Atheism and Theism JJ Haldane - Common Sense Atheism

40 J.J.C. Smart
Platonic objects. (I see no reason why sophisticated robots might not apply
the hypothetico-deductive method.) Quine’s Platonism is thus not in conflict
with modern mechanistic biology as traditional Platonism seems to be. It is
possible that if the world (including space–time) had a discrete grain we could
get by without the real numbers and with difference equations instead of
differential equations. Thus there is some empirical constraint on the mathematics
we need to postulate. Nevertheless because of the slack between
hypothesis and observation mathematics is very much immune to revision,
and this may give it a sort of necessity. However, this necessity would be
epistemological, not ontological.
It should be conceded that the more traditional form of mathematical
Platonism, according to which the mind has direct intuitive contact with the
mathematical entities, is congenial to many mathematicians. 72 Roger Penrose
has indeed used this supposed feature of mathematics to argue towards a new
view of mentality and of how the brain works. 73 Diffidently, because Penrose
after all is an eminent cosmologist and the son of a great neurobiologist, I go
the other way. If Penrose’s view is accepted it could give some comfort for the
theist. It is just conceivable that the brain may need for its full understanding
recondite quantum mechanical principles, such as of non-locality, but it seems
to me that since neurons operate mainly electrochemically the brain is probably
more like a computer or connection machine. Even with the recondite
principles it is hard to be convinced that intuition of Platonic entities is
possible for it.
Another philosophy of mathematics that is a leading contender in the field
is the fictionalism of Hartry Field. 74 He holds that mathematics is a fiction:
all its existential statements are false. The universal ones are true but vacuously
so, since ‘everything is such that’ in this case is equivalent to ‘it is not
the case that something is not such that’. According to Field mathematics
merely facilitates scientific inferences which could be carried out in a more
complicated way nominalistically. (He makes use of space–time points of
which there are as many as there are real numbers.) To show this in detail he
needs to reconstruct physical theories nominalistically and has done so for
certain theories.
Field’s fictionalism would hardly appeal to the pure mathematician,
who would not like to think of himself or herself as a sort of Dickens or
Thackeray. (Or worse, since in novels there are many existential sentences
which are not only pretended to be true but which are true!) Still, that’s not
an argument. Field’s theory is ontologically parsimonious and is in that
way appealing. It is a no nonsense sort of theory. One worry about really
believing set theory, I think, is the fact that the set membership relation
between a set and its members is too intimate: there is something mysterious
about it.