[FOM] Simpson on Tymoczkoism

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Wed Sep 24 04:01:15 EDT 2003

   Stephen Simpson has referred us to an old (1998) post of his, commenting
on the late Thomas Tymoczko's "New Directions in the Philosophy of
Mathematics." Since the issues are important and keep re-arising, this was
very useful.  I'd like to comment briefly on two of his comments.

   FIRST.  In a parenthetical aside, Simpson says
 	"I suspect that Tymoczko's implicit notion of
	    "foundationalism" is very different from
	    f.o.m. as I have defined it in this forum."
Definitely!  "Foundationalism" is term of art among philosophers,
contrasting with "coherentism".  It denotes a certain (family of)
epistemological theory (-ies) according  to which the  propositions that
can, rationally, be believed are organized into a well-founded hierarchy,
with those of lower strata providing the evidence for higher, and with the
bottom stratum consisting of propositions -- perhaps "self-evident" truths,
perhaps descriptions of one's own sensory experience-- that can and must be
believed without further evidence.  A term of art, but also-- since
foundationalist viewpoints have been rejected by many, many epistemologists
in the past half century-- a term of abuse.  Tymoczko, and many of the
writers whose work he anthologized, seem to have suspected those interested
in axiomatic foundations of adhering to a form  of foundationalism in the
epistemology of mathematics: mathematics is what follows from axioms, and
axioms are self-evident.  Obviously an interest in, and conviction of the
philosophical value of, FoM in the sense of this forum does not commit one
to foundationalism.  Stewart Shapiro's excellent "Foundations without
Foundationalism" makes this point in its title: in contents, in addition to
a well-organized survey of results about and issues relating to Higher
Order Logic, it contains a discussion of why the study of "foundations" is
independent of any commitment to "foundationalism."

     SECOND.  Simpson criticizes Tymoczko for
	"never acknowledg[ing] that, in the present historical
	  era, the orthodox and almost universally accepted
	  explication of mathematical rigor is formal
	  provability in ZFC, Zermelo-Frankel set
	  theory with the axiom of choice"
   This seems to me to be only approximately true. A proof which can
(without TOO much gap filling) be reconstructed within ZFC will, almost
universally & among the orthodox, be accepted as a piece of rigorous
mathematics, but CONCEPTUALLY "rigor" and "formalizability within ZFC" are
not the same.  Suppose that H... that SOMEONE (grin!) shows (rigorously)
that some statement of combinatorics is a consequence of, and indeed
equivalent to, Con(ZFC).  Surely this will count as providing a rigorous
proof of the statement!  (As well as, simultaneously, proof that it can't
be proven within ZFC.)
Allen Hazen
Philosophy Department
University of Melbourne
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