Russell on Mathematics, Logic, and Language
Relations were a matter of particular interest to Russell at this time because
the focus of his thought was on the nature of mathematics, in which
relational statements such as ‘n is the successor of m’ play an important
role. Independ ently of Frege, and initially without any knowledge of his
work, Russell had undertaken a logicist project of deriving mathematics
from pure logic. His endeavour was indeed more ambitious than Frege’s
since he hoped to show that not just arithmeti c, but geometry and analysis
also, were derived from general logical axioms. Between 1900 and 1903,
influenced in part by the Italian mathematician Giuseppe Peano, he
worked out his ideas for incorporation into a substantial volume, The
Principles of Mathematics. It was in the course of this work that he encountered
the paradox that bears his name, the paradox generated by the class of all
classes that are not members of themselves. As we have seen, he commu-
nicated this discovery to Frege, to whom he had been directed by Peano.
Russell introduced Frege’s work to an English readership in an appendix to
The Principles. In the light of the paradox, the two great logicists saw that
their project, if it was to succeed, would need considerable modification.
Russell’s attempt to avoid the paradox took the form of a Theory of
Types. According to this theory, it was wrong to treat classes as randomly
classifiable objects. Individuals and classes belonged to different logical
types, and what could be asserted of elements of one type could not be
significantly asserted of another. ‘The class of dogs is not a dog’ was not
true or false but meaningless. Similarly, what can significantly be said of
classes cannot be said of classes of classes, and so on through the hierarchy
of logical types. To avoid the paradox, we must observe the differ ence of
types between different levels of the hierarchy.
But now another difficulty arises. Recall that Frege had, in effect, defined
the number two a s the class of all pairs, and defined all the natural
numbers in a similar manner. But a pair is just a two-membered class, so
the number two, on this account, is a class of classes. If we put limitations
on the formation of classes of classes, how can we define the series of
natural numbers? Russell retained the definition of zero as the class whose
only member is the null class, but he now treated the number one as the
class of all classes equivalent to the class whose members are (a) the
members of the null class, plus (b) any object not a member of that class.
PEIRCE TO STRAWSON
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