FOM: Does Mathematics Need New Axioms?

[Matthew Frank]

Some responses to the recent debate on "Does Mathematics Need New Axioms?"
(especially Steve Simpson's Friday post):

1 Fragmentation in set theory

Friedman and Maddy discussed fragmentation in set theory on Monday and
Tuesday.  I find this issue most interesting when one considers the work
on set theories with an anti-foundation axiom, and the topics that are or
can be most naturally treated in that context.  If we need new axioms,
that might be the sort of axiom we need!

2 The worthwhileness of set theory (and the rest of pure math)

Steel has (repeatedly) asked the practical question "Is the search for,
and study of, new axioms worthwhile?  Should people be working in this
direction?".  I like this way of phrasing the question, and I like the
question "is this area of research worthy of government funding?" even
better.  I am inclined to agree with Simpson's Wednesday claim that
"applications of mathematics seem to be where the action is", and
therefore have doubts about whether most research in pure mathematics is
worthwhile in this strong sense.  Putting the question this way highlights
ethical and political aspects, and makes me suspicious of obvious answers.

3 Formalizing in ZFC and alternative schemes

Simpson has pointed out that "the bulk of existing mathematics is
formalizable in ZFC", but I don't think that this carries much weight for
evaluating set theory against various other foundational schemes.  The
bulk of existing mathematics is also formalizable in second-order
arithmetic, though inelegantly.  More importantly for me, the bulk of
existing mathematics is formalizable in Feferman's predicative systems,
and more elegantly than it is in either second-order arithmetic or ZFC!

4 Constructive mathematics (but not constructive mathematicians)

I recommend against using the word "constructivist" to describe people, as
Simpson does.  There is constructive mathematics, and there are
constructive mathematicians in the sense that there are people who work on
constructive mathematics.  But this work does not necessarily imply an
endorsement of constructivist philosophy, and certainly not of the
subjectivist philosophy which Simpson describes.  My own (pluralist)
reasons for pursuing constructive mathematics are now available in a paper
on my web site, which might be of interest to some fom-ers.

--Matt
http://www.math.uchicago.edu/~mfrank

P.S.  An erratum to my post on Hilbert's second problem:  I am now
convinced that, regardless of what I might consider a solution to
Hilbert's second problem, Hilbert would not have evaluated the recent work
of Friedman and Simpson in that way.  I thank Martin Davis, Mic Detlefsen,
Wilfrid Sieg, and Richard Zach for their comments and criticism on this
issue.