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

[Matthew Frank]

Steve Simpson asked:  What is foundations of math?

I am wary of discussions about these questions.  On the other hand, many
of us aim to do foundational work, and I think it useful for us to be
clear about our (several) goals.

I use the term "foundations of math" to mean:  Axiomatics with related
conceptual and methodological investigation of mathematics.  Under this
usage:

Axiom systems (and by that I mean mathematical axiom systems stated in
precisely specified formal languages) are of foundational interest only if
they exemplify or are connected with particular mathematical methods.  I
think most of the historical tradition of f.o.m. (from Frege on) falls
fairly well under this definition.  Some examples close to my heart:

-- Godel's theorems are of foundational interest as studies of axiom
systems for all ways of doing mathematics satisfying rather minimal
methodological constraints.

-- Work on constructive math is foundational -- whether articulating
axiom systems, making decuctions in them, proving metatheorems, or
conceptual or methodological investigations of those ways of doing
mathematics.  Likewise for intutionistic, (neo)-logicist, nominalist, or
predicative math.

-- Herbert Busemann's (sadly underappreciated) axiomatics for differential
geometry are of foundational interest for exemplifying synthetic methods.

I consider some topics to fall outside my definition:  much philosophy
of math (contra Simpson); most Shelah-Style logic (contra the Wolf prize
committee); Falting's theorem (contra David Marker).  Reducing the
axioms for Euclidean geometry to one predicate lacks a conceptual/
methodological component, so is at most bad f.o.m..  Axioms for homology
theories are not yet, but could become, an interesting topic in f.o.m. if
someone came up with a good formal language in which to deduce things from
the axioms.

I would be interested to hear what other people mean when they say that
they aim to do foundational work.

--Matt

P.S.  Specifically in response to Simpson:  Questions such as "what is a
number/shape/set/function/algorithm" are foundational only when connected
with the elucidation of some axiom system.  I avoid the questions "what
are the appropriate axioms for ...", since I do not like the definite
articles.  For me, foundations of math should offer (for instance) varying
axioms for sets, and elucidate the varying concepts which may be
profitably connected with them.