[FOM] 23.99 Carat Gold

[A.P. Hazen]

   Stephen Simpson, commenting on Tymoczko's book, wrote that Tymoczko
 > 	"never acknowledges that, in the present historical
 > 	  era, the orthodox and almost universally accepted
 > 	  explication of mathematical rigor is formal
 > 	  provability in ZFC, Zermelo-Frankel set
 > 	  theory with the axiom of choice"

    To which I... (You gotta unnerstand, us analytic philosophers is
TRAINED to look for excepshuns to everything-- SOMETIMES, as any
mathematicican can appreciate, counterexamples are important, and sometimes
noting them is, as my wife frequently tells me, just annoying)  ... I
treated this as a conceptual claim, that the concept "rigorous proof" was
now identified with proof formalizable in ZFC.  And so I pointed out the
exception, that one could prove something rigorously by deriving it from
the statement that ZFC was formally consistent, which, by Gödel's 2nd
Incompleteness....

    To which Simpson replies that his "gold standard' of rigorousness is
"98 or 99 percent right."

    To which I reply:
     CERTAINLY!  Taken quantitatively, I suspect it's better than that: I
very much doubt that -- outside of logic and foundations -- very many
correct proofs not formalizable in ZFC get published from one year to the
next.  (Can anybody here cite something like my example from the
mathematical literature of the past decade that wasn't reviewed in the "03"
section of "Math Reviews"?  I know that Harvey thinks that non-ZFC axioms
are going to be required for "normal" mathematics in the future, but I
don't think the bulk of the analysis-algebra-topology-appliedmath...
community is there yet.)
     I think I still have a CONCEPTUAL point, though.  The conception of
rigor that is "orthodox and almost universally accepted" today connects it
to (some formal extension of) classical 1st Order Logic (cf. weasel-clause
1, below) and to a conception of the set-theoretic universe (largely)
encoded in the axioms of ZFC (cf. weasel-clause 2).  So far I think I am in
total agreement with Simpson, but I have a couple of weaselly
reservations... which I think show that the conceptual situation isn't as
simple or neat as his origianl statement, taken naively, might suggest.

   (WC  1) Why an "extension of" 1st Order Logic?  Well, it seems to me
that "reflection arguments" of the form
	P
	Therefore, "P" is true
are logical (they aren't tied to any SPECIFIC mathematical subject matter).
I think anyone using this sort of reasoning in a serious mathematical
context would be in trouble if their reasoning couldn't be formalized
within some well-understood "truth theory" (Tarski?), so I'm not going to
say that an acceptably "rigorous" proof can be in any mystical sense
unformalizable.  But it can go beyond 1st Order Logic.

   (WC 2) My original point.  The conception of rigor that we work with is
tied to a "conceptual scheme" for set theory: this was argued by Robert
McNaughton in articles in the "Philosophical Review" in 1957 and (I think)
"Philosophy of Science" about the same time, since when the "iterative
conception" has been nicely described by, e.g., Shoenfield in the
introduction to the set theory chapter of his "Mathematical Logic" and by
Boolos in "Journal of Philosophy" 1971 (since reprinted in Boolos, "Logic,
Logic and Logic").  The 1st Order axiomatization of ZFC captures enough of
this conception to serve for more than 98 or 99 percent of mathematics, but
-- conceptual messiness! -- not all.

--
  (Professor Simpson has also asked me to clarify some of my other comments
from my "Simpson on Tymoczkoism" post-- I will try, after some more
thought.)
--
Allen Hazen
Philosophy Department
University of Melbourne