RTragesser at compuserve.com
Sun Mar 22 05:24:33 EST 1998
[ FOM moderator's note: This is slightly edited.
Omitted material is indicated by .... ]
 I was not questioning the f.o.m.
importance of HF's results.
.... I want
to say that in general substantial
f.o.m. results have GREATER IMPORT
than g.i.i. can recognize, although
working out that greater import is
a long term problem (see  below).
That is, ironically, g.i.i.--so
easy to ascribe importance -- actually
ascribes less import.
 I found SS's attack on Franzen
quite just. Franzen seemed to
have no idea of the monstrous and
bizarre finite combinatorial problems
which arise in some innocent seeming
mathematics. [see  below]
 ... Judging from secondary
literature on Goedel's PM-I (such
as the Penrose fiasco, and the quest
for natural or real undecidables), it
is clear that the mathematical and
philosophical importance of Goedel's
results is still controversial. Few
would say they are not important. The
question is rather what their import
 In my one serious spell of
mathematical research -- in the
early '70's -- on problems of
the solvability of the freest
groups under some constraints (in
particular, the freest group of
exponent four), I/we joined
what seemed in those days (perhaps
inspired by the Haken-Appel
"success story") the promising tack
of reducing these algebraic
problems to combinatorial problems
(essentially by constructing
rings in the groups). The
combinatorial problems that
emerged were monster problems:
I showed one
of these to Paul Erdos when he
visited campus. He said
he'd never seen anything like
it and advised dropping it
_if_ the principal concern was
the group-theoretical problem.
Many other algebraists
had similar experience with
hopeful combinatorial reductions.
I suggest that wicked combinatorial
structures are more natural than
appears (to Franzen and Feferman),
but they aren't so conspicuous
because they were widely avoided
as dead-end reductions of problems.
After two years of
beating our heads
on that "finitary" problem,
the philosopher in me developed
considerable respect for the
finite. It seemed to me to be
more terrifying than (what I
then knew of) the transfinite.
Thus HF's results are interesting
to me. More deeply, I began to
doubt the value of epistemology
driven philosophies of mathematics
(e.g., finitism, intuitionism).
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