Dummett - Wittgenstein's Philosophy of Mathematics.pdf - Ted Sider

MICHAEL DUMMETT
schools as rivals, there remains the philosophical problem of
deciding which of the various accounts is correct. Wittgenstein's
book is intended as a contribution to the latter task only. It seems
natural to suppose that the philosophical task and the mathe-
matical go hand in hand, for the precise formulation of a con-
ception is not irrelevant to deciding on its correctness, and
unexpected consequences of adopting it may lead one to revise
one's opinion as to its value. Wittgenstein will have none of this:
for him philosophy and mathematics have nothing to say to one
another; no mathematical discovery can have any bearing on
the philosophy of mathematics.2 It would seem that he is theoreti-
cally committed also to the converse, that no philosophical opinion
could, or at least ought to, affect the procedure of the mathe-
matician. This comes out to some extent in his discussion of the
law of excluded middle in mathematics. Against one who insisted
that either the sequence "77777" occurs in the development of
r or it does not, he employs arguments similar to those of the
intuitionists; and yet it appears that he is not wishing to question
the validity in a mathematical proof of, for example, argument
by cases, but only to reprove someone who in the course of
philosophical reflection wishes to insist on the law of excluded
middle.3 Yet this is not to be taken too seriously, for Wittgenstein
would always be able to claim that, while he had not shown
that certain mathematical procedures were wrong, still he had
shown them not to have the interest we were inclined to attach
to them. Certainly in his discussion of Cantor he displays no
timidity about "interfering with the mathematicians."4 I think
that there is no ground for Wittgenstein's segregation of philos-
ophy from mathematics but that this springs only from a general
tendency of his to regard discourse as split up into a number
of distinct islands with no communication between them (state-
ments of natural science, of philosophy, of mathematics, of
religion).
As Frege showed, the nominalist objection to Platonism-that
talk about "abstract entities" is unintelligible-is ill-taken; if we
2 Cf. V, 13, 19; IV, 52; also Investigations, II, xiv; I, 124.
3 IV, 10.
4 I, App. I I.
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