FOM: Working Foundations

[Solomon Feferman]

I was going to call this message "Two cents" to indicate that that's what
I wanted to put in, but chose the above following Steve's request for more
informative subject titles.

As someone who is identified as working primarily and over a long period
of time in the foundations of mathematics, I was of course pleased to see
the creation of this mailing list and the opportunity it provides for
provocative exchanges from a variety of points of view.  And I am
very impressed by the size of the subscriber list and am glad to
see many names on it that I would not have expected to be on it.
At the same time I can think of many more names of people who
ought to be on it but aren't yet. In my own case
I delayed subscribing until just a couple of days ago, because I was very
busy preparing for a meeting at the end of October, and did not want my
e-mail to become overloaded with the deluge of messages I expected to come
--and indeed did come--in my absence.  So I feel like I've walked into the
middle of a very exciting party and know there's a lot I've missed and
that I'm only getting the tag ends of some intense conversations.  (Rick
Sommer was kind enough to provide me with the archives to Oct. 22, and I
know I can access the exchanges between then and Nov.7 when I subscribed,
but there's only so much time--so my two cents may well overlap what
others have said.)  All this by way of background.

Since some have asked for pointers to the literature in one respect or
another, let me point you to my anonymous ftp site

ftp://gauss.stanford.edu/pub/papers/feferman/

which includes my complete list of publications and a baker's dozen or so
of more recent publications.  Another advertisement:  I'm in the process
of completing work on a collection of essays written since the late 70s,
to be published by Oxford Press under the title "In the Light of Logic".
I don't have an exact date of publication yet but I hope it will be early
in '98.  The volume consists of "a selection of my essays of an
expository, historical and philosophical character which in the main are
devoted to the light logic throws on problems in the foundations of
mathematics." 

Now, much of the discussion I have seen concerns the question, just what
is (are?) the foundations of mathematics and what purposes does it serve?
Two of the essays in the aforementioned volume address that directly,
namely:

"Foundational ways", in _Perspectives in Mathematics_, Birkhauser 1984,

and 

"Working foundations '91", in _Bridging the Gap: Philosophy, Mathematics
and Physics_, Boston Studies in the Philosophy of Science 140 (1993).

The latter paper is an update of a paper "Working foundations" appearing
in _Synthese_vol.62, 1985.  The "Foundational ways" paper is a
slimmed-down version of the latter which is good for a quick-read.  My
thesis in these papers is that there is a tremendous amount of logical,
foundational work at a more everyday "local" level than is usual thought
of when considering "the" foundations of mathematics, and that this falls
into five or six characteristic modes.  Moreover, each of these is a
"direct continuation of work that mathematicians themselves have carried
on from the very beginning of our subject up to the present.  The
distinctive role of logic lies in its more conscious, systematic approach
and its different ways of slicing up the subject."  The foundational ways
I then presented with examples from mathematics and logic for each are:

1. Conceptual clarification.
2. Dealing with problematic concepts by interpretations or models.
3. Dealing with problematic concepts by replacement, substitution or
elimination.
4. Dealing with problematic methods and results.
5. Organizational foundations and axiomatization.
6. Reflective expansion.  

Obviously, you have to look at the examples (including, among many
others, NSA) to see how good a case I've made for this way of looking 
at f.o.m.  And of course I welcome comments.

I also look forward to Harvey Friedman's proposed discussion of my recent
paper, "Does mathematics need new axioms?" which can be found at the above 
ftp site and is to appear in _The American Mathematical Monthly_ some time
next year, and hope that will lead to a wider discussion.

--Sol Feferman