FOM: Urbana thoughts; model theory; spirit of generosity?
Stephen G Simpson
simpson at math.psu.edu
Wed Jun 14 20:25:33 EDT 2000
SUBDIVISIONS OF LOGIC
The ASL 2000 meeting included panel discussions on logic in the 20th
and 21st centuries. Each panel consisted of one representative from
each of the four major subdivisions of logic: set theory, model
theory, recursion theory, proof theory.
20th century 21st century
set theory Donald Martin Alekos Kechris
model theory Carol Wood Anand Pillay
recursion theory Martin Davis Richard Shore
proof theory William Tait Sam Buss
Except for Martin Davis, each of these panelists concentrated on
presenting an overview of (at least some of) the major developments
and themes within his/her particular subdivision. Because the four
subdivisions are rather specialized, there was little in the way of
interaction or dialog among the panelists.
In this posting I react to the model-theoretic parts of the panels.
----------
TOP TEN LISTS
Carol Wood presented a fun overview of model theory, organized around
``top ten'' lists and livened with photographs of famous model
theorists. Her talk was essentially an upbeat, almost boosterish
celebration of the achievements of model theory in the 20th century.
Especially interesting were Carol's provocative remarks in support of
her controversial elevation of Macintyre ("Mr Taste") above Shelah
("Mr Waste"). Rather than jump into the middle of this
model-theoretic feud, I will let the model theorists duke it out among
themselves. My only request is that they do it here on the FOM list,
so that the rest of us can enjoy it!
----------
PILLAY THEN AND NOW
I do want to react to some of Anand Pillay's remarks on model theory.
I found it gratifying that many of Anand's remarks now, in the year
2000, were identical to things he had said several years ago, in the
early days of FOM. See especially the FOM archive for October 1997 at
http://www.math.psu.edu/simpson/fom/postings/9710/. I was glad to see
that the issues we discussed, way back then, are still on the minds of
at least one prominent model theorist.
By the way, Harvey has already responded to some of Anand's remarks,
in his recent "Urbana Thoughts" posting of Sat, 10 Jun 2000 00:08:46
-0400. But I want to add my two cents.
----------
INWARD, OUTWARD, GEOMETRIC
Anand set the stage in terms of an opposition between what he called
"outward-looking" model theory (Robinson-style model theory of fields,
etc) and "inward-looking" model theory (Shelah-style classification
theory, etc), which existed for several decades prior to 1990.
According to Anand, these two camps were always reasonably friendly to
each other, due to a "spirit of generosity" exhibited by the
outward-looking model theorists, despite their "deep suspicion" of
Shelah's work.
According to Anand, the last 10 years have seen a fusion or marriage
of inward- and outward-looking model theory, giving birth to a new
"geometric model theory", which represents the future. Some important
concepts here are stability, o-minimality, the Zilber dichotomy
(linear vs non-linear, or RCF vs ACF), simple theories, etc.
----------
SPIRIT OF GENEROSITY ?
In the question period at the end of the panel discussion, I asked
whether the "spirit of generosity" remains alive in the model theory
community. I feel that my question didn't get an adequate answer at
the meeting, so now I repeat the question and request that the model
theorists answer it here on FOM.
Specifically, it seems to me that some of Anand's remarks tend to
unjustly obliterate numerous "non-geometric" topics in model theory
which are now considered unfashionable. Among the unfashionable
topics are infinitary logics, generalized quantifiers, two-cardinal
theorems, models of set theory, models of arithmetic (represented by
recent work of Schmerl, Kossak, et al), models of subsystems of second
order arithmetic :-), etc etc.
According to Harvey's recent "Urbana Thoughts" posting, Anand's actual
position is that some or perhaps all of these topics are not even part
of model theory any more!! And this is confirmed in Anand's FOM
posting of Fri, 17 Oct 1997 10:21:24 -0500, where Anand says that what
used to be called the model theory of fields, recently enriched with
stability theory, should now be called, simply, "model theory"!!!
My question is, is there any sign that the triumphant geometric model
theorists will now be able to find within their hearts a "spirit of
generosity" toward their non-geometric colleagues? Or, will the
non-geometric troglodytes be consigned to outer darkness? Could some
of the model theorists (geometric or otherwise) here on FOM please
undertake to clarify this situation, for us outsiders?
----------
APPLIED MODEL THEORY
Another question. Anand used a number of phrases describing various
aspects of model theory: inward-looking model theory, outward-looking
model theory, geometric model theory, and possibly others. But one
phrase that Anand took pains to condemn was "applied model theory".
This is a phrase that van den Dries used to like, perhaps 15 years
ago, but it has apparently gone out of fashion. Not only Anand Pillay
but also Dave Marker and a number of other model theorists have
expressed deep unhappiness with it. I would like to know why.
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
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