logical principles. This having
failed, I see much of geometric model theory trying to carry out similar
programs for what Harvey calls "tame mathematics", in a form as above. On
another point raised earlier about using logic to get good bounds,I should
say that new doubly exponential bounds HAVE been obtained by Hrushovski in
various diophantine problems, using model-theoretic methods.
I really see an enormous scope for logicians to use their insights and
general point of view to interact with other mathematicians. Harvey, in an
earlier message, asked for some concrete problems logicians could consider.
Here, off the top of my head is an example (this is NOT meant to be THE
problem to work on, it is just an example, and quite possibly it turns out
to be easy): Let V be a complex variety defined over Q. Consider the
structure M = (C,+,.,V(Q)) (i.e. we add a predicate for the rational points
of V). Can one show that either Th(M) is decidable, or (N,+.) can be
interpreted in M? If not, what Turing degrees are possible for Th(M)?
Anand
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