FOM: mathematical logic vs f.o.m.; Shelah
Matthew Frank
mfrank at math.uchicago.edu
Fri Jan 19 15:34:28 EST 2001
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.
More information about the FOM
mailing list