FOM: model theory; Lang conjectures; reverse math; Barry Mazur
Stephen G Simpson
simpson at math.psu.edu
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
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.)
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