[FOM] Re: Platonism 1 (Tait)

[A.P. Hazen]

     Bibliography and Comment on William Tait's recent posting.

     BIBLIOGRAPHY: Tait describes Zermelo's interpretation of set 
theory as describing, not a single, definite, "universe" of sets 
given once and for all, but rather the modelS (note plural ending S) 
constituted by (appropriate) initial segments of the cumulative type 
hierarchy.  For those desiring a more lengthy discussion of this 
view, it is described (and advocated) in the later chapters of Marcus 
Giaquinto's "The Search for Certainty: a philosophical account of 
foundations of mathematics" (Oxford: Clarendon Press, 2002, ISBN 
0-19-875244-X; price hardcover in the U.S. as of early this year 
$45.00).

		(Mini-review: I read Giaquinto's book a few months 
back, and have somewhat mixed feelings about it.  It is written at an 
accessibility level that would allow its use as a text in a BEGINNING 
graduate level, or, with GOOD students, advanced undergraduate level 
unit in the philosophy of mathematics.  It doesn't formally 
presuppose much, but I suspect that students without at least SOME 
prior exposure to formal logic and set theory would find it ... 
strenuous.  I think if I were trying to teach from it, I would want 
to have some more formal reference-- something like, say, Hatcher's 
"The (Logical) Foundations of Mathematics"-- available for students 
to refer to as needed.
		An introductory chapter sets the scene with an 
account of the sorts of things in analysis that motivated the 
development of set theory to "clarify" them (34 pp): Lavine's 
"Understanding the Infinite" has a similar but much more detailed 
discussion.  Second chapter (pp. 35-66) covers the set-theoretic 
Paradoxes, "Frege's Logicism," and type theory.  Third (pp. 67-116) 
discusses semantic paradoxes, "Principia Mathematica," Ramsey. 
Fourth (117-164) introduces Zermelo's axioms and discusses Hilbert's 
Program-- EXPECT good exposition here: I initially bought the book 
because I had been so imressed by the clarity of Giaquinto's article 
on Hilbert's epistemology in "Brit. Jour. for the Philosophy of 
Science" v. 34 (1983), pp. 119-132!  Fifth (165-198) is on Goedel's 
incompleteness theorems: a fairly good informal exposition, 
discussing the technicalities in English but not pretending to be 
rigorous or detailed.  Last chapter (199-231) argues that ZFC, 
interpreted in a Zermelian way, is a philosophically satisfactory 
resolution of the foundational problem, yielding as much "certainty" 
as we can reasonably ask for. (35 pp of end notes follow the text)
		Over all, quite good, but not AS good as I had hoped. 
There were a number of statements that I thought were just mistaken 
(incl. a couple on the technicalities discussed in chapter 5, though 
not on the central points).  More importantly, I was disappointed in 
the exposition of the philosophical positions Giaquinto rejects: 
Brouwer, and in particular Russell, could have been presented in a 
way that made their positions MUCH more plausible, and hence more 
interesting.)

       COMMENT: Tait argues that "Zermelianism" provides a preferable 
philosophical alternative to the sorts of "liberal intuitionism" that 
allow you to use classical logic when quantifying over domains of 
restricted cardinality but says you have to reason constructively 
when you talk about absolutely EVERYTHING.  He is certainly right 
that Zermelo's interpretation legitimates the use of fully classical 
ZFC (as opposed to some system with formally intuitionistic logic for 
unrestricted quantifiers), as each model in the open-ended hierarchy 
is thought of as a classical structure.  On the other hand, in 
speaking about the whole hierarchy of models....  I don't know if 
such talk would be of any mathematical USE, but one can certainly 
FORMULATE statements beginning with quantifiers like
	"For every set in any model in Z's hierarchy...,"
(perhaps most interestingly in something like "For every set, x, in 
any model M, there is some model M+ such that it holds of x in M+ 
that...").  Are such quantifications to be interpreted classically? 
(Does the denial of such a universal quantifiaction imply that THERE 
IS a model containing a set which is a counterexample?)
      If not, we are back in some form of liberalized intuitionism. 
If so, why can't the gung-ho Platonist say
	"Nuts to all this hierarchy of models stuff!  When I say
		'all sets,' I mean every set in ANY segment of the
		cumulative hierarchy: MY 'intended interpretation'
		is the union of all such segments, the WHOLE
		cumulative hierarchy"
?

	A BIT MORE BIBLIOGRAPHY: Charles Parsons has defended views 
about the extensibility of the set-theoretic hierarchy that are at 
least a little bit reminiscent  of Z's (though Parsons allows larger 
models to be "fatter" as well as "taller"): cf. his 'Sets and 
Classes' ("Nous" 8 (1974), pp. 1-12; repr. in Parsons's "Mathematics 
in Philosophy") and his 'Sets and Modality' (in "Mathematics in 
Philosophy").  George Boolos  commented  on the first of these papers 
when it was  given at a conference; his reply is in his "Logic, 
Logic, and Logic," pp. 30-36.

--

Allen Hazen
Philosophy Department
University of Melbourne