[FOM] 23.99 Carat Gold; foundationalism vs. coherentism

Stephen G Simpson simpson at math.psu.edu
Fri Oct 3 20:22:43 EDT 2003

A.P. Hazen Fri, 26 Sep 2003 17:27:24 +1000 writes:

 > (You gotta unnerstand, us analytic philosophers is TRAINED to look
 > for excepshuns to everything-- [...] I treated this as a conceptual
 > claim, that the concept "rigorous proof" was now identified with
 > proof formalizable in ZFC.  And so I pointed out the exception,
 > [...]

Of course I understand that you were, quite properly, noting
exceptions to my claim that the currently widely accepted explication
of mathematical rigor is formal provability in ZFC.  My point was only
that this explication, despite all the exceptions, is not only fairly
compelling, but also basic for much of contemporary f.o.m. research.


On another topic, I want to thank you for your two long and
illuminating messages regarding the foundationalism/coherentism
distinction in philosophy.

A.P. Hazen Sun, 28 Sep 2003 15:31:00 +1000 writes:

 > much FoM work seems, if anything, MORE interesting in a more C-ist
 > framework.
 > 	The problem is with the AXIOMS.  What makes it rational to accept
 > an axiom?  [...]

I'm sorry, maybe it's my "foundationalist filter" (Corfield's term) at
work, but I simply don't see coherentism here.  Yes, the axioms of
subject X (mathematics in this case) require justification other than
proof within the deductive framework of subject X, but this is not at
odds with foundationalism.  Indeed, from the viewpoint of strict
foundationalism, it is *impossible* for all the assertions of subject
X to be justified by means of proof within the deductive framework of
subject X.  Otherwise, there would be infinite regress, which would be
absurd.  Thus, the non-deductive justification of axioms is an
essential feature of the foundationalist landscape.

My knowledge of the history of philosophy is rudimentary.  But it
seems to me that the original foundationalist was Aristotle, with his
detailed blueprints on how to organize sciences deductively from first
principles (archai).  Of course Aristotle recognizes the importance of
choosing the right axioms or starting points for each science.  He
develops various methods for doing so, and he says that this is
ultimately a matter for dialectic.

Foundationalism versus anti-foundationalism is obviously a core
philosophical question.  All sciences beg to be organized
foundationally, but mathematics is the one science where the
foundationalist program has been carried out most fully.  Thus
mathematics emerges as the grand laboratory of foundationalism.  This
is why philosophers through the ages have been so intensely interested
in mathematics, and why philosophy of mathematics has so often been
dominated by arguments concerning foundationalism, pro and con.


Stephen G. Simpson
Professor of Mathematics
Penn State University

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