FOM: What is the role of philosophy in math. logic research?

[Andrew Arana]

I attended the ASL meeting at Urbana last week, and enjoyed both the talks
and the discussions outside of the talks very much. During the last panel
discussion, on the future of mathematical logic in the 21st century, the
panelists emphasized the importance of interplay between math. logic and
computer science (for the computability and proof theorists); and `core'
mathematics (for the model and set theorists). What struck me about this
emphasis was that the panelists failed to mention the importance of
interplay between philosophy and math. logic, or whether they think that
such interplay is even relevant for math. logic. The reason this struck me
is because the founders of math. logic, e.g. Godel, Turing, Tarski,
Church, etc., were motivated toward specific technical results
by particular philosophical problems. For instance, Turing's solution of
the Entsheidungsproblem was a culmination of his analysis of the notion
of computation; and Tarski's work on the formal indefinability of truth
predicates stemmed from his interest in analyzing the notion of truth in
broader contexts, not just formal ones. 

I think it is fair to say that some/many math. logicians have, through the
years, pursued specific technical projects (proving theorems, etc.)
because they were pursuing projects of philosophical interest. The
technical results aid the philosophical project; they cast light on
certain issues, make clear what seemed ambiguous. As examples I gave
Turing and Tarski's work earlier; I think there are many other such
examples. 

I think it is also fair to say that some/many math. logicians
have, through the years, pursued specific technical projects without very
much interest at all in philosophical projects. The panel's emphasis on
applications of math. logic to computer science and core mathematics
suggest the question: do practising math. logicians at the turn of
the 21st century think philosophical problems are a worthy, relevant 
motivation for technical results in math. logic. Clearly mathematicians 
often pursue results for solely `internal' reasons: because of how 
`natural' the questions are, or how attractive the questions seem. Will
this remain the main motivating factor for work in math. logic? What place
do people on this list think philosophy has in motivating specific
technical results in math. logic at the turn of the 21st century?


Andrew Arana
University of Notre Dame
Ph.D. student in mathematics and philosophy
Primary research area: models of Peano arithmetic