[FOM] Re: numbers and sets
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Wed Sep 24 23:36:08 EDT 2003
Benacerraf's problem, of the arbitrariness of the choice of sets to
identify numbers with if arithmetic is to be "reduced" to set theory,
though not generating much mathematical (or F.o.M.) interest, has been a
thorn in the side of Platonistic philosophies of mathematics since he
published "What numbers could not be" in 1965*.
There are numerous choices possible:
(1) Probably the most commonly used in basic set theory books is Von
Neuman's. Eric Steinhart's FoM post of a few days back lists some of its
desirable qualities. His list is a useful starting point for comparisons.
(2) Another is Zermelo's (0 is identified with the null set, successor
with the unit-set-of operation). Quine (preface to second edition of his
"Set Theory and its Logic") reports that it is slightly simpler to derive
the basic principles of arithmetic from set theoretic axioms with this
identification than it is with Von Neumann's, and David Lewis ("Parts of
Classes") makes the conceptual point that it identifies a fundamental
arithmètic operation with a fundamental set theoretic one in a way that
brings out structural similarities between PA and ZF.
(3) Randall Holmes has reminded us of another: Frege's and Russell's
identification of the number n with the set of all n-membered sets. This
has the drawback that the sets identified with numbers can be proved-- in
ZF-- not to exist; Holmes is a leading proponent of a different set theory
that proves they DO exist. I think he is right in saying that, if we have
a set theory that allows us to use this identification, it is more natural
than Von Neuman's or Zermelo's. Penelope Maddy, in her "Realism in
Mathematics," suggested that numbers be thought of, not as sets, but as
PROPERTIES OF sets: the Frege-Russell reduction identifies the numbers with
the extensions of these properties.
Note that, formally, what all these "reductions" provide is a RELATIVE
INTERPRETATION of an axiomatic system of arithmetic (typically PA; Quine is
unusual in treating exponentiation on a par with addition and
multiplication) in the reducer's chosen axiomatic system of set theory.
Clearly this is a necessary condition to meet if one is to make any
philosophical claim about "reducing" arithmetic to set theory, or even the
vaguer claim that set theory provides an "adequate" foundation for
mathematics.
I would like to mention yet another reduction, which I think illustrates
an important conceptual point:
(4) Cardinal numbers are, intuitively, measures of the size of
sets, so in talking about numbers one is talking about properties of sets.
(Cf. Maddy's suggestion mentioned above.) So: let us interpret arithmetic
as being about sets-- interpret the quantifications "For every (natural)
number / There is a (natural) number" as "For every (finite) set / There is
a (finite) set," and at the same time interpret the IDENTITY predicate,
when it occurs between NUMERICAL variables or terms as "--is equipollent
to--." Formally this gives us a RELATIVE INTERPRETATION in the GENERALIZED
sense in which the identity of the interpreted theory is interpreted by
some other predicate of the interpreting, and this, it seems to me, is as
good as the "identity-standard" interpretations mentioned earlier for most
FoM purposes and for philosophical issues of "reduction".
Advantages of (4): It lets you use the "logicist" definitions of the
basic functions that Holmes mentions. It can be used in any of a wide
variety of set-theoretic environments: ZF, NF, Monadic 3rd Order Logic with
an axiom of infinity.... ... What I'd like to suggest HERE is
that, by showing that the advantqages of a set-theoretic "reduction" of
arithmetic do not depend on the identification of individual numbers with
individual sets, it may help to explain why most FoM people coming from the
mathematical side don't find Benacerraf's problem particularly pressing.
Yes, the other reductions make "arbitrary" (in some sense that allows that
it is possible to distinguish better from worse among arbitrary choices)
identifications, but their main intellectual interest does not DEPEND on
these choices.
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* Bibliographical note: Benacerraf's paper was in the "Philosophical
Review" vol. 74 (1964). It (and also his almost equally influential
"Mathematical Truth," orig. in "Journal of Philosophy" vol. 70 (1973)) is
reprinted in the second (1983) edition of P. Benacerraf and H. Putnam, eds,
"Philosophy of Mathematics: selected readings." It is perhaps not as rare
as it should be for an anthologist to include his own work in what is
intended as a collection of basic, fundamental, papers in some area of
philosophy. What is unusual in this case is that virtually the entire
philosophical community would have agreed that the inclusion was justified.
-----
Allen Hazen
Philosophy Department
University of Melbourne
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