FOM: Simpson's Urbana thoughts on model theory
Dave Marker
marker at math.uic.edu
Fri Jun 16 12:06:35 EDT 2000
Simpson writes:
>Harvey's "Urbana Thoughts" posting argues that the phrase "applied
>model theory" is appropriate to describe a certain outward-looking
>orientation in model theory. Why don't the model theorists agree?
>What unfavorable resonance does the phrase "applied model theory" have
>for them?
Steve:
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.
As my opinions have not changed much since I wrote a long note on this
subject October 22, 1997 [I make one revision in the ps below]
http://www.math.psu.edu/simpson/fom/postings/9710/msg00031.html
I will only summarize some points.
* Unlike 20 years ago the distinction between "pure" and "applied"
is often not very clear. Algebraic considerations arise naturally in pure
problems [I give several examples in the message cited above, Baldwin
gives another in his June 11, 2000 posting
http://www.math.psu.edu/simpson/fom/postings/0006/msg00046.html ]
while techniques from stability theory have found suprising applications.
* 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.
Dave Marker
PS: One revision I would make to my earlier message is to lower
the estimate that 80% of research in model theory has an
algebraic or geometric component. Finite model theory is the main
reason for the this revision. An interesting phenomena is that
techniques from infinitary languages, generalized quantifiers and
inductive definitions play a major role here.
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