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PEIRCE TO STRAWSON 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 Independently 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 arithmetic, 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 communicated 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 difference of types between different levels of the hierarchy But now another difficulty arises Recall that Frege had, in effect, defined the number two as 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 50
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