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