FOM: Riemann surfaces; arithmetization of analysis; "list 2"

[Stephen G Simpson]

Martin Davis writes:
 > Weyl's monograph was a straightforward contribution to technical
 > mathematics. However, viewed historically, it was part of a
 > *movement* that most definitely was fom. The movement has been
 > called the "arithmetization of analysis"

Dear Martin, 

That's a perfectly valid way to view the matter.  You are viewing
Weyl's rigorous definition of Riemann surfaces as part of the movement
toward "arithmetization of analysis", initiated by Weierstrass and
Dedekind.  And I completely agree with your opinion that that entire
movement is an important contribution to f.o.m.  We may have slightly
different reasons for saying this, but we are certainly in agreement.

My reason is as follows: A key issue for f.o.m. is to understand which
mathematical concepts are the most basic.  "Arithmetization of
analysis" shows that various concepts of real and complex analysis can
be defined in terms of the natural numbers.  (Actually, the language
of second order arithmetic is needed, but this distinction wasn't so
clear at the time.)  Thus "arithmetization of analysis" may be viewed
as a series of case studies reducing analysis to arithmetic.  Thus
arithmetic, i.e. the theory of the natural numbers, is seen to be much
more basic than might have been realized previously.  This is part of
the background of why second order arithmetic and its subsystems are
so important in f.o.m. research.  (The title of my forthcoming book is
"Subsystems of Second Order Arithmetic".)

On the other hand, this is not how the "list 2" people (Pillay,
Marker, McLarty, Mattes, maybe Lou van den Dries, maybe others here on
the FOM list) view the matter.  The "list 2" people don't give a damn
about arithmetization of analysis, or which mathematical concepts are
truly basic, or any of the other issues that are, in my view, crucial
for f.o.m.  The way they express this is by saying that a great many
concepts should be added to the list of basic mathematical concepts.
For example, they say that the concepts "cohomology" and "Riemannian
manifold" are just as basic as the concepts "set" and "number".
Obviously this destroys the concept of "basic mathematical concept."
That is their intention.

The "list 2" people want to say that Weyl's definition of Riemann
surfaces is first-class f.o.m. simply because it is a conceptual
clarification of what according to them is a "basic mathematical
concept", namely Riemann surfaces.  To these people, the fact that
Weyl reduced Riemann surfaces to sets, numbers, or whatever is
irrelevant.  Lou expressed this as follows:

 > I think Weyl would have resisted the idea that he had *reduced* the
 > notion of Riemann surface to that of the notion of set.  Riemann
 > surfaces will undoubtedly survive any modifications that the
 > present framework for the reduction of mathematics to sets might
 > undergo, or even the disappearance of sets if that ever happens.

The "list 2" people also want to consider a great deal of other pure
mathematics as f.o.m.  For instance, McLarty and Marker say that
Chow's lemma about projective analytic varieties is f.o.m.  Mattes
says that Connes' "non-commutative geometry" (this is just C*-algebras
and algebraic topology in disguise) is f.o.m.  Pillay says that
cohomology theory is f.o.m.  Et cetera, et cetera.  It's hard to think
of any conceptual breakthrough in mathematics that they would not
promote as f.o.m.

To accept this "list 2" viewpoint would be to destroy f.o.m. as a
subject distinct from high-level pure mathematics.

I don't believe that you subscribe to the "list 2" viewpoint.  But
your statement about the nature of f.o.m., if taken as a definition of
f.o.m., would unjustifiably legitimize this viewpoint.  That's why I
dispute your statement.

Sincerely,
-- Steve