FOM: Re: Logic in Ontology and Mathematics

[Allen Hazen]

   I think the latest excange between Jeffrey Ketland and Roger Bishop
Jones has been useful in bringing issues out.
   Quine's "Criterion of Ontological Commitment" says what the ontological
commitments of a THEORY (or a SENTENCE) are.  The question of the
ontological commitments of a PERSON who, in some context, asserts the
sentence or accepts the theory depends, in addition, on whether the
person's assertion is "absolute" (I'll use that dangerous word in the hope
that the following contrasts will clarify what I mean by it) or is in some
way "conditional" or "pretended."
    (As for ontology itself-- I take it that Quine's view is the plausible
one that what THERE IS is what a person would be committed to by accepting
a TRUE theory.)
    Jones says he doesn't want to regard the natural numbers as
"fictional," but does want to interpret his mathematical assertions as
conditional (on some "presupposed" framework) in a way that means HE is not
ontologically committed BY his mathematics TO numbers.  I think many people
probably feel the way he does; it seems to me that identifying just what
this sort of conditional assertion-- more than mere story-telling, less
than genuine "profession of faith"-- amounts to is a real philosophical
problem.  (I note in passing that "Fictionalism"-- the view that assertions
in, for instance, mathematics, are best thought of as FICTION-- has had
considerable discussion in the philosophical journals in recent years.  It
also, of course, has a long history: Hans Vaihinger's early 20th C "The
Philosophy of As If" argued for it.)
    Another way to avoid being ontologically (or otherwise) committed to
what your theory is committed to is to assert it TENTATIVELY.  Many
scientists, perhaps, present their hypotheses in a spirit of "this seems
like the best explanation, but I'm nowhere near being sure it's right."  I
think this is different.  A mathematician isn't tentative in asserting a
proven theorem, and I think the average biologist, say, would agree that IF
their tentatively suggested hypothesis about genetic determination of
blahblah is CORRECT, then THERE IS a blahblah gene.
    Question and Bibliography:
    Question: Quine, at least, and I think many philosophers following him,
thinks that the ontological commitments of a theory are relevant to its
EVALUATION.  You should, other things being equal, be less willing to
accept a theory with "heavy" ontological commitments, should demand more
evidence before accepting it-- with ACCEPTANCE, here, being thought of as
full, literal, belief (so the person who accepts a theory is fully
committed to its commitments).  Subquestion A: is this reasonable?  (My own
feeling is that it is, though ontological commitment is only one of many
relevant factors.)
Subquestion B: For short, call Roger Bishop Jones's attitude to his
mathematical assertions MATHEMATICAL ACCEPTANCE.  Should the ontological
commitments of a mathematical theory have the same sort of relevance to its
MATHEMATICAL ACCEPTANCE as ontological commitments of theories in general
have to their "full belief" acceptance?  (Tentatively, I'd hypothesize that
the answer to this one is YES... which, of course, makes it all the harder
to distinguish mathematical acceptance from full-belief acceptance.)
   Bibliography: On the related issue of whether it is possible to discuss,
literally and without equivocation, all ENTITIES-- concrete, abstract,
mathematical, what have you-- in a single theory (and so whether there is a
GENERAL notion of ontological commitment)...  Richard Cartwright's paper
"Speaking of Everything," in "Nous" (philosophy journal) 28 (1994), pp.
1-20, is admirably clear.
--
Allen Hazen
(interests: General Logic, Philosophy of Mathematics)
Philosophy Department
University of Melbourne