---
tag: M64XSMK07G
kind: statement
role: proposition
name: Bertrand Russell's The Principles of Mathematics 1903
sources:
  - wikipedia: en:36595472@r1359939615
    locator: History of the function concept, section Logicians' function 1850–1950 > Bertrand Russell's The Principles of Mathematics 1903
    note: adopted
    mention: wikipedia://en/36595472#GAOP
    sections: [Statement]
inventory: [YKQQFWVBQZ]
---

## Statement

While 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.

For 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".

As 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*<sup>3</sup> + *x* is what is left when the *x* is taken away, i.e., in the above instance 2( )<sup>3</sup> + ( ). 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*".