FROM DESCRIPTIVE FUNCTIONS TO SETS OF ORDERED PAIRS

Abstract

It has been said that Frege first “mathematicized” logic prior to carrying out his logicist program of reducing mathematics to logic, as illustrated by his account of concepts as functions from objects to truth values. Recently Peter Hylton has tried to understand Bertrand Russell’s propositional functions by first distinguishing them from the more familiar mathematical functions on which Frege’s work is based. My thesis is that Russell deliberately sought to reduce mathematical functions to propositional functions as part of his logicist program of reducing mathematics to logic. The notion of “descriptive functions” is defined in Principia Mathematica via the notion of definite descriptions with *30x01. R'y = (ix)xRy Df. The expression “R'y” is read as “the R of y”. If ‘xRy’ means “x is father of y” then ‘R'y’ is “the x such that x is father of y", or “the father of y”. Combined with the theory of definite descriptions in *14 and then the theory of "relations in extension" in *21, the result is a reduction of mathematical functions to propositional functions. Comments on Frege’s theory of functions from Russell’s Principles of Mathematics, and some of the collected papers, support the thesis that Russell was suggesting this as a reductive account of mathematical functions. I also present the history of the reduction of functions to sets of ordered pairs, and its place in logical accounts of functions.