FOM: Working Foundations (fwd)
Solomon Feferman
sf at Csli.Stanford.EDU
Tue Jan 13 03:25:51 EST 1998
By way of reminder, here are parts of my 12 Nov posting:
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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.
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Note Nos. 5 and 6 in the above list. This shows that f.o.m. is not just
concerned with problematic concepts, etc. as I emphasized in my previous
posting on the definition of the field. But I think the latter are the
driving force.
--Sol Feferman
PS. OUP is now predicting publication of the above-mentioned volume in
Sept. '98.
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