FOM: Urbana thoughts

[Matthew Frank]

I also recently attended and enjoyed the recent ASL conference, where
there were many good talks, and where I had the pleasure of conversing
face to face with several FOMers.  Here are a few comments on the
conference (only in those areas where I think I have something to add, or
where I felt the need to record something here):

Panel:  Does Mathematics Need New Axioms?

Maddy spoke of "accepting or rejecting" new axioms, and Steel spoke of
"adopting" them.  I, as a formalist, see no need to do any of these
things, although we may prove some theorems from these axioms.

Friedman asked (on-line, though not during the panel) for a proof that no
natural axioms settle the continuum hypothesis.  Feferman pointed out that
Freiling's axiom (JSL 51 (1986), 190-200) is a possible counterexample:

for every f: R -> {countable subsets of R},
 there exist x,y such that x not in f(y) and y not in f(x)

I find this axiom quite natural (particularly given Freiling's discussion
of it in probabilistic terms, and the comment that one should expect this
to hold for almost all x and y), and it is equivalent to the negation of
the continuum hypothesis.  I also wonder how this relates to Steel's
comment that "every natural extension of ZFC is equiconsistent with an
extension axiomatized by something like large cardinal axioms":  what sort
of large cardinal axiom is needed for ZFC + Freiling's axiom? 

Soare:  Computability and Differential Geometry

Only when I got the abstract for Soare's talk did I learn that Nabutovsky
and Weinberger had needed some new results in computability (which Soare
then provided) for their proofs.  This convinces me (as I had not been
convinced before) that there will be a fruitful interaction between the
two fields of Soare's talk. However, I disagree with one comment of
Soare's:

Soare mentioned a theorem that a compact Riemannian manifold with
an unsolvable word problem has only finitely many closed contractible
geodesics, characterizing this as logic coming in and geometry coming out.  
I think it is more helpful to look at the theorem as showing that, on a
compact Riemannian manifold with only finitely many closed contractible
geodesics, one can determine whether an arbitrary curve is contractible
(and so the manifold's word problem is solvable).  The theorem is proved
in this form, showing geometry coming in and computability coming out.

Panel:  The Propsects for Mathematical Logic in the 21st Century.

Buss conjectured that P not equal to NP will be proved within twenty
years, and this seems reasonable to me.  Remember that there is a million
dollars waiting for the solver, and that Microsoft's team of
mathematicians is working on it!   I was disappointed to learn from
Kechris's talk that the continuum hypothesis is still considered the
biggest open problem in set theory.

I liked many of Pillay's remarks, and I record some that I suspect might
not make it into print:  He spoke of geometric model theory as "using the
tools of Hilbert and Frege with the sensibility of Poincare".  He
commented that definability was a core notion in every area of logic, and
that this provides some unity in logic.  He also exhorted logicians to be
intellectually responsible, although the details of that are somewhat
unclear (e.g. for Pillay that primarily meant a sensitivity to the
concerns of mathematicians, while for others it might mean a sensitivity
to foundational issues).

In discussion, it seemd that everyone who spoke other than me was over 40,
and so I felt the need for a younger perspective.  I said:

I suspect that, if pure and applied mathematics have been balanced in the
20th century, applied mathematics will be dominant in the 21st century.  
I believe that logic can have a role to play in this applied mathematics,
and than logicians should orient their discipline so that it can help in
that project.  I think it would be a wortwhile goal for logic (and
mathematics generally) to get the point where people say, not "I was never
good at math/logic", but "I always wanted to learn math/logic", or "Wow!  
You're a logician/mathematician!  You're the kind of person who's doing
this [e.g. some specific technological application] for me." 

(I also meant to respond to the question asked about whether there would
be a unity of mathematical logic in the 21st century; I think this unity
will be found in the applications to computer science.  Note that Buss
said explicitly:  "Logicians should be paying more attention to computer
science.")

--Matt