Atheism and Theism 41If Field’s theory is accepted, we must say that there are no true existential
mathematical sentences, and a fortiori no necessary ones. So Field’s theory
does not help in the theist’s possible hope that mathematical necessity throws
some light on what God’s necessary existence might be like.
One philosopher who has strongly felt the mysteriousness of the set mem-
bership relation is David Lewis, who in his Parts of Classes^75 treats the relation
of set to subset as the whole/part relation. (Classically, of course, this is done
by definingsubsetin terms of set membership.) However, the notion of set
membership still obtrudes in one place, the singleton relation, the relation of
a thing to the set of which it is the only member. In an appendix with John
P. Burgess and A.P. Hazen (explaining two methods due to these logicians)
he gets over this problem but at a certain cost of empirical assumption as to
what is in the universe, and also of structuralism, where one talks indiffer-
ently about many different subject matters. He also needs plural quantifica-
tion, which is familiar in ordinary language as in ‘some critics admire only
one another’. This sentence cannot be rendered into first order predicate logic
without talking of sets of critics. George Boolos^76 gives the semantics in
terms of second order logic, but Lewis cannot take this option because he is
trying to replace set theory and he thinks of second order logic as ‘set theory
in sheep’s clothing’, as Quine has put it. (One trouble I have with structural-
ism is that I can think of a structure only in set theoretic terms.) Lewis’s
theory may be the philosophy of mathematics of the future, but because of its
reliance (especially in the Appendix) on some general empirical assumptions
about the world it does not provide the sort of sense of ‘necessity’ which
might help the theist.
Properties may seem less mysterious than sets, because physicists postulate
properties of mass, length, charge, spin, charm, colour (these words not to be
taken in their ordinary sense!) and so on. We might take ‘this has a mass of
2 kg’ as expressing a relation between this, the standard kilogram, the prop-
erty mass, and the number 2. Note that they are not the bad old properties
to which Quine has objected, as if using the predicate ‘tall’ committed one
to the property ‘tallness’. No, they do not come from a bad philosophy of
language and meaning, but from what science tells us. I am myself inclined
only to believe in those properties which fundamental physics and cosmology
need to postulate. This sort of scientific realism about universals was
pioneered in Australia by D.M. Armstrong^77 and has led to various ideas in
the philosophy of mathematics, as by Peter Forrest and Armstrong^78 (who
have their differences) and most notably by John Bigelow in his book The
Reality of Number: A Physicalist’s Philosophy of Mathematics,^79 which needs to
be taken very seriously. There are differences: Bigelow and Forrest believe in
uninstantiated universals, Armstrong only in instantiated ones. But because
of the empirical basis of these theories, it once again does not give any help to