FOM: model theory; Lang conjectures; reverse math; Barry Mazur

Stephen G Simpson simpson at
Tue Oct 21 21:05:42 EDT 1997

Anand Pillay writes:
 > I note that Steve and Harvey often use the expression "applied
 > model theory"
 > 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".

Excuse me, but I think "model theory" was created by Tarski a long
time ago.  Now Anand wants to redefine "model theory" so as to make it
consist solely of stability theory and the model theory of fields.
I'm sorry, but I don't think I'll cooperate with this Putsch, at least
not until a representative group of model theorists has endorsed it.
I'll continue to say "applied model theory", until a better term is

 > 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).

Why in the world would anyone regard the Lang conjectures as having
foundational content?  They have nothing to do with the analysis of
basic mathematical concepts.  Rational numbers may be a basic
mathematical concept; curves may be a basic mathematical concept; but
rational points on curves are definitely NOT a basic mathematical
concept; they are of interest only to specialists in number theory.
If the Lang conjectures are f.o.m., then so is any piece of
mathematics whatsoever.  Obviously this makes no sense.

 > 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.

Well Anand, that's because your point of view *is* nothing but (a
brotherly variant of) the pure mathematician's point of view.  Maybe I
should be glad that Reverse Mathematics appeals to you on some level,
but I'm much happier when people appreciate it for what it is.

Dave Marker writes:
 > -Mazur has a very interesting article on number theoretic issues
 > comming out of Matiyasevich's theorem which appeared a couple of
 > years ago in the JSL.

Yes.  Barry Mazur is exceptional.  But the vast majority of number
theorists (and other pure mathematicians for that matter) are totally
indifferent to f.o.m.  It's a shame, but there it is.

(By the way, this paper of Barry Mazur, if it's the one I'm thinking
of, was published as an outgrowth of an AMS-ASL panel discussion on
Matiyasevich's theorem, here at Penn State in 1990, which I organized.
Among the participants were Angus Macintyre, Lou van den Dries, Barry
Mazur, Serge Lang, Harvey Friedman, and me.  It was a wonderful event,
with a lot of fireworks.  But Angus and Lou never sent in their
manuscripts; I guess they were too busy.)

Crabbily yours,
-- Steve

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