FOM: fom as mathematics

[Joseph Shoenfield]

   In a recent communication, I stated rather casually that fom is a
branch of mathematics.   Here I would like to explain what I mean by that
statement and what consequences I think it has for fom.
     In a reply to my statement, Harvey asserted that fom is a
mathematical subject but not a branch of mathematics.    I do not
understand the difference, and I do not see why fom and statistics are not
branches of mathematics and geometry and algebra are.   He also says that
fom is a branch of the subject of foundations.   This is a truism, but a
useless one.   There are no significant results on foundations which can
be used in the various foundational studies.   By contrast, mathematicians
have though much about mathematics and reach many agreements of how
mathematics should be done by consensus.
     My statement was not intended as a truism, but as a statements that
all of the significant results in fom are mathematical.  I challenge
anyone to find a significant advance in fom in which the principle
ingredient is not the formulation or proof (or both) of a clearly
mathematical theorem.
     If this is correct, it has consequences for the study of fom.   I
suggested one such consequence concerning the study of intuitive ideas
which arise in the consideration of the nature of mathematics.   I said
that the object in studying such concepts should be to replace them by
precise concepts which we can agree capture the essential content of the
intuitive notion.   When we have done this, we come to the most important
part.   This is to formulate and prove mathematical theorems about the
precise concepts which increase our understanding of the intuitive notion.
Sometimes we discover properties of the intuitive notion which we would
probably not have even thought about in an informal discussion.   (I am
sorry that my earlier communication seemed to suggest that the above is
all of fom; Harvey was quite right to say that there is much else.)
     Let me use the above to show what I consider to be the acievement of
reverse mathematics.   The original object was to discover what axioms are
needed to prove the theorems of core mathematics.   To make things
manageable, researchers confined themselves to mathematics expressible in
the language of analysis (= second order arithmetic); this is certainly a 
reasonable restriction.   The main result was that over a very weak system
of analysis, all of the theorems which they considered were equivalent to
one of a small number (I believe 5) theorems. An additional point
(emphasized by Harvey) is that these 5 theorems are linearly ordered by
provable (in the weak system) implication.   I take this to mean that the
intuitive notion of "theorem of core mathematics" gives rise to five
precise notions which are related in a nice way.   The next step, I
believe, should be to prove significant mathematical theorems abou these
concepts.   Thus I think that reverse mathematics has contributed
significantly to fom, but its future progress will decide whether it
becomes a permanent part of the theory of fom.