philosophy, analytic philosophy, history of philosophy


I compare Russell’s theory of mathematical functions, the “descriptive functions” from Principia Mathematica ∗30, with Frege’s well known account of functions as “unsaturated” entities. Russell analyses functional terms with propositional functions and the theory of definite descriptions. This is the primary technical role of the theory of descriptions in P M . In Principles of Mathematics and some unpublished writings from before 1905, Russell offered explicit criticisms of Frege’s account of functions. Consequenly, the theory of descriptions in “On Denoting” can be seen as a crucial part of Russell’s larger logicist reduction of mathematics,aswellasanexcursionintothetheoryof reference.

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Church, Alonzo. 1956. Introduction to Mathematical Logic. Princeton, New Jersey: Princeton University Press.

Frege, Gottlob. 1891. ‘Funktion und Begriff’, translated as ‘Function and Concept’. In ‘Collected Papers’ [1984], 137–156.

Frege, Gottlob. 1893. Grundgesetze der Arithmetik. Jena: Hermann Pohle. Reprinted Hildesheim: Georg Olms Verlag, 1998.

Frege, Gottlob. 1904. ‘Was ist eine Funktion?’, translated as ‘What is a Function?’ In ‘Collected Papers’ [1984], 285–292.

Frege, Gottlob. 1984. Collected Papers on Mathematics, Logic and Philosophy, ed. Brian McGuiness. Oxford: Basil Blackwell.

Hylton, Peter. 1993. ‘Functions and Propositional Functions in Principia Mathematica’. In A.D.Irvine & G.A.Wedeking (eds.) ‘Russell and Analytic Philosophy’, 342–360. Toronto: University of Toronto Press.

Linsky, Bernard. 2002. ‘The Resolution of Russell’s Paradox in Principia Mathematica’. In James E. Tomberlin (ed.) ‘Language and Mind’, No. 16 in Philosophical Perspectives, 395–417. Boston and Oxford: Blackwell.

Linsky, Bernard. 2002. 2004. ‘Russell’s Marginalia in his Copies of Frege’s Works’. Russell: The Journal of Bertrand Russell Studies n.s. 24: 5–36.

Linsky, Bernard. 2004-5. ‘Russell’s Notes on Frege for Appendix A of The Principles of Mathematics’. Russell: The Journal of Bertrand Russell Studies n.s. 24(2), Winter 2004-05: 133–72.

Linsky, Bernard.. 2009. ‘Leon Chwistek’s Theory of Constructive Types’. In Wioletta Miskiewicz Sandra Lapointe, Mathieu Marion & Jan Wolenski (eds.) ‘The Golden Age of Polish Philosophy: Kaziemierz Twardowski’s Philosophical Legacy’, Springer Verlag.

Mancosu, Paolo. 2005. ‘Harvard 1940-1941: Tarski, Carnap and Quine on a Finitistic Language of Mathematics for Science’. History and Philosophy of Logic 26: 327–357.

Pelletier, F.J. & Linsky, B. 2005. ‘What is Frege’s Theory of Descriptions?’ In G. Imaguire & B. Linsky (eds.) ‘On Denoting: 1905-2005’, 195–250. Munich: Philosophia Verlag.

Quine, Willard van Orman. 1963. Set Theory and Its Logic. Cambridge, Mass: Harvard University Press.

Ramsey, Frank Plumpton. 1931. The Foundations of Mathematics and Other Logical Essays, ed. R.B. Braithwaite. London: Routledge, Kegan and Paul.

Russell, Bertrand. 1905. ‘On Denoting’. Mind 14 (Oct. 1905): 479–93. Reprinted in [CP4], 415-427.

Russell, Bertrand. CP4. Foundations of Logic 1903-1905, The Collected Papers of Bertrand Russell, vol. 4, ed. Alasdair Urquhart. London and New York: Routledge, 1994.

Russell, Bertrand. PoM. Principles of Mathematics. Cambridge: Cambridge University Press, 1903.

Whitehead, A.N. & Russell, B.A. PM. Principia Mathematica. Cambridge: Cambridge University Press, 3 vols, 1910-13, 2nd ed, 1925-27.