FOM: General intellectuals etc.

[Anand Pillay]

GENERAL INTELLECTUALS

Harvey and Steve's notions of general intellectual, general intellectual
interest, hierarchy of concepts, seem more and more fraught with
contradictions. Viewing it as what the man in the street or one's dentist
understands (which Harvey now says was just a manner of speaking to those
who don't get it), is actually  directly opposite to the view (which Harvey
says he really holds) that it is that  that which the truly great superior
intellects of all time understand or create. I suspect that the notions
really just serve Harvey and Steve's psychological needs, but maybe we
should not get into that. Lou remarks that what is of "general intellectual
interest" cannot be really discussed sensibly. Maybe he is right, but let's
try and see where the notion gets us. To be a general intellectual would
seem to mean trying to integrate as much as possible of what one sees,
experiences, thinks about, into a reasonably coherent conceptual framework,
or as used to be said "world outlook". What I guess is important here is
not only how successful one is, but also one's interest or desire to do it.
Part of this striving clearly involves attempting to find the unity, where
possible, in apparently disparate ideas or phenomena. In fact this tendency
characterises progress in all intellectual disciplines, mathematics and
logic in particular. It is not concentrating on the difference between say
polynomial equations and curves, surfaces, manifolds etc., which should be
typical of the general intellectual (who has an interest in mathematics),
but realising that these are both ways of talking about the same things.

How does the "hierarchy of concepts" (a notion which I personally find
somewhat cultish) fit into this? What are the "basic" concepts, those which
arise earlier or later, the "lower" ones or the "higher" ones?  The
development of knowledge is not simply a derivation of complex concepts or
propositions from basic ones, but rather the development of concepts which
explain and unify the earlier ones.  I actually think that in mathematics
it is this as much as the accumulation of theorems which represents
mathematical progress. Looked at from a formal point of view this may just
take the form of convenient "abbreviations" but I think it is much more.
The historical development of the abstract notion of a group was a
fundamental advance. Similarly for the rather more concrete notion of a Lie
group which permeates and explains so much in differential geometry,
physics, and many areas of mathematics.

Making sense not only of these and other notions (even cohomology), but
also of what it means for them to be fundamental, is I think very much the
business of mathematical logic, and foundations and philosophy of
mathematics, although it is not necessarily in the spirit of "FOM".

In fact I find the whole discussion of how simple, arcane or whatever
certain concepts are, to be beside the point. If one is to work (or claim
to work) in the foundations of a subject, one has an intellectual duty to
at least try to make sense of at least some of the current and central
concepts, and even to go beyond the "ordinary" mathematician in making
connections.

What about the "general intellectual" or "foundational" interest of
Matijasevich's (or the MRPD) theorem versus Siegel, Faltings etc? One is
right that there is a feeling of Matijasevich being different in spirit.
The number theorists feel the same way. In Manin's very interesting preface
to his Number Theory I, he says that "whether or not a given article
belongs to number theory depends on the author's system of values. If
arithmetic is not there, it will not be considered number-theoretical....
So combinatorics and recursion theory are not usually associated with
number theory..." (I am not sure what he means.)
The issue for us (logicians) is I think that MRDP is something which is
familiar and comes out of the tradition in which we were educated, while
the others are not. I would hope to see the distinctions softened and to
see  number theory courses discuss these various results side-by-side (and
likewise for logic courses).






APPLIED MODEL THEORY

Harvey has gone to some pains to insist on the term "applied
model-theorists", with this having some negative connotation in his eyes
(althought not necessarily in mine). He even goes so far (rather
presumptuously) as to ascribe various motivations to us. He then describes
various work of his (The complete theory of everything) as being really
classical central model theory, which us applied people dismiss out of
hand. The image is of these people doing little technical things while the
true intellectual foundationalists are operating at a much higher, general,
and abstract level.

So maybe it is appropriate for me to say a few general words about model
theory, speaking for myself, not for Dave, Lou or John (Baldwin). First,
Harvey - you told me some results when you were in Urbana. What I remember
is something about the first order validities in a structure in which both
a linear ordering and pairing function are defined. I recall it being an
interesting result. The abstract you sent around (list of your recent
talks) is too vague. In any case, if I remember, this looked like a nice
result firmly in the sphere of modern model theory (which by the way
doesn't need to be marketed as the "theory of everything" ). Anyway I'd
like to know more details.

Back to model theory. I said in a previous message that many interesting
results are coming out of the "convergence" of "pure" model theory and the
model theory of fields traditions. My own background (as also John
Baldwin's) is "pure" model theory, and if I have some degree of expertise
in anything, it is there. The general context here was Shelah's program to
classify complete first order theories, an enterprise with both
mathematical and foundational content. The issue of what the consequences
are of being able (or not) to define a total ordering, pairing function,
random graph, ..., in a model of the theory T, were and remain fundamental
issues. Shelah was interested in trying to classify theories as to whether
or not their models can be described by trees of cardinal invariants. In
the process he concentrated on stable theories (essentially theories which
do not interpret a linear ordering), and a whole array of concepts was
developed, leading to a quite successful conclusion. The notions of a
structure being coordinatized or built up out of "simpler" structures
appear, as do precise notions of how definable sets  in a given structure
can interact (nonorthogonality) or not interact (orthogonality). Inso far
as I was involved in the foundations of O-minimality, it was motivated by
trying to find similar notions in the situation where one DOES have a
linear ordering on the structure.
More recenly other variants of Shelah's machinery have been found outside
the stable context. At some point "geometric" stability theory entered the
picture with attempts to classify structures whose theory is aleph_1
categorical, up to bi-interpretability, giving rise to various Zilber
conjectures. For better or for worse, the motivation for much of the above
was at the abstract level of classifying theories and/or structures under
purely model-theoretic hypotheses.
The amazing recent discovery is that many of these notions (orthogonality,
the Zilber dichotomies) are present in nature, namely in concrete and
classical mathematical, algebraic, number-theoretic structures, although
often in a hidden way. Not only do certain classifications of equations,
differential equations, ..., correspond to the various model-theoretic
categories, but the model-theoretic results have led to new mathematical
results and effective bounds, which I will not say more about now. My point
here is not to say what "general intellectuals" we are, but just to
describe the actual history and state of affairs in a part of what Harvey
and Steve call "applied model theory".

The above developments in model theory have some common features with
reverse mathematics. In the latter, a small number of natural set-existence
postulates are shown to be present in nature, and similarly for the idea
that large cardinals occur in "nature" (namely in finite combinatorics). I
don't see why one  is more foundational than the other, although I may have
my own feeling of what is "deeper", especially from the point of view of
"general intellectual interest".

RATIONAL POINTS

I had mentioned (by request) a (possibly easy) problem to Harvey, if
f_1,..,f_n are polynomial equations over Q in x_1,..,x_m, what can one say
about the structure (C,+,.,P) where P is an m-place predicate for the set
of common rational solutions to the f_i (namely P denotes V(Q) where V is
the  variety determined by the f_i)? Is it decidable, undecidable, what is
the Turing degree..? In fact the issue concerns what is interpreted or
definable in such a structure.  (Harvey also asks if rational complexes are
the same as rational reals. Yes, they are just the rational numbers.)
(One can ask a similar question replacing C by R.)
Working in the complexes is just to make the problem easier to pose. The
real issue is how complicated is the induced structure on V(Q), where by
induced structure on V(Q) we mean the structure whose universe is V(Q) and
which has predicates for solutions of equations over Q in mk inderminates
(k arbitrary). One extreme case is where m=1 and the system is trivial
(x=x) in which case one gets the structure (Q,+,.) which interprets the
natural numbers. Another extreme case is when V is an elliptic curve (or
more generally a so-called "abelian variety"), in which case the structure
appears to be "very nice" (decidable, "superstable"...) (by a theorem of
Faltings).

Anand