Atheism and Theism 39existence was necessary in this sense would be to say that the assertion that
God existed would be completely empty. In the present context I could leave
the matter here, since this philosophy of mathematics does not help the
theist’s search for insight into the way in which God might be said to be a
necessary being. However, Quine has given reasons why the attempt to ex-
hibit set theory (and hence mathematics) as logic should be rejected.^71 (1) Set
theory, unlike propositional logic and first order predicate logic, is incomplete.
No set of axioms will imply all its truths, though of course any truth will be
implied by some set of axioms. Truth in mathematics cannot be identified
with provability, still less with provability from some set of definitions or
conventions. (2) Set theory, unlike logic, has a constant predicate ‘is a mem-
ber of ’. (Logic normally includes the identity predicate, but this is a curious
one and can be eliminated if we have a finite primitive vocabulary, which
could if we liked include all the predicates in the Oxford English Dictionary.)
(3) Set theory is Platonistic. There are assertions in it of the existence of sets
(and so of numbers), which are not particular objects in space or time. These
considerations all make the break between logic and set theory in the same
place and answer Bertrand Russell’s challenge to say where logic ends and
mathematics begins.
The failure of logicism in mathematics should be congenial to the theist, in
that the supposed necessity of existential statements in mathematics lives to
fight another day as a candidate for shedding light on what God’s necessary
existence might be like. It should be welcomed by pure mathematicians who
would not like to think that their life’s work was concocting more and more
recondite ways of saying nothing.
I now pass on briefly to some philosophies of mathematics which do seem
to be the most plausible, even if not completely satisfying, and see how they
might bear on the nature of God’s necessary existence.
Quite attractive is Quine’s form of Platonism. His Platonic objects are sets.
In line with the pioneering work of Frege and of Whitehead and Russell he
holds that set theoretical entities can do duty for all the entities postulated
in classical mathematics. He points out that a physical theory contains
mathematics and empirical physics seemingly inextricably intertwined with
mathematics. Since theories are tested holistically, if we believe physics we
must believe the mathematics needed for it. (Quine concedes that some pure
mathematics may go beyond what is quite needed. This is especially true, of
course, of the more esoteric reaches of set theory. This can be seen as ‘round-
ing out’ and might even be justified ontologically on the score of a sort of
simplicity.) Thus we believe in mathematical objects by the ordinary
hypothetico-deductive method of science: we believe in the entities postu-
lated by the theory that is best explanatory of observations. Thus Quine’s
Platonism does not require talk of mysterious powers of direct intuition of