FOM: Message from Pen Maddy

[Harvey Friedman]

This is an interesting and thoughtful response from Penelope Maddy to my
recent postings that touch on her views. I am submitting this as an FOM
posting, since

i) she is not an FOM subscriber;
ii) her message is timely, relevant, and of FOM quality;
iii) she has given me permission to "post" this.

Date: Mon, 14 Feb 2000 09:09:18 -0800
To: friedman at math.ohio-state.edu
From: Penelope Maddy <pjmaddy at uci.edu>

Dear Harvey,

I'm sorry to be slow in responding to your messages of 2/11 and 2/14 about
the upcoming ASL panel; as a non-FOM subscriber, it's taken me a while to
catch up.  Your 2/11 summary is a reasonable sketch of my views -- as your
generous quotations from *Naturalism* show! -- but perhaps I could add a
few comments.

First off, you are right, I do answer yes to the question "does mathematics
need new axioms?", and I do take it that this has already been established
by the fact that large cardinal axioms serve to settle old questions of
descriptive set theory raised by the early analysts in the 1910s and 20s.
As for CH, though I do think it is 'the first and most famous of the
independent questions' (in the passage you quoted this morning), I don't
think it is yet clear whether or not it is a viable mathematical problem
(see *Nat*, pp. 196, 211-212).  Unlike many, I don't think metaphysical
views about sets or truth (like platonism, realism, idealism, or
fictionalism) are relevant to this question; rather, my "naturalist" looks
to internal mathematical considerations.  (As you point out, this is the
main theme of the book.)

Though I think that descriptions of various mathematicians' reasons for
dismissing the CH are mere sociology, I also agree with you that an
analysis of such reasons is called for, to see if any of them can be
counted as good reasons.  (I try to spell out the relationship between the
naturalist's project and sociology of mathematics in *Nat*, around p. 199.)

My contribution to the panel discussion will focus on Feferman's reasons
for dismissing the search for new axioms, in an attempt to spell out where
and how his views depart from those of my naturalist.

Finally, I agree with Shipman that V=L is not the most interesting of the
new axiom candidates to study.  I took it up first because it's the first
real choice point after ZFC (V=L or 0#?), figuring it'd be easy to spell
out what's wrong with it.  Alas, I haven't found it to be so easy!  As
becomes clear in the last two sections of *Nat*, a fully articulated case
along the lines I suggest there would include discussion of (the more
interesting) large cardinal axioms.

Yours,

Pen

PS:  Feel free to post this if you like.