FOM: mathematical logic versus f.o.m.; Shelah

[Stephen G Simpson]

Dear FOM,

A phenomenon that has bothered me for a long time is the tendency of
many people to misidentify f.o.m. (foundations of mathematics) with
mathematical logic.  In my view, these are very different subjects.
When the distinction is blurred, both subjects suffer, because
research in both is evaluated incorrectly.  Unfortunately, this
destructive kind of error is widespread among mathematicians.

A case in point is the recent news that logician Saharon Shelah will
be sharing this year's Wolf Prize in Mathematics with mathematician
V. I. Arnold.  The prize committee (three mathematicians, identities
unknown) is quoted as saying that Shelah is

  ``the leading mathematician in the field of foundations of
  mathematics and mathematical logic.''

[ I have not seen this quotation in any official Wolf Foundation
source, but I have no reason to doubt its authenticity. ]

Now, Shelah has many huge achievements to his credit, which make him
arguably *the* outstanding researcher in model theory and set theory.
And many other researchers have taken up subjects initiated by Shelah,
including classification theory, proper forcing, pcf theory, etc.  So
it is entirely appropriate to award Shelah the Wolf Prize.

Nevertheless, I question the committee's remark that Shelah is the
leader in foundations of mathematics.  My feeling is that their
statement quoted above stems from a casual and uninformed
misidentification of ``mathematical logic'' with ``foundations of
mathematics''.

This goes back to a subject that we have sometimes discussed here on
the FOM list:

  What is foundations of mathematics?

My answer to this question (from the preface of my book) is:

  Foundations of mathematics (f.o.m.) is the study of the most basic
  concepts and logical structure of mathematics, with an eye to the
  unity of human knowledge.  Among the most basic mathematical
  concepts are: number, shape, set, function, algorithm, mathematical
  axiom, mathematical definition, mathematical proof.  Typical
  questions in foundations of mathematics are: What is a number?  What
  is a shape?  What is a set?  What is a function?  What is an
  algorithm?  What is a mathematical axiom?  What is a mathematical
  definition?  What is a mathematical proof?  What are the most basic
  concepts of mathematics?  What is the logical structure of
  mathematics?  What are the appropriate axioms for numbers?  What are
  the appropriate axioms for shapes?  What are the appropriate axioms
  for sets?  What are the appropriate axioms for functions?  Etc.
  
A different question is:

  What is mathematical logic?

My answer to this is that mathematical logic is a body of technical
tools and subjects which have evolved from within f.o.m. and taken on
a life of their own.  A great deal of mathematical logic is not (or,
is no longer) motivated by f.o.m. considerations.  Perhaps the best
way to delimit mathematical logic is to use the 4-fold scheme of the
Handbook of Mathematical Logic (1977):
   
  Mathematical logic consists of (in no particular order) model
  theory, set theory, recursion theory, proof theory.

Following up on the Wolf Prize committee's statement quoted above, I
would like to consider the question:

  What impact has Shelah had in f.o.m.?

My impression based on published work is that Shelah has little or
nothing to say about foundational issues.  Certainly some of Shelah's
results have foundational significance, but the results of others have
more.  I have e-mailed Shelah asking which aspects of his work he
regards as particularly significant for f.o.m. and urging him to
comment here on the FOM list.  He is already an FOM subscriber.  I
eagerly await his reply.

FOMers, what do you think?

-- Steve