FOM: f.o.m. and the working mathematician; Quinean holism

[Stephen G Simpson]

Neil Tennant writes:
 > Is the phrase 'foundations of' different from the prefix 'meta'?
 > Is foundations of physics different from metaphysics? My guess is
 > not--or at least, that many a metaphysician would not accept the
 > identity claim in question.

It's not clear to me that the prefix `meta' has any specific meaning.
For instance, I don't think metamathematics is to mathematics as
metaphysics is to physics.  I would however insist that the term
`foundations of' has a specific meaning.  Namely, `foundations of X'
means the systematic study of the most basic concepts of subject X,
the logical structure of subject X, etc., with an eye to the unity of
human knowledge.  See my short essay at
www.math.psu.edu/simpson/Hierarchy.html.

 > Question: on either proposal, how is that extra part of foundations of
 > mathematics that cannot itself be mathematized to be made
 > intellectually accessible, arresting and worthwhile to the working
 > mathematician?

I don't know of any way to compel working mathematicians to be
interested in foundations of mathematics.  In fact, many working
mathematicians of the late 20th century are notoriously narrow-minded
and take no interest in any mathematical or scientific topic outside
their own specialized branch of mathematics.

 >  Is it "merely" philosophy of mathematics?

Why "merely"?  Philosophy of mathematics is an important subject.
Foundations of mathematics and philosophy of mathematics are very
closely related.

 > Does the WM have any intellectual responsibility to consider it?

Yes, obviously, it's part of their responsibility as mathematicians.
But many of them routinely evade that responsibility.  They can get
away with it, because nothing more is demanded of them.

 > Would the ordinary work of the WM be enhanced by paying any
 > attention to it?

In many cases, yes.  For example, Atiyah's Bakerian lecture (in front
of a general scientific audience) might have gained a lot by taking
account of the foundational perspective.

 > Could it guide and shape, or help set the agenda, of the community
 > of WMs?

Yes, it could.  For example, most number theorists are completely
uninterested in the foundational issues raised by Matiyasevic's
theorem.  If they took an interest in those issues, number theory
might have a different shape.

 > I note with interest that many of the mathematicians who have
 > contributed to the list so far write as though the discussion of
 > foundations of mathematics cannot proceed without a precise technical
 > definition of its methods and concerns.

Well, I think "precise technical definition" is going a little too
far.  But I would say this: If we are going to discuss f.o.m., then
obviously it's important to have some sort of working definition of
f.o.m. on the table, so that we know what we are talking about, to
avoid talking at cross purposes about everything under the sun.  I
think my concept of f.o.m. given in
www.math.psu.edu/simpson/Hierarchy.html is sufficiently precise to
give us a basis for discussion.

 > The very phrase 'foundations of...' carries with it the implication
 > that the field in question can be built, or reconstructed, on some
 > solid base on which the foundationalist can focus. 
 .....
 > The Quinean holist rejects such a picture for knowledge in general.
 > According to the holist, all knowledge is interdependent, none of it
 > privileged, every 'known' proposition enjoying that status only
 > because of how it lends support to, and finds support from, other
 > propositions in the system.

OK, this is interesting.  Neil seems to be saying that there is an
alternative view of human knowledge, which he dubs Quinean holism,
according to which my concept of f.o.m. makes no sense, because there
is no hierarchy of mathematical concepts.  In other words, according
to this holistic view, no mathematical concept is more basic than any
other mathematical concept.  Is that right?  Under this holistic view,
the concept "positive integer" is not more basic than the concept
"symplectic manifold".  Is that right?  I'm not trying to ridicule
Neil's brand of holism; I'm only trying to understand it and its
implications for f.o.m.

 > Would one respectable 'foundational' enterprise be a definitive
 > articulation of the hierarchy of *definitions* of concepts in all
 > the main fields of mathematics?

In my view, that activity would be foundational (according to my
concept of foundational) only insofar as it would focus on the nature
and role of the most basic mathematical concepts.  When you get into
higher level mathematical concepts such as symplectic manifolds, you
are moving away from f.o.m.

A modest proposal:

Let's make a tentative list of what might be considered the most basic
mathematical concepts.  In this list I would tentatively include
concepts such as the following: number, shape, set, axiom, proof,
algorithm, ....  I would not include specialized mathematical concepts
such as "symplectic manifold", which are to be defined and explained
in terms of more basic concepts.  Can we FOM'ers agree on this much?

Sincerely,
-- Steve Simpson