FOM: mathematical logic vs f.o.m.; Shelah
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
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
-- 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
-- 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
I would be interested to hear what other people mean when they say that
they aim to do foundational work.
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.
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