38 J.J.C. Smart
Thus we say ‘David must have arrived by now’ when we can deduce his
arrival from background knowledge of his desire to come, the length of the
road, the speed of his car, and so on. This seems to account for ordinary
language uses of ‘must’, ‘necessary’, ‘possibly’, etc. Modality is explained
metalinguistically, nor do we need to go far up in the hierarchy of language,
metalanguage, meta-metalanguage, etc. How often do we in real life iterate
modalities or ‘quantify into’ modal contexts in the manner of modal logi-
cians? I do not want to postulate possible worlds other than the actual world
in the manner of David Lewis. This proliferation of possible worlds makes
Carter’s ‘many universes’ hypothesis look parsimonious by comparison. What
Lewis calls ‘ersatz possible worlds’ are not so bad: I talk of them just as a way
of referring to the contextually agreed background assumptions. The defini-
tion (some pages back) of logical necessity in terms of interpretability in any
non-empty universe is not in conflict with my attitude here, because for this
purpose universes can be defined in the universe of natural numbers, which
we can take to be actual and not merely possible. (This is because of the
Löwenheim–Skolem theorem.)
Now perhaps we can account for the sort of necessity that we feel about
‘There is a prime number between 20 and 24’. The proposition is agreed to
follow from unquestioned arithmetical laws, probably not Peano’s axioms
themselves, since most who believe that there is a prime number between 20
and 24 will not have heard of Peano’s axioms. The axioms, Peano’s or other-
wise, may be regarded as necessary because they are so central to our system
of beliefs, and anyway each is trivially, deducible from itself. They are not
definitions, but come rather near to being definitional.
At any rate, the suggestion of mathematical necessity may give some justi-
fiable comfort to the theist. How far this is the case depends on our philo-
sophy of mathematics. It seems to me that there are about five fairly plausible
yet not wholly satisfactory philosophies of mathematics in the field at present,
and how we answer the point about necessary existence in mathematics will
depend on which of these contending philosophies we accept or think of as
the least improbable. Let us take a very brief look at these options. I shall in
fact begin with what I regard as not an option but which has been very
influential in the recent past.
Some Philosophies of Mathematics and their Bearing on TheismShould we say, with Wittgenstein in his Tractatus, that the apparent necessity
of mathematics arises from the fact (or supposed fact) that all mathematical
propositions say the same thing, namely nothing? This would be a way in
which mathematics seems to be removed from the chances and contingencies
of the world, but it would not help the theist, because to say that God’s