FOM: What is f.o.m.?
Solomon Feferman
sf at Csli.Stanford.EDU
Tue Jan 13 03:10:16 EST 1998
In his posting of 11 Jan, 20:15, Steve Simpson says that: "So far as I am
aware, nobody on the FOM list, except me, has proposed a definition of
f.o.m.". He then repeats his proposed definition from Sept or Oct that it
"...is the systematic study of the most basic mathematical concepts and
the logical structure of mathematics, with an eye to the unity of human
knowledge".
Can we really hope for a simple definition like this for f.o.m.? Is it
defined in terms of what it studies, rather than what its concerns are and
how it deals with them? Simpson tries to list the basic mathematical
concepts. Even if we grant those, what is "mathematics" and what is its
"logical structure"? Are the meanings of these to be taken for granted?
Can we do any better in defining f.o.m. than we might do in defining
recursion theory or proof theory? How about the "eye to the unity of
human knowledge"? Is this an essential part of f.o.m.? (For a convincing
argument contra the ideal unity of human knowledge, cf. J. Dupre, *The
Disorder of Things*, in which he makes plain the essential disunity of
science.)
In fact, others have offered ideas as to what f.o.m. is up to. Near the
end of Martin Davis' posting of Jan 5 at 15:21 (near the end) he says
that: "It is the problematic character of mathematical truth at a
(shifting) boundary of what is understood that provides the problems for
workers in FOM". In my posting "Working foundations" of 12 Nov at 13:10
(that Simpson and I corresponded about privately) I described work in
f.o.m. as being concerned with problematic concepts, methods and results
in mathematics dealt with in a more or less systematic way by logical
methods that cut across the traditional mathematical disciplines; and I
repeated from my paper of the same name a listing of six characteristic
"foundational ways" by which this is pursued. In my posting on 10 theses,
of 03 Jan (19:00), I emphasized in theses 9 and 10 the task of conceptual
clarification as one of the main purposes of foundational work. I'm sure
others have offered statements of purpose which are of similar character.
Do we need a Simpson-style definition to guide our discussions and
individual efforts in the field?
--Sol Feferman
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