FOM: CTA meeting; applications; Nabutovsky-Weinberger

[Stephen G Simpson]

Dear FOM and COMP-THY subscribers,

This is a response to Bob Soare's tough on-line COMP-THY response
(July 9) to my on-line FOM remarks (June 21) concerning the CTA
(``Computability Theory and Applications'') meeting.  My remarks and
Soare's response are accessible from
<http://www.math.psu.edu/simpson/cta/>.
 
1. MAILING LISTS AND OFF-LINE PRESSURE

In his on-line response, Soare said ``I apologize if you receive
multiple copies of this message because you may be on several email
lists.''  Soare has also informed me off-line that he posted his
message to several mailing lists, not only COMP-THY.  Unfortunately,
he neglected to say which mailing lists.  Could somebody please tell
me?  If a lively discussion of these issues is taking place somewhere
else in cyberspace, I'd like to know where it is.

Off-line, Soare has written some harsh letters to me where he tries to
act as a spokesman for the recursion theory community and for
reviewers of NSF proposals.  He urges me to ``come in from the cold'',
i.e., to recant or renounce my f.o.m. values.  Needless to say, these
brutal tactics will not work.  I call on Soare to deal openly and
honestly with foundational issues in public electronic forums such as
the FOM and COMP-THY mailing lists.

2. ALLEGED OMISSIONS AND ERRORS

According to Soare, my FOM posting gave a ``full account'' of the
reverse mathematics part of the CTA meeting.  The truth is that that
account was not ``full'' in any meaningful sense.

Furthermore, I didn't even attempt to give what I would call a full
account of *any* part of the CTA meeting.  My only purpose in the June
21 FOM postings was to summarize my impressions of the CTA meeting in
a timely way, and to invite others to do the same.  I hoped and
continue to hope that this will lead to a lively electronic discussion
of issues and programs in recursion theory.  I believed and continue
to believe that such a discussion will be extremely valuable for
researchers and graduate students in this area.

It's a shame that most of the recursion theory community has not taken
up this challenge.  Is Soare now the sole spokesman for all recursion
theorists?

According to Soare,

 > there were some mistakes and omissions in other parts,
 > particularly the new material on applications of computability
 > theory to topology and geometry.

Mistakes and omissions?

As for omissions, I concede that my account of the CTA meeting was
personal and emphasized my own perspective and interests.  However, I
don't see this as a drawback or a reason not to record my impressions.
Other CTA participants are also free to provide their own personal
accounts of the CTA meeting, and I think this would be extremely
valuable.  I would very much like to compare my impressions with those
of other CTA participants such as Steffen Lempp, Harvey Friedman, Leo
Harrington, Julia Knight, Bob Soare, Andrew Arana (a graduate
student), Martin Davis, Manny Lerman, and Gerald Sacks.  I am sure
that all of these people would have interesting insights and
viewpoints to offer.

As for any factual errors that I may have made, I would be very glad
to learn of them.  Soare himself uncovered an error of attribution
(Markov vs Novikov, see below), and for that I am grateful.

However, such errors are clearly not Soare's main concern.

3. TWO CULTURES

It appears that the main thrust of Soare's objection pertains not to
specific facts but rather to a certain philosophical outlook or
perspective that I brought to bear.  Recall that my remarks were made
on FOM, a mailing list devoted to foundations of mathematics
(abbreviated f.o.m.).  That is of course the perspective that I
adopted.

It's not surprising that Soare finds the f.o.m. perspective
uncongenial.  There is a clash of two cultures here: the
f.o.m. culture (see for example the van Heijenoort volume and G"odel's
collected works), and the core math culture.

There are some stark differences between these two cultures.  In the
core math culture, one studies specific mathematical structures for
their own sake, and the cardinal rule is: ``Thou shalt not criticize
closely related branches of core math.''  In the f.o.m. culture, the
cardinal rule is to continually re-assess and re-evaluate everything
in terms of foundational and general intellectual interest.  Such
critical evaluations are often carried out in public.

Soare adheres to the core math culture, while I lean toward the
f.o.m. culture.  Soare probably senses that I tend to have a somewhat
impatient attitude toward research in mathematical logic that
disregards larger f.o.m. issues.  Thus it is no wonder that Soare is
upset with me.

4. THE PURE/APPLIED DISTINCTION

According to Soare, I made an excessively sharp distinction between
``pure'' and ``applied'' recursion theory.

It's true that I employed such a distinction, but in this I was merely
following the lead of the CTA conference organizers.  See for example
the manifesto <http://www.ams.org/meetings/src-cholak.html>, where the
CTA organizers partition the territory into ``classical computability
theory'' (structural aspects of r.e. degrees, r.e. sets, etc.) and
``applications'' (recursive mathematics, etc.).

My actual view is that the pure/applied distinction is almost always
confusing and harmful, both in mathematics generally and in recursion
theory particularly.  What does it mean to ``apply'' one branch of
pure mathematics to another branch of pure mathematics?  Wouldn't it
be better to speak of a ``connection'' rather than an ``application''?

For example, the first ``application'' in Soare's list of
``applications'' at
<http://www.cs.uchicago.edu/~soare/lectures/applications.html> is the
following old result of Jockusch:

  Every recursive 2-coloring of [omega]^2 has an infinite Pi^0_2
  homogeneous set. 

Here Pi^0_2 is best possible in the arithmetical hierarchy.  The proof
uses a priority argument.  This is a beautiful *connection* between
recursion theory and Ramsey theory.  But it is arguably not an
*application* of recursion theory, because the result itself mentions
recursion-theoretic concepts.

If we are going to compile lists of ``applications'' of recursion
theory, such as Soare's compilation at
<http://www.cs.uchicago.edu/~soare/lectures/applications.html>, then
we ought to classify them carefully and distinguish various features,
just as the applied model theorists do.  For each ``application'' of
recursion theory, we need to say what the field of application is,
whether the statement of the result or only the proof involves
recursion-theoretic concepts, which recursion-theoretic concepts
and/or methods and/or results are used, whether the
recursion-theoretic concepts and/or methods and/or results can be
eliminated, what are the costs of eliminating them, etc.  There are
many distinctions to be made, and these distinctions are of essential
scientific importance.  It is misleading to lump everything together
as ``applications'' without further qualification.

According to Soare, Carl Jockusch has suggested replacing the
``pure/applied'' dichotomy by a ``core/interactive/applied''
trichotomy.  This trichotomy may be a useful beginning.  But it is
only a beginning.

5. MY ALLEGED DISMISSAL OF PURE RECURSION THEORY

According to Soare, my FOM posting ``downgraded the interest and
importance of'' some of the pure recursion theory talks at the CTA
meeting, by saying that they were pure recursion theory talks.

The truth is that, when you think about it, this remark of Soare makes
no sense at all, unless we assume that all of pure recursion theory is
ipso facto uninteresting!  I for one would never accept such an
assumption.  Indeed, anyone who reads my impressions of the CTA
meeting at <http://www.math.psu.edu/simpson/cta/> will note that I
praised several of the pure recursion theory talks.

I myself have even published some papers in pure recursion theory.  In
particular, my old paper on interpreting the standard model of
second-order arithmetic into the ordering of the Turing degrees was
very much in evidence at the CTA meeting.

6. MY ALLEGED DISMISSAL OF APPLIED RECURSION THEORY

According to Soare,

 > Simpson misquoted the first paper by Nabutovsky-Weinberger on very
 > recent applications of computability to topology,

Blanket disclaimer: If I misstate any aspect of the first or second
Nabutovsky-Weinberger papers (call them NW1 and NW2), then such errors
are due to inadequacies in my knowledge of differential geometry, not
to an alleged desire to mislead.  Subsequent to the CTA meeting, with
Weinberger's help, I have been studying NW1 and am gradually coming to
understand more of it.  It is an extremely interesting paper.  I will
comment more on it later.

Soare quotes ``one of co-authors of the topology papers'' as saying
that I ``misquoted Markov's theorem, which is about 4-manifolds, not
5-manifolds.''  I don't know what Soare may have said to Nabutovsky
and/or Weinberger in order to elicit this comment.  In any case, there
is a valid criticism here.  There are two classical results: one due
to Markov saying that it is undecidable whether two given compact
4-manifolds are diffeomorphic to each other; the other due to Novikov
saying that it is undecidable whether a given compact 5-manifold is
diffeomorphic to the 5-sphere.  The NW1 paper quotes and uses the
Novikov result.  I also quoted the Novikov result, but I mistakenly
attributed it to Markov.  Mea culpa.  However, I don't think this a
very serious error on my part.  It is an error of attribution, not
substance.

 > and omitted any reference to the results from second paper.

It's true that I didn't say much about the NW2 paper.  That's because
I haven't seen it and don't know what is in it.  At the CTA meeting I
asked Weinberger for a copy of it, but he turned me down on the
grounds that it is a work in progress and not for circulation.

In Soare's Wednesday evening CTA talk, in his COMP-THY posting, and in
his ``survey'' of NW1 and NW2 at
<http://www.cs.uchicago.edu/~soare/lectures/geometry.html>, Soare has
made some vague but very strong claims concerning the alleged role
played by methods and results from his 1987 monograph ``Recursively
Enumerable Sets and Degrees'' in the Nabutovsky-Weinberger papers.  If
these claims turn out to be justified, then that would constitute an
excellent application of Soare's favorite methods and results to
differential geometry.  However, until NW2 becomes available, Soare's
claims will remain difficult or impossible to evaluate.  They are
certainly not justified by NW1 or by Weinberger's published abstract
of NW2 (item 68 in
<http://www.math.uchicago.edu/~shmuel/abstracts.html>).  The
``survey'' given by Soare at
<http://www.cs.uchicago.edu/~soare/lectures/geometry.html> is useless
for this purpose, because it omits the essential geometrical details.

In his COMP-THY posting, Soare said:

 > QUESTION 10.  Paper (2), presented Wednesday evening at Boulder,
 >         contains the main connection between fractals, their local
 >         minima, and computably enumerable sets.  What does Simpson
 >         say about these topics?
 > 
 > ANSWER 10.  Nothing.
 > 
 > QUESTION 11.  But Simpson's report stresses *applications*.  These
 >         results constitute one of the most important and exciting
 >         applications of computably enumerable sets to topology in
 >         25 years.  The talk showed how the Sacks Density Theorem
 >         for the computably enumerable degrees was used to compare
 >         the depth of local minima at different levels.  What
 >         summary does Simpson give of the application of the Sacks
 >         Density Theorem on c.e. degrees to give comparative depth
 >         estimates for local minima of different levels?
 > 
 > ANSWER 11. Nothing.

It's true that I didn't say much about the claims that Soare made in
his Wednesday evening talk and in questions 10 and 11 above.  That's
because, frankly, I don't trust these claims.  I find them somewhat
vague, to say the least.  Soare's ``survey'' at
<http://www.cs.uchicago.edu/~soare/lectures/geometry.html> also does
not contain any clear statements of the claimed applications.  I am
reserving judgment until I get hold of NW2.

Perhaps part of Soare's harshness has to do with my refusal to blindly
and uncritically accept his claims about NW2 and his evaluation of
them.  If that's what is bothering Soare, then so be it.

7. CONCLUSIONS

In the conclusion of his COMP-THY posting, Soare calls for uncritical
acceptance of ``the mathematical merit, beauty, and importance of ALL
the components of computability theory ...''.

I would call instead for a critical, open, honest, serious, public,
interactive, electronic discussion of issues and programs in recursion
theory / computability theory.

Best wishes to all,
-- 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/