FOM: Quasi-empiricism and anti-foundationalism

[Stephen Ferguson]

[Stephen Ferguson - philosophy at St Andrews, UK - recently finished PhD -
"What structuralism could not be"]

Its for exchanges like this that I joined this list! What follows is: some
thoughts about the fate of foundationalism in philosophy generally,
followed by some thoguhts about Lakatos, and where he fits into philosophy
of mathematics today, followed by some questions about philosophical
attitudes to foundations, and what, once free of foundationalism, these
ought to be: all this prompted by the recent Hersh-Friedman exchange.

Looking at the state of modern analytic philosophy, (Cartesian)
foundationalism seems to have been discredited by Quine et al, his
naturalised epistemology and the effects that had on even the people who
disagreed with him.

However, in philosophy of mathematics, foundationalism as an
epistemological doctrine, was live and kicking long after the rest of the
philosophical world had 'moved on'. I'm inclined to think that the
'obvious' connection between f.o.m and foundationalism is largely
responsible for this, and I've always interpreted Lakatos as, quite
rightly, trying to move the debate in philosophy of mathematics along, by
giving it a new set of issues to worry about (methodology, fallibilism,
etc.)

To one extent, he was succesful -- I don't think the easy identification
of f.o.m and foundationalism is common among philosophers (can't speak for
mathematicians) -- and philosophical debate has moved on: just not onto
the topics he suggested. Its all moved on, from epistemology and the
epistemological considerations of ontology, to semantics and semantic
considerations of ontology. (Some may say thats not much of a change!),
e.g. Field, Wright, Shapiro, Hellman, to name but a few.

Unfortunately, I think that makes Lakatos look as if he is attacking a
straw man -- for now there just are not any of these traditional
foundationalists around. The sort of "Foundations without Foundationalism"
attitude, which Friedman expressed in his mailing, and a slogan which
Shapiro takes to articulate much of his own approach to second order logic
and also to philosophy of mathematics, is now I think, becoming the norm.
So whither now for quasi-empiricism?

I think there is a place for considering (in general) how a priori
warrants can be overturned -- for that is what I take it Lakatos has shown
(I guess thats a bit contentious, esp as he claimed math wasn't a priori,
but I think he did so, simply because of the conflation, popular when he
wrote, of infallibility and a priority), and also for considering
epistemology from a methodological or explanatory point of view (the
Hersh/Kitcher approach, as I see it)

Okay, so that gives two programs, which should run, what, inside this now
semantics-dominated philosophy of mathematics culture, or to run against
it?

But there are more questions.

If the discredited philosophical doctrine of foundationalism is
independent of f.o.m, then, what is a philosopher to think about f.o.m?
Lakatos thought that looking at f.o.m would tell you little about
mathematics as a whole -- maybe, if he were being generous, he might have
included it as just another example of a type of mathematics (along with
analysis and algebra, for example); others think that looking at f.o.m
might give you a chance to examine mathematics with some unusual
conceptual clarity, and so f.o.m is more philosophically relevant than
say, Lie algebras. Is there room, in philosophy of mathematics, for a
philosophy of the foundations of mathematics? Is this what Friedman is
asking for, when he says that foundations of mathematics ought to be
primarily judged by nonmathematicians?

Always more questions than answers,

Stephen
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Stephen Ferguson MA, MSc, PhD	Logic and Metaphysics, 
				The University, 
				St Andrews, KY16 9AL
				tel:(01334) 462484
				fax:(01334) 462485

http://www.st-and.ac.uk/~www_spa/STUDENTS/srf1
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