[FOM] The lure of the infinite

[Nik Weaver]

Martin Davis graciously allows that legitimate questions can
be raised about platonism, and concludes:

> Although we may be as far from a satisfactory reply to such
> questions as Berkeley's contemporaries were to his, one can
> confidently predict that so long as mathematicians can obtain
> worthwhile results using set-theoretic methods (and the
> incompleteness theorem does suggest that that will be the
> case), calls for working with more "reliable" methods will
> fall on deaf ears.

Could I just say that I am not actually calling for working
with more "reliable" methods.  Perhaps I haven't made that
sufficiently clear yet.  I tried to express this in
http://www.cs.nyu.edu/pipermail/fom/2006-February/009783.html
and a bit more colorfully in
http://www.cs.nyu.edu/pipermail/fom/2006-February/009811.html
but I guess the point is not getting through.

There seems to be a naturalistic fallacy --- inference of
"ought" from "is" --- here.  Weaver says that only predicative
mathematics has a clear philosophical basis, therefore Weaver
wants people to stop doing impredicative mathematics.

This would be a fairly pointless position to advocate because
most mathematicians already do effectively restrict themselves
to predicative mathematics, even though they are generally not
aware of this.  (A question could be raised as to whether or
not they would be better served by working in some explicitly
predicative system.  I'll return to that in a later message.)

Set theorists do not restrict themselves to predicative
mathematics.  I personally think that set theory is very
interesting.  Although I am a C*-algebraist by training,
it so happens that the result for which I am best known is
a consistency theorem --- it is consistent with ZFC that
there is a counterexample to Naimark's problem.  It's not
as if I want people to stop doing set theory.

Harvey Friedman talks about my desire to "marginalize"
set theory.  Perhaps there is a fear that regardless of
my personal motivation, once the word gets out that core
mathematics can be done predicatively set theorists will be
in danger of being marginalized by the mainstream mathematics
community.  May I be frank: set theory is already marginalized
by the mainstream community.  Not in the sense that core
mathematicians look down on set theorists (maybe some do, but
not many), but in the sense that they see axiomatic set theory
as not being of any use in helping to solve the kinds of problems
they want to solve.  Davis talks about "so long as mathematicians
can obtain worthwhile results using set-theoretic methods" but I
think most mathematicians would be unable to give even one example
of a worthwhile result in their field that was obtained using
set-theoretic methods.  Pinning one's hopes on the prospect of
large cardinals being the vehicle for mainstreaming set theory
seems to me not realistic.

Is all of that clear?  I am not calling on anyone to work with
more "reliable" methods, nor am I out to "ban" impredicative
mathematics or to marginalize set theory.  I actually think
that it is unhealthy and unnecessary for set theorists to seek
affirmation from core mathematicians.  Not that anyone asked my
opinion, but there has been some fairly severe mischaracterization
of my views so I felt I had to say this.

Predicativism --- particularly my version of it, conceptualism
--- is of great interest to me because it resolves a problem
that bothered me for many years.  Mathematics seems totally
solid and objective, yet it rests on a philosophical basis
that is not convincing at all.  How are we to understand this?
Answer: the regions of mathematics that really have this flavor
of solidity actually rest on a firm philosophical basis, namely
predicativism.  The part that doesn't have this flavor (do large
cardinals exist, is the continuum hypothesis true or false) does
not rest on a firm philosophical basis.

I find this explanation intellectually satisfying and I think
the foundations community has dropped the ball in a major way
by marginalizing predicativism.  Yes, initially it was thought
to be too weak to support mainstream mathematics.  Later it
was thought to be subject to an ordinal limitation (the
Feferman-Schutte ordinal Gamma_0) which made it seem a
curiosity.  That is not correct, as I think I have established
decisively in my Gamma_0 paper.  This means that it is time
for a major reevaluation of the significance and interest of
predicativism.

Not to be falsely modest, I think my Gamma_0 paper is a major
advance in our understanding of predicativism.  It is of
fundamental significance in the philosophy of mathematics.
Yet based on the reaction I've received, I believe it is
unpublishable and consequently I have no plans to submit it.
Who's "banning" whom?

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu
http://www.math.wustl.edu/~nweaver