FOM: applied model theory; foundations of mathematics

[Stephen G Simpson]

Lou, thank you for your points 2 through 8, which well articulate the
aims and techniques of applied model theory.  I would summarize as
follows:

   Applied model theory is something like algebraic topology.
   Algebraic topology defines and studies some carefully chosen
   functors (homotopy, cohomology, etc.) from certain topological
   spaces to groups or rings; this has proved fruitful in classifying
   and understanding the topological spaces.  Similarly, applied model
   theory defines and studies some carefully chosen functors from
   certain algebraic structures (stable groups, rings, etc.) to
   certain objects defined in terms of predicate calculus (first order
   theories, categories of definable maps, etc.).  In choosing these
   functors, it's important to avoid the G"odel incompleteness
   phenomenon.  Such functors have proved somewhat fruitful in
   classifying and studying algebraic structures.  These methods are
   very promising, though so far not so successful as the older and
   more familiar algebraic techniques.

Or, to summarize the summary:

   Applied model theory is a kind of little brother to modern algebra.
   Applied model theory has the same aims and goals as modern algebra,
   but it is potentially richer, because it incorporates techniques
   derived from predicate calculus.

With this on the table, I think we can see that the applied model
theory outlook really has very little in common with the foundations
of mathematics outlook.  Applied model theory is just another kind of
pure mathematics.  The only thing in common is the fact that predicate
caluculus is used.  I think Lou would agree with this point.  And I
would also agree with Lou that applied model theory has accomplished
some wonderful things within pure mathematics, and promises to
accomplish more.

Where I would like to take issue with Lou is on his assessment of
foundations of mathematics (f.o.m.), in his point 1.  It seems to me
that Lou is trying to denigrate f.o.m. for no good reason, in a
totally unjustified way.

Lou says:
 > FOM will play only occasionally an *active* role in mathematics,
 > namely when basic notions and ways of reasoning about them are in
 > the process of changing, as happened a century ago.

I completely disagree.  Lou is saying that basic mathematical notions
were changing a century ago but are not changing now.  This seems very
short-sighted.  The fact is that there is a lot of turmoil going on
right now.  Think of the changing role of set theory, new analyses of
the notion of set, new understanding of which set existence axioms are
needed for standard mathematics, and the new science of computational
complexity, necessitating a rethinking of almost all of the most basic
mathematical concepts.  And then there is the Harvey revolution, if he
manages to carry it off.  This is just to name a few.

 > I see no reason why current FOM-specialists would be better
 > equipped to deal with such a situation (when it occurs) than
 > mathematicians with a broad outlook and familiarity with the
 > trouble at hand.

I don't know which FOM-specialists Lou is talking about, but clearly
people trained in f.o.m. have shown themselves to be much better
equipped to understand and deal with the issues that I mentioned.
Pure mathematicians are mostly pretty ill-equipped and have been
reluctant to get involved.  For example, the people who created
computational complexity were trained in classical f.o.m., not
cohomology etc., and most pure mathematicians still don't get it.

 > This is confirmed by what happened in more recent times when
 > subareas of mathematics did undergo an overhaul of their
 > foundations, as shown by the examples of probability theory
 > (Kolmogorov), and algebraic geometry (Zariski, Weil, Grothendieck).

These examples confirm nothing of the sort.  Overhauling a specialized
area such as algebraic geometry is very different from overhauling the
most basic concepts of mathematics and the relationship of mathematics
as a whole with the rest of human knowledge.  The former is in the
domain of pure math; the latter is the domain of f.o.m.

Foundations of probability is a somewhat different case.  That is a
very big applied area, so of course overhauling probability theory had
an impact outside pure mathematics.  Still, I would note that
Kolmogorov was very conversant with current results and techniques in
foundations of mathematics, and I do believe that this contributed to
his work on foundations of probability.  So Lou's point is refuted by
one of his own examples.

 > However, this activity, which concerns "only" subfields of
 > mathematics, is not the "lofty" kind of FOM championed by Harvey
 > and Steve.

Yes, precisely.  Foundations of algebraic geometry has little or
nothing in common with foundations of mathematics.

 > (The influence of such overhauls has nevertheless been extensive
 > and deep.)

Only in a restricted sense.  Overhauls of special areas within pure
mathematics have been influential inside pure mathematics, but not
outside it.

 > Also, I don't see that "state of the art FOM" (like reverse math)
 > has anything to say about applied mathematics, and other ways in
 > which mathematics relates to the rest of the world, notwithstanding
 > the claims that we keep hearing about.

I'll have more to say later about Reverse Mathematics.  A preview: One
of my purposes in doing Reverse Mathematics is to reexamine and
overthrow a very common philosophical misconception: that
set-theoretic foundations is essential to modern mathematics; that
without set-theoretic foundations, modern mathematics would be
impossible, because you would be throwing out too much.  Reverse
Mathematics overthrows this misconception in part by pointing to
alternative foundational schemes and working out their consequences.
In this direction, I think Reverse Mathematics has already had a
considerable impact in the attitude of the philosophical community
toward set-theoretic foundations.  And my book isn't even out yet!
:-)

See my paper on Hilbert's program at 

  http://www.math.psu.edu/simpson/Foundations.html

Also, have a look at the Dreben-Kanamori paper

  http://math.bu.edu/people/aki/

where the impact of Reverse Mathematics is discussed in an appendix.

I have to agree with Lou that Reverse Mathematics hasn't had any
impact on PDE's etc.  But then again, it wasn't intended to.

Best wishes,
-- Steve