FOM: Do realists really know?
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Tue Jun 13 13:45:34 EDT 2000
Peter Schuster wrote:
> Would anybody be so kind to explain why and how
> the law of bivalence, i.e., that any statement is
> either true or else false, does follow from what
> traditionally is called the realist philosophy
> of mathematics, i.e., that mathematical objects
> are existent disregarding whether they are known?
>
> A similar question has recently been put by Fred Richman;
> to my knowledge it has not been answered yet.
This is an excellent question. FOMers might be interested in the following
exerpt from my forthcoming review essay on Penelope Maddy's book
Naturalism in Mathematics. The review essay is to appear in Philosophia
Mathematica. What follows is a section from the critical discussion,
concerning the issue of Bivalence. I have adapted LaTeX source code to
make it slightly more readable on this list.
Neil Tennant
__________________________________________________________________
Exerpt on Bivalence:
Goedel's realism is very much a Platonic affair (concerning the
existence of abstract objects) and offers no argument for the move
from the existence of those objects to the claim that all mathematical
statements about them are determinately truth-valued. Let us call this
latter claim Bivalence. Now it is well-known that the intuitionist
does not accept Bivalence. There is a defensible brand of intuitionism
that nevertheless takes the objects of mathematics to be
mind-independent, and the provision of proofs to be a rule-governed
activity, not susceptible to the creative whims or idiosyncracies of
individual mathematicians. That is to say, mathematical thought is
still constrained by `what is out there', and the individual
mathematician is not free simply to `make things up' as one
goes along.
The brand of intuitionism in question was expressed as follows by
Crispin Wright:
\footnote{Cf. C. Wright, Frege's Conception of Numbers as
Objects, Aberdeen University Press, 1983, pp. xviii-xx.}
... someone could hold both that it is correct to think of the
natural numbers as genuine objects ... and that there are decisive
objections to the realist's way of thinking about the truth or falsity
of statements concerning such objects. ... someone might be
persuaded of the reality---the Selbstst"andigkeit---of the
natural numbers but ... reject the realist conception of the
meaning of statements about them.
The `realist conception' at issue is that Bivalence holds.
Elsewhere I have emphasized and developed further the essential point
here---that ontological realism can be combined with semantic
anti-realism.
\footnote{Cf. N. Tennant, Anti-Realism and Logic, Clarendon Press,
Oxford, 1987, in ch. 2, `Scientific v. Semantic Realism', pp. 7-12.}
Such an `ontologically realist' intuitionist would be in full
agreement with Goedel's sentiment, quoted by Maddy: `the objects and
theorems of mathematics are as objective and independent of our free
choice and our creative acts as is the physical world.'
\footnote{K. Goedel, `Some basic theorems on the foundations of
mathematics and their implications' (The Gibbs Lecture, Brown
University, 1951), in S. Feferman et alia, eds., Collected Works,
vol. III, Oxford University Press, 1995, pp. 304-323; at p. 312,
n. 17.)}
Note how Goedel here talks of theorems, not of truth values of
statements in general. He does not commit himself to anything as
classically full-blooded as, say, `the objects, and a determinate
truth-value for each and every statement, of mathematics are as
objective and independent of our free choice and our creative acts as
is the physical world.' Yet it is this stronger reading that would be
needed by anyone looking to Goedelian realism for a route to the
conviction that CH has a determinate truth-value, possibly independent
of our means of coming to know what that truth-value is.
As the reader will be able to check, every quotation that Maddy
adduces from Goedel's writings (with one exception---see the next
footnote) expresses no more than what an `objective intuitionist'
would readily concede.
The fundamental (and required) justificatory move is never made to
determinacy of truth-value, whether for CH or across the
board.
\footnote{In personal correspondence, Maddy draws attention to her
quotation (on p. 89) from Goedel: `... if the meanings of the
primitive terms of set theory as explained on page 262 and footnote 14
are accepted as sound, it follows that the set-theoretical concepts
and theorems describe some well-determined reality, in which Cantor's
conjecture must be either true or false.' (`What is Cantor's
continuum problem?' (1964), in S. Feferman et alia, eds., Collected
Works, Vol. II, Oxford University Press, 1990, pp. 254-70; at p. 260.)
She comments `This clearly makes the move to determinacy of truth
value for CH.' But I maintain that it does not. Rather, it simply
states determinacy; it does not infer it from claims of existence. A
`well-determined reality' is by definition one in which every
proposition has a determinate truth-value---a fortiori one in which CH
has a determinate truth-value. What is still missing is an argument
from the mere objective existence of mathematical objects, and perhaps
the obtaining of certain primitive relations among them, to the
determinacy of truth-value of every proposition about those objects,
regardless of those propositions' logical (quantificational)
complexity.}
Interestingly, the very paragraph that Maddy quotes from Goedel in
order to show how `it is the mathematical considerations, not the
philosophical ones, that are decisive', seems to embody decisively
philosophical insights into the source of mathematical truth and the
insufficiency of ontological realism to secure realism about
truth-value. The paragraph in question is
However, the question of the objective existence of the objects of
mathematical intuition ... is not decisive for the problem under
discussion here [i.e. the meaningfulness of the continuum
problem]. The mere psychological fact of the existence of an intuition
which is sufficiently clear to produce the axioms of set theory and an
open series of extensions of them suffices to give meaning to the
question of the truth or falsity of propositions like Cantor's
continuum hypothesis.
\footnote{`What is Cantor's continuum problem?', loc. cit., p. 268.}
Thus Goedel, it would appear, grounds the meaningfulness of
CH in the faculty of mathematical intuition, which is relied on
to produce appropriate axioms to settle the matter. Those intuitions
might be only partly `of objects'; they might concern also the
workings of our mathematical concepts, and the meanings of our
mathematical vocabulary. The brute independent existence of those
mathematical objects, could it somehow be secured, would not go far
enough to settle the continuum hypothesis. Something more is needed:
the contact of the mind, through its faculty of intuition, with that
realm of objects. Such contact must issue in linguistically expressed
axioms, and they in turn will settle the question of the
truth-value of CH by means of appropriate proofs.
END OF EXERPT
If you have read this far, and are interested in reading the full version
of my review essay on Maddy's book, please send me a private email
(tennant.9 at osu.edu)
requesting either the .ps file or the .dvi file, which I shall email back
to you an as attachment.
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