FOM: Feferman on "Mathematician's Views ..."
Lee J. Stanley
ljs4 at CS1.CC.Lehigh.EDU
Mon Dec 1 14:13:26 EST 1997
On, Sun, 30 Nov 1997 19:18:47 -0800 (PST),
Solomon Feferman <sf at Csli.Stanford.EDU> wrote, citing
P. J. Davis' review of Dawson's recent biography of Godel (much deleted):
"Although Goedel's theorems are now a significant part of the unstated
metamathematical assumptions of research, they are relegated to a far back
burner. If a number theorist is working, say, on the famous and as yet
unsolved problem of whether there are an unlimited number of twin
primes..., then the strong assumption underlying the work, even in our
post-Goedelian period, is that the answer is either yes or no. It is not
assumed that on the basis of the traditional axioms of arithmetic we
cannot decide for true or false. ..."
And this is certainly a completely reasonable starting point.
What is less reasonable is to seek to maintain some shred of
this attitude when it turns out differently. I am not saying
it has, yet, in number theory as practiced by those who recognize
each other as number theorists. Nor do I care to take a stance
about whether I think it will. However, it has turned out
differently in the study of abelian groups.
Yes, I am thinking of Shelah's
work on the Whitehead problem. The attempt to maintain the shred
of the attitude is to take the stance "Oh, if THAT is how it turned
out, then that means that we were thinking about the wrong question
(and this on the part of some who spent much of their career thinking
about that very question, but only succeeding with partial solutions
of special cases, as they would have said prior to it "turning out that way)."
So the conviction that "if we are talking about *reasonable* matters,
then things SHALL turn out to be true or false" is seen as inextricably
linked to preconceptions (gut feelings?) about what is reasonable,
non-murky, etc. In the preceding sentence, I could have used
the normative *should* rather than the predictive *shall*, since it is this
*should* that it is behind making the "shall" come true by fiat, by
post hoc rejection as "unreasonable", "murky", "not serious", etc of alleged
counterexamples. So perhaps the volitional "will" was best. And this
is, of course, the danger when one allows one's views of mathematical
activity to become overly driven by what one personally finds
reasonable, understandable, attractive, important, <insert your favorite>.
And the danger is even greater when one permits oneself, *on the basis
of such criteria*, to use one's stature to influence the direction of
research.
Tennant's comments about deepening out intuitions about the set concept
and arriving at new axioms are relevant here. This is also related
to the view (mine) that we simply don't know enough models of set theory
yet, nor enough ways of constructing them. We are still gathering data.
This is different, I grant, from the situation, say, with groups. The theory
of sets is rich enough to serve as (a formalized version of) the metatheory.
The theory of groups is not. Surely that grants the theory of sets some
interest. Perhaps a different sort of interest than in the theory of groups
... that is of interest because these groups just seem to keep popping up
wherever we look. Models of set theory are not so easy to come by, and
harder to take in all at once, once you "have" them. That seems to be a
reasonable tradeoff. Some would even see the "downside" as a plus ... more
uncharted territory to scout out. More challenging "chaos" to which
to (gradually) bring (some kind of) order and understanding.
Best Wishes,
Lee Stanley
PS. Thanks for the holiday gift suggestion, Solomon! I may very well
act on it.
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