FOM: a message for young recursion theorists

[Stephen G Simpson]

Dear FOM subscribers,

Please circulate the following to any young recursion theorists whom
you may know.

-- Steve

-----

To: young recursion theorists


Peter Cholak writes:
 > Some of you may know that I am visiting UC--Berkeley for the year.
 > Well after one of the recursion theory seminars here one of the new
 > graduate students asked "What are the current themes and research
 > directions of computability (recursion) theory?" This started a
 > discussion between myself, Ted Slaman, Leo Harrington and several
 > of the graduate students.

It's interesting and significant that this discussion at Berkeley was
initiated not by the recursion theory professors but by "one of the
new graduate students".  It occurs to me that this may be symptomatic
of a certain kind of malaise in this field.  Let me explain what I
mean.

Many young scholars are initially drawn to mathematical logic by
vital, exciting questions concerning foundations of mathematics.  In
the case of recursion theory, I have in mind questions such as: What
is an algorithm?  What is the role of computability in mathematics?
How does this role square with nonconstructive methods and strong set
existence axioms in mathematics?  etc etc.  Historically, questions of
this kind gave rise to the entire field of mathematical logic and
still drive it.  Unfortunately, over time, many young scholars
gradually get immersed in technical issues to the point where they
lose sight of the big picture.  The foundational motivation is
gradually discarded, and what remains is an unmotivated morass.

The situation described above is a crying shame.  Something should be
done about this situation.  In order to do something about this
situation, Harvey Friedman and I started the FOM list.  The acronym
FOM is intended to suggest: foundations of mathematics.

Let me tell you a little about the FOM list.  The FOM list is an
Internet forum for discussing foundations of mathematics.  The
discussion is very lively.  There are approximately 290 subscribers,
including some of the biggest names in the field.  Since we started
the FOM list last September, more than 1800 messages have been posted
and archived.  Unfortunately, only a few recursion theorists have
participated actively in the FOM list.  Nevertheless, there has been
some fairly extensive discussion of recursion theory, and much more is
planned.

If you are a young recursion theorist who is still interested in the
big picture of foundations of mathematics and how recursion theory
comes in, I urge you to subscribe to the FOM list.  To subscribe, or
if you have any questions about the FOM list, please send me e-mail at
simpson at math.psu.edu.

As a "teaser" for young recursion theorists, I present here two short
excerpts from a long FOM posting by Harvey Friedman, 14 Dec 1997
05:47:50.  This is an example of the kind of discussion that can be
found on the FOM list.

In what follows, f.o.m. is an abbreviation for "foundations of
mathematics."

Friedman excerpt number 1:

  ...

  In some parts of mathematical logic, there is some consideration
  paid to f.o.m.; in other parts, virtually none. The same with most
  areas of mathematics.

  However, this disconnect with f.o.m. creates a special problem for
  mathematical logic. Since f.o.m. is no longer the motivating force
  behind most of mathematical logic, there is the real problem of how
  to evaluate it.  Virtually all of research in mathematical logic is
  now housed in mathematics departments, and so it is natural for this
  research to be evaluated as a branch of mathematics. Mathematical
  logicians cannot, in general, cast the importance of what they do in
  terms of f.o.m. So there is a question of how mathematical logicians
  can relate their work to the mathematical community.

  Only a handful of mathematical logicians have been able to solve
  this problem, and their solutions have pretty much been one of the
  following:

  i) move into direct applications to mathematics;
  ii) move into applications to computer science, or into computer science
  itself.

  Since the subject matter of computer science is very fluid, the
  borderline between applications of logic to computer science and
  computer science itself is murky. Often ii) involves moving to a
  computer science department.  ....

  But what of mathematical logic that is not applied nor f.o.m.? This
  is where the real difficulties lie. Here are some of my views on the
  matter.

  1. There are many missed opportunities for a redirection of much of
  this work towards issues in f.o.m.. For instance, for recursion
  theory, there is reverse mathematics, Church's thesis, and also
  developing decision procedures for new classes of mathematical
  statements. Of course, the latter may equally well be classified as
  model theory. ....

Friedman excerpt number 2:

  ....

  The limitations and drawbacks of these approaches are pretty much
  recognized by everyone involved and not involved, but in the current
  atmosphere of disconnect between mathematical logic and f.o.m.,
  these limitations and drawbacks are de-emphasized. In this
  atmosphere, the feeling is that if it leads to complicated and
  intricate work, then it is OK - despite limitations and
  drawbacks. I, for one, have always called for a perpetual rethinking
  of the underlying assumptions behind these, or any technical
  developments. If there is a better motivating idea that would push
  the technical development in an altered direction, then so be it. If
  that means abandoning longstanding technical projects, then fine. If
  people persist in resisting the inevitable, then others will come in
  and steal their thunder.

  3. More bluntly - mathematics, including mathematical logic,
  operates on a kind of code of silence. One simply doesn't want to
  talk openly about significance; particularly about other ways of
  looking at things that may assume greater significance, and involve
  a change in research perspective.  One lapses into: well, if its
  hard, complicated, and intricate, and made some sense some time,
  then it is OK; and it is OK to judge everybody's work in these terms
  - i.e., is it hard, complicated, and intricate? How hard,
  complicated, and intricate?

  4. But whereas 3 is a time honored way that most fields of
  mathematics operate, I don't think that it can really work for the
  mathematical logic that is disconnected from applications and from
  f.o.m.. It is in danger of being marginalized.

-- Steve Simpson

Stephen G. Simpson
Department of Mathematics, Pennsylvania State University
333 McAllister Building, University Park, State College PA 16802
Office 814-863-0775           Fax 814-865-3735
Email simpson at math.psu.edu    Home 814-238-2274
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