FOM: reply to the "list 2" crowd

[Stephen G Simpson]

It seems that the "list 2" crowd is upset.  Perhaps they didn't expect
anyone to catch on to their game.

A brief history of "list 2":

  I proposed to define f.o.m. (= foundations of mathematics) as "the
  systematic study of the most basic mathematical concepts and the
  logical structure of mathematics, with an eye to the unity of human
  knowledge."  See also
  
     http://www.math.psu.edu/simpson/Foundations.html
  
  I presented a tentative list of the most basic mathematical
  concepts: number, shape, set, function, algorithm, mathematical
  axiom, mathematical proof, mathematical definition.

  A bunch of people including Pillay, Marker, Mattes, McLarty,
  ... objected on the grounds that my list of basic mathematical
  concepts is radically incomplete and needs to be supplemented with
  many other concepts, such as: cohomology, projective analytic
  variety, Riemannian manifold, et cetera.  I called this mind-set the
  "list 2" mind-set.

Brief comments on "list 2":

  The "list 2" mind-set has no merit, because the additional concepts
  ("list 2": cohomology, projective analytic variety, et cetera) are
  far from basic qua mathematical concepts.  To accept the "list 2"
  mind-set would be to obliterate the concept "basic mathematical
  concept".  The motivation for the "list 2" mind-set is a desire to
  obliterate f.o.m. as a subject.

Dave Marker writes:
 > ...  In my post on Chow's theorem (ie." analytic subsets of complex
 > projective varieties are alegebraic"). I stated that I did not
 > believe Chow's theorem was a result in the foundations of
 > mathematics but believed it had some foundational point. Nameley
 > that in one very important context (projective algebraic varieties)
 > complex analytic methods are no more powerful than than algebraic
 > methods.  This conservation fact seems very much "foundational" in
 > Hilbert's sprit ...

Dave, the distinction that you are trying to draw is meaningless.
There is no real difference between "Chow's theorem is a result in
f.o.m." and "Chow's theorem has some foundational point".  If you
think there is a subtle difference between these two modes of
speaking, maybe you'd better explain it.  I assume that your term
"foundational" means "related to f.o.m."  If it doesn't, maybe you'd
better explain what you intended.  (Sigh.)

Dave, I'm not sure why you invoked Hilbert.  Since you didn't take the
time to explain yourself, let me now hypothesize as to your reasons
for invoking Hilbert.

  Hypothesis 1.  Perhaps you are harking back to Hilbert's work on
  foundations of geometry.

I have discussed all this in my posting

   57. [[ Stephen G. Simpso Nov 6 119/5757 "FOM: foundations of
     geometry; set-theoretic foundations; Chow"]]

at 

   http://www.math.psu.edu/simpson/fom/postings/9711.57

Even if I do say so myself, that was a lengthy and thoughtful posting
on the issues surrounding Chow's lemma as it has been discussed here
on the FOM list.  I don't know why you haven't cited it in your own
FOM postings on Chow's lemma.

  Hypothesis 2.  Perhaps you are (deliberately?) confusing f.o.m. with
  "anything that might have interested Hilbert".

The obvious answer to this is: Sure, Chow's lemma might have been of
interest to Hilbert, but Hilbert wore many hats.  Hilbert was
intensely interested in, and made major contributions to, not only
f.o.m. but also a lot of other mathematical topics.

  Hypothesis 3 (the most likely of the three).  You are aware that my
  work (see http://www.math.psu.edu/simpson/papers/hilbert/ ) shows
  that certain logical conservation results are relevant to Hilbert's
  program in f.o.m.  And you strain to describe Chow's lemma as a
  "conservation fact in the spirit of Hilbert".  Perhaps you thought
  that this move would fool me into accepting Chow's lemma as
  "foundational", i.e. part of f.o.m.

Sorry Dave, it didn't work.  I wasn't fooled.  I'm not going to
classify arbitrary mathematical topics as f.o.m. just because you
invoke "conservation" and "Hilbert", as if these were magic spells.

 > Pillay nevery said that cohomology theory was part of the
 > foundations of mathematics. He said that understanding the
 > pervasive nature of cohomological methods was an interesting
 > question for the foundation of mathematics.

Once again, there is no real difference between these two possible
things that Pillay could have said.  We have been through all this
before, here on the FOM list.  The reason Pillay says that cohomology
is f.o.m. is because he wants to obliterate f.o.m.  

Josef Mattes writes:
 > > Mattes says that Connes' "non-commutative geometry" (this is just
 > > C*-algebras and algebraic topology in disguise) is f.o.m.
 > 
 > Here we disagree.  But whatever the technical details, again,
 > Connes says: "[There are] many natural spaces for which the
 > classical set-theoretic tools ... loose their pertinence."  Are you
 > saying that this is obviously irrelevant to f.o.m.?

I didn't refer to this silly Connes quotation at all.  This silly
Connes quotation is from Mattes, not from me.

This silly Connes quotation *is* obviously irrelevant to f.o.m.,
because it's obviously irrelevant to everything, because it's
completely out of context.  Mattes presents this silly Connes
quotation with no relevant context or explanation and expects us to
bow our heads and assent to his belief that that f.o.m. is something
other than what it is.

I have one question about Mattes: Is he familiar with the well-known,
standard techniques of how all of mathematics (including analysis,
algebra, topology, geometry, recursion theory, model theory, category
theory, et cetera, et cetera, et cetera) is formalizable in ZFC?

Note that Mattes' list of "recent and current interests" includes
photography and aikido, but not f.o.m.

I'm not saying that Mattes isn't interested in f.o.m.  But his
interest in f.o.m. is of a hostile nature.  This is amply documented
in his writings.

-- Steve

Name: Stephen G. Simpson
Position: Professor of Mathematics
Institution: Penn State University
Research interest: foundations of mathematics
More information: www.math.psu.edu/simpson/