Marker/Simpson correspondence on foundations

[Stephen G Simpson]

OK everybody, here is the famous or infamous Marker/Simpson
correspondence.  In part it is a discussion of the significance of
recent research on the incompleteness phenomenon and applied model
theory.  There is also a discussion of broader issues: what is
foundations of mathematics, should mathematicians be interested in
foundations, technique versus subject matter, etc.

A couple of comments about this Marker/Simpson correspondence:

1. It took place in December 1995 -- a lot longer ago than I had
thought!  Time flies.

2. These are unedited e-mail messages rather than well thought out and
edited opinion papers; also, they were written in 1995, before Harvey
circulated his long-awaited manuscript in summer 1996.

-- Steve Simpson

  From: Stephen G Simpson <simpson>
  To: marker at math.uic.edu
  Cc: InstantP at aol.com (Harvey)
  Subject: hi
  Date: Sun, 3 Dec 1995 11:28:11 -0500
  
  Hi Dave,
  
  How are things?  I'm glad that your research is going well, etc.
  
  Harvey Friedman phoned me yesterday saying he had talked with you
  about Angus Macintyre's views on the
  Frege-Russell-Hilbert-Gödel-... line of research in foundations of
  mathematics.  According to Harvey, Angus gave a talk at a meeting in
  Italy expressing a rather low opinion of this line of research.
  There may have been some influence from Kreisel.
  
  As you know, Harvey and I think highly of the Hilbert-Gödel line and
  consider ourselves to be carrying it on.  So, naturally, we'd like to
  learn more details about the views of Angus and those in his circle.
  Would you be able to point me to some published material on this?  For
  instance, the text of Angus's talk in Italy would be of interest.
  (I'm sorry I don't have more information about the time and place of
  the talk.)  If you give me Angus's email address, I'll ask Angus
  directly.
  
  Harvey and I are thinking of possibly writing some kind of position
  paper about technical work in mathematical logic, recursive function
  theory, etc. and its relation to foundations of mathematics.
  
  Best regards,
  -- 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
  World Wide Web http://www.math.psu.edu/simpson/
  

  From: marker at math.uic.edu
  To: simpson at math.psu.edu
  Subject: Re: hi
  Date: Mon, 4 Dec 1995 10:25:27 -0600
  
  Dear Steve:
  
  Angus' e-mail address is ajm at maths.oxford.ac.uk
  He is currently in the middle of a five year fellowship from the
  Science and Research Council and frequently away from Oxford, but
  usually is in a position to read his mail.
  
  I am affraid I do not know what Angus said at the meeting in 
  Sicily, indeed Harvey is the only person who has mentioned it to me.
  (There was a meeting in Sicily in early September with a number of
  philosphers, logicians and other mathematicians (Manin is the one
  I remember) on "Truth and Mathematics").
  
  I think it would be presumptuous of me to speak for Angus, but I can
  try to quickly summarize some of my own ideas (I should appolovize
  here as I am logged in from home and have very little to edit or correct
  as I go).
  -"Foundations of Mathematics" is a subject somewhere between mathematics
  and philosophy. What is it relevance for mathematicians? Certainly
  Godel's theorem is one of the most significant intelectual results of
  our century and every mathematician must be aware that formal methods
  are necessicarily incomplete. But how does this influence the practice
  of mathematics?  Certainly this has a large impact on any mathematical
  subject that studies arbitrary uncountable structures (general topology,
  large abelian groups...) Any researcher in this area must be aware of
  incompleteness phenomena and set theoretic methods. (Of course this
  type of phenomena have only appeared in the fringes of the subject.)
    On an intelecutal level I find the calibration of interesting 
  mathematical phenomena with set theoretic principles quite interesting.
  Harvey's old results on Borel determinacy is one of my favorites and
  I have a great deal of respect for the current CABAL program calibrating
  determinacy with large cardinals and inner models.
  I just don't think these results have a great deal of influence on mathematical
  practice.  [Perhaps they do in the following sense: earlier I dismissed
  general topology, for example, as a "fringe area", perhaps a good working
  test for a fringe area is one where set theoretic methods have taken over]
  -I must say that I am not particularly taken with Harvey's program of the
  last decade or so, for, while I don't think I can articluate whet the
  difference is, there is a large difference between "looking natural" and
  "being natural".  I find the "Borel diagonalation" results of 15 years ago
  more natural.
  -The Shelah program in model theory was an attempt at a different
  kind of "Foundations".  It was an attempt to build an abstract theory
  of mathematical structures.  Over the past ten years the prevailing
  philosophy of model theorists has changed and is much more in line with the
  thinking of Macintyre, van den Dries and Wilkie. Now the main idea is to 
  try to use logical tools to analyze concrete mathematical structures.
  This has led to two major successes; the breakthroughs on exponentiation
  in real analytic geometry and Hrushovski's proof of the Mordell-Lang
  conjecture for function fields. (An interesting point is that while
  the work on exponentiation uses only Robinson style model theory,
  Hrushovski's work uses a great deal of both classical stability theory and 
  the more recent geometric theory).
  -I, at first, considered the Hrushovski-Zilber work on Zariski geometries
  to be an important step in a different kind of foundations of amathematics.
  (This will appear in Jour AMS, but an extended abstract appeared a couple
  of years ago in the Bull AMS).  They gave some simple properties of a 
  toplogical space, from which one could deduce the presence of an interpretable
  algebraically closed field so that the space was the Zariski topology of 
  the curve over the field.  
  A year later these ideas ended up being central to Hrushovksi's proof of 
  Mordell-Lang.
  
  I apologize again for the sloppy typing.
  I hope these comments are somewhat helpful though they do represent my
  opinions only.
  
  Best Wishes
  Dave
  

  From: Stephen G Simpson <simpson>
  To: marker at math.uic.edu
  Subject: fringe mathematics
  Date: Wed, 6 Dec 1995 18:16:15 -0500
  
  Hi Dave,
  
  Thanks for your letter.  I'll try to find out from Angus exactly
  what he said at the meeting in Sicily.
  
   > Godel's theorem is one of the most significant intelectual results of
   > our century and every mathematician must be aware that formal methods
   > are necessicarily incomplete. But how does this influence the practice
   > of mathematics?  Certainly this has a large impact on any mathematical
   > subject that studies arbitrary uncountable structures (general topology,
   > large abelian groups...) Any researcher in this area must be aware of
   > incompleteness phenomena and set theoretic methods. (Of course this
   > type of phenomena have only appeared in the fringes of the subject.)
  
  Harvey is showing that incompleteness phenomena apply also to
  non-fringe mathematics.  If he can do that, it's very interesting,
  wouldn't you agree?  It should be interesting not only to some or all
  mathematicians, but also of general intellectual interest.
  
  Are you familiar with the Friedman-Robertson-Seymour theorem showing
  that the strength of the graph minor theorem is greater than Pi^1_1
  comprehension?  The graph minor theorem is hardly fringe mathematics.
  
   > perhaps a good working test for a fringe area is one where
   > set theoretic methods have taken over]
  
  Yes, I see what you mean.  It's remarkable the way mathematicians
  often tune out whenever set theoretic methods come in.
  
  But I can't accept your statement about fringe areas.  Taken
  literally, it would seem to imply that the interest of a subject
  matter is to be judged in terms of the methods used to study it.
  Intellectually, I can't accept this criterion.  We have to keep
  methods in their place.  Methods are only a means to an end, not an
  end in themselves.  There is such a thing as intrinsic intellectual
  interest of a subject matter, independently of the methods used to
  study it.  If ugly methods are used to prove an incisive theorem that
  is illuminating or beautiful or of great practical importance, one
  would not for that reason run down the theorem or the subject matter.
  
   > -I must say that I am not particularly taken with Harvey's program
   > of the last decade or so,
  
  I'm not sure what you mean.  During that decade he obtained a lot of
  results that I don't think you are very familiar with.  (Hardly
  anybody is, because he has published very little of it.)  I think it's
  quite possible that his most recent formulations will knock your socks
  off.
  
   > -The Shelah program in model theory was an attempt at a different
   > kind of "Foundations".  It was an attempt to build an abstract theory
   > of mathematical structures.
  
  I don't see how Shelah classification theory has anything to do with
  the Frege-Russel-Hilbert-Gödel-... line that we are discussing.  In
  the F-R-H-G line, we are trying to formalize mathematical practice and
  study properties and limitations of such formalization.  Shelah-style
  classification of arbitrary mathematical structures is a far cry from
  that.  The only connection I can see is that both use the technical
  notions of "first-order formula" and "satisfaction".
  
   > Now the main idea is to try to use logical tools to analyze
   > concrete mathematical structures.
  
  Well, if analysis of concrete mathematical structures is the goal,
  then there is no intellectually sound reason to limit yourself to
  logical tools.  Intellectually, the right thing to do would be to use
  all available tools (insofar as they are useful).  This assumes of
  course that the your concrete mathematical structures are of interest
  apart from the tools.  (If they aren't, then you are talking about
  literally "technical" research, technical in the sense of focusing on
  techniques rather than subject matter or results.  Technical research
  in this sense can be interesting, but it shouldn't be, and usually
  isn't, regarded as high-level stuff.)
  
  I think you can see where this leads.  If you regard, say, Zariski
  geometries as intrinsically interesting, then you should want to use
  all available tools to study Zariski geometries.  But then, how are
  you different from a geometer?  There's nothing wrong with being a
  geometer, but it has less and less to do with foundations.  (Or, same
  question replacing Zariski geometries by function fields and
  "geometer" by "algebraist".)
  
  Sorry to have gotten a bit polemical here, but these things are
  important to me.
  
  Best wishes,
  
  -- Steve
  
  From: marker at math.uic.edu
  To: simpson at math.psu.edu
  Date: Thu, 7 Dec 1995 21:35:03 -0600
  
  Dear Steve:
   
  Let me make some comments on your last note. First, let me say that
  I hope you in no way take my comments as being desparaging of
  Harvey's work or your work.  I have a great deal of respect
  for both you.
   
  -My comments on "fringe mathematics" were intended to be  somewhat
  glib. The central questions in group theory are about finite groups,
  some countable groups and Lie groups. These are the questions which
  have a baring on problems in number theory, dynamical systems and
  physics. Similarly, the important problems in topology are about
  smooth manifolds of reasonably low dimensions.  While researchers
  continue working on more abstract problems about abstract uncountable
  groups and topological spaces, these are far from central areas of
  mathematics.  (You and Harvey should not find this too controversial as
  this is one of the main points behind Harvey's Borel Model theory).
  Moreover it is no longer suprising  (either to logicians, practioners
  in these areas or other mathematicians) to find independence phenomena
  in these areas.
     Certainly I do  not believe that the importance of a subject
  is derived from the methods used. Indeed, the usefulness of
  model theoretic methods in classical mathematics is my main interest
  at this point in time.
   
  -I think that the main point of the reverse
  mathematics program is that there are five fragments of second order
  arithmetic that suffice to classify the strength of most theorems of
  classical countable and continuous mathematics.  This is without doubt
  an important and intriguing point. But does this in anyway effect
  mathematical practice? (this is perhaps related to the
  Kreisel-Macintyre point)
       To answer your question I don't consider
  the Seymour-Robinson theorem as a  "fringe" result (nor do I consider
  it "central"). I think that the result that this requires long
  iteration of hyperjumps a very interesting result from a technical
  point of view and the fact that this is one of the few natural
  results which goes beyond the basic five systems is also of great
  interest...from a metamathematical point of view. But does this fact
  influence the practice of graph theory (of course it might have if,
   like  Borel determinacy, the reverse direction was proved first)?
  Moreover, I doubt that any graph theorist would even find it
  suprising that this difficult result dealing with well
  founded parts of relations requires a very good theory of countable
  ordinals to prove.
   
  -You are right that I have not seen exact statements from Harvey for
  a couple of years (and have not seen even hints of proofs for much
  longer). Still I will be very suprised if they knock my socks off.
  I suspect that if I every see these arguements I will once again
  be impressed with Harvey's insights and technical achievements.
  On the other hand I suspect that I will still view this as coding
  set theoretic phenomena in a natural looking way and I believe that
  there is a big difference between "looking natural" and "being
  natural".  Also again I ask what does it say about mathematical
  practice.  We have known that 1-consistency of large cardinals is
  an independent arithmetic statement.  Now we know that we can code
  this up in a cute way, but this will only be truly important if
  we such phenomena are found in problems we want the answer to.
  (I am ignoring here the rather compelling argument that this
  work must be started now so that in a couple of hundred years
  when such problems arise, some of the necessary machinery will
  be available.)
   
  -I think I take a broader view of the foundations of mathematics
  than you do.  I consider as "foundational" any work that sheds light
  on mathematical practice and mathematical thinking.  I think Shelah
  would consider his classification program as foundational in this
  sense. To me one of the most important points in the foundations of
  mathematics is the duality between algebra and geometry.
  While the fact that Desarguesian projective planes can be coordinatized
  by division rings is arguably just a theorem of geometry, I think it is
  an important foundational result because it clarifies this duality.
  The Hrushovski-Zilber result is just as striking (more so perhaps
  because the connection between topology and algebra is forged
  with highly sophisticated model theoretic machinery).
  Of course the beauty of these results is that in addition to their
  foundational interest they have proven to be mathematically useful.
   
   
  -Finally, when I said that my interest was in using logical tools to
  study interesting mathematical structures, I mis-stated things abit.
  Certainly when I am working on   expansions of the reals (or
  differential fields) I do not limit myself to logical methods. But
  these are just the things I know best.  This is I suspect the dividing
  line that makes me a logician, instead of a real geometer.
  I certainly make no claim of working in foundations.
   
  I hope these comments clarify my opinions.
   
  Best Wishes
   
  Dave
  

  From: Stephen G Simpson <simpson>
  To: marker at math.uic.edu
  Subject: foundations, etc.
  Date: Thu, 14 Dec 1995 15:53:45 -0500
  
  Hi Dave,
  
  Again, thanks for your reply.  I wrote to Angus but he doesn't seem to
  be answering his email.  If you can find out more of what he said in
  Sicily, or point me to related Macintyre or Kreisel-Macintyre
  publications, I'd be most grateful.
  
   >    Certainly I do  not believe that the importance of a subject
   > is derived from the methods used. Indeed, the usefulness of
   > model theoretic methods in classical mathematics is my main interest
   > at this point in time.
  
  Well, sure, but the question is, which do you regard as key, the
  classical mathematics questions, or the model-theoretic methods?  If
  the former, then progress should be measured by success in answering
  those questions, by any and all methods.  In other words, if you are
  studying classical algebraic questions for their own sake, then you
  are competing with algebraists on the basis of results obtained, not
  methods used.
  
  And then there is the deeper question of what wider intellectual
  interest is served by classical algebra; most people other than pure
  mathematicians regard Gödel's incompleteness theorem as interesting
  and algebraic geometry etc as obscure; do you think these people are
  completely wrong?
  
  Still, I have to say I'm impressed by what Hrushovski and other
  applied model theorists (including you, Wilkie, Macintyre, van den
  Dries, etc) have done.  Tell me, has it reached the point where
  algebraic number theorists have to take notice of Hrushovski's
  results?  Of model-theoretic methods?  (These are two distinct
  questions, of course.)
  
   > -I think that the main point of the reverse
   > mathematics program is that there are five fragments of second order
   > arithmetic that suffice to classify the strength of most theorems of
   > classical countable and continuous mathematics.  This is without doubt
   > an important and intriguing point. But does this in anyway effect
   > mathematical practice? (this is perhaps related to the
   > Kreisel-Macintyre point)
  
  My contributions to reverse mathematics have more of a philosophical
  motivation, e.g. what is finitism, what would be the consequences of a
  through-going finitist approach to mathematics, etc.  I don't expect
  this to have much effect in math departments, unless mathematicians in
  math departments resume paying attention to philosophical issues.  I
  think mathematicians should resume paying attention to philosophical
  issues, but then again many pure mathematicians are notoriously
  narrow-minded, as evidenced by the way they turned their backs on some
  of the most interesting parts of mathematics (foundations, complexity
  theory, statistics, much of applied math and numerical analysis, etc).
  
  As to the Kreisel-Macintyre point, I don't know what the point is,
  since I didn't hear the speech or read the paper, if there is one.
  
   > Moreover, I doubt that any graph theorist would even find it
   > suprising that this difficult result dealing with well
   > founded parts of relations requires a very good theory of countable
   > ordinals to prove.
  
  I don't agree.  Most work in this area of graph theory (which
  has applications to VLSI design etc) is very finitistic.
  
   > practice.  We have known that 1-consistency of large cardinals is
   > an independent arithmetic statement.  Now we know that we can code
   > this up in a cute way, but this will only be truly important if
   > we such phenomena are found in problems we want the answer to.
  
  Whe you say "problems we want the answer to,", who is "we"?  We
  geometers?  We algebraists?  Most people (other than some pure
  mathematicians and, possibly, Kreisel and Macintyre) already regard
  Gödel's incompleteness theorems as very interesting and truly
  important, even if Gödel's theorems didn't answer a specific question
  that they had in mind before hearing of Gödel's theorems.
  
   > -I think I take a broader view of the foundations of mathematics
   > than you do.  I consider as "foundational" any work that sheds light
   > on mathematical practice and mathematical thinking.
  
  This is too broad.  Most high-level mathematics has this quality of
  shedding light on broad areas of mathematics.  That doesn't make it
  "foundations."
  
  What I mean by "foundations" (as I defined it at the beginning of this
  discussion) is the Frege-Russell-Hilbert-Gödel-... line: formalization
  of mathematical practice and the study of properties and limitations
  of such formal systems.  Hilbert's 2-volume study of this subject was
  entitled, appropriately, "Foundations of Mathematics."
  
   > I think Shelah
   > would consider his classification program as foundational in this
   > sense. To me one of the most important points in the foundations of
   > mathematics is the duality between algebra and geometry.
  
  These are interesting lines of mathematical research, but they have
  little to do with the FRHG line.  
  
   > Certainly when I am working on   expansions of the reals (or
   > differential fields) I do not limit myself to logical methods. But
   > these are just the things I know best.  This is I suspect the dividing
   > line that makes me a logician, instead of a real geometer.
  
  The fact that someone knows logical methods best does not make him a
  logician.  By definition, a logician is "one who studies logic" (by
  all available methods), not "one who studies geometry (by all
  available methods), but who happens to know logical methods best."
  Subjects are defined by subject matter, not by methods.
  
  -- Steve