FOM: definition of f.o.m.; unity of human knowledge; rabble-rousing

[Stephen G Simpson]

Solomon Feferman writes:
 > 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".

This formulation of mine is not meant to be the last word.  It is
meant as a tentative, working definition of f.o.m. (= foundations of
mathematics, or foundations of mathematical thought) which can serve
to appropriately distinguish f.o.m. from other disciplines, so that we
here on the FOM list can somehow delimit what we are talking about.
Such a working definition is needed when considering broad questions
such as the general intellectual interest of f.o.m.

My definition of f.o.m. is open to constructive criticism and
revision, and I'm glad that Charles Silver and others have initiated
that process.  I have repeatedly encouraged FOMers to post alternative
definitions of f.o.m.  Nobody has done so.

 > 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?

Yes, I think it's appropriate to define subjects in terms of what they
study, i.e. subject matter.  I don't know how else to distinguish
f.o.m. from other subjects.  See below.

Sol cites postings by Martin Davis and himself offering other "ideas
of what f.o.m. is up to", though not definitions of f.o.m.  I
acknowledge the interest of those postings, but I don't think they say
enough to distinguish f.o.m. from other subjects.  Sol quotes Martin as
saying:

 > "It is the problematic character of mathematical truth at a
 > (shifting) boundary of what is understood that provides the
 > problems for workers in FOM".

I agree with this, but I don't see how it sets f.o.m. apart from a
zillion other mathematical research topics, which are also working at
the boundary of what is understood.  Sol says:

 > 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.

Once again, I don't think this formulation adequately distinguishes
f.o.m. from other subjects.  The search for conceptual clarification
and connections between different mathematical disciplines are
features of almost all high-level pure and applied mathematical
research.  The use of "logical methods", i.e. logic, is common to all
of mathematics and indeed all of science.  As an example, consider the
"non-commutative geometry" of Connes, which exhibits all of these
features.  This example was also cited here on the FOM list by Josef
Mattes, who has repeatedly exhibited extreme hostility toward f.o.m.

Sol, let me ask you one question.  What do you think of the "list 2
mind-set" as described in my posting of 11 Jan, 20:15?

By the way, Mattes may be interested to know that when Alain Connes
gave a series of four lectures here at Penn State a few years ago, the
first lecture was devoted entirely to G"odel's incompleteness theorem.

 > 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.)

Dupre's book? Convincing?  Not to me.  Sol recommended Dupre's book to
me off-line, so I got it out of the library and waded through a good
bit of it over the Christmas break.  Dupre deploys the usual
postmodernist arguments, including radical feminism, Foucault, the
specter of "big science", etc etc.  Dupre's main argument seems to run
something like this: "My friends and I can't find any reason to
believe in the unity of science.  Therefore, the unity of science is
nothing but a myth propagated by evil capitalist war-mongers and their
lackeys."

Perhaps Sol would care to summarize Dupre's central argument in a
different way.  If so, I'd like to hear it.

I want to stand up for the unity of human knowledge.  I think this
will resonate very well with Harvey's idea that "foundational studies"
can provide a new unifying focus for universities.  And yes, I do
believe that the place of mathematics within the broad context of
human knowledge is a key issue for f.o.m.  Let's bring this out into
the open.

 > Do we need a Simpson-style definition to guide our discussions and
 > individual efforts in the field? 

It seems that Sol is once again trying to stir up a revolution.  Well,
that's OK with me!  Rabble-rousers are welcome on the FOM list,
especially when they have impeccable f.o.m. credentials, as Sol most
assuredly does.

-- Steve