FOM: How About Listing Foundational Issues?
Robert Tragesser
RTragesser at compuserve.com
Fri Mar 6 08:17:02 EST 1998
Could a list of something like all
"important" foundational issues be given? I
can certainly find these among HF's postings
last year. _But I am also looking for more
than a list_; I'd like to see the following
addressed (I came to this problem by asking
after the source of my embarrassing diatibes
against set foundations):
It continues to be difficult to see
how a claim can be made for or against the
cogency or viability of any proposed foundational
scheme independently of a definite enumeration of
foundational issues and a strong plausibility argument
that the proposed foundational scheme is optimal
for illuminating, if not resolving, _each_ {_sic,_)
of those issues. It might very well be that different
"foundational schemes" are optimal for different
foundational issues, but no one is optimal for
them all. It is difficult to spell out what one might
mean by optimal since there are so many different
variables. But I think one knows it when one sees
it (and I mean sees it, not vaguely imagines it), e.g.,
one might expect that if foundational schemes A and
B are equally insightful about foundational
issues C and D, but B, in treating C, had a
lot of junk of no use in illuminating C, but
A does not, then we might say that A is better
than B for issues C.
Here is a starter list of "global" foundational
issues ("global" = are more or lessed addressed to
mathematics -- qua artifacts not activities -- as a whole.
Mostly, I'll not elaborate the issues but give
the pair of "contraries" which generate issues:
finitary/infinitary
solvable/insolvable
constructible/non-constructible
caculable/incalculable
concrete/abstract
formal/informal
axiomatizable/inaxiomatizable
provable/inprovable
definable/indefinable
good proof/bad (but valid/sound) proof**
variable method/canonical presentation***
geometry/set***
[more/other could be garnered from
e.g., Feferman's "Working Foundations"]
[**This is inspired by (1) Feferman's
approvingly quote Manin, that
a "good" proof is one that makes
us wiser, (2) the closing pages of
Hilbert's "Axiomatic Thought" where
he poses the problem (with an
elaborate example) of coming up
with better proofs, (3) Kreisel's
and S.MacLane's insisting on the
importance of a general proof theory
which addresses rather more than
"validity"; Rota in _Indiscrete
Thoughts_ is rather elaborate about
different proofs being better for
different purposes].
[***--in _both_ cases--I point to a
a tiny but significant example, on
page 13 of E.Artin's _Geometric
Algebra_ where he urges breaking
out of the formal development and
inserting a more geometric argument.
Artin is clear to indicate that
is recommendations are pedagogical
and not foundational. But I
do think that there is a foundational
issue: the role of geometric
methods wrt set.
Robert Tragesser
More information about the FOM
mailing list