FOM: Central Issues in Foundations

[Ryan Conrad Kohl]

    Friedman has provided the fom listserve with an interesting view on
directions for mathematical logic and the foundations of mathematics.
However, a current theme in the fom postings has been on recursion
theory, including Friedman's recent posting on central issues in fom.  I
would like to bring up a topic in fom which is at most indirectly
related to recursion theory.

    Contrary to the definition-theorem-proof method of learning
mathematics in university, mathematics has historically progressed
through a fairly standard sequence of events:
                        1) Something happens in the real world.
                        2) A problem is realized and a solution is
sought.
                        3) Analysis of the problem and oftentimes the
solution takes place to determine underlying concepts.
                        4) A synthesis of underlying concepts follows
the analysis.
    Each of the steps might appeal to certain fields of interest in
lesser or greater degrees.  For example, the aspect of problem solving
might appeal to an engineer, while the aspect of analysis might appeal
to a logician.  But what would motivate a researcher in the foundations
of mathematics?  In the last century, synthesis has been predominant, in
particular concerning the ZF axiom set, with holistic and reductionistic
tendancies back-seat driving the research.  However, the overall method
as a whole would also seem to be of interest to a researcher in fom.  Of
course, fom'ers are not a homogeneous lot, and surely have varying
interests, but fom as a discipline sould certainly have some specified
domain and official interest.  I would say that even the above first
step, that of the observation, conception, and communication of real
world events, should be a focus of fom research.  In this case, most
pure mathematicians will shy from real world events, prefering evidence
from within the field of mathematics.  Should fom research be concerned
with empirical data and studies?  Is the method of mathematics an area
of research in fom?  Should it be?

    I would like to encourage discussion on the fom listserve pertaining
to the philosophical implications of fom, which is a personal interest
of mine.  Naturally, I would not want a dismissal of technical concerns,
which could certainly lead to a naive theory doomed to triviality.
Philosophical interpretations of technical matters in mathematics are
rarely performed in contemporary literature, and perhaps this would be a
good place to popularize this form of activity.

-Ryan Kohl
Penn State University