FOM: Foundationalism

[JoeShipman@aol.com]

Daer Reuben,

Thanks for an interesting posting.  I have some questions about it:

1) Which 3 Fields medalists had incomplete proofs of important results?  Did
they have other sufficiently important results that *were* completely proven,
or is there still some doubt that they have actually proven anything important
(in this paragraph of course "important" means "Fields-medal-quality-
important, arguably as significant as anything else being done in mathematics
in the last few years").  If the named individuals agree with you that their
most important proofs are incomplete, and the Fields committee knew this (two
very big "ifs"), then either you are correct about the dispensability of
foundations or the Fields committee is to be condemned for dishonesty and the
use of improper criteria.

2) Even though the parallel postulate became dubitable, it was still part of a
genuine foundation.  Occasionally foundations need to be repaired!  Your focus
on "dubitability" and the loss of it in modern times when it was not
philosophically correct to invoke Heaven or the mind of God is illuminating,
but don't forget that human mathematicians like Descartes, while believing
that Mathematical Truth really existed and God knew what it was, knew they had
no pipeline to God's mind and had to figure it out on their own, starting from
some bedrock of definitions and propositions to be accepted a priori. 

Why do mathematicians *eventually* come to a consensus (very broad if not
unanimous) about whether a proposition has been proven or not?  What is going
on?  Answer: they satisfy themselves that the proof has reduced the
proposition to things they all agree upon.  If you would deny that mathematics
is "well-founded" in this pragmatic sense, can you please give us some
examples where the "consensus" you say is "really" the hallmark of mathematics
did not arise by virtue of the proposition being logically reducible to
propositions that all the mathematicians participating in the consensus
accepted?

The "things they all agree upon", after sufficient iteration, will be a set of
basic propositions and definitions; since we have had a workable foundation in
"ZFC plus Predicate Calculus" for many years, that empirically is seen to be
sufficient to encompass all of these basic propositions and definitions,
mathematicians have generally stopped worrying about foundations.  When you
observe this and claim that foundations don't matter, you trivialize this
hard-won victory (which was not a victory for indubitability but rather for
mathematical professionalism: the historically-unprecedented-in-scope failures
of consensus in the late 19th and early 20th centuries were repaired).
Philosophers of mathematics can argue about the dubitability of ZFC or
alternative foundations, and  actual mathematicians may ignore the
philosophers, but as the shortcomings of the current foundations become
clearer (Harvey's project is to make them clearer by showing mathematicians
that a lot of things they should really want to know are not settled by ZFC)
the crises of consensus will return, and it will again become clear that
mathematics needs foundations.

Remark: this did NOT happen when the Continuum Hypothesis was shown
independent because 
 a) it was not connected enough to "core mathematics" that many ordinary
mathematicans cared about it, so it could be dismissed as meaningless without
affecting core mathematics; the situation is different with more recent
combinatorial independence results
 b) there was no "proof" of it (or of its negation) using ANY axioms that had
any plausibility, so the issue of "failure of consensus about whether it's
been proven" didn't arise; again the situation is different with more recent
independence results for propositions which are indeed provable from somewhat
plausible axioms (large cardinals).

-- Joe Shipman