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  "body": "\n## Statement\n\nWhile the influence of Cantor and Peano was paramount, in Appendix A \"The Logical and Arithmetical Doctrines of Frege\" of *The Principles of Mathematics*, Russell arrives at a discussion of Frege's notion of *function*, \"...a point in which Frege's work is very important, and requires careful examination\". In response to his 1902 exchange of letters with Frege about the contradiction he discovered in Frege's *Begriffsschrift* Russell tacked this section on at the last moment.\n\nFor Russell the bedeviling notion is that of variable: \"6. Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain *variables*. The notion of the variable is one of the most difficult with which logic has to deal. For the present, I openly wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. . . . We shall find always, in all mathematical propositions, that the words *any* or *some* occur; and these words are the marks of a variable and a formal implication\".\n\nAs expressed by Russell \"the process of transforming constants in a proposition into variables leads to what is called generalization, and gives us, as it were, the formal essence of a proposition ... So long as any term in our proposition can be turned into a variable, our proposition can be generalized; and so long as this is possible, it is the business of mathematics to do it\"; these generalizations Russell named propositional functions. Indeed he cites and quotes from Frege's *Begriffsschrift* and presents a vivid example from Frege's 1891 *Function und Begriff*: That \"the essence of the arithmetical function 2*x*\u003csup\u003e3\u003c/sup\u003e + *x* is what is left when the *x* is taken away, i.e., in the above instance 2( )\u003csup\u003e3\u003c/sup\u003e + ( ). The argument *x* does not belong to the function but the two taken together make the whole\". Russell agreed with Frege's notion of \"function\" in one sense: \"He regards functions – and in this I agree with him – as more fundamental than predicates and relations\" but Russell rejected Frege's \"theory of subject and assertion\", in particular \"he thinks that, if a term *a* occurs in a proposition, the proposition can always be analysed into *a* and an assertion about *a*\".\n",
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