FOM: foundations of geometry; set-theoretic foundations; Chow

[Stephen G Simpson]

Josef Mattes writes:
 > 1.) I'm not completely sure what you mean by "shape". Can I
 > translate it as "basic object of geometry"?

By "shape" I meant the naive concept, e.g. when we say that a
basketball has the same shape as the globe.  It's clearly a basic
mathematical intuition, but it can be made rigorous in many different
ways and studied from many different viewpoints.

The relationship between foundations of geometry (f.o.g.) and
foundations of mathematics (f.o.m.) is not very clear to me.  On the
one hand, geometry may be viewed as being just one branch of
mathematics, in which case f.o.g. needn't necessarily have anything in
common with f.o.m.  After all, Hilbert's two-volume study of
f.o.m. says almost nothing about geometry.  I suppose this is also the
viewpoint of classical Bourbaki-style set-theoretic f.o.m., which
would define geometrical concepts in terms of set theory.  On the
other hand, this seems too stark, especially in view of the fact that
geometry plays a big role in the interface between mathematics and the
rest of human knowledge.  So, I included "shape" in my tentative short
list of basic mathematical concepts, as a kind of place-holder, to
imply among other things that there is room for discussion of f.o.g.
I don't have all the answers here.

 > No question that the three examples you give are foundational. But
 > why are all three out of the area of logical/set-theoretic
 > foundations (F-R-H-G)?

Only the last of the three was set-theoretical.  I chose these three
examples because they illustrated my point about basic mathematical
concepts and f.o.m.  I do regard the Frege-Russell-Hilbert-G"odel line
as the principal line in f.o.m. of this century.

 > Do you really think there has been nothing foundational in, say,
 > geometry?

There has been important work in foundations of geometry going back to
Euclid.  Whether foundations of geometry is now (in the late 20th
century) to be regarded as part of foundations of mathematics is not
so clear to me.  As I said in

  www.math.psu.edu/simpson/Hierarchy.html

if subject Y is a branch of subject X, then foundations of Y may be
very different from foundations of X.

 > is there no theorem in set theory that is not foundational? (I
 > assume you would have included such a theorem in the list of
 > examples for 'not foundational',

That's a point well taken.  Yes, I should have given such an example.
Many parts of set theory are not foundational, e.g. the technical
parts of combinatorial set theory, technical applications of forcing
in general topology, etc.  One should distinguish set theory qua
particular branch of mathematics from set theory qua (generally
accepted) foundation for all of mathematics.

 > some of us are bothered by what we see as an overemphasis on set
 > theory.

Those who are "bothered by an overemphasis on set theory" need to
explain what alternative foundational scheme they have in mind.  As
Harvey pointed out, set theory may not be perfect as a foundational
scheme for all of mathematics, but you need to be aware of what you
would be giving up by abandoning it.

I myself am quite uncomfortable about several aspects of set-theoretic
foundations, and this motivates some of my reverse mathematics work.
I want to answer one of the common arguments in defense of
set-theoretic foundations, by showing that the most significant parts
of mathemathics have a satisfactory foundation based on much weaker
axioms, and that those weaker axioms can be more strongly linked to
the rest of human knowledge.

 > I'm curious to see your foundational exposition which showed that the
 > ergodic theorem is not foundational.

My understanding of the ergodic theorem places it very far from my
list of basic mathematical concepts.  The ergodic theorem is stated in
terms of higher-level concepts involving measure and various technical
conditions on iteration of mappings.  If someone can give an
exposition of the ergodic theorem that ties it more closely to f.o.m.,
that would be a good contribution.

Colin McLarty writes:
 > Here I agree with Steve in good part--except that you can't ask
 > every poster to start from 0 and build every subject they refer
 > to. You have to let the list find its level through some give and
 > take.

Yes, of course, there is no need to spell out a full-scale
foundational exposition, provided you can sketch one in a convincing
way.

Let's take it for granted that you did that for Chow's lemma.  We then
have to note that complex projective varieties are a high-level,
technical mathematical concept, the connection to "shape" or other
basic mathematical concepts being remote and requiring many steps
(complex numbers, points at infinity, polynomials, etc.)  Also, this
technical notion is only one of many competing technical notions:
others are topological space, Riemannian manifold, simplicial complex,
etc, and Chow's lemma is just one of many technical results relating
these technical notions to each other.  So I don't regard Chow's lemma
as foundational, neither f.o.g. nor f.o.m.  A touchstone for this is
that Chow's lemma doesn't have any general intellectual interest.
The contrast with G"odel's incompleteness theorem is quite stark.

About the analogy with Hilbert's 5th problem:

Hilbert viewed Hilbert's 5th problem as foundational because of its
f.o.g. aspect, as in Appendix IV where he considers axioms for planar
motions.  This f.o.g. aspect has been completely lost in, for
instance, Yang's paper on modern mathematical developments arising
from Hilbert's 5th problem.  I can see that Chow's lemma is loosely
analogous to this Yang-like, non-f.o.g. aspect of Hilbert's 5th
problem, but I don't see that Chow's lemma has any f.o.g. aspect of
its own.

-- Steve